Neutron stars represent the densest possible state of matter. Core densities of model neutron stars exceed nuclear densities by factors of 2-4. We do not know the exact form of the equation of state of matter at these densities, because we not measured with any precision the mass and radius of any neutron star.
There are about 1000 known neutron stars.
How does one determine observationally the equation of state (M/R) for a neutron star? Very simply, one needs merely to measure both the mass and radius of a neutron star. Masses can be determined for neutron stars in binary systems, and the masses for the binary radio pulsars are exceedingly well determined. Radii are more difficult: one can determine the angular diameter from an atmospheric model, and then, if one knows the distance, can determine the radius. Alternatively, one can use GR light-bending effects to model the pulse profiles and determine M/R, or can use the expanding cooling atmospheres of bursts to estimate radii. Observations of kHz QPOs, interpreted via the sonic-point model, suggest that we are close to measuring the masses and radii of accreting neutron stars.
However, doubts can be raised about the applicability of the measurements, especially of accreting X-ray binaries, to the "standard" neutron star. These neutron stars may have gained a few tenths of a solar mass of material since their birth.
We seek to determine the mass and radius for the single, isolated neutron star RX J185635-3754. Its spectrum appears thermal, so we have hopes that atmospheric modelling will give us the angular diameter. It is close enough that we expect to be able to measure the parallax. X-ray observations should be able to determine both the surface gravity (M/R2) and the gravitational redshift (M/R). This will give a clean point on the equation of state for neutron stars.
Here, we will describe the data, the current state of the atmospheric modelling, and our expectations from future data.
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