Neutron Stars

At stellar densities that greatly exceed white-dwarf densities, the extreme pressures cause electrons to combine with protons to form neutrons. Thus, any star that collapses to such an extent that its radius becomes significantly less than that characteristic of a white-dwarf is effectively transformed into a gas of neutrons. Eventually, the mean separation between the neutrons becomes comparable with their de Broglie wavelength. At this point, it is possible for the degeneracy pressure of the neutrons to halt the collapse of the star. A star that is maintained against gravity in this manner is called a neutron star.

Neutrons stars can be analyzed in a very similar manner to white-dwarf stars. In fact, the previous analysis can be simply modified by letting $m_p\rightarrow m_p/2$ and $m\rightarrow m_p$ . Thus, we conclude that non-relativistic neutrons stars satisfy the mass-radius law:

$$\frac{R_\ast}{R_\odot}= 0.000011\left(\frac{M_\odot}{M}\right)^{1/3},$$ (8.218)

It follows that the radius of a typical solar mass neutron star is a mere 10km. In 1967, Antony Hewish and Jocelyn Bell discovered a class of compact radio sources, called pulsars, that emit extremely regular pulses of radio waves. Pulsars have subsequently been identified as rotating neutron stars. To date, many hundreds of these objects have been observed.

When relativistic effects are taken into account, it is found that there is a critical mass above which a neutron star cannot be maintained against gravity. According to our analysis, this critical mass, which is known as the Oppenheimer-Volkoff limit, is given by

$$M_{OV} = 4 M_C = 6.9 M_\odot.$$ (8.219)

A more realistic calculation, which does not assume constant density, does not treat the neutrons as point particles, and takes general relativity into account, gives a somewhat lower value of

$$M_{OV} = M_\odot.$$ (8.220)

A star whose mass exceeds the Oppenheimer-Volkoff limit cannot be maintained against gravity by degeneracy pressure, and must ultimately collapse to form a black-hole.