1. ASEN 5050 SPACEFLIGHT DYNAMICS General Perturbations Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 26: General Perturbations 1

2. Announcements Homework #7 is out now! Due Monday morning. –Clarification for Problem 3: you do not have to implement BOTH a variable time-step integrator and a fixed time-step integrator. Pick one. Then fill in that half of the table. I’ll be in my office from 10-1 and 2-3 for office hours. Reading: Chapters 8 and 9 Lecture 26: General Perturbations 2

3. Schedule from here out Lecture 26: General Perturbations 3 10/27: Three-Body Orbits 10/29: General Perturbations (Alan) 10/31: General Perturbations part 2 11/3: Mission Orbits / Designing with perturbations 11/5: Interplanetary 1 11/7: Interplanetary 2 11/10: Entry, Descent, and Landing 11/12: Low-Energy Mission Design 11/14: STK Lab 3 11/17: Low-Thrust Mission Design (Jon Herman) 11/19: Finite Burn Design 11/21: STK Lab 4 Fall Break 12/1: Constellation Design, GPS 12/3: Spacecraft Navigation 12/5: TBD 12/8: TBD 12/10: TBD 12/12: Final Review

4. Schedule from here out Our last lecture will be Friday 12/12. –Final review and final Q&A. –Showcase your final projects – at least any that are finished! Final Exam –Handed out on 12/12 –Due Dec 18 at 1:00 pm – either into D2L’s DropBox or under my door. I heartily encourage you to complete your final project website by Dec 12th so you can focus on your finals. However, if you need more time you can have until Dec 18th. As such the official due date is Dec 18th. The final due date for everything in the class is Dec 18th - no exceptions unless you have a very real reason (medical or otherwise - see CU's policies here: http://www.colorado.edu/engineering/academics/policies/grading). Of course we will accommodate real reasons. If you are a CAETE student, please let me know if you expect an issue with this timeframe. We normally give CAETE students an additional week to complete everything, but the grades are due shortly after the 18th for everyone. So please see if you can meet these due dates. Lecture 26: General Perturbations 4

5. Final Project Get started on it! Worth 20% of your grade, equivalent to 6-7 homework assignments. Find an interesting problem and investigate it – anything related to spaceflight mechanics (maybe even loosely, but check with me). Requirements: Introduction, Background, Description of investigation, Methods, Results and Conclusions, References. You will be graded on quality of work, scope of the investigation, and quality of the presentation. The project will be built as a webpage, so take advantage of web design as much as you can and/or are interested and/or will help the presentation. Lecture 26: General Perturbations 5

6. Final Project Instructions for delivery of the final project: Build your webpage with every required file inside of a directory. –Name the directory “ ” –there are a lot of duplicate last names in this class! –You can link to external sites as needed. Name your main web page “index.html” –i.e., the one that you want everyone to look at first Make every link in the website a relative link, relative to the directory structure within your named directory. –We will move this directory around, and the links have to work! Test your webpage! Change the location of the page on your computer and make sure it still works! Zip everything up into a single file and upload that to the D2L dropbox. Lecture 26: General Perturbations 6

7. Space News Lecture 26: General Perturbations 7 Reflections of sunlight off of seas in Titan’s northern latitudes. Methane and ethane.

8. Implementing Perturbations Any force model may be integrated using a numerical integrator, such as ode45. –Two-body equations of motion –n-body equations of motion –Accelerations caused by solar radiation pressure –Accelerations caused by atmospheric drag –Accelerations caused by any other effect –Put them all together and integrate that collective force model to build the spacecraft’s trajectory. Lecture 26: General Perturbations 8

9. Implementing Perturbations An example of the derivative function for an integrator such as ode45. Lecture 26: General Perturbations 9 function dX = deriv( t, X ) x = X(1); … vz = X(6); a_sun = -GM_sun * r_sun/norm(r_sun)^3 a_mercury = -GM_mercury * r_mercury/norm(r_mercury)^3 … a_neptune = -GM_neptune * r_neptune/norm(r_neptune)^3 a_earth_harmonics = blah blah blah a_SRP = blah blah blah a_earth_albedo = blah blah blah a_drag = blah blah blah a_outgassing = blah blah blah a_thermal_emmission = blah blah blah a = a_sun + a_mercury + … + a_thermal_emmission

10. Perturbation Magnitudes Lecture 26: General Perturbations 10 ISS Orbit

11. Perturbation Magnitudes Lecture 26: General Perturbations 11 GPS Orbit

12. Perturbation Magnitudes Lecture 26: General Perturbations 12 Earth – Mars

13. Lecture 26: General Perturbations 13 Perturbations Special Perturbation Techniques – Numerical integration. Straightforward – however obtaining a good understanding of the effects on the orbit is difficult General Perturbations – Use approximations to obtain analytical descriptions of the effects of the perturbations on the orbit. Assumes perturbative forces are small Early work used general perturbations because of a lack of computational power. Modern work uses special perturbations (numerical integration) because of the wide availability of computers. GP still useful for increasing your understanding. Still used by AF for maintaining space object catalog (> 7000 objects).

14. Lecture 26: General Perturbations 14 General Perturbation Techniques Perturbations can be categorized as secular, short period, long period.

15. Lecture 26: General Perturbations 15 Perturbations secular long-periodic mixed-periodic short-periodic This equation is known as a “Poisson Series”

16. Gaussian VOP Lecture 26: General Perturbations 16

17. Gaussian VOP Note a few limitations: e must be < 1.0 i and e can’t be 0 Hence, this is limited to moderately elliptical, non- equatorial orbits. Lecture 26: General Perturbations 17

18. Lagrangian VOP Different derivation, similar results: Lecture 26: General Perturbations 18

19. Lagrangian VOP Note a few limitations: e must be < 1.0 i and e can’t be 0 Hence, this is limited to moderately elliptical, non- equatorial orbits. Lecture 26: General Perturbations 19

20. VOPs Lecture 26: General Perturbations 20 Lagrangian Gaussian Forcing function in RSW Perturbing Potential Function

21. Lagrangian VOP Constructing perturbing potential functions Consider the spherical harmonic gravitational potential. –Take that potential function, remove the 2-body term, and re-cast it in terms of the classical orbital elements. –This leads to Kaula’s Solution: –Can then evaluate it in the Lagrange planetary equations. Lecture 26: General Perturbations 21

22. L.P.E.s & Kaula’s Solution Lecture 26: General Perturbations 22

23. Using the L.P.E.s Let’s use the Lagrange planetary equations (LPEs) to evaluate the secular trends caused by a 2x2 gravity field. Start with the potential function, R: Lecture 26: General Perturbations 23

24. Using the L.P.E.s Remove all periodic effects Left with: Lecture 26: General Perturbations 24

25. Using the L.P.E.s Convert to orbital elements: We convert latitude: And use trig: Lecture 26: General Perturbations 25

26. Using the L.P.E.s Remove periodic terms again: Yielding: Lecture 26: General Perturbations 26

27. Using the L.P.E.s The value of a/r varies over an orbit, since r varies. Average it over an orbit. Lecture 26: General Perturbations 27

28. Using the L.P.E.s Evaluate this potential in the LPEs: Consider RAAN Lecture 26: General Perturbations 28

29. Using the L.P.E.s After simplifying, we find: Lecture 26: General Perturbations 29 1 st -order secular trend of RAAN over time as a function of the orbit!

30. Using the L.P.E.s We can certainly make this trend more accurate by considering the first six zonals (S.H. order = 0): Lecture 26: General Perturbations 30

31. Using the L.P.E.s Similar techniques reveal other secular trends. Lecture 26: General Perturbations 31

32. Lecture 26: General Perturbations 32 General Perturbation Techniques Which is a “secularly precessing ellipse”. The equatorial bulge introduces a force component toward the equator causing a regression of the node (for prograde orbits) and a rotation of periapse. Note:

33. Lecture 26: General Perturbations 33 General Perturbation Techniques

34. Lecture 26: General Perturbations 34 General Perturbation Techniques

35. Lecture 26: General Perturbations 35 General Perturbation Techniques Periapse also precesses. = 0 at the critical inclination, i  = 63.4  (116.6  )

36. Lecture 26: General Perturbations 36 General Perturbation Techniques

37. Lecture 26: General Perturbations 37 General Perturbation Techniques Application: Sun Synchronous orbits

38. Lecture 26: General Perturbations 38 General Perturbation Techniques Sun Synchronous orbits: –Orbit plane remains at a constant angle (  ’ ) with respect to the Earth-Sun line. –Orbit plane precession about the Earth is equal to period of Earth’s orbit about the Sun.

39. How does this compare to reality? 28.5 deg inclined, somewhat eccentric orbit: Lecture 26: General Perturbations 39

40. Conclusions Consider a perturbation’s trends (secular drift) and periodic effects (fast and slow) What steps are taken to estimate these? Gaussian VOP –Rate of change of orbital elements evaluated using a perturbing potential field Lagrangian VOP –Rate of change of orbital elements evaluated using a forcing function J2 is famous for its secular effects on the node, the argument of periapse, and the mean anomaly. Lecture 26: General Perturbations 40

41. ASEN 5050 SPACEFLIGHT DYNAMICS Mid-Term Exam Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 26: General Perturbations 41

42. Mid-Term Exam Lecture 26: General Perturbations 42 Problem 1 Common errors: –Since this isn’t a circular orbit, the DT is not half of the orbital period. Though the Delta-nu is indeed 180 deg.

43. Mid-Term Exam Hard way: generate the two-body orbit and map the latitude: Lecture 26: General Perturbations 43 ?

44. Mid-Term Exam Problem 1 solution Easy way (and precise): –Argument of latitude: u = ω + ν –At ascending node: u = 0 deg –At descending node: u = 180 deg –ω = 310 deg –At ascending node: ν = -310 deg = 50 deg –At descending node: ν =-130 deg = 230 deg –Compute time past periapse for both –Delta-t = 1.138 hours Lecture 17: Perturbations 44

45. Mid-Term Exam Problem 2 Lecture 26: General Perturbations 45

46. Mid-Term Exam Problem 2 Lecture 26: General Perturbations 46 Periapsis Range: 5473.0 km

47. Mid-Term Exam Problem 2 Lecture 26: General Perturbations 47

48. Mid-Term Exam Problem 2 Lecture 26: General Perturbations 48 Use eccentricity and semi-major axis, and find the eccentric anomaly needed to make radius = 6378.1363 km. Convert to TimePastPeriapse. And also compute TimePastPeriapse of initial state. We find: Time past periapse of initial state: -2049.99566 sec Time past periapse of impact: -676.70889 sec Duration of time to impact: 1373.28677 sec = 0.38147 hours

49. Mid-Term Exam Problem 2 Lecture 26: General Perturbations 49 The missile will strike the surface traveling 8811.91852 m/s in velocity

50. Mid-Term Exam Problem 3 Lecture 26: General Perturbations 50

51. Mid-Term Exam Problem 3 Lecture 26: General Perturbations 51

52. Mid-Term Exam Lecture 26: General Perturbations 52 C/W Equations: Algorithm 48

53. Mid-Term Exam Problem 3 Lecture 26: General Perturbations 53

54. Mid-Term Exam Problem 3 omega: 0.00114400182253 rad/s Satellite A will be located at a position of [0.0, 0.0, 80.8050388] in meters, after 7 minutes Satellite B will be located at a position of [79.17972504, 142.440310957, 0.0] in meters, after 7 minutes Satellite A will have a velocity relative to the shuttle of [0.0, 0.0, 0.177354562] in m/s, after 7 minutes Satellite B will have a velocity relative to the shuttle of [0.369764446, 0.2188365, 0.0] in m/s, after 7 minutes Lecture 26: General Perturbations 54

55. Mid-Term Exam Problem 3 Lecture 26: General Perturbations 55

56. Mid-Term Exam Problem 3 Satellite A's state relative to Satellite B at t=7 minutes, in meters and m/s, before executing any maneuver (A-B): –x0: -79.179725045 m –y0: -142.440310958 m –z0: 80.8050388697 m –vx0: -0.369764446945 m/s –vy0: -0.218836500483 m/s –vz0: 0.17735456256 m/s Lecture 26: General Perturbations 56

57. Mid-Term Exam Problem 3 What velocity is NEEDED? Equation 6-66 Lecture 26: General Perturbations 57

58. Mid-Term Exam Problem 3 What velocity is NEEDED? Equation 6-66 The velocity that Satellite A needs to obtain (m/s): vx: 0.00953422198546 vy: 0.247527429082 vz: -0.0745206696574 Lecture 26: General Perturbations 58

59. Mid-Term Exam Problem 3 What CHANGE in velocity is needed? The Delta-V impulse that A has to perform, in m/s relative to B: Delta-Vx: 0.37929866893 Delta-Vy: 0.466363929565 Delta-Vz: -0.251875232218 Delta-V magnitude: 0.651769842549 Lecture 26: General Perturbations 59

60. Mid-Term Exam Problem 3 Lecture 26: General Perturbations 60

61. Mid-Term Exam Problem 3 Use Algorithm 48 again to propagate state of A relative to B. You should see the position go to zero!!! The rendezvous Delta-V impulse that A has to perform, in m/s relative to B: Delta-Vx: -0.17983714361 Delta-Vy: -0.066363929565 Delta-Vz: 0.11873790189 Delta-V magnitude: 0.225486715161 Lecture 26: General Perturbations 61

62. Mid-Term Exam Problem 3 Lecture 26: General Perturbations 62

63. Mid-Term Exam Problem 3 Answer: yes! If satellite A does not do anything, it will collide with the shuttle in about 45 minutes. Half an orbit later. Lecture 26: General Perturbations 63

64. Mid-Term Exam Problem 4 Inclination = 90+14 = 104 deg P = 30/360*86164.09056 seconds = 7180.341 sec = 1.995 hours P = 2*pi*sqrt(a^3/mu)  a = 8044.32 km Altitude = 1666.18 km Lecture 26: General Perturbations 64

65. Mid-Term Exam Problem 5 Lecture 26: General Perturbations 65

66. Mid-Term Exam Problem 5 Lecture 26: General Perturbations 66

67. Mid-Term Exam Problem 5 Lecture 26: General Perturbations 67

68. Mid-Term Exam Problem 5 mu = muEarth + muMoon = 403503.2405 km3/s2 rp = 362600 km ra = 405400 km a = 384,000 km e = 0.0557292 Period of lunar orbit: 2353711.295 sec = 39228.5216 min = 653.80869 hours = 27.242029 days Lecture 26: General Perturbations 68

69. Mid-Term Exam Problem 5 Lecture 26: General Perturbations 69

70. Mid-Term Exam Problem 5 use equation: r = a(1-e cosE) r = 380000 km + rEarth = 386378.1363 km E = 1.68215420 rad = 96.380336 deg Time past periapse: 609395.909 sec = 10156.5985 min = 169.27664 hours = 7.053193 days Total Duration within 386378 km: 1218791.818 sec = 20313.1970 min = 338.55328 hours = 14.106387 days Lecture 26: General Perturbations 70

71. Mid-Term Exam Problem 5 Lecture 26: General Perturbations 71 Percentage of time the Earth and Moon are within 386378 km: 51.782%

72. Announcements Homework #7 is out now! Due Monday morning. –Clarification for Problem 3: you do not have to implement BOTH a variable time-step integrator and a fixed time-step integrator. Pick one. Then fill in that half of the table. I’ll be in my office from 10-1 and 2-3 for office hours. Reading: Chapters 8 and 9 Lecture 26: General Perturbations 72