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Resonance in the earth-moon system around the sun including earth’s equatorial ellipticity

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Abstract

We have investigated the resonances in the earth-moon system around the sun including earth’s equatorial ellipticity. The resonance resulting from the commensurability between the mean motion of the moon and Γ (angle measured from the minor axis of the earth’s equatorial ellipse to the projection of the moon on the plane of the equator) is analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure of Brown and Shook. We have shown the effects of Γ on the amplitude and the time period of the resonance oscillation using the data of the moon. It is observed that the amplitude decreases and the time period also decreases as Γ increases from 0∘ to 45∘.

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Acknowledgement

We are thankful to the Centre for Fundamental Research in Space Dynamics and Celestial Mechanics (CFRSC) for providing all facilities for this research work.

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Authors and Affiliations

  1. Maharaja Agrasen College, Department of Mathematics, University of Delhi, Delhi, India

    Sushil Yadav

  2. Sri Aurobindo College, Department of Mathematics, University of Delhi, New Delhi, India

    Rajiv Aggarwal

Authors
  1. Sushil Yadav
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  2. Rajiv Aggarwal
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Corresponding author

Correspondence to Rajiv Aggarwal.

Appendices

Appendix A

$$\begin{aligned} &{P_{1} = 2 - 3\sin^{2}\alpha_{m}, } \\ &{P_{2} = \bigl( 2 - 3\sin^{2}\alpha_{m} \bigr)^{2}, } \\ &{P_{3} = \bigl( 1 + \cos^{2}\alpha_{m} \bigr) \bigl( 1 + \cos^{2}\varepsilon \bigr),} \\ &{P_{4} = 2\sin^{2}\varepsilon \sin^{2} \alpha_{m}, } \\ &{P_{5} = - \frac{1}{2}\sin 2\varepsilon \sin 2 \alpha_{m}, } \\ &{P_{6} = \frac{1}{2}\sin^{2}\varepsilon \sin^{2}\alpha_{m},} \\ &{P_{7} = \sin^{2}\varepsilon \sin^{4} \frac{\alpha_{m}}{2},} \\ &{P_{8} = - \sin 2\varepsilon \sin \alpha_{m} \sin^{2}\frac{\alpha_{m}}{2},} \\ &{P_{9} = \frac{1}{2} \bigl( 2 - 3\sin^{2} \varepsilon \bigr)\sin^{2}\alpha_{m}, } \\ &{P_{10} = \sin 2\varepsilon \sin \alpha_{m} \cos^{2}\frac{\alpha_{m}}{2}, } \\ &{P_{11} = \sin^{2}\varepsilon \cos^{4} \frac{\alpha_{m}}{2}, } \\ &{P_{12} = \sin^{2}\alpha_{m}, } \\ &{P_{13} = \sin^{2}\alpha_{m} \bigl( 2 - 3 \sin^{2}\alpha_{m} \bigr), } \\ &{P_{14} = \frac{1}{2}\sin^{2} \alpha_{m}, } \\ &{P_{15} = \cos^{4}\frac{\alpha_{m}}{2}, } \\ &{P_{16} = \sin^{4}\frac{\alpha_{m}}{2}, } \\ &{P_{17} = \frac{3}{8} \bigl( 2 - \sin^{2} \alpha_{m} \bigr) \bigl( 2 + 3\sin^{2}\alpha_{m} \bigr), } \\ &{P_{18} = \frac{3}{8}\sin^{2} \alpha_{m} \bigl( 2 - 3\sin^{2}\alpha_{m} \bigr), } \\ &{P_{19} = P_{5} + P_{6}, } \\ &{P_{20} = P_{7} + P_{8} + P_{9} + P_{10} + P_{11}. } \end{aligned}$$

Appendix B

$$\begin{aligned} &{Q_{1} = \cos \varepsilon \cos \alpha_{m}, } \\ &{Q_{2} = - \sin^{2}\varepsilon \sin^{4} \frac{\alpha_{m}}{2}, } \\ &{Q_{3} = \sin 2\varepsilon \sin \alpha_{m} \sin^{2}\frac{\alpha_{m}}{2}, } \\ &{Q_{4} = - \frac{1}{2} \bigl( 2 - 3\sin^{2} \varepsilon \bigr)\sin^{2}\alpha_{m}, } \\ &{Q_{5} = - \sin 2\varepsilon \sin \alpha_{m} \cos^{2}\frac{\alpha_{m}}{2}, } \\ &{Q_{6} = - \sin^{2}\varepsilon \cos^{4} \frac{\alpha_{m}}{2}, } \\ &{Q_{7} = \sin^{2}\alpha_{m}, } \\ &{Q_{8} = \sin^{2}\alpha_{m} \bigl( 2 - 3 \sin^{2}\alpha_{m} \bigr), } \\ &{Q_{9} = \cos^{4}\frac{\alpha_{m}}{2}, } \\ &{Q_{10} = \sin^{4}\frac{\alpha_{m}}{2}, } \\ &{Q_{11} = \frac{1}{4}\sin^{2} \alpha_{m} \bigl( 2 - 3\sin^{2}\alpha_{m} \bigr), } \\ &{Q_{12} = \sin \alpha_{m}\sin \varepsilon. } \end{aligned}$$

Appendix C

$$\begin{aligned} &{K_{1} = \frac{ ( GM_{E} )}{a^{4} ( 1 - e^{2} )^{4}} - \frac{P_{1}\dot{\phi}^{2}}{4a ( 1 - e^{2} )} - \frac{9 ( GM_{E} )J_{2}^{ ( 2 )}R_{0}^{2}P_{3}}{4a^{6} ( 1 - e^{2} )^{6}}, } \\ &{\begin{aligned} K_{2} & = \biggl[ \frac{4 ( GM_{E} )}{a^{4} ( 1 - e^{2} )^{4}} - \frac{P_{1}\dot{\phi}^{2}}{4a ( 1 - e^{2} )} - \frac{27 (GM_{E} )J_{2}^{ ( 2 )}R_{0}^{2}P_{3}}{2a^{6} ( 1 - e^{2} )^{6}} \\ &\quad {}- \frac{3P_{12}\dot{\phi}^{2}}{8a ( 1 - e^{2} )} \biggr]e,\end{aligned} } \\ &{K_{3} = - \frac{3P_{12}\dot{\phi}^{2}}{4a ( 1 - e^{2} )}, } \\ &{K_{4} = - \frac{3eP_{12}\dot{\phi}^{2}}{8a ( 1 - e^{2} )},} \\ &{K_{5} = - \frac{9 ( GM_{E} )J_{2}^{ ( 2 )}R_{0}^{2}P_{4}}{4a^{6} ( 1 - e^{2} )^{6}} - \frac{27n^{2}J_{2}^{ ( 2 )}R_{0}^{2}P_{19}}{2a^{3} ( 1 - e^{2} )^{3}},} \\ &{\begin{aligned}K_{6} = K_{7} & = - \frac{27 ( GM_{E} )J_{2}^{ ( 2 )}R_{0}^{2}eP_{4}}{4a^{6} ( 1 - e^{2} )^{6}} \\ &\quad {}-\frac{81n^{2}J_{2}^{ ( 2 )}R_{0}^{2}e}{4a^{3} ( 1 - e^{2} )^{3}} \biggl( P_{19} + \frac{1}{2}P_{20} \biggr), \end{aligned}} \\ &{K_{8} = - \frac{81n^{2}J_{2}^{ ( 2 )}R_{0}^{2}e}{8a^{3} ( 1 - e^{2} )^{3}}P_{20},} \\ &{K_{9} = - \frac{3P_{14}\dot{\phi}^{2}}{2a ( 1 - e^{2} )}, } \end{aligned}$$
$$\begin{aligned} &{K_{10} = - \frac{3P_{15}\dot{\phi}^{2}}{2a ( 1 - e^{2} )}, } \\ &{K_{11} = - \frac{3P_{16}\dot{\phi}^{2}}{2a ( 1 - e^{2} )}, } \\ &{K_{12} = - \frac{3e\dot{\phi}^{2}}{4a ( 1 - e^{2} )} ( P_{14} + P_{16} ), } \\ &{K_{13} = - \frac{3e\dot{\phi}^{2}}{4a ( 1 - e^{2} )} ( P_{14} + P_{15} ), } \\ &{K_{14} = - \frac{3e\dot{\phi}^{2}P_{15}}{4a ( 1 - e^{2} )}, } \\ &{K_{15} = - \frac{3e\dot{\phi}^{2}P_{16}}{4a ( 1 - e^{2} )}. } \end{aligned}$$

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Yadav, S., Aggarwal, R. Resonance in the earth-moon system around the sun including earth’s equatorial ellipticity. Astrophys Space Sci 348, 367–375 (2013). https://doi.org/10.1007/s10509-013-1587-y

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