Study of the general nature of motion and stability in elliptical orbit of two connected satellites system under several influences

Introduction

The lineralised and normalized equations of relative motion of two cable connected satellites system, many space configurations of cable-connected that the systems have been proposed and analysed by different authors. Beletsky and Novikova (1969) studied the motion of system of two cable connected satellites in central gravitational field of force relative to its centre of mass. this study assumed that the two satellites are moving in the plane of the centre of mass. Singh and Demin (1972) and Singh (1973) investigated the problem in two and three dimensional cases, Das et. al. (1976). These work conducted the analysis of relative motion of the system for the elliptical orbit of the centre of mass in two dimensional as well as three dimensional cases. Naryan and Singh (1987);Markaev et. al. (2003) studied the planar oscillation of a satellites in circular orbit. Ayub Khan (2011) investigated Chaotic motion in problem of dumbel satellites. Kumar and Srivastava (2006) studied evolutional and non-evolutional motion of a system of two cable connected artificial satellites under some perturbative forces. Kumar and Singh (2010) studied motion and stability of two interconnected satellites system in circular orbit of the centre of mass. This work direct generalization of the works done by Das S.K. and Bhattacharya S.K., Singh AK.P. Kumar R.K. and Singh N.K.

Equation of Motion in Case of Circular Orbit of the Centre of Mass of the System

The normalised and linearised differential equation of motion of one of the two satellites when the centre of mass moves along Keplarian Elliptical orbit in Nechvill's co-ordinate's system have been derived in the form:

o is the Length of cable connecting two satellites of masses 1 m and 2 m respectively.  is the product of gravitational constant with mass of attracting centre (the earth)  is the semi latus ractum

The condition of constraint is given by:

The system of satellites are under the shadow of the earth. When the shadow function 1 0   and in that case the system will have no sunlight effect and hence the Left hand side equation (1) the portubative terms containing 1  will vanished but as soon as the system will express the sun light effect the shadow function 1   . Since general solution of ( 1) is beyond our reach, so, we restrict myself to the case of circular orbit we have 0

we get the equation of motion for two dimensional case.

where dash denotes the differentiation with respect to 't' and 22 r x y  .

The condition of constraint given by (2) Thus after the averaging the periodic terms of (4) we have -

Equation of motion of the system are being described by using averaged values (6) in the form -  cos cos sin " 2 ' 3 4

Multiplying the two equation ( 7) by 2 '.2 ' xy and adding, we get after integrating, we get the Jacobian integral -  

Hence we conclude that the satellites of mass 1 m will move inside the boundary of different curve of zero velocity represental (8) for different values of Jacobian constant.

Equilibrium-solution of the Problem

We have obtained a set of equations given by (4) for motion of the system in the rotating frame of reference. It has been assumed that the system is moving with effective constraints and the connecting cable of two satellites of masses 1 m and 2 m respectively will always remains tight. The equilibrium position of the system is given by the constant values of the coordinates in the rotating frame of reference.

Let  and  denote variations in the co-ordinates for this position of equilibrium then. ""

as the original system of equations admits jacob's integral so there must exists a Jacobi's integral for the system of Equation ( 13) moreover It is not different to deduce the Jacobi's integral for (13) in the form:

where h is the constant of integration. Now (15) can be put in the form:

Conclusion

Where 0 [3] stands for 3rd and higher order terms in the small equations  and  . To test the stability in the sense of Liapunov we take Jacobian integral given by (16) as a Liapunov function   22 , , ,

V     and applying liapunov's theorem on stability. It follows that the only interior for given Equilibrium position [a, b] to be stable is that V must be positive definite and for this the following condition must be satisfied. Thus we conclude that the equilibrium position (a, 0) is stable in the sence of Liapunov.

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