Chaotic Attitude Tumbling of Satellite in Magnetic Field

In this study, the half width of the chaotic separatrix has been estimated by chrikov’s criterion. Through surface of section method, it has been observed that the magnetic torque parameter, the eccentricity of the orbit and the mass distribution parameter play an important in changing the regular motion into chaotic one.


INRODUCTION
Bhardwaj [1] has discussed chaos in non-linear planar oscillation of a satellite under the influence of third-body torque. Sidlichovský [2] has discussed the existence of a chaotic region which is formed by trajectories crossing a critical curve which corresponds to the separatrix of fast pendulum motion. Tiscareno [3] carried out extensive numerical orbit integrations to probe the long-term chaotic dynamics of the 2:3 (Plutinos) and 1:2 (Twotinos) mean motion resonances with Neptune. Kauprianov and Shevchenko [4] studied the problem of observability of chaotic regimes in the rotation of planetary satellites. Contopoulos and Efstathiou [5] studied Escapes and Recurrence in a Simple Hamiltonian System. They studied a simple dynamical system with escapes using a suitably selected surface of section. Selaru et al. [6] studied Chaos in Hill's generalized problem from the solar system to black holes. Carruba et al. [7] have discussed Chaos and Effects of Planetary Migration for the Saturnian Satellite Kiviuq.

Equation of motion:
The equation of motion for the non-linear motion of a satellite under the influence of magnetic torque in an elliptic orbit as obtained as which can also be written as Estimation of resonance width: As r and v are periodic in time and as 2 v δ θ = + , using Fourier like Poisson-Series as discussed in Bhardwaj and Tuli [1] , Equation (1), becomes The half integers 2 m will be denoted by p .
Resonances occur whenever one of the arguments of the sine or cosine functions is nearly stationary i.e., whenever 1   α is an arbitrary constant and the other having a specific value. This is the case of infinite period separatrix which is asymptotic forward and backward in time to the unstable equilibrium.
Near the infinite period separatrix broadened by the high frequency term into narrow chaotic band [9] , for small n, the half width of the chaotic separatrix is given by ( )  For e = 0.0549 the mean eccentricity of Artificial satellite, the critical value of n above which large-scale chaotic behaviour is expected is RO n = 0.347.
The spin orbit phase space: Using Poincare surface of section by looking at the trajectories stroboscopically with period 2π. The section has been drawn with versus v at every periapse passage. Since the orientation denoted by θ is equivalent to the orientation denoted by π +θ , we have, therefore, restricted the interval from 0 to π. In Fig. 1-3, we have plotted d dv θ versusθ , at every periapse passage. It may be observed that the chaotic separatrix surrounds each of the resonance states and each of these chaotic zones is separated from others by non-resonant quasi-periodic rotation trajectories. From Fig. 1-3, it is observed that as n, e, ε increases, the regular curves disintegrate respectively and this disintegration increases with the increase in n, e, ε .

CONCLUSION
It is also observed that the magnetic torque plays a very significant role in changing the motion of revolution into liberation or infinite period separatrix. The half width of the chaotic sepratrices estimated by Chirikov's criterion is not affected by the magnetic torque. It is further observed that in the spin-orbit phase the regular curves start disintegrating due to magnetic torque, the increase in the eccentricity and the irregular mass distribution of the satellite and this disintegration increases with the increase in , n and e. It has been observed that Artificial satellite's spin orbit phase space is dominated by a chaotic zone which increases further due to magnetic torque.