Reducing inter-satellite drift of low Earth orbit constellations using short-periodic corrections

Abstract

Constellation design theory has been studied extensively. However, analysis of longitude-dependent perturbation effects on inter-satellite distance drift has not received much attention. In addition to oblateness-related perturbations, sectoral and tesseral perturbations are non-negligible for low Earth orbits, due to their effect on inter-satellite distance evolution. This paper introduces the idea of reducing the tesseral/sectoral relative drift by including these perturbations in the short-periodic correction of the mean elements. An analytical expression of the correction is derived based on the Lie transform theory. Different from previous works, the ratio between the Earth rotation rate and the satellite’s mean motion is chosen as the small parameter in the Lie theory formulation. The independent variables of the generating-function-related partial differential equations can be reduced to a single variable when using this small parameter. Numerical simulations validate that the sectoral and tesseral effects on the inter-satellite distance drift in satellite constellations can be mitigated by using the proposed correction of the mean elements.

Appendix: Partial derivatives of the generating functions

1.1 Partial derivatives of $W_2$

The partial derivatives of $W_2$ are given by

$$\begin{aligned} \frac{{\partial {W_2}}}{{\partial L}} =&\frac{{{J_2}{R_e}^2}}{{4{a^2}e{\varepsilon ^2}{\eta ^3}}}\left\{ {\sin f\left[ {3\cos 2i\left( {4{{\sin }^2}(f + g) + {\eta ^2}} \right) + 6\cos (2f + 2g) + {\eta ^2} + 2} \right. } \right. \\&\left. { +\, e\cos f\left( {6e\cos f\cos 2i{{\sin }^2}(f + g) + (e\cos f + 3)\left( {3\cos (2f + 2g) + 1} \right) } \right) } \right] \\&\left. { +\, 9e\sin 2f\cos 2i{{\sin }^2}(f + g) + {\eta ^2}{{\sin }^2}i\left[ {3\sin (f + 2g) + \sin (3f + 2g)} \right] } \right\} \\ \frac{{\partial {W_2}}}{{\partial G}} =&\frac{{{J_2}{R_e}^2}}{{4{a^2}{\varepsilon ^2}{\eta ^4}}}\left\{ {2{{\cos }^2}i\left[ {3e\sin (f + 2g) + e\sin (3f + 2g) - 6e\sin f + 3\sin (2f + 2g) - 6f + 6l} \right] } \right. \\&-\, \frac{1}{e}\left[ {{{\sin }^2}i\left[ {\left( {3{e^2} + {\eta ^2}} \right) \left( {3\sin (f + 2g) + \sin (3f + 2g)} \right) + 9e\sin (2f + 2g)} \right] } \right. \\&+\, (3\cos 2i + 1)\left[ {\left( {3{e^2} + {\eta ^2}} \right) \sin f + 3e(f - l)} \right] \\&\left. {\left. { +\, \sin f(e\cos f + 1)(e\cos f + 2)\left( {6\cos 2i{{\sin }^2}(f + g) + 3\cos (2f + 2g) + 1} \right) } \right] } \right\} \\ \frac{{\partial {W_2}}}{{\partial H}} =&- \frac{{{J_2}{R_e}^2\cos i}}{{2{a^2}{\varepsilon ^2}{\eta ^4}}}\left\{ {e\left[ {3\sin (f + 2g) + \sin (3f + 2g) - 6\sin f} \right] + 3\sin (2f + 2g) - 6f + 6l} \right\} \\ \frac{{\partial {W_2}}}{{\partial l}} =&\frac{{{J_2}\sqrt{\mu }{R_e}^2}}{{4{a^{3/2}}{\varepsilon ^2}{\eta ^6}}}\left\{ {{{(e\cos f + 1)}^3}\left[ {6\cos 2i{{\sin }^2}(f + g) + 3\cos (2f + 2g) + 1} \right] - {\eta ^3}(3\cos 2i + 1)} \right\} \\ \frac{{\partial {W_2}}}{{\partial g}} =&\frac{{{J_2}\sqrt{\mu }{R_e}^2{{\sin }^2}i}}{{2{a^{3/2}}{\varepsilon ^2}{\eta ^3}}}\left\{ {e\left[ {3\cos (f + 2g) + \cos (3f + 2g)} \right] + 3\cos (2f + 2g)} \right\} \\ \frac{{\partial {W_2}}}{{\partial h}} =&0.\nonumber \\ \end{aligned}$$

The second-order partial derivatives of $W_2$ are

$$\begin{aligned} \frac{{{\partial ^2}{W_2}}}{{\partial {L^2}}} =&\frac{{{J_2}{R_e}^2}}{{64{a^{5/2}}{e^3}{\varepsilon ^2}{\eta ^3}\sqrt{\mu }}}\left\{ { - 2\sin f\cos 2g{{\sin }^2}i\left[ {24e\left( {5{e^2} + 2} \right) \cos f + \left( {31{e^4} - 80{e^2} + 112} \right) } \right. } \right. \\&\times \, \cos 2f + 4\left( { - 7{e^4} + 8{e^2} + 8} \right) + 3e\left( { - 5{e^3}\cos 6f - 44{e^2}\cos 5f + 4\left( {{e^2} - 32} \right) e\cos 4f} \right. \\&\left. \, {\left. { + 4\left( {{e^2} - 28} \right) \cos 3f} \right) } \right] + \sin 2g{\sin ^2}i\left[ {15{e^4}\cos 7f + 132{e^3}\cos 6f + 48\left( {3{e^3} + e} \right) } \right. \\&+\, 3\left( {128 - 9{e^2}} \right) {e^2}\cos 5f - 12\left( {11{e^2} + 32} \right) e\cos 2f + 48\left( {7 - 3{e^2}} \right) e\cos 4f \\&\left. { +\, 3\left( {55{e^4} - 48{e^2} + 16} \right) \cos f - \left( {25{e^4} + 304{e^2} + 112} \right) \cos 3f} \right] + 2\sin f(3\cos 2i + 1) \\&\left. { \times \,\left[ {5{e^4} + 3{e^3}(e\cos 4f + 8\cos 3f) - 8\left( {{e^2} - 8} \right) {e^2}\cos 2f - 24\left( {{e^2} - 2} \right) e\cos f + 8{e^2} - 24} \right] } \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial L\partial G}} =&\frac{{{J_2}{R_e}^2}}{{8{a^{5/2}}{e^3}\sqrt{\mu }{{\left( {\varepsilon - {e^2}\varepsilon } \right) }^2}}}\left\{ { - 9{e^3}\sin 2f\left( {6\cos 2i{{\sin }^2}(f + g) + 1} \right) + 2\left( {2{e^4} - 3{e^2} + 1} \right) {{\sin }^2}i} \right. \\&\times \, \left[ {3\sin (f + 2g) + \sin (3f + 2g)} \right] - \frac{3}{2}e\sin f\left( {{e^2}\cos 2f + {e^2} + 6e\cos f + 4} \right) \\&\times \, \left[ { - 2\cos 2i\sin (f + g)\left[ {e\sin f\left( {\cos g - 5\cos (2f + g)} \right) - 7\sin (2f + g) + \sin g} \right] } \right. \\&+ e\left( {\sin f\left( {\sin (f + 2g) - 5\sin (3f + 2g)} \right) - 1} \right) + e\cos 2f - \cos (f + 2g) \\&\left. { + \,7\cos (3f + 2g) + 2\cos f} \right] + 4{e^2}\cot i\left[ { - 3\sin f\sin 2i\left( {\left( {{e^2}\cos 2f + {e^2}} \right. } \right. } \right. \\&\left. {\left. { +\, 6e\cos f + 4} \right) {{\sin }^2}(f + g) - {e^2} + 1} \right) - \left( {{e^2} - 1} \right) \sin i\cos i\left( {3\sin (f + 2g)} \right. \\&\left. {\left. { +\, \sin (3f + 2g)} \right) } \right] - \sin f\left[ { - 4{e^4} + 54{e^3}\cos f\cos (2f + 2g) + 2\left( {2{e^2} + 1} \right) } \right. \\&\times {e^2}{\cos ^2}f\left( {6\cos 2i{{\sin }^2}(f + g) + 3\cos (2f + 2g) + 1} \right) + 12\left( {4{e^2} - 1} \right) \\&\left. {\left. { \times \, \cos (2f + 2g) + 22{e^2} + 6\cos 2i\left( { - 2{e^4} + 4\left( {4{e^2} - 1} \right) {{\sin }^2}(f + g) + 3{e^2} - 1} \right) - 6} \right] } \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial L\partial H}} =&\frac{{{J_2}{R_e}^2\cos i}}{{2{a^{5/2}}e{\varepsilon ^2}{\eta ^4}\sqrt{\mu }}}\left\{ {6\sin f\left[ {2\left( {{e^2}{{\cos }^2}f + 2} \right) {{\sin }^2}(f + g) + {\eta ^2}} \right] } \right. \\&\left. { +\, 18e\sin 2f{{\sin }^2}(f + g) - {\eta ^2}\left[ {3\sin (f + 2g) + \sin (3f + 2g)} \right] } \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial L\partial l}} =&\frac{{3{J_2}{R_e}^2{{(e\cos f + \mathrm{{1}})}^3}}}{{8{a^2}e{\varepsilon ^2}{\eta ^6}}}\left\{ { - 2\cos 2i\sin (f + g)\left[ {e\sin f\left( {\cos g - 5\cos (2f + g)} \right) } \right. } \right. \\&\left. { -\, 7\sin (2f + g) + \sin g} \right] + e\left[ {\sin f\left( {\sin (f + 2g) - 5\sin (3f + 2g)} \right) - 1} \right] \\&\left. { +\, e\cos 2f - \cos (f + 2g) + 7\cos (3f + 2g) + 2\cos f} \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial L\partial g}} =&\frac{{{J_2}{R_e}^2{{\sin }^2}i}}{{2{a^2}e{\varepsilon ^2}{\eta ^3}}}\left\{ { - 3\sin f\left( {{e^2}\cos 2f + {e^2} + 6e\cos f + 4} \right) \sin (2f + 2g)} \right. \\&\left. { +\, 3{\eta ^2}\cos (f + 2g) + {\eta ^2}\cos (3f + 2g)} \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial {G^2}}} =&\frac{{{J_2}{R_e}^2}}{{4{a^{5/2}}{e^3}{\varepsilon ^2}{\eta ^5}\sqrt{\mu }}}\left\{ {e\sin f\left[ {e\sin 2i\cot i\left( {6\left( {4{e^2} + 2(e\cos f + 1)(e\cos f + 2){{\sin }^2}(f + g) + 1} \right) } \right. } \right. } \right. \\&\left. { -\, \csc f\left( {3\left( {4{e^2} + 1} \right) \sin (f + 2g) + \left( {4{e^2} + 1} \right) \sin (3f + 2g) + 15e(\sin (2f + g) - 2f + 2l)} \right) } \right) \\&+\, \frac{3}{4}(e\cos f + 1)( - e\cos f - 2)\left[ {2\cos 2i\sin (f + g)\left[ {e\sin (f - g) + 2\left( {\sin g - 7\left( {e\sin (f + g)} \right. } \right. } \right. } \right. \\&\left. {\left. {\left. { + \sin (2f + g)} \right) } \right) - 5e\sin (3f + g)} \right] - e\left( { - 2\cos (2f + 2g) + 5\cos (4f + 2g)} \right. \\&\left. {\left. {\left. { +\, 2\cos 2f + \cos 2g} \right) + 2\left( { - 5e + \cos (f + 2g) - 7\cos (3f + 2g)} \right) - 4\cos f} \right] } \right] \\&+\, \frac{1}{4}\left[ {8{e^2}{{\cos }^2}i\left( { - 3\left( {3{e^2} + 1} \right) \sin (f + 2g) - \left( {3{e^2} + 1} \right) \sin (3f + 2g) + 6\left( {3{e^2} + 1} \right) \sin f} \right. } \right. \\&\left. { - \,12e(\sin (2f + 2g) - 2f + 2l)} \right) + (3\cos 2i + 1)\left( {{e^2}\left( {\left( {3{e^2} + 1} \right) \sin 3f + 48e(f} \right. } \right. \\&\left. {\left. { +\, \sin f\cos f - l)} \right) + 3\left( {9{e^4} + 23{e^2} - 4} \right) \sin f} \right) + 2\sin f\cos 2g{\sin ^2}i \\&\times \, \left( {57{e^4} + 216{e^3}\cos f + 72{e^3}\cos 3f + 3\left( {3{e^2} + 1} \right) {e^2}\cos 4f + 59{e^2}} \right. \\&\left. { +\, 14\left( {3{e^4} + 11{e^2} - 2} \right) \cos 2f - 8} \right) + \sin 2g{\sin ^2}i\left( {144{e^3}\cos 2f + 72{e^3}\cos 4f} \right. \\&\left. {\left. {\left. { -\, 72{e^3} + 3\left( {3{e^2} + 1} \right) {e^2}\cos 5f + 6\left( {9{e^4} - 7{e^2} + 2} \right) \cos f + \left( {33{e^4} + 151{e^2} - 28} \right) \cos 3f} \right) } \right] } \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial G\partial H}} =&-\, \frac{{{J_2}{R_e}^2\sin f\sin 2i\csc i}}{{4{a^{5/2}}e{\varepsilon ^2}{\eta ^5}\sqrt{\mu }}}\left\{ {6\left[ {2\left( {{e^2}{{\cos }^2}f + 3e\cos f + 2} \right) {{\sin }^2}(f + g) + 5{e^2} + {\eta ^2}} \right] } \right. \\&-\, \csc f\left[ {3\left( {5{e^2} + {\eta ^2}} \right) \sin (f + 2g) + 5{e^2}\sin (3f + 2g) + 15e\sin (2f + 2g)} \right. \\&\left. {\left. { -\, 30ef + 30el + {\eta ^2}\sin (3f + 2g)} \right] } \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial G\partial l}} =&\frac{{3{J_2}{R_e}^2}}{{4{a^2}e{\varepsilon ^2}{\eta ^7}}}\left\{ { - \left[ {\cos f{{(e\eta \cos f + \eta )}^2}\left( {6\cos 2i{{\sin }^2}(f + g) + 3\cos (2f + 2g) + 1} \right) } \right. } \right. \\&\left. { +\, 2e{{(e\cos f + 1)}^3}\left( {6\cos 2i{{\sin }^2}(f + g) + 3\cos (2f + 2g) + 1} \right) - e{\eta ^3}(3\cos 2i + 1)} \right] \\&+\, 2e\sin 2i\cot i\left[ {{\eta ^3} - 2{{(e\cos f + 1)}^3}{{\sin }^2}(f + g)} \right] + \sin f(e\cos f + 2){(e\cos f + 1)^2} \\&\left. { \times \, \left[ {4{{\sin }^2}i(e\cos f + 1)\sin (2f + 2g) + e\sin f\left( {6\cos 2i{{\sin }^2}(f + g) + 3\cos (2f + g) + 1} \right) } \right] } \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial G\partial g}} =&\frac{{{J_2}{R_e}^2}}{{4{a^2}{\varepsilon ^2}{\eta ^4}}}\left\{ {4{{\cos }^2}i\left[ {e\left( {3\cos (f + 2g) + \cos (3f + 2g)} \right) + 3\cos (2f + 2g)} \right] } \right. \\&-\, \frac{{2{{\sin }^2}i}}{e}\left[ {\left( {3{e^2} + {\eta ^2}} \right) \left( {3\cos (f + 2g) + \cos (3f + 2g)} \right) + 9e\cos (2f + 2g)} \right. \\&\left. {\left. { -\, 6\sin f(e\cos f + 1)(e\cos f + 2)\sin (2f + 2g)} \right] } \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial {H^2}}} =&- \frac{{{J_2}{R_e}^2}}{{2{a^{5/2}}{\varepsilon ^2}{\eta ^5}\sqrt{\mu }}}\left\{ {e\left[ {3\sin (f + 2g) + \sin (3f + 2g) - 6\sin f} \right] + 3\sin (2f + 2g) - 6f + 6l} \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial H\partial l}} =&- \frac{{3{J_2}{R_e}^2\cos i}}{{{a^2}{\varepsilon ^2}{\eta ^7}}}\left\{ {{\eta ^3} - 2{{(e\cos f + 1)}^3}{{\sin }^2}(f + g)} \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial H\partial g}} =&- \frac{{{J_2}{R_e}^2\cos i}}{{{a^2}{\varepsilon ^2}{\eta ^4}}}\left\{ {e\left[ {3\cos (f + 2g) + \cos (3f + 2g)} \right] + 3\cos (2f + 2g)} \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial {l^2}}} =&- \frac{{3{J_2}{R_e}^2\sqrt{\mu }{{(e\cos f + 1)}^4}}}{{4{a^{3/2}}{\varepsilon ^2}{\eta ^9}}}\left\{ {4{{\sin }^2}i(e\cos f + 1)\sin (2f + 2g)} \right. \\&\left. { +\, e\sin f\left[ {6\cos 2i{{\sin }^2}(f + g) + 3\cos (2f + 2g) + 1} \right] } \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial l\partial g}} =&- \frac{{3{J_2}{R_e}^2\sqrt{\mu }{{\sin }^2}i{{(e\cos f + 1)}^3}\sin (2f + 2g)}}{{{a^{3/2}}{\varepsilon ^2}{\eta ^6}}} \\ \frac{{{\partial ^2}{W_2}}}{{\partial {g^2}}} =&- \frac{{{J_2}{R_e}^2\sqrt{\mu }{{\sin }^2}i}}{{{a^{3/2}}{\varepsilon ^2}{\eta ^3}}}\left\{ {e\left[ {3\sin (f + 2g) + \sin (3f + 2g)} \right] + 3\sin (2f + 2g)} \right\} \\ \frac{{{\partial ^2}{W_2}}}{{\partial L\partial h}} =&\frac{{{\partial ^2}{W_2}}}{{\partial G\partial h}} = \frac{{{\partial ^2}{W_2}}}{{\partial H\partial h}} = \frac{{{\partial ^2}{W_2}}}{{\partial l\partial h}} = \frac{{{\partial ^2}{W_2}}}{{\partial g\partial h}} = 0\\ \frac{{{\partial ^2}{W_2}}}{{\partial h\partial L}} =&\frac{{{\partial ^2}{W_2}}}{{\partial h\partial G}} = \frac{{{\partial ^2}{W_2}}}{{\partial h\partial H}} = \frac{{{\partial ^2}{W_2}}}{{\partial h\partial l}} = \frac{{{\partial ^2}{W_2}}}{{\partial h\partial g}} = \frac{{{\partial ^2}{W_2}}}{{\partial {h^2}}} = 0 \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \frac{{{\partial ^2}{W_2}}}{{\partial G\partial L}} =&\frac{{{\partial ^2}{W_2}}}{{\partial L\partial G}} \;,\; \frac{{{\partial ^2}{W_2}}}{{\partial H\partial L}} = \frac{{{\partial ^2}{W_2}}}{{\partial L\partial H}} \;,\; \frac{{{\partial ^2}{W_2}}}{{\partial H\partial G}} = \frac{{{\partial ^2}{W_2}}}{{\partial G\partial H}} \\ \frac{{{\partial ^2}{W_2}}}{{\partial l\partial L}} =&\frac{{{\partial ^2}{W_2}}}{{\partial L\partial l}} \;,\; \frac{{{\partial ^2}{W_2}}}{{\partial l\partial G}} = \frac{{{\partial ^2}{W_2}}}{{\partial G\partial l}} \;,\; \frac{{{\partial ^2}{W_2}}}{{\partial l\partial H}} = \frac{{{\partial ^2}{W_2}}}{{\partial H\partial l}} \\ \frac{{{\partial ^2}{W_2}}}{{\partial g\partial L}} =&\frac{{{\partial ^2}{W_2}}}{{\partial L\partial g}} \;,\; \frac{{{\partial ^2}{W_2}}}{{\partial g\partial G}} = \frac{{{\partial ^2}{W_2}}}{{\partial G\partial g}} \;,\; \frac{{{\partial ^2}{W_2}}}{{\partial g\partial H}} = \frac{{{\partial ^2}{W_2}}}{{\partial H\partial g}} \\ \frac{{{\partial ^2}{W_2}}}{{\partial g\partial l}} =&\frac{{{\partial ^2}{W_2}}}{{\partial l\partial g}}. \\ \end{aligned} \end{aligned}$$

1.2 Partial derivatives of $W_4$

The partial derivatives of $W_4$ are given by

$$\begin{aligned} \frac{{\partial {W_4}}}{{\partial L}} =&\frac{{3J_2^2{R_e}^4}}{{10240{a^4}{e^3}{\varepsilon ^4}{\eta ^7}}}\left\{ {60{e^6}\sin f{{\sin }^2}i\left[ { - 15{e^2}\cos 2g\cos 2i(e\cos f + 2) - 8\sin f\cos f} \right. } \right. \\&\left. { \times (5\cos 2i + 3)(5e\cos f + e\cos 3f - 12)\sin (2f + 2g)} \right] + 40e\eta (3\cos 2i + 1)\left\{ {\sin 2g{{\sin }^2}i} \right. \\&\times \left[ {24\left( {3{e^2} + 16\cos 2f - 8\cos 4f - 8} \right) + e\left( { - 15{e^2}\cos 7f + 3\left( {41{e^2} - 64} \right) \cos f} \right. } \right. \\&\left. {\left. { + \left( {41{e^2} + 400} \right) \cos 3f - 21\left( {{e^2} + 16} \right) \cos 5f + 132e\cos 2f - 132e\cos 6f} \right) } \right] \\&- 2e\sin f(3\cos 2i + 1)\left[ {8\left( {{e^2} + 6} \right) \cos 2f + 24e\cos f + e(3e\cos 4f - 11e + 24\cos 3f)} \right] \\&- 2\sin f\cos 2g{\sin ^2}i\left[ { - 52{e^3} + \left( {{e^2} - 64} \right) e\cos 2f + 24\left( {{e^2} - 8} \right) \cos f + 16e} \right. \\&\left. {\left. { + 3\left( {\left( {44{e^2} + 64} \right) \cos 3f + e\left( {4\left( {3{e^2} + 28} \right) \cos 4f + e(5e\cos 6f + 44\cos 5f)} \right) } \right) } \right] } \right\} \\&- 60{e^3}{\eta ^2}(f - l)\left\{ {8\left[ {2\left( {45{e^4}\eta + {e^2}\left( { - 60{\eta ^3} + 27\eta + 6} \right) - 18{\eta ^3} + 70} \right) \cos 2g{{\sin }^2}i} \right. } \right. \\&\left. { + 5\cos 4i - 17} \right] + \cos 2i\left[ {5\left( {315{e^6}\eta + 6{e^4}\left( { - 105{\eta ^3} + 40\eta + 3} \right) - 96\left( {{\eta ^3} - 3} \right) } \right. } \right. \\&\left. {\left. {\left. { + 16{e^2}\left( { - 20{\eta ^3} + 9\eta + 2} \right) } \right) \cos 2g{{\sin }^2}i - 416} \right] } \right\} + 960e\sin f\left[ {4{{\sin }^4}i\left( {3\cos (3f + 4g)} \right. } \right. \\&\left. { + 7\cos (5f + 4g)} \right) + 16{\sin ^2}i(3\cos 2i + 1)\left( {\cos (f + 2g) + 2\cos (3f + 2g)} \right) \\&\left. { + \cos f\left( { - 28\cos 2i + 31\cos 4i + 61} \right) } \right] + 128{\eta ^4}{\sin ^2}i\left[ {3{{\sin }^2}i\left( {5\sin (3f + 4g)} \right. } \right. \\&\left. {\left. { + 7\sin (5f + 4g) + 130\sin f} \right) + 75(3\cos 2i + 1)\sin (f + 2g) + 5(3\cos 2i + 1)\sin (3f + 2g)} \right] \\&+ 32{\eta ^5}\left\{ {{e^2}{{\sin }^2}i(5\cos 2i + 3)\left[ {5{e^3}\left( {27\sin (2f - 2g) + 18\sin (2f + 2g) - 18\sin (4f + 2g)} \right. } \right. } \right. \\&\left. { - 7\sin (6f + 2g)} \right) + 18{e^2}\left( {8{{\cos }^3}f(3\cos 2f - 2)\sin 2g - 8{{\sin }^3}f(3\cos 2f + 7)\cos 2g} \right) \\&\left. { - 15e\left( {4\sin (2f + 2g) + 5\sin (4f + 2g)} \right) + 80\sin (3f + 2g)} \right] + 60(3\cos 2i + 1) \\&\left. { \times \left( {4{{\cos }^3}f\sin 2g{{\sin }^2}i + (3\sin f + \sin 3f)\cos 2g{{\sin }^2}i + \sin f(3\cos 2i + 1)} \right) } \right\} \\&+ 240{e^4}\sin f\cos f\left\{ {\cos f\left( { - 224{{\sin }^4}i\cos (4f + 4g) + 788\cos 2i - 89\cos 4i - 91} \right) } \right. \\&+ 2\left[ {{{\sin }^2}i\left( {693\cos 2i\cos (3f + 2g) + 15\cos 2i\cos (5f + 2g) + (3\cos 2i + 1)} \right. } \right. \\&\times \cos (f - 2g) - 3(317\cos 2i + 207)\cos (f + 2g) - 48{\sin ^2}i\cos (5f + 4g) \\&\left. {\left. {\left. { + 375\cos (3f + 2g) + 5\cos (5f + 2g)} \right) + \cos 3f{{(3\cos 2i + 1)}^2}} \right] } \right\} - 120{e^5} \end{aligned}$$

$$\begin{aligned}&\times \left\{ { - 16\sin f{{\sin }^2}i\left[ { - 2\cos f(7\cos 2i + 5)\cos (2f + 2g) - 14{{\cos }^3}f{{\sin }^2}i\cos (4f + 4g)} \right. } \right. \\&\left. { + (9\sin f + \sin 3f)(5\cos 2i + 3)\sin (2f + 2g)} \right] + \sin 2f\left[ { - 92{{\sin }^2}i\cos (4f + 2g)} \right. \\&+ 172\cos 2g{\sin ^2}i + 21\cos 4i + 31 + 4\cos 2i\left( { - 41{{\sin }^2}i\cos (4f + 2g) + 69\cos 2g} \right. \\&\left. {\left. {\left. { \times {{\sin }^2}i - 29} \right) } \right] + \frac{1}{2}\sin 4f\left[ { - 116\cos 2i + 21\cos 4i + 31} \right] } \right\} + 240{e^3}\left\{ {\sin f} \right. \\&\times \left[ {11(4\cos 2i - 3\cos 4i)\cos (5f + 2g) + (372\cos 2i - 383\cos 4i)\cos (3f + 2g)} \right. \\&+ 22\sin f\sin (4f + 2g) + 4{\sin ^2}i\left( {5(3\cos 2i + 1)\cos (f - 2g) - 3(129\cos 2i + 91)} \right. \\&\left. { \times \cos (f + 2g)} \right) + 8{\sin ^4}i\left( {3\cos (f + 4g) + 3\cos (3f + 4g) - 23\cos (5f + 4g)} \right. \\&\left. { + \cos (7f + 4g)} \right) + 2\cos f\left( {16{{\sin }^4}i\cos (4f + 4g) + 8{{\sin }^2}i(15\cos 2i + 1)\cos (2f + 2g)} \right. \\&\left. {\left. {\left. { - 5\cos 4i + 101} \right) } \right] + \sin 2f\left( {344\cos 2i + 19\cos 4i - 11} \right) + \sin 4f\left( {4\cos 2i + 11\cos 4i + 17} \right) } \right\} \\&+ {\eta ^3}\left\{ {15750{e^9}\sin 2f\cos 2g{{\sin }^2}i\cos 2i + 63000{e^8}\sin f\cos 2g{{\sin }^2}i\cos 2i} \right. \\&+ 320{e^2}\left( {(3\cos 2i + 1)\left( {{{\sin }^2}i\left( {8(13 - 6\cos 2f){{\cos }^3}f\sin 2g + 18\sin 3f\cos 2g} \right) } \right. } \right. \\&- \sin f\left( {6{{\cos }^2}f\left( {3\cos 2g{{\sin }^2}i + 3\cos 2i + 1} \right) + 18\cos 3f\cos f\cos 2g{{\sin }^2}i} \right. \\&+ 3\cos 4f\cos 2g{\sin ^2}i + \cos 2f\left( {14\cos 2g{{\sin }^2}i + 3\cos 2i + 1} \right) - 35\cos 2g{\sin ^2}i \\&\left. {\left. {\left. { - 39\cos 2i - 13} \right) } \right) - 72\sin f{{\sin }^2}i(5\cos 2i + 3)\cos (2f + 2g)} \right) + 4\left( {3{e^3}{{\sin }^2}i(5\cos 2i + 3)} \right. \\&\times \left[ {5{e^4}\left( { - 34\sin (2f - 2g) - 4\sin (2f + 2g) + 49\sin (4f + 2g) + 2\sin (6f + 2g)} \right. } \right. \\&\left. { - 7\sin (8f + 2g) + 9\sin (4f - 2g) + 3\sin 2g} \right) + 8{e^3}\left( {110\sin (f + 2g) - 23\sin (5f + 2g)} \right. \\&\left. { - 10\sin (7f + 2g) + 110\sin (f - 2g) + 15\sin (3f - 2g)} \right) + 20{e^2}\left( { - 9\sin (2f - 2g)} \right. \\&\left. { - 4\sin (2f + 2g) + 13\sin (4f + 2g) + 7\sin (6f + 2g) + 2\sin 2g} \right) + 80e\left( {3\sin (f + 2g)} \right. \\&\left. { - 5\sin (3f + 2g) - 3\sin (5f + 2g) + 3\sin (f - 2g)} \right) + 80\left( {5\left( {2\sin (2f + 2g)} \right. } \right. \\&\left. {\left. {\left. { + \sin (4f + 2g)} \right) + 3\sin 2g} \right) } \right] + 320(3\cos 2i + 1)\left[ {2{{\sin }^2}i\left( {\sin (3f + 2g) - 3\sin (f + 2g)} \right) } \right. \\&- 3e\sin f\left( { - 12\sin f{{\cos }^2}f\sin 2g{{\sin }^2}i + 3(\cos f + \cos 3f)\cos 2g{{\sin }^2}i} \right. \\&\left. {\left. {\left. {\left. { + \cos f(3\cos 2i + 1)} \right) } \right] } \right) } \right\} + 480{e^2}\left\{ {\sin f\left[ {4\left( {2\left( {4{{\sin }^4}i\cos (4f + 4g) + 6{{\sin }^4}i\cos (2f + 4g)} \right. } \right. } \right. } \right. \end{aligned}$$

$$\begin{aligned}&+ 2{\sin ^4}i\cos (6f + 4g) + 2{\sin ^2}i\cos (2f + 2g) + 6{\sin ^2}i\cos (4f + 2g) + 2\cos 2i \\&\times \left( {15{{\sin }^2}i\cos (2f + 2g) + 9{{\sin }^2}i\cos (4f + 2g) + 6\cos 2g{{\sin }^2}i + 4} \right) + 4\cos 2g{\sin ^2}i \\&\left. {\left. { + 7\cos 4i + 17} \right) + \cos 2f( - 4\cos 2i + 13\cos 4i + 23)} \right) + {\cos ^2}f( - 28\cos 2i + 31\cos 4i \\&\left. { + 61)} \right] + 2\sin 2f{\sin ^2}i\left[ {4(3\cos 2i + 1)\cos (f + 2g) + 8(3\cos 2i + 1)\cos (3f + 2g)} \right. \\&\left. {\left. { + {{\sin }^2}i\left( {3\cos (3f + 4g) + 7\cos (5f + 4g)} \right) } \right] } \right\} + 8{\eta ^2}\left\{ { - 30{e^3}\left[ {2{{\sin }^2}i\left( {2{{\sin }^2}i\left( { - 3\sin 4g} \right. } \right. } \right. } \right. \\&\left. { + \sin (4f + 4g) + \sin (8f + 4g) - 9\sin (2f + 4g) + 5\sin (6f + 4g) + 8\sin 4f} \right) \\&+ 16\sin f\cos f(3\cos 2i + 1)\cos (2f + 2g) + 8\cos 2f\sin 2g(\cos 2i + 3) \\&\left. {\left. { - (41\cos 2i + 23)\sin (4f + 2g)} \right) + \sin 2f\left( {32\cos 2g{{\sin }^2}2i - 84\cos 2i + 13\cos 4i + 7} \right) } \right] \\&+ {e^2}\left[ {12{{\sin }^4}i\left( {135\sin (f + 4g) + 115\sin (3f + 4g) - 127\sin (5f + 4g) - 75\sin (7f + 4g)} \right) } \right. \\&+ 160{\sin ^2}i(25\cos 2i + 17)\sin (3f + 2g) - 6480\cos f\sin 2g{\sin ^2}2i + 15\sin f \\&\times \left( { - 64\cos 2g{{\sin }^2}i(21\cos 2i + 16) + 412\cos 2i + 49\cos 4i + 243} \right) - 5\sin 3f \\&\left. { \times ( - 564\cos 2i + 117\cos 4i + 383)} \right] + 2e{\sin ^2}i\left[ {5{e^4}(5\cos 2i + 3)\left( { - 3\sin (2f - 2g)} \right. } \right. \\&\left. { + 6\sin (4f + 2g) + \sin (6f + 2g)} \right) - 18{e^3}(5\cos 2i + 3)\left( {\sin (5f + 2g) - 5\left( {9\sin (f + 2g)} \right. } \right. \\&\left. {\left. { - 3\sin (3f + 2g) + \sin (f - 2g)} \right) } \right) + 240\sin f\left( { - {{\sin }^2}i\left( {3\cos (3f + 4g) + 7\cos (5f + 4g)} \right. } \right. \\&\left. {\left. {\left. { + 26\cos f} \right) - (3\cos 2i + 1)\left( {5\cos (f + 2g) + \cos (3f + 2g)} \right) } \right) } \right] + 4\left[ { - 12{{\sin }^4}i} \right. \\&\times \left( {5\sin (3f + 4g) + 7\sin (5f + 4g)} \right) - 80{\sin ^2}i(3\cos 2i + 1)\left( {3\sin (f + 2g)} \right. \\&\left. {\left. {\left. {\left. { + 2\sin (3f + 2g)} \right) - 15\sin f( - 28\cos 2i + 31\cos 4i + 61)} \right] } \right\} } \right. + 480{e^3}\left\{ {\frac{1}{2}\sin f} \right. \\&\times \left[ {11(4\cos 2i - 3\cos 4i)\cos (5f + 2g) + (372\cos 2i - 383\cos 4i)\cos (3f + 2g) + 4{{\sin }^2}i} \right. \\&\left. { \times \left( {5(3\cos 2i + 1)\cos (f - 2g) - 3(129\cos 2i + 91)\cos (f + 2g)} \right) + 22\sin f\sin (4f + 2g)} \right] \\&+ \sin f\left[ {4{{\sin }^4}i(3\cos (f + 4g) + 3\cos (3f + 4g) - 23\cos (5f + 4g) + \cos (7f + 4g))} \right. \\&\left. { + \cos f\left( {16{{\sin }^4}i\cos (4f + 4g) + 8{{\sin }^2}i(15\cos 2i + 1)\cos (2f + 2g) - 5\cos 4i + 101} \right) } \right] \\&\left. {\left. { + \frac{1}{2}\sin 2f(344\cos 2i + 19\cos 4i - 11) + \sin 4f(4\cos 2i + 11\cos 4i + 17)} \right\} } \right\} \\ \frac{{\partial {W_4}}}{{\partial G}} =&\frac{{3J_2^2{R_e}^4}}{{10240{a^4}{e^3}{\varepsilon ^4}{\eta ^8}}}\left\{ { - 128{\eta ^4}{{\sin }^2}i\left\{ {3{{\sin }^2}i\left[ {5\sin (3f + 4g) + 7\sin (5f + 4g) + 130\sin f} \right] } \right. } \right. \end{aligned}$$

$$\begin{aligned}&\left. { + 5(3\cos 2i + 1)\left[ {15\sin (f + 2g) + \sin (3f + 2g)} \right] } \right\} - 15{e^3}(f - l)\left\{ {16\left[ {4\cos 2i\left( {219{e^2}} \right. } \right. } \right. \\&\left. { + \left( {21{e^2} - 34{\eta ^2} + 24} \right) \cos 2g + 138{\eta ^2} + 256} \right) + \cos 4i\left( {3\left( {253{e^2} + 118{\eta ^2} - 88} \right) \cos 2g} \right. \\&\left. {\left. { - 231{e^2} - 186{\eta ^2} + 616} \right) - 69{e^2} - 238{\eta ^2} + 1304} \right] + \cos 2g\left[ {8{\eta ^2}{{\sin }^2}i\left( {48\left( {10{e^2} + 3} \right) {\eta ^3}} \right. } \right. \\&\left. { + 5\left( {{e^2}\left( {5\left( {63{e^2} + 32} \right) {\eta ^3} - 9{e^2} - 16} \right) + 48{\eta ^3}} \right) \cos 2i - 48{e^2}} \right) + {e^2}\left( {105{e^4} + 40{e^2} + 848} \right. \\&\left. { - 4\left( {525{e^4} + 160{e^2} + 96} \right) {\eta ^3}} \right) + {e^2}\left( {2\left( {75{e^4}\left( {14{\eta ^3} - 1} \right) + {e^2}\left( {16 - 640{\eta ^3}} \right) - 384{\eta ^3}} \right) } \right. \\&\left. {\left. {\left. { \times \cos 2i - 5\left( {8\left( {105{e^4} + 80{e^2} + 48} \right) {\eta ^3} - 11{e^2}\left( {3{e^2} + 8} \right) } \right) \cos 4i} \right) + 32\left( {19{\eta ^2} - 44} \right) } \right] } \right\} \\&+ 32{\eta ^5}\left\{ {{e^2}{{\sin }^2}i(5\cos 2i + 3)\left[ {e\left( {e\left( {5e\left( { - 27\sin (2f - 2g) + 7\sin (6f + 2g) + 36\sin f} \right. } \right. } \right. } \right. } \right. \\&\left. {\left. {\left. { \times \cos (3f + 2g)} \right) - 18(5\sin (3f + 2g) + 3\sin (5f + 2g)} \right) + 540\sin f\cos 2g} \right) \\&\left. { + 60\sin (2f + 2g) + 75\sin (4f + 2g)) - 80\sin (3f + 2g)} \right] - 60(3\cos 2i + 1) \\&\left. { \times \left[ {4{{\cos }^3}f\sin 2g{{\sin }^2}i + (3\sin f + \sin 3f)\cos 2g{{\sin }^2}i + \sin f(3\cos 2i + 1)} \right] } \right\} \\&+ 4{\eta ^3}\left\{ { - 7875{e^8}\sin f\cos 2g{{\sin }^2}i\cos 2i(e\cos f + 2) + 24{e^6}\left[ {{{\sin }^2}i(5\cos 2i + 3)} \right. } \right. \\&\times \left( {8( - 49\cos 2f + 10\cos 4f + 31){{\cos }^3}f\sin 2g + (200\sin f - 85\sin 3f - 19\sin 5f} \right. \\&\left. { + 10\sin 7f)\cos 2g} \right) + 8(5\cos 2i - 1){\cos ^2}i\left( {8{{\cos }^3}f(3\cos 2f - 2)\sin 2g - 8{{\sin }^3}f} \right. \\&\left. {\left. { \times (3\cos 2f + 7)\cos 2g} \right) } \right] + 40{e^2}\left[ {4\cos 2i\left( { - 15\sin (f + 2g) + 23\sin (3f + 2g)} \right. } \right. \\&\left. { + 3\sin (5f + 2g) + 3\sin (f - 2g)} \right) - 3\cos 4i\left( {45\sin (f + 2g) + 59\sin (3f + 2g)} \right. \\&\left. { + 3\sin (5f + 2g) + 3\sin (f - 2g)} \right) - 93\sin (f + 2g) - 11\sin (3f + 2g) - 3\sin (5f + 2g) \\&\left. { - 3\sin (f - 2g) + 4\sin 3f{{(3\cos 2i + 1)}^2} + 6\sin f(156\cos 2i + 81\cos 4i + 115)} \right] \\&+ 5{e^3}\left[ {4\cos i(3\cos i + 5\cos 3i)\left( {27{e^4}\sin (2f - 2g) + 6\left( {3{e^4} - 4{e^2} - 24} \right) \sin (2f + 2g)} \right. } \right. \\&\left. { - e\left( {7{e^3}\sin (6f + 2g) + 6\left( {3{e^2} + 5} \right) e\sin (4f + 2g) - 64\sin (3f + 2g)} \right) } \right) - {\sin ^2}i \\&\times (5\cos 2i + 3)\left( { - 2e\sin f\cos 2g\left( {2e\cos f\left( {21{e^2}\cos 6f + 33\left( {3{e^2} + 8} \right) \cos 2f - 212{e^2}} \right. } \right. } \right. \end{aligned}$$

$$\begin{aligned}&\left. {\left. { + \left( {92{e^2} - 84} \right) \cos 4f + 204} \right) + 16( - 32\cos 2f + 9\cos 4f - 25)} \right) - e\sin 2g \\&\times \left( {21{e^3}\cos 8f + 12\left( {3{e^2} + 23} \right) e\cos 2f + 132\left( {{e^2} + 2} \right) e\cos 4f + 4\left( {23{e^2} - 21} \right) } \right. \\&\left. { \times e\cos 6f - 3\left( {3{e^2} + 8} \right) e - 656\cos 3f + 144\cos 5f} \right) + 48( - 32\sin (2f + 2g) \\&\left. {\left. { + 5\sin (4f + 2g) + 3\sin 2g)} \right) } \right] + 320\left( {3\cos 2i + 1} \right) \left[ {3e\sin f\left( {\cos f(3\cos 2i + 1)} \right. } \right. \\&\left. { - 12\sin f{{\cos }^2}f\sin 2g{{\sin }^2}i + 3(\cos f + \cos 3f)\cos 2g{{\sin }^2}i} \right) - 2{\sin ^2}i \\&\left. {\left. { \times \left( {\sin (3f + 2g) - 3\sin (f + 2g)} \right) } \right] } \right\} + 40e\eta \left\{ {4{e^3}\left[ {( - 3\cos 2i - 1)\left( {\frac{3}{2}\sin f\left( { - (131\cos 2f} \right. } \right. } \right. } \right. \\&\left. { + 5(8\cos 4f + \cos 6f + 32))\cos 2g{{\sin }^2}i - (12\cos 2f + \cos 4f + 43)(3\cos 2i + 1)} \right) \\&\left. { - 6{{\cos }^3}f(20\cos 2f + 5\cos 4f + 7)\sin 2g{{\sin }^2}i} \right) + 4{\cos ^2}i\left( {2\sin f(6(\cos 2f + 5)} \right. \\&\times (3\cos 2i + 1) - (14\cos 2f + 3\cos 4f + 19)\cos 2g(3\cos 2i - 1)) - 8{\cos ^3}f(3\cos 2f + 1) \\&\left. {\left. { \times \sin 2g(3\cos 2i - 1)} \right) } \right] + 3{e^2}\left[ {4(3\cos 2i + 1)\left( {{{\sin }^2}i\left( {\sin (2f - 2g) - 10\sin (2f + 2g)} \right. } \right. } \right. \\&\left. { + 42\sin (4f + 2g) + 11\sin (6f + 2g) - 6\sin 2g} \right) + 6(7\sin 2f + \sin 4f)\cos 2i + 2\sin 4f \\&\left. {\left. { + 28\sin f\cos f} \right) + 24\sin 2i\cot i\left( {2(1 - 3\cos 2i)\sin (4f + 2g) + 4\sin 2f(3\cos 2i + 1)} \right) } \right] \\&+ 4e\left[ { - 64(3\cos 2i - 1){{\cos }^2}i\left( {\sin (3f + 2g) - 3\sin (f + 2g)} \right) - 6(3\cos 2i + 1)\left( { - 4\sin f(8\cos 2f} \right. } \right. \\&+ 7\cos 4f - 9)\cos 2g{\sin ^2}i + 4\cos f(6\cos 2f - 7\cos 4f + 9)\sin 2g{\sin ^2}i + 2(\sin f \\&\left. {\left. {\left. { - \sin 3f)(3\cos 2i + 1)} \right) } \right] - 768{{\sin }^2}f{{\sin }^2}i(3\cos 2i + 1)\sin (2f + 2g)} \right\} \end{aligned}$$

$$\begin{aligned}&+ 16{\eta ^2}\left\{ {{e^4}{{\sin }^2}i(5\cos 2i + 3)\left[ { - 5e\left( { - 3\sin (2f - 2g) + 6\sin (4f + 2g) + \sin (6f + 2g)} \right) } \right. } \right. \\&\left. { + 18\left( {\sin (5f + 2g) - 5\left( {9\sin (f + 2g) - 3\sin (3f + 2g) + \sin (f - 2g)} \right) } \right) } \right] + {e^2}\left[ {2\left( {3{{\sin }^4}i} \right. } \right. \\&\times \left( { - 135\sin (f + 4g) + 25\sin (3f + 4g) + 323\sin (5f + 4g) + 75\sin (7f + 4g)} \right) \\&- 12{\sin ^2}2i\left( {5\sin (3f + 4g) + 7\sin (5f + 4g)} \right) - 80{\cos ^2}i(3\cos 2i - 1)\left( {15\sin (f + 2g)} \right. \\&\left. { + \sin (3f + 2g)} \right) + 20{\sin ^2}i\left( {6(87\cos 2i + 47)\sin (f + 2g) - (29\cos 2i + 27)\sin (3f + 2g)} \right. \\&\left. {\left. { + 15(3\cos 2i + 1)\sin (f - 2g)} \right) } \right) + 5\sin f\left( {\cos 2f( - 564\cos 2i + 117\cos 4i + 383)} \right. \\&\left. {\left. { - 3084\cos 2i + 843\cos 4i + 1153} \right) } \right] - 240e\sin f{\sin ^2}i\left[ { - 5(3\cos 2i + 1)\cos (f + 2g)} \right. \\&\left. { - (3\cos 2i + 1)\cos (3f + 2g) - \left( {3\cos (3f + 4g) + 7\cos (5f + 4g) + 26\cos f} \right) {{\sin }^2}i} \right] \\&- 2\left[ { - 12{{\sin }^4}i\left( {5\sin (3f + 4g) + 7\sin (5f + 4g)} \right) - 80{{\sin }^2}i(3\cos 2i + 1)\left( {3\sin (f + 2g)} \right. } \right. \\&\left. {\left. { + 2\sin (3f + 2g)} \right) - 15\sin f( - 28\cos 2i + 31\cos 4i + 61)} \right] + 5{e^3}\left[ {2\left( {2{{\sin }^4}i\left( {81\sin (4f + 4g)} \right. } \right. } \right. \\&\left. { + 99\sin (2f + 4g) + 29\sin (6f + 4g) + 24\sin 4f} \right) + 4{\sin ^2}2i\left( {18\sin (2f + 2g)} \right. \\&\left. { - 6\sin (4f + 4g) - 9\sin (2f + 4g) - \sin (6f + 4g)} \right) - 48{\cos ^2}i(3\cos 2i + 1)\sin (2f + 2g) \\&- 3{\sin ^2}i\left( { - 16(11\cos 2i + 5)\sin (2f + 2g) + (29\cos 2i + 23)\sin (4f + 2g) + 12\sin 2g} \right. \\&\left. {\left. { \times \cos 2i} \right) } \right) + 3\sin 2f\left( {8{{\sin }^4}i\left( {3\cos (2f + 4g) + \cos (6f + 4g)} \right) + 32\cos f{{\sin }^2}i} \right. \\&\left. {\left. {\left. { \times \cos (f + 2g) + 4\cos 2i\left( {12\cos 2g{{\sin }^2}i - 77} \right) + 101\cos 4i + 143} \right) } \right] } \right\} - 960e\sin f \\&\times \left\{ {4{{\sin }^4}i(3\cos (3f + 4g) + 7\cos (5f + 4g)) + 16{{\sin }^2}i(3\cos 2i + 1)(\cos (f + 2g)} \right. \\&\left. { + 2\cos (3f + 2g)) + \cos f( - 28\cos 2i + 31\cos 4i + 61)} \right\} + 32{e^2}\left\{ { - 3{{\sin }^4}i\left[ { - 15\sin (f + 4g)} \right. } \right. \end{aligned}$$

$$\begin{aligned}&\left. { + 105\sin (3f + 4g) + 211\sin (5f + 4g) + 35\sin (7f + 4g)} \right] - 15\sin f\left[ {96\cos f{{\sin }^2}i\cos 2i} \right. \\&\times (2\cos (f + 2g) + 3\cos (3f + 2g)) + 16{\sin ^2}i\left( {{{\sin }^2}i(2\cos (4f + 4g) + 3\cos (2f + 4g)} \right. \\&\left. { + \cos (6f + 4g)) + \cos (2f + 2g) + 3\cos (4f + 2g) + 2\cos 2g} \right) + {\cos ^2}f( - 28\cos 2i + 31\cos 4i \\&\left. { + 61) + 4\cos 2f( - 4\cos 2i + 13\cos 4i + 23) - 132\cos 2i + 273\cos 4i + 563} \right] - 20{\sin ^2}i(3\cos 2i + 1) \\&\times \left[ {78\sin (f + 2g) + 59\sin (3f + 2g) + 6\sin (5f + 2g) + 3\sin (f - 2g)} \right] - 4\cot i\left[ {12{{\sin }^3}i\cos i} \right. \\&\times \left( {60\sin (f + 2g) + 40\sin (3f + 2g) - 5\sin (3f + 4g) - 7\sin (5f + 4g)} \right) - 5\left( {2(2\sin 2i} \right. \\&\left. {\left. {\left. { + 3\sin 4i)\left( {3\sin (f + 2g) + 2\sin (3f + 2g)} \right) + \sin f(42\sin 2i - 93\sin 4i)} \right) } \right] } \right\} \\&+ {e^3}\left\{ { - 160\left[ {2{{\sin }^4}i\left( {3\left( {40\sin (4f + 4g) + \sin (8f + 4g) + 40\sin (2f + 4g)} \right) - 52\sin (6f + 4g)} \right. } \right. } \right. \\&\left. { - 9\sin 4g} \right) + 3{\sin ^2}i\left( {(33\sin 6f - 335\sin 2f)\cos 2g\cos 2i + \sin 2g\left( {\left( { - 365\cos 2f + 506\cos 4f} \right. } \right. } \right. \\&\left. {\left. { + 33\cos 6f + 372} \right) \cos 2i + 276} \right) + 5\sin (2f - 2g) - 442\sin (2f + 2g) + 230\sin (4f + 2g) \end{aligned}$$

$$\begin{aligned}&\left. { + 11\sin (6f + 2g)} \right) + 3\sin 4f\left( {\cos 2i\left( {506\cos 2g{{\sin }^2}i + 4} \right) + 11\cos 4i + 17} \right) + 6\sin 2f \\&\left. { \times (72\cos 2i + 49\cos 4i + 103)} \right] - 160\cot i\left[ {4\sin i\cos i\left( { - 4{{\sin }^2}i\left( {3\sin (4f + 4g)} \right. } \right. } \right. \\&\left. { + 9\sin (2f + 4g) + \sin (6f + 4g)} \right) + (84 - 180\cos 2i)\sin (2f + 2g) + (18 - 54\cos 2i) \\&\left. {\left. {\left. { \times \sin (4f + 2g)} \right) + 12\sin 2f(13\sin 4i - 2\sin 2i)} \right] } \right\} + 16{e^4}\left\{ { - 7\left[ { - 288{{\sin }^4}i\sin (5f + 4g)} \right. } \right. \\&+ 10{\sin ^2}i\left( { - 3(399\cos 2i + 277)\sin (f + 2g) + (365\cos 2i + 191)\sin (3f + 2g)} \right. \\&\left. { + 3(3\cos 2i + 1)\sin (5f + 2g) + (9\cos 2i + 3)\sin (f - 2g)} \right) + 15\sin f(324\cos 2i \\&\left. { - 5\cos 4i + 33) + 10\sin 3f{{(3\cos 2i + 1)}^2}} \right] + 15\sin f\cos f\left[ {96{{\sin }^4}i\cos (5f + 4g)} \right. \\&+ \cos f\left( {224{{\sin }^4}i\cos (4f + 4g) - 788\cos 2i + 89\cos 4i + 91} \right) - 2{\sin ^2}i\left( {3\cos 2i} \right. \\&\times \left( { - 317\cos (f + 2g) + 231\cos (3f + 2g) + 5\cos (5f + 2g) + \cos (f - 2g)} \right) - 621\cos (f + 2g) \\&\left. {\left. { + 375\cos (3f + 2g) + 5\cos (5f + 2g) + \cos (f - 2g)} \right) - 2\cos 3f{{(3\cos 2i + 1)}^2}} \right] \\&+ 4\cot i\left[ {\sin i\cos i\left( { - 288{{\sin }^2}i\sin (5f + 4g) + 30(3\cos 2i - 1)\sin (f - 2g) + 10} \right. } \right. \\&\times \left( {3(61 - 399\cos 2i)\sin (f + 2g) + (365\cos 2i - 87)\sin (3f + 2g) + 3(3\cos 2i - 1)} \right. \\&\left. {\left. {\left. {\left. { \times \sin (5f + 2g)} \right) } \right) + 75\sin f\sin 4i - 30\sin 2i\left( {\sin 3f(3\cos 2i + 1) + 81\sin f} \right) } \right] } \right\} \\&+ 80{e^5}\left\{ {336\sin f{{\cos }^3}f{{\sin }^4}i\cos (4f + 4g) + {{\sin }^2}i\left[ {98{{\sin }^2}i\sin (6f + 4g) - 24\sin (2f + 2g)} \right. } \right. \\&\times \left( {\left( {45{{\sin }^2}f + 5\sin 3f\sin f - 49} \right) \cos 2i + 27{{\sin }^2}f + 3\sin 3f\sin f - 35} \right) \\&\left. { - 21(41\cos 2i + 23)\sin (4f + 2g)} \right] + 6\sin 2f\left( {43\cos 2g{{\sin }^2}i + \cos 2i\left( {69\cos 2g{{\sin }^2}i} \right. } \right. \\&\left. {\left. { - 232} \right) + 42\cos 4i + 62} \right) - 3\cos f\left[ {\sin f\cos 2f(116\cos 2i - 21\cos 4i - 31)} \right. \end{aligned}$$

$$\begin{aligned}&- 4{\sin ^2}i\left( {49{{\sin }^2}i\sin (3f + 4g) + 4\sin f(7\cos 2i + 5)\cos (2f + 2g) - \sin f(41\cos 2i} \right. \\&\left. {\left. { + 23)\cos (4f + 2g)} \right) } \right] + 2\cot i\left[ {\sin i\cos i\left( { - 28{{\sin }^2}i(\sin (6f + 4g) + 6\cos f\sin (3f + 4g))} \right. } \right. \\&\left. {\left. {\left. { - 48(7\cos 2i - 1)\sin (2f + 2g) + 6(41\cos 2i - 9)\sin (4f + 2g)} \right) + 3\sin 2f(21\sin 4i - 58\sin 2i)} \right] } \right\} \\&+ 48{e^6}\left\{ {{{\sin }^2}i(5\cos 2i + 3)( - 645\sin (f + 2g) + 195\sin (3f + 2g) + 29\sin (5f + 2g)} \right. \\&- 85\sin (f - 2g)) - 8{\cos ^2}i(5\cos 2i - 1)( - 45\sin (f + 2g) + 15\sin (3f + 2g) \\&\left. { + \sin (5f + 2g) - 5\sin (f - 2g))} \right\} + {e^7}\left\{ {80(5\cos 2i - 1){{\cos }^2}i\left( { - 3\sin (2f - 2g)} \right. } \right. \\&\left. { + 6\sin (4f + 2g) + \sin (6f + 2g)} \right) + 10{\sin ^2}i(5\cos 2i + 3)\left( {54\sin (2f - 2g)} \right. \\&+ 24\sin (2f + 2g) - 81\sin (4f + 2g) - 26\sin (6f + 2g) - 3\sin (8f + 2g) \\&\left. {\left. { + 3\sin (4f - 2g) + 3\sin 2g} \right) } \right\} + 1800{e^8}\sin f\cos 2g{\sin ^2}i\cos 2i \\&\left. { + 450{e^9}\sin 2f\cos 2g{{\sin }^2}i\cos 2i} \right\} \\ \frac{{\partial {W_4}}}{{\partial H}} =&\frac{{3J_2^2{R_e}^4\csc i}}{{5120{a^4}e{\varepsilon ^4}{\eta ^8}}}\left\{ {30e(f - l)\left\{ {\sin 2i\left[ {32\left( {29{e^2} + 16\left( {{\eta ^2} + 1} \right) } \right) + \left( {{e^2}\left( { - 15{e^4} - 16{e^2}} \right. } \right. } \right. } \right. } \right. \\&\left. {\left. {\left. { + \left( {525{e^4} + 160{e^2} + 96} \right) {\eta ^3} - 416} \right) - 256\left( {{\eta ^2} - 1} \right) } \right) \cos 2g} \right] + \sin 4i\left[ { - 16\left( {21{e^2} + 16{\eta ^2} - 56} \right) } \right. \\&\left. {\left. { + \left( {{e^2}\left( {15{e^4} + 40{e^2} - 5\left( {105{e^4} + 80{e^2} + 48} \right) {\eta ^3} + 1104} \right) + 384\left( {{\eta ^2} - 1} \right) } \right) \cos 2g} \right] } \right\} \\&+ 64\left\{ {12{{\sin }^3}i\cos i\left[ {60\sin (f + 2g) + 40\sin (3f + 2g) - 5\sin (3f + 4g) - 7\sin (5f + 4g)} \right] } \right. \\&\left. { - 5\left[ {2\left( {2\sin 2i + 3\sin 4i} \right) \left( {3\sin (f + 2g) + 2\sin (3f + 2g)} \right) + \sin f\left( {42\sin 2i - 93\sin 4i} \right) } \right] } \right\} \\&+ 80\eta \sin 2i\left\{ { - 2{e^2}\left[ { - 8(3\cos 2f + 1){{\cos }^3}f\sin 2g(3\cos 2i - 1) - 2\sin f\cos 2g(3\cos 2i - 1)} \right. } \right. \\&\left. { \times (14\cos 2f + 3\cos 4f + 19) + 6(9\sin f + \sin 3f)(3\cos 2i + 1)} \right] - 18e\left[ {2(1 - 3\cos 2i)} \right. \\&\left. {\left. { \times \sin (4f + 2g) + 4\sin 2f(3\cos 2i + 1)} \right] + 32(3\cos 2i - 1)\left[ {\sin (3f + 2g) - 3\sin (f + 2g)} \right] } \right\} \\&+ 128{\eta ^2}\sin i\left\{ {10e\left[ {\cos i\left( {{{\sin }^2}i(6\sin (4f + 4g) + 9\sin (2f + 4g) + \sin (6f + 4g) + 24\sin 2f)} \right. } \right. } \right. \\&\left. {\left. { + (9\cos 2i + 3)\sin (2f + 2g)} \right) - 9\sin i\sin 2i\sin (2f + 2g)} \right] + \cos i\left[ {6{{\sin }^2}i\left( {5\sin (3f + 4g)} \right. } \right. \\&\left. {\left. {\left. { + 7\sin (5f + 4g) + 130\sin f} \right) + 10(3\cos 2i - 1)\left( {15\sin (f + 2g) + \sin (3f + 2g)} \right) } \right] } \right\} \end{aligned}$$

$$\begin{aligned}&+ 8{\eta ^3}\left\{ {e\sin i\left( {3\cos i + 5\cos 3i} \right) \left[ { - 72{e^3}\sin (5f + 2g) + 5e\left( {e\left( {{e^2}(9\cos 2f + 18\cos 4f} \right. } \right. } \right. } \right. \\&+ 7\cos 6f)\sin 2g + 2e\sin f\cos 2g( - 20e\cos f + 25e\cos 3f + 7e\cos 5f + 72) \\&\left. {\left. {\left. { + 6(4\sin (2f + 2g) + 5\sin (4f + 2g))} \right) - 8\left( {3{e^2} + 8} \right) \sin (3f + 2g)} \right) + 720\sin (2f + 2g)} \right] \\&\left. { + 120\left[ {\left( {3\sin 4i - 2\sin 2i} \right) \left( {3\sin (f + 2g) + \sin (3f + 2g)} \right) - 6\sin f\left( {2\sin 2i + 3\sin 4i} \right) } \right] } \right\} \\&+ 80e\left\{ {4\sin i\cos i\left[ { - 4{{\sin }^2}i\left( {3\sin (4f + 4g) + 9\sin (2f + 4g) + \sin (6f + 4g)} \right) + ( - 180\cos 2i} \right. } \right. \\&\left. {\left. { + 84)\sin (2f + 2g) + (18 - 54\cos 2i)\sin (4f + 2g)} \right] + 12\sin 2f(13\sin 4i - 2\sin 2i)} \right\} \\&- 32{e^2}\left\{ { - 10{{\sin }^3}i\cos i\left[ { - 1197\sin (f + 2g) + 365\sin (3f + 2g) + 9\sin (5f + 2g) + 9\sin (f - 2g)} \right] } \right. \\&- 288{\sin ^3}i\cos i\sin (5f + 4g) + 15\sin f(5\sin 4i - 162\sin 2i) - 30\sin 3f\sin 2i(3\cos 2i + 1) \\&+ 5\sin i\cos i\left[ { - 3(399\cos 2i + 277)\sin (f + 2g) + (9\cos 2i + 3)\sin (f - 2g) + (365\cos 2i} \right. \\&\left. {\left. { + 191)\sin (3f + 2g) + 3(3\cos 2i + 1)\sin (5f + 2g)} \right] } \right\} - 80{e^3}\left\{ {\sin i\cos i\left[ { - 28{{\sin }^2}i} \right. } \right. \\&\times \left( {\sin (6f + 4g) + 6\cos f\sin (3f + 4g)} \right) - 48(7\cos 2i - 1)\sin (2f + 2g) + 6(41\cos 2i \\&\left. {\left. { - 9)\sin (4f + 2g)} \right] + 3\sin 2f\left( {21\sin 4i - 58\sin 2i} \right) } \right\} + 4{e^4}\sin 2i(5\cos 2i - 1) \\&\times \left\{ {24\left[ {\sin (5f + 2g) - 5\left( {9\sin (f + 2g) - 3\sin (3f + 2g) + \sin (f - 2g)} \right) } \right] } \right. \\&\left. {\left. { - 5e\left[ { - 3\sin (2f - 2g) + 6\sin (4f + 2g) + \sin (6f + 2g)} \right] } \right\} } \right\} \\ \frac{{\partial {W_4}}}{{\partial l}} =&\frac{{3J_2^2\sqrt{\mu }{R_e}^4}}{{10240{a^{7/2}}{\varepsilon ^4}{\eta ^7}}}\left\{ { - 30\left\{ {8\left[ {4\left( {29{e^2} + 16\left( {{\eta ^2} + 1} \right) } \right) \cos 2i + \left( { - 21{e^2} - 16{\eta ^2} + 56} \right) \cos 4i} \right. } \right. } \right. \\&\left. { - 31{e^2} - 48{\eta ^2} + 136} \right] - \cos 2g{\sin ^2}i\left[ {{e^4}\left( {48 - 480{\eta ^3}} \right) + 32{e^2}\left( {43 - 9{\eta ^3}} \right) + 256\left( {{\eta ^2} - 1} \right) } \right. \\&\left. {\left. { + 2\left( {{e^2}\left( {15{e^4} + 40{e^2} - 5\left( {105{e^4} + 80{e^2} + 48} \right) {\eta ^3} + 1104} \right) + 384\left( {{\eta ^2} - 1} \right) } \right) \cos 2i} \right] } \right\} \end{aligned}$$

$$\begin{aligned}&- \frac{{20{{(e\cos f + 1)}^2}}}{{e{\eta ^3}}}\left\{ {24\eta (3\cos 2i + 1)\left[ {2{e^2}{{\cos }^2}f\left( {\cos (f + 2g) + 5\cos (3f + 2g) + 2\cos f} \right. } \right. } \right. \\&+2\cos 2i\sin (f + g)\left( {5\sin (2f + g) + \sin g} \right) ) + 4e\left( {6{{\sin }^2}i\cos (4f + 2g) + \cos 2f(3\cos 2i + 1)} \right) \\&\left. { - 32\sin f{{\sin }^2}i\sin (2f + 2g)} \right] + 96{\eta ^2}\left[ {2e\left( {{{\sin }^4}i\left( {2\cos f\left( {3\cos (3f + 4g) + \cos (5f + 4g)} \right) } \right. } \right. } \right. \\&\left. {\left. { + 8\cos 2f} \right) + 4\cos f{{\sin }^2}i(3\cos 2i + 1)\cos (f + 2g) - 4\cos 2i + \cos 4i + 3} \right) + {\sin ^2}i \\&\times \left( {5(3\cos 2i + 1)\cos (f + 2g) + (3\cos 2i + 1)\cos (3f + 2g) + {{\sin }^2}i\left( {3\cos (3f + 4g)} \right. } \right. \\&\left. {\left. {\left. { + 7\cos (5f + 4g) + 26\cos f} \right) } \right) } \right] + 3{\eta ^3}\left[ { - 525{e^7}\cos 2g{{\sin }^2}i\cos 2i + 32(3\cos 2i + 1)} \right. \\&\times \left( { - 12\sin f{{\cos }^2}f\sin 2g{{\sin }^2}i + 3(\cos f + \cos 3f)\cos 2g{{\sin }^2}i + \cos f(3\cos 2i + 1)} \right) \\&+ 32e{\sin ^2}i(5\cos 2i + 3)\left( {e\left( {e\cos f\left( { - 3\left( {3{e^2}{{\cos }^2}f + 1} \right) \cos (f + 2g) + \left( {7{e^2}{{\cos }^2}f + 5} \right) } \right. } \right. } \right. \\&\left. {\left. {\left. {\left. { \times \cos (3f + 2g) + 6e\sin 2f\sin (2f + 2g)} \right) - 4\cos (3f + 2g)} \right) + 6\cos (2f + 2g)} \right) } \right] \\&+ 24\left[ { - 4{{\sin }^4}i\left( {3\cos (3f + 4g) + 7\cos (5f + 4g)} \right) - 16{{\sin }^2}i(3\cos 2i + 1)\left( {\cos (f + 2g)} \right. } \right. \\&\left. {\left. { + 2\cos (3f + 2g)} \right) + \cos f\left( {28\cos 2i - 31\cos 4i - 61} \right) } \right] + 8e\left[ { - 2{{\sin }^2}i\left( {36(3\cos 2i} \right. } \right. \\&+ 1)\cos (4f + 2g) + 2(90\cos 2i + 6)\cos (2f + 2g) + 12{\sin ^2}i\left( {2\cos (4f + 4g)} \right. \\&\left. {\left. { + 3\cos (2f + 4g) + \cos (6f + 4g)} \right) } \right) - 6\cos 2f\left( { - 4\cos 2i + 13\cos 4i + 23} \right) \\&\left. { - 48\cos 2g{{\sin }^2}i - 144\cos 2g{{\sin }^2}i\cos 2i - 96\cos 2i - 84\cos 4i - 204} \right] - 4{e^2} \end{aligned}$$

$$\begin{aligned}&\times \left[ { - 288{{\sin }^4}i\cos (5f + 4g) + 6\cos 3f{{(3\cos 2i + 1)}^2} + 3\cos f\left( {324\cos 2i - 5\cos 4i + 33} \right) } \right. \\&+ 2{\sin ^2}i\left( {15(3\cos 2i + 1)\cos (5f + 2g) + (9\cos 2i + 3)\cos (f - 2g) + 3(365\cos 2i + 191)} \right. \\&\left. {\left. { \times \cos (3f + 2g) - 3(399\cos 2i + 277)\cos (f + 2g)} \right) } \right] + 24{e^3}\cos f\left[ {\cos f\left( {56{{\sin }^4}i} \right. } \right. \\&\left. { \times \cos (4f + 4g) - 116\cos 2i + 21\cos 4i + 31} \right) + 4{\sin ^2}i\left( {(69\cos 2i + 43)\cos (f + 2g)} \right. \\&\left. {\left. { - (41\cos 2i + 23)\cos (3f + 2g)} \right) } \right] - 48{e^4}(9\sin f + \sin 3f){\sin ^2}i(5\cos 2i + 3)\sin (2f + 2g) \\&\left. {\left. { + 12{e^5}(4\sin 2f + \sin 4f){{\sin }^2}i(5\cos 2i + 3)\sin (2f + 2g) + 45{e^7}\cos 2g{{\sin }^2}i\cos 2i} \right\} } \right\} \\ \frac{{\partial {W_4}}}{{\partial g}} =&- \frac{{3J_2^2\sqrt{\mu }{R_e}^4}}{{2560{a^{7/2}}e{\varepsilon ^4}{\eta ^7}}}\left\{ {15e(l - f)\sin 2g{{\sin }^2}i\left\{ {{e^4}\left( {48 - 480{\eta ^3}} \right) + 32{e^2}\left( {43 - 9{\eta ^3}} \right) + 256\left( {{\eta ^2} - 1} \right) } \right. } \right. \\&\left. { + 2\left[ {{e^2}\left( {15{e^4} + 40{e^2} - 5\left( {105{e^4} + 80{e^2} + 48} \right) {\eta ^3} + 1104} \right) + 384\left( {{\eta ^2} - 1} \right) } \right] \cos 2i} \right\} - 384{\sin ^4}i \\&\times \left[ {5\cos (3f + 4g) + 7\cos (5f + 4g)} \right] - 640{\sin ^2}i(3\cos 2i + 1)\left[ {6\cos (f + 2g) + 4\cos (3f + 2g)} \right] \\&+ 2{\eta ^3}{\sin ^2}i\left\{ {2e(5\cos 2i + 3)\left[ {135{e^4}\cos (2f - 2g) + 192{e^3}{{\cos }^3}f(2 - 3\cos 2f)\cos 2g} \right. } \right. \\&- 192{e^3}{\sin ^3}f(3\cos 2f + 7)\sin 2g + 30\left( { - 3{e^4} + 4{e^2} + 24} \right) \cos (2f + 2g) + 5e \\&\left. { \times \left( {7{e^3}\cos (6f + 2g) + 6\left( {3{e^2} + 5} \right) e\cos (4f + 2g) - 64\cos (3f + 2g)} \right) } \right] + 480(3\cos 2i \\&\left. { + 1)\left[ {3\cos (f + 2g) + \cos (3f + 2g)} \right] } \right\} + 32{\eta ^2}{\sin ^2}i\left\{ {10(3\cos 2i + 1)\left[ {6e\cos (2f + 2g)} \right. } \right. \\&\left. { + 15\cos (f + 2g) + \cos (3f + 2g)} \right] + 4{\sin ^2}i\left[ {5e\left( {6\cos (4f + 4g) + 9\cos (2f + 4g)} \right. } \right. \\&\left. {\left. {\left. { + \cos (6f + 4g)} \right) + 3\left( {5\cos (3f + 4g) + 7\cos (5f + 4g)} \right) } \right] } \right\} + 80\eta {\sin ^2}i(3\cos 2i + 1) \\&\times \left\{ {e\left[ {e\left( {21\cos (f + 2g) + 11\cos (3f + 2g) + 3\cos (5f + 2g) - 3\cos (f - 2g)} \right) } \right. } \right. \\&\left. {\left. { + 18\cos (4f + 2g)} \right] + 16\left[ {\cos (3f + 2g) - 3\cos (f + 2g)} \right] } \right\} + 2e\left\{ { - 5{e^2}\left[ { - 112{{\sin }^4}i} \right. } \right. \\&\times \left( {6\cos f\cos (3f + 4g) + \cos (6f + 4g)} \right) - 6{\sin ^2}i\left( {16(7\cos 2i + 5)\cos (2f + 2g)} \right. \\&\left. {\left. { - 2(41\cos 2i + 23)\cos (4f + 2g)} \right) } \right] - 2e\left[ {10{{\sin }^2}i\left( {6(3\cos 2i + 1)\cos (5f + 2g)} \right. } \right. \\&- 6(3\cos 2i + 1)\cos (f - 2g) + 2(365\cos 2i + 191)\cos (3f + 2g) - 6(399\cos 2i + 277) \\&\left. {\left. { \times \cos (f + 2g)} \right) - 1152{{\sin }^4}i\cos (5f + 4g)} \right] - 80{\sin ^2}i\left[ {9(3\cos 2i + 1)\cos (4f + 2g)} \right. \\&+ (90\cos 2i + 6)\cos (2f + 2g) + 4{\sin ^2}i\left( {3\cos (4f + 4g) + 9\cos (2f + 4g)} \right. \\&\left. {\left. {\left. { + \cos (6f + 4g)} \right) } \right] } \right\} + 48{e^4}{\sin ^2}i(5\cos 2i + 3)\left[ { - 45\cos (f + 2g) + 15\cos (3f + 2g)} \right. \\&\left. { + \cos (5f + 2g) + 5\cos (f - 2g)} \right] - 10{e^5}{\sin ^2}i(5\cos 2i + 3)\left[ {3\cos (2f - 2g)} \right. \\&\left. {\left. { + 6\cos (4f + 2g) + \cos (6f + 2g)} \right] } \right\} \\ \frac{{\partial {W_4}}}{{\partial h}} =&0. \end{aligned}$$

1.3 Partial derivatives of $W_5$

The partial derivatives of $W_5$ are given by

$$\begin{aligned} \begin{aligned} \frac{{\partial {W_5}}}{{\partial l}} =&\frac{{5!}}{{{\varepsilon ^5}}}\left\{ {\sum \limits _{m = 1}^2 {\sum \limits _{p = 1}^{} {\frac{{{\mu ^{1/2}}{R_e}^2{N_{2m}}{F_{2mp}}\left( i \right) }}{{{a^{3/2}}{{\left( {1 - {e^2}} \right) }^{3/2}}}}\cos {\vartheta _{2mp}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 1}^2 {{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) \frac{{e{\mu ^{1/2}}\cos {\vartheta _{3mp}}}}{{{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}} } } \right. \\&\left. { + \sum \limits _{m = 0}^4 {\sum \limits _{p = 2}^{} {{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) \frac{{\left( {3{e^2} + 2} \right) {\mu ^{1/2}}\cos {\vartheta _{4mp}}}}{{2{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 1,3}^{} {{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) \frac{{3{e^2}{\mu ^{1/2}}\cos {\vartheta _{4mp}}}}{{4{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}} } } \right\} \\&- \frac{{{a^{3/2}}}}{{{\mu ^{1/2}}}}\frac{{5!}}{{{\varepsilon ^5}}}\left\{ {\sum \limits _{m = 1}^2 {\sum \limits _{p = 0}^{p = n} {\frac{{\mu {R_e}^2{N_{2m}}}}{{{r^3}}}{F_{2mp}}\left( i \right) \cos \left( {{\psi _{2mp}} + {\varphi _{2m}}} \right) } } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 0}^{p = 3} {\frac{{\mu {R_e}^3{N_{3m}}}}{{{r^4}}}{F_{3mp}}\left( i \right) \cos \left( {{\psi _{3mp}} + {\varphi _{3m}}} \right) } } } \right. \\&\left. { + \sum \limits _{m = 0}^4 {\sum \limits _{p = 0}^{p = 4} {\frac{{\mu {R_e}^4{N_{4m}}}}{{{r^5}}}{F_{4mp}}\left( i \right) \cos \left( {{\psi _{4mp}} + {\varphi _{4m}}} \right) } } } \right\} \\ \frac{{\partial {W_5}}}{{\partial g}} =&\frac{{5!}}{{{\varepsilon ^5}}}\left\{ {\sum \limits _{m = 1}^2 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{2mp,A}}}}{{\partial g}}} } + \sum \limits _{m = 1}^2 {\sum \limits _{p = \left\{ {0,2} \right\} }^{} {\frac{{\partial {W_{2mp,B}}}}{{\partial g}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = \left\{ {0,3} \right\} }^{} {\frac{{\partial {W_{3mp}}}}{{\partial g}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{3mp,A}}}}{{\partial g}}} } } \right. \\&\left. { + \sum \limits _{m = 0}^3 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{3mp,B}}}}{{\partial g}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{4mp,A}}}}{{\partial g}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = \left\{ {0,4} \right\} }^{} {\frac{{\partial {W_{4mp,B}}}}{{\partial g}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{4mp,C}}}}{{\partial g}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 3}^{} {\frac{{\partial {W_{4mp,D}}}}{{\partial g}}} } } \right\} \\ \frac{{\partial {W_5}}}{{\partial h}} =&\frac{{5!}}{{{\varepsilon ^5}}}\left\{ {\sum \limits _{m = 1}^2 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{2mp,A}}}}{{\partial h}}} } + \sum \limits _{m = 1}^2 {\sum \limits _{p = \left\{ {0,2} \right\} }^{} {\frac{{\partial {W_{2mp,B}}}}{{\partial h}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = \left\{ {0,3} \right\} }^{} {\frac{{\partial {W_{3mp}}}}{{\partial h}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{3mp,A}}}}{{\partial h}}} } } \right. \\&\left. { + \sum \limits _{m = 0}^3 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{3mp,B}}}}{{\partial h}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{4mp,A}}}}{{\partial h}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = \left\{ {0,4} \right\} }^{} {\frac{{\partial {W_{4mp,B}}}}{{\partial h}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{4mp,C}}}}{{\partial h}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 3}^{} {\frac{{\partial {W_{4mp,D}}}}{{\partial h}}} } } \right\} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \frac{{{W_5}}}{{dL}}= & {} \frac{{2{a^{1/2}}}}{{{\mu ^{1/2}}}}\frac{{5!}}{{{\varepsilon ^5}}}\left\{ {\sum \limits _{m = 1}^2 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{2mp,A}}}}{{\partial a}}} } + \sum \limits _{m = 1}^2 {\sum \limits _{p = \left\{ {0,2} \right\} }^{} {\frac{{\partial {W_{2mp,B}}}}{{\partial a}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = \left\{ {0,3} \right\} }^{} {\frac{{\partial {W_{3mp}}}}{{\partial a}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{3mp,A}}}}{{\partial a}}} } } \right. \\&+ \sum \limits _{m = 0}^3 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{3mp,B}}}}{{\partial a}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{4mp,A}}}}{{\partial a}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = \left\{ {0,4} \right\} }^{} {\frac{{\partial {W_{4mp,B}}}}{{\partial a}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{4mp,C}}}}{{\partial a}}} } \\&\left. { + \sum \limits _{m = 0}^4 {\sum \limits _{p = 3}^{} {\frac{{\partial {W_{4mp,D}}}}{{\partial a}}} } } \right\} + \frac{1}{e}\frac{{\left( {1 - {e^2}} \right) }}{{{\mu ^{1/2}}{a^{1/2}}}}\frac{{5!}}{{{\varepsilon ^5}}}\left\{ {\sum \limits _{m = 1}^2 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{2mp,A}}}}{{\partial e}}} } + \sum \limits _{m = 1}^2 {\sum \limits _{p = \left\{ {0,2} \right\} }^{} {\frac{{\partial {W_{2mp,B}}}}{{\partial e}}} } } \right. \\&+ \sum \limits _{m = 0}^3 {\sum \limits _{p = \left\{ {0,3} \right\} }^{} {\frac{{\partial {W_{3mp}}}}{{\partial e}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{3mp,A}}}}{{\partial e}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{3mp,B}}}}{{\partial e}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{4mp,A}}}}{{\partial e}}} } \\&\left. { + \sum \limits _{m = 0}^4 {\sum \limits _{p = \left\{ {0,4} \right\} }^{} {\frac{{\partial {W_{4mp,B}}}}{{\partial e}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{4mp,C}}}}{{\partial e}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 3}^{} {\frac{{\partial {W_{4mp,D}}}}{{\partial e}}} } } \right\} \\ \frac{{{W_5}}}{{dG}} =&- \frac{{\sqrt{1 - {e^2}} }}{{{\mu ^{1/2}}{a^{1/2}}e}}\frac{{5!}}{{{\varepsilon ^5}}}\left\{ {\sum \limits _{m = 1}^2 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{2mp,A}}}}{{\partial e}}} } + \sum \limits _{m = 1}^2 {\sum \limits _{p = \left\{ {0,2} \right\} }^{} {\frac{{\partial {W_{2mp,B}}}}{{\partial e}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = \left\{ {0,3} \right\} }^{} {\frac{{\partial {W_{3mp}}}}{{\partial e}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{3mp,A}}}}{{\partial e}}} } } \right. \\&+ \sum \limits _{m = 0}^3 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{3mp,B}}}}{{\partial e}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{4mp,A}}}}{{\partial e}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = \left\{ {0,4} \right\} }^{} {\frac{{\partial {W_{4mp,B}}}}{{\partial e}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{4mp,C}}}}{{\partial e}}} } \\&\left. { + \sum \limits _{m = 0}^4 {\sum \limits _{p = 3}^{} {\frac{{\partial {W_{4mp,D}}}}{{\partial e}}} } } \right\} + \frac{{\cot i}}{{{\mu ^{1/2}}{a^{1/2}}\sqrt{1 - {e^2}} }}\frac{{5!}}{{{\varepsilon ^5}}}\left\{ {\sum \limits _{m = 1}^2 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{2mp,A}}}}{{\partial i}}} } + \sum \limits _{m = 1}^2 {\sum \limits _{p = \left\{ {0,2} \right\} }^{} {\frac{{\partial {W_{2mp,B}}}}{{\partial i}}} } } \right. \\&+ \sum \limits _{m = 0}^3 {\sum \limits _{p = \left\{ {0,3} \right\} }^{} {\frac{{\partial {W_{3mp}}}}{{\partial i}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{3mp,A}}}}{{\partial i}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{3mp,B}}}}{{\partial i}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{4mp,A}}}}{{\partial i}}} } \\&\left. { + \sum \limits _{m = 0}^4 {\sum \limits _{p = \left\{ {0,4} \right\} }^{} {\frac{{\partial {W_{4mp,B}}}}{{\partial i}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{4mp,C}}}}{{\partial i}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 3}^{} {\frac{{\partial {W_{4mp,D}}}}{{\partial i}}} } } \right\} \\ \frac{{{W_5}}}{{dH}} =&- \frac{1}{{{\mu ^{1/2}}{a^{1/2}}\sqrt{1 - {e^2}} \sin i}}\frac{{5!}}{{{\varepsilon ^5}}}\left\{ {\sum \limits _{m = 1}^2 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{2mp,A}}}}{{\partial i}}} } + \sum \limits _{m = 1}^2 {\sum \limits _{p = \left\{ {0,2} \right\} }^{} {\frac{{\partial {W_{2mp,B}}}}{{\partial i}}} } } \right. \\&+ \sum \limits _{m = 0}^3 {\sum \limits _{p = \left\{ {0,3} \right\} }^{} {\frac{{\partial {W_{3mp}}}}{{\partial i}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{3mp,A}}}}{{\partial i}}} } + \sum \limits _{m = 0}^3 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{3mp,B}}}}{{\partial i}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 2}^{} {\frac{{\partial {W_{4mp,A}}}}{{\partial i}}} } \\&\left. { + \sum \limits _{m = 0}^4 {\sum \limits _{p = \left\{ {0,4} \right\} }^{} {\frac{{\partial {W_{4mp,B}}}}{{\partial i}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 1}^{} {\frac{{\partial {W_{4mp,C}}}}{{\partial i}}} } + \sum \limits _{m = 0}^4 {\sum \limits _{p = 3}^{} {\frac{{\partial {W_{4mp,D}}}}{{\partial i}}} } } \right\} \end{aligned}$$

where

$$\begin{aligned} \frac{{\partial {W_{2mp,A}}}}{{\partial a}} =&\frac{{3\sqrt{\mu }{R_e}^2{N_{2m}}{F_{2mp}}\left( i \right) }}{{2{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{3/2}}}}(e\sin f + f - l)\cos (hm + {\varphi _{2m}}) \\ \frac{{\partial {W_{2mp,A}}}}{{\partial e}} =&- \frac{{\sqrt{\mu }{R_e}^2{N_{2m}}{F_{2mp}}\left( i \right) }}{{{a^{3/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\cos (hm + {\varphi _{2m}})\left[ {3e(f - l)} \right. \\&\left. { + \sin f\left( {{e^2}{{\cos }^2}f + 2{e^2} + 3e\cos f + 3} \right) } \right] \\ \frac{{\partial {W_{2mp,A}}}}{{\partial g}} =&0\\ \frac{{\partial {W_{2mp,A}}}}{{\partial h}} =&\frac{{\sqrt{\mu }{R_e}^2{N_{2m}}{F_{2mp}}\left( i \right) }}{{{a^{3/2}}{{\left( {1 - {e^2}} \right) }^{3/2}}}}m(e\sin f + f - l)\sin (hm + {\varphi _{2m}})\\ \frac{{\partial {W_{2mp,A}}}}{{\partial i}} =&- \frac{{\sqrt{\mu }{R_e}^2{N_{2m}}}}{{{a^{3/2}}{{\left( {1 - {e^2}} \right) }^{3/2}}}}(f - l + e\sin f)\cos (hm + {\varphi _{2m}})\frac{{\partial {F_{2mp}}\left( i \right) }}{{\partial i}} \\ \frac{{\partial {W_{2mp,B}}}}{{\partial a}} =&\frac{{3\sqrt{\mu }{R_e}^2{N_{2m}}{F_{2mp}}\left( i \right) }}{{4{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{3/2}}}}\left\{ { - \frac{e}{{\alpha - 1}}\sin \left( { - {\alpha _{2p}}(f + g) + f - hm - {\varphi _{2m}}} \right) } \right. \\&\left. { + \frac{e}{{\alpha + 1}}\sin \left( {{\alpha _{2p}}(f + g) + f + hm + {\varphi _{2m}}} \right) + \frac{2}{\alpha }\sin \left( {{\alpha _{2p}}(f + g) + hm + {\varphi _{2m}}} \right) } \right\} \\ \frac{{\partial {W_{2mp,B}}}}{{\partial e}} =&\frac{{\sqrt{\mu }{R_e}^2{N_{2m}}{F_{2mp}}\left( i \right) }}{{2{a^{3/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ {\frac{1}{{\alpha \left( {{\alpha ^2} - 1} \right) }}\left[ {{\alpha _{2p}}({\alpha _{2p}} + 1)\left( {2{e^2} + 1} \right) \sin \left( { - {\alpha _{2p}}(f + g)} \right. } \right. } \right. \\&\left. { + f - hm - {\varphi _{2m}}} \right) - ({\alpha _{2p}} - 1)\left( {{\alpha _{2p}}\left( {2{e^2} + 1} \right) \sin \left( {{\alpha _{2p}}(f + g) + f + hm + {\varphi _{2m}}} \right) } \right. \\&\left. {\left. { + 6({\alpha _{2p}} + 1)e\sin \left( {{\alpha _{2p}}(f + g) + hm + {\varphi _{2m}}} \right) } \right) } \right] - 2\sin f(e\cos f + 1) \\&\left. { \times (e\cos f + 2)\cos \left( {{\alpha _{2p}}(f + g) + hm + {\varphi _{2m}}} \right) } \right\} \\ \frac{{\partial {W_{2mp,B}}}}{{\partial g}} =&- \frac{{\sqrt{\mu }{R_e}^2{N_{2m}}{F_{2mp}}\left( i \right) }}{{2{a^{3/2}}{{\left( {1 - {e^2}} \right) }^{3/2}}}}\left\{ {\frac{{{\alpha _{2p}}e}}{{\alpha - 1}}\cos \left( { - {\alpha _{2p}}f + f - {\alpha _{2p}}g - hm - {\varphi _{2m}}} \right) } \right. \\&\left. { + \frac{{{\alpha _{2p}}e}}{{\alpha + 1}}\cos \left( {{\alpha _{2p}}f + f + {\alpha _{2p}}g + hm + {\varphi _{2m}}} \right) + 2\cos \left( {{\alpha _{2p}}f + {\alpha _{2p}}g + hm + {\varphi _{2m}}} \right) } \right\} \\ \frac{{\partial {W_{2mp,B}}}}{{\partial h}} =&- \frac{{\sqrt{\mu }{R_e}^2{N_{2m}}{F_{2mp}}\left( i \right) m}}{{2{a^{3/2}}{{\left( {1 - {e^2}} \right) }^{3/2}}}}\left\{ {\frac{e}{{\alpha - 1}}\cos \left( { - {\alpha _{2p}}f + f - {\alpha _{2p}}g - hm - {\varphi _{2m}}} \right) } \right. \\&\left. { + \frac{e}{{\alpha + 1}}\cos \left( {{\alpha _{2p}}f + f + {\alpha _{2p}}g + hm + {\varphi _{2m}}} \right) + \frac{2}{\alpha }\cos \left( {{\alpha _{2p}}f + {\alpha _{2p}}g + hm + {\varphi _{2m}}} \right) } \right\} \\ \frac{{\partial {W_{2mp,B}}}}{{\partial i}} =&- \frac{{\sqrt{\mu }{R_e}^2{N_{2m}}}}{{2{a^{3/2}}{{\left( {1 - {e^2}} \right) }^{3/2}}}}\left\{ { - \frac{e}{{{\alpha _{2p}} - 1}}\sin \left( { - {\alpha _{2p}}(f + g) + f - hm - {\varphi _{2m}}} \right) } \right. \\&\left. { + \frac{e}{{{\alpha _{2p}} + 1}}\sin \left( {{\alpha _{2p}}(f + g) + f + hm + {\varphi _{2m}}} \right) + \frac{2}{{{\alpha _{2p}}}}\sin \left( {{\alpha _{2p}}(f + g) + hm + {\varphi _{2m}}} \right) } \right\} \frac{{\partial {F_{2mp}}\left( i \right) }}{{\partial i}} \\ \frac{{\partial {W_{3mp}}}}{{\partial a}} =&- \frac{{5\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) }}{{8{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ { - \frac{{{e^2}}}{{{\alpha _{3p}} - 2}}\sin \left( {({\alpha _{3p}} - 2)f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) } \right. \\&- \frac{{{e^2}}}{{{\alpha _{3p}} + 2}}\sin \left( {({\alpha _{3p}} + 2)f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) - \frac{{2\left( {{e^2} + 2} \right) }}{{{\alpha _{3p}}}}\sin \left( {{\alpha _{3p}}f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) \\&\left. { + \frac{{4e}}{{{\alpha _{3p}} - 1}}\sin \left( { - {\alpha _{3p}}f + f - {\alpha _{3p}}g - hm - {\varphi _{3m}}} \right) - \frac{{4e}}{{{\alpha _{3p}} + 1}}\sin \left( {{\alpha _{3p}}f + f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) } \right\} \end{aligned}$$

$$\begin{aligned} \frac{{\partial {W_{3mp}}}}{{\partial e}} =&\frac{{\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) }}{{4{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {e\left[ { - \frac{{\left( {3{e^2} + 2} \right) }}{{{\alpha _{3p}} - 2}}\sin \left( {({\alpha _{3p}} - 2)f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) } \right. } \right. \\&- \frac{{6\left( {{e^2} + 4} \right) }}{{{\alpha _{3p}}}}\sin \left( {{\alpha _{3p}}(f + g) + hm + {\varphi _{3m}}} \right) - \frac{{\left( {3{e^2} + 2} \right) }}{{{\alpha _{3p}} + 2}}\sin \left( {({\alpha _{3p}} + 2)f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) \\&\left. { - \frac{{16e}}{{{\alpha _{3p}} + 1}}\sin \left( {{\alpha _{3p}}(f + g) + f + hm + {\varphi _{3m}}} \right) } \right] + \frac{{4\left( {4{e^2} + 1} \right) }}{{{\alpha _{3p}} - 1}}\sin \left( { - {\alpha _{3p}}(f + g) + f - hm - {\varphi _{3m}}} \right) \\&- \frac{4}{{{\alpha _{3p}} + 1}}\sin \left( {{\alpha _{3p}}(f + g) + f + hm + {\varphi _{3m}}} \right) - 4\sin f(e\cos f + 2){(e\cos f + 1)^2} \\&\left. { \times \cos \left( {{\alpha _{3p}}(f + g) + hm + {\varphi _{3m}}} \right) } \right\} \\ \frac{{\partial {W_{3mp}}}}{{\partial g}} =&\frac{{\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) }}{{4{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ { - \frac{{{\alpha _{3p}}{e^2}}}{{{\alpha _{3p}} - 2}}\cos \left( {({\alpha _{3p}} - 2)f + \alpha g + hm + {\varphi _{3m}}} \right) } \right. \\&- \frac{{{\alpha _{3p}}{e^2}}}{{{\alpha _{3p}} + 2}}\cos \left( {({\alpha _{3p}} + 2)f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) - 2\left( {{e^2} + 2} \right) \cos \left( {{\alpha _{3p}}(f + g) + hm + {\varphi _{3m}}} \right) \\&\left. { - \frac{{4{\alpha _{3p}}e}}{{{\alpha _{3p}} - 1}}\cos \left( { - {\alpha _{3p}}(f + g) + f - hm - {\varphi _{3m}}} \right) - \frac{{4{\alpha _{3p}}e}}{{{\alpha _{3p}} + 1}}\cos \left( {{\alpha _{3p}}(f + g) + f + hm + {\varphi _{3m}}} \right) } \right\} \\ \frac{{\partial {W_{3mp}}}}{{\partial h}} =&\frac{{\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) m}}{{4{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ { - \frac{{{e^2}}}{{{\alpha _{3p}} - 2}}\cos \left( {({\alpha _{3p}} - 2)f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) } \right. \\&- \frac{{{e^2}}}{{{\alpha _{3p}} + 2}}\cos \left( {({\alpha _{3p}} + 2)f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) - \frac{{2\left( {{e^2} + 2} \right) }}{{{\alpha _{3p}}}}\cos \left( {{\alpha _{3p}}(f + g) + hm + {\varphi _{3m}}} \right) \\&\left. { - \frac{{4e}}{{{\alpha _{3p}} - 1}}\cos \left( { - {\alpha _{3p}}(f + g) + f - hm - {\varphi _{3m}}} \right) - \frac{{4e}}{{{\alpha _{3p}} + 1}}\cos \left( {{\alpha _{3p}}(f + g) + f + hm + {\varphi _{3m}}} \right) } \right\} \\ \frac{{\partial {W_{3mp}}}}{{\partial i}} =&\frac{{\sqrt{\mu }{R_e}^3{N_{3m}}}}{{4{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ { - \frac{{{e^2}}}{{{\alpha _{3p}} - 2}}\sin \left( {({\alpha _{3p}} - 2)f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) } \right. \\&- \frac{{{e^2}}}{{{\alpha _{3p}} + 2}}\sin \left( {({\alpha _{3p}} + 2)f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) - \frac{{2\left( {{e^2} + 2} \right) }}{{{\alpha _{3p}}}}\sin \left( {{\alpha _{3p}}f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) \\&\left. { + \frac{{4e}}{{{\alpha _{3p}} - 1}}\sin \left( { - {\alpha _{3p}}f + f - {\alpha _{3p}}g - hm - {\varphi _{3m}}} \right) - \frac{{4e}}{{{\alpha _{3p}} + 1}}\sin \left( {{\alpha _{3p}}f + f + {\alpha _{3p}}g + hm + {\varphi _{3m}}} \right) } \right\} \frac{{\partial {F_{3mp}}\left( i \right) }}{{\partial i}} \\ \frac{{\partial {W_{3mp,A}}}}{{\partial a}} =&\frac{{5\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) }}{{24{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ {e\left[ {(9e\sin f + e\sin 3f + 12f - 12l)\cos (g + hm + {\varphi _{3m}})} \right. } \right. \\&\left. {\left. { + 4e{{\cos }^3}f\sin (g + hm + {\varphi _{3m}}) + 6\sin (2f + g + hm + {\varphi _{3m}})} \right] + 12\sin (f + g + hm + {\varphi _{3m}})} \right\} \\ \frac{{\partial {W_{3mp,A}}}}{{\partial e}} =&- \frac{{\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) }}{{12{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {3\left[ {4\left( {4{e^2} + 1} \right) (f - l)\cos (g + hm + {\varphi _{3m}}) + e\left( {3{e^2} + 2} \right) } \right. } \right. \\&\times \sin (f - g - hm - {\varphi _{3m}}) + 6e\left( {{e^2} + 4} \right) \sin (f + g + hm + {\varphi _{3m}}) + 2\left( {4{e^2} + 1} \right) \\&\times \sin (2f + g + hm + {\varphi _{3m}}) + 4\sin f(e\cos f + 2){(e\cos f + 1)^2} \\&\left. {\left. { \times \cos (f + g + hm + {\varphi _{3m}})} \right] + e\left( {3{e^2} + 2} \right) \sin (3f + g + hm + {\varphi _{3m}})} \right\} \\ \frac{{\partial {W_{3mp,A}}}}{{\partial g}} =&- \frac{{\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) }}{{12{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ {e\left[ { - (9e\sin f + e\sin 3f + 12f - 12l)\sin (g + hm + {\varphi _{3m}})} \right. } \right. \\&\left. {\left. { + 4e{{\cos }^3}f\cos (g + hm + {\varphi _{3m}}) + 6\cos (2f + g + hm + {\varphi _{3m}})} \right] + 12\cos (f + g + hm + {\varphi _{3m}})} \right\} \\ \frac{{\partial {W_{3mp,A}}}}{{\partial h}} =&- \frac{{\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) }}{{12{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ {em\left[ { - (9e\sin f + e\sin 3f + 12f - 12l)\sin (g + hm + {\varphi _{3m}})} \right. } \right. \\&\left. {\left. { + 4e{{\cos }^3}f\cos (g + hm + {\varphi _{3m}}) + 6\cos (2f + g + hm + {\varphi _{3m}})} \right] + 12m\cos (f + g + hm + {\varphi _{3m}})} \right\} \\ \frac{{\partial {W_{3mp,A}}}}{{\partial i}} =&- \frac{{\sqrt{\mu }{R_e}^3{N_{3m}}}}{{12{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ {e\left[ {(9e\sin f + e\sin 3f + 12f - 12l)\cos (g + hm + {\varphi _{3m}})} \right. } \right. \\&\left. {\left. { + 4e{{\cos }^3}f\sin (g + hm + {\varphi _{3m}}) + 6\sin (2f + g + hm + {\varphi _{3m}})} \right] + 12\sin (f + g + hm + {\varphi _{3m}})} \right\} \frac{{\partial {F_{3mp}}\left( i \right) }}{{\partial i}} \\ \frac{{\partial {W_{3mp,B}}}}{{\partial a}} =&\frac{{5\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) }}{{24{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ {e\left[ {\left( {9e\sin f + e\sin 3f + 12f - 12l} \right) \cos (g - hm - {\varphi _{3m}})} \right. } \right. \\&\left. { + 4e{{\cos }^3}f\sin (g - hm - {\varphi _{3m}}) + 6\sin (2f + g - hm - {\varphi _{3m}})} \right] \\&\left. { + 12\sin (f + g - hm - {\varphi _{3m}})} \right\} \\ \frac{{\partial {W_{3mp,B}}}}{{\partial e}} =&- \frac{{\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) }}{{12{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {12\left( {4{e^2} + 1} \right) (f - l)\cos (g - hm - {\varphi _{3m}})} \right. \\&+ e\left[ {18\left( {{e^2} + 4} \right) \sin (f + g - hm - {\varphi _{3m}}) + 24e\sin (2f + g - hm - {\varphi _{3m}})} \right. \\&\left. { + \left( {3{e^2} + 2} \right) \left( {\sin (3f + g - hm - {\varphi _{3m}}) + 3\sin (f - g + hm + {\varphi _{3m}})} \right) } \right] \\&+ 12\sin f(e\cos f + 2){(e\cos f + 1)^2}\cos (f + g - hm - {\varphi _{3m}}) \\&\left. { + 6\sin (2f + g - hm - {\varphi _{3m}})} \right\} \\ \frac{{\partial {W_{3mp,B}}}}{{\partial g}} =&- \frac{{\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) }}{{12{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ {e\left[ { - \left( {9e\sin f + e\sin 3f + 12f - 12l} \right) \sin (g - hm - {\varphi _{3m}})} \right. } \right. \\&\left. { + 4e{{\cos }^3}f\cos (g - hm - {\varphi _{3m}}) + 6\cos (2f + g - hm - {\varphi _{3m}})} \right] \\&\left. { + 12\cos (f + g - hm - {\varphi _{3m}})} \right\} \\ \frac{{\partial {W_{3mp,B}}}}{{\partial h}} =&- \frac{{\sqrt{\mu }{R_e}^3{N_{3m}}{F_{3mp}}\left( i \right) }}{{12{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ {em\left[ {\left( {9e\sin f + e\sin 3f + 12f - 12l} \right) \sin (g - hm - {\varphi _{3m}})} \right. } \right. \\&\left. { - 4e{{\cos }^3}f\cos (g - hm - {\varphi _{3m}}) - 6\cos (2f + g - hm - {\varphi _{3m}})} \right] \\&\left. { - 12m\cos (f + g - hm - {\varphi _{3m}})} \right\} \\ \frac{{\partial {W_{3mp,B}}}}{{\partial i}} =&- \frac{{\sqrt{\mu }{R_e}^3{N_{3m}}}}{{12{a^{5/2}}{{\left( {1 - {e^2}} \right) }^{5/2}}}}\left\{ {e\left[ {\left( {9e\sin f + e\sin 3f + 12f - 12l} \right) \cos (g - hm - {\varphi _{3m}})} \right. } \right. \\&\left. { + 4e{{\cos }^3}f\sin (g - hm - {\varphi _{3m}}) + 6\sin (2f + g - hm - {\varphi _{3m}})} \right] \\&\left. { + 12\sin (f + g - hm - {\varphi _{3m}})} \right\} \frac{{\partial {F_{3mp}}\left( i \right) }}{{\partial i}} \\ \frac{{\partial {W_{4mp,A}}}}{{\partial a}} =&\frac{{7\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{12{a^{9/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left[ {e\sin f\left( {{e^2}\cos 2f + 5{e^2} + 9e\cos f + 18} \right) } \right. \\&\left. { + 3\left( {3{e^2} + 2} \right) (f - l)} \right] \cos (hm + {\varphi _{4m}}) \\ \frac{{\partial {W_{4mp,A}}}}{{\partial e}} =&- \frac{{\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) \cos (hm + {\varphi _{4m}})}}{{48{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{9/2}}}}\left\{ {30\left( {5{e^4} + 36{e^2} + 8} \right) \sin f} \right. \\&+ e\left[ {3{e^3}\sin 5f + 120\left( {3{e^2} + 4} \right) (f - l) + 30{e^2}\sin 4f} \right. \\&\left. {\left. { + 5\left( {5{e^2} + 24} \right) e\sin 3f + 240\left( {{e^2} + 1} \right) \sin 2f} \right] } \right\} \\ \frac{{\partial {W_{4mp,A}}}}{{\partial g}} =&0 \\ \frac{{\partial {W_{4mp,A}}}}{{\partial h}} =&\frac{{\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{6{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {e\sin f\left( {{e^2}\cos 2f + 5{e^2} + 9e\cos f + 18} \right) } \right. \\&\left. { + 3\left( {3{e^2} + 2} \right) (f - l)} \right\} m\sin (hm + {\varphi _{4m}}) \\ \frac{{\partial {W_{4mp,A}}}}{{\partial i}} =&- \frac{{\sqrt{\mu }{R_e}^4{N_{4m}}}}{{6{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\;\left[ {e\sin f\left( {{e^2}\cos 2f + 5{e^2} + 9e\cos f + 18} \right) } \right. \\&\left. { + 3\left( {3{e^2} + 2} \right) (f - l)} \right] \cos (hm + {\varphi _{4m}})\frac{{\partial {F_{4mp}}\left( i \right) }}{{\partial i}} \\ \frac{{\partial {W_{4mp,B}}}}{{\partial a}} =&- \frac{{7\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{16{a^{9/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ { - \frac{{{e^3}}}{{{\alpha _{4p}} - 3}}\sin \left( {({\alpha _{4p}} - 3)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) } \right. \\&- \frac{{{e^3}}}{{{\alpha _{4p}} + 3}}\sin \left( {({\alpha _{4p}} + 3)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - \frac{{6{e^2}}}{{{\alpha _{4p}} - 2}}\sin \left( {({\alpha _{4p}} - 2)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) \\&- \frac{{6{e^2}}}{{{\alpha _{4p}} + 2}}\sin \left( {({\alpha _{4p}} + 2)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) + \frac{{3\left( {{e^2} + 4} \right) e}}{{{\alpha _{4p}} - 1}}\sin \left( { - {\alpha _{4p}}f + f - {\alpha _{4p}}g - hm - {\varphi _{4m}}} \right) \\&\left. { - \frac{{3\left( {{e^2} + 4} \right) e}}{{{\alpha _{4p}} + 1}}\sin \left( {{\alpha _{4p}}f + f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - \frac{{4\left( {3{e^2} + 2} \right) }}{{{\alpha _{4p}}}}\sin \left( {{\alpha _{4p}}f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) } \right\} \\ \frac{{\partial {W_{4mp,B}}}}{{\partial e}} =&\frac{{\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{8{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{9/2}}}}\left\{ { - \frac{{\left( {4{e^2} + 3} \right) {e^2}}}{{{\alpha _{4p}} - 3}}\sin \left( {({\alpha _{4p}} - 3)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) } \right. \\&- \frac{{\left( {4{e^2} + 3} \right) {e^2}}}{{{\alpha _{4p}} + 3}}\sin \left( {({\alpha _{4p}} + 3)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - \frac{{6\left( {5{e^2} + 2} \right) e}}{{{\alpha _{4p}} - 2}} \\&\times \sin \left( {({\alpha _{4p}} - 2)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - \frac{{20\left( {3{e^2} + 4} \right) e}}{{{\alpha _{4p}}}}\sin \left( {{\alpha _{4p}}f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) \\&- \frac{{6\left( {5{e^2} + 2} \right) e}}{{{\alpha _{4p}} + 2}}\sin \left( {({\alpha _{4p}} + 2)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) + \frac{{3\left( {4{e^4} + 27{e^2} + 4} \right) }}{{{\alpha _{4p}} - 1}}\sin \left( { - {\alpha _{4p}}f + f} \right. \\&\left. { - {\alpha _{4p}}g - hm - {\varphi _{4m}}} \right) - \frac{{3\left( {4{e^4} + 27{e^2} + 4} \right) }}{{{\alpha _{4p}} + 1}}\sin \left( {{\alpha _{4p}}f + f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) \\&+ \sin f( - e\cos f - 2)\left[ {{e^3}\cos \left( {({\alpha _{4p}} - 3)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) + {e^3}\cos \left( {({\alpha _{4p}} + 3)f} \right. } \right. \\&\left. { + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) + 6{e^2}\cos \left( {({\alpha _{4p}} - 2)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) + 6{e^2}\cos \left( {({\alpha _{4p}} + 2)f} \right. \\&\left. { + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) + 3\left( {{e^2} + 4} \right) e\cos \left( { - {\alpha _{4p}}f + f - {\alpha _{4p}}g - hm - {\varphi _{4m}}} \right) + 3\left( {{e^2} + 4} \right) \\&\left. {\left. { \times e\cos \left( {{\alpha _{4p}}(f + g) + f + hm + {\varphi _{4m}}} \right) + 4\left( {3{e^2} + 2} \right) \cos \left( {{\alpha _{4p}}(f + g) + hm + {\varphi _{4m}}} \right) } \right] } \right\} \end{aligned}$$

$$\begin{aligned} \frac{{\partial {W_{4mp,B}}}}{{\partial g}} =&\frac{{\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{8{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ { - \frac{{{\alpha _{4p}}{e^3}}}{{{\alpha _{4p}} - 3}}\cos \left( {({\alpha _{4p}} - 3)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) } \right. \\&- \frac{{{\alpha _{4p}}{e^3}}}{{{\alpha _{4p}} + 3}}\cos \left( {({\alpha _{4p}} + 3)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - \frac{{6{\alpha _{4p}}{e^2}}}{{{\alpha _{4p}} - 2}}\cos \left( {({\alpha _{4p}} - 2)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) \\&- \frac{{6{\alpha _{4p}}{e^2}}}{{{\alpha _{4p}} + 2}}\cos \left( {({\alpha _{4p}} + 2)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - \frac{{3{\alpha _{4p}}\left( {{e^2} + 4} \right) e}}{{{\alpha _{4p}} - 1}}\cos \left( { - {\alpha _{4p}}f + f - {\alpha _{4p}}g - hm - {\varphi _{4m}}} \right) \\&\left. { - \frac{{3{\alpha _{4p}}\left( {{e^2} + 4} \right) e}}{{{\alpha _{4p}} + 1}}\cos \left( {{\alpha _{4p}}f + f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - 4\left( {3{e^2} + 2} \right) \cos \left( {{\alpha _{4p}}f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) } \right\} \\ \frac{{\partial {W_{4mp,B}}}}{{\partial h}} =&\frac{{\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) m}}{{8{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ { - \frac{{{e^3}}}{{{\alpha _{4p}} - 3}}\cos \left( {({\alpha _{4p}} - 3)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) } \right. \\&- \frac{{{e^3}}}{{{\alpha _{4p}} + 3}}\cos \left( {({\alpha _{4p}} + 3)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - \frac{{6{e^2}}}{{{\alpha _{4p}} - 2}}\cos \left( {({\alpha _{4p}} - 2)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) \\&- \frac{{6{e^2}}}{{{\alpha _{4p}} + 2}}\cos \left( {({\alpha _{4p}} + 2)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - \frac{{3\left( {{e^2} + 4} \right) e}}{{{\alpha _{4p}} - 1}}\cos \left( { - {\alpha _{4p}}f + f - {\alpha _{4p}}g - hm - {\varphi _{4m}}} \right) \\&\left. { - \frac{{3\left( {{e^2} + 4} \right) e}}{{{\alpha _{4p}} + 1}}\cos \left( {{\alpha _{4p}}f + f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - \frac{{4\left( {3{e^2} + 2} \right) }}{{{\alpha _{4p}}}}\cos \left( {{\alpha _{4p}}f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) } \right\} \\ \frac{{\partial {W_{4mp,B}}}}{{\partial i}} =&\frac{{\sqrt{\mu }{R_e}^4{N_{4m}}}}{{8{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\frac{{\partial {F_{4mp}}\left( i \right) }}{{\partial i}}\left\{ { - \frac{{{e^3}}}{{{\alpha _{4p}} - 3}}\sin \left( {({\alpha _{4p}} - 3)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) } \right. \\&- \frac{{{e^3}}}{{{\alpha _{4p}} + 3}}\sin \left( {({\alpha _{4p}} + 3)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - \frac{{6{e^2}}}{{{\alpha _{4p}} - 2}}\sin \left( {({\alpha _{4p}} - 2)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) \\&- \frac{{6{e^2}}}{{{\alpha _{4p}} + 2}}\sin \left( {({\alpha _{4p}} + 2)f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) + \frac{{3\left( {{e^2} + 4} \right) e}}{{{\alpha _{4p}} - 1}}\sin \left( { - {\alpha _{4p}}f + f - {\alpha _{4p}}g - hm - {\varphi _{4m}}} \right) \\&\left. { - \frac{{3\left( {{e^2} + 4} \right) e}}{{{\alpha _{4p}} + 1}}\sin \left( {{\alpha _{4p}}f + f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) - \frac{{4\left( {3{e^2} + 2} \right) }}{{{\alpha _{4p}}}}\sin \left( {{\alpha _{4p}}f + {\alpha _{4p}}g + hm + {\varphi _{4m}}} \right) } \right\} \frac{{\partial {F_{4mp}}\left( i \right) }}{{\partial i}} \\ \frac{{\partial {W_{4mp,C}}}}{{\partial a}} =&\frac{{7\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{160{a^{9/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {2{e^3}\sin (5f + 2g + hm + {\varphi _{4m}}) + 40\sin (2f + 2g + hm + {\varphi _{4m}})} \right. \\&+ 5e\left[ {2\left( {{e^2}\sin (f - 2g - hm - {\varphi _{4m}}) + 3\left( {{e^2} + 4} \right) \sin (f + 2g + hm + {\varphi _{4m}})} \right. } \right. \\&+ \left( {{e^2} + 4} \right) \sin (3f + 2g + hm + {\varphi _{4m}}) + 6e(f - l)\cos (2g + hm + {\varphi _{4m}}) \\&\left. {\left. {\left. { + 6e\sin (2f + 2g + hm + {\varphi _{4m}})} \right) + 3e\sin (4f + 2g + hm + {\varphi _{4m}})} \right] } \right\} \\ \frac{{\partial {W_{4mp,C}}}}{{\partial e}} =&- \frac{{\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{80{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{9/2}}}}\left\{ {8{e^4}\left[ {5\left( {\sin (f - 2g - hm - {\varphi _{4m}}) + 3\sin (f + 2g + hm + {\varphi _{4m}})} \right. } \right. } \right. \\&\left. {\left. { + \sin (3f + 2g + hm + {\varphi _{4m}})} \right) + \sin (5f + 2g + hm + {\varphi _{4m}})} \right] + 75{e^3}\left[ {4\sin (2f + 2g} \right. \\&\left. { + hm + {\varphi _{4m}}) + \sin (4f + 2g + hm + {\varphi _{4m}})} \right] + 60\left( {5{e^2} + 2} \right) e(f - l)\cos (2g + hm + {\varphi _{4m}}) \\&+ 6{e^2}\left[ {5\sin (f - 2g - hm - {\varphi _{4m}}) + \sin (5f + 2g + hm + {\varphi _{4m}}) + 45\left( {3\sin (f + 2g} \right. } \right. \\&\left. {\left. { + hm + {\varphi _{4m}}) + \sin (3f + 2g + hm + {\varphi _{4m}})} \right) } \right] + e\left[ {400\sin (2f + 2g + hm + {\varphi _{4m}})} \right. \\&\left. { + 30\sin (4f + 2g + hm + {\varphi _{4m}})} \right] + 80\sin f{(e\cos f + 1)^3}(e\cos f + 2)\cos (2f + 2g \\&\left. { + hm + {\varphi _{4m}}) + 40\left[ {3\sin (f + 2g + hm + {\varphi _{4m}}) + \sin (3f + 2g + hm + {\varphi _{4m}})} \right] } \right\} \\ \frac{{\partial {W_{4mp,C}}}}{{\partial g}} =&- \frac{{\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{80{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {4{e^3}\cos (5f + 2g + hm + {\varphi _{4m}}) + 80\cos (2f + 2g + hm + {\varphi _{4m}})} \right. \\&+ 5e\left[ { - 4{e^2}\cos (f - 2g - hm - {\varphi _{4m}}) + 12\left( {{e^2} + 4} \right) \cos (f + 2g + hm + {\varphi _{4m}})} \right. \\&+ 4\left( {{e^2} + 4} \right) \cos (3f + 2g + hm + {\varphi _{4m}}) - 24e(f - l)\sin (2g + hm + {\varphi _{4m}}) \\&\left. {\left. { + 24e\cos (2f + 2g + hm + {\varphi _{4m}}) + 6e\cos (4f + 2g + hm + {\varphi _{4m}})} \right] } \right\} \\ \frac{{\partial {W_{4mp,C}}}}{{\partial h}} =&- \frac{{\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{80{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {2{e^3}m\cos (5f + 2g + hm + {\varphi _{4m}}) + 40m\cos (2f + 2g + hm + {\varphi _{4m}})} \right. \\&+ 5e\left[ {2m\left( { - {e^2}\cos (f - 2g - hm - {\varphi _{4m}}) + 3\left( {{e^2} + 4} \right) \cos (f + 2g + hm + {\varphi _{4m}})} \right. } \right. \\&+ \left( {{e^2} + 4} \right) \cos (3f + 2g + hm + {\varphi _{4m}}) - 6e(f - l)\sin (2g + hm + {\varphi _{4m}}) \\&\left. {\left. {\left. { + 6e\cos (2f + 2g + hm + {\varphi _{4m}})} \right) + 3em\cos (4f + 2g + hm + {\varphi _{4m}})} \right] } \right\} \\ \frac{{\partial {W_{4mp,C}}}}{{\partial i}} =&- \frac{{\sqrt{\mu }{R_e}^4{N_{4m}}}}{{80{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {2{e^3}\sin (5f + {\vartheta _{4mp}}) + 5e\left[ {2\left( {{e^2}\sin (f - {\vartheta _{4mp}}) + 3\left( {{e^2} + 4} \right) \sin (f + {\vartheta _{4mp}})} \right. } \right. } \right. \\&\left. { + \left( {{e^2} + 4} \right) \sin (3f + {\vartheta _{4mp}}) + 6e(f - l)\cos {\vartheta _{4mp}} + 6e\sin (2f + {\vartheta _{4mp}})} \right) \\&\left. {\left. { + 3e\sin (4f + {\vartheta _{4mp}})} \right] + 40\sin (2f + {\vartheta _{4mp}})} \right\} \frac{{\partial {F_{4mp}}\left( i \right) }}{{\partial i}} \\ \frac{{\partial {W_{4mp,D}}}}{{\partial a}} =&\frac{{7\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{160{a^{9/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {e\left[ {e\left( {2\left[ {5\left( {4e\sin f + e\sin 3f + 6f - 6l} \right) + e\sin 5f} \right] } \right. } \right. } \right. \\&\times \cos (2g - hm - {\varphi _{4m}}) + 32e{\cos ^5}f\sin (2g - hm - {\varphi _{4m}}) + 60\sin (2f + 2g - hm - {\varphi _{4m}}) \\&\left. { + 15\sin (4f + 2g - hm - {\varphi _{4m}})} \right) + 40\left[ {3\sin (f + 2g - hm - {\varphi _{4m}})} \right. \\&\left. {\left. {\left. { + \sin (3f + 2g - hm - {\varphi _{4m}})} \right] } \right] + 40\sin (2f + 2g - hm - {\varphi _{4m}})} \right\} \\ \frac{{\partial {W_{4mp,D}}}}{{\partial e}} =&\frac{{\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{80{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{9/2}}}}\left\{ { - 40{e^4}\sin (3f + 2g - hm - {\varphi _{4m}}) - 8{e^4}\sin (5f + 2g - hm - {\varphi _{4m}})} \right. \\&- 40{e^4}\sin (f - 2g + hm + {\varphi _{4m}}) - 300{e^3}\sin (2f + 2g - hm - {\varphi _{4m}})\\&- 75{e^3}\sin (4f + 2g - hm - {\varphi _{4m}}) - 60\left( {5{e^2} + 2} \right) e(f - l)\cos (2g - hm - {\varphi _{4m}})\\&- 270{e^2}\sin (3f + 2g - hm - {\varphi _{4m}}) - 6{e^2}\sin (5f + 2g - hm - {\varphi _{4m}})\\&- 30{e^2}\sin (f - 2g + hm + {\varphi _{4m}}) - 30\left( {4{e^4} + 27{e^2} + 4} \right) \sin (f + 2g - hm - {\varphi _{4m}})\\&- 400e\sin (2f + 2g - hm - {\varphi _{4m}}) - 30e\sin (4f + 2g - hm - {\varphi _{4m}})\\&- 80\sin f{(e\cos f + 1)^3}(e\cos f + 2)\cos (2f + 2g - hm - {\varphi _{4m}}) \\&\left. { - 40\sin (3f + 2g - hm - {\varphi _{4m}})} \right\} \end{aligned}$$

$$\begin{aligned} \frac{{\partial {W_{4mp,D}}}}{{\partial g}} =&- \frac{{\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) }}{{80{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {e\left[ {e\left( { - 4\left[ {5\left( {4e\sin f + e\sin 3f + 6f - 6l} \right) + e\sin 5f} \right] } \right. } \right. } \right. \\&\times \sin (2g - hm - {\varphi _{4m}}) + 64e{\cos ^5}f\cos (2g - hm - {\varphi _{4m}}) + 120\cos (2f + 2g - hm - {\varphi _{4m}}) \\&\left. { + 30\cos (4f + 2g - hm - {\varphi _{4m}})} \right) + 80\left( {3\cos (f + 2g - hm - {\varphi _{4m}})} \right. \\&\left. {\left. {\left. { + \cos (3f + 2g - hm - {\varphi _{4m}})} \right) } \right] + 80\cos (2f + 2g - hm - {\varphi _{4m}})} \right\} \\ \frac{{\partial {{{\mathcal {W}}}_{4mp,D}}}}{{\partial h}} =&\frac{{\sqrt{\mu }{R_e}^4{N_{4m}}{F_{4mp}}\left( i \right) m}}{{80{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {30e\left( {{e^2} + 4} \right) \cos (f + 2g - hm - {\varphi _{4m}}) + 20\left( {3{e^2} + 2} \right) } \right. \\&\times \cos (2f + 2g - hm - {\varphi _{4m}}) + e\left[ {10\left( {{e^2} + 4} \right) \cos (3f + 2g - hm - {\varphi _{4m}}) + e} \right. \\&\times \left( {2e\cos (5f + 2g - hm - {\varphi _{4m}}) - 10e\cos (f - 2g + hm + {\varphi _{4m}})} \right. \\&- 60f\sin (2g - hm - {\varphi _{4m}}) + 15\cos (4f + 2g - hm - {\varphi _{4m}}) \\&\left. {\left. {\left. { + 60l\sin (2g - hm - {\varphi _{4m}})} \right) } \right] } \right\} \\ \frac{{\partial {W_{4mp,D}}}}{{\partial i}} =&- \frac{{\sqrt{\mu }{R_e}^4{N_{4m}}}}{{80{a^{7/2}}{{\left( {1 - {e^2}} \right) }^{7/2}}}}\left\{ {e\left[ {e\left( {2\cos {\vartheta _{4mp}}(5(4e\sin f + e\sin 3f + 6f - 6l) + e\sin 5f)} \right. } \right. } \right. \\&\left. { - 32e{{\cos }^5}f\sin {\vartheta _{4mp}} + 60\sin (2f - {\vartheta _{4mp}}) + 15\sin (4f - {\vartheta _{4mp}})} \right) \\&\left. {\left. { + 40(3\sin (f - {\vartheta _{4mp}}) + \sin (3f - {\vartheta _{4mp}}))} \right] + 40\sin (2f - {\vartheta _{4mp}})} \right\} \frac{{\partial {F_{4mp}}\left( i \right) }}{{\partial i}}. \end{aligned}$$