JFK/RB, 2005-11-30

Fundamentals of Orbit

OPS-G Forum05.05.2006

Uwe Feucht

BackgroundBackground

This Presentation is compiled from:

Lecture on Satellite Technique, TU Umea/TU Lulea

Spacecraft Operations Course, DLR

ContentContent

1. Mathematical Description of Satellite Orbits2. Orbit Perturbations3. Orbit Analysis4. Orbit Change Maneuvers5. Orbit Maintenance6. Typical Types of Orbits7. Typical Flight Dynamics Tasks

1. Mathematical Description of Satellite Orbits

88

8

8

88

pb

ava·e PA w

A.N.

r

S

E

Geometry in the Orbital Plane

Orbital Parameters (1)

a = semi-major axise = (numerical) eccentricityω = argument of perigeeb = semi-minor axisp = parameter (of a cone) r = (orbit-) radiusE = Earth centerA = apogeeP = perigeeν = true anomalyA.N. =ascending node u = ω + v

= argument of latitudeS = satellite position

A.N.~

Ω ωP

i

p

pp

N

Equator

B

12

3

Orbit

Geometry in Space

1. Mathematical Description of Satellite Orbits

Orbital Parameters (2)

i = inclinationΩ = right ascension of the

ascending nodeω = argument of perigee ~ = Vernal equinox (cut of

Earth equator and ecliptic)A.N. =ascending nodeP = perigeeN = north poleB = orbital plane

vrr =&,

Commonly used Orbital Parameters

a = semi-major axise = eccentricityi = inclinationΩ = right ascension of the ascending

nodeω = argument of perigeeM/ν = mean/true anomaly

Keplerian-Elements

(used for visualization, not applicable for computation because of singularities for e = 0, i = 0°, i = 180°)

State Vector(Position-, Velocity Vector)

Orbital Parameters (3)

1. Mathematical Description of Satellite Orbits

Looking at a real orbit shows that at each instant the satellite motion can be described by a different set of orbital elements.

These instantaneous parameters are called osculating elements.

An average over the osculating parameters yield the mean elements

Mean vs. osculating Elements

1. Mathematical Description of Satellite Orbits

1. Mathematical Description of Satellite Orbits

i

N

a

e

i

1. Mathematical Description of Satellite Orbits

N

Ω

γ

ν

Ω

ω

ν

1. Mathematical Description of Satellite Orbits

Satellite Velocities

1st cosmic velocity: skmR

VE

C /905.71 ==µ

2nd cosmic velocity (escape): skmVR

V CE

C /180.112212 ===

µ

⎟⎠⎞

⎜⎝⎛ −=

arV 12µVelocity on elliptical path:

Velocity on circular path:a

VCµ

=

GTO:Perigee: ~10.0 km/sApogee: ~ 1.7 km/s

LEO: ~ 7.6 km/sGEO: ~ 3.0 km/sMoon: ~ 1.0 km/s

Sources of Perturbations

2. Orbit Perturbations

Earth Gravitational Field

Air Drag

Solar Radiation

Sun/Moon Influence

Thruster Activity

others (e.g. planets, albedo)

2. Orbit Perturbations

Effect on nodal lineDue to Earth ellipsoid, rotation of the nodal line around the pole axis

Orbital Elements(Example)

a= 7400 kmi = 57°J2 = 0.1 (unrealistic!)Duration: 1 day

Earth Gravitational Field (1)

5.321)cos(

aiJC ⋅⋅⋅−≈Ω&

0

1000

2000

3000

4000

5000

6000

90 100 110 120 130 140 150 160 170 180

Inclination [deg]

Alti

tude

[km

]

2. Orbit Perturbations

yearaiC π21)cos( 5.3 =⋅⋅−≈Ω&

Earth Gravitational Field (2)

Rotation of the orbital plane around the pole axis

=

Mean motion of the Earth around the Sun

Requirement:

Node Regression:

Sun-synchronous Orbits

2. Orbit Perturbations

Air Drag

• Generally decrease of semi-major axis

• For elliptical orbits decrease of apogee height

• For circular orbits decrease of orbital height

• Decrease of orbital period (increase of satellite velocity)

• Depending on Solar activity (Solar Flux)

3. Orbit Analysis

High Eccentric Orbits

• Motion of a satellite with respect to the pericenter

• Used for– transfer orbits to GEO

(e=0.7)– transfer orbits to Moon

(e=0.966)– scientific missions

3. Orbit Analysis

Circular Orbits (e ≈ 0.0)• Motion of a satellite with

respect to equator crossings– draconic Motion

• Used for– remote sensing satellite

orbits (LEO)– manned missions

o space stations MIR and ISS

o STS (Space Shuttle)– orbit selection of remote

sensing satellites

GEO

TO

a

ar

µ

µ

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

GEO

aTO

v

12v = 1.603 km/s

= 3.075 km/s

4. Orbit Change Maneuvers

In-Plane Maneuver: Change of Perigee or Apogee Height

ra = 42164 km (TO apogee radius)

aTO= 24400 km (TO semi-major axis)

aGEO=42164 km (GEO semi-majoraxis)

⇒ ∆v = 1.472 km/s

Example: Lift of perigee (e.g. from TO into GEO)

VTO

VGEO∆V

TOGEO

4. Orbit Change Maneuvers

Out-Of-PlaneManeuver

V2

∆V

1

2

V1

∆i

Example: GTO (Geostationary Transfer Orbit)a = 24400 km, r= 42164 km, ∆i = 7° (Ariane

Launch)

2i sinv2v

12vvv 21

∆⋅⋅=∆

⎟⎠⎞

⎜⎝⎛ −===

arµ

vv

⇒ v = 1.603 km/s

⇒ ∆v = 0.195 km/s

5. Orbit Maintenance

Purpose of orbit maintenance maneuvers

• Compensate orbit decay

• Keep ground track stable

• Keep time relation of orbit stable

• Keep orbit form stable

• Achieve mission target orbit

5. Orbit Maintenance

Orbit Characteristics• Sun-synchronous• Repeat cycle: 11 days• Requirement

– Tolerance interval for nodal longitude: ∆s = ±200 m

avavt ⋅∆⋅=∆

21:IncrementVelocity

TerraSar-X Orbit Maintenance Maneuver

0

40

80

120

160

0 1 2 3 4 5MET [y]

∆a

[m]

0 10 20 30 40 50 60 70 80 90 100

-200

-100

0

100

200

Time [days]

∆λ

[m]

Use of predicted flux values fororbit propagation

5. Orbit Maintenance

a

r = a = 42164 km h = 35786 km U = 24 hours

GEO

LEOr = a = 6678... ca. 7878 km h = 300... ca. 1500 km U = 90 min

IO

GTO a = 24370 km h = 200... 35786 km U = 10 hours

6. Typical Types of Orbits

Example: Ground Track for 20° Inclination

6. Typical Types of Orbits

7. Flight Dynamics Typical Tasks

Ground Station Coverage (1)Ground Track for Polar Orbit with 87° Inclination (a)

Ground Station Coverage (2)Ground Track for Polar Orbitwith 87° Inclination (b)

7. Flight Dynamics Typical Tasks

Ground Station Coverage (3)Visibility Plot

7. Flight Dynamics Typical Tasks

x y

z

x y

z

vx

vz

vy

r

rx

rz

ry

Meßpunkte zi

berechneteBahn (r,v)

Position vector r

Velocity vector V

Orbit estimation by averaging:

MeasurementPoints zi

Computedorbit

Orbit Determination Principles (1)

[ ] Minfzi

ii =−∑ 2),( Vr

7. Flight Dynamics Typical Tasks

Orbit Determination Principles (2)Satellite Tracking

ρ ⇒ distance measurement (Ranging)

ρ ⇒ relative velocity measurement(Doppler)

.

Angle Measurements (Auto-track)

h(Elevation)

A(Azimuth)

7. Flight Dynamics Typical Tasks

Range, Doppler

Station Keeping

Geostationary Orbit altitude: 36000 km

Orbit determination and corrections performed by control center Orbit perturbations

caused by Earth, Sun and Moon

Control boxTV-SAT 2

DFS 2

ASTRA 1A .. 1D

EUTELSAT II-F1

HOT BIRD 1EUTELSAT II-F4

0.6° W

7.0° O13.0° O19.2° O

28.5° O

100 - 150 km (± 0.1°)

7. Flight Dynamics Typical Tasks