This page is designed to generate models of white dwarf stars according to Chandrasekhar's original equations for a totally degenerate self-gravitating electron-gas sphere (i.e., a white dwarf star). It does not take into account inverse beta decay, rotation, or anything more sophisticated than his original 1935 paper "The highly collapsed configurations of a stellar mass", Monthly Notices of the Royal Astronomical Society, Vol. 95, p.207-225 or his book "An Introduction to the Study of Stellar Structure" which will be used for references. Given the simplicity of this model, the structure of the star is governed by only two parameters -- one related to the central density (and, by extension, to the overall mass of the star) and another related to the composition. The other two parameters control the numerical integration process.
The parameter 1/y02 is dimensionless and is related to the density of the white dwarf at its core. It is defined in Chandrasekhar's book in Chapter 11, Equation 24 and the text and equations immediately following this. As this value approaches zero, core density approaches infinity and the radius of the star approaches zero, and as this value approaches one, core density approaches zero (and the size of the star approaches infinity). Note that with white dwarfs, size and mass vary inversely -- that is, very massive stars are very small and vise-versa. This is due to the nature of degeneracy pressure which supports the star against gravitational collapse.
The parameter mu is related to the composition of the star. It is the mean number of nucleons (protons + neutrons) per free electron. In a white dwarf it is generally safe to assume that atoms are completely ionized, so using carbon 12 as an example (six protons and six neutrons per nucleus and six free electrons) mu would have a value of (6+6)/6 = 2.0.
The Chandrasekhar limit -- the maximum possible mass of a white dwarf that is predicted by these equations -- can be found as 1/y02 approaches 0. In this limit, the density at the core goes to inifinity as the size of the star goes to zero, however the overall mass of the star approaches a finite limit. This limit is the Chandrasekhar limit - about 1.44 solar masses (if mu = 2 -- the mass is dependant on the value chosen for mu).
Also note that this model does not take into account inverse beta decay. As the mass of the star (and hence the centeral density) becomes very large, electrons and protons combine to form neutrons. In a star close to the Chandrasekhar limit, this process can lead to star with fewer electrons than might otherwise be expected, so the model becomes increasingly inaccurate as one approaches the 1/y02=0. 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 here.
In a gas at "normal" conditions, the pressure is provided almost exclusively by the thermal motions of the particles, so pressure is a function of temperature and density. According to this logic, at absolute zero, when motion ceases, the pressure would drop to zero. Degeneracy pressure, however, is due to quantum-mechanical effects that are unrelated to the thermal motions of the particles. As a result, degeracy pressure is unrelated to temperature and will have a non-zero value even at zero temperature.
The upshot of this is that in a white dwarf, the degenracy pressure dominates and the thermal pressure is essentially irrelevant. In fact, the thermal pressure is neglected in these calculations, meaning that the star is implicitly assumed to be at a temperature of absolute zero. While it may be very hot by human standards -- tens of thousands of degrees or hotter -- it is still far cooler than would be necessary for the thermal pressure term to dominate the degeneracy pressure. For all practical purposes, it may as well be at absolute zero.
In addition, since the density and pressure predicted by this model are unrelated to temperature, the temperature of the star cannot be predicted from the equation of state used in this model, and experimental knowledge of the temperature of a white dwarf is not particularly useful in verifying the model.
This page makes use of a Fortran 95 program written by Jonathan Tomshine to solve Equation 28, Chapter 11, in Chandrasekhar's book for φ(η) (phi and eta are defined in equation 26). This second-order differential equation is decomposed into two coupled first-order differential equations. This pair is then solved simultaneously with a backward (implicit) Euler integration from the center of the star to the surface (where density goes to zero, defined by Equation 29) using a fixed step-size. The integration is halted when the surface is crossed. This tentative profile is then used as an ititial guess in a finite difference solution method. The finite difference method increases the accuracy of the final calculations and locates the surface of the star -- where density goes to zero -- more preceisely (the problem is reparameterized in order to make the location of the surface one of the degrees of freedom).
The computer code can be downloaded here, however there is no makefile included. To compile, something like the following command should be used (g95 should work, as should the Intel Fortran compiler):
g95 io.f90 lu.f90 wd_backward_euler.f90 wd_relax.f90 wd.f90 -o wd
Command line options are located in the about.txt file -- without supplying options, the code will produce no output! Direct questions/comments/criticisims to me, Jonathan Tomshine.