Perturbed Robe’s Restricted Problem of 2+2 Bodies when the primaries form a Roche Ellipsoid - Triaxial System

Bhavneet Kaur¹, Rajiv Aggarwal², Sushil Yadav³

¹Lady Shri Ram College for Women, Delhi University, Delhi, India

²Shri Aurobindo College, Delhi University, Delhi, India

³Maharaja Agrasen College, Delhi University, Delhi, India

*Corresponding Author Email: bhavneet.lsr@gmail.com, rajiv_agg1973@yahoo.com, sushilyadav1973@rediffmail.com

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 and respectively) are small solid spheres of density and 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 is describing a circle around . The masses and mutually attract each other, do not influence the motion of and 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 (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 and lying on -axis are unstable for and and stable for and , using the data of submarines in the Earth -Moon system. The equilibrium solutions of and respectively when the displacement is given in the direction of or 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.

INTRODUCTION:

Robe (1977) has investigated a new kind of restricted three-body problem in which one of the primaries of mass is a rigid spherical shell filled with a homogeneous incompressible fluid of density . The smaller primary is a mass point outside the shell. The third body of mass , supposed moving inside the shell, is a small solid sphere of density , with the assumption that the mass and the radius of the third body are infinitesimal. He assumed that the mass describes a Keplerian orbit around the mass . He has proved that the centre of the first primary, is the only equilibrium solution for all values of the density parameter , mass parameter , eccentricity parameter . Further, he has discussed the linear stability of this equilibrium solution. He has explicitly discussed two cases. In the first case, the orbit of around is circular and in the second case, the orbit is elliptic, but the shell is empty (i.e. no fluid inside it) or the densities of and are equal. In each case, the domain of stability has been investigated for the whole range of parameters occurring in the problem.

In the above problem, the existence of only one equilibrium solution namely, centre of the first primary is discussed. Hallan & Rana(2001) studied the existence and the linear stability of all the equilibrium solutions in the Robe’s restricted three-body problem. They have proved that besides the centre, there are other equilibrium solutions too which exist only when the density parameter and the second primary moves around the first in a circular orbit.

Szebehely(1967) considered the effect of small perturbations in the Coriolis force on the stability of equilibrium points in the classical restricted problem, keeping the centrifugal force constant and proved that the collinear points remain unstable. The range of stability of the triangular equilibrium points increase or decrease depending upon whether the change in the Coriolis force is positive or negative thereby establishing that the Coriolis force is a stabilizing force.In the same problem, Bhatnagar & Hallan(1978) studied the effect of the changes and in the Coriolis and centrifugal forces respectively on the stability of equilibrium points and concluded that the collinear points remain unstable. For the triangular equilibrium points, the range of stability increases or decreases depending upon whether the point lie in one or the other of the two parts in which the line divides the plane.

The Robe’s problem has been varied by the introduction of perturbing forces by many researchers. Shrivastava and Garain (1991) examined the effect of small perturbations in the Coriolis and centrifugal forces on the position of the equilibrium point. In obtaining position of the equilibrium point, they considered the circular case with equal densities (i.e.). While evaluating the buoyancy force, bothRobe (1977) and Shrivastava and Garain (1991)assumed that the pressure field of the fluid has spherical symmetry around the centre of the shell and they considered only one of the three components of the pressure field, which is, that due to the own gravitational field of the fluid . The other two components which are those arising from the centrifugal force and attraction of were ignored. Plastino & Plastino (1995) took into account all these components of the pressure field. They assumed the hydrostatic equilibrium figure of the first primary as a Roche ellipsoid (Chandrashekhar(1987)).

Hallan&Mangang(2007) examined Robe’s circular restricted three-body problem by considering the buoyancy force as done byPlastino&Plastino (1995). They assumed the hydrostatic equilibrium figure of the first primary as an oblate spheroid and considered the full potential due to the second primary. They found conditions for the existence of an infinite number of equilibrium points and analysed their stability.

Singh and Mohammed (2012) studied the Robe’s circular restricted three-body problem under oblate and triaxial primaries. Singh and Sandah (2012) studied the existence and linear stability of equilibrium points in the Robe’s restricted three-body problem with oblateness. Singh and Mohammed (2013) studied the out-of-plane equilibrium points and their stability in the Robe’s problem with oblateness and triaxiality. Hallan&Rana(2003) studied the effect of perturbations in the Coriolis and centrifugal forces on the location and stability of the equilibrium points in the Robe’s circular restricted problem with density parameter having arbitrary value.

Many authors have worked on problem of 2+2 bodies. Significant work is that of Whipple(1984). He studied equilibrium solutions of the restricted problem of 2+2 bodies. He further studied the linear stability of all the equilibrium solutions.Bhavneet & Rajiv(2012) extended the Robe’s restricted three-body problem to bodies. Bhavneet & Rajiv(2013) studied the Robe’s restricted problem of 2+2 bodies when the bigger primary is a Roche ellipsoid. They studied the equilibrium solutions of the infinitesimal masses and analysed their linear stability. Bhavneet & Rajiv(2013) studied Robe’s restricted problem of 2+2 bodies when the bigger primary is a Roche ellipsoid and the smaller primary is an oblate body. They studied the equilibrium solutions of the infinitesimal masses and analysed their linear stability.

Our aim in this paper is to study the effect of perturbations in the Coriolis and centrifugal forces on the location and stability of equilibrium points in Robe’s restricted problem of 2+2 bodies considering the effect of the full buoyancy force of the fluid of the first primary and taking it as Roche ellipsoid and the second primary as a triaxial rigid body.

Statement of the Problem and Equations of Motion:

Let the first primary of mass be a Roche ellipsoid filled with homogeneous incompressible fluid of density and the second primary a triaxial rigid body of mass outside the ellipsoid. The two infinitesimal bodies (of mass and respectively) are small solid spheres of density and respectively inside the ellipsoid. Let denote the distance between the centres of mass of and . Let describe a circular orbit of radius R around with constant angular velocity . The masses and mutually attract each other, do not influence the motions of and but are influenced by them.We assume that (i) and never reach the surface of the Ellipsoid (ii)their position vectors at any time are not the same. Consider a uniformly rotating coordinate system with origin of the coordinate system at the centre of the bigger primary , pointing towards and being the orbital plane of around coinciding with the equatorial plane of . The coordinate system is as shown in the Figure 1. Let the synodic system of coordinates initially coincident with the inertial system rotate with angular velocity . This is the same as the angular velocity of which is describing a circle around . Let initially the principal axes of be parallel to the synodic axes and their axes of symmetry be perpendicular to the plane of motion. Since is revolving without rotation about with the same angular velocity as that of the synodic axes, the principal axes of will remain parallel to them throughout the motion. The equations of motion of and in the Robe’s circular restricted problem of bodies in the Roche ellipsoid -Triaxial body setup when perturbations are given to the Coriolis and centrifugal forces are given by

CONCLUSION:

In this paper, we have studied the effect of perturbations in the Coriolis and centrifugal forces on the location and stability of the equilibrium points in Robe’s restricted problem of 2+2 bodies. Plastino & Plastino (1995) revised the Robe’s restricted three-body problem under the assumption that the fluid body assumes the shape of a Roche ellipsoid. They took into consideration all the three components of the pressure field in deriving the expression for the buoyancy force viz, due to the own gravitational field of the fluid, that originating in the attraction of and that arising from the centrifugal force.They proved that that the centre of the Roche Ellipsoid viz, is the only equilibrium solution of the system.The equilibrium solutions in our case as compared to Plastino & Plastino (1995) are perturbed along , and axis.We have assumed throughout the paper that the perturbed coordinates of and are such that and remain within the Ellipsoid.There exist two equilibrium solutions of the system lying each on axis or axis or axis. Hence, there exist six equilibrium solutions of the system, provided they lie within Roche ellipsoid. We observe that there are no other equilibrium solutions except these six. We observe that the triaxiality of the second primary and small perturbations in the Coriolis and centrifugal forces have significant effect on the equations of motion. The number of equilibrium solutions in this problem are the same as in the problem with no perturbation (Bhavneet & Rajiv(2013)), but positions of the equilibrium solutions have changed. The change in the Coriolis force does not affect the position of the equilibrium solutions. The stability behavior of the equilibrium solution depends on the sign of the small perturbations in the Coriolis and centrifugal forces .The equilibrium solutions of ( and similarly of ) lying on , , axis are unstable for ; and stable for ; ,using the data of submarines in the Earth -Moon system. Similarly, the equilibrium solutions of and respectively when the displacement is given in the direction of or axis are conditionally stable. Earth is the only celestial body we know of which has fluid and the test particles within the fluid are considered as the submarines. In the solar or the extra solar system, we donot know of any other celestial body having liquid inside it. Our entire work is applicable in solar or extra solar system when and if such a system is discovered.

Table 1ϵ>0 and ϵ'<0


0.02	-0.01	-4.60675	0.000121742
0.03	-0.02	-4.68675	0.000200461
0.04	-0.03	-4.76675	0.00027918
0.05	-0.04	-4.84675	0.000357899
0.06	-0.05	-4.92675	0.000436617
0.07	-0.06	-5.00675	0.000515336
0.08	-0.07	-5.08675	0.000594055
0.09	-0.08	-5.16675	0.000672774
0.1	-0.09	-5.24675	0.000751492

Table 2ϵ>0 and ϵ'>0


0.02	0.01	-4.60675	-0.0000356952
0.03	0.02	-4.68675	-0.000114414
0.04	0.03	-4.76675	-0.000193133
0.05	0.04	-4.84675	-0.000271851
0.06	0.05	-4.92675	-0.00035057
0.07	0.06	-5.00675	-0.000429289
0.08	0.07	-5.08675	-0.000508008
0.09	0.08	-5.16675	-0.000586726
0.1	0.09	-5.24675	-0.000665445

Table 3ϵ<0 and ϵ'>0


-0.02	0.01	-4.28675	-0.0000356952
-0.03	0.02	-4.20675	-0.000114414
-0.04	0.03	-4.12675	-0.000193133
-0.05	0.04	-4.04675	-0.000271851
-0.06	0.05	-3.96675	-0.00035057
-0.07	0.06	-3.88675	-0.000429289
-0.08	0.07	-3.80675	-0.000508008
-0.09	0.08	-3.72675	-0.000586726
-0.1	0.09	-3.64675	-0.000665445

Table 4ϵ<0 and ϵ'<0


-0.02	-0.01	-4.28675	0.000121742
-0.03	-0.02	-4.20675	0.000200461
-0.04	-0.03	-4.12675	0.00027918
-0.05	-0.04	-4.04675	0.000357899
-0.06	-0.05	-3.96675	0.000436617
-0.07	-0.06	-3.88675	0.000515336
-0.08	-0.07	-3.80675	0.000594055
-0.09	-0.08	-3.72675	0.000672774
-0.1	-0.09	-3.64675	0.000751492

REFERENCES:

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2. Bhavneet Kaur, Rajiv Aggarwal, Robe’s restricted problem of 2+2 bodies when the bigger primary is a Roche ellipsoid, Acta Astronautica, Volume 89, August-September 2013, Pages 31-37, ISSN 0094-5765, http://dx.doi.org/10.1016/j.actaastro.2013.03.022.

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12. Singh, J. and Sandah, A.U.:2012, ’Existence and Linear Stability of Equilibrium Points in the Robe’s Restricted Three-Body Problem with Oblateness’, Advances in Mathematical Physics, Volume 2012, 18 pages.

13. Singh, J. and Mohammed, H.L.:2013, ’Out-of-plane equilibrium points and their stability in the Robe’s problem with oblateness and triaxiality’, Astrophysics and Space Science 345, 265-271.

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Received on 23.08.2016 Accepted on 08.09.2016

Int. J. Tech. 2016; 6(2): 150-160.

DOI: 10.5958/2231-3915.2016.00024.9