Celestial Mechanics: Osculating Orbital Elements using Gass' Variational Equations

jsandberg

1. The problem statement, all variables and given/known data
The orbital elements (a,e,i,w,Omega,f) are to be integrated forward in time using two methods: 1) Cowell's method, which is direct interation, and 2) Gauss' Variational diffeential equations. The full assignment description, initial conditions, and my code are given in the attachment.


2. Relevant equations
See page 488 of this reference:

http://books.google.com/books?id=OjH...ements&f=false

These equations are used in the attached code.


3. The attempt at a solution
As can be seen from the attached code, my results do not match for the Cowell method and the Gauss' Variational method. These results should match, however. I can't figure out what I'm doing wrong! Any suggestions would be fantastic.

Attached Files

Celestial Mechanics.zip (96.5 KB, 46 views)

D H

Whoa! That is a lot of stuff to look at! Several hundred lines of matlab code, plus problem assignment, etc. is a bit much. Can you briefly summarize what you have done and why you did it that way?

jsandberg

Yes! I am so sorry, I have never posted on a forum before and don't know how it works. Let me try to summarize.

I am to find the time-varying elements (a,e,i,w,Omega,f) over 20 orbital periods for a satellite orbiting the earth. The equations are given on page 488 of this textbook:
http://books.google.com/books?id=OjH...page&q&f=false

I am using an ode45 integration with given initial conditions- basically, I think my orb_el_prop function is not working correctly. If you download the zipped files and run Part1Main.m (no need to look through Part 1 code unless you'r curious), the plots should look the same as for my Part2Main. If you run Part2Main.m, you'll see the plots don't match.

I hope that makes more sense. Thanks!

D H

In part 2 you really should not be integrating position and velocity. You should be calculating them based on orbital elements. Also, you have some math errors in your part 2 orb_el_prop.m. Per that file, ar and ah will be identically equal to one (within numerical error) and atheta will be zero.

jsandberg

Thank you! To calculate the orbial elements, I need norm(r), which means I need the position vector. How do I calculate this if I'm not integrating the position?

D H

You use your integrated osculating orbital elements.

Addendum
jsandberg, I have to run out for a while; I may not be back until quite a bit later this evening. I have asked other homework helpers to dive up, but to be honest, numerically integrating Lagrange's Planetary Equations is a bit esoteric a topic. You can help by briefly specifying what you are trying to do here. It will help both you and other helpers who might dive in if you explain what your ar, ah, and atheta are supposed to be.

It will also help if you type out (rather than refer us to a book) the relevant equations. We have a LaTeX mechanism by which you can post mathematics in LaTeX.