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Bessel Function FROM Kepler Equation?

  1. Apr 7, 2012 #1
    I have been reading some material about the relationship between Kepler's Equation for Elliptical Motion and Bessel Functions.

    Kepler's Equation for Elliptical Motion is the following:

    E = M + e sin(E)​

    It can also be stated in terms of Bessel Functions of the first kind of integer order n:

    [tex]E = M + \sum_{1}^{\infty }(\frac{2}{n})J_{n}(ne)sin(nM)[/tex]


    [tex]J_{n}(x) = \frac{1}{\pi}\int_{0}^{\pi }cos(x sin(\theta) - n\theta )d\theta[/tex]

    (the integral form)

    My question is:

    Can this relationship be stated the other way around (i.e. - Bessel Functions in terms of Kepler's Equation)?

    So far, I have not come across one, but I am thinking it might be useful, if any insights gleaned through Kepler's Equation could then be applied to Bessel's Function.
  2. jcsd
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