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# 4.2 The Chandrasekhar limit

We have now derived an expression for the electron number density inside a star in the non-relativistic case, but is the material inside a white dwarf really non-relativistic? In the non-relativistic limit, the Fermi kinetic energy of degenerate electrons is given by (Equation 6), where pF is the Fermi momentum given by (Equation 8). Therefore, the Fermi kinetic energy of the degenerate, non-relativistic electrons in the core of a white dwarf is

So, using Equation 16, this becomes

Equation label:(17)

The expression above relates the Fermi kinetic energy to a set of physical constants and the mass of the white dwarf, M. Substituting for these values therefore we have keV. Evaluating this expression for a white dwarf with mass 0.4 M☉ reveals that the Fermi kinetic energy in this case is about 107 keV which is already more than 20% of the electron rest-mass energy ( keV). This makes it doubtful that the non-relativistic treatment is reliable for such a white dwarf. In more massive white dwarfs, the Fermi kinetic energy is even higher (because ) and the non-relativistic treatment will be progressively less reliable. Equation 17 implies that when M = 1.3 M☉, indicating that the ultra-relativistic treatment is certainly required instead when masses become this large.

## Activity 5

1. By equating the core pressure in a star to the degenerate pressure of ultra-relativistic electrons, derive an expression for the mass of a star supported by ultra-relativistic electrons.

2. Evaluate the mass, assuming Ye = 0.5, in SI units and solar masses. Note that , , and

### Discussion

1. For the ultra-relativistic case, equating Pc to PUR gives where

   Note that, in contrast to the non-relativistic case, the density term is the same on both sides, so cancels out, leaving

   Collecting M on one side and swapping left and right sides gives

   Taking the (3/2)-power and substituting for KUR gives

2. Putting in the numbers, we have

   So Converting to solar units, this is .

The mass of a star when the ultra-relativistic limit is reached may be viewed as the maximum stellar mass that can be supported by degenerate electron pressure. The approximate calculation in the previous activity shows that this maximum mass is of the order of the mass of the Sun.

The maximum mass for a white dwarf is called the Chandrasekhar limit; the most realistic computations estimate it as .

If the mass is increased further, then the pressure required to support the star also increases. But the ultra-relativistic degenerate electrons will not be able to increase their pressure to support it. Something has to give.... That something is the material supporting the star. There is a stable configuration of material at higher mass, but it does not consist of degenerate electrons. Rather, the electrons and protons of higher-mass objects are forced to combine as neutrons, and the stable object above the maximum white-dwarf mass is called a neutron star. Such objects will be studied later in this course.

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