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 cap e sub cap f equals p sub cap f squared solidus left parenthesis two times m sub e right parenthesis (Equation 6), where p_F is the Fermi momentum given by p sub cap f equals left parenthesis three times n sub e solidus eight times pi right parenthesis super one solidus three times h (Equation 8). Therefore, the Fermi kinetic energy of the degenerate, non-relativistic electrons in the core of a white dwarf is

cap e sub cap f equals left square bracket left parenthesis three times n sub e divided by eight times pi right parenthesis super one solidus three times h right square bracket squared times one divided by two times m sub e

So, using Equation 16, this becomes

cap e sub cap f equals left square bracket three divided by eight times pi times left parenthesis 16 times pi cubed divided by 81 right parenthesis times left parenthesis five times m sub e divided by h squared right parenthesis cubed times cap g cubed times left parenthesis m sub cap h divided by cap y sub e right parenthesis super four times cap m squared right square bracket super two solidus three times h squared divided by two times m sub e

cap e sub cap f equals 25 divided by two times left parenthesis two times pi squared divided by 27 right parenthesis super two solidus three times left parenthesis cap g divided by h right parenthesis squared times left parenthesis m sub cap h divided by cap y sub e right parenthesis super eight solidus three times cap m super four solidus three times m sub e full stop

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 cap e sub cap f equals 364 times left parenthesis cap m solidus cap m sub circled dot operator right parenthesis super four solidus three 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 ( m sub e times c squared equals 511 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 cap e sub cap f proportional to cap m super four solidus three ) and the non-relativistic treatment will be progressively less reliable. Equation 17 implies that cap e sub cap f almost equals m sub e times c squared when M = 1.3 M_☉, indicating that the ultra-relativistic treatment is certainly required instead when masses become this large.

Activity 5

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

Discussion

For the ultra-relativistic case, equating P_c to P_UR 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 K_UR gives
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 cap m sub Ch almost equals 1.4 times cap m sub circled dot operator .

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.

OpenLearn - White dwarfs and neutron stars
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