A Study of the Axisymmetric Restricted Five-Body Problem within the Frame of Variable Mass: The Concave Case

Abstract

In the framework of axisymmetric problem of restricted five bodies with variable mass, our goal is to study the existence and stability of libration points. In the axisymmetric restricted five-body problem, we have assumed that the mass of the fifth particle varies according to Jeans’ law. We have further supposed that the four bodies having masses ${{m}_{1}},\;{{m}_{2}},\;{{m}_{3}}$ and ${{m}_{4}}$ (with ${{m}_{3}} = {{m}_{4}} = \tilde {m}$) form an axisymmetric concave configuration. The equations of motion of a test particle of infinitesimal mass $m$ have been illustrated, which is similar to the axisymmetric restricted five-body problem when the problem of variable mass evolves with the problem of constant mass. In this paper, we have determined in-plane as well as out-of-plane libration points along with their stability. Further, we have shown the regions of motion where the fifth body can move freely. We have also shown that the angle parameters $\alpha$ and $\beta$ and the parameters, arising due to variation of mass $\gamma \;(0 < \gamma < 1)$ and $\sigma \;(0 < \sigma < 2.5)$, affect the existence and number of these libration points. Moreover, the bivariate version of the Newton-Raphson iterative scheme is applied in an attempt to unveil the analysis of the basins of convergence linked with the libration points as a function of the mass parameter.

APPENDIX

First order partial derivatives are:

$${{U}_{\xi }} = \left( {1 + \frac{{{{\sigma }^{2}}}}{4}} \right)\xi - \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \frac{{{{m}_{i}}{{{\tilde {\xi }}}_{i}}}}{{\rho _{i}^{3}}},$$

(A.1a)

$${{U}_{\eta }} = \left( {1 + \frac{{{{\sigma }^{2}}}}{4}} \right)\eta - \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \frac{{{{m}_{i}}{{{\tilde {\eta }}}_{i}}}}{{\rho _{i}^{3}}},$$

(A.1b)

$${{U}_{\zeta }} = \frac{{{{\sigma }^{2}}\zeta }}{4} - \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \frac{{{{m}_{i}}\tilde {\zeta }}}{{\rho _{i}^{3}}},$$

(A.1c)

the second-order partial derivatives of the effective potential function $U(\xi ,\eta ,\zeta )$, which are used to illustrate the stability along with BoCs associated with the LPs, are:

$${{U}_{{\xi \xi }}} = \left( {1 + \frac{{{{\sigma }^{2}}}}{4}} \right) - \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \left( {\frac{{{{m}_{i}}}}{{\rho _{i}^{3}}} - \frac{{3{{m}_{i}}\tilde {\xi }_{i}^{2}}}{{\rho _{i}^{5}}}} \right),$$

(A.1d)

$${{U}_{{\eta \eta }}} = \left( {1 + \frac{{{{\sigma }^{2}}}}{4}} \right) - \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \left( {\frac{{{{m}_{i}}}}{{\rho _{i}^{3}}} - \frac{{3{{m}_{i}}\tilde {\eta }_{i}^{2}}}{{\rho _{i}^{5}}}} \right),$$

(A.1e)

$${{U}_{{\zeta \zeta }}} = \frac{{{{\sigma }^{2}}}}{4} - \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \left( {\frac{{{{m}_{i}}}}{{\rho _{i}^{3}}} - \frac{{3{{m}_{i}}\tilde {\zeta }_{i}^{2}}}{{\rho _{i}^{5}}}} \right),$$

(A.1f)

$${{U}_{{\xi \eta }}} = \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \left( {\frac{{3{{m}_{i}}{{{\tilde {\xi }}}_{i}}{{{\tilde {\eta }}}_{i}}}}{{\rho _{i}^{5}}}} \right) = {{\Omega }_{{\eta \xi }}},$$

(A.1g)

$${{U}_{{\eta \zeta }}} = \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \left( {\frac{{3{{m}_{i}}{{{\tilde {\eta }}}_{i}}{{{\tilde {\zeta }}}_{i}}}}{{\rho _{i}^{5}}}} \right) = {{\Omega }_{{\zeta \eta }}},$$

(A.1h)

$${{U}_{{\zeta \xi }}} = \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \left( {\frac{{3{{m}_{i}}{{{\tilde {\zeta }}}_{i}}{{{\tilde {\xi }}}_{i}}}}{{\rho _{i}^{5}}}} \right) = {{\Omega }_{{\xi \zeta }}}.$$

(A.1i)

$${{\Xi }_{{11}}} = - \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \left( {\frac{{{{m}_{i}}}}{{\rho _{{0i}}^{3}}} - \frac{{3{{m}_{i}}\tilde {\xi }_{{0i}}^{2}}}{{\rho _{{0i}}^{5}}}} \right) + \left( {1 + \frac{{{{\sigma }^{2}}}}{4}} \right),$$

(A.2)

$${{\Xi }_{{22}}} = - \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \left( {\frac{{{{m}_{i}}}}{{\rho _{{0i}}^{3}}} - \frac{{3{{m}_{i}}\tilde {\eta }_{{0i}}^{2}}}{{\rho _{{0i}}^{5}}}} \right) + \left( {1 + \frac{{{{\sigma }^{2}}}}{4}} \right),$$

(A.3)

$${{\Xi }_{{12}}} = \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \frac{{3{{m}_{i}}{{{\tilde {\xi }}}_{{0i}}}{{{\tilde {\eta }}}_{{0i}}}}}{{\rho _{{0i}}^{5}}} = {{\Xi }_{{21}}},$$

(A.4)

$${{\Xi }_{{13}}} = \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \left( {\frac{{3{{m}_{i}}{{{\tilde {\zeta }}}_{{0i}}}{{{\tilde {\xi }}}_{{0i}}}}}{{\rho _{{0i}}^{5}}}} \right) = {{\Xi }_{{31}}},$$

(A.5)

$${{\Xi }_{{23}}} = \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \left( {\frac{{3{{m}_{i}}{{{\tilde {\eta }}}_{{0i}}}{{{\tilde {\zeta }}}_{{0i}}}}}{{\rho _{{0i}}^{5}}}} \right) = {{\Xi }_{{32}}},$$

(A.6)

$${{\Xi }_{{33}}} = - \frac{{{{\gamma }^{{3/2}}}}}{\Delta }\sum\limits_{i = 1}^4 \left( {\frac{{{{m}_{i}}}}{{\rho _{{0i}}^{3}}} - \frac{{3{{m}_{i}}\tilde {\zeta }_{{0i}}^{2}}}{{\rho _{{0i}}^{5}}}} \right) + \frac{{{{\sigma }^{2}}}}{4},$$

(A.7)

where

$$\rho _{{0i}}^{2} = \tilde {\xi }_{{0i}}^{2} + \tilde {\eta }_{{0i}}^{2} + \tilde {\zeta }_{{0i}}^{2},$$

$${{\tilde {\xi }}_{{0i}}} = ({{\xi }_{0}} - {{\xi }_{i}}),$$

$${{\tilde {\eta }}_{{0i}}} = ({{\eta }_{0}} - {{\eta }_{i}}),$$

$${{\tilde {\zeta }}_{{0i}}} = ({{\zeta }_{0}} - {{\zeta }_{i}}).$$