Stability of Equilibrium Points in the Generalised Photogravitational Restricted Three Body Problem When It Is Coplanar

Stability of Equilibrium Points in the Generalised Photogravitational Restricted Three Body Problem When It Is Coplanar

Ajit Kumar1, Minni Rani2

1,2Department of Applied Sciences, Ganga Institute of Technology & Management,

Kablana, Jhajjar, Haryana, India

Abstract: The Restricted Three Body Problem Is Generalised To Include The Effects Of An Inverse Square Distance Radiation Pressure Force On The Infinitesimal Mass Due To The Primaries, Which Are Both Radiating. In This Paper We Investigate The Stability Of Coplanar Equilibrium Points, Based On Equations In Variations. We Have Found The Characteristic Equation For The Complex Normal Frequencies Which Is A Sixth Order Polynomial .Thus We Conclude That Coplanar Equilibrium Points Are Unstable Due To Positive Real Part In Complex Roots.

Keywords: Stability; coplanar points; generalised; photogravitational; RTBP

1. INTRODUCTION

   Radzievskii (1950) showed that in the restricted photogravitational three body problem, allowing for the

   restricted three body problem. Both the primaries are radiating and smaller primary is supposed to be an oblate spheroid.

   We linearise the equations of motion. We have found the characteristic equation. The partial derivatives are evaluated at the equilibrium point L6. We have found the roots of characteristic equation. We conclude that due to positive real part in complex roots, the out-of-plane equilibrium points are unstable.

2. STABILITY OF EQUILIBRIUM POINTS

   The equations of motion (1) of the infinitesimal mass are given by Douskos and Markellos(2006).

   x

   x

   x 2ny

   gravitational attraction and light pressure of primaries, coplanar equilibrium points (L6, L7) exist in addition to three collinear and two triangular ones. Chernikov(1970) described the photogravitational restricted three body problem. Perezhogin (1976) discussed the stability of coplanar equilibrium points in the absence of a repulsive force from the smaller of the primaries. An investigation of the stability of collinear and triangular solutions in this problem was made by Kunitsyn and Tureshbaev (1983), (1985). A.T. Tureshbaev (1986) investigated the stability of the relative equilibrium positions (coplanar libration points) for a particle

   in a gas-dust cloud subject to the gravitational field and radiation pressure of a binary star. Lukyanov (1987)

   y 2nx

   y

   y

   z

   z

   where

   2

   (1)

   2

   2

   obtained regions of stability for libration points L6 and L7 for

   1 n2 x2 y2 1 q1 q2 1 A2

   • 3z

     A2 an

     any values of three parameters ( , q1 , q 2 ). Perezhogin and Tureshbaev (1989) showed that stability for the majority of initial conditions and formal stability occur almost everywhere in the domain of first order stability of coplanar

     libration points.

     2

     d n2

     r1

     1 3 A2

     2

     r2

     2r 2

     2r 4

     2

     Sharma, R.K. and Subba Rao, P.V. (1976) discussed the three dimensional restricted three body problem with oblateness. C.N. Douskos and V.V. Markellos (2006) found the existence of non-planar equilibrium points in the three dimensional restricted three body problem with oblateness.

     Hence, we thought to examine the stability of equilibrium points L6, L7 in the generalised photogravitational coplanar

     2 x 2 y 2 z2

     r

     r

     r

     r

     1

     1

     2

     2

     2 x 12 y 2 z2

     6 31 1 q

     A 32

     2 5 3 5 3 2 q oq 2 q

     oq 2

     1 2

     1 1 2 2

     x0

     1

     2

     q2

     b oq2

     A

     A

     3

     3

     2

     3

     3

     3

     A2 2

     9(1 )q1 A2

     92

     184

     92 2

     184

     368

     184 2

     2 2

     z0 3 A2

     2

     3 3

     3 3 3

     3 q1 oq1 oq2

     q2

     A2

     9 3 9 3 2 2 2

     A2 = oblateness co-efficient of smaller primary

     3 q1 oq1 q2 oq2

     2 2

     n = Mean motion

     q1 = radiation co-efficient of bigger primary q2 = radiation co-efficient of smaller primary

     A2

     1 1 2

     1 1 2

     1 57 159 102 2 q oq 2 oq 2

     µ = m2 m1 m2

     288 3

     2280 3

     2 2

     The equilibrium point L6 is given by Ishwar et.al(2010) (using Mathematica)

     We transfer the origin to equilibrium point (x0, z0) for examining the linear stability of the out-of-plane point L6 .

     576 3 288 3 6264 3 5688 3 1704 3 q1 oq1

     q2

     We linearise the equations of motion (1). We obtain

     225 3

     225

     3 2 2

     2

     2

     x – 2ny xx .x xz .z y 2nx yy .y

     (2)

     8

     q1 oq1

     8

     8

     q2 oq2

     A2

     z

     zx .x zz .z

     where the partial derivatives are evaluated at the equilibrium

     point and

     zx xz .

     1

     5724

     22896 2

     34344

     22896 5724

     q 2 2

     The characteristic equation is given by

     2

     (xx yy zz ) xxyy xxzz yyzz xz 4n zz

     137376

     91584 22896

     q1 oq1

     6

     yy

     4

     xz 2

     xxzz 0

     2 2 2

     (3)

     22896 91584

     2

     2 2

     3 1062 2097 10352 q1 oq12 oq2 2 A2 oA 2

     3

     2

     2

     i.e. 6 a4 b2 c 0

     c 46 92 462 1211 3409 32002 10023q1

     a ( xx yy zz )

     188

     3

     376

     3

     1882

     3

     1736

     3

     5584

     3

     59602

     3

     7043 q

     1

     1

     where

     b xx yy xx zz yy zz xz

     b xx yy xx zz yy zz xz

     • 4n zz

     • 4n zz

     2 2 A2

     c

     2

     140492

     140493

     2

     yy xz

     xx zz

     2

     2

     A2 2898 7875

     q1

     65448 10908 43632 43632 109082

     2

     2

     The values of co-efficients a, b and c of equation (3) are

     95886

     2

     q

     q

     2

     523062 2 3

     (using Mathematica)

     1133388

     2

     1220652 653958

     139518 A2q1

     a 2n2

     2 3A2

     q

     q

     2

     2

     30208

     7552

     15104 3

     302082

     75523

   • q1

   3 3 3 3

   A2 q2

   16560 3

   4632 3

   21696 3 12240 32

   2472 33 /p>

   44

   9 3

   882

   9 3

   443

   q2

   q2

   9 3

   A2 q1

   A2 q1

   q 5

   q 5

   2 A2 2

   88

   1762

   883

   • q1q2

     58 2

     58 2

     32 3

     383

     q1q2

     1

     2

     2q 9 3 9 3 9 3 3 3

     2 1 2

     3 5 3

     3 3A 2 A 2 A 2

     a

     –

     2 3 a

     2 2 2

     3

     1

     121 3 246 32 125 33 q q

     2

     3 2a3 9ab 27c 3 3

     a2b2 4b3 4a3c 18abc 27c 2 3

     8

     8

     1 2 171 3

     171 3

     A2 q1q2

     7552

     1888

     A2

     3776 3

     75522

     18883

     8

     8

     1

     2 3 b

     3 3 3 3

     2q2

     96 3 2 1

     q 192 3

     96 3

     3q 3

     2 3

     A2 2

     2a3 9ab 27c 3 3

     a2b2 4b3 4a3c 18abc 27c2

     A2 q2 2

     Substituting in characteristic equation (3), we find six roots

     2a3 9ab 27c 3 3

     1

     a2b2 4b3 4a3c 18abc 27c2 3

     (using Mathematica).

     Solution of equation

     6 a4

     b2

   • c 0 is given

     1

     32 3

     1

     as (using Mathematica)

     2

     2

     a a

     –

     1 3 2

     1

     2

     32 3 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c2 3

     1

     1

     a

     2 3 a2

     –

     1

     2

     3 3

     2 2 3 3

     2 3 ia

     3 2a

     9ab 27c 3 3 a b

   • 4b

     4a c 18abc 27c

     2

     2 3

     1

     3 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c2 3

     1

     1

     1

     1

     –

     2 3 b

     1

     b

     1

     2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c 2 3

     23 3

     2 2 3 3

     2 3

     1

     2

     3

     3

     –

     2a

     9ab 27c 3 3 a b

   • 4b

   4a c 18abc 27c

   2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c 2 3

   i 3b

   1

   2 1

   2 3 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c2 3

   32

   32

   3

   1

   2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c2 3

   1

   62 3

   1

   i 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c2 3

   1

   22 3 3

   a

   1 1

   3

   3

   –

   –

   2

   2

   a2 2

   3

   3

   –

   32

   2a 9ab 27c 3 3

   1

   2 2

   2 2

   3

   3

   3

   3

   2

   2

   a b 4b 4a c 18abc 27c

   a a2

   3 2

   1

   1

   3

   32 3 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c 2

   2

   3

   2

   3

   3

   3

   ia 2

   2 3

   3 2a3 9ab 27c 3 3

   a2b2 4b3 4a3c 18abc 27c2

   1

   1

   2 3

   2

   ia2

   1

   b

   2 3 3 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c 2 3

   1

   223 2a3 9ab 27c 3 3

   a2b2 4b3 4a3c 18abc 27c2 3

   b

   4 2 13

   i 3b

   2 3 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c 2

   2 1

   3

   3

   3

   3

   2 2

   2 2

   3

   3

   3

   3

   2

   2

   3

   3

   6

   6

   2 2a 9ab 27c 3 3 a b 4b 4a c 18abc 27c

   i 3b

   1

   2a 9ab 27c 3 3 a b 4b 4a c 18abc 27c

   2 3 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c 2

   3 2 2 3 3

   1

   62 3

   2 3 2 13

   1

   i 2a 9ab 27c 3 3 a b 4b 4a c 18abc 27c

   3 2 2 3 3

   2 13 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c 2 3

   1

   22 3 3

   1

   62 3

   1

   i 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c 2 3

   1

   2

   1

   22 3 3

   a

   –

   a2

   3 2

   1

   32 3 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c2 3

   ia2

   2

   1

   2 3 3 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c 2 3

   b

   2 1

   2 3 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c2 3

   5

   5

   i 3b

   2 1

   2 3 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c2 3

   1

   2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c2 3

   1

   62 3

   1

   i 2a3 9ab 27c 3 3 a2b2 4b3 4a3c 18abc 27c 2 3

   1

   22 3 3

   We may examine the stability of other quilibrium point L7 in the same manner as L6 . We will find that L7 is also unstable. Thus we conclude that non-planar equilibrium points are unstable in linear sense due to positive real part in complex roots.

3. CONCLUSION

We conclude that equilibrium points are unstable due to positive real part in complex roots when they are out of plane.

REFERENCES

[1]. Chernikov, Yu.A. : The photogravitational restricted three body problem, Soviet Astronomy AJ. vol 14, No.1, 1970, 176-181.

[2]. Douskos, C.N., & Markellos V.V. : Out-of-plane equilibrium points in the restricted three body problem with oblateness, A&A 446, 2006, 357-360.

[3]. Kunitsyn, A.L., and Tureshbaev, A.T. : The collinear libration points in the photogravitational three body problem, Sov. Astron. Lett. 1983, 9: 228.

[4]. Kunitsyn, A.L., and Tureshbaev, A.T. : Stability of triangular libration points in the photogravitational three-body problem, Sov. Astron. Lett. 1985, 11:60.

[5]. Lukyanov, L.G. : Stability of coplanar libration points in the restricted three- body problem, Sov. Astron. 1987, 31(6): 677-681.

[6]. Perezhogin, A.A. : Stability of the sixth and seventh libration points in the photogravitational restricted circular three- body problem, Sov. Astron. letters 1976, 2, No.5: 174-175.

[7]. Perezhogin, A.A. & Tureshbaev, A.T. : Stability of coplanar libration points in the photogravitational restricted three- body problem, Sov. Astron. 1989, 33(4):445-448.

[8]. Radzievsky, V.V.: The restricted problem of three bodies taking account of light pressure, Astron Zh, 1950, 27: 250.

[9]. Shankaran, Sharma, J.P. & Ishwar, B.: Equilibrium points in the generalised photogravitational non-planar restricted three body problem, IJEST, 2010, under publication.

[10]. Sharma, R.K., & Subba Rao, P.V. : Stationary solutions and their characteristic exponents in the restricted three- body problem when the more massive primary is an oblate spheroid, Celes. Mech., 1976, 13:137.

[11]. Tureshbaev, A.T. : Stability of coplanar libration points in the photogravitational three- body problem II, Sov. Astron. Lett. 1986, 12(5): 303-304