Four pages - Page one University of the Witwatersrand Advanced Techniques in Physics : 2002 Examination : June 2002 |
White dwarfs are cold stars where the nucleo-synthesis process is completed. They consist of heavy stable nuclei and their electrons. In a simple model, they can be considered centrally symmetric and non-rotating (neglect magnetic fields). Their structure is then determined by the hydrostatic equilibrium between gravitational pressure seeking to compress the stellar material and Pauli pressure which resists this.
Depending on the mass of the star, the nucleo-synthesis process is assumed to have run to completion leaving the star dominantly composed of a single stable nucleus terminating a fusion cycle, such as C, or Fe. The electrons are modelled as a free Fermi gas, and they dominate the Pauli pressure term.
Under these conditions, we can find two coupled first order differential equations for the radial mass and density distribution of the white dwarf.
As in electrostatics, the gravitational force per unit volume
at a given radial distance from the centre of the star is dependent
on the amount of matter enclosed by a sphere of that same radius.
Therefore
(1) |
(2) |
(3) |
(4) | |||
The initial condition will determine the final mass and radius of the star, , by integration of the two coupled first order differential equations for and .
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(8) |
(9) |
(10) |
(11) |
Noting that , we see that
(12) |
(14) | |||
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(15) | |||
(16) |
Converting to dimensionless variables , and for
the stellar radius, density and mass,
(18) |
(19) | |||
Note that solutions for different values of may all be scaled from = 1.
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(20) |
(21) |