Dynamical Instability of Gaseous Sphere in the Reissner-Nordstrom Limit

In this paper, we study the dynamical instability of gaseous sphere under radial oscillations approaching the Reissner-Nordstr\"om limit. For this purpose, we derive linearized perturbed equation of motion following the Eulerian and Lagrangian approaches. We formulate perturbed pressure in terms of adiabatic index by employing the conservation of baryon numbers. A variational principle is established to evaluate characteristic frequencies of oscillations which lead to the criteria for dynamical stability. The dynamical instability of homogeneous sphere as well as relativistic polytropes with different values of charge in Newtonian and post-Newtonian regimes is explored. We also find their radii of instability in terms of the Reissner-Nordstrom radius. We conclude that dynamical instability occurs if the gaseous sphere contracts to the Reissner-Nordst\"orm radius for different values of charge.