StarSimu - Example: a model of white dwarf

Example: a model of white dwarf

Here is the internal structure of a white dwarf star computed with StarSimu. The figure shows the variation with radial distance of the relevant physical quantities, such as mass, pressure and density (all referred to their maximum values). Note that the decimal digits are indicated by commas rather than by dots.

The internal structure of a white dwarf

About white dwarfs

White dwarfs are the final stage in the evolution of low mass stars with initial masses smaller than 3 solar units. The position of white dwarfs in the HR-diagram shows that they have a very small radius compared to normal main-sequence stars. On the other hand, the masses of white dwarfs are in the order of one solar mass. These two facts imply that these stars have very large mean densities.

After a low mass star has exhausted its nuclear fuel, it may contract into the white dwarf state. However, most stars that reach the white dwarf state must have lost a significant amount of mass. Indeed there is an upper limit for the mass of white dwarfs. This upper limit, the "Chandrasekhar limit", is the largest mass a white dwarf can have about 1.44 solar masses.

The best known white dwarf is Sirius B, whose existence was postulated by Bessel in 1834 to explain the sinusoidal motion of Sirius in the sky. Eighteen years later Sirius B was discovered by Clark. Subsequently, a large number of white dwarfs were discovered. Their frequency in our Solar neighbourhood is estimated at about 0.001 white dwarfs per cubic light year, which corresponds to an average distance of 10 light years. In other words, white dwarfs are quite common in our galaxy.

This is a schematic view of your PC's screen when you run StarSimu:

StarSimu's screenshot

Some remarks

As an example of input data we suggest to adopt a polytrope exponent Y = 4/3 and a constant K = 4.94 * 10^14 (hereinafter, we'll refer to this parameter choice as the "standard case"). This model describes remarkably well a star supported by the pressure of a relativistic completely degenerate electron gas, a physical situation occurring in white dwarfs with a density rho >> 10^6 g/cm³ in the whole structure.

Since in a star the density decreases going from the center to the surface, in a white dwarf the above condition is fulfilled at each point along the radius only if the central density is - as you can check with StarSimu itself - 3 or 4 orders of magnitude higher than the density at the surface. For example, you can adopt a central density of 10^11 g/cm³ and a dr of 0,8 km, like in the figure above.

On the other hand, we may take 10^12 as an upper limit for the central density. Indeed, at higher densities the atomic nuclei also become degenerate, and we end up with a neutron star, having a different equation of state. Therefore, in the standard case the choice of the central density is not totally arbitrary, but it is limited by the condition 10^10 < rho_c < 10^12.

With StarSimu you can check the value of the Chandrasekhar limit by computing a series of models with increasing central density. Indeed, the mass of a white dwarf depends in a quasi-linear way on the logarithm of the central density for densities below 10^8, but tends asymptotically to a constant value of 1.44 solar masses for higher densities.

Masses higher than the Chandrasekhar limit are not possible for white dwarfs. If the mass is between 1.44 and about 2.8 solar masses, the star becomes a neutron star, which only in a very rough approximation can be described by a polytropic model of exponent Y = 5/3 and with K = 5.4 * 10^10. If the mass is higher than 2.8 solar units, the star become a black hole.

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