Expansion of Lagrange planetary equations

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

$${\cal R} = \frac{\tilde{\mu}'}{a'}\,{\cal S},$$ (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

$$\frac{d \ln a}{dt} =2\,n\,\epsilon'\alpha\,\frac{\partial {\cal S}}{\partial\skew{5}\bar{\lambda}},$$ (H.2)

$$\frac{d\skew{5}\bar{\lambda}}{dt} =n - 2\,n\,\epsilon'\alpha^{\,2}\,\frac{\partial {\cal S}}{\parti... ...\partial {\cal S}}{\partial h} + k\,\frac{\partial {\cal S}}{\partial k}\right)$$

$$\phantom{=}+ \frac{n\,\epsilon'\alpha\,\cos I}{2\,\cos^{\,2}(I/2)... ...}{\partial p} + q\,\frac{\partial {\cal S}}{\partial q}\right),\displaybreak[0]$$ (H.3)

$$\frac{dh}{dt} = -\frac{n\,\epsilon'\alpha\,(1-e^{\,2})^{1/2}}{[1+(1-e^{\,2})^{1... ...} + n\,\epsilon'\alpha\,(1-e^{\,2})^{1/2}\,\frac{\partial {\cal S}}{\partial k}$$

$$\phantom{=}+ \frac{n\,\epsilon'\alpha\,\cos I}{2\,\cos^{\,2}(I/2)... ...partial {\cal S}}{\partial p} + q\,\frac{\partial {\cal S}}{\partial q}\right),$$ (H.4)

$$\frac{dk}{dt} = -\frac{n\,\epsilon'\alpha\,(1-e^{\,2})^{1/2}}{[1+(1-e^{\,2})^{1... ...}} -n\,\epsilon'\alpha\,(1-e^{\,2})^{1/2}\,\frac{\partial {\cal S}}{\partial h}$$

$$\phantom{=}- \frac{n\,\epsilon'\alpha\,\cos I}{2\,\cos^{\,2}(I/2)... ...partial {\cal S}}{\partial p} + q\,\frac{\partial {\cal S}}{\partial q}\right),$$ (H.5)

$$\frac{dp}{dt} = - \frac{n\,\epsilon'\alpha\,\cos I}{2\,\cos^{\,2}(I/2)\,(1-e^{\... ...rac{\partial{\cal S}}{\partial h}-h\,\frac{\partial{\cal S}}{\partial k}\right)$$

$$\phantom{=}+ \frac{n\,\epsilon'\alpha\,\cos I}{(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal S}}{\partial q},$$ (H.6)

$$\frac{dq}{dt} = - \frac{n\,\epsilon'\alpha\,\cos I}{2\,\cos^{\,2}(I/2)\,(1-e^{\... ...rac{\partial{\cal S}}{\partial h}-h\,\frac{\partial{\cal S}}{\partial k}\right)$$

$$\phantom{=}- \frac{n\,\epsilon'\alpha\,\cos I}{(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal S}}{\partial p},$$ (H.7)

where

$$\epsilon' =\frac{\tilde{\mu}'}{\mu}= \frac{m'}{M+m},$$ (H.8)

$$\alpha =\frac{a}{a'}.$$ (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

$$\skew{5}\bar{\lambda}(t) = \skew{5}\bar{\lambda}_0+n^{(0)}\,t+ \skew{5}\bar{\lambda}^{(1)}(t),$$ (H.10)

$$a(t) = a^{(0)}\left[1+\epsilon'\,a^{(1)}(t)\right],$$ (H.11)

$$n(t) = n^{(0)}\left[1-(3/2)\,\epsilon'\,a^{(1)}(t)\right],$$ (H.12)

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

$$\frac{d a^{(1)}}{dt} =\epsilon'\,n^{(0)}\left[2\,\alpha\,\frac{\partial {\cal S}}{\partial\skew{5}\bar{\lambda}^{(0)}}\right],$$ (H.13)

$$\frac{d\skew{5}\bar{\lambda}^{(1)}}{dt} =\epsilon'\,n^{(0)}\left[- \frac{3}{2}\, a^{(1)}- 2\,\alpha^{\,2}... ...l {\cal S}}{\partial h} + k\,\frac{\partial {\cal S}}{\partial k}\right)\right.$$

$$\phantom{=}\left.+ \frac{\alpha\,\cos I}{2\,\cos^{\,2}(I/2)\,(1-e... ...ial p} + q\,\frac{\partial {\cal S}}{\partial q}\right)\right],\displaybreak[0]$$ (H.14)

$$\frac{dh}{dt} = \epsilon'\,n^{(0)}\left[-\frac{\alpha\,(1-e^{\,2})^{1/2}}{[1+(1... ...{(0)}} + \alpha\,(1-e^{\,2})^{1/2}\,\frac{\partial {\cal S}}{\partial k}\right.$$

$$\phantom{=}\left.+ \frac{\alpha\,\cos I}{2\,\cos^{\,2}(I/2)\,(1-e... ... {\cal S}}{\partial p} + q\,\frac{\partial {\cal S}}{\partial q}\right)\right],$$ (H.15)

$$\frac{dk}{dt} =\epsilon'\,n^{(0)}\left[ -\frac{\alpha\,(1-e^{\,2})^{1/2}}{[1+(1... ...^{(0)}} -\alpha\,(1-e^{\,2})^{1/2}\,\frac{\partial {\cal S}}{\partial h}\right.$$

$$\phantom{=}\left.- \frac{\alpha\,\cos I}{2\,\cos^{\,2}(I/2)\,(1-e... ... {\cal S}}{\partial p} + q\,\frac{\partial {\cal S}}{\partial q}\right)\right],$$ (H.16)

$$\frac{dp}{dt} = \epsilon'\,n^{(0)}\left[- \frac{\alpha\,\cos I}{2\,\cos^{\,2}(I... ...rtial{\cal S}}{\partial h}-h\,\frac{\partial{\cal S}}{\partial k}\right)\right.$$

$$\phantom{=}\left.+ \frac{\alpha\,\cos I}{(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal S}}{\partial q}\right],$$ (H.17)

$$\frac{dq}{dt} = \epsilon'\,n^{(0)}\left[- \frac{\alpha\,\cos I}{2\,\cos^{\,2}(I... ...rtial{\cal S}}{\partial h}-h\,\frac{\partial{\cal S}}{\partial k}\right)\right.$$

$$\phantom{=}\left.- \frac{\alpha\,\cos I}{(1-e^{\,2})^{1/2}}\,\frac{\partial {\cal S}}{\partial p}\right],$$ (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

$$\epsilon'\ll e,\,I\ll 1,$$ (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)

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

$$\frac{d a^{(1)}}{dt} =\epsilon'\,n\left[2\,\alpha\,\frac{\partial ({\cal S}_0+{\cal S}_1)}{\partial\skew{5}\bar{\lambda}}\right],$$ (H.21)

$$\frac{d\skew{5}\bar{\lambda}^{(1)}}{dt} =\epsilon'\,n\left[- \frac{3}{2}\, a^{(1)}- 2\,\alpha^{\,2}\,\fra... ...l h} + k\,\frac{\partial {\cal S}_1}{\partial k}\right)\right],\displaybreak[0]$$ (H.22)

$$\frac{dh}{dt} = \epsilon'\,n\left[-\alpha\, h\,\frac{\partial {\cal S}_0}{\part... ...\lambda}} + \alpha\,\frac{\partial ({\cal S}_1+{\cal S}_2)}{\partial k}\right],$$ (H.23)

$$\frac{dk}{dt} = \epsilon'\,n\left[-\alpha\,k\, \frac{\partial {\cal S}_0}{\part... ...{\lambda}} -\alpha\,\frac{\partial ({\cal S}_1+{\cal S}_2)}{\partial h}\right],$$ (H.24)

$$\frac{dp}{dt} =\epsilon'\,n\left[ - \frac{\alpha}{2}\, p\,\frac{\partial {\cal ... ...\skew{5}\bar{\lambda}} + \alpha\,\frac{\partial {\cal S}_2}{\partial q}\right],$$ (H.25)

$$\frac{dq}{dt} = \epsilon'\,n\left[- \frac{\alpha}{2}\, q\,\frac{\partial {\cal ... ...l\skew{5}\bar{\lambda}}- \alpha\,\frac{\partial {\cal S}_2}{\partial p}\right].$$ (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]

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

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

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

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

$$\frac{d a^{(1)'}}{dt} =\epsilon\,n'\left[2\,\alpha^{-1}\,\frac{\partial ({\cal S}_0'+{\cal S}_1')}{\partial\skew{5}\bar{\lambda}'}\right],$$ (H.29)

$$\frac{d\skew{5}\bar{\lambda}^{(1)'}}{dt} =\epsilon\,n'\left[- \frac{3}{2}\, a^{(1)'}+2\,\frac{\partial ({\... ..._1'}{\partial h'} + k'\,\frac{\partial {\cal S}_1'}{\partial k'}\right)\right],$$ (H.30)

$$\frac{dh'}{dt} = \epsilon\,n'\left[-\alpha^{-1}\, h'\,\frac{\partial {\cal S}_0'... ...} + \alpha^{-1}\,\frac{\partial ({\cal S}_1'+{\cal S}_2')}{\partial k'}\right],$$ (H.31)

$$\frac{dk'}{dt} = \epsilon\,n'\left[-\alpha^{-1}\,k'\, \frac{\partial {\cal S}_0'... ...'} -\alpha^{-1}\,\frac{\partial ({\cal S}_1'+{\cal S}_2')}{\partial h'}\right],$$ (H.32)

$$\frac{dp'}{dt} =\epsilon\,n'\left( - \frac{\alpha^{-1}}{2}\, p'\,\frac{\partial ... ...\bar{\lambda}'} + \alpha^{-1}\,\frac{\partial {\cal S}_2'}{\partial q'}\right),$$ (H.33)

$$\frac{dq'}{dt} = \epsilon\,n'\left(- \frac{\alpha^{-1}}{2}\, q'\,\frac{\partial ... ...}\bar{\lambda}'}- \alpha^{-1}\,\frac{\partial {\cal S}_2'}{\partial p'}\right),$$ (H.34)

where

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