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Homework Help: Circularisation of Orbit

  1. May 19, 2015 #1
    1. The problem statement, all variables and given/known data

    A body on an orbit with semi-major axis a and eccentricity e undergoes tidal circularisation.

    Show that the orbit will circularise at a semi-major axis, acirc, given by

    acirc = 2rperi = 2a (1 − e).

    2. Relevant equations

    No equations given, but I think the following could be useful

    E = -GMm/2a
    e2 = 1 - b2/a2

    3. The attempt at a solution

    An earlier part of the question hints at L conservation

    Equating centripetal force and grav force for the circular orbit gives:
    L = m (GMR)0.5

    Finding the velocity at the closest point in orbit r = a(1-e)

    E = -GMm/2a = 1/2 mv2 - GMm/a(1-e)

    simplifies to
    v2 = GM(1+e)/rp

    Equating L2
    L2 = GMm2 rp (1+e) = GMm2 rc

    rc = rp (1+e)

    This is close to the final answer, but not quite!
    Somethings gone wrong somewhere but I'm sure what.. I've checked my working several times.
    Sorry a lot of my working lines are missing, it's quite tricky to type them all out.
  2. jcsd
  3. May 19, 2015 #2


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    2017 Award

    Staff: Mentor

    You have to find out which quantities are conserved during the process. Energy, angular momentum, or something else?

    That cannot be the final semi-major axis. Consider the trivial case of e=0, for example, where the semi-major axis will certainly not double.
    It could be twice the semi-major axis.
  4. May 19, 2015 #3
    Oh, well spotted with the trivial case.
    Yea it looks like there's something wrong the question, and i think my method was fine.
    Thanks for your time.
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