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

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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 2

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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 3

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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 19: Mid-Term Review 4

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

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Concept Quiz 12 Lecture 19: Mid-Term Review 6

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Concept Quiz 12 Lecture 19: Mid-Term Review 7 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.

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

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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 9

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Course Topics Two-Body Problem –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 10

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Course Topics Orbital Maneuvering –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 11

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Course Topics Proximity Operations –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 12

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

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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 14

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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 15

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

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

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

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

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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 20

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

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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 22

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

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

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

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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 26

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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 27

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

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

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Lecture 19: Mid-Term Review 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, 30

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Lecture 19: Mid-Term Review 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: 31

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

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Lecture 19: Mid-Term Review 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. 33

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

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

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

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

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

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

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Lecture 19: Mid-Term Review Clohessy-Wiltshire (CW) Equations (Hill’s Equations) We can also determine V needed for rendezvous. Given x 0, y 0, z 0, we want to determine necessary to make x = y = z = 0. Set first 3 equations to zero, and solve for. Assumptions: 1.Satellites only a few km apart 2.Target in circular orbit 3.No external forces (drag, etc.) 40

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

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

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

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

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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) Lecture 19: Mid-Term Review 45 Target: some constant values in the RSW frame Initial state in the RSW frame

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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) Lecture 19: Mid-Term Review 46 Target: some constant values in the RSW frame Initial state in the RSW frame This is easy if the targets are zero (Eq. 6-66) This is harder if they’re not! Velocity needed to get onto transfer

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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) Lecture 19: Mid-Term Review 47 Target: some constant values in the RSW frame Initial state in the RSW frame This is easy if the targets are zero (Eq. 6-66) This is harder if they’re not! Velocity needed to get onto transfer The Delta-V is the difference of these velocities

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

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

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

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

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

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

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

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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 55

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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 4 th edition Lecture 19: Mid-Term Review 56

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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 57

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

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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 59

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

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

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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 62

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

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Changing Inclination Let’s start with circular orbits Lecture 10: Orbit Transfers 64 V0V0 VfVf

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Changing Inclination Let’s start with circular orbits Lecture 10: Orbit Transfers 65 V0V0 VfVf

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Changing Inclination Let’s start with circular orbits Lecture 10: Orbit Transfers 66 V0V0 VfVf ΔiΔi Are these vectors the same length? What’s the ΔV? Is this more expensive in a low orbit or a high orbit?

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Changing Inclination More general inclination-only maneuvers Lecture 10: Orbit Transfers 67 Line of Nodes Where do you perform the maneuver? How do V 0 and V f compare? What about the FPA?

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

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Changing The Node Lecture 10: Orbit Transfers 69

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Changing The Node Lecture 10: Orbit Transfers 70 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 u initial and u final in the book for circular orbits. Lambert’s Problem gives easier solutions.

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Changing Argument of Perigee Lecture 10: Orbit Transfers 71

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Changing Argument of Perigee Lecture 10: Orbit Transfers 72

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

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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 74

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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 75

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

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

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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 78 ✔ ✔ ✔

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