4B. Non-Keplerian Motion - LTAS-SDRG :: Welcome ! · 2017-10-26 · J.E. Prussing, B.A. Conway,...

Gaeumltan Kerschen

Space Structures amp

Systems Lab (S3L)

4B Non-Keplerian Motion

Astrodynamics (AERO0024)

2 Two-body

problem 41 Dominant

perturbations

Orbital elements

(aeiΩω) are constant

Real satellites may undergo

perturbations

This lecture

1 Effects of these perturbations on the orbital elements

2 Computation of these effects

3

STK Different Propagators

4

Why Different Propagators

Analytic propagation

Better understanding of the perturbing forces

Useful for mission planning (fast answer) eg lifetime

computation

Numerical propagation

The high accuracy required today for satellite orbits can only be

achieved by using numerical integration

Incorporation of any arbitrary disturbing acceleration

(versatile)

5

4 Non-Keplerian Motion

2

2

(1 )

sin

sin 1 cos

a e

N

i e

42 Analytic treatment

43 Numerical methods

nt 1nt

r

44 Geostationary satellites

6



421 Variation of parameters

422 Non-spherical Earth

423 J2 propagator in STK

424 Atmospheric drag

425 Third-body perturbations

426 SGP4 propagator in STK

427 Solar radiation pressure

2

2

(1 )

sin

sin 1 cos

a e

N

i e

7

Analytic Treatment Definition

Position and velocity at a requested time are computed

directly from initial conditions in a single step

Analytic propagators use a closed-form solution of the

time-dependent motion of a satellite

Mainly used for the two dominant perturbations drag and

earth oblateness


8

Analytic Treatment Pros and Cons

Useful for mission planning and analysis (fast and insight)

Though the numerical integration methods can generate more

accurate ephemeris of a satellite with respect to a complex force

model the analytical solutions represent a manifold of solutions for

a large domain of initial conditions and parameters

But less accurate than numerical integration

Be aware of the assumptions made


9

Assumption for Analytic Developments

The magnitude of the disturbing force is assumed to be

much smaller than the magnitude of the attraction of the

satellite for the primary

3 perturbedr

r r a

perturbed a r


10

Variation of Parameters (VOP)

Originally developed by Euler and improved by Lagrange

(conservative) and Gauss (nonconservative)

It is called variation of parameters because the orbital

elements (ie the constant parameters in the two-body

equations) are changing in the presence of perturbations

The VOP equations are a system of first-order ODEs that

describe the rates of change of the orbital elements


a i e M

11

Disturbing Acceleration (Specific Force)

ˆ ˆ ˆRperturbed R T NT N a F e e e

2


Rotating basis whose

origin is fixed to the

satellite

12

Perturbation Equations (Gauss)


2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)

The generating solution is that of the 2-body problem

ˆ ˆR Tr r rR r T Fr F e e (4)

Time rate-of-change

of the work done by

the disturbing force

13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos

a e h a ha R T e R T

h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e

JE Prussing BA Conway Orbital Mechanics Oxford University Press


2 sin 2 cos1 (1 )cos cos

1 cos

T ea ei R

e e

2 2(1 ) 2 cos cos sin 2 cos with

1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15


Limited to eccentricities less than 1

Singular for e=0 sin i=0 (use of equinoctial elements)

In what follows we apply the Gauss equations to Earth

oblateness and drag Analytical expressions for third-body

and solar radiation forces are far less common because

their effects are much smaller for many orbits


16

Non-spherical Earth J2

Focus on the oblateness through the first zonal harmonic

J2 (tesseral and sectorial coefficients ignored)

The J2 effect can still be viewed a small perturbation when

compared to the attraction of the spherical Earth


17


2 2 222

4

3 1 3sin sinsin sin cos sin sin cos

2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r

1 1 ˆˆ ˆ with cos

Ur r r

F r φ λ

Chapter 4A

18

Physical Interpretation of the Perturbation

The oblateness means that the force of gravity is no longer

within the orbital plane non-planar motion will result

The equatorial bulge exerts a force that pulls the satellite

back to the equatorial plane and thus tries to align the

orbital plane with the equator

Due to its angular momentum the orbit behaves like a

spinning top and reacts with a precessional motion of the

orbital plane (the orbital plane of the satellite to rotate in

inertial space)


19



20

Effect of Perturbations on Orbital Elements

Secular rate of change average rate of change over many

orbits

Periodic rate of change rate of change within one orbit

(J2 ~ 8-10km with a period equal to the orbital period)


21


Periodic

Secular


22

Secular Effects on Orbital Elements

Nodal regression regression of the nodal line

Apsidal rotation rotation of the apse line

Mean anomaly

No secular variations for a e i


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23

Secular Effects Node Line

2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

For posigrade orbits the node line drifts westward

(regression of the nodes) And conversely

0 90 0i

For polar orbits the node line is stationary

90 0i


Vallado Fundamental of

Astrodynamics and

Applications Kluwer 2001

25

Exploitation Sun-Synchronous Orbits

The orbital plane makes a constant angle with the radial

from the sun


26


The orbital plane must rotate in inertial space with the

angular velocity of the Earth in its orbit around the Sun

360ordm per 36526 days or 09856ordm per day

The satellite sees any given swath of the planet under

nearly the same condition of daylight or darkness day after

day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28

Secular Effects Apse Line

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

The perigee advances in the direction of the motion

of the satellite And conversely

0 634 or 1166 180 0i i

The apse line does not move

634 or 1166 0i i



Astrodynamics and


30

Exploitation Molniya Orbits

A geostationary satellite cannot view effectively the far

northern latitudes into which Russian territory extends

(+ costly plane change maneuver for the launch vehicle )

Molniya telecommunications satellites are launched from

Plesetsk (628ordmN) into 63ordm inclination orbits having a

period of 12 hours

3

2 the apse line is 53000km longellip

aT


31

Analytic Propagators in STK 2-body J2

2-body constant orbital elements

J2 accounts for secular variations in the orbit elements

due to Earth oblateness periodic variations are

neglected


32

J2 Propagator Underlying Equations


33

2-body and J2 Propagators Applied to ISS

Two-body propagator J2 propagator


34

HPOP and J2 Propagators Applied to ISS

Nodal regression of the ISS

35

Effects of Atmospheric Drag Semi-Major Axis

2a

Lecture 2

2

2

2

2

aa

gt0

Because drag causes the dissipation of mechanical energy

from the system the semimajor axis contracts


Drag paradox the effect of atmospheric drag is to increase

the satellite speed and kinetic energy

36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37

Effects of Atmospheric Drag Orbit Plane

2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e

The orientation of the orbit plane is not changed by drag


38

Effects of Atmospheric Drag Apogee Perigee

Apogee height changes drastically perigee height remains

relatively constant


Vallado Fundamental of Astrodynamics and Applications Kluwer 2001

39

Effects of Atmospheric Drag Eccentricity



40

Early Reentry of Skylab (1979)

Increased solar activity which

increased drag on Skylab led to

an early reentry

Earth reentry footprint could not

be accurately predicted (due to

tumbling and other parameters)

Debris was found around

Esperance (31ndash34degS 122ndash

126degE) The Shire of Esperance

fined the United States $400 for

littering a fine which to this day

remains unpaid


41

Lost of ASCA Satellite (2000)

July 15 2000 a strong solar

flare heated the Earthrsquos

atmosphere increasing the

air density to a value 100

times greater than that for

which its ADCS had been

designed to cope The

magnetorquers were unable

to compensate and the

satellite was lost

httpheasarcgsfcnasagovdocsascasafemodehtml


42

Effects of Third-Body Perturbations

The only secular perturbations are in the node and in the

perigee

For near-Earth orbits the dominance of the oblateness

dictates that the orbital plane regresses about the polar

axis For higher orbits the regression will be about some

mean pole lying between the Earthrsquos pole and the ecliptic

pole

Many geosynchronous satellites launched 30 years ago

now have inclinations of up to plusmn15ordm collision avoidance

as the satellites drift back through the GEO belt


43



The Sunrsquos attraction

tends to turn the

satellite ring into the

ecliptic The orbit

precesses about the

pole of the ecliptic


44

STK Analytic Propagator (SGP4)

The J2 propagator does not include drag

SGP4 which stands for Simplified General Perturbations

Satellite Orbit Model 4 is a NASANORAD algorithm


45


Several assumptions propagation valid for short durations

(3-10 days)

TLE data should be used as the input (see Lecture 03)

It considers secular and periodic variations due to Earth

oblateness solar and lunar gravitational effects and

orbital decay using a drag model


46

SGP4 Applied to ISS RAAN


47

SGP4 Applied to ISS Semi-Major Axis


48

Further Reading on the Web Site


49

Effects of Solar Radiation Pressure

The effects are usually small for most satellites

Satellites with very low mass and large surface area are

more affected



Astrodynamics and

Applications Kluwer

2001

51

Secular Effects Orders of Magnitude


52

Periodic Effects Orders of Magnitude


53


2

2

(1 )

sin

sin 1 cos

a e

N

i e


431 Orbit prediction

432 Numerical integration

433 Single-step methods Runge-Kutta

434 Multi-step methods

435 Integrator and step size selection

436 ISS example

nt 1nt

r

54

2-body analytic propagator (constant orbital elements)

J2 analytic propagator (secular variations in the orbit

elements due to Earth oblateness

HPOP numerical integration of the equations of motion

(periodic and secular effects included)

STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55

Real-Life Example German Aerospace Agency


56


57



58

Real-Life Example Envisat

httpnngesocesadeenvisat

ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60

NASA began the first complex numerical integrations

during the late 1960s and early 1970s

Did you Know

1969 1968


61

What is Numerical Integration

1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62

State-Space Formulation

3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63

How to Perform Numerical Integration

2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s

h hf t h f t hf t f t f t R

s

Taylor series expansion

( )ntu

1( )nt u


64

First-Order Taylor Approximation (Euler)

1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution

The stepsize has to be extremely

small for accurate predictions

and it is necessary to develop

more effective algorithms

along the tangent


65

Numerical Integration Methods

1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0

Implicit the solution method becomes

iterative in the nonlinear case

0 0 Explicit un+1 can be deduced directly from

the results at the previous time steps

=0

for 1

j j

j

Single-step the system at time tn+1

only depends on the previous state tn

State vector

0

for 1

j j

j

Multi-step the system at time tn+1 depends

several previous states tntn-1etc


66

Examples Implicit vs Explicit

Trapezoidal rule (implicit)

nt 1nt

Euler forward (explicit)

nt 1nt

Euler backward (implicit)

nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67

A variety of methods has been applied in astrodynamics

Each of these methods has its own advantages and

drawbacks

Accuracy what is the order of the integration scheme

Efficiency how many function calls

Versatility can it be applied to a wide range of problems

Complexity is it easy to implement and use

Step size automatic step size control

Why Different Methods


68

Runge-Kutta Family Single-Step

Perhaps the most well-known numerical integrator

Difference with traditional Taylor series integrators the RK

family only requires the first derivative but several

evaluations are needed to move forward one step in time

Different variants explicit embedded etc


69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau

The Runge-Kutta methods are fully described by the

coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip

hellip hellip hellip

bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k

Slope at the beginning

Slope at the midpoint (k1 is

used to determine the value

of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)

The local truncation error for a 4th order RK is O(h5)

The accuracy is comparable to that of a 4th order Taylor

series but the Runge-Kutta method avoids the

calculation of higher-order derivatives

Easy to use and implement

The step size is fixed


75

RK4 in STK


76

Embedded Methods

They produce an estimate of the local truncation error

adjust the step size to keep local truncation errors

within some tolerances

This is done by having two methods in the tableau one with

order p and one with order p+1 with the same set of

function evaluations

( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods

The two different approximations for the solution at each

step are compared

If the two answers are in close agreement the approximation is

accepted

If the two answers do not agree to a specified accuracy the step

size is reduced

If the answers agree to more significant digits than required the

step size is increased


78

Ode45 in Matlab Simulink

Runge-Kutta (45) pair of Dormand and Prince

Variable step size

Matlab help This should be the first solver you try


79


edit ode45


80


Be very careful with the default parameters

options = odeset(RelTol1e-8AbsTol1e-8)


81

RKF 7(8) Default Method in STK

Runge-Kutta-Fehlberg integration method of 7th order

with 8th order error control for the integration step size


83

Multi-Step Methods (Predictor-Corrector)

They estimate the state over time using previously

determined back values of the solution

Unlike RK methods they only perform one evaluation for

each step forward but they usually have a predictor and a

corrector formula

Adams() ndash Bashforth - Moulton Gauss - Jackson

() The first with Le Verrier to predict the existence and position of Neptune


84

Multi-Step Methods Principle

( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the

previous values Four function values

interpolated by a

third-order polynomial

t

u


85

Multi-Step Methods Initiation

with ( ) ( )t f tu u 0 0( )t u u

What is the inherent problem

t

u


86


Because these methods require back values they are not

self-starting

One may for instance use of a single-step method to

compute the first four values


87

Gauss-Jackson in STK

One of the most recommendable fixed-stepsize multistep

methods for orbit computations

88

Integrator Selection


Montenbruck and Gill

Satellite orbits Springer

2000

89


Pros

Very fast

Cons

Special starting procedure

Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower

Multi-step Single step


90

Why is the Step Size So Critical

Theoretical arguments

1 The accuracy and the stability of the algorithm are

directly related to the step size

2 Nonlinear equations of motion

Data for Landsat 4 and 6 in circular orbits around 800km

indicates that a one-minute step size yields about 47m

error

A three-minute step size produces about a 900m error


91


More practical arguments

1 The computation time is directly related to the

step size

2 The particular choice of step size depends on the

most rapidly varying component in the disturbing

functions (eg 50 x 50 gravity field)


92

Appropriate Step Size

The problem of determining an appropriate step size is a

challenge in any numerical process

Fixed step size (rule of thumb for standard

applications)

But an algorithm with variable step size is really

helpful The step size is chosen in such a way that

each step contributes uniformly to the total integration

error

100

orbitTt


93

Three Examples XMM OUFTI-1 ISS

Can you plot the step size vs true anomaly


94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97

ldquoDifficultrdquo Orbits

Automatic time step is especially nice on highly eccentric

orbits (Molniya XMM) These orbits are best computed

using variable step sizes to maintain some given level of

accuracy

Without this variable step size we waste a lot of time near

apoapsis when the integration is taking too small a step

Likewise the integrator may not be using a small enough step

size at periapsis where the satellite is traveling fast


98

HPOP Propagator ISS Example

1 Earthrsquos oblateness only

2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99

Earthrsquos Oblateness Only Ω

HPOP

J2

2-body

436 ISS example

100

Earthrsquos Oblateness Only i Ω a

HPOP with central body (20 + WGS84_EGM96)

(without dragSRPSun and Moon)

436 ISS example

101

Drag Only i Ω a

HPOP with drag ndash Harris Priester

(without oblatenessSRPSun and Moon)

436 ISS example

102

Drag Relationship with Eclipses

103

SRP Only i Ω a

HPOP with SRP

(without oblatenessdragSun and Moon)

436 ISS example

104

SRP Relationship with Eclipses

105

All Perturbations Together

436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107

Practical Example GEO Satellites

Nice illustration of

1 Perturbations of the 2-body problem

2 Secular and periodic contributions

3 Accuracy required by practical applications

4 The need for orbit correction and thrust forces

And it is a real-life example (telecommunications

meteorology)

44 GEO satellites

108

Three Main Perturbations for GEO Satellites

44 GEO satellites


2 SRP

3 Sun and Moon

109

Station Keeping of GEO Satellites

The effect of the perturbations is to cause the spacecraft

to drift away from its nominal station If the drift was

allowed to build up unchecked the spacecraft could

become useless

A station-keeping box is defined by a longitude and a

maximum authorized distance for satellite excursions in

longitude and latitude

For instance TC2 -8ordm plusmn 007ordm EW plusmn 005ordm NS

44 GEO satellites

110

East-West and North-South Drift

44 GEO satellites

NS drift

EW drift

What are the perturbations generating these drifts

111

East-West Drift

44 GEO satellites

A GEO satellite drifts in longitude due to the influence of

two main perturbations

1 The elliptic nature of the Earthrsquos equatorial cross-

section J22 (and not from the NS oblateness J2)

2

ΔV

ΔV vsat

vsat SRP

112

East-West Drift due to Equatorial Ellipticity

44 GEO satellites

113


44 GEO satellites

114

East-West Drift HPOP (20) vs HPOP (22)

44 GEO satellites

115

East-West Drift Stable Equilibirum

HPOP with 22

(without Sun and moonSRPdrag)

116


HPOP with 22


117


HPOP with 22


118

North-South Drift

The perturbations caused by the Sun and the Moon are

predominantly out-of-plane effects causing a change in

the inclination and in the right ascension of the orbit

ascending node

Similar effects on the orbit to those of the Earthrsquos

oblateness (but here with respect to the ecliptic)

A GEO satellite therefore drifts in latitude with a

fundamental period equal to the orbit period

119

North-South Drift

Period

HPOP with Sun and Moon

(without oblatenessSRPdrag)

120

North-South Drift

Period



121

Thrust Forces for Stationkeeping

GEO spacecraft require continual stationkeeping to stay

within the authorized box using onboard thrusters

44 GEO satellites

122

42 ANALYTIC TREATMENT









43 NUMERICAL METHODS






436 ISS example

44 GEOSTATIONARY SATELLITES

Gaeumltan Kerschen


Systems Lab (S3L)



2 Two-body

problem 41 Dominant

perturbations

Orbital elements

(aeiΩω) are constant

Real satellites may undergo

perturbations

This lecture

1 Effects of these perturbations on the orbital elements

2 Computation of these effects

3


4





computation





(versatile)

5


2

2

(1 )

sin

sin 1 cos

a e

N

i e



nt 1nt

r


6










2

2

(1 )

sin

sin 1 cos

a e

N

i e

7







earth oblateness


8










9





3 perturbedr

r r a

perturbed a r


10










a i e M

11



2




satellite

12



2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)



Time rate-of-change

of the work done by


13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



3


4





computation





(versatile)

5


2

2

(1 )

sin

sin 1 cos

a e

N

i e



nt 1nt

r


6










2

2

(1 )

sin

sin 1 cos

a e

N

i e

7







earth oblateness


8










9





3 perturbedr

r r a

perturbed a r


10










a i e M

11



2




satellite

12



2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)



Time rate-of-change

of the work done by


13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



4





computation





(versatile)

5


2

2

(1 )

sin

sin 1 cos

a e

N

i e



nt 1nt

r


6










2

2

(1 )

sin

sin 1 cos

a e

N

i e

7







earth oblateness


8










9





3 perturbedr

r r a

perturbed a r


10










a i e M

11



2




satellite

12



2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)



Time rate-of-change

of the work done by


13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



5


2

2

(1 )

sin

sin 1 cos

a e

N

i e



nt 1nt

r


6










2

2

(1 )

sin

sin 1 cos

a e

N

i e

7







earth oblateness


8










9





3 perturbedr

r r a

perturbed a r


10










a i e M

11



2




satellite

12



2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)



Time rate-of-change

of the work done by


13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



6










2

2

(1 )

sin

sin 1 cos

a e

N

i e

7







earth oblateness


8










9





3 perturbedr

r r a

perturbed a r


10










a i e M

11



2




satellite

12



2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)



Time rate-of-change

of the work done by


13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



7







earth oblateness


8










9





3 perturbedr

r r a

perturbed a r


10










a i e M

11



2




satellite

12



2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)



Time rate-of-change

of the work done by


13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



8










9





3 perturbedr

r r a

perturbed a r


10










a i e M

11



2




satellite

12



2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)



Time rate-of-change

of the work done by


13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



9





3 perturbedr

r r a

perturbed a r


10










a i e M

11



2




satellite

12



2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)



Time rate-of-change

of the work done by


13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



10










a i e M

11



2




satellite

12



2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)



Time rate-of-change

of the work done by


13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



11



2




satellite

12



2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)



Time rate-of-change

of the work done by


13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



12



2a

Chapter 2

2

2

2

2

aa

2

h

r

2 2

2

1 sin sin

1 cos

h r e er r

e h h

(1)

(2)

(3)



Time rate-of-change

of the work done by


13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



13



(1)

(2)

(3)

(4)

2 2 2

2

2 sin 2sin

2 sin 1 cos


h r h r

aRe T e

h

2(1 )h a e

3

22 sin 1 cos

1

aa Re T e

e

Chapter 2

14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



14


3

22 Resin 1 cos

1

aa T e

e

2(1 )

sin cos cosa e

e R T E

2

2cos(1 )

1 cos

Na ei

e




1 cos

T ea ei R

e e


1 cos

e R e e T eaM nt

e e

2

2sin(1 )

sin 1 cos

Na e

i e

15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



15









16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



16







17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



17


2 2 222

4


2r T N

J R ii i i

r

F e e e


2 2

2

3sin 11

2

satRU J

r r


Ur r r

F r φ λ

Chapter 4A

18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



18










inertial space)


19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



19



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



20



orbits




21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



21


Periodic

Secular


22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



22




Mean anomaly



2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a

2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a

23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



23


2

2

2 2 7 20

1 3cos

2 (1 )

T

avg

J Rdt i

T e a



0 90 0i


90 0i



Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)




Astrodynamics and


25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



25



from the sun


26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



26







day


27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



27

Existing Satellites

SPOT-5

(820 kms 987ordm)

NOAAPOES

(833 kms 987ordm)

28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



28


2

22

2 2 7 20

1 34 5sin

4 (1 )

T

avg

J Rdt i

T e a



0 634 or 1166 180 0i i


634 or 1166 0i i



Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)




Astrodynamics and


30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



30







period of 12 hours

3


aT


31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



31





neglected


32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



32



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



33




34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



34



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



35


2a

Lecture 2

2

2

2

2

aa

gt0






36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



36


21 10

2 2D r D

A AN R T C v C

m m a

0D

Aa a C

m

2

Df i f i

C Aa a t t

m

is assumed constant


3

22 Resin 1 cos

1

aa T e

e

Circular orbit

37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



37


2

2sin(1 )

sin 1 cos

Na e

i e

2

2cos(1 )

1 cos

Na ei

e



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



38



relatively constant



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



39




40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



40




an early reentry









remains unpaid


41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



41











satellite was lost



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



42



perigee





pole





43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



43




tends to turn the


ecliptic The orbit

precesses about the



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



44






45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



45



(3-10 days)






46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



46



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



47



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



48



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



49




more affected



Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)




Astrodynamics and

Applications Kluwer

2001

51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



51



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



52



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



53


2

2

(1 )

sin

sin 1 cos

a e

N

i e







436 ISS example

nt 1nt

r

54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



54






STK Propagators


Accurate

Versatile

Errors accumulation

for long intervals

Computationally

intensive

55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



55



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



56


57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



57



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



58



ENVpredhtml


Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



Why do the

predictions degrade

for lower altitudes

60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



60



Did you Know

1969 1968


61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



61


1n nt t t

3 perturbedr

r r a

Given

Compute

( ) ( )n nt tr r

1 1( ) ( )n nt t r r


62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



62


3 perturbedr

r r a

( )f tu u

ru

r

6-dimensional

state vector


63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



63


2( )( ) ( ) ( ) ( ) ( )

2

ss

n n n n n s


s


( )ntu

1( )nt u


64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



64


1

( ) ( ) ( )

( )

n n n

n n n n

t t t t t

t f t

u u u

u u uEuler step

0 1 2 3 4 5 60

5

10

15

20

25

30

35

40

Time t (s)

x(t

)=t2

Exact solution





along the tangent


65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



65


1 1 1

1 0

m m

n j n j j n j

j j

t

u u u

0 0





=0

for 1

j j

j



State vector

0

for 1

j j

j




66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



66



nt 1nt


nt 1nt


nt 1nt

1n n nt u u u

1 1n n nt u u u

1

12

n n

n n t

u uu u

r

r

r


67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



67



drawbacks








68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



68








69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



69


with ( ) ( )t f tu u 0 0( )t u u

1

1

s

n n i i

i

t b

u u k

1 1

1

1

2

n n

i

i n ij j n i

j

f t c t

f t a t c t i s

k u

k u k

Slopes at

various points

within the

integration step


70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



70


Butcher Tableau


coefficients

c1

c2

cs

a21

as1

b1

as2

b2

ass-1

bs-1 hellip

hellip


bs

1

1

1

1

1

0

s

i

i

i

i ij

j

b

c

c a


71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



71

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

Butcher Tableau

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k


72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



72

RK4 (Explicit)

1 2 3 41

2 2

6n n t

k k k ku u

1

2 1

3 2

4 3

2 2

2 2

n n

n n

n n

n n

f t

t tf t

t tf t

f t t t

k u

k u k

k u k

k u k




of u Euler)


now used)

Slope at the end

Estimated slope

(weighted average)


73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



73

RK4 (Explicit)

nt 2nt t nt t

1k3k

2k

4k

Estimate at new

time

u


74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



74

RK4 (Explicit)








75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



75

RK4 in STK


76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



76

Embedded Methods







( 1) ( 1) ( 1)

1

1

sp p p

n n i i

i

t b

u u k

( ) ( ) ( )

1

1

sp p p

n n i i

i

t b

u u k


77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



77

Embedded Methods


step are compared


accepted


size is reduced




78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



78



Variable step size



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



79


edit ode45


80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



80





81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



81





83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)




83






corrector formula




84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



84


( ) ( )t f tu u1

1( ) ( ) ( ) n

n

t

n nt

t t f t dt

u u u

unknown

Replace it by a

polynomial that

interpolates the


interpolated by a


t

u


85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



85


with ( ) ( )t f tu u 0 0( )t u u


t

u


86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



86



self-starting




87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



87




88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



88





2000

89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



89


Pros

Very fast

Cons


Fixed time steps

Error control

Pros

Plug and play

Error control

Cons

Slower



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



90








error



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



91




step size





92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



92





applications)




error

100

orbitTt


93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



93




94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



94

XMM Report in STK


95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



95

XMM (e~08)

0 50 100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

True anomaly (deg)

Ste

p s

ize

(s)


96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



96

ISS(e~0)


0 50 100 150 200 250 300 350 40030

35

40

45

50

55

60

65

70

True anomaly (deg)

Ste

p s

ize

(s)

97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



97





accuracy






98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



98



2 Drag only

3 Sun and moon only

4 SRP only

5 All together

436 ISS example

99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



99


HPOP

J2

2-body

436 ISS example

100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



100




436 ISS example

101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



101

Drag Only i Ω a



436 ISS example

102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



102


103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



103

SRP Only i Ω a

HPOP with SRP


436 ISS example

104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



104


105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



105


436 ISS example

106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



106


2

2

(1 )

sin

sin 1 cos

a e

N

i e


107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



107








meteorology)

44 GEO satellites

108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



108


44 GEO satellites


2 SRP

3 Sun and Moon

109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



109





become useless





44 GEO satellites

110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



110


44 GEO satellites

NS drift

EW drift


111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



111

East-West Drift

44 GEO satellites





2

ΔV

ΔV vsat

vsat SRP

112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



112


44 GEO satellites

113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



113


44 GEO satellites

114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



114


44 GEO satellites

115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



115


HPOP with 22


116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



116


HPOP with 22


117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



117


HPOP with 22


118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



118

North-South Drift




ascending node





119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



119

North-South Drift

Period



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



120

North-South Drift

Period



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



121




44 GEO satellites

122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



122
















436 ISS example


Gaeumltan Kerschen


Systems Lab (S3L)



Gaeumltan Kerschen


Systems Lab (S3L)



4B. Non-Keplerian Motion - LTAS-SDRG :: Welcome ! · 2017-10-26 · J.E. Prussing, B.A. Conway,...

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Transcript of 4B. Non-Keplerian Motion - LTAS-SDRG :: Welcome ! · 2017-10-26 · J.E. Prussing, B.A. Conway,...