ASEN 5050 SPACEFLIGHT DYNAMICS General Perturbations Alan Smith University of Colorado – Boulder...

ASEN 5050SPACEFLIGHT DYNAMICS

General Perturbations

Alan Smith

University of Colorado – Boulder

Lecture 25: Perturbations 1

“It causeth my head to ache” - Isaac Newton

Announcements• Homework #7 is out now! Due Monday morning.

– Clarification for Problem 3: you do not have to implement BOTH a variable time-step integrator and a fixed time-step integrator. Pick one. Then fill in that half of the table.

• Reading: Chapters 8 and 9

Lecture 24: General Perturbations 2

Schedule from here out


• 10/27: Three-Body Orbits• 10/29: General Perturbations (Alan)• 10/31: General Perturbations part 2

• 11/3: Mission Orbits / Designing with perturbations• 11/5: Interplanetary 1• 11/7: Interplanetary 2

• 11/10: Entry, Descent, and Landing• 11/12: Low-Energy Mission Design• 11/14: STK Lab 3

• 11/17: Low-Thrust Mission Design (Jon Herman)• 11/19: Finite Burn Design• 11/21: STK Lab 4

• Fall Break

• 12/1: Constellation Design, GPS• 12/3: Spacecraft Navigation• 12/5: TBD

• 12/8: TBD• 12/10: TBD• 12/12: Final Review

Schedule from here out

• Our last lecture will be Friday 12/12.– Final review and final Q&A. – Showcase your final projects – at least any that are finished!

• Final Exam– Handed out on 12/12– Due Dec 18 at 1:00 pm – either into D2L’s DropBox or under my door.

• I heartily encourage you to complete your final project website by Dec 12th so you can focus on your finals. However, if you need more time you can have until Dec 18th. As such the official due date is Dec 18th.

• The final due date for everything in the class is Dec 18th - no exceptions unless you have a very real reason (medical or otherwise - see CU's policies here: http://www.colorado.edu/engineering/academics/policies/grading). Of course we will accommodate real reasons.

• If you are a CAETE student, please let me know if you expect an issue with this timeframe. We normally give CAETE students an additional week to complete everything, but the grades are due shortly after the 18th for everyone. So please see if you can meet these due dates.


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

• Rockets are tricky.

• Sad day for Orbital and its customers yesterday. Hopefully we will pick up all the pieces and come out of it even stronger.– No injuries reported, thankfully. And the ISS is not low on

any critical supplies.


ASEN 5050SPACEFLIGHT DYNAMICS

General Perturbations

Alan Smith

University of Colorado – Boulder


“It causeth my head to ache” - Isaac Newton

Lecture 25: Perturbations9

Perturbations

Special Perturbation Techniques – Numerical integration.

Straightforward – however obtaining a good understanding of the effects on the orbit is difficult.

General Perturbations – Use approximations to obtain analytical descriptions of the effects of the perturbations on the orbit. Assumes perturbing forces are small

Early work used general perturbations because of a lack of computational power. Modern work uses special perturbations (numerical integration) because of the wide availability of computers. GP still useful for increasing your understanding. Still used by AF for maintaining space object catalog (> 7000 objects).

Trend Analyses

• While it’s useful to be able to numerically integrate high-fidelity equations of motion, it’s also useful to have an expectation of possible effects for mission design and analysis

• Drag– Clear reduction in a and e; can also impact i and the others slightly.

• SRP– Depends, but in some circumstances it can increase e

• Earth’s oblateness– Dramatic change in Ω, ω, and M

• Third-body effects– Precession of the nodes, same effect as oblateness.


Trend Analyses

• Secular Trends• Long periodic effects• Short periodic effects

• Mean elements• Osculating elements



General Perturbation Techniques

Perturbations can be categorized as secular, short period, long period.

Method of Perturbations

• What is a small parameter?– J2

– Cl,m, Sl,m

– a / a3

– Thrust / acc2B

– Etc

• We need a way to generate the approximation. We’ll use the method of Variation of Parameters (VOP)


Variation of Parameters

• VOP was developed by Euler and improved by Lagrange (1873)

• Describes the variations in the orbital elements over time to first order constants.

• Need: – Unperturbed system as a reference (2-body solution).

– A way to generalize the constants in the system to be time-varying parameters.



• Consider a system of six first-order differential equations:

• These c parameters are osculating elements since they’re no longer constant.



• Any six elements may be used, including the conventional Keplerian orbital elements. But this isn’t necessary.

• These parameters are a solution to:

– If no perturbation, then we’d have conic sections.



• Any six elements may be used, including the conventional Keplerian orbital elements. But this isn’t necessary.

• These parameters are a solution to:

– In the presence of perturbations, what is this?



• Unperturbed and perturbed relationships for the velocity:

• We impose a constraint:the condition of osculation



• If we take the derivative of the perturbed velocity using the condition of osculation, we obtain:



• We substitute this equation into our earlier:



• We have:

• We can simplify this using our reference – that our unperturbed equations of motion hold for any given instant in time:



• We have:



• We have:

• This is a system of three equations of six variables. We need three more equations!

• Condition of osculation:



• Now we have 6 equations with 6 variables



• Now we have 6 equations with 6 variables

• Not well-suited for computation. We need this to be of the form:



• Two well-known ways to convert

to

• Lagrangian VOP and Gaussian VOP– Gauss’ is easier to present first, though Vallado presents

Lagrange’s first. We’ll do Gauss then LagrangeLecture 25: Perturbations 26

Gaussian VOP

• Start by taking the dot product of the first equation with

and the dot product of the second with

and add them together:


Gaussian VOP

• The parameters are mutually independent, so the term inside of the parentheses simplifies to the Kronecker delta function (1 for i=j, 0 else)

• Hence: Or... Lecture 25: Perturbations 28

Gaussian VOP

• The hard part!

• Gauss chose to perform these partial derivatives in the RSW coordinate frame. The disturbing force is thus:



Gaussian VOP

Gaussian VOP

• Vallado provides page after page of derivations of these partials. The result:


Gaussian VOP


Gaussian VOP

• Note a few limitations:

• e must be < 1.0

• i and e can’t be 0

• Hence, this is limited to moderately elliptical, non-equatorial orbits.


Lagrangian VOP

• Let’s restart and derive the Lagrange planetary equations of motion (LPEs), or simply the Lagrangian VOP

• Lagrange was the first person to perform this transformation.


Lagrangian VOP

• Take the dot product of the top vector with

• And the dot product of the bottom vector with

• Yields:


Comparison

•Gauss:

•Lagrange:


Disturbing potential in6 parameters

Lagrangian VOP

• We have:

• Lagrange brackets:


Lagrangian VOP

• We have:


Lagrangian VOP

• Some characteristics of Lagrange brackets:


Lagrangian VOP

• Thus many of the values are easy. The remainder are tricky.– Diagonal terms = 0

– L = Skew-symmetric

• Since the brackets are independent of time, we can evaluate them anywhere along the orbit.– Convenient to evaluate them at periapse!

– Convert state to PQW (perifocal) frame.


Lagrangian VOP

• Convert to PQW:


Lagrangian VOP

• Convert to PQW:


Note!

Lagrangian VOP

• Convert to PQW:


Lagrangian VOP

• Convenient independence:


Lagrangian VOP

• We can also determine the PQW position and velocity vectors at periapse:


Lagrangian VOP

• Three cases arise:


Lagrangian VOP

• After a lot of math:


Lagrangian VOP

• Note a few limitations:

• e must be < 1.0

• i and e can’t be 0

• Hence, this is limited to moderately elliptical, non-equatorial orbits.

• That seemed a LOT like the Gaussian VOP result


VOPs


• Lagrangian Gaussian

Forcing function in RSWForcing function in RSWPerturbing Potential FunctionPerturbing Potential Function

Lagrangian VOP

• Constructing perturbing potential functions

• Consider the spherical harmonic gravitational potential.– Take that potential function, remove the 2-body term, and

re-cast it in terms of the classical orbital elements.

– This leads to Kaula’s Solution:

– Can then evaluate it in the Lagrange planetary equations.


L.P.E.s & Kaula’s Solution


Using the L.P.E.s

• Let’s use the Lagrange planetary equations (LPEs) to evaluate the secular trends caused by a 2x2 gravity field.

• Start with the potential function, R:


Using the L.P.E.s

• Remove all periodic effects

• Left with:


Using the L.P.E.s

• Convert to orbital elements:

• We convert latitude:

• And use trig:


Using the L.P.E.s

• Remove periodic terms again:

• Yielding:


Using the L.P.E.s

• The value of a/r varies over an orbit, since r varies.• Average it over an orbit.


Using the L.P.E.s

• Evaluate this potential in the LPEs:

• Consider RAAN


Using the L.P.E.s

• After simplifying, we find:


1st-order secular trend of RAAN over time as a function of the orbit!

Using the L.P.E.s

• We can certainly make this trend more accurate by considering the first six zonals (S.H. order = 0):


Using the L.P.E.s

• Similar techniques reveal other secular trends.



Oblateness Perturbations



Which is a “secularly precessing ellipse”. The equatorial bulge introduces a force component toward the equator causing a regression of the node (for prograde orbits) and a rotation of periapse.

Note:







Periapse also precesses.

= 0 at the critical inclination,

i = 63.4 (116.6)





Application: Sun Synchronous orbits



Sun Synchronous orbits:

– Orbit plane remains at a constant angle (’) with respect to the Earth-Sun line.

– Orbit plane precession about the Earth is equal to period of Earth’s orbit about the Sun.

Perturbation Magnitudes


ISS Orbit

Perturbation Magnitudes


GPS Orbit


FIN

Date post:	20-Jan-2016
Category:	Documents
Upload:	neal-bennett
View:	230 times
Download:	1 times

ASEN 5050 SPACEFLIGHT DYNAMICS General Perturbations Alan Smith University of Colorado – Boulder...

Documents