Motion around triangular points in the restricted three-body problem with radiating heterogeneous primaries surrounded by a belt

Sunusi Haruna,Jagadish Singh

doi:10.1038/s41598-020-75174-7

Abstract

The present paper studies the locations and linear stability of the triangular equilibrium points when both primaries are radiating and considered as heterogeneous spheroid with three layers of different densities. Additionally, we include the effects of small perturbations in the Coriolis and centrifugal forces and potential from a belt (circumbinary disc). It is observed that the positions of the triangular equilibrium points are substantially affected by all parameters (except a perturbation in Coriolis force) involved in the system.The stabilty of motion is found only when 0 < mu < mu_{c}, where mu_{c} is the critical mass value which depends on the combined effect of radiation pressures and heterogeneity of the primaries, small perturbations and the potential from a belt.It is also seen that the Coriolis force and the belt have stabilizing effect,while the centrifugal force, radiation and heterogeineity of the primaries have destabilizing behaviour.The net effect is that the size of the region of stability decreases when the value of these parameters increases where mu is the mass ratio and k_{1} ,k_{2} characterize heterogeneity of both primaries. A practical application of this model could be the study of motion of a dust grain near the heterogeneous and luminous binary stars surrounded by a belt.Finally, we carried out and discuss numerical experiments aiming at computing the positions of triangular points and critical masses of three binary systems: Archid, Xi Booties and Kruger 60.

Highlights

The present paper studies the locations and linear stability of the triangular equilibrium points when both primaries are radiating and considered as heterogeneous spheroid with three layers of different densities
A useful property of the CR3BP which is the possible position of equilibrium occurs when the three bodies form an equilateral triangle which can be observed in the motion of Trojan asteroid around the triangular points L4 and L5
The first term μ0 represents Routh’s critical mass value; the second term μh is the effect arising from heterogeneity of both primaries, the third term μr denotes the effect of the radiation pressure of both primaries; the fourth term μp is the measure of the impact of small perturbations in Coriolis and centrifugal forces; the last term is due to the gravitational potential from the circumbinary disc

Summary

Equations of motion

Let m1, m2 , and m3 be the masses of the primary, secondary, and the infinitesimal body (third body ) respectively. The third body is moving under the influence of both primaries which are heterogeneous with three layers, sources of radiation and surrounded by a belt. In the aforesaid coordinate system and dimensionless variables, the equations of motion of the third body in the frame work of the CR3BP when both the primaries are heterogeneous, radiating and surrounded by a belt under the influence of small perturbations in the Coriolis and centrifugal forces, can be written, following the work of Suraj et al.[12] and Singh and Taura[11], as x − 2nαy = x y + 2nαx = y. The parameter a controls the flatness of the profile and is known as flatness parameter, while b controls the size of the core of the density profile and is called the core parameter

Location of triangular equilibrium points

Linear stability of triangular points

Discussion

Conclusion

Author contributions

Additional information