Inner core temperatures of white dwarfs

[Ray Emery]

We often read that core temperatures during the helium-fusing stage of low/medium mass stars might be of the order of 100 million K. But when the core becomes exposed (after the outer layers are ejected) and settles as a white dwarf, surface temps can be up to, say 100,000 K. My question: what is the temperature of the inner core at this point? Is there an enormous temperature gradient between the inner core and the surface (gaseous) layer? Does the idea of temperature as the velocity of moving particles no longer apply to the inner core (Pauli exclusion principle applying as regards tightly packed protons)?

Could someone give me a hand-waving answer to this, or point me towards one? Thankee.

[Dominic Ford]

White dwarfs have good thermal conductivity, so the core won’t be much hotter than the surface.

Young white dwarfs start life at ~10 million K. But that phase doesn’t last long. Heat dissipation scales as T^4 (Stefan’s Law), so the initial cooling is very rapid.

A rough calculation suggests that if a white dwarf takes 10 billion years to cool to 10,000 K, the same white dwarf would take only about a decade to cool from 10^7 to 10^6 K, and a few hours to cool from 10^8 to 10^7 K.

[Ray Emery]

Dominic – seems fair enough! Presumably once the core eventually solidifies/crystallises, it will become a much poorer conductor of such heat as remains (probably not much).  Cheers!

[Dr Paul Leyland]

The temperature may be quite low but the density is so high that hydrogen fusion still occurs in white dwarfs. I have seen an estimate that about 5% of the luminosity of a 10,000K WD is due to this process.  I will see if I can dig out the reference if anyone is interested.

[Ray Emery]

I knew it would be complicated; most things are! Could well be in the “not well understood” category? If you find the reference, I would be interested to give it a look over. Thanks!

[Dr Paul Leyland]

Here it is: https://ui.adsabs.harvard.edu/abs/1997ApJ…482..420L/abstract

Hydrogen burning continues in the fully convective body of a low mass white dwarf until the temperature drops below 2000K.

White dwarfs in binary systems very often undergo hydrogen fusion. The BAA-VSS observes them all the time. I admit this is stretching the terminology for “energy output in white dwarfs” but still …

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