Introduction

The purpose of this appendix is to derive simplified evolution equations for the osculating orbital elements of a two-planet solar system, starting from the Lagrange planetary equations, Equations (G.125)–(G.130), and exploiting the fact that the planetary masses are all very small compared to the solar mass, as well as the fact that the planetary eccentricities and inclinations (in radians) are small compared to unity. Our approach is mostly based on that of Murray and Dermott 1999.

Let the first planet have position vector ${\bf r}$, mass $m$, and the standard osculating elements $a$, $\skew{5}\bar{\lambda}_0$, $e$, $I$, $\varpi$, and ${\mit \Omega }$. (See Section 4.12.) It is convenient to define the alternative elements $\skew{5}\bar{\lambda}= \skew{5}\bar{\lambda}_0+\int_0^t n(t')\,dt'$, $h=e\,\sin\varpi$, $k=e\,\cos\varpi$, $p=\sin I\,\sin {\mit\Omega}$, and $q=\sin I\,\cos{\mit\Omega}$, where $n=(\mu/a^3)^{1/2}$ is the mean orbital angular velocity, $\mu= G\,(M+m)$, and $M$ is the solar mass. Thus, the osculating elements of the first planet become $a$, $\skew{5}\bar{\lambda}$, $h$, $k$, $p$, and $q$. Let $a'$, $\skew{5}\bar{\lambda}'$, $h'$, $k'$, $p'$, and $q'$ be the corresponding osculating elements of the second planet. Furthermore, let the second planet have position vector ${\bf r}'$, mass $m'$, and mean orbital angular velocity $n'=(\mu'/a'^{\,3})^{1/2}$, where $\mu'=G\,(M+m')$.