ASTR 498N                   Lecture 19                     White Dwarfs

 

(online at www.astro.umd.edu/~drabin/)

 

The Chandrasekhar limit

 

We mentioned the Chandrasekhar limiting mass for white dwarfs in the last lecture.  We can derive M_Ch using tools we have already developed.  Of course, the structure and evolution of white dwarfs is properly a subject in itselfas is, for that matter, the study of neutron stars or black holes.  We must be satisfied here with the bare minimum.

 

Consider first a gas in which the electrons are fully degenerate but nonrelativistic.  The pressure is

 

                     

 

In the solutions to Homework 4 we showed that

 

                     

 

where η_e  is the number of electrons per nucleon and X is the mass fraction of hydrogen.

 

Substitute this expression for n_e into the expression for P_NR and equate it to an independent estimate of the central pressure.   The linear model from the last lecture gives

                        (among friends)

                   

so we write

 

                     

 

We can solve this for the central density and write it in terms of the fundamental mass introduced in Lecture 16,   (where  ), as

 

                     

 

Are the electrons in fact nonrelativistic in a white dwarf of, say, one solar mass?  If we set the Fermi momentum equal to m_ec,

 

                     

 

we can solve for n_e to find that the electrons become ultrarelativistic when the density approaches

 

                     

 

Comparing this with our expression above for ρ_c ,  we see that the electrons in a white dwarf will be relativistic unless M is small compared to M_*.  When M is comparable to M_*, we must equate the relativistic expression for degenerate electron pressure,

 

                     

 

with our estimate of central pressure:

         

                     

 

Notice that the central density cancels, leaving

 

                     

 

Consider a sequence of masses beginning well below M_*, such that the electrons are nonrelativistic.  As the mass increases, the central density increases.  As the electrons shift over to relativistic behavior, the density increases more and more steeply as mass increases, until at M_Ch, the central density becomes infinite.  Physically, the star cannot be supported by degeneracy pressure and must collapse.  Its final fate is determined by different physics.

 

The linear model (for which  ) is not as centrally concentrated as a white dwarf.  As we discussed in the last lecture, the equation of state for a relativistic electron-degenerate gas is described by a polytrope of index 3.  For this polytrope,  and  compared to  for the linear model, which leads to a more accurate prediction of the Chandrasekhar mass,

                     

 

For the typical value η_e  0.5, M_Ch  1.45 M.

 

It’s not difficult, with a bit more mathematics, to develop a general expression for the pressure of a degenerate electron gas that is valid for any value of the Fermi momentum.  Then we can watch the central density increase as ~M² for low masses (nonrelativistic) and then increase sharply to a vertical asymptote at M_Ch.

 

 

 

 

Mass and Radius

 

We’ll use the nonrelativistic degenerate of state for our estimates.  If we equate the central density in the linear model to the expression for central density derived above, we have

 

                     

 

Note that the characteristic density is the proton mass divided by the cube of the electron Compton wavelength, :

         

                     

 

In the expression for central density, solve for the radius to find

 

                     

 

which shows that the characteristic size of a white dwarf of characteristic mass is determined by the “gravitational fine-structure constant”  and the electron Compton wavelength.  Scaled to solar values, we have

 

                      

 

Thus, a white dwarf is about the size of the Earth.  This mass-radius relationship is only approximate since it is based on nonrelativistic degeneracy.  Accounting for relativistic degeneracy modifies the relationship for the more massive white dwarfs; and, of course, the relationship terminates at the Chandrasekhar mass. 

 

Q  Why can’t we parallel the preceding steps for fully relativistic degeneracy?

 

 

HR diagram

 

Using  to eliminate R in the mass-radius relationship, we derive

 

                     

 

This expression is compared with observations in the figure below.

 

 

Mass, luminosity, core temperature and cooling

 

The “mechanical” equations of stellar structure (hydrostatic equilibrium and conservation of mass) allow us to estimate the order-of-magnitude pressure and temperature inside a star, but not the luminosity.  The luminosity is determined by how easily energy is transported within the star.  For radiative transport, the familiar equation is

 

                     

 

where L_r is the luminosity interior to r.

 

Recall that the thermal conductivity of a degenerate electron gas is very high because the electron mean free path is very long:  almost all the states into which it might scatter are already occupied.  Our working model of the structure of a white dwarf is, therefore, a nearly isothermal, electron-degenerate sphere surrounded by a thin insulating skin of classical ideal gas.  The luminosity is set by how well the skin insulates.

 

To estimate the properties of the skin, make the idealization that the boundary between the interior and the skin occurs where the degenerate electron pressure and the classical electron pressure are equal.  Outside this point, we assume that the gas is classical.

 

Combine the hydrostatic equilibrium equation, , with the radiative transport equation to give

 

                     

 

where we have set m = M (thin skin) and L_r = L.  Substitute an opacity of the Kramers form,  to get

 

                     

 

Integrating from the surface, where P = T = 0,

 

                     

 

This so-called “radiative zero” solution is an approximation to the outer envelope of any star as long as the opacity is Kramers and gas is nondegenerate.  Using the ideal gas equation of state, we can also write

 

                     

 

The criterion that, at the lower boundary of the skin, the classical gas pressure and the nonrelativistic degenerate electron pressure are equal, reads

 

                     

 

where the temperature at the boundary is equal to the isothermal core temperature T_c (note the use of μ_e, the mean molecular weight per electron; μ_e = 2 for pure ionized helium).  If we solve this expression for density, equate it to the preceding expression for density, and solve for L/M, we find

 

                     

 

With typical values for a carbon white dwarf, we find, in solar units,

 

                     

 

Thus, following the completion of helium burning, when T_c  10⁸ K, the white dwarf has about one solar luminosity. 

 

The energy source for the luminosity of a white dwarf is the thermal energy of the classical ions in the interior (the contribution of gravitational contraction is minor).  That is,

 

                     

 

Setting  gives

 

                         where       

 

This simple differential equation for the temperature may be integrated,

 

                                

 

and substituted in the MLT_c relation to give the luminosity as a function of time, as illustrated below: