Resonant Curve Due to Perturbations in Geo-Synchronous Satellite Including the Earth’s Equatorial Ellipticity and Resistive Force

Abstract

In this paper, we investigated the resonant curves in the motion of the geo-synchronous satellite under the gravitational effect of the Sun, the Moon and the Earth. We have taken into consideration the effect of Earth’s equatorial ellipticity and resistive force on the resonant curve. By defining the perturbations in $r$ (radial distance of the satellite from the center of the Earth) and ${{\dot {\theta }}_{E}}$ (spin rate of Earth), we reduced equations of motion of satellite to a second order linear non-homogeneous differential equation in $\Delta r$ (radial deviation). With the help of different graphs, we have analyzed the effect of $\dot {\gamma }$ (rate of change of Earth’s equatorial ellipticity parameter) and $\dot {\alpha }$ (angular velocity of the bary-center system around the Sun) on the oscillatory amplitudes. We observed that the oscillatory amplitudes decrease when the value of $\gamma$ (Earth’s equatorial ellipticity parameter) and $\alpha$ (orbital angle of the bary-center system around the Sun) increase. Finally, we have studied the phase portrait, phase space and bifurcation theory using the method of Poincaré section.

Appendices

APPENDIX A

Values of ${{L}_{i}}$’s used in Eq. (1)

$${{L}_{1}} = \frac{r}{2}\cos \phi (1 - \cos \lambda ) - \frac{{{{r}_{{20}}}}}{{4{\kern 1pt} \mu }}\sin (2\phi )\sin (2\lambda )\sin \nu ,$$

$${{L}_{2}} = \frac{r}{2}\cos \phi (1 + \cos \lambda ) + \frac{{{{r}_{{20}}}}}{{4\mu }}\sin (2\phi )\sin (2\lambda )\sin \nu ,$$

$${{L}_{3}} = \frac{{ - {{r}_{{20}}}}}{\mu }\cos \nu \left( {1 + \frac{{{{{\cos }}^{2}}\phi }}{2}} \right),$$

$${{L}_{4}} = \frac{{ - R}}{4}{{\cos }^{2}}\phi (1 + \cos \lambda ),$$

$${{L}_{5}} = \frac{{ - R}}{4}{{\cos }^{2}}\phi (1 - \cos \lambda ),$$

$${{L}_{6}} = \frac{{ - {{r}_{{20}}}}}{{4\mu }}{{\cos }^{2}}\phi \cos \nu (1 - \sin \lambda ),$$

$${{L}_{7}} = \frac{{ - {{r}_{{20}}}}}{{4\mu }}{{\cos }^{2}}\phi \cos \nu (1 + \sin \lambda ),$$

$${{L}_{8}} = \frac{{ - R}}{4}{{\cos }^{2}}\phi ,$$

$${{L}_{9}} = \frac{{ - R}}{4}{{\cos }^{2}}\phi - \frac{R}{2}{{\sin }^{2}}\lambda {{\sin }^{2}}\phi ,$$

$${{L}_{{10}}} = \frac{R}{8}{{\cos }^{2}}\phi {{\cos }^{2}}\lambda ,$$

$${{L}_{{11}}} = \frac{R}{8}{{\cos }^{2}}\phi {{\cos }^{2}}\lambda ,$$

$${{L}_{{12}}} = \frac{{ - R}}{4}{{\cos }^{2}}\phi {{\cos }^{2}}\lambda ,$$

$${{L}_{{13}}} = \frac{{ - R}}{4}{{\cos }^{2}}\phi {{\cos }^{2}}\lambda ,$$

$$\begin{gathered} {{L}_{{14}}} = - r\sin \lambda \sin \phi - \frac{{{{r}_{{20}}}\sin \nu }}{\mu } \\ \times \;\left( {1 + {{{\sin }}^{2}}\lambda {{{\sin }}^{2}}\phi + \frac{{{{{\cos }}^{2}}\phi {{{\cos }}^{2}}\lambda }}{2}} \right), \\ \end{gathered} $$

$${{L}_{{15}}} = \frac{{ - R}}{4}\sin 2\phi \sin \lambda (1 - \cos \lambda ),$$

$${{L}_{{16}}} = \frac{{ - R}}{4}\sin 2\phi \sin \lambda (1 + \cos \lambda ),$$

$${{L}_{{17}}} = \frac{{ - {{r}_{{20}}}}}{{4{\kern 1pt} \mu }}{{\cos }^{2}}\phi \cos \lambda \sin \nu (1 + \cos \lambda ),$$

$${{L}_{{18}}} = \frac{{{{r}_{{20}}}}}{{4\mu }}{{\cos }^{2}}\phi \cos \lambda \sin \nu (1 + \cos \lambda ),$$

$${{L}_{{19}}} = \frac{{{{r}_{{20}}}}}{{4\mu }}{\kern 1pt} \sin 2\phi \sin \lambda (\sin \nu + \cos \nu ),$$

$${{L}_{{20}}} = \frac{{ - {{r}_{{20}}}}}{{4{\kern 1pt} \mu }}\sin 2\phi \sin \lambda (\sin \nu + \cos \nu ),$$

$${{L}_{{21}}} = \frac{{ - R}}{4}\sin 2\phi \sin 2\lambda ,$$

$${{L}_{{22}}} = {{\sin }^{2}}\lambda \left( {{{{\sin }}^{2}}\phi - \frac{{{{{\cos }}^{2}}\phi }}{2}} \right).$$

APPENDIX B

Values of ${{M}_{i}}$’s used in Eq. (2)

$${{M}_{1}} = \frac{{ - {{r}_{{20}}}}}{{2\mu }}\cos \phi \cos \lambda \sin \nu ,$$

$${{M}_{2}} = \frac{{ - {{r}_{{20}}}}}{{4\mu }}\cos \phi \cos \lambda \sin \nu ,$$

$${{M}_{3}} = \frac{{ - {{r}_{{20}}}}}{{4\mu }}\cos \phi \cos \lambda \sin \nu ,$$

$${{M}_{4}} = \frac{R}{4}\sin \phi \left( {\frac{{\sin 2\lambda }}{2} - \sin \lambda } \right),$$

$${{M}_{5}} = \frac{R}{4}\sin \phi \left( {\frac{{\sin 2\lambda }}{2} + \sin \lambda } \right),$$

$${{M}_{6}} = \frac{{ - R}}{4}\sin \phi \sin 2\lambda ,$$

$${{M}_{7}} = \frac{R}{4}\cos \phi ({{\cos }^{2}}\lambda - 1),$$

$${{M}_{8}} = \frac{R}{8}\cos \phi \cos \lambda (1 - \cos \lambda ),$$

$${{M}_{9}} = \frac{{ - R}}{4}\cos \phi \left( {1 + \frac{{{{{\cos }}^{2}}\lambda }}{2} + \cos \lambda } \right),$$

$${{M}_{{10}}} = \frac{R}{4}\cos \phi (1 - \cos \lambda ),$$

$${{M}_{{11}}} = \frac{{ - {{r}_{{20}}}}}{{4{\kern 1pt} \mu }}{{\cos }^{2}}\lambda \cos \phi \sin \nu ,$$

$${{M}_{{12}}} = \frac{{{{r}_{{20}}}}}{{2{\kern 1pt} \mu }}{\kern 1pt} \left( {\cos \nu - \frac{{{{{\cos }}^{2}}\lambda \sin \nu }}{2}} \right),$$

$${{M}_{{13}}} = \frac{{{{r}_{{20}}}}}{{2{\kern 1pt} \mu }}\cos \phi \cos \nu ,$$

$${{M}_{{14}}} = \frac{{{{r}_{{20}}}}}{{2{\kern 1pt} \mu }}{\kern 1pt} \left( {\sin \lambda \sin \phi \cos \nu + \frac{{\sin 2\lambda \sin \nu }}{2}} \right),$$

$${{M}_{{15}}} = \frac{{{{r}_{{20}}}}}{{2\mu }}\left( {\sin \lambda \sin \phi \cos \nu - \frac{{\sin 2\lambda \sin \nu }}{2}} \right).$$

APPENDIX C

Values of ${{P}_{i}}$’s used in Eq. (6)

$${{P}_{1}} = - \frac{{12J_{2}^{{(2)}}{{g}_{0}}R_{0}^{4}{{{\cos }}^{2}}\phi }}{{r_{c}^{3}}}\frac{B}{{{{B}^{2}} + 4{{{\dot {\gamma }}}^{2}}}},$$

$${{P}_{2}} = \frac{{12J_{2}^{{(2)}}{{g}_{0}}R_{0}^{4}{{{\cos }}^{2}}{{\phi }_{m}}}}{{r_{{em}}^{4}}}\left( {\frac{{2{{{\dot {\theta }}}_{E}}\dot {\gamma }}}{{{{B}^{2}} + 4{{{\dot {\gamma }}}^{2}}}} - \frac{3}{4}} \right),$$

$$\begin{gathered} {{P}_{3}} = \frac{{6{{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{1}}}}{{R({{B}^{2}} + {{{\dot {\alpha }}}^{2}})}} \\ - \;\frac{{6{{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{3}}\cos \phi {{M}_{{13}}}}}{{R({{B}^{2}} + {{{\dot {\alpha }}}^{2}})}} + \frac{{3{{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}{{L}_{3}}}}{R}, \\ \end{gathered} $$

$$\begin{gathered} {{P}_{4}} = \frac{{6{{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{2}}}}{{R({{B}^{2}} + {{{(2{{{\dot {\theta }}}_{E}} + \dot {\alpha })}}^{2}})}} \\ - \;\frac{{6{{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi (2{{{\dot {\theta }}}_{E}} + \dot {\alpha }){\kern 1pt} {{M}_{{11}}}}}{{R{\kern 1pt} ({{B}^{2}} + {{{(2{{{\dot {\theta }}}_{E}} + \dot {\alpha })}}^{2}})}} + \frac{{3{{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{6}}}}{R}, \\ \end{gathered} $$

$$\begin{gathered} {{P}_{5}} = \frac{{6{\kern 1pt} {{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{3}}}}{{R({{B}^{2}} + {{{(2{{{\dot {\theta }}}_{E}} - \dot {\alpha })}}^{2}})}} \\ - \;\frac{{6{{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi (2{{{\dot {\theta }}}_{E}} - \dot {\alpha }){{M}_{{12}}}}}{{R({{B}^{2}} + {{{(2{{{\dot {\theta }}}_{E}} - \dot {\alpha })}}^{2}})}} + \frac{{3{{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{7}}}}{R}, \\ \end{gathered} $$

$${{P}_{6}} = \frac{{6{{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{4}}}}{{R{\kern 1pt} ({{B}^{2}} + {{{({{{\dot {\theta }}}_{E}} + 2\dot {\alpha })}}^{2}})}},$$

$${{P}_{7}} = \frac{{6{{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{5}}}}{{R({{B}^{2}} + {{{({{{\dot {\theta }}}_{E}} - 2\dot {\alpha })}}^{2}})}},$$

$${{P}_{8}} = \frac{{6{{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{6}}}}{{R{\kern 1pt} ({{B}^{2}} + \dot {\theta }_{E}^{2})}},$$

$${{P}_{9}} = \frac{{ - 12{\kern 1pt} {{r}_{c}}\dot {\theta }_{E}^{2}{{{\dot {\alpha }}}^{2}}\cos \phi {{M}_{7}}}}{{R{\kern 1pt} ({{B}^{2}} + 4\dot {\theta }_{E}^{2})}} + \frac{{3{\kern 1pt} {{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{8}}}}{R},$$

$${{P}_{{10}}} = \frac{{ - 12{\kern 1pt} {{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi {\kern 1pt} ({{{\dot {\theta }}}_{E}} + \dot {\alpha }){{M}_{8}}}}{{R{\kern 1pt} ({{B}^{2}} + 4({{{\dot {\theta }}}_{E}} + \dot {\alpha }{{)}^{2}})}} + \frac{{3{{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{4}}}}{R},$$

$${{P}_{{11}}} = \frac{{ - 12{\kern 1pt} {{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi {\kern 1pt} ({{{\dot {\theta }}}_{E}} - \dot {\alpha }){{M}_{9}}}}{{R{\kern 1pt} ({{B}^{2}} + 4({{{\dot {\theta }}}_{E}} - \dot {\alpha }{{)}^{2}})}} + \frac{{3{{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{5}}}}{R},$$

$${{P}_{{12}}} = \frac{{ - 12{\kern 1pt} {{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{3}}\cos \phi {{M}_{{10}}}}}{{R{\kern 1pt} ({{B}^{2}} + 4{{{\dot {\alpha }}}^{2}})}} + \frac{{3{{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{9}}}}{R},$$

$${{P}_{{13}}} = \frac{{ - 6{\kern 1pt} {{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi {\kern 1pt} ({{{\dot {\theta }}}_{E}} + \dot {\alpha }){{M}_{{14}}}}}{{R{\kern 1pt} ({{B}^{2}} + {{{({{{\dot {\theta }}}_{E}} + \dot {\alpha })}}^{2}})}} + \frac{{3{{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{1}}}}{R},$$

$${{P}_{{14}}} = \frac{{ - 6{\kern 1pt} {{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi {\kern 1pt} ({{{\dot {\theta }}}_{E}} - \dot {\alpha }){{M}_{{15}}}}}{{R{\kern 1pt} ({{B}^{2}} + {{{({{{\dot {\theta }}}_{E}} - \dot {\alpha })}}^{2}})}} + \frac{{3{{r}_{c}}{\kern 1pt} {{{\dot {\alpha }}}^{2}}{{L}_{2}}}}{R},$$

$${{P}_{{15}}} = \frac{{3{\kern 1pt} {{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{{10}}}}}{R},$$

$${{P}_{{16}}} = \frac{{3{\kern 1pt} {{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{{11}}}}}{R},$$

$${{P}_{{17}}} = \frac{{3{\kern 1pt} {{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{{12}}}}}{R},$$

$${{P}_{{18}}} = \frac{{3{\kern 1pt} {{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{{13}}}}}{R},$$

$$\begin{gathered} {{P}_{{19}}} = \frac{{6{\kern 1pt} {{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi }}{{R({{B}^{2}} + {{{\dot {\alpha }}}^{2}})}}(\dot {\alpha }{\kern 1pt} {{M}_{1}} + B{\kern 1pt} {{M}_{{13}}}) \\ + \;\frac{{3{{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{{14}}}}}{R}, \\ \end{gathered} $$

$$\begin{gathered} {{P}_{{20}}} = \frac{{6{{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi {\kern 1pt} }}{{R{\kern 1pt} ({{B}^{2}} + {{{(2{{{\dot {\theta }}}_{E}} + \dot {\alpha })}}^{2}})}}((2{{{\dot {\theta }}}_{E}} + \dot {\alpha }){\kern 1pt} {{M}_{2}} + B{{M}_{{11}}}) \\ + \;\frac{{3{{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{{17}}}}}{R}, \\ \end{gathered} $$

$$\begin{gathered} {{P}_{{21}}} = \frac{{6{{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi }}{{R{\kern 1pt} ({{B}^{2}} + {{{(2{{{\dot {\theta }}}_{E}} - \dot {\alpha })}}^{2}})}}((2{{{\dot {\theta }}}_{E}} - \dot {\alpha }){\kern 1pt} {{M}_{3}} + B{{M}_{{12}}}) \\ + \;\frac{{3{{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{{18}}}}}{R}, \\ \end{gathered} $$

$${{P}_{{22}}} = \frac{{6{\kern 1pt} {{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi {\kern 1pt} ({{{\dot {\theta }}}_{E}} + 2\dot {\alpha }){{M}_{4}}}}{{R{\kern 1pt} ({{B}^{2}} + {{{({{{\dot {\theta }}}_{E}} + 2\dot {\alpha })}}^{2}})}} + \frac{{3{\kern 1pt} {{r}_{c}}{\kern 1pt} {{{\dot {\alpha }}}^{2}}{{L}_{{15}}}}}{R},$$

$${{P}_{{23}}} = \frac{{6{\kern 1pt} {{r}_{c}}{{{\dot {\theta }}}_{E}}{{{\dot {\alpha }}}^{2}}\cos \phi {\kern 1pt} ({{{\dot {\theta }}}_{E}} + 2\dot {\alpha }){{M}_{5}}}}{{R{\kern 1pt} ({{B}^{2}} + {{{({{{\dot {\theta }}}_{E}} - 2\dot {\alpha })}}^{2}})}} + \frac{{3{\kern 1pt} {{r}_{c}}{\kern 1pt} {{{\dot {\alpha }}}^{2}}{{L}_{{16}}}}}{R},$$

$${{P}_{{24}}} = \frac{{6{\kern 1pt} {{r}_{c}}\dot {\theta }_{E}^{2}{{{\dot {\alpha }}}^{2}}\cos \phi {{M}_{6}}}}{{R{\kern 1pt} ({{B}^{2}} + \dot {\theta }_{E}^{2})}} + \frac{{3{{r}_{c}}{{{\dot {\alpha }}}^{2}}{{L}_{{21}}}}}{R},$$

$${{P}_{{25}}} = \frac{{6{\kern 1pt} {{r}_{c}}{\kern 1pt} {{{\dot {\theta }}}_{E}}{\kern 1pt} {{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{7}}}}{{R{\kern 1pt} ({{B}^{2}} + 4\dot {\theta }_{E}^{2})}},$$

$${{P}_{{26}}} = \frac{{6{\kern 1pt} {{r}_{c}}{\kern 1pt} {{{\dot {\theta }}}_{E}}{\kern 1pt} {{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{8}}}}{{R{\kern 1pt} ({{B}^{2}} + 4({{{\dot {\theta }}}_{E}} + \dot {\alpha }{{)}^{2}})}},$$

$${{P}_{{27}}} = \frac{{6{\kern 1pt} {{r}_{c}}{\kern 1pt} {{{\dot {\theta }}}_{E}}{\kern 1pt} {{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{9}}}}{{R{\kern 1pt} ({{B}^{2}} + 4({{{\dot {\theta }}}_{E}} - \dot {\alpha }{{)}^{2}})}},$$

$${{P}_{{28}}} = \frac{{6{\kern 1pt} {{r}_{c}}{\kern 1pt} {{{\dot {\theta }}}_{E}}{\kern 1pt} {{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{{10}}}}}{{R{\kern 1pt} ({{B}^{2}} + 4{{{\dot {\alpha }}}^{2}})}},$$

$${{P}_{{29}}} = \frac{{6{\kern 1pt} {{r}_{c}}{\kern 1pt} {{{\dot {\theta }}}_{E}}{\kern 1pt} {{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{{14}}}}}{{R{\kern 1pt} ({{B}^{2}} + {{{({{{\dot {\theta }}}_{E}} + \dot {\alpha })}}^{2}})}} + \frac{{3{\kern 1pt} {{r}_{c}}{\kern 1pt} {{{\dot {\alpha }}}^{2}}{{L}_{{19}}}}}{R},$$

$${{P}_{{30}}} = \frac{{6{\kern 1pt} {{r}_{c}}{\kern 1pt} {{{\dot {\theta }}}_{E}}{\kern 1pt} {{{\dot {\alpha }}}^{2}}\cos \phi B{{M}_{{15}}}}}{{R{\kern 1pt} ({{B}^{2}} + {{{({{{\dot {\theta }}}_{E}} - \dot {\alpha })}}^{2}})}} + \frac{{3{\kern 1pt} {{r}_{c}}{\kern 1pt} {{{\dot {\alpha }}}^{2}}{{L}_{{20}}}}}{R}.$$