How long could a white dwarf remain heated?

[Dartz]

One thing I've been researching is the possibility of habitable zones around white dwarf stars, and planets in them. The one question I can't find an answer for, is how long the star will take to cool off. If a planet were in a white dwarfs habitable zone, how long would it be before the star cooled and the planet froze?

[vh]

trillions of years, potentially.

[smjjames]

One estimate on wikipedia (under black dwarfs) is 10¹⁵ years, however ifWIMPs (not the same thing as the Higgs Boson) exist, it could be as long as 10²⁵ years for some white dwarfs.

[vh]

Ok i tried to do it with some physics:

using this chart: http://www.astro.sunysb.edu/mzingale/wdconvect/initial_model_sm.png
we can assume that a white dwarf is on average, 5e8 Kelvins.

the formula for heat capacity of a degenerate electron gas is

Cv=pi^3/2*k^2*T*N/Ef
where k is the boltzman constant, T is the temperature, N is the number of electrons, and Ef is the fermi energy

T=5e8k
mass is 1e30kg (assuming white dwarf of about 0.5 sol masses for now)
assuming white dwarf is made of 100% carbon, which should be a sufficient estimate, and a carbon atom weighs about 12amu or 2e-23 grams which means 5e55 carbon atoms in out white dwarf. since each carbon atom has 6 electrons, that is 3e56 electrons in our white dwarf
N=3e56
to find fermi energy:
Ef=h-bar^2/2m*(3*pi^2*n)^2/3 where h-bar is dirac's constant, m is the mass of an electron, n is the density of electrons.

Density=1e9kg/cm=3e32 electrons/cm^3
n=3e35
dirac's constant=h/2*pi where h is planck's constant
m=9.1e-31 kg

Ef=6.603e-14 kg*m^-2*s^-2

Cv=pi^3/2*(1.38065×10^-23)^2*5e8*3e56/(6.603e-14)

Cv=6.713e33 j/kgK

He=6.6713e33mT joules, where He is heat energy, m is mass in kilograms, and T is temperature in kelvins

for our white dwarf, m is 1e30, T is 5e8K, so the heat energy stored is about 3.34e72 joules.

The energy lost my a white dwarf, radiated into space can be calculated by

F=oT^4 where F is flux, T is temperature in kelvins, and o is the stefan boltzmann constant.

according to HR-diagram, white dwarf is about 1e4 K
stefan boltzman constant: 5.6704×10^-8 watts per square meter kelvin to the fourth
F=5.67e8 W/m^2

Assuming 5e6m radius for white dwarf, 4pi*r^2=3.1416e14m^2

white dwarf outputs 1.78e23 watts, about 1/2000th solar luminosity.

the loss in energy causes the temperature to lower and emit less energy which causes the temperature to lower more slightly and so on.

the formula is

L(t)=8.03e-291/(2.68e-218+1.356e-250*t)^(4/3)

(that took an incredible amount of time to figure out)
where L is the luminosity in percentage of original luminosity, and t is the time.

the graph looks like this.



sooo uhh, about 10 million times longer than predicted by wikipedia. close enough.

[mjduniverse]

23,000,000,000 Years
And If That Is True... There Is Probably NO BLACK DWARFS

[karakris]

Due to the Peculiar Physics of White Dwarfs - the Universe might not even CONTINUE TO EXIST long enough for White Dwarfs to become "black heat" type Black Dwarfs.

However - there could well be Back Dwarfs.  Brown Dwarfs cool down VERY MUCH quicker -  Some evidence exists that there are Black Dwarfs from this origin, with Surface Temperatures as low as 150 degrees Celsius.

Scientific Thought seems to speculate that there could be some as cold as 25 degrees Celsius or lower.
However - how would you see them, and damned hard to avoid, unless you use Mass Detectors.

These would be made almost entirely of Iron.
However - Damned dangerous things - the Gravity will crush you flat.

Habitable Planets around White Dwarves seems resonable - but they must be a long way out.  White Dwarves radiate as much as 30,000 to 1 Million times the Light and Heat of our Sol - AND some of it is Neutrons and X-Rays.