ASEN 5050SPACEFLIGHT DYNAMICS

Orbit Transfers

Prof. Jeffrey S. Parker

University of Colorado – Boulder

Lecture 10: Orbit Transfers 1

Announcements• Homework #4 is due Friday 9/26 at 9:00 am

– You’ll have to turn in your code for this one.– Again, write this code yourself, but you can use other code to validate it.

• Concept Quiz #8 is active after this lecture; due before Wednesday’s lecture.

• 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’s office hours are at 2:00.

• Reading: Chapter 6 (SIX, we jumped a few)Lecture 10: Orbit Transfers 2

Space News

• Sunday: MAVEN arrived at Mars!

Lecture 10: Orbit Transfers 3

Space News

Lecture 10: Orbit Transfers 4

Today: Cassini is flying by Titan for the 106th time. 1400 km altitude, 5.6 km/s Vp

Space News

• Then Tuesday: MOM arrives at Mars!

• MOI: Tuesday at 20:00 Mountain– It will enter occultation at 20:04

– MOI will end at 20:24

– We’ll know if it’s successful around 20:30

– Notice that I write “Tuesday” here. It’ll be Wednesday in India and that keeps throwing me off Aw, time conversions!

– Not sure if there will be media coverage. Try http://www.spaceflightnow.com/ or NASA TV.

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ASEN 5050SPACEFLIGHT DYNAMICS

Orbital Maneuvers

Prof. Jeffrey S. Parker

University of Colorado - Boulder

Lecture 10: Orbit Transfers 6

Lecture 10: Orbit Transfers7

Orbital ManeuversHohmann Transfer – Walter Hohmann (1880-1945) showed

minimum energy transfer between two orbits used two tangential burns.

Lecture 10: Orbit Transfers8

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

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V is highest at perigee, thus energy-changing maneuvers are the most efficient at perigee!

Energy Changes

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Hohmann Transfer

• Example: LEO to GEO:

• LEO: altitude 185 km, radius 6563.136 km• GEO: altitude 35,786 km, radius 42,164 km

• VLEO = 7.7932 km/s VGEO = 3.0747 km/s• Vp

T = 10.2521 km/s VaT = 1.5958 km/s

• ΔV1 = 2.4590 km/s ΔV2 = 1.4788 km/s

• Total ΔV = 3.9378 km/s

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Hohmann Transfer

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GEO

Moon Radius

Hohmann Transfer

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GEO

Moon Radius

General radii transfers

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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

• Equations you need:

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SIMPLE, because all maneuvers are tangential, co-planar.

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Bi-elliptic Transfer

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

(Algorithm 37, Example 6-2)

Bi-elliptic Transfer

• LEO – GEO via 100,000 km altitude ΔV

• ΔV1 = 2.903 km/s• ΔV2 = 0.799 km/s• ΔV3 = 0.605 km/s• Total ΔV: 4.307 km/s

– More than Hohmann!Lecture 10: Orbit Transfers 17

Bi-elliptic LEO-GEO

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Moon Radius

Bi-elliptic LEO-GEO

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Moon Radius

Hohmann

Bi-elliptic Transfer

• LEO – 250,000 km via 2.4 million km altitude ΔV

• ΔV1 = 3.192 km/s• ΔV2 = 0.329 km/s• ΔV3 = 0.327 km/s• Total ΔV: 3.849 km/s

– More than Hohmann (4.058 km/s)!Lecture 10: Orbit Transfers 20

Bi-elliptic 185 km – 250,000 km

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Moon Radius

Hohmann

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Hohmann vs Bi-elliptic

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One-Tangent Burns

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Orbit Transfer Comparison

Changing Orbital Elements

• Δa Hohmann Transfer• Δe Hohmann Transfer• Δi Plane Change• ΔΩ Plane Change• Δω Coplanar Transfer• Δν Phasing/Rendezvous

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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

Changing Inclination

• Let’s start with circular orbits

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V0

Vf

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?

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?

Changing Inclination

• More general inclination-only maneuvers

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Changing The Node

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Changing The Node

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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 uinitial and ufinal in the book for circular orbits.

Lambert’s Problem gives easier solutions.

Changing Argument of Perigee

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Changing Argument of Perigee

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Changing Argument of Perigee

Lecture 10: Orbit Transfers 36

Which ΔV is cheaper?

Lecture 8: Orbital Maneuvers37

Circular Rendezvous (coplanar)

Target spacecraft; interceptor spacecraft

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Circular Rendezvous (coplanar)

How do we build these?

• Determine your phase angle, φ• Determine how long you want to spend performing the

transfer– How many revolutions?

• Build the transfer

• Compute the ΔVLecture 8: Orbital Maneuvers 39

How do we build these?

• Compute the ΔV

Lecture 8: Orbital Maneuvers 40

Lecture 8: Orbital Maneuvers41

Example 6-8

This should be +20°

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Example 6-8

This should really be an absolute value (one maneuver is in-track, one is anti-velocity)

This should really be an absolute value (one maneuver is in-track, one is anti-velocity)

Should be positive

Conclusions

• Better to use as many revolutions as possible to save fuel.

• Trade-off is transfer duration

• If you perform the transfer quickly, be sure to check your periapse altitude.

Lecture 8: Orbital Maneuvers 43

Lecture 8: Orbital Maneuvers44

Circular Coplanar Rendezvous (Different Orbits)

Lecture 8: Orbital Maneuvers45

Circular Coplanar Rendezvous (Different Orbits)

Use Hohmann Transfer

The “wait time”, or time until the interceptor and target are in the correct positions:

Synodic Period:

π – αL

Lecture 8: Orbital Maneuvers46

Example 6-9

Lecture 8: Orbital Maneuvers47

Example 6-9

I think this should be pi – alpha, not alpha – pi (see Fig 6-17)I think this should be pi – alpha, not alpha – pi (see Fig 6-17)

Announcements• Homework #4 is due Friday 9/26 at 9:00 am

– You’ll have to turn in your code for this one.– Again, write this code yourself, but you can use other code to validate it.

• Concept Quiz #8 is active after this lecture; due before Wednesday’s lecture.

• 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’s office hours are at 2:00.

• Reading: Chapter 6 (SIX, we jumped a few)Lecture 10: Orbit Transfers 48