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# OPS Forum Fundamentals of orbits 05.05.2006

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Uwe Feucht, a flight dynamics expert, will present fundamental concepts related to the mathematics and physics of orbital calculations and discuss typical flight dynamics tasks in the support of space …

Uwe Feucht, a flight dynamics expert, will present fundamental concepts related to the mathematics and physics of orbital calculations and discuss typical flight dynamics tasks in the support of space flight.

Topics to be covered include:
Mathematical description of satellite

orbits
Orbit perturbations
Orbit analysis
Orbit change manoeuvres
Orbit maintenance
Typical types of orbits

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• 1. Fundamentals of Orbit OPS-G Forum 05.05.2006 Uwe Feucht JFK/RB, 2005-11-30
• 2. Background This Presentation is compiled from: Lecture on Satellite Technique, TU Umea/TU Lulea Spacecraft Operations Course, DLR
• 3. 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
• 4. 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
• 5. 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
• 6. 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)
• 7. 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
• 8. 1. Mathematical Description of Satellite Orbits a e N i i
• 9. 1. Mathematical Description of Satellite Orbits N Ω Ω ω γ ν ν
• 10. 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
• 11. 2. Orbit Perturbations Sources of Perturbations Earth Gravitational Field Air Drag Solar Radiation Sun/Moon Influence Thruster Activity others (e.g. planets, albedo)
• 12. 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
• 13. 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]
• 14. 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)
• 15. 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
• 16. 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
• 17. 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
• 18. 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)
• 19. 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
• 20. 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]
• 21. 5. Orbit Maintenance
• 22. 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
• 23. 6. Typical Types of Orbits Example: Ground Track for 20° Inclination
• 24. 7. Flight Dynamics Typical Tasks Ground Station Coverage (1) Ground Track for Polar Orbit with 87° Inclination (a)
• 25. 7. Flight Dynamics Typical Tasks Ground Station Coverage (2) Ground Track for Polar Orbit with 87° Inclination (b)
• 26. 7. Flight Dynamics Typical Tasks Ground Station Coverage (3) Visibility Plot
• 27. 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
• 28. 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)
• 29. 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°)