Radial oscillations of zero-temperature white dwarfs and neutron stars below nuclear densities

The radial oscillations of zero-temperature degenerate stars with subnuclear densities have been calculated for the fundamental and first two excited modes using recent equations of state. The calculations were made under the assumptions that (a) the dynamical time scale T TN, where TN is the nuclear relaxation time, in which case the adiabatic index Y = YE (calculated along the equation of state) and (b) T TN with y = Yc calculated at constant composition. For case a, it is shown that white dwarfs are dynamically stable for Pc <% 1.2 x 10 g em - while neutron stars are dynamically stable for Pc > 1 0 g em -3. The instability for white dwarfs in case a arises principally from the phase transitions rather than from the effects of general relativity, confirming the conjecture of Baym et al. Low-density neutron stars have for the fundamental (unstable) mode "e-folding" times which are a factor 100 less than the typically expected dynamical time scale. It is shown that this is consistent with the unusual decay of the radial eigenfunction, a qualitative explanation of which is given. For densities below the neutron drip point it is shown that Yc = YE except at a finite number of phase transition points. For densities just above the neutron drip point, it is shown analytically that Ye must be greater than 4/3 in the critical region where YE drops below 4/3. For case b, white dwarfs are dynamically stable for Pc <% 4 X 1010 g , while neutron stars are dynamically stable for Pc > 7 x 1012 g as had been suggested previously by the author. Case b is shown to be more appropriate for infinitesimal perturbations. It is also pointed out that arguments given earlier to indicate that high-density white dwarfs and low-density neutron stars may not exist, because of instabilities over long time scales, are lacking in rigor. Subject headings: dense matter - equation of state - stars: neutron - stars: pulsation - stars: white dwarfs