ASEN 5050 SPACEFLIGHT DYNAMICS Mid-Term Review Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 19: Mid-Term Review 1

ASEN 5050SPACEFLIGHT DYNAMICS

Mid-Term Review

Prof. Jeffrey S. Parker

University of Colorado – Boulder

Lecture 19: Mid-Term Review 1

Announcements• Alan’s office hours are on FRIDAY this week! 1:00 pm.

• No Concept Quiz active after this lecture

• STK LAB 2, due 10/17

• Homework #6 will be due Friday 10/24 (2014 not 2013)– CAETE by Friday 10/31

• 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: review. Friday: GRAIL and more perturbation fun.Lecture 19: Mid-Term Review 2

Final Project

• Get started on it!

• Worth 20% of your grade, equivalent to 6-7 homework assignments.

• Find an interesting problem and investigate it – anything related to spaceflight mechanics (maybe even loosely, but check with me).

• Requirements: Introduction, Background, Description of investigation, Methods, Results and Conclusions, References.

• You will be graded on quality of work, scope of the investigation, and quality of the presentation. The project will be built as a webpage, so take advantage of web design as much as you can and/or are interested and/or will help the presentation.


Final Project• Instructions for delivery of the final project:

• Build your webpage with every required file inside of a directory. – Name the directory “<LastName_FirstName>”– there are a lot of duplicate last names in this class!– You can link to external sites as needed.

• Name your main web page “index.html”– i.e., the one that you want everyone to look at first

• Make every link in the website a relative link, relative to the directory structure within your named directory.

– We will move this directory around, and the links have to work!

• Test your webpage! Change the location of the page on your computer and make sure it still works!

• Zip everything up into a single file and upload that to the D2L dropbox.


Space News


• MAVEN’s first scientific announcement!• UV views of Mars’ escaping atmosphere

Concept Quiz 12


Concept Quiz 12


Honestly, I didn’t make this clear enough.

There IS indeed correlation that the geoid follows mass (strictly speaking it is defined by the gravity of matter!)But it IS NOT perfectly correlated and sometimes it appears quite the opposite.


Mid-Term Review




Mid-Term Logistics

• Please write your name on the exam• Carry “infinite” precision in your math at all steps.

At the very end you are welcome to round your answers, but don’t round too much!

• Document which equations / process you use, and then verify that your math is correct using a computer.

• Try to learn something I’m not JUST doing this exam to torture you.


Course Topics

• Two-Body Problem– Newton’s Law of Gravitation– Orbital elements– Converting Cartesian to/from Keplerian Orbital Elements– Using Eccentric Anomaly to determine when an orbit is at some

radius from the central body and, given that, estimate how much time has elapsed since periapse, etc.

– Vis-Viva Equation

• Coordinate Systems– IJK, XYZ, Perifocal, RSW, SEZ, VNC, etc.

• Time Systems– UT1, UTC, TAI, GPS, ET


Course Topics

• Orbital Maneuvering– Changing each orbital element– Plane Changes are expensive

• Orbital Transfers– Hohmann Transfer (2 tangential burns)– Phase Changing

• Intercepting an object and rendezvousing with it– Combining maneuvers, such as an optimal LEO – GEO transfer

• Where do you perform the inclination change?• Do you perform all of the inclination change there?

– One-tangent burn• To speed up the transfer

– Lambert’s Problem• Minimum-energy orbital transfer between two arbitrary position vectors.


Course Topics

• Proximity Operations– Clohessy-Wiltshire / Hill’s Equations

– We’ll talk about this more in a few minutes

• Groundtracks– Plotting the sub-point of a satellite over time

– Repeat groundtracks and other practical groundtracks• Build me an orbit that exactly repeats its groundtrack every 12

days, after 121 revolutions.


Course Topics

• Additional Topics– The shape of the Earth

• Geocentric latitude vs. geodetic latitude

– Solar day vs. Sidereal day• 86400 seconds in a solar day, 86164.1 ish in a sidereal day. Be

careful!


Review Requests (first 43)

• CW/H x14• Groundtracks x2• Coordinate systems x5• Transformations x7• f and g series x4• Canonical units x2• Odd orbital elements• Geocentric/geodetic• Kepler’s Equation derivation• Lambert’s Problem x3• Synodic period• Plane changes / maneuvers not covered in HW x5• EVERYTHING x4• NOTHING x5


Coordinate Systems

Geocentric Coordinate System (IJK)

- aka: Earth Centered Inertial (ECI), or the Conventional Inertial System (CIS)

- J2000 – Vernal equinox on Jan 1, 2000 at noon

- non-rotatingIntersection of ecliptic and celestial eq


Coordinate Systems

Earth-Centered Earth-Fixed Coordinates (ECEF)

Topocentric Horizon Coordinate System (SEZ)


Coordinate Systems

Perifocal Coordinate System (PQW)


Coordinate SystemsSatellite Coordinate Systems:

RSW – Radial-Transverse-Normal

NTW – Normal-Tangent-Normal; VNC is a rotated version

CVR

S


Coordinate Transformations

Coordinate rotations can be accomplished through rotations about the principal axes.



To convert from the ECI (IJK) system to ECEF, we simply rotate around Z by the GHA:

ignoring precession, nutation, polar motion, motion of equinoxes.



To convert from ECEF to SEZ:


To set up a SYSTEM you also need to specify a center, which can be anything but is usually the reference site.


• One of the coolest shortcuts for building transformations from one system to any other, without building tons of rotation matrices:

The unit vector in the S-direction, expressed in I,J,K coordinates

(sometimes this is easier, sometimes not)


(Vallado, 1997)

Latitude/Longitude

Geocentric latitude


(Vallado, 1997)

Latitude/Longitude

For geodetic latitude use:

where e=0.081819221456


Latitude/Longitude

Rotate into ECEF



To convert between IJK and PQW:

To convert between PQW and RSW:

Thus, RSW IJK is:

R

S

P


CW / H

• Useful to answer questions like:– If I deploy a satellite from my current position in orbit, and

the deployment imparts some small Delta-V, where does the satellite go, relative to me?

– If I’m approaching a space station, what Delta-V should I execute to rendezvous with the station after 10 minutes?

– Also helps evaluate trajectories rapidly, since you don’t have to numerically integrate them.


Lecture 19: Mid-Term Review

Coordinate Systems

Satellite Coordinate System (RSW) -- (Radial-Transverse-Normal)

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To convert between IJK and PQW:

To convert between PQW and RSW:

Thus, RSW IJK is:

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Clohessy-Wiltshire (CW) Equations(Hill’s Equations)

Use RSW coordinate system (may be different from NASA)

Target satellite has two-body motion:

The interceptor is allowed to have thrusting

Then

So,

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Need more information to solve this:

Lecture 14 (Slide 27+) takes you through all of the steps needed to convertthis acceleration into one that isonly dependent on the relativevector and omega:

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We assume circular motion:

Thus,

Assume = 0 (good for impulsive V maneuvers, not for continuous thrust targeting).

Equations also assume circular orbits and close proximity!

CW or Hill’s Equations

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These equations can be solved (see book: Algorithm 48) leaving:

So, given of interceptor, can determine

of interceptor at future time.33



34



35



36



Hubble’s Drift from Shuttle

• RSW Coordinate Frame




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We can also determine V needed for rendezvous. Given x0, y0, z0, we want to determine necessary to make x = y = z = 0.Set first 3 equations to zero, and solve for .

Assumptions:1. Satellites only a few km apart2. Target in circular orbit3. No external forces (drag, etc.)

40



41



42



43

NOTE: This is not the Delta-V, this is the new required relative velocity!

CW/H Examples

• Scenario 1:

• Scenario 1b:


Deployment: x0, y0, z0 = 0, Delta-V = non-zero

Use eqs to compute x(t), y(t), and z(t)RSW coordinates, relative to deployer.

Initial state: non-zero

Use eqs to compute x(t), y(t), and z(t)RSW coordinates, relative to reference.

Note: in the CW/H equations, the reference state doesn’t move – it is the origin!

Note: in the CW/H equations, the reference state doesn’t move – it is the origin!

CW/H Equations

• Rendezvous– For rendezvous we usually specify the coordinates relative to the

target vehicle and set x, y, and z to zero– Though if there’s a docking port, then that will be offset from

the center of mass of the vehicle.– Define RSW targets: x, y, z (often zero)

Lecture 19: Mid-Term Review 45Target: some constant values in the RSW frame

Initial state in the RSW frame

CW/H Equations






This is easy if the targets are zero (Eq. 6-66)This is harder if they’re not!

Velocity needed to get onto transfer

CW/H Equations






This is easy if the targets are zero (Eq. 6-66)This is harder if they’re not!

Velocity needed to get onto transfer

The Delta-V is the difference of these

velocities

CW / H

• Scenario 3: A jetpack-wielding astronaut leaves the shuttle and then returns.

Lecture 19: Mid-Term Review 48Shuttle: reference frame

CW / H



Result of deployment

CW / H




Rendezvous trajectory (Eq 6-66)

CW / H




Rendezvous trajectory (Eq 6-66)

Delta-V is the difference of these velocities.

Example from last year’s mid-term!


Mid-Term Review

• Problem 4


Mid-Term Review

• Problem 4


Mid-Term Review

• Problem 4

• First, we need the velocity of the shuttle relative to the experiment before the maneuver:– Experiment’s velocity relative to shuttle, from Algorithm 48 of

part (a)

– Shuttle’s velocity relative to experiment is just the opposite of that:


Mid-Term Review

• Problem 4

• Second, we need the velocity that the shuttle must obtain to perform the rendezvous.– Equation 6-66 in Vallado’s 4th edition


Mid-Term Review

• Problem 4

• The initial conditions are the state of the shuttle relative to the experiment (opposite signs of the experiment’s position relative to the shuttle from part (a))

• t = 10*60 sec = 600 sec (you can keep omega the same as before, or update it; doesn’t make much difference)


Mid-Term Review

• Problem 4, part (b)

• Velocity required to achieve the transfer:

• Delta-V for the rendezvous:


Mid-Term Review

• Problem 4, part (c)

• Use Algorithm 48 once again and now plug in the initial position and velocity of the shuttle relative to the experiment.

• The position of the shuttle after 10 minutes (15 minutes after the deployment) should be zero (GOOD CHECK!)

• The velocity of the shuttle after 10 minutes (15 minutes after the deployment) will not be zero.

• The Delta-V is that which will remove the relative velocity of the shuttle relative to the experiment.


Mid-Term Review

• Problem 4, part (c)


Orbit Maneuvering

• Hohmann Transfers• Bi-elliptic Transfers• Circular Rendezvous• Coplanar Rendezvous• Changing orbital elements

– a, e, rp, ra, P, M, AOP are all coplanar

– i, RAAN are plane changes

• Lambert’s Problem


Changing Orbital Elements

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

Lecture 10: Orbit Transfers 62

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 The Node


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






Which ΔV is cheaper?

Maneuver Combinations

• General rules to consider:– Energy changes are more efficient when traveling FAST

• Periapse

– Plane changes are more efficient when traveling SLOW• Apoapse

– Combinations take advantage of vector addition• 3+4 = 5 not 7 • Some inclination change at periapse is optimal• Some energy change at apoapse is optimal (or necessary)• Delta-V vector = V final vector – V initial vector

– One’s initial and final orbit do not always intersect; if they don’t you have to build a transfer orbit.


Questions

• These are the sorts of questions I hope you can answer:– Given orbit X, when does the satellite reach radius R?

– Where in orbit Y is the satellite at its maximum latitude?

– Compute element XYZ given element ABC

– And all of those concept quiz type questions.

– There will be math and concepts.



GRAIL





Perturbations




Perturbation Discussion Strategy

• Introduce the 3-body and n-body problems– We’ll cover halo orbits and low-energy transfers later

• Numerical Integration• Introduce aspherical gravity fields

– J2 effect, sun-synchronous orbits• Solar Radiation Pressure• Introduce atmospheric drag

– Atmospheric entries• Other perturbations• General perturbation techniques• Further discussions on mean motion vs. osculating

motion.


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ASEN 5050 SPACEFLIGHT DYNAMICS Mid-Term Review Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 19: Mid-Term Review 1