Analysis of the Effect of Earth’s Equatorial Ellipticity on the Existence and Stability of the Lagrangian Points in the Earth–Moon–Sun System

Abstract

We have investigated existence of Lagrangian points and analyzed stability in the Earth–Moon–Sun system including the effect of Earth’s equatorial ellipticity parameter $\gamma$. First we have expressed the equations of motion of the Moon in the spherical coordinate system using the potential of the Earth. We observed that there exist collinear and non-collinear Lagrangian points for different values of Earth’s equatorial ellipticity $\gamma$. We also analyzed the effect of $\gamma$ on the stability of Lagrangian points including the effect of $\gamma$. We observed that all the Lagrangian points are unstable for different values of $\gamma$. Finally, we have drawn and analyzed zero velocity curves by taking different values of Jacobi constants.

Appendices

APPENDIX A

$${{K}_{1}} = \cos {{\phi }_{m}} - \cos {{\phi }_{m}}\cos \epsilon ,$$

$${{K}_{2}} = \cos {{\phi }_{m}} + \cos {{\phi }_{m}}\cos \epsilon ,$$

$${{K}_{3}} = 2\sin {{\phi }_{m}},$$

$${{K}_{4}} = \frac{{{{{\cos }}^{2}}{{\phi }_{m}}}}{2} - {{\cos }^{2}}{{\phi }_{m}}\cos \epsilon + \frac{{{{{\cos }}^{2}}{{\phi }_{m}}{{{\cos }}^{2}}\epsilon }}{2},$$

$${{K}_{5}} = \frac{{{{{\cos }}^{2}}{{\phi }_{m}}}}{2} + {{\cos }^{2}}{{\phi }_{m}}\cos \epsilon + \frac{{{{{\cos }}^{2}}{{\phi }_{m}}{{{\cos }}^{2}}\epsilon }}{2},$$

$${{K}_{6}} = {{\cos }^{2}}{{\phi }_{m}} - {{\cos }^{2}}{{\phi }_{m}}{{\cos }^{2}}\epsilon ,$$

$${{K}_{7}} = {{\cos }^{2}}{{\phi }_{m}} - {{\cos }^{2}}{{\phi }_{m}}{{\cos }^{2}}\epsilon - 2{{\sin }^{2}}\epsilon {{\sin }^{2}}{{\phi }_{m}},$$

$${{K}_{8}} = - 2\sin \epsilon \sin {{\phi }_{m}}\cos {{\phi }_{m}}(1 - \cos \epsilon ),$$

$${{K}_{9}} = 2\sin \epsilon \sin {{\phi }_{m}}\cos {{\phi }_{m}}(1 + \cos \epsilon ),$$

$${{K}_{{10}}} = - 4\sin \epsilon \sin {{\phi }_{m}}\cos {{\phi }_{m}}\cos \epsilon ,$$

$${{K}_{{11}}} = {{\cos }^{2}}{{\phi }_{m}} + {{\cos }^{2}}{{\phi }_{m}}{{\cos }^{2}}\epsilon + {{\sin }^{2}}\epsilon {{\sin }^{2}}{{\phi }_{m}}.$$

APPENDIX B

$${{M}_{1}} = \frac{{ - \cos {{\phi }_{m}}}}{8} + \frac{{\cos {{\phi }_{m}}\cos \epsilon }}{4},$$

$${{M}_{2}} = \frac{{ - \cos {{\phi }_{m}}}}{4},$$

$${{M}_{3}} = \frac{{ - \cos {{\phi }_{m}}}}{8} - \frac{{\cos {{\phi }_{m}}\cos \epsilon }}{4},$$

$${{M}_{4}} = \frac{{\sin \epsilon \cos \epsilon }}{4}(1 + \sin {{\phi }_{m}}),$$

$${{M}_{5}} = \frac{{\sin \epsilon \cos \epsilon }}{4}(\sin {{\phi }_{m}} - 1),$$

$${{M}_{6}} = \frac{{ - 3\cos {{\phi }_{m}}{{{\cos }}^{2}}\epsilon }}{8},$$

$${{M}_{7}} = \frac{{\cos {{\phi }_{m}}{{{\cos }}^{2}}\epsilon }}{8},$$

$${{M}_{8}} = \frac{{\cos {{\phi }_{m}}{{{\cos }}^{2}}\epsilon }}{8},$$

$${{M}_{9}} = \frac{{ - \sin \epsilon \cos \epsilon \sin {{\phi }_{m}}}}{2}.$$