The Analysis of the Photogravitational R4BP Under the Combined Effect of Stokes Drag and Oblateness with Variable Mass

Abstract

In this paper, we have studied the existence, locations and stability of the equilibrium points as well as zero velocity curves (ZVCs) under the combined effect of oblateness, radiation pressure and the dissipative force (Stokes drag) in the restricted four-body problem (R4BP) with variable mass when the bigger primary $m_{1}$ is a source of radiation, second primary $m_{2}$ is an oblate/prolate spheroid and third primary $m_{3}$ is a point mass. Jeans’ law and space time transformations of Meshcherskii have been used to derive the equations of motion of the infinitesimal body whose mass is varying. The dynamical behaviour of an infinitesimal body has been investigated under the influence of radiation pressure of bigger primary and oblateness of second primary with Stokes drag. The numerical investigation shows that all the equilibrium points are non-collinear and the collinear equilibrium points do not exist due to the presence of Stokes drag. The effect of oblateness parameter A, radiation parameter $q\,(0<q<1)$, the proportionality constant $\alpha \,(0<\alpha \le 2.2)$ occurs in Jeans’ law, parameter due to variation in mass $\gamma \,(0<\gamma <1)$ and dissipative constant $k\,(0< k<1)$ have been investigated on the existence, locations of equilibrium points, and their stability. Further, it has been shown that the regions of motion increase for the increasing values of the parameters A, q and $\alpha$ whereas these regions decrease for the increasing values of the dissipative constant k. We have also explored that all the equilibrium points are unstable for all values of the parameters used.

Appendix

See Appendix Tables 1, 2, 3, 4, 5, 6, 7, 8, 9.

Table 1 Equilibrium points in the $\xi \eta$-plane for $\mu =0.019$, $\gamma =0.1$, $\sigma =0.015$ and $\alpha =0.2$, $q=0.99$, $k=0.00001$ when the oblateness parameter A is increasing

Table 2 Equilibrium points in the $\xi \eta$-plane for $\mu =0.019$, $\gamma =0.15$, $\sigma =0.25$, $A=0.00015$, $k=0.00001$ and $\alpha =0.2$ when the radiation pressure q is increasing

Table 3 Equilibrium points in the $\xi \eta$-plane for $\mu =0.019$, $\gamma =0.35$, $\sigma =0.05$ and $k=0.0015$, $A=0.00015$, $q=0.99$ when $\alpha$ is increasing

Table 4 Equilibrium points in the $\xi \eta$-plane for $\mu =0.019$, $\alpha =0.2$, $\gamma =0.15$, $\sigma =0.035$, $A=0.00015$, $q=0.95$ when k is increasing

Table 5 Equilibrium points in the $\xi \eta$-plane for $\mu =1/3$, $\alpha =0.2$, $\gamma =0.1$, $\sigma =0.2$, $A=0.00025$ and $q=0.985$ when the dissipative constant k is increasing

Table 6 The characteristic roots of eight non-collinear equilibrium points for $\mu =0.019$, $\alpha =0.2$, $\gamma =0.1$, $k=0.00001$, $A=0.0001$, $k=0.00001$, $q=0.99$ and $\sigma =0.015$

Table 7 The characteristic roots of eight non-collinear equilibrium points for $\mu =0.019$, $\alpha =0.2$, $\gamma =0.1$, $k=0.00001$, $k=0.00001$, $q=0.99$, $\sigma =0.05$ and $A=-0.001$

Table 8 The characteristic roots when ten non-collinear equilibrium points exists for $\mu =1/3$, $\alpha =0.2$, $\gamma =0.1$, $k=0.00001$, $A=0.00025$, $k=0.00001$, $q=0.985$ and $\sigma =0.2$

Table 9 The characteristic roots of six equilibrium points in the Gliese 667C-Gliese 667Ce-Gliese 667Cf-Spacecraft system for 0.0000261538, $\alpha =0.2$, $\gamma =0.1$, $k=0.0140248$, $A=0.0001$, 0.999949 and $\sigma =0.05$