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Given the mass and the effective temperature of a white dwarf star, find the radius, the luminosity, the age, the habitable zone radius and the Roche limit.
Given: M/M๏, T
This is my own curvefit to the mass-radius relation for carbon-oxygen core white dwarf stars. The line fitting the center portion of the domain runs through 40 Eridani B (0.5 solar mass) and Sirius B (1.0 solar mass) and roughly bisects the remaining data. It also fits very closely to theoretical models of white dwarfs supported against their own gravity by the electron degeneracy pressure in a relativistic Fermi gas.
If 0.25 < M/M๏ < 0.45, then
R/R๏ = 0.07279307 (M/M๏)² − 0.0752974 (M/M๏) + 0.03327478
If 0.45 < M/M๏ < 1.2, then
R/R๏ = −0.010421 (M/M๏) + 0.018821
If 1.2 < M/M๏ ≤ 1.41, then
R/R๏ = −0.0814246 (M/M๏)² + 0.1899852 (M/M๏) − 0.1044496
The luminosity is calculated from the radius and from the effective temperature using the ratio form of the Stefan-Boltzmann law.
T๏ = 5784K
L/L๏ = (R/R๏)² (T/T๏)⁴
The age of a carbon-oxygen core white dwarf (i.e. the time since the star became a white dwarf), in years, is given by
t = 10^[6.7 − (5/7) log(L/L๏)]
Although there aren't any helium-core white dwarf stars in the universe as yet, since stars insufficiently massive to fuse helium into heavier elements have not yet had time to leave the main sequence, the only difference in the age relation in their case is to replace the 6.7 by 7.0.
The distance to the habitable zone, from the center of a white dwarf star, can be calculated directly from its age as follows:
rᵤ = 48977 t^(−0.7)
Where rᵤ is returned in astronomical units. (1 AU = 1.4959787e11 meters.)
rᵥ = 0.0657104 ∛[(M/M๏)/ρᵥ]
Where rᵥ is returned in astronomical units when ρᵥ (the density of a test object in orbit around the white dwarf) is input in kg m⁻³.
By using trial parameters, you can determine that only younger low-mass white dwarf stars might be used as suns for artificial space colonies. The older (cooler) and more massive (smaller size) white dwarfs will have habitable zone radii shorter than those of their Roche limits.
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