Neutron star properties from astrophysical observations

Recently, simultaneous measurements of the masses and the radii of neutron stars from X-ray observations by the Neutron Star Interior Composition Explorer (NICER) have been reported. In addition, gravitational wave detections from the mergers of neutron star binaries allowed us to estimate the tidal deformability of neutron stars. Based on these recent observations and our works, we review their implications for the masses, radii and tidal deformabilities of neutron stars. We also discuss the basic principles of the polytropic equations of state, including sound velocities, and the implications of those equations to the properties of neutron stars.