Mean Element Propagations Using Numerical Averaging

Abstract

The long-term evolution characteristics and stability of an orbit are well characterized using a mean element propagation of the perturbed two body variational equations of motion. The averaging process eliminates short period terms leaving only secular and long period effects. In this study, a non-traditional approach is taken that averages the variational equations using adaptive numerical techniques and then numerically integrating the resulting equations of motion. Doing this avoids the Fourier series expansions and truncations required by the traditional analytic methods. The resultant numerical techniques can be easily adapted to propagations at most solar system bodies.

Appendix: : Direct Equinoctal Elements And Partials

The direct equinoctial elements as functions of the classical elements $\left \{ {a,e,i,{\Omega } ,\omega ,M} \right \}$ can be defined as

$$ \begin{array}{l} a\equiv a, \\ h\equiv e\sin \left( {\omega +{\Omega}} \right), \\ k\equiv e\cos \left( {\omega +{\Omega}} \right), \\ p\equiv \tan \left( {\frac{i}{2}} \right)\sin {\Omega} , \\ q\equiv \tan \left( {\frac{i}{2}} \right)\cos {\Omega} , \\ \lambda \equiv M+\omega +{\Omega} . \end{array} $$

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Some intermediate quantities are now defined. The equinoctial reference frame is composed of three orthogonal unit vectors $\left \{ {\text {\textbf {f}},\text {\textbf {g}},\text {\textbf {w}}} \right \}$ where

• 1. f and g are in the satellite orbit plane,

• 2. w is along the orbit normal,

• 3. The angle between f and the ascending node is equal to the longitude of the ascending node Ω.

Using these properties the unit vectors $\left \{ {\text {\textbf {f}},\text {\textbf {g}},\text {\textbf {w}}} \right \}$are obtained using

$$ \begin{array}{l} \text{\textbf{f}}=\frac{1}{1+p^{2}+q^{2}}\left[ {{\begin{array}{l} {1-p^{2}+q^{2}}\\ {2pq}\\ {-2p} \end{array}} } \right], \\ \text{\textbf{g}}=\frac{1}{1+p^{2}+q^{2}}\left[ {{\begin{array}{l} {2pq} \hfill \\ {1+p^{2}-q^{2}}\\ {2q} \end{array}} } \right], \\ \text{\textbf{w}}=\frac{1}{1+p^{2}+q^{2}}\left[ {{\begin{array}{l} {2p}\\ {-2q}\\ {1-p^{2}-q^{2}} \end{array}} } \right]. \end{array} $$

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From the position r and velocity $\dot {\textbf {r}}$ vectors the following components $\left \{ {X,Y,\dot {{X}},\dot {{Y}}} \right \}$can be computed

$$ \begin{array}{c} \text{\textbf{r}}\equiv X \textbf{f}+Y \textbf{g}\\ \dot{\textbf{r}}\equiv \dot{X}\textbf{f}+\dot{{Y}} \textbf{g} \end{array}\\ \to \begin{array}{c} X=\textbf{r}\cdot \textbf{f}, Y= \textbf{r}\cdot \textbf{g}\\ \dot{{X}}= \textbf{r} \cdot \textbf{f},\dot{Y}= \textbf{r}\cdot \textbf{g} \end{array}. $$

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Finally the partials identified in Gauss’ (7) are

$$\begin{array}{@{}rcl@{}} \frac{\partial a}{\partial{\dot{\textbf{r}}}}&=&\frac{2a^{2} \dot{\textbf{r}}}{\mu} , \\ \frac{\partial h}{\partial{\dot{\textbf{r}}}}&=&\frac{\left({2X\dot{Y}-\dot{X}Y}\right)\textbf{f}-X \dot{X}\textbf{g}}{\mu} +\frac{k\left( {qY-pX} \right)\textbf{w}}{AB}, \\ \frac{\partial k}{\partial{\dot{\textbf{r}}}}&=&\frac{\left({2X\dot{Y}-\dot{X}Y}\right)\textbf{f}-Y\dot{Y}\textbf{g}} {\mu}-\frac{h\left( {qY-pX} \right)\textbf{w}}{AB}, \\ \frac{\partial p}{\partial{\dot{\textbf{r}}}}&=&\frac{CY\textbf{w}}{2AB}, \\ \frac{\partial q}{\partial{\dot{\textbf{r}}}}&=&\frac{CX\textbf{w}}{2AB}, \\ \frac{\partial \lambda} {\partial{\dot{\textbf{r}}}}&=&-\frac{2 \textbf{r}}{A}+\frac{k\frac{\partial h}{\partial \dot{\textbf{r}}}-h\frac{\partial k}{\partial{\dot{\textbf{r}}}}}{1+B} +\frac{\left({qY-pX}\right)\textbf{w}}{A}, \end{array} $$

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Where the following definitions apply

$$\begin{array}{@{}rcl@{}} A=na^{2}=\sqrt {\mu a}, \\ B=\sqrt {1-h^{2}-k^{2}}, \\ C=1+p^{2}+q^{2}. \end{array} $$

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