The nonlinear model of the small asymmetric perturbations of the equilibrium distribution of density of the rapidly rotating, gravitating, magnetized polytrope

The equilibrium, gravitating, inhomogeneous, rapidly rotating magnetized configurations are under investigation in the present paper. The equation of state is taken in the form of the polytrope $P=K\rho^{1+\gamma}$, $\gamma>1$. By using the expanding in the series in the parameter $\kappa=1/\gamma$ and owing to the method of the Burman–Lagrange series the problem of the determination the equilibrium distribution of density is reduced to the system of the nonlinear equations. The existence is shown of the “critical” points $(\gamma_c,v_c)$ (the special values of the parameters $v=\omega^2/(2\pi G\rho_0)$ and $\gamma$), in which even a small internal magnetic field asymmetric about the axis of rotation leads to the considerable rise of the asymmetry of the configuration in question. The maximum of the correlation $v_c=v_c(\gamma)$ $(v_c^{(\mathrm{max})}\simeq0.21$, $\gamma\simeq7.8)$ is firstly obtained. The possible applications of these results to the theory of the gravitational radiation from pulsars are also discussed.