Upload
willa-jackson
View
218
Download
0
Tags:
Embed Size (px)
Citation preview
ASEN 5050SPACEFLIGHT DYNAMICS
Conversions, f & g, Orbit Transfers
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 9: Conversions, Orbit Transfers 1
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)
Lecture 9: Conversions, Orbit Transfers 2
Concept Quiz 7
Lecture 9: Conversions, Orbit Transfers 5
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.
Lecture 9: Conversions, Orbit Transfers 7
Space News• Sunday: MAVEN arrives at Mars!
• MOI: this Sunday at 19:37 Mountain
• LASP is holding a viewing party
Lecture 9: Conversions, Orbit Transfers 8
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
Lecture 9: Conversions, Orbit Transfers 11
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
Lecture 9: Conversions, Orbit Transfers 13
CV
ASEN 5050SPACEFLIGHT DYNAMICSCoordinate Transformations
Prof. Jeffrey S. Parker
University of Colorado - Boulder
Lecture 9: Conversions, Orbit Transfers 14
Express the position and velocity in the perifocal system ( goes through periapse, in the direction of , perpendicular to and , in the orbit plane.)
and From Orbital Elements
Lecture 9: Conversions, Orbit Transfers 17
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. 2-6Lecture 9: Conversions, Orbit Transfers 18
f and g SeriesStart 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:
Lecture 9: Conversions, Orbit Transfers 20
f and g SeriesNow 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:
Lecture 9: Conversions, Orbit Transfers 21
f and g SeriesLook at the cross-product:
Which can only be true if: A good test!
Lecture 9: Conversions, Orbit Transfers 22
f and g SeriesSo, to summarize, given an initial position and velocity, we cancalculate a future position and velocity given the change in the trueanomaly :
Which you can test using Example 2-4 in the textbook.Lecture 9: Conversions, Orbit Transfers 24
f and g Series: State Transition MatrixWe can re-express our f and g series representation:
in terms of a state-variable relationship:
Lecture 9: Conversions, Orbit Transfers 25
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.
Lecture 9: Conversions, Orbit Transfers 26
ASEN 5050SPACEFLIGHT DYNAMICS
Orbital Maneuvers
Prof. Jeffrey S. Parker
University of Colorado - Boulder
Lecture 9: Conversions, Orbit Transfers 27
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.
Lecture 9: Conversions, Orbit Transfers 28
Terminology
• Coplanar maneuvers: no change to the orbit plane; the maneuvers can only change a, e, .
• 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).
Lecture 9: Conversions, Orbit Transfers 29
Lecture 9: Conversions, Orbit Transfers30
Orbital Maneuvers
Tangential Burns: in velocity/anti-velocity vector direction
- Do not change velocity orientation, just magnitude
- Do not change flight path angle
Lecture 9: Conversions, Orbit Transfers31
Orbital Maneuvers
Nontangential: plane changes, orbit rotations
Lecture 9: Conversions, Orbit Transfers32
Orbital ManeuversHohmann Transfer – Walter Hohmann (1880-1945) showed
minimum energy transfer between two orbits used two tangential burns.
Lecture 9: Conversions, Orbit Transfers34
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
Lecture 9: Conversions, Orbit Transfers 35
V is highest at perigee
Lecture 9: Conversions, Orbit Transfers37
Orbital ManeuversBi-elliptic Transfer – Uses two Hohmann transfers. Can save v
in some cases. rb must be greater than rfinal, but can otherwise be optimized.
Lecture 9: Conversions, Orbit Transfers38
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)
Lecture 9: Conversions, Orbit Transfers 42
Changing Inclination• Δi Plane Change• Inclination-Only Change vs. Free Inclination Change
Lecture 9: Conversions, Orbit Transfers 43
Changing Inclination
• Let’s start with circular orbits
Lecture 9: Conversions, Orbit Transfers 44
V0
Vf
Changing Inclination
• Let’s start with circular orbits
Lecture 9: Conversions, Orbit Transfers 45
V0
Vf
Changing Inclination
• Let’s start with circular orbits
Lecture 9: Conversions, Orbit Transfers 46
V0
Vf
Δi
Are these vectors the same length?
What’s the ΔV?
Is this more expensive in a low orbit or a high orbit?
Changing Inclination
• More general inclination-only maneuvers
Lecture 9: Conversions, Orbit Transfers 47
Line of Nodes
Where do you perform the maneuver?
How do V0 and Vf compare?
What about the FPA?
Changing Inclination
• More general inclination-only maneuvers
Lecture 9: Conversions, Orbit Transfers 48