Delaunay variables approach to the elimination of the perigee in Artificial Satellite Theory

Abstract

Analytical integration in Artificial Satellite Theory may benefit from different canonical simplification techniques, like the elimination of the parallax, the relegation of the nodes, or the elimination of the perigee. These techniques were originally devised in polar-nodal variables, an approach that requires expressing the geopotential as a Pfaffian function in certain invariants of the Kepler problem. However, it has been recently shown that such sophisticated mathematics are not needed if implementing both the relegation of the nodes and the parallax elimination directly in Delaunay variables. Proceeding analogously, it is shown here how the elimination of the perigee can be carried out also in Delaunay variables. In this way the construction of the simplification algorithm becomes elementary, on one hand, and the computation of the transformation series is achieved with considerable savings, on the other, reducing the total number of terms of the elimination of the perigee to about one third of the number of terms required in the classical approach.

Appendix

The integration “constant” $V_2$ is

$$\begin{aligned} V_2&= \frac{n\,\alpha ^4}{512a^2}\,\frac{J_2^2}{\eta ^7}\left\{ \frac{\left( 1-15 c^2\right) ^2 \left( 2-15 c^2\right) }{2(1-5c^2)^3}\,s^4\,e^4\sin 4\omega \right. \nonumber \\&\left. +\left[ 12\,\frac{6-43c^2+125c^4}{1-5c^2} -(1-15c^2)\,\frac{25-126c^2+45c^4}{(1-5c^2)^2}\,e^2\right] s^2\,e^2\sin 2\omega \right\} .\nonumber \\ \end{aligned}$$

(45)

The term $W_3$ of the generating function of the elimination of the perigee is

$$\begin{aligned} W_3&= V_3-J_2^3\frac{\alpha ^6 n }{a^4 \eta ^{11}}\left\{ \frac{14-15s^2}{4-5 s^2}s^2 \left[ \left( \frac{2925 s^8-7710 s^6+7064 s^4-2496 s^2+224}{2048(4-5s^2)^2}e^2 \right. \right. \right. \nonumber \\&\left. -\frac{(14-15 s^2)\,(2-3 s^2)}{2048(4-5s^2)}s^2\right) e^2\sin 2f+\left( -\frac{(14-15s^2)\,(2-3 s^2)}{512(4-5 s^2)}s^2 \right. \nonumber \\&\left. \left. +\frac{6075 s^8-15960 s^6+14556s^4-5104 s^2+448}{1024 \left( 4-5 s^2\right) ^2}e^2 \right) e\sin {f} \right] \nonumber \\&-\frac{(14-15 s^2)^2}{(4-5 s^2)^2}s^4\left[ \frac{135s^4-242 s^2+112}{2048(4-5 s^2)}e^3\sin (f+4 g)+\frac{3(2-3 s^2)}{2048}\,e^3 \sin (3 f+4 g) \right. \nonumber \\&\left. +\left( \frac{30 s^4-55 s^2+26}{2048(4-5 s^2)}e^2+\frac{5(2-3s^2)}{2048}\right) e^2\sin (2f+4g) +\frac{(2-3 s^2)}{4096}\,e^4 \sin (4f+4 g) \right] \nonumber \\&+\frac{3s^2 }{4-5 s^2}\left[ -\frac{14-15 s^2}{128}\,(2-3 s^2)^2\left( \frac{1}{8} e^2 \sin (4 f+2 g)+e \sin (3 f+2 g)\right) \right. \nonumber \\&-\left( \frac{3(14-15 s^2)}{1024(4-5s^2)}\,(85 s^6-22 s^4-96 s^2+48)\,e^2 \right. \nonumber \\&\left. +\frac{-3105 s^6+7251 s^4-5598 s^2+1424}{128}\right) e\sin (f+2 g) \nonumber \\&+\frac{(14-15 s^2)\,(-45 s^4-36 s^2+56)\,(2-3s^2)}{1024(4-5 s^2)}\left( \frac{1}{4} e^4 \sin (2 f-2 g)+e^3 \sin (f-2 g)\right) \nonumber \\&+\left( -\frac{3(14-15 s^2)}{4096(4-5 s^2)}\,(85 s^6-22s^4-96 s^2+48)\,e^4 \right. \nonumber \\&\left. \left. \left. +\frac{1890s^6-4317 s^4+3258 s^2-808}{512}e^2-\left( \frac{14}{16}-\frac{15}{16}s^2\right) \left( 1-\frac{3}{2}s^2\right) ^2\right) \sin (2f+2g)\right] \right\} \nonumber \\ \end{aligned}$$

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where the integration constant $V_3$ is

$$\begin{aligned} V_3&= -\frac{\alpha ^6 n J_2^3}{a^4 \eta ^{11} \left( 4-5 s^2\right) ^3}\left\{ \!-\!\left( \frac{275}{16384}s^4\!-\!\frac{1445}{49152}s^2\!+\!\frac{317}{24576}\right) \frac{(14\!-\!15 s^2)^3}{(4-5 s^2)^2}e^6s^6\sin 6\omega \qquad \right. \nonumber \\&+\left[ \left( \frac{225}{4096}s^6+\frac{2655}{32768}s^4-\frac{513}{2048}s^2+\frac{491}{4096}\right) \frac{(14-15 s^2)^2}{4-5 s^2}\,e^2 -\frac{617625}{8192}s^8 \right. \nonumber \\&\left. +\frac{4334325}{16384}s^6\!-\!\frac{2843175}{8192}s^4\!+\!\frac{826239}{4096}s^2\!-\!\frac{22429}{512}\right] e^4s^4\sin 4\omega \!+\!\left[ \!-\!\left( \frac{1164375}{8192}s^{12} \right. \right. \nonumber \\&-\frac{8703375}{16384}s^{10}+\frac{6658575}{8192}s^8-\frac{2712045}{4096}s^6+\frac{644695}{2048}s^4-\frac{11333}{128}s^2 \nonumber \\&\left. +\frac{49}{4}\right) \frac{e^4(14-15s^2)}{(4-5s^2)^2}+\left( \frac{14338125}{2048}s^{12}-\frac{126428625}{4096}s^{10}+\frac{56930775}{1024}s^8 \right. \nonumber \\&\left. -\frac{26745515}{512}s^6+\frac{6896989}{256}s^4-\frac{115555}{16}s^2+\frac{1575}{2}\right) \frac{e^2}{4-5s^2}+\frac{16407375}{4096}s^{10} \nonumber \\&\left. \left. -\frac{17059525}{1024}s^8\!+\!\frac{7037775}{256}s^6\!-\!\frac{1442021}{64}s^4\!+\!\frac{146825}{16}s^2\!-\!\frac{11883}{8}\right] e^2s^2\sin 2\omega \right\} \nonumber \\ \end{aligned}$$

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