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Projectiles and the Kepler problem

  1. Oct 2, 2009 #1
    1. The problem statement, all variables and given/known data

    Within the gravitational field produced by a celestial body of mass M we want to send a projectile of mass m from (x1,y1) to (x2,y2). The M mass is placed at the origin of coordinates. If the flight time is T, what is the initial velocity vector that we have to give to the projectile? Is this vector unique?

    2. Relevant equations

    The usual equations of the Kepler problem, ie

    [tex]l=mr^2\dot{\theta}[/tex]

    [tex]E=\frac{1}{2}m\dot{r}^2+\frac{l^2}{2mr^2}-\frac{k}{r}[/tex]

    [tex]r=\frac{\frac{l^2}{mk}}{1+\epsilon\cos\theta}[/tex]

    Maybe the most relevant one is

    [tex]dt=\frac{m}{l}r^2\,d\theta[/tex]

    since this equation includes time explicitly.

    3. The attempt at a solution

    Using [tex]dt=\frac{m}{l}r^2\,d\theta[/tex] and the equation for the orbit I integrated from [tex]t=0[/tex] to [tex]t=T[/tex] but the angular integration, that although can be done analytically, it turns out to be quite difficult, so I think this is not the right way to go.

    The special case of a circular orbit is solved easily but I think the problem has to be solved in general.

    Thanks in advance!
     
  2. jcsd
  3. Oct 3, 2009 #2
    In this problem, conservation principles are not quite useful since they are time-independent. The main difficulty of this problem is to incorporate T into the solution of the problem.

    Any ideas?
     
  4. Oct 3, 2009 #3

    D H

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    Lambert targeting.
     
  5. Oct 3, 2009 #4
    Hi! Thanks for replying.

    So my problem is about Lambert targeting? Can you tell me where I can read about it?

    Thanks!
     
  6. Oct 4, 2009 #5

    D H

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    Staff Emeritus
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    [strike]As a starting point only, here is the wikipedia article on Lambert's problem: http://en.wikipedia.org/wiki/Lambert's_problem[/strike]

    Fundamentals of Astrodynamics and Applications by David Vallado is an excellent reference. Some of the section regarding Lambert's problem is available at Google books: http://books.google.com/books?id=PJLlWzMBKjkC&pg=RA1-PA448#v=onepage&q=&f=false

    Fundamentals of Astrodynamics by Bate, Mueller, and White also covers the topic.

    Chapter 2 ("Guidance Algorithm") of http://dspace.mit.edu/bitstream/handle/1721.1/34137/67775726.pdf?sequence=1" [Broken] does a nice job of describing Lambert's problem.


    Edit:
    Wikipedia reference struck. This is an example of why wikipedia is a lousy reference. The article has no references, is poorly written, and misses the very important concept of the "short way" versus the "long way".
     
    Last edited by a moderator: May 4, 2017
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