The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness

Abstract

This paper deals with the existence of libration points and their linear stability when the more massive primary is radiating and the smaller is an oblate spheroid. Our study includes the effects of oblateness of $\bar{J}_{2i}$ (i=1,2) with respect to the smaller primary in the restricted three-body problem. Under combining the perturbed forces that were mentioned before, the collinear points remain unstable and the triangular points are stable for 0<μ<μ _c , and unstable in the range $\mu_{c} \le\mu\le\frac{1}{2}$, where $\mu_{c} \in(0,\frac{1}{2})$, it is also observed that for these points the range of stability will decrease. The relations for periodic orbits around five libration points with their semimajor, semiminor axes, eccentricities, the frequencies of orbits and periods are found, furthermore for the orbits around the triangular points the orientation and the coefficients of long and short periodic terms also are found in the range 0<μ<μ _c .

Appendix

1.1 A.1 The polynomials of ξ _i (i=1,2,3)

• The polynomials of ξ ₁:

  

• The polynomials of ξ ₂:

  

• The polynomials of ξ ₃:

  

where: s=n ².

1.2 A.2 The values of μ at L _i (i=1,2,3)

• At point L ₁

  

• At point L ₂

  

• At point L ₃

  $$\mu = 1 + \frac{(8 + 12B_{1} - 15B_{2} - 8s)\xi _{3}^{2}}{8q_{1}} + O[\xi_{3}]^{3} $$