Celestial Mechanics: Osculating Orbital Elements using Gass' Variational Equations

In summary: See the "LaTeX" link on any forum page.In summary, the conversation is about a homework assignment involving the numerical integration of orbital elements using two methods - Cowell's method and Gauss' Variational differential equations. The student is having trouble with their results not matching and is seeking suggestions on what they may be doing wrong. They provide a link to a reference book for the equations and code they are using. Another helper points out some potential math errors and suggests using integrated osculating orbital elements instead. The student asks for clarification on how to calculate the position vector without integrating it. The helper suggests that the student specify what they are trying to do and provide the relevant equations in their explanation.
  • #1
jsandberg
7
0

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.
 

Attachments

  • Celestial Mechanics.zip
    96.5 KB · Views: 338
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  • #2


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?
 
  • #3


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!
 
  • #4


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.
 
  • #5


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?
 
  • #6


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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What is Celestial Mechanics?

Celestial Mechanics is the branch of physics that studies the motion and behavior of objects in space, particularly celestial bodies such as planets, moons, and comets.

What are Osculating Orbital Elements?

Osculating Orbital Elements are a set of parameters that describe the shape, orientation, and position of an orbiting body's trajectory in space at a specific point in time. These elements are constantly changing due to the gravitational influence of other bodies in the system.

What are Gass' Variational Equations?

Gass' Variational Equations are a mathematical model used to calculate the changes in an orbiting body's osculating orbital elements over time. They take into account the perturbations caused by other celestial bodies and can be used to predict the future trajectory of an object.

How are Osculating Orbital Elements and Gass' Variational Equations used?

Osculating Orbital Elements and Gass' Variational Equations are commonly used in the field of astrodynamics to study and predict the motion of objects in space. They are also used in spacecraft trajectory planning and in the analysis of satellite orbits.

What are some practical applications of studying Celestial Mechanics and Osculating Orbital Elements?

Studying Celestial Mechanics and Osculating Orbital Elements has many practical applications, including predicting and avoiding collisions between orbiting objects, planning space missions, and understanding the effects of gravity on objects in space. It is also important for accurately predicting the movements of natural celestial bodies, such as predicting the timing and location of eclipses.

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