The circular restricted three-body problem in curvilinear coordinates

Abstract

We derive the exact equations of motion for the circular restricted three-body problem in cylindrical curvilinear coordinates together with a number of useful analytical relations linking curvilinear coordinates and classical orbital elements. The equations of motion can be seen as a generalization of Hill’s problem after including all neglected nonlinear terms. As an application of the method, we obtain a new expression for the averaged third-body disturbing function including eccentricity and inclination terms. We employ the latter to study the dynamics of the guiding center for the problem of circular coorbital motion providing an extension of some results in the literature.

Appendix

1.1 Curvilinear elements initialization

We here provide simple analytical expressions for obtaining initial curvilinear elements starting from a set of orbital elements. In order to keep the expressions as simple as possible the initial conditions must be referred to the time of orbital plane crossing $(\nu _{0}=-\omega _{0}$) in which case we have:

$$\begin{aligned} \rho _{0}= & {} \frac{a_{0}\left( 1-e_{0}^{2}\right) }{1+e_{0}\cos \omega _{0}}-1\\ \theta _{0}= & {} \Omega _{0}\\ z_{0}= & {} 0\\ \dot{\rho }_{0}= & {} \frac{-e_{0}\sin \omega _{0}}{\sqrt{a_{0}\left( 1-e_{0}^{2}\right) }}\\ \dot{\theta }_{0}= & {} \frac{\sin i_{0}}{\left| \sin i_{0}\right| } \frac{\cos i_{0}\left( 1+e_{0}\cos \omega _{0}\right) ^{2}}{a_{0}^{3/2} \left( 1-e_{0}^{2}\right) ^{3/2}}-1\\ \dot{z}_{0}= & {} \frac{\sin i_{0}\left( 1+e_{0}\cos \omega _{0}\right) }{\sqrt{a_{0}\left( 1-e_{0}^{2}\right) }}. \end{aligned}$$