# Exactly how long does it take for the exposed core of a star to cool from its starting temperature (several billion K) to ~50,000 K?

OK, I didn't know how I should word this question. But the basic point is that most white dwarfs that we have classified fall in temperature ranges from ~50,000 K to 6000 K. However, at the end of a low-mass star's life, the core temperatures reach billions or maybe even trillions of Kelvin. So when the star dies and sheds its outer layers, exposing the core, exactly how long does it take for the core to cool down to the "standard white dwarf temperature?"

The answer is of order 1 million years to cool from a standard end of He burning temperature of just over $$10^8$$ K to the top end of the white dwarf temperature range you give in your question. The details would depend exactly on the mass and composition of the white dwarf and there are also some theoretical uncertainties in neutrino cooling rates.
For example, white dwarfs with photospheric temperatures of 50,000 K to 6,000 K have interior temperatures of about $$5\times 10^7$$ K to $$3\times 10^6$$ K. This interior temperature applies to the bulk ($$>99$$%) of the white dwarf because electron degeneracy leads to high thermal conductivity and the surface layer where the temperature falls and the electron degeneracy disappears is very thin because of the high gravity.
So the cooling timescale for cores that become white dwarfs may not be as short as you think. The thin, outer, non-degenerate layer acts like an insulating jacket around a hot water tank. The cooling timescale (the time to radiate all internal heat at its current luminosity) changes from tens of millions of years at interior temperatures of $$3\times 10^7$$ K to billions of years at interior temperatures of $$3\times 10^6$$ K, thanks to the strong dependence of luminosity on temperature.
However, this story only works once the core has cooled to around $$3\times 10^7$$ K from its initial temperatures. Above that temperature, the main cooling mechanism is via neutrino emission and the insulating jacket does not stop them. Neutrino emission is very temperature dependent $$(\sim \propto T^{10})$$. So the initial cooling phase to $$\sim 3\times 10^7$$ K is very rapid once nuclear reactions have ceased; about a million years. After that the neutrino losses become nmuch smaller quite quickly and certainly photon emission is dominant by the time the interior temperatures are $$10^7$$ K and the photospheric temperature is <30,000 K.
A similar process occurs in neutron stars. Very rapid cooling by neutrinos for $$\sim 10^5$$ years, followed by slower photospheric cooling later. The difference is that most of the white dwarfs we see are in the photospheric cooling phase, whereas the neutron stars we can see are cooling with neutrinos. Neutron stars have such small radii that they become effectively invisible once they reach the photospheric cooling phase.