asen 5050 spaceflight dynamics conversions, f & g, orbit transfers
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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) - PowerPoint PPT PresentationTRANSCRIPT
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ASEN 5050SPACEFLIGHT DYNAMICS
Conversions, f & g, Orbit Transfers
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 9: Conversions, Orbit Transfers 1
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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)
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Concept Quiz 7
Lecture 9: Conversions, Orbit Transfers 3
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Concept Quiz 7
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Concept Quiz 7
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The class is 50-50 split on this!
Talk it over with your neighbor and convince him/her why you’re right.
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Concept Quiz 7
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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.
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Space News• Sunday: MAVEN arrives at Mars!
• MOI: this Sunday at 19:37 Mountain
• LASP is holding a viewing party
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Space News
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Space News
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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
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Space News
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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
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CV
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ASEN 5050SPACEFLIGHT DYNAMICSCoordinate Transformations
Prof. Jeffrey S. Parker
University of Colorado - Boulder
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Principal Axis Rotations
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Principal Axis Rotations
Lecture 9: Conversions, Orbit Transfers 16Equation 3-15
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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
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PQW
PQW
PQW
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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.
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f and g Series
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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:
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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:
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f and g SeriesLook at the cross-product:
Which can only be true if: A good test!
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f and g SeriesWhich gives:
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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
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f and g Series: State Transition MatrixWe can re-express our f and g series representation:
in terms of a state-variable relationship:
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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.
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ASEN 5050SPACEFLIGHT DYNAMICS
Orbital Maneuvers
Prof. Jeffrey S. Parker
University of Colorado - Boulder
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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.
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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).
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Orbital Maneuvers
Tangential Burns: in velocity/anti-velocity vector direction
- Do not change velocity orientation, just magnitude
- Do not change flight path angle
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Lecture 9: Conversions, Orbit Transfers31
Orbital Maneuvers
Nontangential: plane changes, orbit rotations
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Lecture 9: Conversions, Orbit Transfers32
Orbital ManeuversHohmann Transfer – Walter Hohmann (1880-1945) showed
minimum energy transfer between two orbits used two tangential burns.
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Lecture 9: Conversions, Orbit Transfers33
Hohmann Transfer(math, compute DV and DT)
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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)
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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
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V is highest at perigee
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Energy Changes
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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.
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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)
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Lecture 9: Conversions, Orbit Transfers39
Hohmann vs Bi-elliptic
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One-Tangent Burns
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Orbit Transfer Comparison
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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 (later discussion)
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Changing Inclination• Δi Plane Change• Inclination-Only Change vs. Free Inclination Change
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Changing Inclination
• Let’s start with circular orbits
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V0
Vf
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Changing Inclination
• Let’s start with circular orbits
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V0
Vf
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Changing Inclination
• Let’s start with circular orbits
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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?
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Changing Inclination
• More general inclination-only maneuvers
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Line of Nodes
Where do you perform the maneuver?
How do V0 and Vf compare?
What about the FPA?
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Changing Inclination
• More general inclination-only maneuvers
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Changing The Node
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Changing Argument of Perigee
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