Nuclear Astrophysics - Chapter 7 Physics and Astronomy Department at the University of Washington

Neutron Stars

7.1 White dwarfs

We consider an old star, below the mass necessary for a supernova, that exhausts its fuel and begins to cool and contract. At a sufficiently low temperature the electrons will fill the lowest possible quantum levels. There are two spin states per level and levels per unit volume with momentum between k and k+dk. Thus the equation for the Fermi momentum is

a result we had before in HW4. Now essentially all the mass of the star is in nucleons, so we can relate the electron number density n to the mass density by

where is the ratio of nucleons to electrons in the state and is the nucleon mass. Thus

All of this is valid for a cold gas, that is, one where the kinetic energy (e.g, Fermi energy) of the electrons is much larger than kT

Now we can easily calculate the kinetic energy of the gas

and also the pressure. To do the pressure, first consider a one dimensional box. The pressure is the force per unit area on the walls, which we can evaluate from the transfered momentum resulting from collisions of particles of momenta k on a surface area A over a time dt

where the first term is the momentum transfer on reflecting off a surface, n A v dt is the number of particles that may hit the area A in time dt, and the 1/2 is needed because half of the particles in that volume are going the wrong way to collide with the box wall. Thus the 1D pressure is

where is the energy. Now in 3D , and repeating this calculation one takes only the x projection of the momentum transfer and velocity. Thus

So integrating over the Fermi sea yields

If we substituted our expression for the Fermi momentum into the equation above, we would have the pressure as a function of density, the equation of state.

The point where defines a rough partition between nonrelativistic and relativistic regimes. This corresponds to the critical density

where we inserted the needed factors of c.

Thus if the electrons are nonrelativistic. We can integrate the expressions above to get



Note that the equation for the pressure is of the form

where

Any star where the equation of state has this form is called a polytrope. This particular polytrope is a reasonably good approximation of a small mass (that is, nonrelativistic electrons) white dwarf. It can be shown that the equations of stellar evolution can be integrated semi-analytically for a polytrope, given a value for the star's central density. The result for the specific polytrope above is



The details are given in Weinberg's Gravitation and Cosmology.

Similarly, these integrations can be done in the relativistic limit where and thus . This yields



Thus this is a polytrope with

One remarkable result for the polytrope is that the resulting stellar mass is unique, independent of ,

while the radius is

This then gives the mass-radius relation for a large-mass white dwarf (relativistic electron gas).

What these results suggest is, as is increased in small mass white dwarfs, the mass grows as ; but as the gas becomes relativistic, it runs up against an asymptote for the maximum mass of . Consistent with this, as is further increased, the radius decreases as . Thus in the relativistic regime, a higher just leads to a rescaling of the overall density, as the mass M remains the same. Thus it appears that there is a maximum mass for stable, cold white dwarfs

Actually things are a bit more complicated. We expect initially , as the matter initially should be made up of the -like nuclear products of stellar burning. But when becomes sufficiently high, , electron capture on nuclei becomes favorable: recall for the homework problem on Co, we got a value of about 6 . If the material becomes more neutron rich, increases, which then reduces the maximum mass M. Thus our naive exercise of keeping fixed while producing successively more compact white dwarfs by increasing is not correct. The naive maximum mass for a white dwarf ( 2) is 1.47 ; if we use = 56/26 (appropriate for iron), the result is 1.27 . The result of a detailed calculation that takes into account electron capture is 1.2 . The radius of the star achieving this maximum is finite, about km.

One can calculate the gravitational potential for a large-mass (relativistic) white dwarf

More exactly, by plugging in the numerical values for the maximum mass and radius (from a calculation that includes electron capture) one finds a value of . That is, the gravitational potential energy of an electron at the surface of such a star would be less than 0.1% its rest mass energy.

7.2 Neutron stars

The above discussion points out the the final state of a star that has exhausted its nuclear fuel but lies above 1.2 is not a white dwarf: the electron gas cannot exert enough pressure to support the star. Thus a more dramatic collapse must ensue in which higher densities are achieved: much more gravitational work is done on the collapsing matter, heating it. This energy powers the subsequent shock wave, neutrino emission, and mantle ejection of a supernova, which we have already discussed. We will now suppose that the supernova explosion results in the ejection of a sufficient fraction of the star's mantle that the remain dense core has a mass below the Chandrasekhar limit (to be defined for this case below). This object, clearly denser than a white dwarf, forms the stable object known as a neutron star.

The neutron star is very similar to the white dwarf except that the degeneracy pressure of the nuclear matter, not the electron gas, supports the star. The equations are similar to those for a white dwarf. As the neutrons are relativistic, the total energy density is

where the critical density is defined as before

that is, we have substituted = 1 and . Likewise the pressure is

In principle these equations can be combined to eliminate and thus to yield an equation of state of the form

To get some analytic results, we proceed as in the white dwarf case and consider the nonrelativistic limit, . The results are



Again, these results follow from the white dwarf ones with the substitutions and .

The relativistic limit (or ) is easily evaluated to give





Now one can integrate the fundamental equation of stellar stability (gravitational force balanced by the pressure gradient) for a star that satisfies this equation of state in its interior with the boundary condition to find

This is an interesting result because our nonrelativistic result

yields the same mass for the choice , which is clearly still nonrelativistic and thus within the range of this equation's validity. Clearly we could increase somewhat, remaining in the nonrelativistic regime, and get greater masses. It follows that, for the ideal gas, T=0 equations of state we are using, the maximum neutron star mass must be achieved for some intermediate value of . Numerical calculations yield

This mass is known as the Oppenheimer-Volkoff limit.

It follows that masses above this limit cannot be stable: they will collapse into a black hole. One of the most interesting issues in the supernova game is to understand what fraction of the collapses produce a neutron star, and which fraction a black hole.

The above discussion is summarized in the figure, taken from Weinberg, showing the white dwarf and neutron star mass trajectories for the ideal equations of state we considered, as well as the results of better calculations that take into account the effects of , that is, the fact that the equations of state are not those of pure iron or pure neutrons.

In addition to the simplification of the chemical composition, we have also neglected the effects of rotation and, most important, the effects of the strong interaction on the equation of state. The latter is crucial in determining the true maximum neutron star mass. A number of neutron star masses have been determined from observations, usually with substantial error bars. The minimum possible maximum mass, based on observation must be in the range 1.2-1.6 . This is derived by requiring that the observations not be in obvious conflict with this upper bound. Calculations with detailed equations of state tend to give maximum masses in the range , compatible with observation. Model independent bounds on the maximum mass, which come from considerations like maintaining causal equations of state, allow maximum masses up to .

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Last update: July 10, 1999