1. ASEN 5050 SPACEFLIGHT DYNAMICS Mid-Term Review
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 19: Mid-Term Review

2. Announcements Alan’s office hours are on FRIDAY this week! 1:00 pm.
No Concept Quiz active after this lecture
STK LAB 2, due 10/17
Homework #6 will be due Friday 10/24 (2014 not 2013)
CAETE by Friday 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.
Today: review. Friday: GRAIL and more perturbation fun.
Lecture 19: Mid-Term Review

3. 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 19: Mid-Term Review

4. Final Project Instructions for delivery of the final project:
Build your webpage with every required file inside of a directory.
Name the directory “<LastName_FirstName>”
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 19: Mid-Term Review

5. Space News MAVEN’s first scientific announcement!
UV views of Mars’ escaping atmosphere
Lecture 19: Mid-Term Review

6. Concept Quiz 12
Lecture 19: Mid-Term Review

7. Concept Quiz 12 Honestly, I didn’t make this clear enough.
There IS indeed correlation that the geoid follows mass (strictly speaking it is defined by the gravity of matter!)
But it IS NOT perfectly correlated and sometimes it appears quite the opposite.
Lecture 19: Mid-Term Review

8. ASEN 5050 SPACEFLIGHT DYNAMICS Mid-Term Review
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 19: Mid-Term Review

9. Mid-Term Logistics Please write your name on the exam
Carry “infinite” precision in your math at all steps. At the very end you are welcome to round your answers, but don’t round too much!
Document which equations / process you use, and then verify that your math is correct using a computer.
Try to learn something  I’m not JUST doing this exam to torture you.
Lecture 19: Mid-Term Review

10. Course Topics Two-Body Problem Coordinate Systems Time Systems
Newton’s Law of Gravitation
Orbital elements
Converting Cartesian to/from Keplerian Orbital Elements
Using Eccentric Anomaly to determine when an orbit is at some radius from the central body and, given that, estimate how much time has elapsed since periapse, etc.
Vis-Viva Equation
Coordinate Systems
IJK, XYZ, Perifocal, RSW, SEZ, VNC, etc.
Time Systems
UT1, UTC, TAI, GPS, ET
Lecture 19: Mid-Term Review

11. Course Topics Orbital Maneuvering Orbital Transfers
Changing each orbital element
Plane Changes are expensive
Orbital Transfers
Hohmann Transfer (2 tangential burns)
Phase Changing
Intercepting an object and rendezvousing with it
Combining maneuvers, such as an optimal LEO – GEO transfer
Where do you perform the inclination change?
Do you perform all of the inclination change there?
One-tangent burn
To speed up the transfer
Lambert’s Problem
Minimum-energy orbital transfer between two arbitrary position vectors.
Lecture 19: Mid-Term Review

12. Course Topics Proximity Operations Groundtracks
Clohessy-Wiltshire / Hill’s Equations
We’ll talk about this more in a few minutes
Groundtracks
Plotting the sub-point of a satellite over time
Repeat groundtracks and other practical groundtracks
Build me an orbit that exactly repeats its groundtrack every 12 days, after 121 revolutions.
Lecture 19: Mid-Term Review

13. Course Topics Additional Topics The shape of the Earth
Geocentric latitude vs. geodetic latitude
Solar day vs. Sidereal day
86400 seconds in a solar day, ish in a sidereal day. Be careful!
Lecture 19: Mid-Term Review

14. Review Requests (first 43)
CW/H x14
Groundtracks x2
Coordinate systems x5
Transformations x7
f and g series x4
Canonical units x2
Odd orbital elements
Geocentric/geodetic
Kepler’s Equation derivation
Lambert’s Problem x3
Synodic period
Plane changes / maneuvers not covered in HW x5
EVERYTHING x4
NOTHING x5
Lecture 19: Mid-Term Review

15. Coordinate Systems Geocentric Coordinate System (IJK)
- aka: Earth Centered Inertial (ECI), or the Conventional Inertial System (CIS)
- J2000 – Vernal equinox on Jan 1, 2000 at noon
- non-rotating
Intersection of ecliptic and celestial eq
Lecture 19: Mid-Term Review

16. Coordinate Systems Earth-Centered Earth-Fixed Coordinates (ECEF)
Topocentric Horizon Coordinate System (SEZ)
Lecture 19: Mid-Term Review

17. Coordinate Systems Perifocal Coordinate System (PQW)
Lecture 19: Mid-Term Review

18. Coordinate Systems Satellite Coordinate Systems:
RSW – Radial-Transverse-Normal
NTW – Normal-Tangent-Normal; VNC is a rotated version R V C S
Lecture 19: Mid-Term Review

19. Coordinate Transformations
Coordinate rotations can be accomplished through rotations about the principal axes.
Lecture 19: Mid-Term Review

20. Coordinate Transformations
To convert from the ECI (IJK) system to ECEF, we simply rotate around Z by the GHA:
ignoring precession, nutation, polar motion, motion of equinoxes.
Lecture 19: Mid-Term Review

21. Coordinate Transformations
To convert from ECEF to SEZ:
To set up a SYSTEM you also need to specify a center, which can be anything but is usually the reference site.
Lecture 19: Mid-Term Review

22. Coordinate Transformations
One of the coolest shortcuts for building transformations from one system to any other, without building tons of rotation matrices:
The unit vector in the S-direction, expressed in I,J,K coordinates
(sometimes this is easier, sometimes not)
Lecture 19: Mid-Term Review

23. Latitude/Longitude Geocentric latitude Lecture 19: Mid-Term Review
(Vallado, 1997)
Geocentric latitude
Lecture 19: Mid-Term Review

24. Latitude/Longitude For geodetic latitude use: where e=0.081819221456
(Vallado, 1997)
Lecture 19: Mid-Term Review

25. Latitude/Longitude
Rotate into ECEF
Lecture 19: Mid-Term Review

26. Coordinate Transformations
To convert between IJK and PQW:
To convert between PQW and RSW:
Thus, RSW  IJK is: R S P
Lecture 19: Mid-Term Review

27. CW / H Useful to answer questions like:
If I deploy a satellite from my current position in orbit, and the deployment imparts some small Delta-V, where does the satellite go, relative to me?
If I’m approaching a space station, what Delta-V should I execute to rendezvous with the station after 10 minutes?
Also helps evaluate trajectories rapidly, since you don’t have to numerically integrate them.
Lecture 19: Mid-Term Review

28. Coordinate Systems
Satellite Coordinate System (RSW) -- (Radial-Transverse-Normal)
Lecture 19: Mid-Term Review

29. Coordinate Transformations
To convert between IJK and PQW:
To convert between PQW and RSW:
Thus, RSW  IJK is:
Lecture 19: Mid-Term Review

30. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
Use RSW coordinate system (may be different from NASA)
Target satellite has two-body motion:
The interceptor is allowed to have thrusting
Then
So,
Lecture 19: Mid-Term Review

31. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
Need more information to solve this:
Lecture 14 (Slide 27+) takes you through all of the steps needed to convert this acceleration into one that is only dependent on the relative vector and omega:
Lecture 19: Mid-Term Review

32. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
We assume circular motion:
Thus,
Assume = 0 (good for impulsive DV maneuvers, not for continuous thrust targeting).
Equations also assume circular orbits and close proximity!
CW or Hill’s Equations
Lecture 19: Mid-Term Review

33. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
These equations can be solved (see book: Algorithm 48) leaving:
So, given of interceptor, can determine
of interceptor at future time.
Lecture 19: Mid-Term Review

34. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
Lecture 19: Mid-Term Review

35. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
Lecture 19: Mid-Term Review

36. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
Lecture 19: Mid-Term Review

37. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
Lecture 19: Mid-Term Review

38. Hubble’s Drift from Shuttle
RSW Coordinate Frame
Lecture 19: Mid-Term Review

39. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
Lecture 19: Mid-Term Review

40. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
We can also determine DV needed for rendezvous. Given x0, y0, z0, we want to determine necessary to make x = y = z = 0.
Set first 3 equations to zero, and solve for
Assumptions:
Satellites only a few km apart
Target in circular orbit
No external forces (drag, etc.)
Lecture 19: Mid-Term Review

41. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
Lecture 19: Mid-Term Review

42. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
Lecture 19: Mid-Term Review

43. Clohessy-Wiltshire (CW) Equations (Hill’s Equations)
NOTE: This is not the Delta-V, this is the new required relative velocity!
Lecture 19: Mid-Term Review

44. CW/H Examples Scenario 1: Scenario 1b:
Use eqs to compute x(t), y(t), and z(t)
RSW coordinates, relative to deployer.
Note: in the CW/H equations, the reference state doesn’t move – it is the origin!
Deployment: x0, y0, z0 = 0, Delta-V = non-zero
Use eqs to compute x(t), y(t), and z(t)
RSW coordinates, relative to reference.
Initial state: non-zero
Lecture 19: Mid-Term Review

45. Initial state in the RSW frame
CW/H Equations
Rendezvous
For rendezvous we usually specify the coordinates relative to the target vehicle and set x, y, and z to zero
Though if there’s a docking port, then that will be offset from the center of mass of the vehicle.
Define RSW targets: x, y, z (often zero)
Initial state in the RSW frame
Lecture 19: Mid-Term Review
Target: some constant values in the RSW frame

46. CW/H Equations Rendezvous
For rendezvous we usually specify the coordinates relative to the target vehicle and set x, y, and z to zero
Though if there’s a docking port, then that will be offset from the center of mass of the vehicle.
Define RSW targets: x, y, z (often zero)
This is easy if the targets are zero (Eq. 6-66)
This is harder if they’re not!
Velocity needed to get onto transfer
Initial state in the RSW frame
Lecture 19: Mid-Term Review
Target: some constant values in the RSW frame

47. CW/H Equations Rendezvous
For rendezvous we usually specify the coordinates relative to the target vehicle and set x, y, and z to zero
Though if there’s a docking port, then that will be offset from the center of mass of the vehicle.
Define RSW targets: x, y, z (often zero)
This is easy if the targets are zero (Eq. 6-66)
This is harder if they’re not!
Velocity needed to get onto transfer
Initial state in the RSW frame
The Delta-V is the difference of these velocities
Lecture 19: Mid-Term Review
Target: some constant values in the RSW frame

48. CW / H
Scenario 3: A jetpack-wielding astronaut leaves the shuttle and then returns.
Lecture 19: Mid-Term Review
Shuttle: reference frame

49. CW / H
Scenario 3: A jetpack-wielding astronaut leaves the shuttle and then returns.
Result of deployment
Lecture 19: Mid-Term Review
Shuttle: reference frame

50. CW / H
Scenario 3: A jetpack-wielding astronaut leaves the shuttle and then returns.
Rendezvous trajectory (Eq 6-66)
Result of deployment
Lecture 19: Mid-Term Review
Shuttle: reference frame

51. CW / H
Scenario 3: A jetpack-wielding astronaut leaves the shuttle and then returns.
Delta-V is the difference of these velocities.
Rendezvous trajectory (Eq 6-66)
Result of deployment
Lecture 19: Mid-Term Review
Shuttle: reference frame

52. Example from last year’s mid-term!
Lecture 19: Mid-Term Review

53. Mid-Term Review
Problem 4
Lecture 19: Mid-Term Review

54. Mid-Term Review
Problem 4
Lecture 19: Mid-Term Review

55. Mid-Term Review Problem 4
First, we need the velocity of the shuttle relative to the experiment before the maneuver:
Experiment’s velocity relative to shuttle, from Algorithm 48 of part (a)
Shuttle’s velocity relative to experiment is just the opposite of that:
Lecture 19: Mid-Term Review

56. Mid-Term Review Problem 4
Second, we need the velocity that the shuttle must obtain to perform the rendezvous.
Equation 6-66 in Vallado’s 4th edition
Lecture 19: Mid-Term Review

57. Mid-Term Review Problem 4
The initial conditions are the state of the shuttle relative to the experiment (opposite signs of the experiment’s position relative to the shuttle from part (a))
t = 10*60 sec = 600 sec (you can keep omega the same as before, or update it; doesn’t make much difference)
Lecture 19: Mid-Term Review

58. Mid-Term Review Problem 4, part (b)
Velocity required to achieve the transfer:
Delta-V for the rendezvous:
Lecture 19: Mid-Term Review

59. Mid-Term Review Problem 4, part (c)
Use Algorithm 48 once again and now plug in the initial position and velocity of the shuttle relative to the experiment.
The position of the shuttle after 10 minutes (15 minutes after the deployment) should be zero (GOOD CHECK!)
The velocity of the shuttle after 10 minutes (15 minutes after the deployment) will not be zero.
The Delta-V is that which will remove the relative velocity of the shuttle relative to the experiment.
Lecture 19: Mid-Term Review

60. Mid-Term Review
Problem 4, part (c)
Lecture 19: Mid-Term Review

61. Orbit Maneuvering Hohmann Transfers Bi-elliptic Transfers
Circular Rendezvous
Coplanar Rendezvous
Changing orbital elements
a, e, rp, ra, P, M, AOP are all coplanar
i, RAAN are plane changes
Lambert’s Problem
Lecture 19: Mid-Term Review

62. Changing Orbital Elements
Δa  Hohmann Transfer
Δe  Hohmann Transfer
Δi  Plane Change
ΔΩ  Plane Change
Δω  Coplanar Transfer
Δν  Phasing/Rendezvous
Lecture 10: Orbit Transfers

63. Changing Inclination Δi  Plane Change
Inclination-Only Change vs. Free Inclination Change
Lecture 10: Orbit Transfers

64. Changing Inclination Let’s start with circular orbits Vf V0
Lecture 10: Orbit Transfers

65. Changing Inclination Let’s start with circular orbits Vf V0
Lecture 10: Orbit Transfers

66. Changing Inclination Vf V0 Let’s start with circular orbits Δi
Are these vectors the same length?
What’s the ΔV?
Is this more expensive in a low orbit or a high orbit? Vf V0 Δi
Lecture 10: Orbit Transfers

67. Changing Inclination More general inclination-only maneuvers
Where do you perform the maneuver?
How do V0 and Vf compare?
What about the FPA?
Line of Nodes
Lecture 10: Orbit Transfers

68. Changing Inclination More general inclination-only maneuvers
Lecture 10: Orbit Transfers

69. Changing The Node
Lecture 10: Orbit Transfers

70. Changing The Node Where is the maneuver located?
Neither the max latitude nor at any normal feature of the orbit!
There are somewhat long expressions for how to find uinitial and ufinal in the book for circular orbits.
Lambert’s Problem gives easier solutions.
Lecture 10: Orbit Transfers

71. Changing Argument of Perigee
Lecture 10: Orbit Transfers

72. Changing Argument of Perigee
Lecture 10: Orbit Transfers

73. Changing Argument of Perigee
Which ΔV is cheaper?
Lecture 10: Orbit Transfers

74. Maneuver Combinations
General rules to consider:
Energy changes are more efficient when traveling FAST
Periapse
Plane changes are more efficient when traveling SLOW
Apoapse
Combinations take advantage of vector addition
3+4 = 5 not 7 
Some inclination change at periapse is optimal
Some energy change at apoapse is optimal (or necessary)
Delta-V vector = V final vector – V initial vector
One’s initial and final orbit do not always intersect; if they don’t you have to build a transfer orbit.
Lecture 19: Mid-Term Review

75. Questions These are the sorts of questions I hope you can answer:
Given orbit X, when does the satellite reach radius R?
Where in orbit Y is the satellite at its maximum latitude?
Compute element XYZ given element ABC
And all of those concept quiz type questions.
There will be math and concepts.
Lecture 19: Mid-Term Review

76. ASEN 5050 SPACEFLIGHT DYNAMICS GRAIL
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 19: Mid-Term Review

77. ASEN 5050 SPACEFLIGHT DYNAMICS Perturbations
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 19: Mid-Term Review

78. Perturbation Discussion Strategy✔
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
Solar Radiation Pressure
Introduce atmospheric drag
Atmospheric entries
Other perturbations
General perturbation techniques
Further discussions on mean motion vs. osculating motion. ✔ ✔
Lecture 19: Mid-Term Review