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B Mathematical Proof of Kepler's First Law of Orbits

  1. May 20, 2016 #1
    Hello friends (I hope :biggrin:),

    For a maths project I am working on, I need to be able to prove the equation for an elliptical orbit, related to Kepler's first law:

    b115356a043a8d816886221bdd807dfb.png and p = a(1-e2) (or should be as p can be replaced by that value)

    r = distance from sun to any point on the orbit
    p = semi latus rectrum
    a = semi-major axis
    e = eccentricity
    θ = true anomaly (angle between a and r anticlockwise I think)

    Can someone please help me to understand where these equations come from and also confirm that I have got my current facts straight?

  2. jcsd
  3. May 20, 2016 #2

    George Jones

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    Have you looked in classical mechanics and/or astrophysics books?
  4. May 20, 2016 #3
    I haven't been able to come by any at the moment that help me derive the equation itself, which is what I don't understand. However, if you could name me any books that do that would be exceptionally helpful :smile:

    Thank you
  5. May 20, 2016 #4

    George Jones

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    Examples at the level of about second-year university include "Analytical Mechanics" by Fowles and Cassiday, and "Foundations of Astrophysics" by Ryden and Peterson. I haven't looked, but I suspect that derivations of Kepler's first law also can be found on-line. Just Google "Kepler's first law".
  6. May 20, 2016 #5
    I have tried online but to no avail unfortunately, apart from it bringing me here actually :biggrin:. Thank you very much though, I shall look for those books as soon as possible.

  7. May 20, 2016 #6


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    There's plenty of material online which derives Kepler's laws from different perspectives.

    Look at:
    https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion (check out the bibliography at the end of the article)

    https://en.wikipedia.org/wiki/Ellipse (for details of the equation of the ellipse in polar form)

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