The Photo-Gravitational R3BP when the Primaries are Heterogeneous Spheroid with Three Layers

Abstract

The present paper deals with the stationary solution of the planar restricted three body problem when both the primaries are heterogeneous oblate spheroid with three layers of different densities and sources of radiation as well. It derives equations of motion and examines the existence and the linear stability of libration points. It is seen that there are five libration points, two triangular points and three collinear ones. It is further observed that the collinear points are unstable, while the triangular points are stable for the mass parameter 0 ≤ μ < μ _{c r i t} . Here μ _{c r i t} (the critical mass parameter) depends upon the combined effects of radiation and that due to the three different layers in the primaries. It is further seen that long or short periodic elliptical orbits emanate from the triangular points in the aforementioned range of μ.

Appendix

List of co-efficient of equations of Section “Location of Collinear Points” :

$$\begin{array}{@{}rcl@{}} \upsilon^{(1)}_{0}&=&3k_{2}q_{2}\\ \upsilon^{(1)}_{1}&=&12k_{2}q_{2}\\ \upsilon^{(1)}_{2}&=&2q_{2}\mu+18k_{2}q_{2}\\ \upsilon^{(1)}_{3}&=&8q_{2}\mu+12k_{2}q_{2}\\ \upsilon^{(1)}_{4}&=&-2+2\mu-3k_{3}+3\mu k_{3}+2q_{1}-2\mu q_{1}+3k_{1}q_{1}+12\mu q_{2}+3k_{2}q_{2}\\ \upsilon^{(1)}_{5}&=&-10+8\mu-15k_{3}+12\mu k_{3}+4q_{1}-4\mu q_{1}+8\mu q_{2}\\ \upsilon^{(1)}_{6}&=&2\left(-10+6\mu_{1}5k_{3}+9\mu k_{3}+q_{1}-\mu q_{1}+\mu q_{2}\right.\\ \upsilon^{(2)}_{7}&=&-20+8\mu-30k_{3}+12\mu k_{3}\\ \upsilon^{(2)}_{8}&=&-10+2\mu-15k_{3}+3\mu k_{3}\\ \upsilon^{(2)}_{9}&=&-2-3k_{3} \end{array} $$

$$\begin{array}{@{}rcl@{}} \upsilon^{(2)}_{0}&=&-3k_{2}q_{2}\\ \upsilon^{(2)}_{1}&=&12k_{2}q_{2}\\ \upsilon^{(2)}_{2}&=&-2\mu q_{2}-18k_{2}q_{2}\\ \upsilon^{(2)}_{3}&=&8\mu q_{2}+12k_{2}q_{2}\\ \upsilon^{(2)}_{4}&=&-2+2\mu-3k_{3}+3\mu k_{3}+2q_{1}-2\mu q_{1}+3k_{1}q_{1}-12\mu q_{1}-3k_{2}q_{2}\\ \upsilon^{(2)}_{5}&=&10-8\mu+15k_{3}-12\mu k_{3}-4q_{1}+4\mu q_{1}+8\mu q_{2}\\ \upsilon^{(2)}_{6}&=&-20+12\mu-30k_{3}+18\mu k_{3}+2q_{1}-2\mu q_{1}-2\mu q_{2}\\ \upsilon^{(2)}_{7}&=&20-8\mu+30k_{3}-12\mu k_{3}\\ \upsilon^{(2)}_{8}&=&-10+2\mu-15k_{3}+3\mu k_{3}\\ \upsilon^{(2)}_{9}&=&2+3k_{3}\\ \upsilon^{(3)}_{0}&=&-3k_{1}q_{1}\\ \upsilon^{(3)}_{1}&=&-12k_{1}q_{1}\\ \upsilon^{(3)}_{2}&=&-2q_{1}+2\mu q_{1}-18k_{1}q_{1}\\ \upsilon^{(3)}_{3}&=&-8q_{1}+8\mu q_{1}-12k_{1}q_{1}\\ \upsilon^{(3)}_{4}&=&2\mu+3\mu k_{3}-12q_{1}+12\mu q_{1}-3k_{1}q_{1}-2\mu q_{2}-3k_{2}q_{2}\\ \upsilon^{(3)}_{5}&=&2+8\mu+3k_{3}+12\mu k_{3}-8q_{1}+8\mu q_{1}-4\mu q_{2}\\ \upsilon^{(3)}_{6}&=&8+12\mu+12k_{3}+18\mu k_{3}-2q_{1}+2\mu q_{1}-2\mu q_{2}\\ \upsilon^{(3)}_{7}&=&12+8\mu+18k_{3}+12\mu k_{3}\\ \upsilon^{(3)}_{8}&=&8+2\mu+12k_{3}+3\mu k_{3}\\ \upsilon^{(3)}_{9}&=&2+3k_{3} \end{array} $$