ASEN 5050 SPACEFLIGHT DYNAMICS Conversions, f & g, Orbit Transfers Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 9: Conversions, Orbit

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)


Concept Quiz 7


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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.

Concept Quiz 7


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


Space News


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


Space News


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


CV

ASEN 5050SPACEFLIGHT DYNAMICSCoordinate Transformations


University of Colorado - Boulder


Principal Axis Rotations


Principal Axis Rotations

Lecture 9: Conversions, Orbit Transfers 16Equation 3-15

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


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 Series


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:


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:


f and g SeriesLook at the cross-product:

Which can only be true if: A good test!


f and g SeriesWhich gives:


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:


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 DYNAMICS

Orbital Maneuvers


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, .

• 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 Transfers30

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 ManeuversHohmann 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

Energy Changes



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.


Bi-elliptic Transfer

Much longer flight times for bi-elliptic transfer, but sometimes less energy.

(Algorithm 37, Example 6-2)


Hohmann vs Bi-elliptic


One-Tangent Burns


Orbit Transfer Comparison

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


V0

Vf




V0

Vf




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?


• More general inclination-only maneuvers


Line of Nodes

Where do you perform the maneuver?

How do V0 and Vf compare?

What about the FPA?


• More general inclination-only maneuvers


Changing The Node


Changing Argument of Perigee


Documents

ASEN 5050 SPACEFLIGHT DYNAMICS Conversions, f & g, Orbit Transfers Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 9: Conversions, Orbit