Semi-analytical model for third-body perturbations including the inclination and eccentricity of the perturbing body

Abstract

A general third-body perturbation problem, considering the perturbing body in an elliptic and inclined orbit, is investigated by using a semi-analytical theory. Previous works have contributed to deriving the averaged third-body-perturbed dynamics, but did not provide a transformation between osculating and mean elements in the general case. In this paper, an analytical transformation between osculating and mean elements is developed explicitly using von Zeipel’s method, in addition to develo** the long-term dynamical equations. The resulting dynamical model is improved, because the disturbing function is averaged as a whole, instead of separating the disturbing function into many terms and averaging them independently. The simulation results indicate that the new singly averaged dynamical model behaves much better than the doubly averaged dynamics in propagating the long-term evolution of the orbital elements. Moreover, it is shown that the perturbing body’s inclination and eccentricity have a vital influence on the evolution of the satellite’s inclination and eccentricity.

Appendix A: Coefficients in the generating function \(S_{22}^{{{*}}*}\)

The coefficients in the generating function $S_{22}^{{{*}}*}$ are given by

$$\begin{aligned} A_1^{**}&=- \frac{1}{{360{{\left( {\sqrt{1 - {e_3}^2} + 1} \right) }^3}}}\left\{ { - 30{e_3}^{10}\left( {9\cos 4{f_3} + 4\cos 6{f_3}} \right) + 144{e_3}^9{{\cos }^3}{f_3}} \right. \\&\quad \times \, \left[ {8\left( {\sqrt{1 - {e_3}^2} + 3} \right) \cos 2{f_3} + 8\sqrt{1 - {e_3}^2} - 1} \right] - 5{e_3}^8\left[ { - 8\left( {\sqrt{1 - {e_3}^2} + 6} \right) \cos 6{f_3}} \right. \\&\quad \left. { +\, 9\left( {20\sqrt{1 - {e_3}^2} + 29} \right) \cos 4{f_3} + 36\left( {22\sqrt{1 - {e_3}^2} + 31} \right) \cos 2{f_3}} \right] - 24{e_3}^7\cos {f_3} \\&\quad \times \,\left[ { - \left( {41\sqrt{1 - {e_3}^2} + 142} \right) \cos 2{f_3} + 36\left( {\sqrt{1 - {e_3}^2} + 2} \right) \cos 4{f_3} - 77\sqrt{1 - {e_3}^2} - 64} \right] \\&\quad +\, 5{e_3}^6\left[ {\sqrt{1 - {e_3}^2} (501\cos 4{f_3} - 8\cos 6{f_3}) + 12\left( {103\sqrt{1 - {e_3}^2} + 96} \right) \cos 2{f_3}} \right. \\&\quad \left. { +\, 684\cos 4{f_3} - 24\cos 6{f_3}} \right] + 8{e_3}^5\cos {f_3}\left[ { - \left( {1027\sqrt{1 - {e_3}^2} + 1593} \right) \cos 2{f_3}} \right. \\&\quad \left. { +\, 36\left( {2\sqrt{1 - {e_3}^2} + 3} \right) \cos 4{f_3} - 229\sqrt{1 - {e_3}^2} + 279} \right] + 15{e_3}^4\left[ {4\left( {59\sqrt{1 - {e_3}^2} + 113} \right) } \right. \\&\quad \left. { \times \,\cos 2{f_3} - 131\left( {\sqrt{1 - {e_3}^2} + 1} \right) \cos 4{f_3}} \right] + 80{e_3}^3\left[ {\left( {35\sqrt{1 - {e_3}^2} + 47} \right) \cos 3{f_3}} \right. \\&\quad \left. {\left. { +\, 54\cos {f_3}} \right] - 7560\left( {\sqrt{1 - {e_3}^2} + 1} \right) {e_3}^2\cos 2{f_3} - 8640\left( {\sqrt{1 - {e_3}^2} + 1} \right) {e_3}\cos {f_3}} \right\} \\ B_1^{**}&= - \frac{3}{8}{e_3}^2{M_3}\sin 2{\omega _3} - \frac{1}{{96}}\left\{ {12\left( {2{e_3}^2 + 3} \right) \cos (2{f_3} + 2{\omega _3}) + {e_3}\left[ {3{e_3}\cos (4{f_3} + 2{\omega _3})} \right. } \right. \\&\quad \left. {\left. { + \,4\left( { - 9{e_3}{f_3}\sin 2{\omega _3} + 27\cos ({f_3} + 2{\omega _3}) + 5\cos (3{f_3} + 2{\omega _3})} \right) } \right] } \right\} \\ C_1^{**}&= - \frac{3}{2}{e_3}^2{M_3}\cos 2{\omega _3} - \frac{1}{{24}}\left\{ { - {e_3}\left[ {3{e_3}\left( {8\sin (2{f_3} + 2{\omega _3}) + \sin (4{f_3} + 2{\omega _3}) + 12{f_3}} \right. } \right. } \right. \\&\quad \left. {\left. {\left. { \times \,\cos 2{\omega _3}} \right) + 4\left( {27\sin ({f_3} + 2{\omega _3}) + 5\sin (3{f_3} + 2{\omega _3})} \right) } \right] - 36\sin (2{f_3} + 2{\omega _3})} \right\} \\ A_2^{**}&= \left\{ {\dfrac{{{e_3}^2}}{4}\left( {20{e_3}^2 + 3} \right) {{\left( {1 - {e_3}^2} \right) }^{3/2}}\cos 2{\omega _3} + \dfrac{{{e_3}^2\sin 2{\omega _3}}}{{32{{\left( {\sqrt{1 - {e_3}^2} + 1} \right) }^3}}}\left[ { - 1170\left( {\sqrt{1 - {e_3}^2} + 1} \right) } \right. } \right. \\&\quad \times \, {e_3}^4 - 168{e_3}^8 - 16\left( {3\sqrt{1 - {e_3}^2} + 13} \right) {e_3}^2 + 288\left( {\sqrt{1 - {e_3}^2} + 1} \right) + 3\left( {57\sqrt{1 - {e_3}^2} } \right. \\&\quad \left. {\left. {\left. { +\, 335} \right) {e_3}^6} \right] } \right\} {M_3} + \frac{{{e_3}}}{{887040{{\left( {\sqrt{1 - {e_3}^2} + 1} \right) }^3}}}\left\{ {5544{e_3}^9\left[ {840{f_3}\sin 2{\omega _3} +420} \right. } \right. \\&\quad \times \, \cos (2{f_3} - 2{\omega _3}) + 210\cos (4{f_3} + 2{\omega _3}) + 140\cos (6{f_3} + 2{\omega _3}) + 45\cos (8{f_3} \\&\quad \left. { +\, 2{\omega _3}) + 6\cos (10{f_3} + 2{\omega _3}) + 90\cos (4{f_3} - 2{\omega _3}) + 10\cos (6{f_3} - 2{\omega _3})} \right] + 48{e_3}^8 \\ \end{aligned}$$

$$\begin{aligned} \begin{aligned}&\quad \times \, \left[ {38115\sqrt{1 - {e_3}^2} \cos ({f_3} - 2{\omega _3}) + 9\sqrt{1 - {e_3}^2} \left( { - 11( - 105\sin {f_3} - 140\sin 3{f_3}} \right. } \right. \\&\quad +\, 14\sin 5{f_3} + 65\sin 7{f_3} + 35\sin 9{f_3})\sin 2{\omega _3} + 55( - 21\cos {f_3} + 42\cos 3{f_3} \\&\quad \left. { +\, 28\cos 5{f_3} + 17\cos 7{f_3} + 7\cos 9{f_3})\cos 2{\omega _3} + 70\cos (11{f_3} + 2{\omega _3})} \right) + 16 \\&\quad \times \, \left( {80(7\cos 2{f_3} + 21\cos 4{f_3} + 19){{\cos }^7}{f_3}\cos 2{\omega _3} + 7{{\sin }^3}{f_3}(2229\cos 2{f_3}} \right. \\&\quad \left. {\left. { +\, 750\cos 4{f_3} + 155\cos 6{f_3} + 15\cos 8{f_3} + 2131)\sin 2{\omega _3}} \right) } \right] + 693{e_3}^7\left[ {30\left( {27\sqrt{1 - {e_3}^2} } \right. } \right. \\&\quad \left. { -\, 67} \right) \cos (4{f_3} - 2{\omega _3}) + 3\sqrt{1 - {e_3}^2} \left( { - 2280{f_3}\sin 2{\omega _3} - 240\cos (2{f_3} - 2{\omega _3})} \right. \\&\quad -\, 570\cos (4{f_3} + 2{\omega _3}) - 80\cos (6{f_3} + 2{\omega _3}) + 135\cos (8{f_3} + 2{\omega _3}) + 48\cos (10{f_3} + 2{\omega _3})\\&\quad \left. { +\, 80\cos (6{f_3} - 2{\omega _3})} \right) - 335\left( {120{f_3}\sin 2{\omega _3} + 48\cos (2{f_3} - 2{\omega _3}) + 30\cos (4{f_3} + 2{\omega _3})} \right. \\&\quad \left. {\left. { +\, 16\cos (6{f_3} + 2{\omega _3}) + 3\cos (8{f_3} + 2{\omega _3})} \right) } \right] + 88{e_3}^6\left[ {21\sqrt{1 - {e_3}^2} \left( {160( - 40} \right. }\right. \\&\quad \times \, \cos 2{f_3} + 17\cos 4{f_3} + 33){\cos ^5}{f_3}\cos 2{\omega _3} + 8{\sin ^3}{f_3}(1059\cos 2{f_3} + 480 \\&\left. { \times \cos 4{f_3} + 85\cos 6{f_3} + 536)\sin 2{\omega _3}} \right) + 160\left( { - 2500\cos 2{f_3} + 77\cos 4{f_3}} \right. \\&\quad \left. { +\, 1077} \right) {\cos ^5}{f_3}\cos 2{\omega _3} - 8{\sin ^3}{f_3}\left( {78681\cos 2{f_3} + 9420\cos 4{f_3} - 385\cos 6{f_3}} \right. \\&\quad \left. {\left. { +\, 118924} \right) \sin 2{\omega _3}} \right] + 270270\left( {\sqrt{1 - {e_3}^2} + 1} \right) {e_3}^5\left[ {48\cos (2{f_3} - 2{\omega _3})+120{f_3}} \right. \\&\quad \times \, \sin 2{\omega _3} + + 30\cos (4{f_3} + 2{\omega _3}) + 16\cos (6{f_3} + 2{\omega _3}) + 3\cos (8{f_3} + 2{\omega _3}) \\&\quad \left. { +\, 6\cos (4{f_3} - 2{\omega _3})} \right] + 6336{e_3}^4\left\{ { - 2730\cos ({f_3} + 2{\omega _3}) + 910\cos (3{f_3} + 2{\omega _3})} \right. \\&\quad +\, 861\cos (5{f_3} + 2{\omega _3}) + 225\cos (7{f_3} + 2{\omega _3}) + 525\cos (3{f_3} - 2{\omega _3})+15 \\&\quad \times \,\left[ {\sqrt{1 - {e_3}^2} \left( { - 210\cos ({f_3} + 2{\omega _3}) + 70\cos (3{f_3} + 2{\omega _3}) + 63\cos (5{f_3} + 2{\omega _3})} \right. } \right. \\&\quad \left. {\left. {\left. { +\, 15\cos (7{f_3} + 2{\omega _3}) + 35\cos (3{f_3} - 2{\omega _3})} \right) + 7\left( {45\sqrt{1 - {e_3}^2} + 41} \right) \cos ({f_3} - 2{\omega _3})} \right] } \right\} \\&\quad -\, 36960{e_3}^3\left[ {3\left( {4\left( {3\sqrt{1 - {e_3}^2} + 1} \right) \cos (2{f_3} - 2{\omega _3}) + \sqrt{1 - {e_3}^2} \left[ {4\left( {\cos (6{f_3} + 2{\omega _3})} \right. } \right. } \right. }\right. \\&\quad \left. {\left. {\left. { -\, 3{f_3}\sin 2{\omega _3}} \right) - 3\cos (4{f_3} + 2{\omega _3})} \right] } \right) - 39\cos (4{f_3} + 2{\omega _3}) + 4\left( { - 39{f_3}\sin 2{\omega _3}} \right. \\&\quad \left. {\left. {+\,\cos (6{f_3} + 2{\omega _3})} \right) } \right] - 88704{e_3}^2\left[ {3\sqrt{1 - {e_3}^2} \left( {45\cos ({f_3} + 2{\omega _3}) + \cos (5{f_3} + 2{\omega _3})} \right. } \right. \\&\quad \left. { -\, 15\cos (3{f_3} + 2{\omega _3})} \right) + 5\left( {3\sqrt{1 - {e_3}^2} - 5} \right) \cos ({f_3} - 2{\omega _3}) - 5\left( { - 63\cos ({f_3} + 2{\omega _3})} \right. \\&\quad \left. {\left. { + \,21\cos (3{f_3} + 2{\omega _3}) + \cos (5{f_3} + 2{\omega _3})} \right) } \right] - 1995840\left( {\sqrt{1 - {e_3}^2} + 1} \right) {e_3}\left[ {4{f_3}} \right. \\&\quad \left. { \times \, \sin 2{\omega _3} + \cos (4{f_3} + 2{\omega _3})} \right] - 10644480\left( {\sqrt{1 - {e_3}^2} + 1} \right) \left[ {\cos (3{f_3} + 2{\omega _3})} \right. \\&\quad \left. {\left. { -\, 3\cos ({f_3} + 2{\omega _3})} \right] } \right\} - \frac{1}{{240}}{\left( {1 - {e_3}^2} \right) ^{3/2}}\left\{ {20{e_3}^4\left[ {9\left( {\sin (2{f_3} - 2{\omega _3}) + 60{f_3}\cos 2{\omega _3}} \right. } \right. } \right. \\&\quad \left. {\left. { +\, 4\sin (2{f_3} + 2{\omega _3}) + \sin (4{f_3} + 2{\omega _3})} \right) + \sin (6{f_3} + 2{\omega _3})} \right] + {e_3}^3\left( {96{{\cos }^3}{f_3}} \right. \\&\quad \left. { \times \,(3\cos 2{f_3} - 2)\sin 2{\omega _3} - 96{{\sin }^3}{f_3}(3\cos 2{f_3} + 7)\cos 2{\omega _3}} \right) + 15{e_3}^2\left[ {12{f_3}} \right. \\&\quad \left. { \times \,\cos 2{\omega _3} - 4\sin (2{f_3} + 2{\omega _3}) - 5\sin (4{f_3} + 2{\omega _3})} \right] + 160{e_3}\sin (3{f_3} + 2{\omega _3}) \\&\quad \left. { - \,360\sin (2{f_3} + 2{\omega _3})} \right\} \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} B_2^{**}&= - \frac{1}{{64}}\left[ { - 6{e_3}^2\cos (4{f_3} + 4{\omega _3}) - 9{e_3}^2\cos (2{f_3} + 4{\omega _3}) - {e_3}^2\cos (6{f_3} +4{\omega _3})} \right. \\&\quad +\, 6{e_3}^2\cos 2{f_3} - 18{e_3}\cos (3{f_3} + 4{\omega _3}) - 6{e_3}\cos (5{f_3} + 4{\omega _3}) + 24{e_3}\cos {f_3} \\&\quad \left. { -\, 9\cos (4{f_3} + 4{\omega _3})} \right] \\ C_2^{**}&=\frac{3}{4}\left( {2{e_3}^2 + 3} \right) {M_3} - \frac{1}{{16}}\left[ {{e_3}^2\left( {6\sin (4{f_3} + 4{\omega _3}) + 9\sin (2{f_3} + 4{\omega _3})} \right. } \right. \\&\quad \left. { +\, \sin (6{f_3} + 4{\omega _3}) + 24{f_3} + 12\sin 2{f_3}} \right) + 6{e_3}\left( {3\sin (3{f_3} + 4{\omega _3})} \right. \\&\quad \left. {\left. { +\, \sin (5{f_3} + 4{\omega _3}) + 14\sin {f_3}} \right) + 9\sin (4{f_3} + 4{\omega _3}) + 36{f_3}} \right] \\ A_3^{**}&= \left\{ {\frac{{{e_3}^2\cos 2{\omega _3}}}{{16{{\left( {\sqrt{1 - {e_3}^2} + 1} \right) }^3}}}\left[ {168{e_3}^8 + 16\left( {3\sqrt{1 - {e_3}^2} + 13} \right) {e_3}^2 - 288\left( {\sqrt{1 - {e_3}^2} + 1} \right) - 3} \right. } \right. \\&\quad \left. { \times \, \left( {57\sqrt{1 - {e_3}^2} + 335} \right) {e_3}^6 + 1170\left( {\sqrt{1 - {e_3}^2} + 1} \right) {e_3}^4} \right] - \frac{{{e_3}^2}}{8}{\left( {1 - {e_3}^2} \right) ^{3/2}}\left( {20{e_3}^2 + 3} \right. \\&\quad \left. { \times \,\sin 2{\omega _3}} \right\} {M_3} - \frac{1}{2}{\left( {1 - {e_3}^2} \right) ^{3/2}}\left\{ {\frac{1}{{12}}{e_3}^4\left[ { - 60{f_3}\sin 2{\omega _3} - 9\cos (2{f_3} - 2{\omega _3}) + 36} \right. } \right. \\&\quad \left. { \times \,\cos (2{f_3} + 2{\omega _3}) + 9\cos (4{f_3} + 2{\omega _3}) + \cos (6{f_3} + 2{\omega _3})} \right] + \frac{2}{5}{e_3}^3\left[ {(3\cos 2{f_3} - 2)} \right. \\&\quad \left. { \times \,{{\cos }^3}{f_3}\cos 2{\omega _3} + {{\sin }^3}{f_3}(3\cos 2{f_3} + 7)\sin 2{\omega _3}} \right] - \frac{1}{{16}}{e_3}^2\left[ {12{f_3}\sin 2{\omega _3} + 4} \right. \\&\quad \left. {\left. { \times \,\cos (2{f_3} + 2{\omega _3}) + 5\cos (4{f_3} + 2{\omega _3})} \right] + \frac{2}{3}{e_3}\cos (3{f_3} + 2{\omega _3}) - \frac{3}{2}\cos (2{f_3} + 2{\omega _3})} \right\} \\&\quad -\, \frac{{{e_3}}}{{443520{{\left( {\sqrt{1 - {e_3}^2} + 1} \right) }^3}}}\left\{ {5544{e_3}^9\left[ { - 6\sin (10{f_3} + 2{\omega _3}) - 5( - 84\sin 2{f_3} + 24} \right. } \right. \\&\quad \times \,\sin 4{f_3} + 26\sin 6{f_3} + 9\sin 8{f_3} - 168{f_3})\cos 2{\omega _3} - 15(28\cos 2{f_3} + 20\cos 4{f_3} \\&\quad \left. { +\, 10\cos 6{f_3} + 3\cos 8{f_3})\sin 2{\omega _3}} \right] + 48{e_3}^8\left( {38115\sqrt{1 - {e_3}^2} \sin ({f_3} - 2{\omega _3}) - 9} \right. \\&\quad \times \,\sqrt{1 - {e_3}^2} \left[ {\left( { - 1155\sin {f_3} - 1540\sin 3{f_3} + 154\sin 5{f_3} + 715\sin 7{f_3} + 385} \right. } \right. \\&\quad \left. { \times \,\sin 9{f_3} + 70\sin 11{f_3}} \right) \cos 2{\omega _3} + 5\left( { - 231\cos {f_3} + 462\cos 3{f_3} + 308\cos 5{f_3}} \right. \\&\quad \left. {\left. { + \,187\cos 7{f_3} + 77\cos 9{f_3} + 14\cos 11{f_3}} \right) \sin 2{\omega _3}} \right] - 1280\left( {19+7\cos 2{f_3}} \right. \\&\quad \left. { +\, 21\cos 4{f_3}} \right) {\cos ^7}{f_3}\sin 2{\omega _3} + 112{\sin ^3}{f_3}\left( {2229\cos 2{f_3} + 750\cos 4{f_3}} \right. \\&\quad \left. {\left. { +\, 155\cos 6{f_3} + 15\cos 8{f_3} + 2131} \right) \cos 2{\omega _3}} \right) - 693{e_3}^7\left[ {3\left( {40{f_3}\left( {57\sqrt{1 - {e_3}^2} } \right. } \right. } \right. \\&\quad \left. { +\, 335} \right) \cos 2{\omega _3} + \sqrt{1 - {e_3}^2} (48\sin (10{f_3} + 2{\omega _3}) + 5(48\sin 2{f_3} - 168\sin 4{f_3}\\&\quad -\, 32\sin 6{f_3} + 27\sin 8{f_3})\cos 2{\omega _3} + 15( - 16\cos 2{f_3} - 20\cos 4{f_3} + 9\cos 8{f_3}) \\&\quad \left. { \times \,\sin 2{\omega _3})} \right) - 335\left( { - 48\sin (2{f_3} - 2{\omega _3}) + 30\sin (4{f_3} + 2{\omega _3}) + 16\sin (6{f_3} + 2{\omega _3})} \right. \\&\quad \left. {\left. { +\, 3\sin (8{f_3} + 2{\omega _3}) - 6\sin (4{f_3} - 2{\omega _3})} \right) } \right] + 88{e_3}^6\left[ {21\sqrt{1 - {e_3}^2} \left( {8{{\sin }^3}{f_3}} \right. } \right. \\&\quad \times \,(1059\cos 2{f_3} + 480\cos 4{f_3} + 85\cos 6{f_3} + 536)\cos 2{\omega _3} - 160{\cos ^5}{f_3} \\&\quad \left. { \times \,( -\, 40\cos 2{f_3} + 17\cos 4{f_3} + 33)\sin 2{\omega _3}} \right) - 160( - 2500\cos 2{f_3} + 77\cos 4{f_3} \\&\quad +\, 1077){\cos ^5}{f_3}\sin 2{\omega _3} - 8{\sin ^3}{f_3}\left( {78681\cos 2{f_3} + 9420\cos 4{f_3} - 385} \right. \\ \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned}&\quad \left. {\left. { \times \,\cos 6{f_3} + 118924} \right) \cos 2{\omega _3}} \right] + 270270\left( {\sqrt{1 - {e_3}^2} + 1} \right) {e_3}^5\left[ {48\sin (2{f_3} - 2{\omega _3})} \right. \\&\quad -\, 30\sin (4{f_3} + 2{\omega _3}) - 16\sin (6{f_3} + 2{\omega _3}) - 3\sin (8{f_3} + 2{\omega _3}) + 6\sin (4{f_3} - 2{\omega _3}) \\&\quad \left. { +\, 120{f_3}\cos 2{\omega _3}} \right] + 6336{e_3}^4\left\{ {2730\sin ({f_3} + 2{\omega _3}) - 910\sin (3{f_3} + 2{\omega _3}) - 861} \right. \\&\quad \times \, \sin (5{f_3} + 2{\omega _3}) - 225\sin (7{f_3} + 2{\omega _3}) + 525\sin (3{f_3} - 2{\omega _3})+15\left[ {\sqrt{1 - {e_3}^2} } \right. \\&\quad \times \,\left( {210\sin ({f_3} + 2{\omega _3}) - 70\sin (3{f_3} + 2{\omega _3}) - 63\sin (5{f_3} + 2{\omega _3}) - 15} \right. \\&\quad \left. {\left. {\left. { \times \, \sin (7{f_3} + 2{\omega _3}) + 35\sin (3{f_3} - 2{\omega _3})} \right) + 7\left( {45\sqrt{1 - {e_3}^2} + 41} \right) \sin ({f_3} - 2{\omega _3})} \right] } \right\} \\&\quad +\, 36960{e_3}^3\left\{ { - 12\sin (2{f_3} - 2{\omega _3}) - 39\sin (4{f_3} + 2{\omega _3}) + 4\sin (6{f_3} + 2{\omega _3})} \right. \\&\quad +\,3\left( {\sqrt{1 - {e_3}^2} \left[ {4\sin (6{f_3} + 2{\omega _3}) - 3\left( {4\sin (2{f_3} - 2{\omega _3}) + \sin (4{f_3} + 2{\omega _3})} \right) } \right] } \right. \\&\quad \left. {\left. { +\, 4\left( {3\sqrt{1 - {e_3}^2} + 13} \right) {f_3}\cos 2{\omega _3}} \right) } \right\} + 88704{e_3}^2\left[ {3\sqrt{1 - {e_3}^2} \left( {45\sin ({f_3} + 2{\omega _3})} \right. } \right. \\&\quad \left. { -\, 15\sin (3{f_3} + 2{\omega _3}) + \sin (5{f_3} + 2{\omega _3})} \right) + 5\left( {5 - 3\sqrt{1 - {e_3}^2} } \right) \sin ({f_3} - 2{\omega _3}) \\&\quad \left. { -\, 5\left( { - 63\sin ({f_3} + 2{\omega _3}) + 21\sin (3{f_3} + 2{\omega _3}) + \sin (5{f_3} + 2{\omega _3})} \right) } \right] \\&\quad +\, 1995840\left( {\sqrt{1 - {e_3}^2} + 1} \right) {e_3}\left[ {\sin (4{f_3} + 2{\omega _3}) - 4{f_3}\cos 2{\omega _3}} \right] \\&\quad \left. { +\, 10644480\left( {\sqrt{1 - {e_3}^2} + 1} \right) \left[ {\sin (3{f_3} + 2{\omega _3}) - 3\sin ({f_3} + 2{\omega _3})} \right] } \right\} \\ B_3^{**}&= \frac{3}{{16}}\left( {2{e_3}^2 + 3} \right) {M_3} - \frac{1}{{64}}\left\{ {{e_3}^2\left[ { - 6\sin (4{f_3} + 4{\omega _3}) - 9\sin (2{f_3} + 4{\omega _3}) + 12\sin 2f} \right. } \right. \\&\quad \left. { -\, \sin (6{f_3} + 4{\omega _3}) + 24{f_3}_3} \right] - 6{e_3}\left[ {3\sin (3{f_3} + 4{\omega _3}) + \sin (5{f_3} + 4{\omega _3})} \right. \\&\quad \left. {\left. { -\, 14\sin {f_3}} \right] - 9\sin (4{f_3} + 4{\omega _3}) + 36{f_3}} \right\} \\ C_3^{**}&= - \frac{1}{8}\left\{ {{e_3}^2\left[ { - 6\cos (4{f_3} + 4{\omega _3}) - 9\cos (2{f_3} + 4{\omega _3}) - \cos (6{f_3} + 4{\omega _3}) - 6\cos 2{f_3}} \right] } \right. \\&\quad \left. { - \,6{e_3}\left[ {3\cos (3{f_3} + 4{\omega _3}) + \cos (5{f_3} + 4{\omega _3}) + 4\cos {f_3}} \right] - 9\cos (4{f_3} + 4{\omega _3})} \right\} \end{aligned} \end{aligned}$$