International Journal of Technology

Bhavneet Kaur ¹, Rajiv Aggarwal ², Sushil Yadav ³

Affiliations
1 Lady Shri Ram College for Women, Delhi University, Delhi, IN
2 Shri Aurobindo College, Delhi University, Delhi, IN
3 Maharaja Agrasen College, Delhi University, Delhi

Abstract

The aim of this paper is to study the effect of perturbations in the Coriolis and centrifugal forces on the location and stability of the equilibrium solutions in the Robe's restricted problem of 2+2 bodies under the assumption that the hydrostatic equilibrium figure of the first primary is a Roche ellipsoid and the shape of the second primary is triaxial. The third and the fourth bodies (of mass m₃ and m₄ respectively) are small solid spheres of density ρ₃ and ρ₄ respectively inside the ellipsoid, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m₂ is describing a circle around m₁. The masses m₃ and m₄ mutually attract each other, do not influence the motion of m₁ and m₂ but are influenced by them. We have taken into consideration all the three components of the pressure field in deriving the expression for the buoyancy force viz (i) due to the own gravitational field of the fluid (ii)that originating in the attraction of m₂ (iii) that arising from the centrifugal force. The linear stability of this configuration is examined. It is observed that there exist only six equilibrium solutions of the system, provided they lie within the Roche ellipsoid. The equilibrium solutions of m₃

and m₄ lying on x₁-axis are unstable for ∈ > 0,∈' > 0 and ∈ <; 0,∈' > 0 and stable for ∈ > 0,∈′ < 0 and ∈ < 0,∈′ < 0, using the data of submarines in the Earth-Moon system. The equilibrium solutions of m₃ and m₄ respectively when the displacement is given in the direction of x₂ or x₃ − axis are conditionally stable.We observe that the conditions of stability are influenced by the small perturbations in the Coriolis and centrifugal forces.

Keywords

Robe’s Restricted Problem, Roche Ellipsoid, Triaxial, Stability.

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Dinesh Kumar ¹, Rajiv Aggarwal ², Mamta Jain ³

Affiliations
1 Shyama Prasad Mukherji College (For Women), University of Delhi, New Delhi, IN
2 Sri Aurobindo College, University of Delhi, New Delhi, IN
3 Bhagini Nivedita College, University of Delhi, New Delhi, IN

Abstract

This paper deals with the existence and stability of the libration points in the restricted-three body problem when the smaller primary is a finite straight segment and bigger primary is an oblate spheroid. We have observed that five libration points do exist in this problem, out of which three are collinear and two are non-collinear with the primaries. The collinear libration points are unstable for all values of mass parameter μ and the non-collinear libration points are stable if μ < μ_c, where μ_c is a critical value of mass parameter μ. The range of stability depends length of the finite straight segment and oblateness factor.

Keywords

Restricted Three-Body Problem, Libration Points, Stability, Finite Straight Segment, Oblate Spheroid.

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