Expansion of planetary disturbing functions

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

$${\cal S} = {\cal S}_D + \alpha\,{\cal S}_E,$$ (H.36)

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

where

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

and

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

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

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

Now,

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

Let

$$\zeta = \frac{r-a}{a},$$ (H.42)

$$\zeta' = \frac{r'-a'}{a'},$$ (H.43)

$$\delta =\cos \psi- \cos(\vartheta-\vartheta'),$$ (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

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

where

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

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

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

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

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

Hence,

$${\cal S}_D \simeq \left[(1-\zeta'+\zeta'^{\,2}) + (\zeta-\zeta'-2\,\zeta\,\z... ...1}{2}(\zeta^{\,2}-2\,\zeta\,\zeta'+\zeta'^{\,2})\,\alpha^{\,2}\,D^{\,2}\right]F$$

$$\phantom{=}+ \delta\,\alpha\,F^{\,3}.$$ (H.49)

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

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

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

where

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

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

$${\cal S}_D \simeq\frac{1}{2}\sum_{j=-\infty,\infty}\left\{ \left[(1-\zeta'+\... ...}) + (\zeta-\zeta'-2\,\zeta\,\zeta' + 2\,\zeta'^{\,2})\,\alpha\,D\right.\right.$$

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

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

Equation (4.87) gives

$$\zeta \equiv \frac{r}{a}-1 \simeq - e\,\cos\,{\cal M}+ \frac{e^{\,2}}{2}\,(1-\cos\,2{\cal M})$$

$$\simeq - e\,\cos(\skew{5}\bar{\lambda}-\varpi) + \frac{e^{\,2}}{2}\,\left[1-\cos(2\,\skew{5}\bar{\lambda}-2\,\varpi)\right]$$ (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

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

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

Hence,

$$\cos(\omega+\theta) \equiv \cos\omega\,\cos \theta - \sin\omega\,\sin \theta$$

$$\simeq\cos(\omega+{\cal M}) +e\,\left[\cos(\omega+2{\cal M})-\cos\omega\right]$$

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

and

$$\sin(\omega+\theta) \equiv \sin\omega\,\cos \theta + \cos\omega\,\sin \theta$$

$$\simeq\sin(\omega+{\cal M}) +e\,\left[\sin(\omega+2{\cal M})-\sin\omega\right]$$

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

Thus, Equations (4.72)–(4.74) yield

$$\frac{X}{r} \simeq \cos(\omega+{\mit\Omega}+{\cal M}) + e\left[\cos(\omega+{\mit\Omega}+2{\cal M}) - \cos(\omega+{\mit\Omega})\right]$$

$$\phantom{=}+ \frac{e^{\,2}}{8}\left[9\,\cos(\omega+{\mit\Omega}+3... ...os(\omega+{\mit\Omega}-{\cal M}) - 8\,\cos(\omega+{\mit\Omega}+{\cal M})\right]$$

$$\phantom{=}+s^{\,2}\left[\cos(\omega-{\mit\Omega}+{\cal M}) - \cos(\omega+{\mit\Omega}+{\cal M})\right],$$ (H.59)

$$\frac{Y}{r} \simeq \sin(\omega+{\mit\Omega}+{\cal M}) + e\left[\sin(\omega+{\mit\Omega}+2{\cal M}) - \sin(\omega+{\mit\Omega})\right]$$

$$\phantom{=}+ \frac{e^{\,2}}{8}\left[9\,\sin(\omega+{\mit\Omega}+3... ...in(\omega+{\mit\Omega}-{\cal M}) - 8\,\sin(\omega+{\mit\Omega}+{\cal M})\right]$$

$$\phantom{=}-s^{\,2}\left[\sin(\omega-{\mit\Omega}+{\cal M}) +\sin(\omega+{\mit\Omega}+{\cal M})\right],$$ (H.60)

$$\frac{Z}{r} \simeq 2\,s\,\sin(\omega+{\cal M}) + 2\,e\,s\left[\sin(\omega+2{\cal M})-\sin\omega)\right],$$ (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,

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

so

$$\cos\psi \simeq (1-e^{\,2}-e'^{\,2}-s^{\,2}-s'^{\,2})\,\cos(\skew{5}\bar{\... ...+e\,e'\,\cos(2\,\skew{5}\bar{\lambda}-2\,\skew{5}\bar{\lambda}'-\varpi+\varpi')$$

$$\phantom{=}+ e\,e'\,\cos(\varpi-\varpi') + 2\,s\,s'\,\cos(\skew{5}\bar{\lambda}-\skew{5}\bar{\lambda}'-{\mit\Omega}+{\mit\Omega}')$$

$$\phantom{=}+ e\,\cos(2\,\skew{5}\bar{\lambda}-\skew{5}\bar{\lambd... ...da}'-\varpi)+ e'\,\cos(\skew{5}\bar{\lambda}-2\,\skew{5}\bar{\lambda}'+\varpi')$$

$$\phantom{=} - e'\,\cos(\skew{5}\bar{\lambda}-\varpi')+\frac{9\,e^... ...\frac{e^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}'-2\,\varpi)$$

$$\phantom{=}+\frac{9\,e'^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}-3\,... ...rac{e'^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}'-2\,\varpi')$$

$$\phantom{=}-e\,e'\,\cos(2\,\skew{5}\bar{\lambda}-\varpi-\varpi')-e\,e'\,\cos(2\,\skew{5}\bar{\lambda}'-\varpi-\varpi')$$

$$\phantom{=}+s^{\,2}\,\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lam... ...s'^{\,2}\,\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}' - 2\,{\mit\Omega}')$$

$$\phantom{=}-2\,s\,s'\,\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}'-{\mit\Omega}-{\mit\Omega}').$$ (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),

$$\delta \simeq s^{\,2}\left[\cos(\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}'-2\,{\mit\Omega}) - \cos(\skew{5}\bar{\lambda}-\skew{5}\bar{\lambda}')\right]$$

$$\phantom{=}+2\,s\,s'\left[\cos(\skew{5}\bar{\lambda}-\skew{5}\bar... ...\skew{5}\bar{\lambda}+\skew{5}\bar{\lambda}'-{\mit\Omega}-{\mit\Omega}')\right]$$

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

Now,

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

However, from Equation (4.86),

$$\cos(j\,\vartheta) \equiv \cos[j\,(\varpi+\theta)]$$

$$\simeq (1-j^{\,2}\,e^{\,2})\,\cos(j\,\skew{5}\bar{\lambda}) + e^{... ...{j^{\,2}}{2}- \frac{5\,j}{8}\right)\cos[(2-j)\,\skew{5}\bar{\lambda}-2\,\varpi]$$

$$\phantom{=}+ e^{\,2}\left(\frac{j^{\,2}}{2} + \frac{5\,j}{8}\right)\cos[(2+j)\,\skew{5}\bar{\lambda}-2\,\varpi]$$

$$\phantom{=} - j\,e\,\cos[(1-j)\,\skew{5}\bar{\lambda}-\varpi] + j\,e\,\cos[(1+j)\,\skew{5}\bar{\lambda}-\varpi)],$$ (H.66)

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

$$\sin(j\,\vartheta) \equiv \sin[j\,(\varpi+\theta)]$$

$$\simeq (1-j^{\,2}\,e^{\,2})\,\sin(j\,\skew{5}\bar{\lambda}) + e^{... ...{5\,j}{8}- \frac{j^{\,2}}{2}\right)\sin[(2-j)\,\skew{5}\bar{\lambda}-2\,\varpi]$$

$$\phantom{=}+ e^{\,2}\left(\frac{5\,j}{8} + \frac{j^{\,2}}{2}\right)\sin[(2+j)\,\skew{5}\bar{\lambda}-2\,\varpi]$$

$$\phantom{=} + j\,e\,\sin[(1-j)\,\skew{5}\bar{\lambda}-\varpi] + j\,e\,\sin[(1+j)\,\skew{5}\bar{\lambda}-\varpi].$$ (H.67)

Hence, we obtain

$$\cos[j\,(\vartheta-\vartheta')] \simeq (1-j^{\,2}\,e^{\,2}-j^{\,2}\,e'^{\,2}) \,\cos[j\,(\skew{5}\bar{\lambda}-\skew{5}\bar{\lambda}')]$$

$$\phantom{=}+ e^{\,2}\left(\frac{5\,j}{8} +\frac{j^{\,2}}{2}\right)\cos[(2+j)\,\skew{5}\bar{\lambda}-j\,\skew{5}\bar{\lambda}'-2\,\varpi]$$

$$\phantom{=}+ e^{\,2}\left(\frac{j^{\,2}}{2} -\frac{5\,j}{8}\right)\cos[(2-j)\,\skew{5}\bar{\lambda}+j\,\skew{5}\bar{\lambda}'-2\,\varpi]$$

$$\phantom{=}+j\,e\,\cos[(1+j)\,\skew{5}\bar{\lambda}-j\,\skew{5}\b... ...pi] - j\,e\,\cos[(1-j)\,\skew{5}\bar{\lambda}+j\,\skew{5}\bar{\lambda}'-\varpi]$$

$$\phantom{=}+ e'^{\,2}\left(\frac{j^{\,2}}{2} -\frac{5\,j}{8}\right)\cos[j\,\skew{5}\bar{\lambda}+(2-j)\,\skew{5}\bar{\lambda}'-2\,\varpi']$$

$$\phantom{=}+ e'^{\,2}\left(\frac{5\,j}{8} +\frac{j^{\,2}}{2}\right)\cos[j\,\skew{5}\bar{\lambda}-(2+j)\,\skew{5}\bar{\lambda}'+2\,\varpi']$$

$$\phantom{=}-j\,e'\,\cos[j\,\skew{5}\bar{\lambda}+(1-j)\,\skew{5}\... ...] + j\,e'\,\cos[j\,\skew{5}\bar{\lambda}-(1+j)\,\skew{5}\bar{\lambda}'+\varpi']$$

$$\phantom{=}-j^{\,2}\,e\,e'\,\cos[(1+j)\,\skew{5}\bar{\lambda}+ (1-j)\,\skew{5}\bar{\lambda}'-\varpi-\varpi']$$

$$\phantom{=}-j^{\,2}\,e\,e'\,\cos[(1-j)\,\skew{5}\bar{\lambda}+(1+j)\,\skew{5}\bar{\lambda}'-\varpi-\varpi']$$

$$\phantom{=}+ j^{\,2}\,e\,e'\,\cos[(1+j)\,\skew{5}\bar{\lambda}-(1+j)\,\skew{5}\bar{\lambda}'-\varpi+\varpi']$$

$$\phantom{=}+ j^{\,2}\,e\,e'\,\cos[(1-j)\,\skew{5}\bar{\lambda}-(1-j)\,\skew{5}\bar{\lambda}'-\varpi+\varpi'].$$ (H.68)

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

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

where

$${\cal S}^{(j)} \simeq \left[\frac{b^{(j)}_{1/2}}{2} + \frac{1}{8}\,\left(e^{\,2}... ...eft(-4\,j^{\,2}+2\,\alpha\,D+\alpha^{\,2}\,D^{\,2}\right)\,b_{1/2}^{(j)}\right.$$

$$\phantom{=}\left. -\frac{\alpha}{4}\,\left(s^{\,2}+s'^{\,2}\right... ...2}^{(j+1)}\right)\right]\cos[j\,(\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda})]$$

$$\phantom{=}+\frac{e\,e'}{4}\left(2+6\,j+4\,j^{\,2}-2\,\alpha\,D-\... ...(j+1)}\,\cos(j\,\skew{5}\bar{\lambda}'-j\,\skew{5}\bar{\lambda}+\varpi'-\varpi)$$

$$\phantom{=}+ s\,s'\,\alpha\,b_{3/2}^{(j+1)}\,\cos(j\,\skew{5}\bar{\lambda}'-j\,\skew{5}\bar{\lambda}+{\mit\Omega}'-{\mit\Omega})$$

$$\phantom{=}+\frac{e}{2}\,\left(-2\,j-\alpha\,D\right)\,b^{(j)}_{1/2}\,\cos[j\,\skew{5}\bar{\lambda}'+(1-j)\,\skew{5}\bar{\lambda}-\varpi]$$

$$\phantom{=}+\frac{e'}{2}\,(-1+2\,j+\alpha\,D)\,b_{1/2}^{(j-1)}\,\cos[j\,\skew{5}\bar{\lambda}'+(1-j)\,\skew{5}\bar{\lambda}-\varpi']$$

$$\phantom{=}+ \frac{e^{\,2}}{8}\,\left(-5\,j+4\,j^{\,2}-2\,\alpha\... ...}^{(j)}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-2\,\varpi]$$

$$\phantom{=}+\frac{e\,e'}{4}\,\left(-2+6\,j-4\,j^{\,2}+2\,\alpha\,... ...)}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-\varpi'-\varpi]$$

$$\phantom{=}+ \frac{e'^{\,2}}{8}\,\left(2-7\,j+4\,j^{\,2}-2\,\alph... ...(j-2)}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-2\,\varpi']$$

$$\phantom{=}+\frac{s^{\,2}}{2}\,\alpha\,b^{(j-1)}_{3/2}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-2\,{\mit\Omega}]$$

$$\phantom{=}- s\,s'\,\alpha\,b_{3/2}^{(j-1)}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-{\mit\Omega}'-{\mit\Omega}]$$

$$\phantom{=}+\frac{s'^{\,2}}{2}\,\alpha\,b_{3/2}^{(j-1)}\,\cos[j\,\skew{5}\bar{\lambda}'+(2-j)\,\skew{5}\bar{\lambda}-2\,{\mit\Omega}'].$$ (H.70)

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

$${\cal S}_E \simeq \left(-1+\frac{e^{\,2}}{2}+\frac{e'^{\,2}}{2}+ s^{\,2}+s'^{\,2}\right)\cos(\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda})$$

$$\phantom{=}-e\,e'\,\cos(2\,\skew{5}\bar{\lambda}'-2\,\skew{5}\bar... ...\,\cos(\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda}-{\mit\Omega}'+{\mit\Omega})$$

$$\phantom{=}-\frac{e}{2}\,\cos(\skew{5}\bar{\lambda}'-2\,\skew{5}\... ...}'-\varpi)-2\,e'\,\cos(2\,\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda}-\varpi')$$

$$\phantom{=}-\frac{3\,e^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'-3\,... ...\frac{e^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,\varpi)$$

$$\phantom{=}+3\,e\,e'\,\cos(2\,\skew{5}\bar{\lambda}-\varpi'-\varp... ...rac{e'^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,\varpi')$$

$$\phantom{=}-\frac{27\,e'^{\,2}}{8}\,\cos(3\,\skew{5}\bar{\lambda}... ...i')-s^{\,2}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,{\mit\Omega})$$

$$\phantom{=}+2\,s\,s'\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\l... ... s'^{\,2}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,{\mit\Omega}'),$$ (H.71)

and

$${\cal S}_I \simeq \left(-1+\frac{e^{\,2}}{2}+\frac{e'^{\,2}}{2}+ s^{\,2}+s'^{\,2}\right)\cos(\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda})$$

$$\phantom{=}-e\,e'\,\cos(2\,\skew{5}\bar{\lambda}'-2\,\skew{5}\bar... ...\,\cos(\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda}-{\mit\Omega}'+{\mit\Omega})$$

$$\phantom{=}-2\,e\,\cos(\skew{5}\bar{\lambda}'-2\,\skew{5}\bar{\la... ...i')-\frac{e'}{2}\,\cos(2\,\skew{5}\bar{\lambda}'-\skew{5}\bar{\lambda}-\varpi')$$

$$\phantom{=}-\frac{27\,e^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'-3\... ...\frac{e^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,\varpi)$$

$$\phantom{=}+3\,e\,e'\,\cos(2\,\skew{5}\bar{\lambda}-\varpi'-\varp... ...rac{e'^{\,2}}{8}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,\varpi')$$

$$\phantom{=}-\frac{3\,e'^{\,2}}{8}\,\cos(3\,\skew{5}\bar{\lambda}'... ...i')-s^{\,2}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,{\mit\Omega})$$

$$\phantom{=}+2\,s\,s'\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\l... ... s'^{\,2}\,\cos(\skew{5}\bar{\lambda}'+\skew{5}\bar{\lambda}-2\,{\mit\Omega}').$$ (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

$${\cal S}_0 =\frac{1}{2} \sum_{j=-\infty,\infty} b_{1/2}^{(j)}\,\cos [j\,(\sk... ...- \alpha\,\cos(\skew{5}\bar{\lambda} - \skew{5}\bar{\lambda}'),\displaybreak[0]$$ (H.73)

$${\cal S}_1 = \frac{1}{2}\sum_{j=-\infty,\infty}\left\{k\,(-2\,j-\alpha\,D)\,... ...}_{1/2} \,\cos[ (1-j)\,\skew{5}\bar{\lambda}+ j\,\skew{5}\bar{\lambda}']\right.$$

$$\phantom{=}+ h\,(-2\,j-\alpha\,D)\,b^{(j)}_{1/2} \,\sin[(1-j)\,\skew{5}\bar{\lambda}+j\,\skew{5}\bar{\lambda}' ]$$

$$\phantom{=}+k'\,(-1+2\,j+\alpha\,D)\,b^{(j-1)}_{1/2} \,\cos[ (1-j)\,\skew{5}\bar{\lambda}+j\,\skew{5}\bar{\lambda}' ]$$

$$\phantom{=}\left. +h'\,(-1+2\,j+\alpha\,D)\,b^{(j-1)}_{1/2} \,\sin[ (1-j)\,\skew{5}\bar{\lambda}+j\,\skew{5}\bar{\lambda}' ]\right\}$$

$$\phantom{=}+ \frac{\alpha}{2}\left\{-k\,\cos(2\,\skew{5}\bar{\lam... ...')+ 3\,k\,\cos \skew{5}\bar{\lambda}' + 3\,h\,\sin\skew{5}\bar{\lambda}'\right.$$

$$\phantom{=}\left.-4\,k'\,\cos(\skew{5}\bar{\lambda}-2\,\skew{5}\b... ...,\sin(\skew{5}\bar{\lambda}-2\,\skew{5}\bar{\lambda}')\right\},\displaybreak[0]$$ (H.74)

$${\cal S}_2 = \frac{1}{8}\,(h^{\,2}+k^{\,2}+h'^{\,2}+k'^{\,2})\,(2\,\alpha\,D + \alpha^{\,2}\,D^{\,2})\,b^{(0)}_{1/2}$$

$$\phantom{=} - \frac{1}{8}\,(p^{\,2}+q^{\,2}+p'^{\,2}+q'^{\,2})\,\alpha\,b_{3/2}^{(1)}$$

$$\phantom{=}+\frac{1}{4}\,(k\,k'+h\,h')\,(2-2\,\alpha\,D - \alpha^{\,2}\,D^{\,2})\,b_{1/2}^{(1)}+\frac{1}{4}\,(p\,p'+q\,q')\,\alpha\,b_{3/2}^{(1)}.$$ (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

$${\cal S}_0' =\frac{\alpha}{2} \sum_{j=-\infty,\infty} b_{1/2}^{(j)}\,\cos [j\... ...pha^{-1}\,\cos(\skew{5}\bar{\lambda}' - \skew{5}\bar{\lambda}),\displaybreak[0]$$ (H.76)

$${\cal S}_1' = \frac{\alpha}{2}\sum_{j=-\infty,\infty}\left\{k\,(-2\,j-\alpha\... ..._{1/2} \,\cos[ j\,\skew{5}\bar{\lambda}' +(1-j)\,\skew{5}\bar{\lambda} ]\right.$$

$$\phantom{=}+ h\,(-2\,j-\alpha\,D)\,b^{(j)}_{1/2} \,\sin[j\,\skew{5}\bar{\lambda}' +(1-j)\,\skew{5}\bar{\lambda}]$$

$$\phantom{=}+k'\,(-1+2\,j+\alpha\,D)\,b^{(j-1)}_{1/2} \,\cos[ j\,\skew{5}\bar{\lambda}'+(1-j)\,\skew{5}\bar{\lambda}]$$

$$\phantom{=}\left. +h'\,(-1+2\,j+\alpha\,D)\,b^{(j-1)}_{1/2} \,\sin[ j\,\skew{5}\bar{\lambda}' +(1-j)\,\skew{5}\bar{\lambda}]\right\}$$

$$\phantom{=}+ \frac{\alpha^{-1}}{2}\left\{-k'\,\cos(2\,\skew{5}\ba... ...})+ 3\,k'\,\cos \skew{5}\bar{\lambda} + 3\,h'\,\sin\skew{5}\bar{\lambda}\right.$$

$$\phantom{=}\left.-4\,k\,\cos(\skew{5}\bar{\lambda}'-2\,\skew{5}\b... ...,\sin(\skew{5}\bar{\lambda}'-2\,\skew{5}\bar{\lambda})\right\},\displaybreak[0]$$ (H.77)

$${\cal S}_2' = \frac{1}{8}\,(h^{\,2}+k^{\,2}+h'^{\,2}+k'^{\,2})\,\alpha\,(2\,\alpha\,D + \alpha^{\,2}\,D^{\,2})\,b^{(0)}_{1/2}$$

$$\phantom{=}- \frac{1}{8}\,(p^{\,2}+q^{\,2}+p'^{\,2}+q'^{\,2})\,\alpha^{\,2}\,b_{3/2}^{(1)}$$

$$\phantom{=}+\frac{1}{4}\,(k\,k'+h\,h')\,\alpha\,(2-2\,\alpha\,D - \alpha^{\,2}\,D^{\,2})\,b_{1/2}^{(1)}$$

$$\phantom{=}+\frac{1}{4}\,(p\,p'+q\,q')\,\alpha^{\,2}\,b_{3/2}^{(1)}.$$ (H.78)