Does anyone know a cooling formula for white dwarfs?

Question

I am looking for a general cooling formula for white dwarfs that can possibly be used in relation to age.

Answer 1

I'm not sure if there is a simple formula, like what San is suggesting, that will work. The first problem, of course, is knowing the thermal properties of electron-degenerate matter.

The astrophysical literature makes this look like a pretty complex problem. Here's a paper that deals with the influence of the core composition on cooling rates, including how crystallization affects it: http://www.journals.uchicago.edu/ApJ/journal/issues/ApJ/v486n1/34903/34903.html

If nothing else, it has a plot of luminosity versus time for various size white dwarfs that you may find interesting - see figure 4. It also has references to previous work, if you want to get seriously into it.

Answer 2

Can't help you either on a ready-made formula, but who can resist.

You could just set it up as a radiative heat transfer problem, since this is the limiting resistance.

dT = - dQ/mCp = - dt * (E * SB * A * T^4) / ( m * Cp)

dT/T^4 = - (( E * SB * A ) / ( m * Cp )) dt

integrating,
T^(-3) / (-3) = - (( E * SB * A ) / ( m * Cp )) * t + C1
or 1/T^3 = ((3 * E * SB * A) / (m * Cp)) * t + 3C1
or T = (((3 * E * SB * A) / (m * Cp)) * t + C2) ^ (-1/3)

at t=o, Ti = ( C2) ^ (-1/3) or C2 = 1 / Ti^3

So T varies inversely with elapsed time t to the 1/3 power, if I did the algebra right.

Just throw in some vanilla values for emissivity and heat capacity for the degenerate matter to find the ball park. Assume maybe one solar mass for m, and say earth-sized to calc A. Stefan-Boltzmann constant is 5.67E-8 W/m^2 K^4. Ti might be around 20000 K or so. Have fun.

Answer 3

I've not come across a definite formula. I have read that the denser & smaller a white dwarf is, the longer it'll take to cool.

Answer 4

I,ve been looking for this and data on WD objects for a while and don't find much. They seem to have all temps at all ages so I figure the effort is a waste of time. You can try the ADS/NASA data base. I found some data there but not anything that helps much.