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Orbit of a star in a sherical potential

  1. Sep 8, 2009 #1
    1. The problem statement, all variables and given/known data
    A star is at radius r = 10kpc with v_t=100km/s and v_r=50km/s. The spherical potential is
    \phi = V^2ln(r) with V=200km/s
    1. what is r_min r_max?
    2. Integrate the orbit numerically

    3. The attempt at a solution

    at r_min and r_max v_r is 0 therefore I can write

    E_min = E_max = L^2/(2r^2) + V^2ln(r) = E

    but how can I solve this to r = ??

    I thought actually I can use angular momentum conservation and say L^2/(2r_min^2) = L^2/(2r^2) which does not really work out? But I think I have to include angular momentum conservation somehow...

    I tried numerical integration with the following equation

    dr/dt = sqrt(2*(E-\phi) - L^2/(2r^2))

    but the value below the root is negative ???

    I also found

    (dr/r^2d\theta)^2 = 2E/L^2 - 1/r^2 + 2V^2ln(r)/L^2

    but here I have a very similar problem.
    So my question concerning the numerical integration is actually which equation should I use and how to integrate it. I would like to do the integration in a C/C++ for loop by myself instead of using mathematica or other tools... is that possible?
    thanks and best regards
  2. jcsd
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