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ASEN 5050 SPACEFLIGHT DYNAMICS Conversions, f & g, Orbit Transfers. Prof. Jeffrey S. Parker University of Colorado – Boulder. Announcements. Homework #3 is due right now You must write your own code. For this HW, please turn in your code (preferably in one text/Word/PDF document)
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ASEN 5050SPACEFLIGHT DYNAMICSConversions, f & g, Orbit Transfers Prof. Jeffrey S. Parker University of Colorado – Boulder
Announcements • Homework #3 is due right now • You must write your own code. • For this HW, please turn in your code (preferably in one text/Word/PDF document) • After this assignment, you may use Vallado’s code, but if you do you must give him credit for work done using his code. If you don’t, it’s plagiarism. • Homework #4 is due Friday 9/26 at 9:00 am • You’ll also have to turn in your code for this one. • No Quiz over the weekend! Enjoy your weekend. • I’ll be at the career fair Monday, so I’m delaying Monday’s office hours to 2:00. • Reading: Chapter 6 (SIX, we jumped a few)
Concept Quiz 7 ✔ ✔ ✔ ✔ ✔
Concept Quiz 7 The class is 50-50 split on this! Talk it over with your neighbor and convince him/her why you’re right.
Quizzes • Speaking of quizzes, I had a bug in my grade book and only the first two quiz scores were shown. As of now, Quiz 1-6 should be shown.
Space News • Sunday: MAVEN arrives at Mars! • MOI: this Sunday at 19:37 Mountain • LASP is holding a viewing party
Space News • Then Wednesday: MOM arrives at Mars! • MOI: Wednesday at 21:00 Mountain • It will enter occultation at 21:04 • MOI will end at 21:24 • We’ll know if it’s successful around 21:30
Challenge #4 • If you were to plot the position and velocity of a satellite over time using VNC (Velocity-Normal-Conormal) coordinates, what would you find? • Say, an elliptical orbit V C
ASEN 5050SPACEFLIGHT DYNAMICSCoordinate Transformations Prof. Jeffrey S. Parker University of Colorado - Boulder
Principal Axis Rotations Equation 3-15
and From Orbital Elements Express the position and velocity in the perifocal system ( goes through periapse, in the direction of , perpendicular to and , in the orbit plane.) PQW PQW PQW
and From Orbital Elements Now we simply need to rotate into the geocentric equatorial system. The order of the rotations does matter Algorithm 10 in book. Ex. 2-6
f and g Series Start by crossing the position vector into the initial velocity vector: The second term is zero, and the other terms are normal to the plane: Differentiating this last equation:
f and g Series Now cross the initial position vector into the position vector: The first term is zero, and the other terms are normal to the plane: Differentiating this last equation:
f and g Series Look at the cross-product: Which can only be true if: A good test!
f and g Series Which gives:
f and g Series So, to summarize, given an initial position and velocity, we can calculate a future position and velocity given the change in the true anomaly Dn: Which you can test using Example 2-4 in the textbook.
f and g Series: State Transition Matrix We can re-express our f and g series representation: in terms of a state-variable relationship:
f and g Series • What uses do these functions have? • Given two states, find the time of flight between them. • Given two states, find an orbit that connects them. • Big fan of this application. • Using an iterative technique, such as Newton Raphson, can determine a future state given a current state and a transfer time or transfer angle.
ASEN 5050SPACEFLIGHT DYNAMICSOrbital Maneuvers Prof. Jeffrey S. Parker University of Colorado - Boulder
Orbital Maneuvers • Orbital maneuvers are used to do many things: • Change a satellite’s orbit • Size • Shape • Orientation • Change the phase of a satellite in its orbit • Rendezvous and/or proximity operations • Avoid collisions (debris) • Change the satellite’s groundtrack • Etc.
Terminology • Coplanar maneuvers: no change to the orbit plane; the maneuvers can only change a, e, w. • Impulsive maneuvers: instantaneous change in velocity: ΔV • Requires an infinitely powerful engine • Finite maneuvers: maneuvers that require a duration of time to achieve • Ballistic: the trajectory of an object under the effects of only external forces (no maneuver firings).
Orbital Maneuvers Tangential Burns: in velocity/anti-velocity vector direction • Do not change velocity orientation, just magnitude • Do not change flight path angle
Orbital Maneuvers Nontangential: plane changes, orbit rotations
Orbital Maneuvers Hohmann Transfer – Walter Hohmann (1880-1945) showed minimum energy transfer between two orbits used two tangential burns.
Hohmann Transfer (math, compute DV and DT)
Hohmann Transfer Can also be done using elliptical orbits, but must start at apogee or perigee to be a minimum energy transfer. (Algorithm 36, Example 6-1)
Hohmann Transfer • We just argued that the Hohmann Transfer is (usually) the most energy-efficient orbital transfer. • Why? • Consider Elliptical—Elliptical transfer • Tangential Burns • Energy efficiency considerations V is highest at perigee
Orbital Maneuvers Bi-elliptic Transfer – Uses two Hohmann transfers. Can save Dv in some cases. rb must be greater than rfinal, but can otherwise be optimized.
Bi-elliptic Transfer Much longer flight times for bi-elliptic transfer, but sometimes less energy. (Algorithm 37, Example 6-2)
Changing Orbital Elements • Δa Hohmann Transfer • Δe Hohmann Transfer • Δi Plane Change • ΔΩ Plane Change • Δω Coplanar Transfer • Δν Phasing/Rendezvous (later discussion)
Changing Inclination • Δi Plane Change • Inclination-Only Change vs. Free Inclination Change
Changing Inclination • Let’s start with circular orbits Vf V0
Changing Inclination • Let’s start with circular orbits Vf V0
Changing Inclination • Let’s start with circular orbits 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
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
Changing Inclination • More general inclination-only maneuvers
