1. ASEN 5050 SPACEFLIGHT DYNAMICS Intro to Perturbations Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 17: Intro to Perturbations 1

2. Announcements STK LAB 2 –Alan will be in ITLL 2B10 Fri 2-3 –STK Lab 2 will be due 10/17, right when the mid-term exam starts. Homework #5 is due right now! –CAETE by Friday 10/17 Homework #6 will be due Friday 10/17 10/24 –CAETE by Friday 10/24 10/31 Mid-term Exam will be handed out Friday, 10/17 and will be due Wed 10/22. (CAETE 10/29) –Take-home. Open book, open notes. –Once you start the exam you have to be finished within 24 hours. –It should take 2-3 hours. Lecture 17: Intro to Perturbations 2

3. Concept Quiz 11 Lecture 17: Intro to Perturbations 3

4. Concept Quiz 11 Lecture 17: Intro to Perturbations 4

5. Concept Quiz 11 Lecture 17: Intro to Perturbations 5

6. Concept Quiz 11 Lecture 17: Intro to Perturbations 6

7. Space News Lecture 17: Intro to Perturbations 7 Anyone watch the ISS? Anyone see the lunar eclipse? There’s an event in 1.5 weeks that is directly related to the lunar eclipse we just had. Anyone have an idea what the event is?

8. LADEE’s Mission to the Moon Earth phasing orbits, followed by lunar phasing orbits Lecture 17: Intro to Perturbations Credit: NASA/Goddard 8

9. LADEE’s Mission to the Moon Lunar Orbit Lecture 17: Intro to Perturbations Credit: NASA/Ames / ADS 9

10. LADEE’s Mission to the Moon Lunar orbit perturbations Lecture 17: Intro to Perturbations Credit: NASA/Ames / ADS 10

11. ASEN 5050 SPACEFLIGHT DYNAMICS Perturbations Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 17: Intro to Perturbations 11

12. Orbital Perturbations You’ll notice that LADEE’s orbit is not strictly conical. So far, we’ve only considered orbital solutions to the two-body problem –Point-masses In reality, nothing is ever in orbit about a point-mass without any other perturbations –(even in an orbit about a black hole!) The two-body relationship is typically the dominant orbital dynamic. Everything else is a small perturbation –Realistic gravitational masses –Other gravitating bodies –Atmospheric drag –Solar radiation pressure –Spacecraft effects –Even relativity and other subtle effects. Lecture 17: Intro to Perturbations 12

13. Perturbation Discussion Strategy We know the 2-body problem *really well!* Introduce the 3-body and n-body problems –We’ll cover halo orbits and low-energy transfers later Numerical Integration Introduce aspherical gravity fields –J2 effect, sun-synchronous orbits Introduce atmospheric drag –Atmospheric entries General perturbation techniques Further discussions on mean motion vs. osculating motion. Lecture 17: Intro to Perturbations 13

14. Gravitational Perturbations Start by considering the effects of other gravitating bodies. Recall the two-body equation of motion: which is a differential equation describing the motion of m sat WRT m . How would this change if we had multiple gravitating bodies? Lecture 17: Intro to Perturbations 14

15. 3-Body Problem Lecture 17: Intro to Perturbations 15

16. 3-Body Problem We want to know the position vector of the satellite relative to the Earth over time Lecture 17: Intro to Perturbations 16

17. 3-Body Problem We want to know the position vector of the satellite relative to the Earth over time Lecture 17: Intro to Perturbations 17 Be cognizant of the signs – the signs are defined according to how the vectors are drawn!

18. 3-Body Problem We want to know the position vector of the satellite relative to the Earth over time Lecture 17: Intro to Perturbations 18 Direct Effect Indirect Effect

19. n-Body Problem The equation of motion for the position vector of a satellite in the presence of n bodies. … relative to Body “1” (Earth?) Lecture 17: Intro to Perturbations 19

20. Full 2-Body Problem How about the perturbations that result in being in orbit about a non-spherical body? Images from Park, Werner, and Bhaskaran, “Estimating Small-Body Gravity Field from Shape Model and Navigation Data”, Journal of Guidance, Control, and Dynamics, Vol. 33, No. 1, Jan – Feb 2010.

21. Dynamical Analysis System Two-Body Problem Full Three- Body RTBP: 3 rd Body Massless CRTBP: Circular orbits PCRTBP (Synodic Frame) Degrees of Freedom1218 12 Integrals of Motion121012139 Linear Momentum66664 Angular Momentum33331 Energy11111 Kepler’s Laws20222 Jacobi IntegralN/A 11 x y M2M2 M1M1 z

22. Dynamical Analysis System Two-Body Problem Full Three- Body RTBP: 3 rd Body Massless CRTBP: Circular orbits PCRTBP (Synodic Frame) Degrees of Freedom1218 12 Integrals of Motion121012139 Linear Momentum66664 Angular Momentum33331 Energy11111 Kepler’s Laws20222 Jacobi IntegralN/A 11 x y M3M3 M2M2 M1M1 z

23. Dynamical Analysis System Two-Body Problem Full Three- Body RTBP: 3 rd Body Massless CRTBP: Circular orbits PCRTBP (Synodic Frame) Degrees of Freedom1218 12 Integrals of Motion121012139 Linear Momentum66664 Angular Momentum33331 Energy11111 Kepler’s Laws20222 Jacobi IntegralN/A 11 x y M 3 ~ 0 M2M2 M1M1 z M 1 and M 2 follow conic trajectories about their COM

24. Dynamical Analysis System Two-Body Problem Full Three- Body RTBP: 3 rd Body Massless CRTBP: Circular orbits PCRTBP (Synodic Frame) Degrees of Freedom1218 12 Integrals of Motion121012139 Linear Momentum66664 Angular Momentum33331 Energy11111 Kepler’s Laws20222 Jacobi IntegralN/A 11 y x M 3 ~ 0 M2M2 M1M1 z M 1 and M 2 follow circular orbits about their COM Synodic Frame

25. Dynamical Analysis System Two-Body Problem Full Three- Body RTBP: 3 rd Body Massless CRTBP: Circular orbits PCRTBP: Planar Degrees of Freedom1218 12 Integrals of Motion121012139 Linear Momentum66664 Angular Momentum33331 Energy11111 Kepler’s Laws20222 Jacobi IntegralN/A 11 x M 3 ~ 0 M2M2 M1M1 y Planar motion Synodic Frame

26. Building Solutions to the n-Body Problem We have more degrees of freedom than we have integrals of motion! Conic sections are no longer solutions. Most common method used to build solutions to the n-Body problem is to take initial conditions and integrate them forward in time. –Build a trajectory using knowledge of the equations of motion. Lecture 17: Intro to Perturbations 26

27. Numerical Integration Say we have a state (pos, vel) and some equations of motion. Lecture 17: Intro to Perturbations 27 Accelerations due to 2-body, n-body, etc.

28. Numerical Integration We want to recover the spacecraft’s trajectory using knowledge of the derivative of its state over time. If we were to accurately integrate the derivative function over time, using the spacecraft’s initial state as the constant of motion, then we could recover its trajectory. Lots of ways to do this. Some are better than others! Lecture 17: Intro to Perturbations 28

29. Numerical Integration Euler integration Lecture 17: Intro to Perturbations 29 Actual Trajectory

30. Numerical Integration How do we improve it? Take smaller time-steps Take smarter steps Lecture 17: Intro to Perturbations 30 Actual Trajectory

31. Euler Integration Example Lecture 17: Intro to Perturbations 31

32. Euler Integration Example Lecture 17: Intro to Perturbations 32

33. Euler Integration Example Lecture 17: Intro to Perturbations 33

34. Euler Integration Example Lecture 17: Intro to Perturbations 34

35. Euler Integration Example Lecture 17: Intro to Perturbations 35

36. Euler Integration Example Lecture 17: Intro to Perturbations 36

37. Higher order terms Here’s what we just tried: What about this modification?: That would be better! –But really hard to implement in a general sense. Lecture 17: Intro to Perturbations 37

38. Higher order terms Here’s what we just tried: How about a correction term. Here’s a second-order scheme, usually referred to as a midpoint method: Lecture 17: Intro to Perturbations 38 Actual Trajectory

39. Midpoint Integration Example Lecture 17: Intro to Perturbations 39 Euler Midpoint

40. Midpoint Integration Example Lecture 17: Intro to Perturbations 40 Euler Midpoint Note: this does take 2x as many derivative function calls, but the improvement is better than just doubling Euler’s!

41. Runge-Kutta Integrators Runge-Kutta integration 4 th order Runge-Kutta “RK4” or “The Runge-Kutta method” Lecture 17: Intro to Perturbations 41 Weighted average correction system, related to Simpson’s Rule

42. Example RK4 Lecture 17: Intro to Perturbations 42

43. Example RK4 Lecture 17: Intro to Perturbations 43

44. Example RK4 Lecture 17: Intro to Perturbations 44

45. Example RK4 Lecture 17: Intro to Perturbations 45

46. Example RK4 Lecture 17: Intro to Perturbations 46

47. Example RK4 Lecture 17: Intro to Perturbations 47

48. Example RK4 Lecture 17: Intro to Perturbations 48

49. Example RK4 Lecture 17: Intro to Perturbations 49

50. Example RK4 Lecture 17: Intro to Perturbations 50

51. Example RK4 Shorter Time Interval Lecture 17: Intro to Perturbations 51

52. Example RK4 Shorter Time Interval Lecture 17: Intro to Perturbations 52

53. Example RK4 Shorter Time Interval Lecture 17: Intro to Perturbations 53

54. Example RK4 Shorter Time Interval Lecture 17: Intro to Perturbations 54 Oversampled Undersampled Use a variable time-step!

55. Variable Time-Steps These may be implemented in many ways. Method 1. 1.Compute X i+1 using one full step 2.Compute X i+1 using two half-steps 3.Compare them. a.If they are similar to within a tolerance, then use the second state and move on. b.If they are similar to within some fraction of the tolerance, then increase the time-step. c.If they are sufficiently dissimilar, then reduce the time-step and try again. Do not move on until they are similar within the tolerance. Lecture 17: Intro to Perturbations 55

56. Variable Time-Steps These may be implemented in many ways. Method 2. 1.Compute X i+1 using a 4 th order scheme 2.Compute X i+1 using a 5 th order scheme 3.Compare them. a.If they are similar to within a tolerance, then use the second state and move on. b.If they are similar to within some fraction of the tolerance, then increase the time-step. c.If they are sufficiently dissimilar, then reduce the time-step and try again. Do not move on until they are similar within the tolerance. Lecture 17: Intro to Perturbations 56

57. Variable Time-Steps These may be implemented in many ways. Some algorithms change the time-steps every iteration; some either cut it in half or double it. Some use a formula. When plotting the trajectories, it’s generally best to plot at a constant time-step. –Often the integrator takes many steps between each outputted time-step. –Often the integrator does not actually evaluate the trajectory AT a desired time-step. Either force it to (expensive) Or interpolate (less expensive) Lecture 17: Intro to Perturbations 57

58. Runge-Kutta Take 2 4 th order Runge-Kutta “RK4” or “The Runge-Kutta method” Lecture 17: Intro to Perturbations 58

59. Runge-Kutta Take 2 The weights may be placed into a table, called a Butcher Table Lecture 17: Intro to Perturbations 59

60. Runge-Kutta Take 2 The weights may be placed into a table, called a Butcher Table Lecture 17: Intro to Perturbations 60

61. Runge-Kutta Take 2 The general explicit Runge-Kutta scheme: Lecture 17: Intro to Perturbations 61 (oh yes, I did just copy and paste that from wikipedia. They have a great write-up.)

62. Runge-Kutta Take 2 The general explicit Runge-Kutta scheme: Lecture 17: Intro to Perturbations 62 The method is consistent if:

63. Butcher Tables Lecture 17: Intro to Perturbations 63 RK4 EulerMidpoint

64. Butcher Tables To build an adaptive time-step Butcher Table: –Assemble an order p Butcher Table –Append an order p-1 Butcher Table to it. Lecture 17: Intro to Perturbations 64

65. Butcher Tables Runge-Kutta-Fehlberg method consists of a 5 th order method and a 4 th order check. Hence: Lecture 17: Intro to Perturbations 65

66. Drawbacks of Explicit RK Methods Explicit Runge-Kutta methods are easy to derive and require relatively few computations to implement. They are usually quite good for astrodynamic applications. Don’t be afraid to use them However, for stiff problems they are unstable. They are particularly unstable for solving partial differential equations. To combat this, implicit Runge-Kutta methods have been formulated. Lecture 17: Intro to Perturbations 66

67. Implicit Runge-Kutta Methods Similar form: Lecture 17: Intro to Perturbations 67 A may be a full matrix!

68. Implicit Runge-Kutta Methods These require an iterative scheme, since, for instance, k 1 depends on itself! Example: the backward Euler method: Lecture 17: Intro to Perturbations 68 This uses the slope of the next state to integrate the current state forward in time. Notice that k 1 depends on itself.

69. Implicit Runge-Kutta Methods Gauss-Legendre method may be scaled to include as many stages as desired. The 4 th order (2 stages) method has this Butcher table: All collocation methods are implicit Runge-Kutta methods, but not all implicit Runge-Kutta methods are collocation methods. Lecture 17: Intro to Perturbations 69

70. Announcements STK LAB 2 –Alan will be in ITLL 2B10 Fri 2-3 –STK Lab 2 will be due 10/17, right when the mid-term exam starts. Homework #5 is due right now! –CAETE by Friday 10/17 Homework #6 will be due Friday 10/17 10/24 –CAETE by Friday 10/24 10/31 Mid-term Exam will be handed out Friday, 10/17 and will be due Wed 10/22. (CAETE 10/29) –Take-home. Open book, open notes. –Once you start the exam you have to be finished within 24 hours. –It should take 2-3 hours. Lecture 17: Intro to Perturbations 70