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Alternative forms of Lagrange planetary equations

It can be seen, from Equations (G.115)-(G.120), that in the limit of small eccentricity, $ e$ , and small inclination, $ I$ , certain terms on the right-hand sides of the Lagrange planetary equations become singular. This problem can be alleviated by defining the alternative orbital elements,

    $\displaystyle h$ $\displaystyle =e\,\sin\varpi,$ (G.121)
    $\displaystyle k$ $\displaystyle = e\,\cos\varpi,$ (G.122)
    $\displaystyle p$ $\displaystyle = \sin I\,\sin{\mit\Omega},$ (G.123)
and   $\displaystyle q$ $\displaystyle = \sin I\,\cos{\mit\Omega}.$ (G.124)

If we write the Lagrange planetary equations in terms of these new elements then we obtain

    $\displaystyle \frac{da}{dt}$ $\displaystyle =\frac{2}{n\,a}\,\frac{\partial {\cal R}}{\partial\skew{5}\bar{\lambda}},$   $\displaystyle \displaybreak[0]$ (G.125)
    $\displaystyle \frac{d\skew{5}\bar{\lambda}}{dt}$ $\displaystyle =n - \frac{2}{n\,a}\,\frac{\partial {\cal R}}{\partial a}+ \frac{...
...\partial {\cal R}}{\partial h} + k\,\frac{\partial {\cal R}}{\partial k}\right)$    
      $\displaystyle \phantom{=}+ \frac{\cos I}{2\,n\,a^{\,2}\,\cos^2(I/2)\,(1-e^{\,2}...
...partial {\cal R}}{\partial p} + q\,\frac{\partial {\cal R}}{\partial q}\right),$   $\displaystyle \displaybreak[0]$ (G.126)
    $\displaystyle \frac{dh}{dt}$ $\displaystyle =-\frac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}\,[1+(1-e^{\,2})^{1/2}]}\, h...
...}} + \frac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}}\,\frac{\partial {\cal R}}{\partial k}$    
      $\displaystyle \phantom{=}+ \frac{\cos I}{2\,n\,a^{\,2}\,\cos^2(I/2)\,(1-e^{\,2}...
...partial {\cal R}}{\partial p} + q\,\frac{\partial {\cal R}}{\partial q}\right),$ (G.127)
    $\displaystyle \frac{dk}{dt}$ $\displaystyle = -\frac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}\,[1+(1-e^{\,2})^{1/2}]}\,k...
...a}} -\frac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}}\,\frac{\partial {\cal R}}{\partial h}$    
      $\displaystyle \phantom{=}- \frac{\cos I}{2\,n\,a^{\,2}\,\cos^2(I/2)\,(1-e^{\,2}...
...partial {\cal R}}{\partial p} + q\,\frac{\partial {\cal R}}{\partial q}\right),$ (G.128)
    $\displaystyle \frac{dp}{dt}$ $\displaystyle = - \frac{\cos I}{2\,n\,a^{\,2}\,\cos^2(I/2)\,(1-e^{\,2})^{1/2}}\...
...rac{\partial{\cal R}}{\partial h}-h\,\frac{\partial{\cal R}}{\partial k}\right)$    
      $\displaystyle \phantom{=}+ \frac{\cos I}{n\,a^{\,2}\,(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal R}}{\partial q},$ (G.129)
and   $\displaystyle \frac{dq}{dt}$ $\displaystyle =- \frac{\cos I}{2\,n\,a^{\,2}\,\cos^2(I/2)\,(1-e^{\,2})^{1/2}}\,...
...rac{\partial{\cal R}}{\partial h}-h\,\frac{\partial{\cal R}}{\partial k}\right)$    
      $\displaystyle \phantom{=}- \frac{\cos I}{n\,a^{\,2}\,(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal R}}{\partial p}.$ (G.130)

Note that the new equations now contain no singular terms in the limit $ e, I\rightarrow 0$ .

It is sometimes convenient to write the Lagrange planetary equations in terms of the mean anomaly, $ {\cal M} = \skew{5}\bar{\lambda}-\varpi$ , and the argument of the perigee, $ \omega=\varpi-{\mit\Omega}$ , rather than $ \skew{5}\bar{\lambda}$ and $ \omega$ . Making the appropriate substitutions, the equations take the form (Brouwer and Clemence 1961)

    $\displaystyle \frac{da}{dt}$ $\displaystyle = \frac{2}{n\,a}\,\frac{\partial{\cal R}}{\partial{\cal M}},$ (G.131)
    $\displaystyle \frac{d{\cal M}}{dt}$ $\displaystyle = n -\frac{2}{n\,a}\,\frac{\partial {\cal R}}{\partial a}- \frac{1-e^{\,2}}{n\,a^{\,2}\,e}\,\frac{\partial{\cal R}}{\partial e},$ (G.132)
    $\displaystyle \frac{de}{dt}$ $\displaystyle = \frac{1-e^{\,2}}{n\,a^{\,2}\,e}\,\frac{\partial {\cal R}}{\part...
...ac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}\,e}\,\frac{\partial{\cal R}}{\partial \omega},$ (G.133)
    $\displaystyle \frac{dI}{dt}$ $\displaystyle = \frac{\cot I}{n\,a^{\,2}\,(1-e^{\,2})^{1/2}}\,\frac{\partial{\c...
...2})^{-1/2}}{n\,a^{\,2}\,\sin I}\,\frac{\partial{\cal R}}{\partial{\mit\Omega}},$ (G.134)
    $\displaystyle \frac{d\omega}{dt}$ $\displaystyle = \frac{(1-e^{\,2})^{1/2}}{n\,a^{\,2}\,e}\,\frac{\partial {\cal R...
...ac{\cot I}{n\,a^{\,2}\,(1-e^{\,2})^{1/2}}\,\frac{\partial{\cal R}}{\partial I},$ (G.135)
and   $\displaystyle \frac{d{\mit\Omega}}{dt}$ $\displaystyle = \frac{(1-e^{\,2})^{-1/2}}{n\,a^{\,2}\,\sin I}\,\frac{\partial{\cal R}}{\partial I}.$     (G.136)


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Next: Expansion of orbital evolution Up: Derivation of Lagrange planetary Previous: Lagrange planetary equations
Richard Fitzpatrick 2016-03-31