Basics of Orbital Mechanics - UPM Orbital Elements – Orbit Orientation (page 1) Ω: Right...

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Basics of Orbital MechanicsInternational Space Development Conference

Washington, DC19-22 May 2005

Gary Bingaman & Seth PotterThe Boeing Company

Phantom WorksHuntington Beach, CA

[email protected]


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Orbital Mechanics• An orbit is the path that a body takes under predominantly

the influence of gravity• Orbital mechanics is

– the science of defining and determining that path– the engineering art of using or modifying that path to perform a

specific function or accomplish a particular mission


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Topics• Fundamental Laws and Constants• Orbit Shapes• Orbital Elements• Basic Equations• Orbital Perturbations• Orbital Maneuvers• Special Orbits


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Newton’s Three Laws of Motion• First Law

– Every body continues in its state of rest or uniform motion in astraight line unless it is compelled to change that state by forces impressed upon it.

• Second Law– The rate of change of momentum is proportional to the force

impressed and is in the same direction as that force.• Third Law

– To every action there is always opposed an equal reaction.


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Kepler’s Three Laws of Orbital Motion• First Law

– Each planet moves along an elliptical orbit with the sun as one focus.

• Second Law– The line joining the sun and the planet sweeps out equal areas in equal times.

• Third Law– The squares of the periods of the planets are proportional to the cubes of

their mean distances from the sun.

r = a * (1 - ε 2)1 + ε * cos θ

constant = r2 *dθdt

P = * a3/22πµ1/2

r


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• Universal Law of Gravitation:

• Applying Newton’s equation: Fg = mg

• Replacing: GM = µ• Results in: µ = gr2

Gravitational ConstantsFg =

GMmr2

Fg = force acting on a body in a gravitational fieldG = universal gravitational constant

6.6742 * 10-11 Nm2 / kg2

M = mass of the large central bodym = mass of the small orbiting bodyr = distance between the centers of the bodies

g = acceleration due to gravity at radius rµ= gravitational parameter – unique for each central body

3.986 * 1014 m3 / sec2 (1.4077 * 1016 ft3 / sec2)

g = GMr2


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Conic Sections• Orbital motion occurs along conic sections

Circle(unique shape)

Ellipse(many shapes)

Parabola(unique shape)

Hyperbola(many shapes)

Conic Ellipticity UseCircle ε = 0 Ideal (theoretical)Ellipse 0 < ε < 1 Planetary orbits

Parabola ε = 1 Escape orbitsHyperbola ε > 1 Transfer orbits


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Ellipse is the Most Common Orbit

r = radius from the center of the prime focusrp = perigee radius (closest point)ra = apogee radius (farthest point)a = semi-major axisε = eccentricity

focus(prime) focus

rp ra

2a

ε = ra + rp

ra - rp


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Isaac Newton’s Velocity Equation

rp = perigee radius (closest point)ra = apogee radius (farthest point)a = semi-major axisε = eccentricityµ = gravitational parameterr = radius at any point in the orbitv = velocity at any point in the orbit

rp ra

2a

r

V

r2

a1-v2 = µ * [ ]


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Orbital Energy

• Newton’s velocity equation:

• Rearranged:

• Note:– Total energy is a function only of the semi-major axis– As the satellite moves along its orbit, the velocity and the

radius continually change– There is a continuous exchange between kinetic and

potential energies

r2

a1-v2 = µ * [ ]

2a-µ

2v2

- rµ=

TotalEnergy

KineticEnergy

PotentialEnergy+=


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Inertial Reference System

Ecliptic

Equatorial

Plane

Earth’s Orbit

Sun

Earth

PolarAxis

N

VernalEquinox

X

Z

YOrbiting Satellite


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Keplerian Orbital Elements –Orbit Size and Shape

a = semi-major axisr0 = radius of central bodyh = altituder = radiusε = eccentricity (oblateness)

rarp

hahp

a: Semi-Major Axis

a = 2

ra + rprp = r0 + hp

ra = r0 + ha

rarp

2a

ε: Orbital Eccentricity

ε = 0 circular orbitε = 1 parabolic orbitε = ra + rp

ra - rp


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Keplerian Orbital Elements –Orbit Orientation (page 1)

Ω: Right Ascension of Ascending Node

ΩEquatorial PlaneOrb

it Plan

e

Satellite

Nodal Crossing Point (ascending)

Vernal Equinox

Equatorial crossing point – measured in the equatorial plane between the vernal

equinox and the ascending node

• Vernal equinox (first point of Aries) is the location on the celestial sphere where the Sun apparently crosses the Earth’s equator going North (in the Spring)

• Right Ascension is measured in an inertial frame

• Longitude is measured in a relative frame

• Right Ascension is related to Longitude through time

• Ascending node is the point where the satellite crosses the equator going North


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Keplerian Orbital Elements –Orbit Orientation (page 2)

i: Orbital Inclination

Angle between equatorial plane and orbit plane

0o < i < 90o posigrade orbit(West to East)

90o < i < 180o retrograde orbit(East to West)

Equator i

Satellite

Orbit Plane

ω: Argument of Perigee

Equatorial Plane

Nodal Crossing Point (ascending)

Perigee Point

Orbit Plane

Satellite

ω

Angle between the ascending node and the perigee pointAscending node is the point in the orbit where the satellite crosses the equator going North


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Keplerian Orbital Elements –Satellite Position

θ: True Anomaly

Satellite

rθ

Perigee

Earth centered angle between perigee and the current satellite location

r = a * (1 - ε 2)1 + ε * cos θ


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Basic Equations

µ * a * ( 1 - ε 2)r2 * v2γ = sin-1 √1 - [ ]

r2

a1-v = √µ * [ ]

a = 2

ra + rp

ε = ra + rp

ra - rp

semi-major axis:

eccentricity:

velocity:

flight path angle:

orbital period: p = µ1/22 * π * a3/2


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Sources of Orbital Perturbations

Dominant effects• Circular Altitude = 250 nm• Inclination = 28.5º

• Nodal regression = 0.986º/day• Apsidal rotation = -3.587º/day

Dominant effects• Circular Altitude = 250 nm• Inclination = 28.5º

• Nodal regression = 0.986º/day• Apsidal rotation = -3.587º/day

Spherical

Second Harmonic

Third Harmonic

Fourth Harmonic

J2

• Gravitational Anomalies– Non-spherical central body (Earth)

• Oblateness• Higher order terms

– Third bodies• Moon, Sun, other

• Atmospheric drag• Solar radiation

– Direct pressure– Atmospheric function that changes the

effects of atmospheric dragOrbit Solar Solar

Circular Min. Max.Altitude Lifetime Lifetime

(nm) (days) (days)100 2 1150 50 10210 550 80250 1200 400300 3400 2600350 13400 12600400 42000 41000450 127000 128000500 341000 340000

@ W/CdA = 10 lb/ft^210

J3

J4


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Orbit Types• Classification by Altitude:

– Low Earth Orbit (LEO)• ~<1000 nm

– Medium Earth Orbit (MEO)– High Earth Orbit (HEO)

• ~>10000 nm– Geostationary Earth Orbit (GEO)

• (19320 nm)

• Classification by Orientation to Equator:

– Equatorial (inclination = 0º)– Low Inclination– High Inclination

• Sun synchronous – dependant on altitude

– i = 96.3º @ 100 nm– i = 99.1º @ 500 nm

– Polar (inclination = 90º)– Critically Inclined – no apsidal

rotation• i = 63.4º - posigrade• i = 116.6º - retrograde

• Classification by Shape– Circular– Elliptical

• Molniya– Critically inclined highly elliptical

• Hohmann Transfer• Other

– Parabolic (marginal escape trajectory)

– Hyperbolic (escape trajectory)

• Classification by Direction of Motion:

– Prograde, Direct, or Posigrade: satellite moves eastward

• inclination < 90º– Retrograde: satellite moves

westward• inclination > 90º

– Polar (inclination = 90º)


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Orbit Transfer

• Two Impulse– High thrust rockets– One perigee burn– One apogee burn– Close to ideal delta V

• Multi-Impulse– High or medium

thrust rockets– More than one

perigee burn– More than one

apogee burn– Steering losses

• Continuous Burn– Low thrust rockets– Spiral trajectory– Steering losses

Two Impulse Multi-Impulse Continuous BurnThrust to Weight Ratio >0.1 <0.1 <<0.001Total Delta V (ft/sec) 14000 to 17000 14000 to 19200 ~19200Travel Time < 1 day several days > 1 week

LEO to GEO Illustrative ExampleIdeal MethodMinimum Energy Method

“Primary” Method


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Hohmann Transfer Orbit(Minimum Energy)

Hohmann Transfer Orbit

Initial OrbitFinal Orbit

Burn 1

Burn 2

Not to ScaleCircular Velocity in Initial OrbitPerigee Velocity in Transfer OrbitDelta Velocity – Burn 1

Apogee Velocity in Transfer OrbitCircular Velocity in Final OrbitDelta Velocity – Burn 2


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Plane Change

∆v = 2 * v * sin [ ]2Θ

Θ

vv

For a 60 degree plane change the plane change ∆v equals the circular orbital velocityFor a 60 degree plane change the plane change ∆v equals the circular orbital velocity


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Combined Hohmann Transfer and Plane Change

∆V = √(( V12 ) + ( V22 ) - ( 2 * V1 * V2 * COS( Θ )))

V1 = velocity in orbit 1V2 = velocity in orbit 2Θ = inclination change between the two orbits ( ∆i )

V1

V2Θ


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Molniya Orbit• Inclined at 63.43 degrees

– No apsidal rotation– Apsidal rotation (deg. / rev):

• Highly elliptical• Original purpose

– Orbit used by Russia for communications satellites• Launch sites are too high in latitude to achieve economical GEO orbits• Place satellites into highly elliptical orbits with apogee located over

focal location – long dwell time• With no apsidal rotation the apogee would remain over the desired focal

location• Balance between altitudes and ellipticity provides daily repeating

ground track

= 540 * J2 * [ ]2 * [2- (5/2) * cos 2(i)](a * (1 - ε2))

Reequatorialdωrev


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Molniya Orbit – View From Space 1Orbital Period = 11 hr 57 mini = 63.43ºhp = 500 km (270 nm)ha = 39850 km (21518 nm)ε = 0.74


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Molniya Orbit – View From Space 2Orbital Period = 11 hr 57 mini = 63.43ºhp = 500 km (270 nm)ha = 39850 km (21518 nm)ε = 0.74


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Molniya Orbit – Ground Track

Time spent in this part of orbit:10 hr 10 min

Orbital Period = 11 hr 57 mini = 63.43ºhp = 500 km (270 nm)ha = 39850 km (21518 nm)ε = 0.74


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Walker Constellation Example – t/p/f = 6/3/1, RAAN Spread = 360º (view 1)

6 satellites3 planes (2 satellites per plane)Phase factor = 1Evenly spaced ascending nodes around entire equator


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Walker Constellation Example – t/p/f = 6/3/1, RAAN Spread = 360º (view 2)

6 satellites3 planes (2 satellites per plane)Phase factor = 1Evenly spaced ascending nodes around entire equator


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Walker Constellation Definitionst / p / f = total number of satellites / number of planes / phase factor

phase angle in degrees = ( ) * f

where: f = integer number 0 ≤ f ≤ p - 1

(Phase angle measured between ascending node and satellite when satellite in next most Western plane is at ascending node)

Phase angle

Ascending node

Equator

Constellation planes

SatelliteSatellite at ascending

node

360ºt


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Walker Constellation Example – t/p/f = 6/3/1, RAAN Spread = 360º (ground tracks)


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Repeating Ground TracksResonant Orbits

• Provide repeating periodic coverage of a specific area– Null Earth rotation: 1.1408 * 10-5 deg./sec– Null nodal regression:

• Drift of the ascending node– Toward the West for 0 ≤ i ≤ 90º– Toward the East for 90 ≤ i ≤ 180º

• Nodal regression (deg. / rev):

– Designed to repeat in an integer number of orbits• Daily• Every “x” number of days

= -540 * J2 * [ ]2 * cos(i)(a * (1 - ε2))

ReequatorialdΩrev


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Repeating Ground Tracks 1

-90

-75

-60

-45

-30

-15

0

15

30

45

60

75

90

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

Longitude (deg)

Latit

ude

• 14 orbits to repeat ground track in 1 day• hcirc = 446.48 nm• i = 51.6º



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-90

-75

-60

-45

-30

-15

0

15

30

45

60

75

90

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

Longitude (deg)

Latit

ude




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-90

-75

-60

-45

-30

-15

0

15

30

45

60

75

90

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

Longitude (deg)

Latit

ude

• 41 orbits to repeat ground track in 3 days• hcirc = 511.12 nm• i = 51.6º

• 41 orbits to repeat ground track in 3 days• hcirc = 511.12 nm• i = 51.6º


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Geosynchronous Orbits• Satellite always remains over the same location

over the Earth’s equator (geostationary)• Inclination and eccentricity are close to zero• Useful for communication or weather satellites• Limited number of satellite locations (slots)

available

Altitude ≈ 35785 km (19322 nm)


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Geosynchronous Satellite with 2 deg. Inclination and 0.001 Eccentricity


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Sun Synchronous Orbit Illustration

Ecliptic

Equatorial

Plane

Earth’s Orbit

Sun

Earth

PolarAxis

VernalEquinox

X

YOrbiting Satellite

N

Z


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Sun Synchronous Orbit Parameters• Satellite orbit maintains the same relationship to

the sun– Satellite always observes the Earth at the same local time

Nodal regression (deg. / rev):

For sun synchronous orbit:

≈ 360º / 365.2425 days = 0.9856º per day

= -540 * J2 * [ ]2 * cos(i)(a * (1 - ε2))

ReequatorialdΩrev

dΩdt


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Summary• Orbital mechanics is a function of Kepler’s Laws

of orbital motion as modified by Newton’s Laws of motion and gravity

• Knowledge of gravity forces are used to predict future satellite states given initial state vector –six Keplerian orbital elements

• Mission requirements dictate, or provide options for, the choice of orbital parameters – size, shape, and orientation


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Bibliography• Larson, Wiley J., and Wertz, James R., (editors), Space Mission Analysis

and Design, Second Edition, published jointly by Microcosm, Inc., El Segundo, CA, and Kluwer Academic Publishers, Dordecht, The Netherlands, 1992; Chapters 6 and 7.

• http://liftoff.msfc.nasa.gov/academy/rocket_sci/orbmech/orbmech.html

• http://homepages.primex.co.uk/~omen/omen13.htm#ORBITAL%20TERMS

• Satellite Tool Kit Help Menu

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