4.3 The white dwarf mass-radius relationship
The mean density of a star is equal to its mass divided by its volume, . It can be related to the core density if the density profile of the star is known. So-called polytropic stellar models are ones in which the pressure at some radius is proportional to the density at that radius to some power. For a star described by a polytropic model with (a reasonable approximation), the mean density turns out to be one-sixth of the core density: . It is therefore possible to write a relation between the radius, mass and core density of a star.
Where the core density is dictated by non-relativistic degenerate electrons, we have already seen how the core density and mass are related (Equation 15). We can eliminate the density from this equation and find an expression for the radius of the object as a function of its mass, as shown by the following activity.
Activity 6
a.Rearrange the definition of the mean density to get an expression for the radius in terms of the mass and mean density.
b.Use the result for a polytropic model with (i.e. ) to re-express the radius in terms of mass and core density.
c.Substitute in the expression for the core density of non-relativistic degenerate electrons (Equation 15) to derive an expression for the radius of a white dwarf as a function of mass. Give your final answer in units normalised to the solar mass and solar radius. Note that , , and
d.Re-express the final result in solar masses and Earth radii, where the Earth’s radius is m.
Comment
a.The radius as a function of mass and average density is
b.If the average density is 1/6 of the core density, then
c.Substituting in the core density from Equation 15 gives
Rearranging the terms with negative powers:
Now consolidate the mass terms and the numerical factors:
Now, for convenience, express the mass in solar units:
Then putting in the numbers:
And so
Because , the white dwarf radius is
d.Because the radius of the Earth is ,
The result of the previous activity
is remarkable for two reasons. First, it shows that the radius of a white dwarf with the same mass as the Sun is two orders of magnitude smaller, comparable in radius to the Earth, as illustrated in Figure 3.
Second, notice the mass-dependence of the radius: . Unlike low-mass main-sequence stars fusing hydrogen in helium, where , note the important result that the sign of the exponent is negative for white dwarfs!
While the more massive a main-sequence star is, the larger its radius, the more massive a white dwarf is, the smaller its radius!
The derivation above of the white dwarf mass-radius relationship was based on the assumption that the electrons are non-relativistic. We showed this to be incorrect as the mass increases and it would imply infinite density (zero radius) at the Chandrasekhar limit, 1.4 M☉. There is another formula which gives a more accurate picture of the radius of white dwarfs, but it is an empirical formula, simply fitted to observed data. This formula, derived by Michael Nauenberg in 1972, is
The formula derived in Equation 18, while unable to correctly describe white dwarfs near the Chandrasekhar limit, does encapsulate real physical ideas that are valid for low-mass stars. There is a place in science for both formulae!