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Expansion of planetary disturbing functions

Equations (10.8), (10.9), (H.1), and (H.27) give

    $\displaystyle {\cal S}$ $\displaystyle = {\cal S}_D + \alpha\,{\cal S}_E,$ (H.36)
and   $\displaystyle {\cal S}'$ $\displaystyle = \alpha\,{\cal S}_D +\alpha^{-1}{\cal S}_I,$ (H.37)

where

$\displaystyle {\cal S}_D = \frac{a'}{\vert{\bf r}'-{\bf r}\vert},$ (H.38)

and

$\displaystyle {\cal S}_E$ $\displaystyle = - \left(\frac{r}{a}\right)\left(\frac{a'}{r'}\right)^2\cos\psi,$ (H.39)
$\displaystyle {\cal S}_I$ $\displaystyle =-\left(\frac{r'}{a'}\right)\left(\frac{a}{r}\right)^2\,\cos\psi.$ (H.40)

In the preceding equations, $ \psi$ is the angle subtended between the directions of $ {\bf r}$ and $ {\bf r}'$ .

Now,

$\displaystyle {\cal S}_D = a' \left(r'^{\,2}-2\,r'\,r\,\cos\psi + r^{\,2}\right)^{-1/2}.$ (H.41)

Let

    $\displaystyle \zeta$ $\displaystyle = \frac{r-a}{a},$ (H.42)
    $\displaystyle \zeta'$ $\displaystyle = \frac{r'-a'}{a'},$ (H.43)
and   $\displaystyle \delta$ $\displaystyle =\cos \psi- \cos(\vartheta-\vartheta'),$ (H.44)

where $ \vartheta = \varpi+\theta$ and $ \vartheta'=\varpi'+\theta'$ . Here, $ \theta $ and $ \theta'$ are the true anomalies of the first and second planets, respectively. We expect $ \zeta$ and $ \zeta'$ to both be $ {\cal O}(e)$ [see Equation (H.54)], and $ \delta$ to be $ {\cal O}(e^{\,2})$ [see Equation (H.64)]. We can write

$\displaystyle {\cal S}_D = (1+\zeta')^{-1}\,\left[1-2\,\tilde{\alpha}\,\cos(\vartheta-\vartheta')+\tilde{\alpha}^2 -2\,\tilde{\alpha}\,\delta\right]^{-1/2},$ (H.45)

where

$\displaystyle \tilde{\alpha} = \left(\frac{1+\zeta}{1+\zeta'}\right)\alpha.$ (H.46)

Expanding in $ e$ , and retaining terms only up to $ {\cal O}(e^{\,2})$ , we obtain

$\displaystyle {\cal S}_D \simeq (1+\zeta')^{-1}\left[F + (\tilde{\alpha}-\alpha...
...c{1}{2}\,(\tilde{\alpha}-\alpha)^2\,D^{\,2}\,F\right]+ \delta\,\alpha\,F^{\,3},$ (H.47)

where $ D\equiv \partial/\partial\alpha$ , and

$\displaystyle F(\alpha,\vartheta-\vartheta') = \frac{1}{[1-2\,\alpha\,\cos(\vartheta-\vartheta')+\alpha^{\,2}]^{1/2}}.$ (H.48)

Hence,

$\displaystyle {\cal S}_D$ $\displaystyle \simeq \left[(1-\zeta'+\zeta'^{\,2}) + (\zeta-\zeta'-2\,\zeta\,\z...
...1}{2}(\zeta^{\,2}-2\,\zeta\,\zeta'+\zeta'^{\,2})\,\alpha^{\,2}\,D^{\,2}\right]F$    
  $\displaystyle \phantom{=}+ \delta\,\alpha\,F^{\,3}.$ (H.49)

Now, we can expand $ F$ and $ F^{\,3}$ as Fourier series in $ \vartheta-\vartheta'$ :

$\displaystyle F (\alpha,\vartheta-\vartheta')$ $\displaystyle = \frac{1}{2}\sum_{j=-\infty,\infty}b_{1/2}^{(j)}(\alpha)\,\cos[j\,(\vartheta-\vartheta')],$ (H.50)
$\displaystyle F^{\,3}(\alpha,\vartheta-\vartheta')$ $\displaystyle = \frac{1}{2}\sum_{j=-\infty,\infty}b_{3/2}^{(j)}(\alpha)\,\cos[j\,(\vartheta-\vartheta')],$ (H.51)

where

$\displaystyle b_s^{(j)}(\alpha) = \frac{1}{\pi}\int_0^{2\pi}\frac{\cos(j\,\psi)\,d\psi}{[1-2\,\alpha\,\cos\psi+\alpha^{\,2}]^s}.$ (H.52)

Incidentally, the $ b_s^{(j)}$ are known as Laplace coefficients. Thus,

$\displaystyle {\cal S}_D$ $\displaystyle \simeq\frac{1}{2}\sum_{j=-\infty,\infty}\left\{ \left[(1-\zeta'+\...
...}) + (\zeta-\zeta'-2\,\zeta\,\zeta' + 2\,\zeta'^{\,2})\,\alpha\,D\right.\right.$    
  $\displaystyle \phantom{=}\left.\left. +\frac{1}{2}(\zeta^{\,2}-2\,\zeta\,\zeta'...
...)+\delta\,\alpha\,b_{3/2}^{(j)}(\alpha)\right\}\cos[j\,(\vartheta-\vartheta')],$ (H.53)

where $ D$ now denotes $ d/d\alpha$ .

Equation (4.87) gives

$\displaystyle \zeta$ $\displaystyle \equiv \frac{r}{a}-1 \simeq - e\,\cos\,{\cal M}+ \frac{e^{\,2}}{2}\,(1-\cos\,2{\cal M})$    
  $\displaystyle \simeq - e\,\cos(\skew{5}\bar{\lambda}-\varpi) + \frac{e^{\,2}}{2}\,\left[1-\cos(2\,\skew{5}\bar{\lambda}-2\,\varpi)\right]$ (H.54)

to $ {\cal O}(e^{\,2})$ . Here, $ {\cal M} = \skew{5}\bar{\lambda}-\varpi$ is the first planet's mean anomaly. Obviously, there is an analogous equation for $ \zeta'$ . Moreover, from Equation (4.86), we have

$\displaystyle \sin \theta$ $\displaystyle \simeq \sin {\cal M} + e\,\sin 2{\cal M} + \frac{e^{\,2}}{8}\left(9\,\sin 3{\cal M} - 7\,\sin {\cal M}\right),$ (H.55)
$\displaystyle \cos \theta$ $\displaystyle \simeq \cos {\cal M} + e\,(\cos 2{\cal M}-1)+ \frac{e^{\,2}}{8}\left(9\,\cos 3{\cal M} - 9\,\cos{\cal M}\right).$ (H.56)

Hence,

$\displaystyle \cos(\omega+\theta)$ $\displaystyle \equiv \cos\omega\,\cos \theta - \sin\omega\,\sin \theta$    
  $\displaystyle \simeq\cos(\omega+{\cal M}) +e\,\left[\cos(\omega+2{\cal M})-\cos\omega\right]$    
  $\displaystyle \phantom{=}+ \frac{e^{\,2}}{8}\left[-8\,\cos(\omega+{\cal M})- \cos(\omega-{\cal M}) + 9\,\cos(\omega+3{\cal M})\right],$ (H.57)

and

$\displaystyle \sin(\omega+\theta)$ $\displaystyle \equiv \sin\omega\,\cos \theta + \cos\omega\,\sin \theta$    
  $\displaystyle \simeq\sin(\omega+{\cal M}) +e\,\left[\sin(\omega+2{\cal M})-\sin\omega\right]$    
  $\displaystyle \phantom{=}+ \frac{e^{\,2}}{8}\left[-8\,\sin(\omega+{\cal M})+ \sin(\omega-{\cal M}) + 9\,\sin(\omega+3{\cal M})\right].$ (H.58)

Thus, Equations (4.72)-(4.74) yield

    $\displaystyle \frac{X}{r}$ $\displaystyle \simeq \cos(\omega+{\mit\Omega}+{\cal M}) + e\left[\cos(\omega+{\mit\Omega}+2{\cal M}) - \cos(\omega+{\mit\Omega})\right]$    
      $\displaystyle \phantom{=}+ \frac{e^{\,2}}{8}\left[9\,\cos(\omega+{\mit\Omega}+3...
...os(\omega+{\mit\Omega}-{\cal M}) - 8\,\cos(\omega+{\mit\Omega}+{\cal M})\right]$    
      $\displaystyle \phantom{=}+s^{\,2}\left[\cos(\omega-{\mit\Omega}+{\cal M}) - \cos(\omega+{\mit\Omega}+{\cal M})\right],$ (H.59)
    $\displaystyle \frac{Y}{r}$ $\displaystyle \simeq \sin(\omega+{\mit\Omega}+{\cal M}) + e\left[\sin(\omega+{\mit\Omega}+2{\cal M}) - \sin(\omega+{\mit\Omega})\right]$    
      $\displaystyle \phantom{=}+ \frac{e^{\,2}}{8}\left[9\,\sin(\omega+{\mit\Omega}+3...
...in(\omega+{\mit\Omega}-{\cal M}) - 8\,\sin(\omega+{\mit\Omega}+{\cal M})\right]$    
      $\displaystyle \phantom{=}-s^{\,2}\left[\sin(\omega-{\mit\Omega}+{\cal M}) +\sin(\omega+{\mit\Omega}+{\cal M})\right],$ (H.60)
and   $\displaystyle \frac{Z}{r}$ $\displaystyle \simeq 2\,s\,\sin(\omega+{\cal M}) + 2\,e\,s\left[\sin(\omega+2{\cal M})-\sin\omega)\right],$     (H.61)

where $ s\equiv\sin (I/2)$ is assumed to be $ {\cal O}(e)$ . Here, $ X$ , $ Y$ , $ Z$ are the Cartesian component of $ {\bf r}$ . There are, of course, completely analogous expressions for the Cartesian components of $ {\bf r}'$ .

Now,

$\displaystyle \cos\psi = \frac{X}{r}\,\frac{X'}{r'} + \frac{Y}{r}\,\frac{Y'}{r'} + \frac{Z}{r}\,\frac{Z'}{r'},$ (H.62)

so

$\displaystyle \cos\psi$ $\displaystyle \simeq (1-e^{\,2}-e'^{\,2}-s^{\,2}-s'^{\,2})\,\cos(\skew{5}\bar{\...
...+e\,e'\,\cos(2\,\skew{5}\bar{\lambda}-2\,\skew{5}\bar{\lambda}'-\varpi+\varpi')$    
  $\displaystyle \phantom{=}+ e\,e'\,\cos(\varpi-\varpi') + 2\,s\,s'\,\cos(\skew{5}\bar{\lambda}-\skew{5}\bar{\lambda}'-{\mit\Omega}+{\mit\Omega}')$    
  $\displaystyle \phantom{=}+ e\,\cos(2\,\skew{5}\bar{\lambda}-\skew{5}\bar{\lambd...
...da}'-\varpi)+ e'\,\cos(\skew{5}\bar{\lambda}-2\,\skew{5}\bar{\lambda}'+\varpi')$    
  $\displaystyle \phantom{=} - e'\,\cos(\skew{5}\bar{\lambda}-\varpi')+\frac{9\,e^...
...\frac{e^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}'-2\,\varpi)$    
  $\displaystyle \phantom{=}+\frac{9\,e'^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}-3\,...
...rac{e'^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}'-2\,\varpi')$    
  $\displaystyle \phantom{=}-e\,e'\,\cos(2\,\skew{5}\bar{\lambda}-\varpi-\varpi')-e\,e'\,\cos(2\,\skew{5}\bar{\lambda}'-\varpi-\varpi')$    
  $\displaystyle \phantom{=}+s^{\,2}\,\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lam...
...s'^{\,2}\,\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}' - 2\,{\mit\Omega}')$    
  $\displaystyle \phantom{=}-2\,s\,s'\,\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}'-{\mit\Omega}-{\mit\Omega}').$ (H.63)

It is easily demonstrated that $ \cos(\vartheta-\vartheta')$ represents the value taken by $ \cos\psi$ when $ s=s'=0$ . Hence, from Equations (H.44) and (H.63),

$\displaystyle \delta$ $\displaystyle \simeq s^{\,2}\left[\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}'-2\,{\mit\Omega}) - \cos(\skew{5}\bar{\lambda}-\skew{5}\bar{\lambda}')\right]$    
  $\displaystyle \phantom{=}+2\,s\,s'\left[\cos(\skew{5}\bar{\lambda}-\skew{5}\bar...
...\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}'-{\mit\Omega}-{\mit\Omega}')\right]$    
  $\displaystyle \phantom{=}+ s'^{\,2}\left[\cos(\skew{5}\bar{\lambda}+\skew{5}\ba...
...}'-2\,{\mit\Omega}')-\cos(\skew{5}\bar{\lambda}-\skew{5}\bar{\lambda}')\right].$ (H.64)

Now,

$\displaystyle \cos[ j\,(\vartheta-\vartheta') ]\equiv \cos( j\,\vartheta)\,\cos(j\,\vartheta') + \sin( j\,\vartheta)\,\sin (j\,\vartheta').$ (H.65)

However, from Equation (4.86),

$\displaystyle \cos(j\,\vartheta)$ $\displaystyle \equiv \cos[j\,(\varpi+\theta)]$    
  $\displaystyle \simeq (1-j^{\,2}\,e^{\,2})\,\cos(j\,\skew{5}\bar{\lambda}) + e^{...
...{j^{\,2}}{2}- \frac{5\,j}{8}\right)\cos[(2-j)\,\skew{5}\bar{\lambda}-2\,\varpi]$    
  $\displaystyle \phantom{=}+ e^{\,2}\left(\frac{j^{\,2}}{2} + \frac{5\,j}{8}\right)\cos[(2+j)\,\skew{5}\bar{\lambda}-2\,\varpi]$    
  $\displaystyle \phantom{=} - j\,e\,\cos[(1-j)\,\skew{5}\bar{\lambda}-\varpi] + j\,e\,\cos[(1+j)\,\skew{5}\bar{\lambda}-\varpi)],$ (H.66)

because $ {\cal M} = \skew{5}\bar{\lambda}-\varpi$ . Likewise,

$\displaystyle \sin(j\,\vartheta)$ $\displaystyle \equiv \sin[j\,(\varpi+\theta)]$    
  $\displaystyle \simeq (1-j^{\,2}\,e^{\,2})\,\sin(j\,\skew{5}\bar{\lambda}) + e^{...
...{5\,j}{8}- \frac{j^{\,2}}{2}\right)\sin[(2-j)\,\skew{5}\bar{\lambda}-2\,\varpi]$    
  $\displaystyle \phantom{=}+ e^{\,2}\left(\frac{5\,j}{8} + \frac{j^{\,2}}{2}\right)\sin[(2+j)\,\skew{5}\bar{\lambda}-2\,\varpi]$    
  $\displaystyle \phantom{=} + j\,e\,\sin[(1-j)\,\skew{5}\bar{\lambda}-\varpi] + j\,e\,\sin[(1+j)\,\skew{5}\bar{\lambda}-\varpi].$ (H.67)

Hence, we obtain

$\displaystyle \cos[j\,(\vartheta-\vartheta')]$ $\displaystyle \simeq (1-j^{\,2}\,e^{\,2}-j^{\,2}\,e'^{\,2}) \,\cos[j\,(\skew{5}\bar{\lambda}-\skew{5}\bar{\lambda}')]$    
  $\displaystyle \phantom{=}+ e^{\,2}\left(\frac{5\,j}{8} +\frac{j^{\,2}}{2}\right)\cos[(2+j)\,\skew{5}\bar{\lambda}-j\,\skew{5}\bar{\lambda}'-2\,\varpi]$    
  $\displaystyle \phantom{=}+ e^{\,2}\left(\frac{j^{\,2}}{2} -\frac{5\,j}{8}\right)\cos[(2-j)\,\skew{5}\bar{\lambda}+j\,\skew{5}\bar{\lambda}'-2\,\varpi]$    
  $\displaystyle \phantom{=}+j\,e\,\cos[(1+j)\,\skew{5}\bar{\lambda}-j\,\skew{5}\b...
...pi] - j\,e\,\cos[(1-j)\,\skew{5}\bar{\lambda}+j\,\skew{5}\bar{\lambda}'-\varpi]$    
  $\displaystyle \phantom{=}+ e'^{\,2}\left(\frac{j^{\,2}}{2} -\frac{5\,j}{8}\right)\cos[j\,\skew{5}\bar{\lambda}+(2-j)\,\skew{5}\bar{\lambda}'-2\,\varpi']$    
  $\displaystyle \phantom{=}+ e'^{\,2}\left(\frac{5\,j}{8} +\frac{j^{\,2}}{2}\right)\cos[j\,\skew{5}\bar{\lambda}-(2+j)\,\skew{5}\bar{\lambda}'+2\,\varpi']$    
  $\displaystyle \phantom{=}-j\,e'\,\cos[j\,\skew{5}\bar{\lambda}+(1-j)\,\skew{5}\...
...] + j\,e'\,\cos[j\,\skew{5}\bar{\lambda}-(1+j)\,\skew{5}\bar{\lambda}'+\varpi']$    
  $\displaystyle \phantom{=}-j^{\,2}\,e\,e'\,\cos[(1+j)\,\skew{5}\bar{\lambda}+ (1-j)\,\skew{5}\bar{\lambda}'-\varpi-\varpi']$    
  $\displaystyle \phantom{=}-j^{\,2}\,e\,e'\,\cos[(1-j)\,\skew{5}\bar{\lambda}+(1+j)\,\skew{5}\bar{\lambda}'-\varpi-\varpi']$    
  $\displaystyle \phantom{=}+ j^{\,2}\,e\,e'\,\cos[(1+j)\,\skew{5}\bar{\lambda}-(1+j)\,\skew{5}\bar{\lambda}'-\varpi+\varpi']$    
  $\displaystyle \phantom{=}+ j^{\,2}\,e\,e'\,\cos[(1-j)\,\skew{5}\bar{\lambda}-(1-j)\,\skew{5}\bar{\lambda}'-\varpi+\varpi'].$ (H.68)

Equations (H.53), (H.54), (H.64), and (H.68) yield

$\displaystyle {\cal S}_D= \sum_{j=-\infty,\infty}\,{\cal S}^{(j)},$ (H.69)

where

$\displaystyle {\cal S}^{(j)}$ $\displaystyle \simeq \left[\frac{b^{(j)}_{1/2}}{2} + \frac{1}{8}\,\left(e^{\,2}...
...eft(-4\,j^{\,2}+2\,\alpha\,D+\alpha^{\,2}\,D^{\,2}\right)\,b_{1/2}^{(j)}\right.$    
  $\displaystyle \phantom{=}\left. -\frac{\alpha}{4}\,\left(s^{\,2}+s'^{\,2}\right...
...2}^{(j+1)}\right)\right]\cos[j\,(\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda})]$    
  $\displaystyle \phantom{=}+\frac{e\,e'}{4}\left(2+6\,j+4\,j^{\,2}-2\,\alpha\,D-\...
...(j+1)}\,\cos(j\,\skew{5}\bar{\lambda}'-j\,\skew{5}\bar{\lambda}+\varpi'-\varpi)$    
  $\displaystyle \phantom{=}+ s\,s'\,\alpha\,b_{3/2}^{(j+1)}\,\cos(j\,\skew{5}\bar{\lambda}'-j\,\skew{5}\bar{\lambda}+{\mit\Omega}'-{\mit\Omega})$    
  $\displaystyle \phantom{=}+\frac{e}{2}\,\left(-2\,j-\alpha\,D\right)\,b^{(j)}_{1/2}\,\cos[j\,\skew{5}\bar{\lambda}'+(1-j)\,\skew{5}\bar{\lambda}-\varpi]$    
  $\displaystyle \phantom{=}+\frac{e'}{2}\,(-1+2\,j+\alpha\,D)\,b_{1/2}^{(j-1)}\,\cos[j\,\skew{5}\bar{\lambda}'+(1-j)\,\skew{5}\bar{\lambda}-\varpi']$    
  $\displaystyle \phantom{=}+ \frac{e^{\,2}}{8}\,\left(-5\,j+4\,j^{\,2}-2\,\alpha\...
...}^{(j)}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-2\,\varpi]$    
  $\displaystyle \phantom{=}+\frac{e\,e'}{4}\,\left(-2+6\,j-4\,j^{\,2}+2\,\alpha\,...
...)}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-\varpi'-\varpi]$    
  $\displaystyle \phantom{=}+ \frac{e'^{\,2}}{8}\,\left(2-7\,j+4\,j^{\,2}-2\,\alph...
...(j-2)}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-2\,\varpi']$    
  $\displaystyle \phantom{=}+\frac{s^{\,2}}{2}\,\alpha\,b^{(j-1)}_{3/2}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-2\,{\mit\Omega}]$    
  $\displaystyle \phantom{=}- s\,s'\,\alpha\,b_{3/2}^{(j-1)}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-{\mit\Omega}'-{\mit\Omega}]$    
  $\displaystyle \phantom{=}+\frac{s'^{\,2}}{2}\,\alpha\,b_{3/2}^{(j-1)}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-2\,{\mit\Omega}'].$ (H.70)

Likewise, Equations (H.39), (H.40), (H.54), and (H.63) give

$\displaystyle {\cal S}_E$ $\displaystyle \simeq \left(-1+\frac{e^{\,2}}{2}+\frac{e'^{\,2}}{2}+ s^{\,2}+s'^{\,2}\right)\cos(\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda})$    
  $\displaystyle \phantom{=}-e\,e'\,\cos(2\,\skew{5}\bar{\lambda}'-2\,\skew{5}\bar...
...\,\cos(\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda}-{\mit\Omega}'+{\mit\Omega})$    
  $\displaystyle \phantom{=}-\frac{e}{2}\,\cos(\skew{5}\bar{\lambda}'-2\,\skew{5}\...
...}'-\varpi)-2\,e'\,\cos(2\,\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda}-\varpi')$    
  $\displaystyle \phantom{=}-\frac{3\,e^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'-3\,...
...\frac{e^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,\varpi)$    
  $\displaystyle \phantom{=}+3\,e\,e'\,\cos(2\,\skew{5}\bar{\lambda}-\varpi'-\varp...
...rac{e'^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,\varpi')$    
  $\displaystyle \phantom{=}-\frac{27\,e'^{\,2}}{8}\,\cos(3\,\skew{5}\bar{\lambda}...
...i')-s^{\,2}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,{\mit\Omega})$    
  $\displaystyle \phantom{=}+2\,s\,s'\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\l...
... s'^{\,2}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,{\mit\Omega}'),$ (H.71)

and

$\displaystyle {\cal S}_I$ $\displaystyle \simeq \left(-1+\frac{e^{\,2}}{2}+\frac{e'^{\,2}}{2}+ s^{\,2}+s'^{\,2}\right)\cos(\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda})$    
  $\displaystyle \phantom{=}-e\,e'\,\cos(2\,\skew{5}\bar{\lambda}'-2\,\skew{5}\bar...
...\,\cos(\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda}-{\mit\Omega}'+{\mit\Omega})$    
  $\displaystyle \phantom{=}-2\,e\,\cos(\skew{5}\bar{\lambda}'-2\,\skew{5}\bar{\la...
...i')-\frac{e'}{2}\,\cos(2\,\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda}-\varpi')$    
  $\displaystyle \phantom{=}-\frac{27\,e^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'-3\...
...\frac{e^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,\varpi)$    
  $\displaystyle \phantom{=}+3\,e\,e'\,\cos(2\,\skew{5}\bar{\lambda}-\varpi'-\varp...
...rac{e'^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,\varpi')$    
  $\displaystyle \phantom{=}-\frac{3\,e'^{\,2}}{8}\,\cos(3\,\skew{5}\bar{\lambda}'...
...i')-s^{\,2}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,{\mit\Omega})$    
  $\displaystyle \phantom{=}+2\,s\,s'\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\l...
... s'^{\,2}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,{\mit\Omega}').$ (H.72)

We can distinguish two different types of term that appear in our expansion of the disturbing functions. Periodic terms vary sinusoidally in time as our two planets orbit the Sun (i.e., they depend on the mean ecliptic longitudes, $ \skew{5}\bar{\lambda}$ and $ \skew{5}\bar{\lambda}'$ ), whereas secular terms remain constant in time (i.e., they do not depend on $ \skew{5}\bar{\lambda}$ or $ \skew{5}\bar{\lambda}'$ ). We expect the periodic terms to give rise to relatively short-period (i.e., of order a typical orbital period) oscillations in the osculating orbital elements of our planets. On the other hand, we expect the secular terms to produce an initially linear increase in these elements with time. Because such an increase can become significant over a long period of time, even if the rate of increase is small, it is necessary to evaluate the secular terms in the disturbing function to higher order than the periodic terms. Hence, in the following, we shall evaluate periodic terms to $ {\cal O}(e)$ , and secular terms to $ {\cal O}(e^{\,2})$ .

Making use of Equations (H.20), (H.36), and (H.69)-(H.71), as well as the definitions (G.121)-(G.124) (with $ p\simeq I\,\sin {\mit\Omega}$ and $ q\simeq I\,\cos\omega$ ), the order by order expansion (in $ e$ ) of the first planet's disturbing function becomes $ {\cal S}={\cal S}_0+{\cal S}_1+{\cal S}_2$ , where

    $\displaystyle {\cal S}_0$ $\displaystyle =\frac{1}{2} \sum_{j=-\infty,\infty} b_{1/2}^{(j)}\,\cos [j\,(\sk...
...bar{\lambda}')] - \alpha\,\cos(\skew{5}\bar{\lambda} - \skew{5}\bar{\lambda}'),$   $\displaystyle \displaybreak[0]$ (H.73)
    $\displaystyle {\cal S}_1$ $\displaystyle = \frac{1}{2}\sum_{j=-\infty,\infty}\left\{k\,(-2\,j-\alpha\,D)\,...
...}_{1/2} \,\cos[ (1-j)\,\skew{5}\bar{\lambda}+ j\,\skew{5}\bar{\lambda}']\right.$    
      $\displaystyle \phantom{=}+ h\,(-2\,j-\alpha\,D)\,b^{(j)}_{1/2} \,\sin[(1-j)\,\skew{5}\bar{\lambda}+j\,\skew{5}\bar{\lambda}' ]$    
      $\displaystyle \phantom{=}+k'\,(-1+2\,j+\alpha\,D)\,b^{(j-1)}_{1/2} \,\cos[ (1-j)\,\skew{5}\bar{\lambda}+j\,\skew{5}\bar{\lambda}' ]$    
      $\displaystyle \phantom{=}\left. +h'\,(-1+2\,j+\alpha\,D)\,b^{(j-1)}_{1/2} \,\sin[ (1-j)\,\skew{5}\bar{\lambda}+j\,\skew{5}\bar{\lambda}' ]\right\}$    
      $\displaystyle \phantom{=}+ \frac{\alpha}{2}\left\{-k\,\cos(2\,\skew{5}\bar{\lam...
...')+ 3\,k\,\cos \skew{5}\bar{\lambda}' + 3\,h\,\sin\skew{5}\bar{\lambda}'\right.$    
      $\displaystyle \phantom{=}\left.-4\,k'\,\cos(\skew{5}\bar{\lambda}-2\,\skew{5}\b...
...lambda}')+4\,h'\,\sin(\skew{5}\bar{\lambda}-2\,\skew{5}\bar{\lambda}')\right\},$   $\displaystyle \displaybreak[0]$ (H.74)
and   $\displaystyle {\cal S}_2$ $\displaystyle = \frac{1}{8}\,(h^{\,2}+k^{\,2}+h'^{\,2}+k'^{\,2})\,(2\,\alpha\,D + \alpha^{\,2}\,D^{\,2})\,b^{(0)}_{1/2}$    
      $\displaystyle \phantom{=} - \frac{1}{8}\,(p^{\,2}+q^{\,2}+p'^{\,2}+q'^{\,2})\,\alpha\,b_{3/2}^{(1)}$    
      $\displaystyle \phantom{=}+\frac{1}{4}\,(k\,k'+h\,h')\,(2-2\,\alpha\,D - \alpha^{\,2}\,D^{\,2})\,b_{1/2}^{(1)}+\frac{1}{4}\,(p\,p'+q\,q')\,\alpha\,b_{3/2}^{(1)}.$ (H.75)

Likewise, the order by order expansion of the second planet's disturbing function becomes $ {\cal S}'={\cal S}_0'+{\cal S}_1'+{\cal S}_2'$ , where

    $\displaystyle {\cal S}_0'$ $\displaystyle =\frac{\alpha}{2} \sum_{j=-\infty,\infty} b_{1/2}^{(j)}\,\cos [j\...
...\lambda})] - \alpha^{-1}\,\cos(\skew{5}\bar{\lambda}' - \skew{5}\bar{\lambda}),$   $\displaystyle \displaybreak[0]$ (H.76)
    $\displaystyle {\cal S}_1'$ $\displaystyle = \frac{\alpha}{2}\sum_{j=-\infty,\infty}\left\{k\,(-2\,j-\alpha\...
..._{1/2} \,\cos[ j\,\skew{5}\bar{\lambda}' +(1-j)\,\skew{5}\bar{\lambda} ]\right.$    
      $\displaystyle \phantom{=}+ h\,(-2\,j-\alpha\,D)\,b^{(j)}_{1/2} \,\sin[j\,\skew{5}\bar{\lambda}' +(1-j)\,\skew{5}\bar{\lambda}]$    
      $\displaystyle \phantom{=}+k'\,(-1+2\,j+\alpha\,D)\,b^{(j-1)}_{1/2} \,\cos[ j\,\skew{5}\bar{\lambda}'+(1-j)\,\skew{5}\bar{\lambda}]$    
      $\displaystyle \phantom{=}\left. +h'\,(-1+2\,j+\alpha\,D)\,b^{(j-1)}_{1/2} \,\sin[ j\,\skew{5}\bar{\lambda}' +(1-j)\,\skew{5}\bar{\lambda}]\right\}$    
      $\displaystyle \phantom{=}+ \frac{\alpha^{-1}}{2}\left\{-k'\,\cos(2\,\skew{5}\ba...
...})+ 3\,k'\,\cos \skew{5}\bar{\lambda} + 3\,h'\,\sin\skew{5}\bar{\lambda}\right.$    
      $\displaystyle \phantom{=}\left.-4\,k\,\cos(\skew{5}\bar{\lambda}'-2\,\skew{5}\bar{\lambda})+4\,h\,\sin(\skew{5}\bar{\lambda}'-2\,\skew{5}\bar{\lambda})\right\},$   $\displaystyle \displaybreak[0]$ (H.77)
and   $\displaystyle {\cal S}_2'$ $\displaystyle = \frac{1}{8}\,(h^{\,2}+k^{\,2}+h'^{\,2}+k'^{\,2})\,\alpha\,(2\,\alpha\,D$    
      $\displaystyle \phantom{=}+ \alpha^{\,2}\,D^{\,2})\,b^{(0)}_{1/2} - \frac{1}{8}\,(p^{\,2}+q^{\,2}+p'^{\,2}+q'^{\,2})\,\alpha^{\,2}\,b_{3/2}^{(1)}$    
      $\displaystyle \phantom{=}+\frac{1}{4}\,(k\,k'+h\,h')\,\alpha\,(2-2\,\alpha\,D - \alpha^{\,2}\,D^{\,2})\,b_{1/2}^{(1)}$    
      $\displaystyle \phantom{=}+\frac{1}{4}\,(p\,p'+q\,q')\,\alpha^{\,2}\,b_{3/2}^{(1)}.$ (H.78)


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