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I read that when white dwarfs do not proceed with nuclear fusion, the heat radiation from it is solely based on heat it retained in the past But then, it floats in an almost 0 K empty space. So, why does it take a million (or many many more) years to cool down (to a black dwarf?)?

Shouldn't this temperature difference between them and the environment (space) cool them down much faster? Or maybe the emptiness of the space causes this very slow temperature outflow?

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    $\begingroup$ See also astronomy.stackexchange.com/questions/48442/… $\endgroup$
    – ProfRob
    Commented Jul 6, 2023 at 7:43
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    $\begingroup$ Why do you think that is slow? A few million years isn't a very long time for a star that burned fuel for 10 billion years. $\endgroup$ Commented Jul 6, 2023 at 19:18
  • $\begingroup$ @12431234123412341234123 uh, actually isn't it theorized that a white dwarf will take around 100 quadrillion years to cool? That's around 7 orders of magnitude longer than it burned fuel... $\endgroup$
    – Michael
    Commented Jul 7, 2023 at 3:39
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    $\begingroup$ What do you mean with cool down? How cool is cool, 10 kK? 3kK? 300 K? $\endgroup$ Commented Jul 7, 2023 at 17:48

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Since space is empty, heat can only be transmitted by radiation. Stefan-Boltzmann's radiation law states that the energy flow is proportional to the surface area, times the temperature to the fourth power. It is true that very hot objects transmit a lot of energy per square meter of surface due to the fourth power: a 10 times hotter object sends out 10,000 times more energy. But the surface area is small. A typical white dwarf is about 100 times smaller than the Sun, and hence has 10,000 times smaller surface area. So as soon as the white dwarf cools to a temperature comparable to the Sun it will have a much harder time radiating away the remaining heat.

The sun could in theory radiate away all its heat in 30 million years if it did not produce more (the thermal radiation timescale), but the dwarf would need (very roughly) 10,000 times that time just due to the surface area issue - literally hundreds of billions of years.

This calculation gets complicated by the internal structure of white dwarfs. They are solid rather than plasma, with dense degenerate matter sharing electrons between atoms. This makes their heat conductivity much bigger than stellar heat conductivity (speeding up cooling). But they also slowly crystalize when they are cool enough, and this releases heat that slows the cooling.

The actual cooling dynamics has been studied a lot since the temperature of white dwarfs give valuable information about their age.

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    $\begingroup$ Also the specific heat of that degenerate matter is stupidly high. $\endgroup$
    – Joshua
    Commented Jul 6, 2023 at 18:26
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    $\begingroup$ @Joshua (and upvoters) the specific heat of degenerate gases is approximately zero. However, the carbon and oxygen ions are not degenerate and behave like a perfect gas (3kT/2 per ion), then a solid (3kT per ion) as they cool and finally the heat capacity goes very small below the Debye temperature. The heat capacity is therefore smaller than for normal matter, since the electrons never contribute. $\endgroup$
    – ProfRob
    Commented Jul 6, 2023 at 22:22
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One obvious reason is that the bulk of a white dwarf is much hotter than indicated by the surface temperature. About 99% of a white dwarf's mass is at temperatures about 100 times higher than the surface (photospheric) temperature. There is thus a very large thermal reservoir available to keep the surface warm.

The 1% on top acts like a insulating blanket - like a jacket on a hot water tank - so that the white dwarf cannot lose energy at a rate faster than the blackbody radiation from its (relatively) cool surface.

Once that is established, the cooling time, given by the ratio of the thermal energy content to the rate at which heat is radiated, is obviously going to be very long for a given surface temperature. The thermal energy is proportional to the mass multiplied by the internal temperature (very large), whereas the cooling luminosity is proportional to the surface area (small for a white dwarf) multiplied by the fourth power of the surface temperature of the white dwarf.

To add more complexity, the time for white dwarfs to become "black dwarfs" (objects too cool to emit visible light, say cooler than a surface temperature of 2000 K) may not be as long as you think - "only" about the current age of the universe.

The reason for is that white dwarfs crystallise at interior temperatures of about $3\times 10^6$ K, when their surfaces are of order 30,000 K. However, the Debye temperatures of their crystalline interiors has almost the same value. Below this temperature a quantum mechanical effect (quantisation of the oscillation modes of the crystal - a.k.a. phonons) drastically reduces the heat capacity of their interiors leading to a greatly reduced cooling timescale and an accelerated fall in temperature (both the interior and the surface) with time.

The exact temperatures for crystallisation and "Debye cooling" depend on density and hence on the white dwarf mass, but some example cooling curves are shown below. They show the catastrophic cooling that takes place once the white dwarf enters the Debye cooling regime. This plot shows that the oldest white dwarfs in our Galaxy should already be "black dwarfs" if they have more than a solar mass.

Typical Galactic white dwarfs are actually a bit below the lowest mass in this plot (more like 0.6 solar masses), so the sudden drop in temperature will occur after about 15-20 Gyr for most white dwarfs.

White dwarf cooling curves.

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Space is a thermos

You say "Shouldn't this temperature difference [...] cool them down much faster?" And that's a common point of confusion, because space isn't cold (at least not the way we think of it), space is just empty. It's more correct to say space does not have a temperature, at least in the way we humans think of temperature.

Particles in space are extremely diffuse, which means they have almost no power to transfer heat by convection. An object in space is much like an object inside a thermos flask; it pretty much just retains its own temperature, cooling only very slowly due to black-body radiation (which means infrared for things at human-adjacent temperatures, but visible light for things like stars).

We can (and do) talk about the temperature of the particles in space, but there just aren't enough of them to change the temperature of macro-scale objects. If you got hit by a few dozen extremely cold atoms, you wouldn't even notice it. Contrary to Hollywood's fanciful special effects, space is not an ice bath.

Surface area

The second thing that makes white dwarfs cool slowly is that they are unbelievably dense -- the mass of the entire Sun packed into a ball the size of Earth. This is past the point where atoms as we understand them can exist. That means there is very little surface area for the amount of hot matter packed in there.

So putting those together, you have a whole sun's worth of hot matter trying to push energy out through a tiny surface area, and only able to give off that energy by radiating from its surface. That means there's an enormous amount of thermal energy trapped inside, all waiting in line to get out. A line trillions of years long*.

*Which really puts my commute in perspective.

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    $\begingroup$ White dwarfs will become invisible black dwarfs in 10-20 billion years. Not trillions of years. $\endgroup$
    – ProfRob
    Commented Jul 6, 2023 at 22:24
  • $\begingroup$ And say you had 'cold' matter that just happened to fall into the white dwarf. Far from cooling it down, it would actually increase the temperature. A 0.6 solar mass white dwarf would be about the size of the Earth and something orbiting just above it's surface would be moving 3536 km/s. If it came from further away it would be going much faster. A 1kg block of ice in that orbit would contain 6.5 trillion J of kinetic energy. If it crashed into the white dwarf that would be converted to heat, about enough to raise its temperature to 1.5 million degrees celsius. $\endgroup$ Commented Jul 7, 2023 at 3:44
  • $\begingroup$ @ProfRob "Scientists have calculated that a white dwarf will take at least a hundred million billion years to cool down and become a black dwarf, according to astronomer Ethan Siegel." That's 100 quadrillion years, no? $\endgroup$
    – Michael
    Commented Jul 7, 2023 at 3:50
  • $\begingroup$ @Michael I define a black dwarf as something that emits almost no visible light. A practical definition of that is one whose temperature falls below 2000K. Cooling curves are shown in my answer. I have no idea how Ethan Siegel defines a black dwarf, but since nothing can reach 0 K, a definition there must be. $\endgroup$
    – ProfRob
    Commented Jul 7, 2023 at 7:04
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    $\begingroup$ @ProfRob I didn't say "black dwarf", and I'm not looking for 0 Kelvin, but rather thermal equilibrium with the space around it, which would require trillions if not quadrillions of years. The point at which it cannot cool any more. If you go by "just cool enough to stop emitting in the visible spectrum" then yes, it's "only" billions of years. But I was talking about how long it takes for all the energy trapped inside the dwarf to get out, with no reference to its emission spectrum. $\endgroup$ Commented Jul 7, 2023 at 21:23

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