Sirius B & W ...

Something there is more immortal even than the stars,
(Many the burials, many the days and nights, passing away,)
Something that shall endure longer even than lustrous Jupiter
Longer than sun or any revolving satellite,
Or the radiant sisters the Pleiades.

-- Walt Whitman, On the Beach at Night

My plan is for this to be my final entry in the "Sirius B & White Dwarfs" series of blog entries.

In my last entry I discussed the nature and origin of electron degeneracy pressure. In this entry I will (briefly) discuss a program that I wrote to solve for the structure of a white dwarf star -- that is, to calculate the mass and density vs. radius profiles of such a star, starting from the core and working towards the surface.



Some of the most critical work in this area was done by Chandrasekhar and is addressed in his book (which I mentioned in my last entry) or in his 1935 paper The Highly Collapsed Configurations of a Stellar Mass, Monthly Notices of the Royal Astronomical Society, Vol. 95, p.207-225.

Deriving the equation of state (i.e., pressure / density relationship) for a degenerate electron gas is the first and probably most theoretically challenging step towards solving for the structure of a white dwarf. This equation of state is represented parametrically by Chandrasekhar in equation 3 of his paper. After deriving the equation of state, solving for the structure of the star is conceptually simple. One simply has to solve a pair of first-order ordinary differential equations simultaneously (Equation 6 in Chandrasekhar's paper). Those equations are the equation of hydrostatic equilibrium:



...and a second differential equation that is used to simply add up all the mass interior to a given radius:



In a "normal" model of stellar structure (i.e., a model of a star where fusion is still actively occuring), there are additional differential equations that describe the rate of energy generation and transport to the surface, but since a white dwarf is a dead star -- i.e., there is no fusion going on -- these terms do not apply. Furthermore, since the equation of state for a degenerate electron gas does not depend on temperature (since the electrons are already moving faster -- due to quantum-mechanical considerations -- than they should be going based on the range of temperatures encountered in white dwarfs), there is no differential equation to account for temperature either. Even if temperature were considered, the degenerate electron gas is such a good conductor of heat that the star is essentially isothermal anyway. All things considered, this is a very simple model.

After defining the problem with the equation of state (which I have not presented, but is available in the paper that I linked to above) and the two differential equations, Chandrasekhar performs some substitutions and combines all these equations into a single equation -- Equation 16 in his paper:



This equation is written with dimensionless variables, but eta is related to the radius and phi is related to the density. For exact definitions, please see Chandrasekhar's paper (it's free online -- just click the link above). As this is a second-order differential equation, it requires two boundary conditions for unique solution. These are given immediately after Equation 16, but basically amount to "density goes to zero at the surface" and "d-phi/d-eta = 0 at the core due to symmetry".

From my standpoint, I wished to solve Equation 16 above in order to have a complete density vs. radius and mass vs. radius model of a white dwarf star. I accomplished this numerically in Fortran 95, and put the results online here. By supplying the two parameters that describe the star and two additional parameters that control the integration, one can create profiles of white dwarfs of any given mass. If all goes well, a plot similar to the one above should appear after submitting the form. Please see the bottom of the front page for some additional technical details regarding the code. Also, the actual code can be downloaded from the bottom of that page. Compiling it will require a Fortran 95 compiler, and g95 should work (it's what I used).

As one tries various values for 1/y0^2 (a quantity that is related to the central density), it should become apparent that as this quantity approaches zero, the size of the star also approaches zero while the overall mass approaches a finite limit -- about 1.4 solar masses. This, of course, is the Chandrasekhar limit -- the maximum possible mass for a white dwarf.

I will not go into further details here, but suffice to say that this will conclude my series on the physical nature of white dwarfs. There is quite a lot of additional theory, of course. The results that my program can generate are 70+ years old, and the field has progressed a great deal in that time.

One point that does deserve some mention is what happens to white dwarfs that approach the Chandrasekhar limit. The equation of state used in this model does not account for the possibility of inverse beta decay. As the mass of the star (and hence the centeral density) becomes very large, electrons and protons combine to form neutrons (inverse beta decay). In a star close to the Chandrasekhar limit, this process can lead to a star with fewer electrons than might otherwise be expected, so the model becomes increasingly inaccurate as one approaches the 1/y0^2=0 limit. If this process progresses to completion, the entire star is converted to neutrons leading to a neutron star rather than a white dwarf. Such an entity requires a different equation of state and is not dealt with in this model.