1. Dr. Andrew Ketsdever Lesson 3 MAE 5595
Orbital Mechanics II: Transfers, Rendezvous, Patched Conics, and Perturbations
Dr. Andrew Ketsdever
Lesson 3
MAE 5595

2. Orbital Transfers Hohmann Transfer
Efficient means of increasing/decreasing orbit size
Doesn’t truly exist
Assumptions
Initial and final orbits in the same plane
Co-apsidal orbits (Major axes are aligned)
ΔV is instantaneous
ΔV is tangential to initial and final orbits (velocity changes magnitude but not direction)

3. Hohmann Transfer

4. Hohmann Transfer

5. Conceptual Walkthrough alt1 = 300 km alt2 = 1000 kmV1
ΔV1
Slides Courtesy of Major David French, USAFA/DFAS

6. 2 2
Vt1

7. 2 2
ΔV2
Vt2

8. 2 2 V2

9. Time of Flight 2 2

10. Hohmann Transfer

11. Orbital Transfers One Tangent Burn Transfer
First burn is tangent to the initial orbit
Second burn is at the final orbit
Transfer orbit intersects final orbit
An infinite number of transfer orbits exist
Transfer orbit may be elliptical, parabolic or hyperbolic
Depends on transfer orbit energy
Depends on transfer time scale

12. One-Tangent Burn

13. One-Tangent Burn

14. Spiral Transfer Expect to multiply by as much as a
factor of 2 for some missions

15. Orbital Transfer Plane Changes Simple Combined
Only changes the inclination of the orbit, not its size
Combined
Combines the ΔV maneuver of a Hohmann (tangential) transfer with the ΔV maneuver for a plane change
Efficient means to change orbit size and inclination

16. Plane Changes
Simple
Combined

17. Rendezvous Co-Orbital Rendezvous Co-Planar Rendezvous
Interceptor and Target initially in the same orbit with different true anomalies
Co-Planar Rendezvous
Interceptor and Target initially in different orbits with the same orbital plane (inclination and RAAN)

18. Co-Orbital Rendezvous

19. Co-Orbital Rendezvous
Target Leading

20. Co-Orbital Rendezvous
Target Leading

21. 3 step process for determining phasing orbit size
Co-Orbital Rendezvous
Target Leading
3 step process for determining phasing orbit size

22. Co-Orbital Rendezvous
Target Leading
ωTGT 1

23. Co-Orbital Rendezvous
Target Leading
ωTGT
Φtravel 2

24. Co-Orbital Rendezvous
Target Leading
ωTGT
Φtravel 3

25. Co-Orbital Rendezvous
Target Trailing

26. Co-Orbital Rendezvous
Target Trailing

27. Co-Orbital Rendezvous
Target Trailing
ωTGT
Φtravel

28. Co-Planar Rendezvous

29. Coplanar Rendezvous

31. 5 step process for determining wait time (WT)2 2
5 step process for determining wait time (WT)

32. 1 2
ωTGT 2
ωINT

33. 2 2
TOF 2

34. 3 2
ωTGT
TOF
αlead 2
ωINT

35. 4 2
ωTGT
Φfinal
αlead 2
ωINT

36. 5 2
ωTGT
Φfinal
αlead 2
ωINT
Φinitial

37. Interplanetary Travel
In our two-body universe (based on the restricted, two-body EOM), we can not account for the influence of other external forces
In reality we can account for many body problems, but for our purposes of simplicity we will stick to two-body motion in the presence of gravity
Need a method to insure that only two-bodies are acting during a particular phase of the spacecraft’s motion
Spacecraft – Earth (from launch out to the Earth’s SOI)
Spacecraft – Sun (From Earth SOI through to the Target SOI)
Spacecraft – Planet (From Target Planet SOI to orbit or surface)

38. Patched Conic Approximation
Spacecraft – Earth
Circular or Elliptical low-Earth orbit (Parking)
Hyperbolic escape
Geo-centric, equatorial coordinate system
Spacecraft – Sun
Elliptical Transfer Orbit
Helio-centric, ecliptic coordinate system
Spacecraft – Target
Hyperbolic arrival
Circular or Elliptical orbit
Target-centric, equatorial coordinate system

39. Patched Conic Approximation
Geo: Hyperbolic escape
Helio: Elliptical transfer
Targeto: Hyperbolic arrival

40. Orbital Perturbations
Several factors cause perturbations to a spacecraft’s attitude and/or orbit
Drag
Earth’s oblateness
Actuators
3rd bodies
Gravity gradient
Magnetic fields
Solar pressure

41. Orbital Drag Orbital drag is an issue in low-Earth orbit
Removes energy from the s/c orbit (lowers)
Orbital decay due to drag depends on several factors
Spacecraft design
Orbital velocity
Atmospheric density
Altitude, Latitude
Solar activity

42. 3rd Bodies
Geosynchronous Equatorial Orbits are influenced by the Sun and Moon

43. 3rd Bodies Right ascension of the ascending node: Argument of perigee
i = orbit inclination
n = number of orbit revs per day

44. Gravity Gradient, Magnetic Field, Solar Pressure
I = s/c moment of inertia about axis
R = s/c distance from center of Earth
 = angle between Z axis and local vertical
D = s/c electric field strength (Am2)
B = local magnetic field strength (T); varies with R-3
 = 1367 W/m2 at Earth’s orbit
c = speed of light
= reflectivity
 = angle of incidence

45. Varying Disturbance Torques
NOTE: The magnitudes of the torques is
dependent on the spacecraft design.
Drag
Torque (au)
Gravity
Solar
Press.
Magnetic
LEO
GEO
Orbital Altitude (au)

46. Actuators Passive Active Gravity Gradient Booms Electrodynamic Tethers
Magnetic Torque Rods
Thrusters

47. Oblate Earth
The Earth is not a perfect sphere with the mass at the center (point mass)
In fact, the Earth has a bulge at the equator and a flattening at the poles
Major assumption of the restricted, two-body EOM
The J2 effects
RAAN
Argument of perigee
Magnitude of the effect is governed by
Orbital altitude
Orbital eccentricity
Orbital inclination
Earth's second-degree zonal spherical harmonic coefficient

48. J2 Effects

49. Sun Synchronous Orbit
Select appropriate inclination of orbit to achieve a nodal regression rate of ~1º/day (Orbit 360º in 365 days)

50. J2 Effects

51. Molniya Orbit
Select orbit inclination so that the argument of perigee regression rate is essentially zero
Allows perigee to remain in the hemisphere of choice
Allows apogee to remain in the hemisphere of choice
VIDEO

52. J2 Increasing?
Initial decrease thought to be from a mantle rebound from melted ice since the last Ice Age
Recent increase can only be caused by a significant movement of mass somewhere in the Earth J2
C. Cox and B. F. Chao, "Detection of large-scale mass redistribution in the terrestrial system since 1998," Science, vol 297, pp 831, 2 August 2002.