A reliable description of the radial oscillations of compact stars

We develop a numerical algorithm for the solution of the Sturm-Liouville differential equation governing the stationary radial oscillations of nonrotating compact stars. Our method is based on the Numerovs method that turns the Sturm-Liouville differential equation in an eigenvalue problem. In our development we provide a strategy to correctly deal with the star boundaries and the interfaces between layers with different mechanical properties. Assuming that the fluctuations obey the same equation of state of the background, we analyze various different stellar models and we precisely determine hundreds of eigenfrequencies and of eigenmodes. If the equation of state does not present an interface discontinuity, the fundamental radial eigenmode becomes unstable exactly at the critical central energy density corresponding to the largest gravitational mass. However, in the presence of an interface discontinuity, there exist stable configurations with a central density exceeding the critical one and with a smaller gravitational mass.

Gravitational wave echoes from strange stars

It has recently been claimed, with a $4.2 sigma$ significance level, that gravitational wave echoes at a frequency of about $72$ Hz have been produced in the GW170817 event. The merging of compact stars can lead to the emission of gravitational waves echoes if the post-merger object features a photon-sphere capable of partially trapping the gravitational waves. If the post-merger source is a black hole, a second internal reflection surface, associated to quantum effects near the black hole horizon, must be present to avoid the gravitational wave capture. Alternatively, gravitational wave echoes can be produced by ultracompact stars crossing the photon-sphere line in the mass-radius diagram during the neutron star merging. In this case, the second reflection surface is not needed. A recently proposed preliminary analysis using an incompressible (and so unphysical) equation of state suggests that gravitational wave echoes at a frequency of tens of Hz can be produced by an ultracompact star. Since strange stars are extremely compact, we examine the possibility that strange stars emit gravitational wave echoes at such a frequency. Using parameterized models of the equation of state of ultra-stiff quark matter we find that a strange star can emit gravitational wave echoes, but the corresponding frequencies are of the order of tens of kHz, thus not compatible with the $72$ Hz signal.