Why Don't Cowell's and Gauss' Methods Match for Osculating Orbital Elements?

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Discussion Overview

The discussion revolves around the integration of orbital elements (a, e, i, w, Omega, f) using two methods: Cowell's method and Gauss' Variational differential equations. Participants are exploring the discrepancies in results obtained from these two approaches over a specified time frame for a satellite orbiting the Earth.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes their attempt to integrate orbital elements over 20 periods using MATLAB, noting that the results from Cowell's method do not match those from Gauss' method.
  • Another participant requests a summary of the approach taken, indicating the complexity of the provided code.
  • A participant suggests that the integration of position and velocity in the second part of the assignment may not be appropriate and points out potential mathematical errors in the code.
  • There is a question raised about how to calculate the position vector needed for the orbital elements without integrating position directly.
  • A suggestion is made to use the integrated osculating orbital elements for calculations.
  • A later reply emphasizes the need for clarity in the participant's explanation of specific variables (ar, ah, atheta) and encourages the use of LaTeX for mathematical expressions.

Areas of Agreement / Disagreement

Participants express differing views on the methods of integration and the correctness of the code, indicating that multiple competing perspectives remain. The discussion has not reached a consensus on the best approach or the source of the discrepancies.

Contextual Notes

Participants note potential mathematical errors and the need for clearer definitions of variables, but these issues remain unresolved within the discussion.

jsandberg
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Homework Statement


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.


Homework Equations


See page 488 of this reference:

http://books.google.com/books?id=Oj...snippet&q=osculating orbital elements&f=false

These equations are used in the attached code.


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.
 

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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?
 


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=OjH7aVhiGdcC&printsec=frontcover&dq=Battin&cd=1#v=onepage&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!
 


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.
 


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?
 


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.
 
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