6.2 The radius and mass of neutron stars

When deriving equations to describe neutron stars, we should be wary of oversimplifying the assumptions about conditions inside them. In reality, Newtonian gravity should be replaced by Einstein’s general relativistic treatment, because for a neutron star of mass M_NS and radius R_NS, a neutron’s gravitational potential energy, , is comparable to its rest-mass energy, m_nc².

The momenta of neutrons also approach the relativistic limits, requiring special relativity. Nevertheless, the non-relativistic approximation can lead to useful insights into the size of the star, and an understanding of the dominant physics. In particular, it is interesting to compare calculations of the radius of a white dwarf, supported by degenerate electrons, and a neutron star, supported by degenerate neutrons.

It turns out that the radius of a degenerate star scales according to the mass of the particle whose degenerate pressure supports the star. (The mass of the particle also determines its de Broglie wavelength, so you could also say that the de Broglie wavelength determines the radius of the star.) Because the pressure in a neutron star is provided by neutrons that are almost 2000 times heavier than an electron, neutron stars are a factor of almost 2000 smaller in radius. That is, while white dwarfs are comparable in size to the Earth, neutron stars have radii of only a few km! The typical masses and radii of different types of compact object are shown in Figure 5.

Figure 5 The mass-radius relationship for degenerate stellar remnants.

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This graph illustrates the relationship between the mass and radius of degenerate stellar remnants. The horizontal axis is labelled ‘M over M subscript solar’ and ranges from 0 to 3 in increments of 0.5 units increasing to the right. The vertical axis is labelled ‘radius in kilometres’ and ranges from 1 to 10000 in logarithmic spacing increasing upwards. A grey curve labelled ‘Schwarzschild radius of black hole’ starts from (0.75, 3), curves up slightly, and ends at (3, 10). A blue curve labelled ‘neutron star’ starts from (0.2, 30), and ends at (2.5, 10). The blue and grey curves intersect at a point labelled ‘TOV limit’ at (2.5, 10). A red curve labelled ‘white dwarf’ starts from (0.1, 10000), and ends at (1.4, 800). (1.4, 800) is labelled ‘Chandrasekhar limit’.

Figure 5 The mass-radius relationship for degenerate stellar remnants.

In principle, neutron stars have a minimum stable mass of , although in practice the minimum mass observed is around 1.2 M_☉. An absolute upper mass limit of is set by the requirement that the sound speed inside a neutron star is less than the speed of light. However, it is more difficult to specify the actual upper mass limit for neutron stars than for white dwarfs for a couple of reasons.

First, interactions between neutrons are significant. They repel one another at close separations (), which makes them less compressible, but their energies are also so high that they produce baryons, containing one or more strange quarks, and pions, which reduce the pressure and make them more compressible. The net effect is to enable neutron stars that are more massive than one might expect on the basis of simpler assumptions.

Second, the gravitational fields are so strong that Einstein’s theory of gravity – general relativity – must be used instead of Newtonian gravity and, under general relativity, gravity also depends on pressure. Whereas, in a normal star, internal pressure resists gravity, under general relativity it strengthens the gravity, which reduces the maximum mass that will be stable.

The competing mechanisms of these two effects mean that it is extremely difficult to calculate the actual upper mass limit for neutron stars, and no consensus exists on the matter. Various calculations and theories suggest that the maximum mass is ; this is referred to as the Tolman–Oppenheimer–Volkoff (TOV) limit.