Orbital Mechanics II: Transfers, Rendezvous, Patched Conics, and Perturbations

1. 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. 2 Conceptual Walkthroughalt1 = 300 kmalt2 = 1000 km 2 V1 ΔV1 Slides Courtesy of Major David French, USAFA/DFAS

6. 2 2 Vt1

7. 2 2 ΔV2 Vt2

8. 2 2 V2

9. 2 Time of Flight 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 • 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 • 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. 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

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 • Gravity Gradient Booms • Electrodynamic Tethers • Active • 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