Mass and radius constraints for neutron stars using the cooling tail method

Neutron stars (NS) are the most compact objects that can be directly observed. They can be used to study properties of matter at supranuclear densities. This in turn gives us information to separate between numerous theoretical equations of states of dense matter. Thermonuclear (type-I) X-ray bursts from low mass X-ray binaries can be used to address this issue. Some of these bursts can be so energetic that they cause the whole photosphere of the NS to expand. The cooling of these photospheric radius expansion bursts can be compared to theoretical atmosphere models to obtain the mass and radius measurements of the NS. These measurements can then be used to differentiate between the different equations of state. We present a set of differential equations needed to compute these atmospheric models. We introduce an exact treatment of Compton scattering via the relativistic integral equation and an angle-dependent redistribution function. Using these equations, we can construct a set of atmosphere models in plane-parallel approximation in a local thermodynamical equilibrium for hot NSs. The emergent spectra is then fitted by a diluted blackbody to obtain the dilution factor w and the colour-correction factor f_c. On the other hand, the observed spectra from X-ray bursting neutron stars are close to thermal and can be fitted with a blackbody with two free parameters: the observed blackbody temperature T_bb and the normalization K. By equating the dilution factor w and the normalization K, we obtain a relation between the theoretical atmosphere models and the observations. This connection is the main idea of the so called cooling tail method. We then introduce a small correction to this method and discuss the consequences. A common problem encountered using this method is that different bursts from a given system can yield completely different mass and radius measurements. This fact casts a doubt on the robustness of the entire method. We study the burst emission from 4U 1608--52 at various persistent fluxes. We find a strong dependence of the burst properties on the flux before the burst. Bursts that ignite during the hard state at a low accretion rate show strong evolution of the apparent blackbody radius which is consistent with the model predictions of the neutron star atmosphere models. On the other hand, bursts occurring during the soft state at a higher accretion rate show constant apparent radius, which is inconsistent with the models. We then use the hard state bursts only to constrain the neutron star mass and radius from our set of sources. By taking only the physically relevant results into account, we also get information of the chemical composition of the atmosphere. This then gives us a way to conclude if the atmosphere is hydrogen- or helium-rich. After we know the chemical composition, we constrain the NS radius to be between 12 and 16 km. This implies a stiff equation of state of neutron star matter