Perturbed Restricted Problem of Three Bodies with Elongated Smaller Primary

Abstract

This paper presents the study of the perturbed restricted problem of three bodies with an elongated smaller primary and an oblate radiating bigger primary. The albedo effect and small perturbations in the Coriolis and centrifugal forces have also been considered. The equilibrium points of the system and their linear stability are elaborated, and the significant variation in equilibrium points and their stability is observed. It is observed that the critical value of mass parameter $\mu _c$ increases due to all the considered parameters except oblateness and segment-length. It is found that the critical mass parameter $\mu _c$ decreases due to the effect of segment-length and oblateness. In addition, periodic orbits are constructed in the neighbourhood of equilibrium points. The effects of perturbing parameters in the periodic orbits are studied. The Poincaré Surface Section is constructed and then used to produce periodic orbits associated with its resonance. Considerable impact on the zero velocity curves and the periodic orbits are observed near the elongated primary. These effects decrease when the infinitesimal particle moves farther away from the elongated primary.

A. Appendix

1.1 A. 1. Polynomials of Collinear Equilibrium Points

$$\begin{aligned}{} & {} \mathbf{f_{L_{1}}(x)} =[3 A (1 - \mu ) q - 3 A d^2 (1 - \mu ) q - 6 A (1 - \mu ) \mu q + 2 (1 - \mu ) \mu ^2 q + 3 A (1 - \mu ) \mu ^2 q - 2 d^2 (1 - \mu ) \mu ^2 q \\{} & {} - 4 (1 - \mu ) \mu ^3 q + 2 (1 - \mu )\mu ^4 q + 2 \mu ^5 Q] \\{} & {} + [2 \mu ^4\delta _2 n^2 - 2 d^2 \mu ^4\delta _2 n^2 - 4 \mu ^5\delta _2n^2 + 2 \mu ^6 \delta _2n^2 + 6 A (1 - \mu ) q - 4 (1 - \mu ) \mu q - 6 A (1 - \mu ) \mu q \\{} & {} + 4 d^2 (1 - \mu ) \mu q + 12 (1 - \mu ) \mu ^2 q - 8 (1 - \mu ) \mu ^3 q - 8 \mu ^4 Q]\textbf{x} \\{} & {} + [-8 \mu ^3 \delta _2\delta _2n^2 + 8 d^2 \mu ^3\delta _2 n^2 + 20 \mu ^4 \delta _2n^2 - 12 \mu ^5 \delta _2n^2 + 2 (1 - \mu ) q + 3 A (1 - \mu ) q - 2 d^2 (1 - \mu ) q \\{} & {} - 12 (1 - \mu ) \mu q + 12 (1 - \mu ) \mu ^2 q + 12 \mu ^3 Q] \mathbf{x^2} \\{} & {} + [12 \mu ^2 \delta _2n^2 - 12 d^2 \mu ^2 \delta _2n^2 - 40 \mu ^3 \delta _2n^2 + 30 \mu ^4 \delta _2n^2 + 4 (1 - \mu ) q - 8 (1 - \mu ) \mu q - 8 \mu ^2 Q] \mathbf{x^3} \\{} & {} + [-8 \mu \delta _2n^2 + 8 d^2 \mu \delta _2n^2 + 40 \mu ^2 \delta _2n^2 - 40 \mu ^3 \delta _2n^2 + 2 (1 - \mu ) q + 2 \mu Q] \mathbf{x^4} \\{} & {} + [2\delta _2 n^2 - 2 d^2 \delta _2n^2 - 20 \mu \delta _2n^2 + 30 \mu ^2 \delta _2n^2] \mathbf{x^5} + [4 \delta _2n^2 - 12 \mu n^2] \mathbf{x^6} + 2 \delta _2n^2 \mathbf{x^7};\\ \mathbf{f_{L_{2}}(x)}= & {} [3 A (1 - \mu ) q - 3 A d^2 (1 - \mu ) q - 6 A (1 - \mu ) \mu q + 2 (1 - \mu ) \mu ^2 q + 3 A (1 - \mu ) \mu ^2 q - 2 d^2 (1 - \mu )\mu ^2 q\\{} & {} - 4 (1 - \mu ) \mu ^3 q + 2 (1 - \mu ) \mu ^4 q - 2 \mu ^5 Q] \\{} & {} + [2 \mu ^4 \delta _2n^2 - 2 d^2 \mu ^4 \delta _2n^2 - 4 \mu ^5 \delta _2n^2 + 2 \mu ^6 \delta _2n^2 + 6 A (1 - \mu ) q - 4 (1 - \mu ) \mu q - 6 A (1 -\mu ) \mu q \\{} & {} + 4 d^2 (1 - \mu ) \mu q + 12 (1 - \mu ) \mu ^2 q - 8 (1 - \mu ) \mu ^3 q + 8 \mu ^4 Q] \textbf{x} \\{} & {} + [-8 \mu ^3 \delta _2n^2 + 8 d^2 \mu ^3 \delta _2n^2 + 20 \mu ^4 \delta _2n^2 - 12 \mu ^5 \delta _2n^2 + 2 (1 - \mu ) q + 3 A (1 - \mu ) q - 2 d^2 (1 - \mu ) q - \\{} & {} 12 (1 - \mu ) \mu q + 12 (1 - \mu ) \mu ^2 q - 12 \mu ^3 Q] \mathbf{x^2} \\{} & {} + [12 \mu ^2 n^2 - 12 d^2 \mu ^2 \delta _2n^2 - 40 \mu ^3 n^2 + 30 \mu ^4 \delta _2n^2 + 4 (1 - \mu ) q - 8 (1 - \mu ) \mu q + 8 \mu ^2 Q] \mathbf{x^3}\\{} & {} + [-8 \mu \delta _2n^2 + 8 d^2 \mu \delta _2 n^2 + 40 \mu ^2 \delta _2n^2 - 40 \mu ^3 \delta _2n^2 + 2 (1 - \mu ) q - 2 \mu Q] \mathbf{x^4} \\{} & {} + [2 \delta _2n^2 - 2 d^2 \delta _2n^2 - 20 \mu \delta _2n^2 + 30 \mu ^2 \delta _2n^2] \mathbf{x^5} + [4 \delta _2n^2 - 12 \mu \delta _2n^2] \mathbf{x^6} + 2 \delta _2n^2 \mathbf{x^7};\\{} & {} \mathbf{f_{L_{3}}(x)} = [-3 A (1 - \mu ) q + 3 A d^2 (1 - \mu ) q + 6 A (1 - \mu ) \mu q - 2 (1 - \mu ) \mu ^2 q - 3 A (1 - \mu ) \mu ^2 q + 2 d^2 (1 - \mu ) \mu ^2 q \\{} & {} + 4 (1 - \mu ) \mu ^3 q - 2 (1 - \mu ) \mu ^4 q - 2 \mu ^5 Q] \\{} & {} + [2 \mu ^4 \delta _2n^2 - 2 d^2 \mu ^4 \delta _2n^2 - 4 \mu ^5 \delta _2n^2 + 2 \mu ^6 \delta _2n^2 - 6 A (1 - \mu ) q + 4 (1 - \mu ) \mu q + 6 A (1 - \mu ) \mu q \\{} & {} - 4 d^2 (1 - \mu ) \mu q - 12 (1 - \mu ) \mu ^2 q + 8 (1 - \mu ) \mu ^3 q + 8 \mu ^4 Q]\textbf{x} \\{} & {} + [-8 \mu ^3 \delta _2n^2 + 8 d^2 \mu ^3 \delta _2n^2 + 20 \mu ^4 \delta _2n^2 - 12 \mu ^5 \delta _2n^2 - 2 (1 - \mu ) q - 3 A (1 - \mu ) q + 2 d^2 (1 - \mu ) q \\{} & {} + 12 (1 - \mu ) \mu q - 12 (1 - \mu ) \mu ^2 q - 12 \mu ^3 Q] \mathbf{x^2} \\{} & {} + [12 \mu ^2 \delta _2n^2 - 12 d^2 \mu ^2 \delta _2n^2 - 40 \mu ^3 \delta _2n^2 + 30 \mu ^4 \delta _2n^2 - 4 (1 - \mu ) q + 8 (1 - \mu ) \mu q + 8 \mu ^2 Q] \mathbf{x^3} \\{} & {} + [-8 \mu \delta _2n^2 + 8 d^2 \mu \delta _2n^2 + 40 \mu ^2 \delta _2n^2 - 40 \mu ^3 \delta _2n^2 - 2 (1 - \mu ) q - 2 \mu Q] \mathbf{x^4} \\{} & {} + [2 \delta _2n^2 - 2 d^2 \delta _2n^2 - 20 \mu \delta _2n^2 + 30 \mu ^2 \delta _2n^2] \mathbf{x^5} + [4 \delta _2n^2 - 12 \mu \delta _2n^2] \mathbf{x^6} + 2 \delta _2n^2 \mathbf{x^7}. \end{aligned}$$

1.2 A. 2. Terms Used for Non-collinear Equilibrium Points ($\theta _{1,2}$)

$$\begin{aligned}{} & {} \phi _1 = 4(2\delta _2n^2 + (2 + 3A)(-1 + \mu )q)^2 + \mu (2(8 - 24\mu + d^2(-19 + 5\mu ))\delta _2n^2 -(2 + 3A)(-8+ 19d^2) \times \\{} & {} (-1 + \mu )q)Q + 4(4 + d^2)\mu ^2Q^2; \\ \phi _2= & {} 16\delta _2^2n^4 - 8(-1 + \mu )(-4 + 6\mu + 3A(-2 + 5 \mu ))\delta _2n^2q + 4(2 + 3A)^2(-1 + \mu )^2q^2 \\{} & {} + 4\mu (-8 + d^2 (-7 + 10 \mu ))\delta _2n^2Q - (-8(2 + 9A) + (46 + 87A)d^2)(-1 + \mu )\mu q Q + 4 (4 - 3 d^2) \mu ^2Q^2;\\ \psi= & {} 3(4(2 + 5A)(-1 + \mu )q(2\delta _2n^2 + (2 + 3A)(-1 + \mu )q) - \mu (2(8 + 5d^2)\delta _2n^2 + (-8(2 + 7A) \\{} & {} + (46 + 105A)d^2)(-1 + \mu )q)Q - 4(-4 + d^2) \mu ^2 Q^2). \end{aligned}$$