We develop a numerical algorithm for the solution of the Sturm-Liouville differential equation governing the stationary radial oscillations of nonrotating compact stars. Our algorithm is based on the Numerov’s 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.

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 wave echoes if the postmerger object features a photon sphere capable of partially trapping the gravitational waves. If the postmerger source is a black hole, a second internal reflection surface, associated with 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 hertz 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 parametrized models of the equation of state of ultrastiff quark matter we find that a strange star can emit gravitational wave echoes, but the corresponding frequencies are on the order of tens of hilohertz, and thus not compatible with the 72 Hz signal.

Strange stars are one of the possible compact stellar objects formed in the core collapse of supernovae. These hypothetical stars are made by deconfined quark matter and are selfbound. In our study, we focus on the torsional oscillations of a non bare strange star, i.e., a strange star with a thin crust made of standard nuclear matter. We construct a theoretical model assuming that the inner parts of the star are in two different phases, namely the color flavour locked phase and the crystalline colour superconducting phase. Since the latter phase is rigid, with a large shear modulus, it corresponds to a first stellar crust. Above this crust a second small crust made by standard nuclear matter is suspended thanks to a strong electromagnetic dipolar moment. We focus on the electromagnetically coupled oscillations of the two stellar crusts. Notably, we find that if a small fraction of the energy of a glitch event like a typical Vela glitch is conveyed in torsional oscillations, the small nuclear crust will likely break. This is due to the fact that in this model the maximum stress, due to torsional oscillations, is likely located near the star surface.

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 Numerov's 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.

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.