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

Radium wrote:
I've read about the sun's life cycle. Apparently, when the sun becomes
a white dwarf, it will take at least a trillion years to completely
cool off. Why such a long time?

It seems that the sun would exist much longer dead [i.e. as a white
dwarf] than alive [burning hydrogen and helium].

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 = lifetime
E = total thermal energy in the star
L = luminosity of star = energy radiated per unit time

tau = E/L

The internal energy is just the product of the mass, the specific heat
(energy to heat/cool 1 kg of stellar matter by 1K), and the temperature,
so

m = mass of star
T = surface temperature of star
C = specific heat of stellar matter

tau = mCT/L

The luminosity is easy: it's the product of the surface area of the star
and the energy radiated per unit surface area, the latter given by the
Steffan-Boltzmann black-body radiation formula:

r = radius of star
sigma = Steffan-Boltzmann constant

L = sigma T^4 * (surface area)
= sigma T^4 * 4*pi*r^2

(Again, there are some approximations he we're ignoring neutrino
luminosity, we're assuming the surface to be in local thermodynamic
equilibrium (so we can characterize it by a single temperature), and
we're taking the stellar surface to be a black body. [In practice
astrophysicists usually turn things around, and define the "effective
temperature" of a star to be that temperature which -- given the
actual, non-black-body, stellar spectrum -- would give the observed
luminosity in the formula we just derived.])

[Another big approximation is that we're equating the surface temperature
(for radiation) and the mean temperature of the whole star (for internal
thermal energy).]

So, putting all the pieces together,

tau = E/L
= mCT / (sigma T^4 * 4*pi*r^2)
= mC/(4*pi*sigma*T^3*r^2)

Now we need to know what these various factors are, or more precisely
how they vary between sun-now and sun-as-white-dwarf:
* To a reasonable approximation, the sun as a white dwarf will have the
same mass as the sun today.
* The sun's specific heat depends on details of chemical composition
and ionization, but I guess it won't vary by more than an order of
magnitude (and maybe by no more than a factor of 2) between sun-now
and sun-as-white-dwarf.
* The sun-as-white-dwarf will be a *lot* smaller in diameter than
sun-now, maybe by a factor of 50 or so. That means 50^4 times less
surface area to radiate energy away!
* Finally, sun-as-white-dwarf will be somewhat hotter than sun-now,
and thus radiate more energy per square meter of surface area per
unit time. I'm rusty enough on my stellar astrophysics that I no
longer recall the typical surface temperatures of *old* white dwarfs,
but I'll make a SWAG of a factor of 3 hotter than the sun, i.e. 80
times more radiated energy per square meter of surface area per
unit time.

This means

tau_wd
------- = (m_wd/m_now) * (C_wd/C_now) * (T_wd/T_now)^-3 * (r_wd/r_now)^-3
tau_now
= 1 * 1 * 3^-3 * (1/50)^-3
= 5000

If we take the lifetime of sun-now to be on the order of 10 Gyr
(we're about 1/2 way through that at the moment), then this gives
a lifetime of 5 Tyr. Given the number of SWAGs involved, this is
a reasonable approximation to "at least a trillion years".

So, you can see that the biggest factor is that the sun-as-white-dwarf
will have a much smaller radius than the sun now, and hence a much
lower luminosity, and hence it will take much longer to radiate away
its energy.

--
-- "Jonathan Thornburg -- remove -animal to reply"
Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),
Golm, Germany, "Old Europe" http://www.aei.mpg.de/~jthorn/home.html
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powerless means to side with the powerful, not to be neutral."
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