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Elliptic expansions

The well-known Bessel functions of the first kind, $ J_n(x)$ , where $ n$ is an integer, are defined as the Fourier coefficients in the expansion of $ \exp(\,{\rm i}\,x\,\sin\phi)$ :

$\displaystyle {\rm e}^{\,{\rm i}\,x\,\sin\phi}\equiv \sum_{n=-\infty,\infty}J_n(x)\,{\rm e}^{\,{\rm i}\,n\,\phi}.$ (A.116)

It follows that

$\displaystyle J_n(x) = \frac{1}{2\pi}\oint {\rm e}^{-{\rm i}\,(n\,\phi-x\,\sin\phi)}\,d\phi = \frac{1}{\pi}\int_0^\pi \cos\left(x\,\sin\phi-n\,\phi\right)\,d\phi$ (A.117)

(Gradshteyn and Ryzhik 1980a). The Taylor expansion of $ J_n(x)$ about $ x=0$ is

$\displaystyle J_n(x) = \left(\frac{x}{2}\right)^n\sum_{k=0,\infty} \frac{(-x^{\,2}/4)^k}{k!\,(n+k)!}$ (A.118)

for $ n\geq 0$ (Gradshteyn and Ryzhik 1980b). Moreover,

    $\displaystyle J_{-n}(x)$ $\displaystyle =(-1)^n\,J_n(x),$ (A.119)
and   $\displaystyle J_{-n}(-x)$ $\displaystyle = J_n(x).$ (A.120)

In particular,

    $\displaystyle J_0(x)$ $\displaystyle = 1-\frac{x^{\,2}}{4} + {\cal O}(x^{\,4}),$ (A.121)
    $\displaystyle J_1(x)$ $\displaystyle =\frac{x}{2}-\frac{x^{\,3}}{16} + {\cal O}(x^{\,5}),$ (A.122)
    $\displaystyle J_2(x)$ $\displaystyle =\frac{x^{\,2}}{8}+{\cal O}(x^{\,4}),$ (A.123)
and   $\displaystyle J_3(x)$ $\displaystyle =\frac{x^{\,3}}{48}+{\cal O}(x^{\,5}).$ (A.124)

Let us write

$\displaystyle {\rm e}^{\,{\rm i}\,E} = \sum_{n=-\infty,\infty} A_n\,{\rm e}^{\,{\rm i}\,n\,{\cal M}},$ (A.125)

where $ E$ is the eccentric anomaly, and $ {\cal M}$ the mean anomaly, of a Keplerian elliptic orbit. (See Section 4.11.) It follows that

$\displaystyle A_n = \frac{1}{2\pi}\oint {\rm e}^{\,{\rm i}\,(E-n\,{\cal M})}\, d{\cal M}.$ (A.126)

Integrating by parts, we obtain

$\displaystyle A_n = \frac{1}{2\pi\,n}\oint{\rm e}^{\,{\rm i}\,(E-n\,{\cal M})} \,dE.$ (A.127)

However, according to Equation (4.59), the relationship between the eccentric and the mean anomalies is

$\displaystyle E-e\,\sin E={\cal M},$ (A.128)

where $ e$ is the orbital eccentricity. Hence,

$\displaystyle A_n = \frac{1}{2\pi\,n} \oint {\rm e}^{-{\rm i}\,[(n-1)\,E - n\,e\,\sin E]}\,dE.$ (A.129)

Comparison with Equation (A.117) reveals that

$\displaystyle A_n = \frac{J_{n-1}(n\,e)}{n}.$ (A.130)

For the special case $ n=0$ , L'Hopital's rule, together with Equations (A.119) and (A.122), yields

$\displaystyle A_0 = e\,J_{-1}'(0)=-e\,J_1'(0) = -\frac{e}{2},$ (A.131)

where $ '$ denotes a derivative.

The real part of Equation (A.125) gives

$\displaystyle \cos E$ $\displaystyle = A_0 + \sum_{n=1,\infty}(A_n+A_{-n})\,\cos(n\,{\cal M})$    
  $\displaystyle =-\frac{e}{2} + \sum_{n=1,\infty} \left[\frac{J_{n-1}(n\,e)-J_{-n-1}(-n\,e)}{n}\right]\cos (n\,{\cal M})$    
  $\displaystyle = -\frac{e}{2} + \sum_{n=1,\infty}\left[\frac{J_{n-1}(n\,e)-J_{n+1}(n\,e)}{n}\right]\cos(n\,{\cal M}),$ (A.132)

where use has been made of Equations (A.120), (A.130), and (A.131). Likewise, the imaginary part of Equation (A.125) yields

$\displaystyle \sin E = \sum_{n=1,\infty}\left[\frac{J_{n-1}(n\,e)+ J_{n+1}(n\,e)}{n}\right]\sin(n\,{\cal M}).$ (A.133)

It follows from Equations (A.121)-(A.124) that

    $\displaystyle \sin E$ $\displaystyle = \left(1-\frac{e^{\,2}}{8}\right)\sin {\cal M}+\frac{e}{2}\left(1-\frac{e^{\,2}}{3}\right)\sin 2{\cal M} +\frac{3\,e^{\,2}}{8}\,\sin 3{\cal M}$    
      $\displaystyle \phantom{=}+ \frac{e^{\,3}}{3}\,\sin 4{\cal M} + {\cal O}(e^{\,4}),$ (A.134)
and   $\displaystyle \cos E$ $\displaystyle =-\frac{e}{2} +\left(1-\frac{3\,e^{\,2}}{8}\right)\cos {\cal M}+\frac{e}{2}\left(1-\frac{2\,e^{\,2}}{3}\right)\cos 2{\cal M}$    
      $\displaystyle \phantom{=} +\frac{3\,e^{\,2}}{8}\,\cos 3{\cal M}+ \frac{e^{\,3}}{3}\,\cos 4{\cal M}+ {\cal O}(e^{\,4}).$     (A.135)

Hence, from Equation (A.128),

$\displaystyle E$ $\displaystyle = {\cal M} + e\,\sin{\cal M} + \frac{e^{\,2}}{2}\,\sin 2{\cal M} + \frac{e^{\,3}}{8}\left(3\,\sin 3{\cal M}-\sin{\cal M}\right)$    
  $\displaystyle \phantom{=}+\frac{e^{\,4}}{6}\left(2\,\sin 4{\cal M}-\sin 2{\cal M}\right) + {\cal O}(e^5).$ (A.136)

According to Equation (4.69),

$\displaystyle \frac{r}{a} = 1-e\,\cos E,$ (A.137)

where $ r$ is is the radial distance from the focus of the orbit and $ a$ is the orbital major radius. Thus,

$\displaystyle \frac{r}{a}$ $\displaystyle =1-e\,\cos{\cal M} -\frac{e^{\,2}}{2}\left(\cos 2{\cal M}-1\right) - \frac{3\,e^{\,3}}{8}\left( \cos 3{\cal M}- \cos{\cal M}\right)$    
  $\displaystyle \phantom{=} - \frac{e^{\,4}}{3}\left(\cos 4{\cal M}-\cos 2{\cal M}\right)+{\cal O}(e^5).$ (A.138)

Equations (4.39) and (4.67) imply that

$\displaystyle \frac{d\theta}{d{\cal M}} = (1-e^{\,2})^{1/2}\left(\frac{a}{r}\right)^2,$ (A.139)

where $ \theta $ is the true anomaly. Hence, it follows from Equations (A.128) and (A.137), and the fact that $ \theta=0$ when $ {\cal M}=0$ , that

$\displaystyle \theta = (1-e^{\,2})^{1/2}\int_0^{\cal M} \left(\frac{dE}{d{\cal M}}\right)^2\,d{\cal M}.$ (A.140)

From Equation (A.136),

$\displaystyle \frac{dE}{d{\cal M}}$ $\displaystyle =1+e\left(1-\frac{e^{\,2}}{8}\right)\cos{\cal M} + e^{\,2}\left(1-\frac{e^{\,2}}{3}\right) \cos 2{\cal M} +\frac{9\,e^{\,3}}{8}\,\cos 3{\cal M}$    
  $\displaystyle \phantom{=}+\frac{4\,e^{\,4}}{3}\,\cos 4{\cal M} + {\cal O}(e^5).$ (A.141)


$\displaystyle \left(\frac{dE}{d{\cal M}}\right)^2$ $\displaystyle = 1+\frac{e^{\,2}}{2}+\frac{3\,e^{\,4}}{8} + 2\,e\left(1+\frac{3\... M} + \frac{5\,e^{\,2}}{2}\left(1+\frac{2\,e^{\,2}}{15}\right) \cos 2{\cal M}$    
  $\displaystyle \phantom{=}+\frac{13\,e^{\,3}}{4}\,\cos 3{\cal M} + \frac{103\,e^{\,4}}{24}\,\cos 4{\cal M} +{\cal O}(e^5),$ (A.142)


$\displaystyle \theta$ $\displaystyle = {\cal M} + 2\,e\,\sin{\cal M} + \frac{5\,e^{\,2}}{4}\,\sin 2{\c...
... e^{\,3}\left( \frac{13}{12}\,\sin 3{\cal M} - \frac{1}{4}\,\sin{\cal M}\right)$    
  $\displaystyle \phantom{=}+e^{\,4}\left(\frac{103}{96}\,\sin 4{\cal M} - \frac{11}{24}\,\sin 2{\cal M}\right)+{\cal O}(e^5).$ (A.143)

next up previous
Next: Matrix eigenvalue theory Up: Useful mathematics Previous: Conic sections
Richard Fitzpatrick 2016-03-31