When the sun becomes a white dwarf why will it take SO long to cool off?

I wrote
Ok, let's work out what determines the lifetime of a star:
To within a factor of two ot so (we're neglecting all time dependence),
the lifetime is just the ratio of the internal (thermal) energy in the
star, to the luminosity (energy radiated per unit time):
[[...]]
tau = E/L
= mCT / (sigma T^4 * 4*pi*r^2)
= mC/(4*pi*sigma*T^3*r^2)

As several people have pointed out, there are a couple of mistakes
in my analysis:
* My analysis assumes no internal energy sources. This is ok for a
white dwarf (rotation, magnetic fields, etc, are all only very minor
contributions, and so can be neglected in this "only good up to factors
of order unity" analysis). But it's spectacularly wrong for a
main-sequence star like the sun, which has a huge thermonuclear-fusion
heat source in its core.
* I had two different wrong powers of the radius in my derivation,
although I did use the correct power in the equation I quoted above.
[An old physics joke: Always make an even number of
mistakes in a calculation. That way they'll cancel
out and the final answer will be correct!]

The basic physics is that stated by several posters in this thread:
a white dwarf has a huge amount of thermal energy, and (relatively)
a tiny surface area and hence a tiny luminosity, so it takes a long
time to cool.

ciao,

--
-- "Jonathan Thornburg (remove -animal to reply)"
Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),
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