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Assume an underground city existed below the surface of Antarctica. The ice sheet is ~2 miles thick on average. At what range/depth would the earth's thermal heat allow livable temperature for humans, assuming the civilization is technologically medieval (so no modern heating/cooling systems). Note: "Livable" does not mean comfortable. Extreme hot or cold would be fine as long as humans can reasonably survive in it.

Please ignore the obvious issues with an underground city (such as lighting, respiration, excavation, food sources, etc) and focus solely on the temperature aspect.

I assume there would be little subterranean temperature change between the polar day and night (despite each being six-months long).

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    $\begingroup$ Appreciated, @JBH. I've edited the original post. In response to your first comment, you are correct. What is the ideal depth range for ideal temperature conditions? $\endgroup$ – Silvirs Aug 19 at 22:43
  • $\begingroup$ Thanks. +1, I'll be interested to see what the geologists on the site come up with. $\endgroup$ – JBH Aug 19 at 23:20
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Not deep at all.

Assuming that "Livable" does not mean comfortable, people just have to hide from the elements to be able to survive. Indigenous people of Siberia and Alaska are surviving the temperatures plunging below -60°C. If you have shelter and fuel (and oxygen, of course) it doesn't matter how cold it is outside the shelter.

Venturing out into extreme cold is different, of course, and one can bundle up only so much. -80°C may not be livable for an extended period of time for a human, so if that is the requirement, we need to get warmer.

Within the Antarctic ice sheet, temperature gradient steadily rising towards the bottom, sometimes reaching ice melting point down below: Revealing interior temperature of Antarctic ice sheet. So getting 1000-2000 meters under the ice would definitely get people into livable, if not comfortable range.

But what if we want to get comfortable? In this case, we need to dig into Antarctic continent under the ice shield. Geothermal gradient suggests that for every 1 km of depth, temperature is rising by 25–30°C. To get the most comfortable +20°C, we need to get about 750-800 meters below the rocky surface under the ice.

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    $\begingroup$ I want to point out that the geothermal gradient is true no matter where in the world you are. So this answer can be used for any underground city anywhere on Earth. $\endgroup$ – Ryan_L Aug 20 at 2:04

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