If the Sun disappeared, how long would it take for the Earth to freeze beyond excavatable depths?

If the Sun disappeared

• How long would it take for the earth to freeze beyond excavatable depths?
• How long until it freezes to the core? (Is this different than without removing the Sun?)
• Would we be able to live for a while by relocating downwards?
• If so around what portion of the population and for how long?
• Could we find a way to keep some oxygen with us or to extract what would turn liquid/solid on the surface?
• Would the molten/ductile rock expand or contract?

Step wise approach:

Right now when the surface is very cold, the frost depth here (Alberta, -40 min winter temps) can reach 8 feet. Below about 30 feet seasonal temperature changes are under a degree.

Around depths of 100 feet temperatures start to rise, increasing about 25C/km or 1F/70 feet.

First approximation then for zero solar input (effective temp -270K) would be that you would need (270+20)/25 = ~12 km of depth to have roughly room temperature conditions.

We currently can't dig holes that deep.

The world's deepest is 3.9 km in South Africa. Note that at 25C/km the temperature at that depth is well over boiling point. Air is compressed at the surface, then chilled. Air expansion in the working levels of the mine expands and cools further.

This assumes that the earth's insulation value is linear with depth.

Now: Time: We can put a lower bound on it:

A year's temperature variation right now reaches 30 feet, with most of that being in a fraction of that. So suppose that it penetrates at 30 feet per year. I think this is too much, but it serves as a bound. At 30 feet per year it would take 33 years per thousand feet, or about 110 years per kilometer. At present depts of 3.9 km, we have about 4 centuries.

In practice you wouldn't do it like this. Go deep enough to get away from the worst, built an insulated cave, and run heat exchange tubing deeper to heat your cozy refuge until whoever borrowed the sun puts it back.

Assuming the earth to be a perfect blackbody, the radiated power is given by:

$$\rm{P = A \cdot \sigma \cdot T^4}$$

where T is the blackbody temperature and sigma is the Stefan Boltzman constant.

If the sun disappears, then the Earth's core supplies the heat. The flux of this heat is estimated at $$91.6 \rm{\frac{mW}{m^2}}$$. Equating the two, we have

$$\rm{91.6\frac{mW}{m^2} \cdot A = A \cdot \sigma T^4}$$

Which gives $$\rm{T = 35.65 K}$$ at the surface.

However, the temperature would increase as you go down to the core, the source of the radioactive decay. It's 4-6k deg C in the core, so there will always be a spot inside the Earth which is toasty.

• Radiogenic heat comes mostly from the mantle, not from the core. – Jean-Marie Prival Oct 7 at 10:36
• @Jean-MariePrival I was always surprised on it. Uranium and thorium have much higher density than iron (and light metal oxides). And, although molten uranium doesn't mix well with molten iron, there is a 2ppm maximal limit of molten uran in molten iron. On these reason, it is hard for me to believe this. – peterh - Reinstate Monica Oct 8 at 15:45
• @peterh-ReinstateMonica I think it has to do with these elements being lithophile rather than siderophile, so when diapirs of molten iron sunk to the core through molten rock, they stayed in the rock because of chemical affinity. Now it is very difficult for them to migrate even if they are dense, because the mantle has a very high viscosity. I guess you could calculate a very, very low sedimentation velocity with Stoke's law. – Jean-Marie Prival Oct 8 at 16:46
• @Jean-MariePrival The earth science considers it enough fluid to call it liquid, and they had billions of years for the task. Ok, the inner core is solid, but so they should be accumulated in a some km high layer at the bottom of the outer core. But chemical affinity is a very strong and very interesting reason. – peterh - Reinstate Monica Oct 8 at 17:24

If the sun disappeared the Earth would be flung out of its orbit, and with a bit of luck would collide with another planet to put us all out of our misery. For the Earth to freeze beyond excavatable depths would take hundreds of years; there are mines in S.Africa which are miles deep and are heated to almost unbearable temperatures by heat rising from the Earth's interior, but food stocks would soon run out. To freeze right to the centre of the Earths core would certainly take millions, probably billions of years, because some of the core's heat comes from the decay of radioactive elements. Another almost everlasting source of heat is tides generated by the moon. There are tides in the solid crust as well a in the sea, and the constant flexion of the earth's crust and mantle would generate some heat, but not enough to keep plants and animals alive.

Fortunately, as I'm sure you understand, this is never going to happen. You could survive for a short while by relocating downwards, but only for a short while, months rather than years, and that's assuming you had food and water down there and you lived in S.Africa and gained access to their deepest mine. Advanced life on Earth depends on the sun. Nothing could grow without it, and therefore when stocks of food ran out they could not be replaced. All traffic, cars, buses trains etc would come to a halt within a few days. To avoid dying a slow and miserable death, the best plan would be to blow your brains out! Rock contracts when it cools, by the way, but water expands when it freezes.