Intermediary LEO propagation including higher order zonal harmonics

Abstract

Two new intermediary orbits of the artificial satellite problem are proposed. The analytical solutions include higher order effects of the geopotential, and are obtained by means of a torsion transformation applied to the quasi-Keplerian system resulting after the elimination of the parallax simplification, for the first intermediary, and after the elimination of the parallax and perigee simplifications, for the second one. The new intermediaries perform notably well for low Earth orbits propagation, are free from special functions, and result advantageous, both in accuracy and efficiency, when compared to the standard Cowell integration of the $J_2$ problem, thus providing appealing alternatives for onboard, short-term, orbit propagation under limited computational resources.

Appendix

Calling $\xi$ to any of the canonical variables and using

$$\begin{aligned} \epsilon =-\frac{1}{2}\frac{\alpha ^2}{p^2}J_{2}, \end{aligned}$$

(44)

as given in Eq. (10), under the simplifications used by the LEO intermediary, the direct transformation is written

$$\begin{aligned} \xi =\xi ^{\prime }+\epsilon {\varDelta }_1\xi ^{\prime } \end{aligned}$$

(45)

and the inverse transformation

$$\begin{aligned} \xi ^{\prime }=\xi +\epsilon {\varDelta }_2\xi +\frac{1}{2}\epsilon ^2 \delta _2\xi . \end{aligned}$$

(46)

The corrections are conveniently expressed in polar nodal variables and are given below, where the eccentricity functions $\kappa$ and $\sigma$ defined in Eq. (41) are used for convenience. Besides, the abbreviation $\tilde{J}_{m}=J_{m}{/}J_{2}^2=\mathcal {O}(1)$, $m>2$, is used in the second order corrections.

Note that the first order corrections were first provided by Deprit (1981a), and are given here for the sake of completeness—but in the arrangement proposed by Gurfil and Lara (2014) whose evaluation is much more efficient. Besides, second order terms were previously given in Lara (2015b) but limited to the $J_2$ perturbation.

1.1 First order corrections

The first-order corrections are formally the same both in the direct and inverse transformations but with different signs, namely ${\varDelta }_1={\varDelta }$ and ${\varDelta }_2=-{\varDelta }$ where:

$$\begin{aligned}&{\varDelta }{r} = p\left( 1-\frac{3}{2}s^2-\frac{1}{2}s^2\cos 2\theta \right) , \end{aligned}$$

(47)

$$\begin{aligned}&{\varDelta }\theta = \left[ 1-6c^2+\left( 1-2c^2\right) \cos 2\theta \right] \sigma -\left[ \frac{1}{4}-\frac{7}{4}c^2+\left( 1-3c^2\right) \kappa \right] \sin 2\theta , \end{aligned}$$

(48)

$$\begin{aligned}&{\varDelta }\nu = c\left[ (3+\cos 2\theta ) \sigma -\left( \frac{3}{2}+2\kappa \right) \sin 2\theta \right] , \end{aligned}$$

(49)

$$\begin{aligned}&{\varDelta }{R} = \frac{{\varTheta }}{r} (1+\kappa )s^2\sin 2\theta , \end{aligned}$$

(50)

$$\begin{aligned}&{\varDelta }{\varTheta }= -{\varTheta }s^2\left[ \left( \frac{3}{2}+2\kappa \right) \cos 2\theta +\sigma \sin 2\theta \right] , \end{aligned}$$

(51)

$$\begin{aligned}&{\varDelta }{N} = 0, \end{aligned}$$

(52)

and the right member of each of Eqs. (47)–(52) as well as p in Eq. (44) must be expressed in prime variables when computing ${\varDelta }_1\xi ^{\prime }\equiv {\varDelta }\xi ^{\prime }$, or in original ones when computing ${\varDelta }_2\xi \equiv -{\varDelta }\xi$.

1.2 Second order, simplified inverse corrections

The necessary corrections are given in following formulas, where all the symbols are functions of the original variables. The correction $\delta _2{R}$ is not used by the accelerated intermediary and is provided just for convenience of those interested in checking Eq. (25).

$$\begin{aligned} \delta _2{r}= & {} p \left\{ -3+10c^2+c^4 -\left( 4-32c^2\right) s^2\cos 2\theta -s^4\cos 4\theta \right. \nonumber \\&-\frac{3}{2}\frac{p}{\alpha }\tilde{J}_{3}\left[ \left( 1-5c^2\right) s\sin \theta +\frac{5}{6}s^3\sin 3\theta \right] \nonumber \\&\left. -\tilde{J}_{4}\left[ \frac{9}{8}\left( 3-30c^2+35c^4\right) +\frac{5}{2}\left( 1-7c^2\right) s^2\cos 2\theta -\frac{7}{8}s^4\cos 4\theta \right] +\mathcal {O}(e)\right\} \end{aligned}$$

(53)

$$\begin{aligned} \delta _2{R}= & {} \frac{{\varTheta }}{p}\left\{ s^2\left( 2-22c^2\right) \sin 2\theta +s^4\sin 4\theta +\frac{3}{2}\frac{p}{\alpha }\tilde{J}_{3}\left[ \left( 1-5c^2\right) s\cos \theta -\frac{5}{2}s^3\cos 3\theta \right] \right. \nonumber \\&\left. +\tilde{J}_{4} \left[ 5\left( 1-7c^2\right) s^2\sin 2\theta -\frac{7}{2}s^4\sin 4\theta \right] +\mathcal {O}(e) \right\} \end{aligned}$$

(54)

$$\begin{aligned} \delta _2{\varTheta }= & {} {\varTheta }\left\{ -\left[ \frac{1}{4}\left( 7-25c^2\right) +6\left( 1-3c^2\right) \kappa \right] s^2-\left[ \frac{3}{2}\left( 1-9c^2\right) +\left( 4-44c^2\right) \kappa \right] s^2\cos 2\theta \right. \nonumber \\&-\sigma \left( 2-28c^2\right) s^2\sin 2\theta +\frac{3}{4}s^4\cos 4\theta -\frac{3}{2}\sigma s^4\sin 4\theta \nonumber \\&+\frac{p}{\alpha }\tilde{J}_{3} \left[ \frac{3}{2}\left( 1-5c^2\right) s\left( \sigma \cos \theta +(2+\kappa )\sin \theta \right) -\frac{5}{4}(4+9\kappa )s^3\sin 3\theta +\frac{15}{4}\sigma s^3\cos 3\theta \right] \nonumber \\&-\tilde{J}_{4} \left[ \frac{5}{2}\left( 1-7c^2\right) s^2 \left( 2\sigma \sin 2\theta +(1+4\kappa )\cos 2\theta \right) -\frac{7}{8}(5+16\kappa )s^4\cos 4\theta \right. \nonumber \\&\left. \left. -\frac{7}{2}\sigma s^4\sin 4\theta \right] +\mathcal {O}\left( e^2\right) \right\} \end{aligned}$$

(55)