NOTES ON ORBIT DETERMINATION MODELS C.K. Shum, Ohio … · - Drag coefficient for the satellite...

C.K. Shum, Orbit Determination Notes 1 October 1997

NOTES ON ORBIT DETERMINATION MODELS

C.K. Shum, Ohio State University

October 1997

SPATIAL REPRESENTATION OF ALTIMETER CROSSOVERERROR DUE TO GEOPOTENTIAL PERTURBATION

Single satellite crossovers:

€

Δx = 2Δν

€

=2l =1

∞

∑m = 0

l

∑p=0

l

∑ lmpD lmps

Φ (C lm sinmλ−S lm cosmλ)

• Zonals unobservable (to this level of approximation)

Dual satellite crossovers:

€

Δx=Δyi −Δy j +Δvi −Δv j

€

=l=1

∞

∑m= 0

l

∑p= 0

l

∑ Dlmp˜ Φ lmp

c( ) i C lm cosmλ+ S lmsinmλ( )

€

−l=1

∞

∑m= 0

l

∑p= 0

l

∑ Dlmp˜ Φ lmp

c( ) i C lm cosmλ+ S lmsinmλ( )

€

ml=1

∞

∑m= 0

l

∑p= 0

l

∑ Dlmp˜ Φ lmp

s( ) i C lm sinmλ−S lmcosmλ( )

€

±l=1

∞

∑m= 0

l

∑p= 0

l

∑ Dlmp˜ Φ lmp

s( ) i C lm sinmλ−S lmcosmλ( )

for satellites i and j


SPATIAL REPRESENTATION OF THE RADIAL ORBIT ERRORDUE TO GEOPOTENTIAL PERTURBATION

For 0=q , radial orbit error [Tapley and Rosborough, 1985]

€

Δr(0) =l=1

∞

∑m= 0

l

∑p= 0

l

∑ lmpD lmp

c˙ Φ lmC cosmλ−lmS sinmλ( )

€

±l=1

∞

∑m= 0

l

∑p= 0

l

∑ lmpD lmp

s˙ Φ lmC sinmλ−lmS cosmλ( )

where

Dlmp - function of satellite altitude and inclination

Φ&c

lmp and Φ&

s

lmp - latitude functions

+ sign denotes satellite is on ascending pass

- sign denotes satellite is on descending pass

Geographical mean radial orbit error:

€

Δγ =l=1

∞

∑m= 0

l

∑p= 0

l

∑ lmpD lmp

c˙ Φ lmC cosmλ+lmS sinmλ( )

Geographical variability error about the mean:

€

Δv =±l=1

∞

∑m= 0

l

∑p= 0

l

∑ lmpD lmp

s˙ Φ lmC sinmλ−lmS cosmλ( )


ATMOSPHERIC DENSITY CORRECTION

€

ρ=ρm Δρ

where mρ - normal density model ρΔ - a correction factor

€

Δρ= oC 1+ 1C s− s s

solar flux effect

€

o 1+ 2C pK[ ] geomagnetic effect

€

o 1+ 3C cosφ cos3 Dlmp˜ Φ lmps( )−1 2( )[ ] diurnal bulge heating effect

€

o 1+ 4C oδεsinφ

seasonal-latitudinal effect

€

o 1+ 5C rro−1

altitude dependent

€

o 1+C 6sin ω + M( )+C 7cos ω + M( )[ ] once per revolution

€

o 1+∑ iA cos2πf it+ iB sin2πf it( )[ ] long period

SIMPLIFIED DENSITY CORRECTION MODEL

€

6C = 6C ⋅ DC

€

7C = 7C ⋅ DC

€

Δρ= DC +C 6sin M +ω( )+C 7cos M+ω( )


PRECISION ORBIT DETERMINATION METHODS

Dynamical Equations of Motion:

€

˙ ̇ r =− GMr r3 +∑ f r ,v ,c ,t( )

vr , - Position and Velocity Vectors

( )tcvrf ,,,∑ - Perturbation Forces

Gravitational:

• Non-spherical Earth• Luni-solar and planetary• Solid Earth tides• Ocean tides• General relativity

Nongravitational:

• Atmospheric drag• Direct solar radiation pressure• Earth albedo radiation pressure• Empirical forces

c - Constant Parameters• Dynamical• Kinematical


LEAST SQUARES SYSTEM SOLUTION TECHNIQUES

( ) yWHPHWHx TT 110

−−+=)

Conventional Approach:

• Gauss and Gauss-Jordan Elimination

• Gauss-Siedel Iteration

• Choiesky Decomposition

Orthogonal Matrix Approach:

• Householder Accumulation

• Gram-Schmidt Orthogonalization

• Givens-Gentleman Square-Root-Free Accumulation


GIVENS-GENTLEMAN ACCUMULATION TECHNIQUES

Find mmQ ×−− orthogonal matrix such that,

[ ]

=

ebR

yHWQ0

21

M

where

nnR ×− upper triangular matrix

1×−nb column vector

( ) 1×−− nme column vector

The resulting equation

bxR =)

can be solved by back-substitutions


ORDINARY DIFFERENTIAL EQUATION NUMERICALINTEGRATION TECHNIQUES

Single-Step Integrators

• Runge-Kutta 2(4), and 7(8) IntegratorFixed or variable step,For first-order equations

Multi-Step Integrators

• Sampine-Gordon IntegratorVariable order or variable mesh,For first-order equations

• Adams-Bashforth IntegratorFixed mesh and order,For first-order equations

• Krogh-Sampine-Gordon IntegratorFixed mesh and order,For second-order equations


N-BODY PERTURBATION

Expressing in a geocentric coordinate system, the central bodyand the N-body forces can be expressed as

Δ

Δ−

∇∑+∇

+= 31i

ii

i GMUGMU

MmF

where F is the force acting on the satellite due to N-bodyattraction; m and M are masses of the satellite and the Earth,respectively; iΔ is the position vector between the satellite andthe perturbing mass, iM ; and U∇ refers to the gradient of thegeopotential.


EARTH’S GRAVITY FIELD

02 =∇ U

( )[ ]λλ mSmCPrR

rGMU nmnmnm

ne

mnsincossin

00+Φ

= ∑∑

=

∞

=

where ( )ΦsinnmP are associated Legendre function of Φsin of degreen and order m ; nmC and nmS are normalized spherical harmoniccoefficients whose values are functions of Earth’s massdistribution; and eR is Earth’s mean equatorial radius. GM , nmC

and nmS can be parameters within the estimated state vector inthe orbit determination process.


SOLID EARTH TIDES

)()(12

2rV

rRkrU n

ne

nn

+∞

=

=Δ ∑

where )(rUΔ is the change in potential at position r , nk are Lovenumbers of degree n , and nV is the disturbing tidal potential ofdegree n .

The changes in geopotential caused by the luni-solar tides can beexpressed in terms of time-dependent geopotential coefficients;that is nmC and nmS which can be expressed as follows

211

)!()!)(2(4

+

−+=Δ

+

mnmnnq

GMRkC nm

nen

nm

211

)!()!)(2(4

+

−+=

+

mnmnnu

GMRkS nm

nen

nm

and

( ) jmjPmnmn

rGM

q nmnj

j

jnm Ψ

+

−=

+=∑ coscos

)!()!(2

11

3

2θ

( ) jmjPmnmn

rGM

u nmnj

j

jnm Ψ

+

−=

+=∑ sincos

)!()!(2

11

3

2θ

where the index 3,2,1=j denotes Earth, moon and sun.


OCEAN TIDES

The disturbing ocean tide potential, UΔ , can be expressed asfollows

( )⋅Φ

+

+∑∑∑∑=Δ

+−

+=

∞

=sin

1214

1'

00nm

nen

n

mnew P

rR

nkRGU

µρπ

( )±± ±±⋅ nmnm mtC µµµ ελβη ~)(sin~

where G is the gravitational constant; wρ is the mean density ofthe sea water; eR is the mean equatorial radius; '

nk is the loaddeformation coefficient for degree n ; m is the order of thecoefficient; µ is the ocean tide constituent index; ]'[)( 1pNphst τβ =

are the Doodson arguments which define lunar and solarephemeris and t is the time; ][ 621 nnnn ⋅⋅⋅= are integer multipliersof Doodson arguments; ±

nmCµ

~ are the amplitudes of ocean tideconstituents; and ±

nmµε~ are the phase angles.

Let

( )λβηµµ

µ

mtCSC

nm

nm

±⋅

=

±

±

±

)(cossin~

( )βηµ

µµ

⋅

−

+=

−+

−+

cosnmnm SS

CCBA

( )βηµ

µ

⋅

−

++

−+

−+

sinnmCC

SS


21

)2)(12()!()!(4

0

2

−+−

+=

m

enm nmn

mnM

wRFδ

ρπ

+

+

12'1

nk n

where M is the mass of the Earth; ±C , ±S are ocean tidecoefficients; and m0δ is the Kronecker delta function, 1=δ for 0=m ;

0=δ , otherwise.

The total potential, U that includes gravitational and tidalpotential and thus be expressed

( )

++

+

= ∑∑∑∑

=

∞

=

λµλµφµµ

mnmBFSmnmAFCPrR

rGMU nmnmnmnmnm

ne

n

mnsincossin

00


ATMOSPHERIC DRAG MODEL

Drag Acceleration of a Composite Satellite:

++−= nCvvCv

mA

vvmACa LD

pD 'cos 221 γρ

where

ρ - Atmospheric density

DC - Drag coefficient for the satellite main body

'DC and LC = Drag and lift coefficients for the satellite solar panels

A and pA - Cross-sectional areas of main body and solarpanels perpendicular to v

v - Velocity of satellite relative to the atmosphere

γcos = nvv⋅

n - Unit vector normal to the solar panels

m - Satellite mass

Semi-Empirical Thermospheric Density Models

• Jacchia Models (1965, 1971, 1977)• Mass spectrometer-Incoherent Scatter Models (MSIS)• Drag Temperature Model (DTM)• ESRO-4 Models


CHARACTERISTICS OF UPPER ATMOSPHERE

Diurnal Variation

• Solar heating, atmospheric bulge

Solar Activities

• 27-day variation: solar rotation

• 11 year: solar cycle

• Solar decimeter flux (10.7 cm wavelength)

Geomagnetic Activities

• Geomagnetic index ),( pp AK

Seasonal-Latitudinal

• Seasonal migration of Helium

Semi-Annual Variation

• Year to year, vary in amplitudes

• Probably not due to solar activities

Others


ATMOSPHERIC DENSITY CORRECTION

ρρρ Δ= m

where mρ - normal density model ρΔ - a correction factor

−+=Δ

sssCCo 11ρ solar flux effect

[ ]KC p21+o geomagnetic effect

( )( )[ ]21~coscos1 33 −Φ+ s

lmplmpDC φo diurnal bulge heating effect

+ φ

εδ sin1 4

oCo seasonal-latitudinal effect

−+ 11 5

orrCo altitude dependent

( ) ( )[ ]MCMC ++++ ωω cossin1 76o once per revolution

( )[ ]tftf iiii BA ππ 2sin2cos1 +∑+o long period

SIMPLIFIED DENSITY CORRECTION MODEL

CCC D⋅= 66

CCC D⋅= 77

( ) ( )

++++=Δ ωωρ MM CCCD cossin 76


VALIDATION OF DENSITY CORRECTION MODEL

DENSITY CORRECTION MODEL STUDY

Simulation Summary

SEASAT 9-day orbit trajectory fit Jacchia 1971 drag model filledwith nominal exponential model estimating density correction

parameters.

Estimated Parameters Along-Track RMS (m) Radial RMS(m)

Global DC 9.18 0.67Daily DC 1.37 0.66Global 321 ,, CCC and 4C 6.04 0.443-Day 321 ,, CCC and 4C 0.48 0.02Daily DC , 3-Day 6C and

7C0.36 0.02


DENSITY CORRECTION MODEL ON STARLETTE ORBIT

Arc Length: March 18 – April 17, 1983

30-day arc

Date Type: Laser Range

Density Model: DTM

Estimated Parameters: 6,,,, CCCvr RDrv and 7C

Case Density Model Correction Range RMS (m)

1 10-day DC 127.0

2 10-day DC , 30-day 6C and 7C 111.0

3 10-day DC , 15-day 6C and 7C 83.6

4 5-day DC , 15-day 6C and 7C 78.5


DENSITY CORRECTION MODEL ON SEASAT ORBITS

Arc Lengths: July 28 – August 14, 198518-day arc

September 15 – October 8, 198524-day arc

Data Weights: Laser range - 1 mUnified S-Band - 1 cm/secTranet doppler - 1 cm/sec

Estimated Parameters: 76 ,,,,, CCCCvr RDrv and resonant geopotentialcoefficients

RMS

Case Arc MeasurementsUsed

DensityModel

Laser(cm)

USB(cm/s)

Tranet(cm/s)

Crossover(cm)

1 18day

USB, laser DTM, dailyDC

75.3 1.41 -- 92.9

2 18day

USB, laser DTM, dailyDC , 6C and

7C

73.3 1.39 -- 89.8

3 18day

USB, laser Exponential,_-day DC ,

6C and 7C

67.2 1.39 -- 95.9

4 24day

USB, laserand Tranet

J71, _-dayDC

77.0 1.58 1.59 80.0

5 24day

USB, laserand Tranet

J71, _-dayDC , 6C and

7C

75.8 1.56 1.27 76.0


SATELLITE RADIATION PRESSURE

[ ]dsunIFs

θβγθ cos)1()f( −+∫∫−=

n = unit vector normal surface element ds

u = unit vector along satellite-sun direction

θ = angle between n and u

I = incident flux ~1350 watts/m2

[ ] θθγβγθβγθ cos)()1()1((f)cos2)f( 2 KB −+−+=

γ = total reflectivity

β = specular reflectivity

(f)B = diffuse reflection and reradiation function

= 2/3 for Lambert’s cosine law

TTTT

bbff

bbffK 44

44

)(εε

εεθ

+

−=

bf εε , = front and back emissively

bf TT , = front and back temperature

from Georgivic, 1976.


CONVENTIONAL RADIATION PRESSURE MODELS

uAIF )1( η+−=

I = incident flux ~ 4.5 x 10-6 N/m2

A = cross-sectional area (may vary)

u = satellite-sun unit vector

η = “reflectivity” coefficient includes lumped effect ofspecular and diffuse reflection and unbalanced thermalradiation

All forces assumed to be along u ; off axis forces assumednegligible or having zero mean.

Solar panels can be modeled more exactly given knowledge oforientation, but approximations are still made about specular vs.diffuse reflection and about temperature imbalance.

For Topex:

A ~ 7 m2 (ignoring panels)

η ~ 0.2 – 0.5 (lumped reflectivity/emissivity)

Acceleration mFa = , mass = 2000 kg

a ~ 2-5 x 10-8 m/sec2

Therefore, 1% imbalance ≥ 2 x 10-10 m/sec2


Simplifying Hall and Fote [1981] and ignoring internallygenerated heat, surface temperature is determined by

4)()/()( TAtdtdTTCV σεφα −=

T = surface temperature

)(TCV = heat capacity

α = absorptivity

ε = hemispherical emittance

σ = Stefan-Boltzman constant

A = surface area

)(tφ = radiant power = θsinEA

E = solar irradiance (~1350 watts/m2)

θ = angle of irradiance


EARTH RADIATION PRESSURE MODEL

Earth Radiation Pressure is the radiation pressure exerted on asatellite due to:

• Sunlight reflected off the earth (shortwave)

• Heat emission of the earth (longwave

THINGS THAT “LOOK” LIKE EARTH RADIATION PRESSURE

• Solar Radiation Pressure Reflectivity

• Odd Zonal Geopotential Coefficients

• Some Tide Coefficients

• Drag Coefficients (under some circumstances)


MODELING THE TERRESTRIAL SPHERE

I. Nominal Diffuse Model

• Earth-atmosphere system assumed to be purely diffusereflector and emitter

• Albedo and longwave emissivity vary with:

o Latitude )(sin)(sin)( 2210 1 φφ PaPtaaa ++=

o Season )sin()cos()( 2101 tctccta ωω ++=

II. Enhanced Diffuse Model:

• Longitudinal albedo variations included. (4 x 4 sphericalharmonic expression)

III. Advanced Earth Radiation Pressure model:

• Anisotropy in reflected component included.

• Diurnal variations in albedo included.


EARTH ORIENTATION AND TIME MODEL(For updated models see IERS Standards 2005)

Conventional Inertial System (CIS):

J2000

Conventional Terrestrial System (CTS):

Lageos-derived system (CSR 2000) [Richard Eanes, 2000]

Precession:

1976 IAU formula [Lieske, 1977]

Nutation:

1980 IAU nutation formula [Wahr, 1981b; Seidelman, 1982]

Polar Motion:

Smoothed values from Lageos earth rotation series (CSR 2000)

UT1-TAI

Smoothed values from Lageos plus short-period (<40 days) tidalvariations from Yoder et al. [1981]


FORCE MODEL SUMMARY: GRAVITATIONAL

Earth*:

EGM-96, TEG-4, GGSM’s, EIGENGM = 398600.4414 km3 s-2

ae = 6378136.50 m

Luni-Solar:

Point mass sun, moon and planets with DE-200 ephemeris[Standish, 1982]

Solid Tides:

Neutral 1066 A earth model [Gilbert and Dziewonski, 1975] usingWahr [1981a] formulation

Frequency independent k2 = 0.3 and δ = 0 used

Honkasalo correction to C20 applied, 820 10391.1 −×−=ΔC per unit k2

Ocean Tides:

Schwiderski [1980] based on MERIT Standards [Melbourne et al.,1983]

General Relativity:

One-body (earth), 1== βγ

* GSFC, Texas, GFZ Gravity models


FORCE MODEL SUMMARY: NONGRAVITATIONAL

Atmosphere:

Drag Temperature Model (DTM) density model [Barlier, 1977],DTM200, MSIS, etc. with daily solar flux and geomagnetic indices[Solar-Geophysical Data, Boulder, NOAA]

Cross-sectional area model: Constant or area variation, mass andarea are satellite dependent

Solar Radiation Pressure:

Conical shadow model of radius 6371 km

Constant cross-sectional area model

Earth Radiation Pressure:

Zonal and periodic terms for earth albedo and emissivity


LASER MEASUREMENT MODEL

Model:

Instantaneous range

Laser Station Coordinates:

Lageos-derived coordinates CSR 2002 [R. Eanes, 2000]

Earth Tides:

h2 = 0.609, l2 = 0.0852, δ = 0 [Farrell, 1972]

(no ocean loading)

Center of Mass:

Correction from reflector to the center of mass (satellitedependent)


ALTIMETER MEASUREMENT MODEL

Model:

Ocean surface model composed of gravitational and rotationalpotential, Schwiderski [1980] ocean tide model and Levitus[1982] sea surface topography model.

Gravitational Potential:

TEG-4 (200x200) [Tapley et al., 2002]

oW = 6263861 m2/sec2

GM = 3698600.4414 km3/sec2

ea = 6378137 m

Ocean Tide Model:

Dynamical contribution using MERIT ocean tide only [Melbourne,et al., 1983]. Geometric correction at the ocean surface used thefull model Schwiderski [1980].

Sea Surface Topography Model:

Spherical harmonic representation (10x10) computed by Engelis[1985] based on the model developed by Levitus [1982]

Data Smoothing*:

Raw altimeter data smoothed to represent spectrum consistent todegree and order 50* Not implemented for the current gravity field solution



Observed:

nbhhhhh SSTacs ++++++= ζ

- sh is the instantaneous distance between the satellite altimeterand the ocean surface.

- nbhh ac ,,, are corrections and random error.

- SSTh is the sea surface topography ( SSTh can be represented by atime series).

( ) CSC lmmmml

SST Pmmh '

00

''

01)(sinsincos ++= ∑∑

−=

φλλ

- ζ is the ocean tide height

)),(cos(),( λφδχλφζζ kkkkk

−+Θ=∑

[ ] )(sin)sin()cos(00

φλλ lmkklmkklm

l

mlkPmm SC ±Θ+±Θ=

±±−

+=

∞

=∑∑∑∑

- kΘ , kχ are the Doodson argument and phase correction forconstituent k

- ),( λφζ k is the amplitude of the tide height at geographic location),( λφ

- ),( λφδ k is the phase of the tide height at geographic location ),( λφ

- SCSC klmklmklmklm

−−++ ,,, are the unnormalized prograde and retrogradetide coefficients



(continued)

Computed:

ec rrh rrr−=

- err can be computed from:

€

w=wo ; ∇w∇w

=) u h

- w is the potential for the global ocean surface, ow is a constant

- w∇ is the gradient of the potential, w , and is normal to w .

- hu) represents the unit vector for the altimeter measurement.

Observation Equation:

chhyr

−=

Dynamically consistent model if w is used as the perturbingpotential affecting the satellite’s orbit.

NOTES ON ORBIT DETERMINATION MODELS C.K. Shum, Ohio … · - Drag coefficient for the satellite...

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Transcript of NOTES ON ORBIT DETERMINATION MODELS C.K. Shum, Ohio … · - Drag coefficient for the satellite...