Neutron stars are extreme objects in the universe. Their birth is marked by the death of a star with mass between 8 and 25 solar masses in a cataclysmic explosive event, releasing 10^53 erg of energy and can suppress the brightness of an entire galaxy. After the contraction of the iron core of the progenitor, the remnant is a neutron star with 1.4 solar masses and 10 km of radius, which means an average density 10^14 g/cm^3.Extreme objects allow us to study extreme physics and extreme physics means involving various branches of science and connect them in new and differential ways, leading to advances in our understanding of the phenomenon. More specifically, neutron stars are natural laboratories to test two of the most fundamental theories for the construction of reality: the theory of general relativity of gravitation in the strong field regime and the physics of matter at extremely high densities and temperatures. Both tests are impossible to be performed in laboratories on Earth with today's technology.But it is not only the theoretical point of view that the study of neutron stars leads to advances in the understanding of Nature. It also leads to technological advances, especially in the field of observational astrophysics. We have always to improve our detectors and telescopes in order to understand more clearly the astrophysical signatures of these compact objects in all its varieties, which would also lead us to a greater understanding of nature.The group of Professor Jorge Horvath, my supervisor at the doctorate, has been working on this topic for a long time, but always more focused on theory. Now, my contribution to the group is to connect the theory to observations. During my PhD I've been searching for the observational link to theory, having spent one year at Kapteyn Astronomical Institute in Groningen, The Netherlands, working with Professor Mariano Mendez on x-ray emitting systems containing a neutron star. There, I acquired the necessary experience to conduct this observational project, collaborating in a way that complements the group's goals.Because the rapid technological developments of the last decade, a vast amount of data was obtained, but much remains to be analyzed. We now have in hand, already available for treatment, data from five very special systems that will drive us to a real breakthrough in the restriction of masses and radii of neutron stars, allowing us to select a smaller set of possible equations of state for the description superdense matter inside the neutron stars, as best described throughout this manuscript.
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