As you've surmised, the radii and densities of neutron stars are extremely hard to measure. Several models have been created that attempt to calculate how the neutrons in a neutron star interact under these extreme conditions (this is called an "equation of state.") Of course, we can't create a neutron star in a lab! As a result, this is primarily done using numerical simulations. These models project that a neutron star weighing 1.5 solar masses would have a radius between 10 and 15 kilometers, but estimates better than these have not yet been developed.
While this is true for most neutron stars, a very few can be more accurately estimated. Occasionally, neutron stars are part of a binary pair, and sometimes the neutron star accretes matter from the other member of the pair onto its surface. When enough matter is accreted, this process results in a thermonuclear chain reaction, and we observe a burst of X-rays (these neutron stars are called X-Ray Bursters.) This radiation emitted follows a blackbody spectrum, so with an estimate of the temperature of the burst and the luminosity emitted, we can estimate the radius. These are usually found to be comparable to the 10 km projected by equation of state models.
With these considerations about our uncertainties in neutron star measurements, the average neutron star has a density around 5 x 1017
on average. This is not uniform though! Models estimate that the density is as low as 109
on the surface and as high as 8 x 1017
at the core. For comparison, water has a density of 1000 kg/m3
In this answer, I talked about the surface of a neutron star. You may be wondering how this surface is defined. Most modern theories of neutron star formation suggest that the core of degenerate neutrons is surrounded by an extremely strong, extremely dense crust of ions and electrons; this crust is about one mile thick. Here, the surface pressure is not enough to break apart the atoms into neutrons. Instead, the nuclei and electrons split, but they do not form neutrons. This is why the surface has such a "small" density compared to the interior. Above the crust, there is an atmosphere, made up of the excess gas from the original star that was not ejected in the formation of the neutron star (during a core-collapse supernova). However, because of the strong gravity of the neutron star, this atmosphere is held very close to the surface of the star: it is approximately 1 cm in thickness!
If you want more details, the Wikipedia page for neutron stars
is pretty good. Also, this page
describes the process through which protons and electrons become neutrons, known as electron capture or inverse beta decay. If either of these don't answer any questions you may have, feel free to follow up!
Thanks for the question!
(published on 03/02/2011)