Variational and symplectic integrators for satellite relative orbit propagation including drag

Abstract

Orbit propagation algorithms for satellite relative motion relying on Runge–Kutta integrators are non-symplectic—a situation that leads to incorrect global behavior and degraded accuracy. Thus, attempts have been made to apply symplectic methods to integrate satellite relative motion. However, so far all these symplectic propagation schemes have not taken into account the effect of atmospheric drag. In this paper, drag-generalized symplectic and variational algorithms for satellite relative orbit propagation are developed in different reference frames, and numerical simulations with and without the effect of atmospheric drag are presented. It is also shown that high-order versions of the newly-developed variational and symplectic propagators are more accurate and are significantly faster than Runge–Kutta-based integrators, even in the presence of atmospheric drag.

Appendices

Appendix A: transformation from earth-centered inertial to LVLH

The relative position between two satellites in elliptic orbits can be expressed in the LVLH reference frame using the relations (Alfriend et al. 2009)

$$\begin{aligned} \xi= & {} \dfrac{\delta \varvec{r}^{T} \varvec{r}_{L}}{r_{L}} \end{aligned}$$

(103)

$$\begin{aligned} \varrho= & {} \dfrac{\delta \varvec{r}^{T} \left( \varvec{h}_{L} \times \varvec{r}_{L} \right) }{\Vert \varvec{h}_{L} \times \varvec{r}_{L} \Vert } \end{aligned}$$

(104)

$$\begin{aligned} \psi= & {} \dfrac{\delta \varvec{r}^{T} \varvec{h}_{L}}{h_{L}}, \end{aligned}$$

(105)

where $\varvec{h}_{L} = \varvec{r}_{L} \times \varvec{v}_{L}$. Next, the relative velocity expressed also in the LVLH reference frame is obtained by differentiating the previous equations:

$$\begin{aligned} {\dot{\xi }}= & {} \dfrac{\delta \varvec{v}^{T} \varvec{r}_{L} + \delta \varvec{r}^{T} \varvec{v}_{L}}{r_{L}} - \dfrac{\left( \delta \varvec{r}^{T} \varvec{r}_{L} \right) \left( \delta \varvec{r}^{T}\varvec{v}_{L} \right) }{r_{L}^{3}} \end{aligned}$$

(106)

$$\begin{aligned} {\dot{\varrho }}= & {} \dfrac{\delta \varvec{v}^{T} \left( \varvec{h}_{L} \times \varvec{r}_{L} \right) + \delta \varvec{r}^{T} \left( \dot{\varvec{h}}_{L} \times \varvec{r}_{L} + \varvec{h}_{L} \times \varvec{v}_{L}\right) }{\Vert \varvec{h}_{L} \times \varvec{r}_{L} \Vert } \nonumber \\&- \dfrac{\delta \varvec{r}^{T} \left( \varvec{h}_{L} \times \varvec{r}_{L} \right) \left( \varvec{h}_{L} \times \varvec{r}_{L} \right) ^{T} \left( \dot{\varvec{h}}_{L} \times \varvec{r}_{L} + \varvec{h}_{L} \times \varvec{v}_{L}\right) }{\Vert \varvec{h}_{L} \times \varvec{r}_{L} \Vert ^{3}} \end{aligned}$$

(107)

$$\begin{aligned} {\dot{\psi }}= & {} \dfrac{\delta \varvec{v}^{T} \varvec{h}_{L} + \delta \varvec{r}^{T} \dot{\varvec{h}}_{L}}{h_{L}} - \dfrac{\delta \varvec{r}^{T}\varvec{h}_{L}\left( \varvec{h}_{L}^{T} \dot{\varvec{h}}_{L} \right) }{h_{L}^{3}}, \end{aligned}$$

(108)

where $\dot{\varvec{h}}_{L} = \varvec{r}_{L} \times \dot{\varvec{v}}_{L}$.

Appendix B: elements of the Jacobian matrix

The exact calculation of the elements of the Jacobian matrix are used to improve the speed of the Newton–Raphson method in the semi-implicit symplectic integrator presented in Sect. 3.2.1. First, the system of equations of interest is re-arranged in the form $f(x) = 0$ and then the expression is differentiated with respect to the unknown elements. For the half-indexed momenta, Eqs. (41) and (42) are re-arranged as

$$\begin{aligned} f_{1}= & {} R_{1/2} - R - \frac{h}{2}\left\{ \frac{\varTheta _{1/2}^{2}}{r^{3}} -\frac{\mu }{r^{2}} - \frac{\kappa }{r^{4}}\left[ 1 - 3\sin ^{2}\theta \left( 1 - \frac{N_{1/2}^{2}}{\varTheta _{1/2}^{2}} \right) \right] \right\} \nonumber \\&\quad +\, \frac{h}{2}B_{L}Rv_{L} = 0 \end{aligned}$$

(109)

$$\begin{aligned} f_{2}= & {} \varTheta _{1/2} - \varTheta - \frac{h}{2}\left[ -\frac{\kappa \sin 2\theta }{r^{3}} \left( 1 - \frac{N_{1/2}^{2}}{\varTheta _{1/2}^{2}} \right) \right] + \frac{h}{2}B_{L}\varTheta v_{L} = 0, \end{aligned}$$

(110)

and then each of these equations is differentiated with respect to the unknown elements $R_{1/2}$ and $\varTheta _{1/2}$,

$$\begin{aligned} \frac{\partial f_{1}}{\partial R_{1/2}}= & {} 1 \end{aligned}$$

(111)

$$\begin{aligned} \frac{\partial f_{1}}{\partial \varTheta _{1/2}}= & {} -h\left( \frac{\varTheta _{1/2}}{r^{3}} + \frac{3\kappa \sin ^{2}\theta }{r^{4}}\frac{N_{1/2}^{2}}{\varTheta _{1/2}^{3}} \right) \end{aligned}$$

(112)

$$\begin{aligned} \frac{\partial f_{2}}{\partial R_{1/2}}= & {} 0 \end{aligned}$$

(113)

$$\begin{aligned} \frac{\partial f_{2}}{\partial \varTheta _{1/2}}= & {} 1 + h\frac{\kappa \sin 2\theta }{r^{3}}\frac{N_{1/2}^{2}}{\varTheta _{1/2}^{3}}. \end{aligned}$$

(114)

Then, for the new positions, Eqs. (44)–(46) are re-arranged as

$$\begin{aligned} g_{1}= & {} r_{+} - r - hR_{1/2} = 0 \end{aligned}$$

(115)

$$\begin{aligned} g_{2}= & {} \theta _{+} - \theta - \frac{h}{2}\left[ \frac{\varTheta _{1/2}}{r^{2}} + \frac{2\kappa \sin ^{2}\theta }{r^{3}\varTheta _{1/2}}\left( \frac{N_{1/2}^{2}}{\varTheta _{1/2}^{2}}\right) \right] \nonumber \\&\quad -\, \frac{h}{2}\left[ \frac{\varTheta _{1/2}}{r_{+}^{2}} + \frac{2\kappa \sin ^{2}\theta _{+}}{r_{+}^{3}\varTheta _{1/2}}\left( \frac{N_{1/2}^{2}}{\varTheta _{1/2}^{2}} \right) \right] = 0 \end{aligned}$$

(116)

$$\begin{aligned} g_{3}= & {} \nu _{+} - \nu - \frac{h}{2}\left[ -\frac{2\kappa \sin ^{2}\theta }{r^{3}\varTheta _{1/2}}\left( \frac{N_{1/2}}{\varTheta _{1/2}} \right) \right] \nonumber \\&\quad -\, \frac{h}{2}\left[ -\frac{2\kappa \sin ^{2}\theta _{+}}{r_{+}^{3}\varTheta _{1/2}}\left( \frac{N_{1/2}}{\varTheta _{1/2}} \right) \right] = 0. \end{aligned}$$

(117)

After differentiating each of these equations with respect to the unknown elements $r_{+}$, $\theta _{+}$ and $\nu _{+}$, we obtain the expressions:

$$\begin{aligned} \frac{\partial g_{1}}{\partial r_{+}}= & {} 1 \end{aligned}$$

(118)

$$\begin{aligned} \frac{\partial g_{1}}{\partial \theta _{+}}= & {} 0 \end{aligned}$$

(119)

$$\begin{aligned} \frac{\partial g_{1}}{\partial \nu _{+}}= & {} 0 \end{aligned}$$

(120)

$$\begin{aligned} \frac{\partial g_{2}}{\partial r_{+}}= & {} h\left[ \frac{\varTheta _{1/2}}{r_{+}^{3}} + \frac{3\kappa \sin ^{2}\theta _{+}}{\varTheta _{1/2}r_{+}^{4}} \left( \frac{N_{1/2}}{\varTheta _{1/2}} \right) ^2\right] \end{aligned}$$

(121)

$$\begin{aligned} \frac{\partial g_{2}}{\partial \theta _{+}}= & {} 1 - h\frac{\kappa \sin 2\theta _{+}}{\varTheta _{1/2}r_{+}^{3}}\left( \frac{N_{1/2}}{\varTheta _{1/2}} \right) ^2 \end{aligned}$$

(122)

$$\begin{aligned} {\frac{\partial g_{2}}{\partial \nu _{+}} }= & {} {0} \end{aligned}$$

(123)

$$\begin{aligned} \frac{\partial g_{3}}{\partial r_{+}}= & {} {-h\frac{3\kappa \sin ^2\theta _{+}N_{1/2}}{r_{+}^{4}\varTheta _{1/2}^{2}}} \end{aligned}$$

(124)

$$\begin{aligned} \frac{\partial g_{3}}{\partial \theta _{+}}= & {} {h\frac{2\kappa \sin \theta _{+}\cos \theta _{+}N_{1/2}}{r_{+}^{3}\varTheta _{1/2}^{2}}} \end{aligned}$$

(125)

$$\begin{aligned} \frac{\partial g_{3}}{\partial \nu _{+}}= & {} 1. \end{aligned}$$

(126)