Fundamentals of Orbit
OPS-G Forum
05.05.2006
Uwe Feucht
JFK/RB, 2005-11-30

Background
This Presentation is compiled from:
Lecture on Satellite Technique, TU Umea/TU Lulea
Spacecraft Operations Course, DLR

Content
1. Mathematical Description of Satellite Orbits
2. Orbit Perturbations
3. Orbit Analysis
4. Orbit Change Maneuvers
5. Orbit Maintenance
6. Typical Types of Orbits
7. Typical Flight Dynamics Tasks

1. Mathematical Description of Satellite Orbits
Orbital Parameters (1)
Geometry in the Orbital Plane a = semi-major axis
e = (numerical) eccentricity
ω = argument of perigee
b = semi-minor axis
b 8
S p = parameter (of a cone)
p
r
r = (orbit-) radius
E = Earth center
a·e v
A = apogee
A 8 8 8
w
8
P
a
E P = perigee
8
ν = true anomaly
A.N. A.N. =ascending node
u = ω+v
= argument of latitude
S = satellite position

1. Mathematical Description of Satellite Orbits
Orbital Parameters (2)
Geometry in Space
p
3
N
i = inclination
Ω = right ascension of the
ascending node
B
ω = argument of perigee
~ = Vernal equinox (cut of
Or
bit Earth equator and
P
ecliptic)
Ω ω
A.N.
i
p
A.N. =ascending node
~ p Equator 2
1
P = perigee
N = north pole
B = orbital plane

1. Mathematical Description of Satellite Orbits
Orbital Parameters (3)
Commonly used Orbital Parameters
Keplerian-Elements a =
semi-major axis
e =
eccentricity
i =
inclination
Ω =
right ascension of the ascending
node
ω = argument of perigee
M/ν = mean/true anomaly
(used for visualization, not applicable for computation because of
singularities for e = 0, i = 0°, i = 180°)
State Vector r, r = v
&
(Position-, Velocity Vector)

1. Mathematical Description of Satellite Orbits
Mean vs. osculating Elements
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

1. Mathematical Description of Satellite Orbits
a
e
N
i
i

1. Mathematical Description of Satellite Orbits
N
Ω
Ω
ω
γ
ν
ν

1. Mathematical Description of Satellite Orbits
Satellite Velocities
⎛ 2 1⎞ GTO:
Velocity on elliptical path: V= µ⎜ − ⎟ Perigee: ~10.0 km/s
⎝ r a⎠ Apogee: ~ 1.7 km/s
µ LEO: ~ 7.6 km/s
Velocity on circular path: VC = GEO: ~ 3.0 km/s
a Moon: ~ 1.0 km/s
µ
1st cosmic velocity: VC1 = = 7.905 km / s
RE
2µ
2nd cosmic velocity (escape): VC2 = = 2 VC1 = 11.180 km / s
RE

2. Orbit Perturbations
Sources of Perturbations
Earth Gravitational Field
Air Drag
Solar Radiation
Sun/Moon Influence
Thruster Activity
others (e.g. planets, albedo)

2. Orbit Perturbations
Earth Gravitational Field (1)
Effect on nodal line
Due to Earth ellipsoid,
rotation of the nodal line
around the pole axis
& 1
Ω ≈ −C ⋅ J2 ⋅ cos(i ) ⋅ 3.5
a
Orbital Elements
(Example)
a= 7400 km
i = 57°
J2 = 0.1 (unrealistic!)
Duration: 1 day

2. Orbit Perturbations
Earth Gravitational Field (2)
Sun-synchronous Orbits
6000
Requirement: 5000
Rotation of the orbital plane
4000
around the pole axis
Altitude [km]
= 3000
Mean motion of the Earth Node Regression:
2π
around the Sun 2000
& 1
Ω ≈ −C ⋅ cos(i ) ⋅ 3.5 =
1000 a year
0
90 100 110 120 130 140 150 160 170 180
Inclination [deg]

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

4. Orbit Change Maneuvers
In-Plane Maneuver: Change of Perigee or Apogee Height
Example: Lift of perigee (e.g. from TO into GEO)
ra = 42164 km (TO apogee radius)
aTO= 24400 km (TO semi-major axis) ∆V VGEO
aGEO=42164 km (GEO semi-major VTO
axis)
⎛2 1 ⎞
⎜ −
vTO = µ⎜ ⎟
⎟
= 1.603 km/s
⎝ ra aTO ⎠ GEO TO
µ
vGEO = = 3.075 km/s
aGEO
⇒ ∆v = 1.472 km/s

4. Orbit Change Maneuvers
Out-Of-Plane
Maneuver
v v ⎛ 2 1⎞
v1 = v 2 = v = µ ⎜ − ⎟
⎝r a⎠
∆i
∆v = 2 ⋅ v ⋅ sin V1
2 ∆V
2 ∆i
V2
1
Example: GTO (Geostationary Transfer Orbit) ⇒ v = 1.603 km/s
a = 24400 km, r= 42164 km, ∆i = 7° (Ariane ⇒ ∆v = 0.195 km/s
Launch)

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
TerraSar-X Orbit Maintenance Maneuver
200
Orbit Characteristics 100
∆ λ [m ]
• Sun-synchronous 0
• Repeat cycle: 11 days
-100
• Requirement
– Tolerance interval for -200
nodal longitude: ∆s = 0 10 20 30 40 50 60
Time [days]
70 80 90 100
±200 m
160
1 v
Velocity Increment : ∆vt = ⋅ ∆a ⋅ 120
2 a
∆ a [m]
80
Use of predicted flux values for 40
orbit propagation
0
0 1 2 3 4 5
M ET [y]

5. Orbit Maintenance

6. Typical Types of Orbits
GEO
r = a = 42164 km
h = 35786 km
U = 24 hours
IO
LEO
r = a = 6678... ca. 7878 km
a h = 300... ca. 1500 km
U = 90 min
GTO
a = 24370 km
h = 200... 35786 km
U = 10 hours

6. Typical Types of Orbits
Example: Ground Track for 20° Inclination

7. Flight Dynamics Typical Tasks
Ground Station Coverage (1)
Ground Track for Polar Orbit with 87° Inclination (a)

7. Flight Dynamics Typical Tasks
Ground Station Coverage (2)
Ground Track for Polar Orbit
with 87° Inclination (b)

7. Flight Dynamics Typical Tasks
Ground Station Coverage (3)
Visibility Plot

7. Flight Dynamics Typical Tasks
Orbit Determination Principles (1)
Measurement
Meßpunkte zi
z z Points zi
vy
ry vz
vx
r
rz
berechnete
Computed
rx Bahn (r,v)
orbit
x y x y
Orbit estimation by averaging:
Position
vector r ∑ [zi − fi (r, V )]2 = Min
i
Velocity vector
V

7. Flight Dynamics Typical Tasks
Orbit Determination Principles (2)
Satellite Tracking
.
ρ ⇒ relative velocity measurement
(Doppler)
ρ ⇒ distance measurement (Ranging)
Range, Doppler
Angle Measurements (Auto-track)
A
h (Azimuth)
(Elevation)

7. Flight Dynamics Typical Tasks
Station Keeping
13.0° O 7.0° O
19.2° O
28.5° O 0.6° W
EUTELSAT II-F1
ASTR EUTELSAT II-F4
A 1A .. HOT BIRD 1 TV-SAT 2
DFS 2 1D
Control box
Geostationary Orbit Orbit determination
altitude: 36000 km and corrections
performed by
control center Orbit perturbations
caused by Earth,
Sun and Moon
100 - 150 km
(± 0.1°)