Tre aspetti sulla stabilità di Stelle Compatte

This thesis is aim to explore three different aspects of Compact Stars, the densest and the smallest stars known. Compact Stars are a very interesting subject of study because of their extreme conditions, making them a laboratory to test our knowledge on the properties of dense matter. The first aspect is the possible Gravitational wave echoes emission from the merging of Compact Stars. In this thesis we examine the possibility that the ultracompact object produced in the GW170817 event is a Strange Star, and we evaluate the frequency of the corresponding Gravitational wave echoes. GW170817 event is the first merging observation compatible with the coalescence of a binary Compact Stars system and the nature of its final product is still unclear. It has recently been claimed that in the GW170817 event a gravitational wave echoes at a frequency of about 72 Hz have been produced with a 4.2 sigma significance level. 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. Our study can in principle give us some fundamental information on Compact stars' equation of state. Indeed, if the postmerger object was a strange stars it would be proved the hypothesis of Bodmer and Witten that collapsed nuclei are the true fundamental state of matter. However, the frequency obtained even considering a final object described by an ultrastiff strange star is about two order of magnitude bigger than the experimental measurement. This could be a signature that the remnant of GW170817 is a Black Hole. The second aspect is the possible presence of Mirror matter into a Neutron Star. Mirror matter is a non conventional dark matter candidate, firstly suggested by Lee and Yang in 1956 which proposed the existence of a hidden sector of particles and interactions to restore the parity symmetry. Mirror matter communicates with our world only through gravity. In this thesis we assume that neutrons could oscillate into Mirror neutrons inside a Neutron Star. This phenomenon has interesting implications. Whether the measurements of radii will improve in the following years, this kind of phenomenon could lead to the detection of stars with equal masses but different radii corresponding to stars composed of different fraction of Mirror matter. Furthermore, whether the oscillation of neutron is possible, a large number of stars could produce Mirror matter and eventually collapse in a Black Hole without any violent explosion. The last aspect is the radial oscillation of Compact Stars. In this thesis we develop a numerical algorithm for the solution of the Sturm-Liouville differential equation governing the stationary radial oscillations of Compact Stars. Our method is based on the Numerov Method that turns the Sturm-Liouville differential equation in an eigenvalue problem. In our model we provide a strategy to deal with the star boundaries. Assuming that the fluctuations obey the same equation of state of the background, we analyze different stellar models using several equation of state. With our model it is possible to test the paradigm that the last stable configuration corresponds to the null eigenfrequency. We obtain this result using continuous equation of state providing a stability test of our method. In addition, in order to correctly treat equations of state with discontinuous speed of sound, we develop an innovative numerical method that can be used to analyze the stability of Compact Stars.