A computational approach to the powered Swing-By in the elliptic restricted problem

Abstract

This work performs a computational investigation of the energy variations given by a powered Swing-By maneuver realized in an elliptical system. It extends previous works by giving the freedom to choose the location and the direction of the thrust vector, aspects that were not considered before in the literature. Those variations are obtained numerically as a function of the parameters related to the thrust (magnitude, direction and location of the application) and the orbital parameters of the primaries (eccentricity and true anomaly). The maneuver is realized around the smaller primary, and the energy variations are measured with respect to the main body of the system. The initial orbit of the space vehicle is defined by its periapsis distance, angle and approach velocity with respect to the smaller primary. The study is applied to a system composed of two primaries that are in elliptic orbits around the center of mass of the system. The eccentricity is varied as a free parameter, to measure its effects. The results show that the best maneuvers apply the thrust at a point inside the sphere of influence of the secondary body, but not in the periapsis of the orbit. The best direction of the thrust is not aligned with the motion of the space vehicle. The techniques studied here are applied in situations where it is desired to increase the energy of the space vehicle. Empirical equations are obtained for the energy variations, based on the simulations made in the present paper. The numerical approach makes the results more accurate and not limited to particular regions of the eccentricity.

Appendices

Appendix 1: Coefficients of the empirical equations of $\Delta E_{\max }$

$${\text{Coef}}_{n} = z_{10} \delta V^{9} + z_{9} \delta V^{8} + z_{8} \delta V^{7} + z_{7} \delta V^{6} + z_{6} \delta V^{5} + z_{5} \delta V^{4} + z_{4} \delta V^{3} + z_{3} \delta V^{2} + z_{2} \delta V + z_{1}.$$

(9)

See Table 4.

Table 4 Coefficients of Eq. 2, for $n = 1,2,3,4$ and coefficients of Eq. 3, for $n = 5$

Appendix 2: Maps of the differences between $\Delta E$ and $\Delta E_{{\theta = 0^{ \circ } }}$ (in canonical units–c.u.)

See Figs. 18, 19 and 20.

Fig. 18

Differences between the ${\Delta }E$ for free $\theta$ and ${\Delta }E$ for $\theta = 0^{ \circ }$ (in canonical units—c.u.), for the situations where $e = 0.1, \nu = 0^{ \circ } \left( {t = 0} \right), \psi = 270^{ \circ }$° and $\delta V$ goes from 0.1 to 2.0 c.u

Fig. 19

Difference between the ${\Delta }E$ for free $\theta$ and ${\Delta }E$ for $\theta = 0^{ \circ }$ (in canonical units—c.u.), for the situations where $e = 0.1$, $\psi = 270^{ \circ }$, $\delta V = 0.5$ c.u. and different true anomalies

Fig. 20

Difference between the ${\Delta }E$ for free $\theta$ and ${\Delta }E$ for $\theta = 0^{ \circ }$ (in canonical units—c.u.), for $\nu = 0^{ \circ } \left( {t = 0} \right)$, $\psi = 270^{ \circ }$, $\delta V = 0.3$ c.u. and different eccentricities