Resonance in Satellite’s Motion Under Air Drag

This article studies the attitude motion of a satellite in a circular orbit under the influence of central body of mass M and its moon of mass m, whose orbit is assumed to be circular and coplanar with the orbit of the satellite. The body is assumed to be tri-axial body with principal moments of inertia A < B < C at its centre of mass, C is the moment of inertia about the spin axis which is perpendicular to the orbital plane. These principal axes are taken as the co-ordinate axes x, y, z; the z axis being perpendicular to the orbital plane. We have studied the rotational motion of satellite in the circular orbit under the influence of aerodynamic torque. Using BKM method, it is observed that the amplitude of the oscillation remains constant upto the second order of approximation. The main and the parametric resonance have been shown to exist and have been studied by BKM method. The analysis regarding the stability of the stationary planar oscillation of a satellite near the resonance frequency shows that the discontinuity occurs in the amplitude of the oscillation at a frequency of the external periodic force which is less than the frequency of the natural oscillation.


INTRODUCTION
The determination and prediction of the orbit of a satellite in the near-earth environment is complicated by the fact that the satellite is influenced by the dissipative effects of the earth's atmosphere. For many artificial satellites, this fluctuation in the drag is a fundamental source of error in the orbital predictions. The study of dynamics of rotating bodies has been studied by Inarrea and Lanchares [1] under the influence of aerodynamic drag. Abd. et al. [2] have constructed a second order atmospheric drag theory based on the usage of TD88 model. Maciejewski & Przybylska [3] analyzed the integrability of a dynamical system under the influence of gravitational and magnetic fields. Barkin, Ferrandiz [4] discussed Resonant Rotation of Two-layer Moon and Mercury. Callegari, Ferraz-Mello, Michtchenko [5] , discussed the Dynamics of Two Planets in the 3/2 Mean-motion Resonance in Application to the Planetary System of the Pulsar PSR B1257+12. Beaugé et al. [6,7] studied Planetary migration and extra solar planets in the 2/1 meanmotion resonance. They reviewed recent results on the dynamics of multiple-planet extra-solar systems, including main sequence stars and the pulsar PSR B1257+12 and comparatively, our own Solar System. They discussed Resonances and stability of extra-solar planetary systems. Massimiliano [8,9] , numerically detected the web of three-planet resonances (i.e., resonances among mean anomalies, nodes and perihelia of three planets) with respect to the variation of the semi-major axis of Saturn and Jupiter, in a model including the planets from Jupiter to Neptune. Zhou et al. [10,11] showed that the occurrence of apsidal secular resonance depends only on the mass ratio semi-major rate and eccentricity rate between the two planets. Yokoyama et al. [12] have shown that once captured in the resonance, the inclination of the satellite reaches very high values. But none of them have studied the effect of aerodynamic torque on the attitude motion of a satellite in circular orbit. Using BKM method, we have discussed that the amplitude of the oscillation remains constant upto the second order of approximation.

Equation of motion:
Let r be the instantaneous radius vector of the centre of mass of the satellite, θ be the angle that the long axis of the satellite makes with a fixed line EF lying in the orbital plane and /2 the angle between the radius vector r and the long axis.
Euler's equation of motion about z-axis, taking v (true anomaly as an independent variable) is obtained as: a Ω − Ω ω = and where the amplitude a and the phase ψ are determined by differential equations In our problem Bessel's function of order k.
Thus in the first approximation, the solution is given by ψ η ( 1) where the amplitude a and the phase ψ are given by From the equations (8) & (10), we observe that the amplitude of the oscillation remains constant even upto the second order of approximation of the aerodynamic torque parameter ε and the equation (7) gives us the main resonance at 1 ω = ± and the parametric resonance at 1 , k I 2 k 1 ω = ± ∈ + .
Resonant planar oscillation of a satellite: We proceed to construct the asymptotic solution of the system in the general case, which is valid at and near the resonance k ≅ ω exploiting the well known BKM method.
For ε = 0 the generating solution is given by Solving the equation (14) taking k = 1, we get Thus the solution in the first approximation is given by a cos(v ) η = + θ where the amplitude ' a ' and the phase' θ 'are the solutions of the system 2 4 2 2 2 2 da 2 ( 1) cos 4 ( 1) cos 4( 1) sin cos dv 1 The equations (16) cannot be integrated in a closed form due to the dependence of the right hand side on 'a' and " θ '. The stationary state of the oscillation is defined by However equation (16), may be represented, correct to the second order in the form: , which is tabulated in Table 1 and the jump was shown with 2D and 3D resonance curves.

CONCLUSION
The attitude motion of a satellite in circular orbit under aerodynamic torque is obtained. Using BKM method, we have observed that the amplitude of the oscillation remains constant upto the second order of approximation. The main resonance occurs at 1 ω = ± and the parametric resonance at I k k ∈ + ± = , 1 2 1 ω At main resonance the stability of the stationary planar oscillation of the satellite show that the discontinuity occurs in the amplitude of the oscillation at a frequency of the external periodic force which is less than the frequency of the natural oscillation.