Radial modes of levitating atmospheres around Eddington-luminosity neutron stars

We consider an optically thin radiation-supported levitating atmosphere suspended well above the stellar surface, as discussed recently in the Schwarzschild metric for a star of luminosity close to the Eddington value. Assuming the atmosphere to be geometrically thin and polytropic, we investigate the eigenmodes and calculate the frequencies of the oscillations of the atmosphere in Newtonian formalism. The ratio of the two lowest eigenfrequencies is $\sqrt{\gamma+1}$, i.e., it only depends on the adiabatic index.