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#

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
, mass
, and the standard osculating elements
,
,
,
,
,
and
. (See Section 4.12.) It is convenient to define the alternative elements
,
,
,
, and
, where
is the mean orbital angular velocity,
, and
is the solar mass. Thus, the osculating elements of the first planet become
,
,
,
,
, and
. Let
,
,
,
,
, and
be the corresponding osculating elements of the second planet. Furthermore, let the
second planet have position vector
, mass
, and mean orbital angular velocity
,
where
.

** Next:** Expansion of Lagrange planetary
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Richard Fitzpatrick
2016-03-31