Neutron stars are the densest objects in the universe and may contain hyperon-dominated matter, condensed mesons, or even deconfined or strange quark matter. Because of their low temperatures and high chemical potentials, the physical conditions in their interiors differ greatly from the dense conditions of the early universe or those achieved at hadron colliders. This region of the QCD phase diagram can only be probed through astrophysical observations that measure the mass and radius of neutron stars. For decades, this effort has been hampered by a number of model uncertainties as well as by the lack of accurate measurements of different spectroscopic phenomena from a single source that would break the degeneracies between the neutron star parameters of interest. I will discuss how we can now overcome these problems by combining recent developments in our understanding of neutron star atmospheres with observations of distinct phenomena from the same neutron star source. In particular, I will present a unique measurement of the mass and radius of the neutron star in EXO 0748-676. The high inferred mass and large radius of this neutron star rule out all the soft equations of state of neutron star matter.
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