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Next: Expansion of planetary disturbing Up: Expansion of orbital evolution Previous: Introduction


Expansion of Lagrange planetary equations

The first planet's disturbing function can be written in the form [see Equation (10.8)]

$\displaystyle {\cal R} = \frac{\tilde{\mu}'}{a'}\,{\cal S},$ (H.1)

where $ \tilde{\mu}'=G\,m'$ , and $ {\cal S}$ is $ {\cal O}(1)$ . Thus, because $ \mu=n^2\,a^3$ , the Lagrange planetary equations, Equations (G.125)-(G.130), applied to the first planet, reduce to

    $\displaystyle \frac{d \ln a}{dt}$ $\displaystyle =2\,n\,\epsilon'\alpha\,\frac{\partial {\cal S}}{\partial\skew{5}\bar{\lambda}},$ (H.2)
    $\displaystyle \frac{d\skew{5}\bar{\lambda}}{dt}$ $\displaystyle =n - 2\,n\,\epsilon'\alpha^{\,2}\,\frac{\partial {\cal S}}{\parti...
...\partial {\cal S}}{\partial h} + k\,\frac{\partial {\cal S}}{\partial k}\right)$    
      $\displaystyle \phantom{=}+ \frac{n\,\epsilon'\alpha\,\cos I}{2\,\cos^{\,2}(I/2)...
...partial {\cal S}}{\partial p} + q\,\frac{\partial {\cal S}}{\partial q}\right),$   $\displaystyle \displaybreak[0]$ (H.3)
    $\displaystyle \frac{dh}{dt}$ $\displaystyle = -\frac{n\,\epsilon'\alpha\,(1-e^{\,2})^{1/2}}{[1+(1-e^{\,2})^{1...
...} + n\,\epsilon'\alpha\,(1-e^{\,2})^{1/2}\,\frac{\partial {\cal S}}{\partial k}$    
      $\displaystyle \phantom{=}+ \frac{n\,\epsilon'\alpha\,\cos I}{2\,\cos^{\,2}(I/2)...
...partial {\cal S}}{\partial p} + q\,\frac{\partial {\cal S}}{\partial q}\right),$ (H.4)
    $\displaystyle \frac{dk}{dt}$ $\displaystyle = -\frac{n\,\epsilon'\alpha\,(1-e^{\,2})^{1/2}}{[1+(1-e^{\,2})^{1...
...}} -n\,\epsilon'\alpha\,(1-e^{\,2})^{1/2}\,\frac{\partial {\cal S}}{\partial h}$    
      $\displaystyle \phantom{=}- \frac{n\,\epsilon'\alpha\,\cos I}{2\,\cos^{\,2}(I/2)...
...partial {\cal S}}{\partial p} + q\,\frac{\partial {\cal S}}{\partial q}\right),$ (H.5)
    $\displaystyle \frac{dp}{dt}$ $\displaystyle = - \frac{n\,\epsilon'\alpha\,\cos I}{2\,\cos^{\,2}(I/2)\,(1-e^{\...
...rac{\partial{\cal S}}{\partial h}-h\,\frac{\partial{\cal S}}{\partial k}\right)$    
      $\displaystyle \phantom{=}+ \frac{n\,\epsilon'\alpha\,\cos I}{(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal S}}{\partial q},$ (H.6)
and   $\displaystyle \frac{dq}{dt}$ $\displaystyle = - \frac{n\,\epsilon'\alpha\,\cos I}{2\,\cos^{\,2}(I/2)\,(1-e^{\...
...rac{\partial{\cal S}}{\partial h}-h\,\frac{\partial{\cal S}}{\partial k}\right)$    
      $\displaystyle \phantom{=}- \frac{n\,\epsilon'\alpha\,\cos I}{(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal S}}{\partial p},$ (H.7)

where

    $\displaystyle \epsilon'$ $\displaystyle =\frac{\tilde{\mu}'}{\mu}= \frac{m'}{M+m},$ (H.8)
and   $\displaystyle \alpha$ $\displaystyle =\frac{a}{a'}.$     (H.9)

The Sun is much more massive than any planet in the solar system. It follows that the parameter $ \epsilon'$ is very small compared to unity. Expansion of Equations (H.2)-(H.7) to first order in $ \epsilon'$ yields

    $\displaystyle \skew{5}\bar{\lambda}(t)$ $\displaystyle = \skew{5}\bar{\lambda}_0+n^{(0)}\,t+ \skew{5}\bar{\lambda}^{(1)}(t),$ (H.10)
    $\displaystyle a(t)$ $\displaystyle = a^{(0)}\left[1+\epsilon'\,a^{(1)}(t)\right],$ (H.11)
and   $\displaystyle n(t)$ $\displaystyle = n^{(0)}\left[1-(3/2)\,\epsilon'\,a^{(1)}(t)\right],$     (H.12)

where $ \lambda^{(1)}\sim {\cal O}(\epsilon')$ , $ a^{(1)}\sim {\cal O}(1)$ , $ n^{(0)}=(\mu/[a^{(0)}]^3)^{1/2}$ , and

    $\displaystyle \frac{d a^{(1)}}{dt}$ $\displaystyle =\epsilon'\,n^{(0)}\left[2\,\alpha\,\frac{\partial {\cal S}}{\partial\skew{5}\bar{\lambda}^{(0)}}\right],$ (H.13)
    $\displaystyle \frac{d\skew{5}\bar{\lambda}^{(1)}}{dt}$ $\displaystyle =\epsilon'\,n^{(0)}\left[- \frac{3}{2}\, a^{(1)}- 2\,\alpha^{\,2}...
...l {\cal S}}{\partial h} + k\,\frac{\partial {\cal S}}{\partial k}\right)\right.$    
      $\displaystyle \phantom{=}\left.+ \frac{\alpha\,\cos I}{2\,\cos^{\,2}(I/2)\,(1-e...
...ial p} + q\,\frac{\partial {\cal S}}{\partial q}\right)\right],\displaybreak[0]$ (H.14)
    $\displaystyle \frac{dh}{dt}$ $\displaystyle = \epsilon'\,n^{(0)}\left[-\frac{\alpha\,(1-e^{\,2})^{1/2}}{[1+(1...
...{(0)}} + \alpha\,(1-e^{\,2})^{1/2}\,\frac{\partial {\cal S}}{\partial k}\right.$    
      $\displaystyle \phantom{=}\left.+ \frac{\alpha\,\cos I}{2\,\cos^{\,2}(I/2)\,(1-e...
... {\cal S}}{\partial p} + q\,\frac{\partial {\cal S}}{\partial q}\right)\right],$ (H.15)
    $\displaystyle \frac{dk}{dt}$ $\displaystyle =\epsilon'\,n^{(0)}\left[ -\frac{\alpha\,(1-e^{\,2})^{1/2}}{[1+(1...
...^{(0)}} -\alpha\,(1-e^{\,2})^{1/2}\,\frac{\partial {\cal S}}{\partial h}\right.$    
      $\displaystyle \phantom{=}\left.- \frac{\alpha\,\cos I}{2\,\cos^{\,2}(I/2)\,(1-e...
... {\cal S}}{\partial p} + q\,\frac{\partial {\cal S}}{\partial q}\right)\right],$ (H.16)
    $\displaystyle \frac{dp}{dt}$ $\displaystyle = \epsilon'\,n^{(0)}\left[- \frac{\alpha\,\cos I}{2\,\cos^{\,2}(I...
...rtial{\cal S}}{\partial h}-h\,\frac{\partial{\cal S}}{\partial k}\right)\right.$    
      $\displaystyle \phantom{=}\left.+ \frac{\alpha\,\cos I}{(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal S}}{\partial q}\right],$ (H.17)
and   $\displaystyle \frac{dq}{dt}$ $\displaystyle = \epsilon'\,n^{(0)}\left[- \frac{\alpha\,\cos I}{2\,\cos^{\,2}(I...
...rtial{\cal S}}{\partial h}-h\,\frac{\partial{\cal S}}{\partial k}\right)\right.$    
      $\displaystyle \phantom{=}\left.- \frac{\alpha\,\cos I}{(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal S}}{\partial p}\right],$ (H.18)

with $ \skew{5}\bar{\lambda}^{(0)} = \skew{5}\bar{\lambda}_0+n^{(0)}\,t$ , and $ \alpha=(a/a')^{(0)}$ . In the following, for ease of notation, $ \skew{5}\bar{\lambda}^{(0)}$ , $ a^{(0)}$ , and $ n^{(0)}$ are written simply as $ \skew{5}\bar{\lambda}$ , $ a$ , and $ n$ , respectively.

According to Table 4.1, the planets in the solar system all possess orbits whose eccentricities, $ e$ , and inclinations, $ I$ (in radians), are small compared to unity, but large compared to the ratio of any planetary mass to that of the Sun. It follows that

$\displaystyle \epsilon'\ll e,\,I\ll 1,$ (H.19)

which is our fundamental ordering of small quantities. Assuming that $ I, e', I'\sim {\cal O}(e)$ , we can perform a secondary expansion in the small parameter $ e$ . It turns out that when the normalized disturbing function, $ {\cal S}$ , is expanded to second order in $ e$ it takes the general form (see Section H.3)

$\displaystyle {\cal S} = {\cal S}_0(\alpha, \skew{5}\bar{\lambda},\skew{5}\bar{...
...(\alpha,\skew{5}\bar{\lambda},\skew{5}\bar{\lambda}',h,h', k,k', p, p', q, q'),$ (H.20)

where $ {\cal S}_n$ is $ {\cal O}(e^{\,n})$ . If we expand the right-hand sides of Equations (H.13)-(H.18) to first order in $ e$ then we obtain

    $\displaystyle \frac{d a^{(1)}}{dt}$ $\displaystyle =\epsilon'\,n\left[2\,\alpha\,\frac{\partial ({\cal S}_0+{\cal S}_1)}{\partial\skew{5}\bar{\lambda}}\right],$ (H.21)
    $\displaystyle \frac{d\skew{5}\bar{\lambda}^{(1)}}{dt}$ $\displaystyle =\epsilon'\,n\left[- \frac{3}{2}\, a^{(1)}- 2\,\alpha^{\,2}\,\fra...
...al S}_1}{\partial h} + k\,\frac{\partial {\cal S}_1}{\partial k}\right)\right],$   $\displaystyle \displaybreak[0]$ (H.22)
    $\displaystyle \frac{dh}{dt}$ $\displaystyle = \epsilon'\,n\left[-\alpha\, h\,\frac{\partial {\cal S}_0}{\part...
...\lambda}} + \alpha\,\frac{\partial ({\cal S}_1+{\cal S}_2)}{\partial k}\right],$ (H.23)
    $\displaystyle \frac{dk}{dt}$ $\displaystyle = \epsilon'\,n\left[-\alpha\,k\, \frac{\partial {\cal S}_0}{\part...
...{\lambda}} -\alpha\,\frac{\partial ({\cal S}_1+{\cal S}_2)}{\partial h}\right],$ (H.24)
    $\displaystyle \frac{dp}{dt}$ $\displaystyle =\epsilon'\,n\left[ - \frac{\alpha}{2}\, p\,\frac{\partial {\cal ...
...\skew{5}\bar{\lambda}} + \alpha\,\frac{\partial {\cal S}_2}{\partial q}\right],$ (H.25)
and   $\displaystyle \frac{dq}{dt}$ $\displaystyle = \epsilon'\,n\left[- \frac{\alpha}{2}\, q\,\frac{\partial {\cal ...
...l\skew{5}\bar{\lambda}}- \alpha\,\frac{\partial {\cal S}_2}{\partial p}\right].$ (H.26)

Note that $ h$ , $ k$ , $ p$ , $ q$ are $ {\cal O}(e)$ , whereas $ \alpha$ and $ \skew{5}\bar{\lambda}$ are $ {\cal O}(1)$ .

By analogy, writing the second planet's disturbing function as [see Equation (10.9]

$\displaystyle {\cal R}' = \frac{\tilde{\mu}}{a}\,{\cal S}',$ (H.27)

where $ \tilde{\mu} = G\,m$ , and $ {\cal S}'$ is $ {\cal O}(1)$ , and assuming that $ {\cal S}'$ takes the form

$\displaystyle {\cal S}' = {\cal S}_0'(\alpha, \skew{5}\bar{\lambda},\skew{5}\ba...
...(\alpha,\skew{5}\bar{\lambda},\skew{5}\bar{\lambda}',h,h', k,k', p, p', q, q'),$ (H.28)

where $ {\cal S}_n'$ is $ {\cal O}(e^n)$ , the Lagrange planetary equations, applied to the second planet, yield

    $\displaystyle \frac{d a^{(1)'}}{dt}$ $\displaystyle =\epsilon\,n'\left[2\,\alpha^{-1}\,\frac{\partial ({\cal S}_0'+{\cal S}_1')}{\partial\skew{5}\bar{\lambda}'}\right],$ (H.29)
    $\displaystyle \frac{d\skew{5}\bar{\lambda}^{(1)'}}{dt}$ $\displaystyle =\epsilon\,n'\left[- \frac{3}{2}\, a^{(1)'}+2\,\frac{\partial ({\...
..._1'}{\partial h'} + k'\,\frac{\partial {\cal S}_1'}{\partial k'}\right)\right],$ (H.30)
    $\displaystyle \frac{dh'}{dt}$ $\displaystyle = \epsilon\,n'\left[-\alpha^{-1}\, h'\,\frac{\partial {\cal S}_0'...
...} + \alpha^{-1}\,\frac{\partial ({\cal S}_1'+{\cal S}_2')}{\partial k'}\right],$ (H.31)
    $\displaystyle \frac{dk'}{dt}$ $\displaystyle = \epsilon\,n'\left[-\alpha^{-1}\,k'\, \frac{\partial {\cal S}_0'...
...'} -\alpha^{-1}\,\frac{\partial ({\cal S}_1'+{\cal S}_2')}{\partial h'}\right],$ (H.32)
    $\displaystyle \frac{dp'}{dt}$ $\displaystyle =\epsilon\,n'\left( - \frac{\alpha^{-1}}{2}\, p'\,\frac{\partial ...
...\bar{\lambda}'} + \alpha^{-1}\,\frac{\partial {\cal S}_2'}{\partial q'}\right),$ (H.33)
and   $\displaystyle \frac{dq'}{dt}$ $\displaystyle = \epsilon\,n'\left(- \frac{\alpha^{-1}}{2}\, q'\,\frac{\partial ...
...}\bar{\lambda}'}- \alpha^{-1}\,\frac{\partial {\cal S}_2'}{\partial p'}\right),$ (H.34)

where

$\displaystyle \epsilon =\frac{\tilde{\mu}}{\mu}= \frac{m}{M+m'}.$ (H.35)


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Next: Expansion of planetary disturbing Up: Expansion of orbital evolution Previous: Introduction
Richard Fitzpatrick 2016-03-31