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There have been several previous questions about geothermal warming of a rogue planet with regard to depth: how much overlying atmosphere, water, rock do you need for a given geothermal flux to maintain a livable temperature (potentially across the entire globe). I'm asking instead about local surface heating. Specifically:

A rogue planet has atmospheric composition same as Earth.

There is a lake 1 km wide whose surface is maintained at 20 Celsius by geothermal heat flux. (The required heat flux would probably be larger than exists anywhere on Earth today. I'm okay with that.) This will have a warming effect on the air in the vicinity of the lake.

How far from the lake, will the air temperature 2 m above the ground, still be at least -60 Celsius? (I'm using that as a proxy for 'humans can walk around in suitable protective clothing instead of needing an oxygen mask'.)

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    $\begingroup$ You're leaving out a lot of relevant details. What is the temperature of the ground surrounding the lake? What is the pressure of the atmosphere? What is the composition of the atmosphere? Depending on the mix of gasses humans have worked at pressures as low as 30 kPa to as high as 7026 kPa. Both pressure and gas mix will have an impact on conductivity. $\endgroup$
    – sphennings
    Commented Oct 6, 2023 at 21:01
  • $\begingroup$ @sphennings: A rogue rocky planet has no atmospheric composition at all, because all the air is frozen solid covering the surface. The atmospheric pressure is zero. $\endgroup$
    – AlexP
    Commented Oct 6, 2023 at 21:02
  • $\begingroup$ If there's no atmosphere how can humans walk around in suitable protective clothing instead of needing an oxygen mask? $\endgroup$
    – sphennings
    Commented Oct 6, 2023 at 21:03
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    $\begingroup$ Also if there's no atmosphere what are you measuring the temperature of 2m above the surface? $\endgroup$
    – sphennings
    Commented Oct 6, 2023 at 21:05
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    $\begingroup$ Also if there's no atmosphere, a liquid lake would boil. $\endgroup$
    – jdunlop
    Commented Oct 6, 2023 at 22:46

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A rogue planet has atmospheric composition same as Earth.

Frame challenge: without stellar radiation, the surface temperature of an Earth analogue would be low enough that all non-trace gases would precipitate. So in terms of an atmosphere, you don't have an atmosphere.

But according to Wikipedia:

However, in 1998, David J. Stevenson theorized that some planet-sized objects adrift in interstellar space might sustain a thick atmosphere that would not freeze out. He proposed that these atmospheres would be preserved by the pressure-induced far-infrared radiation opacity of a thick hydrogen-containing atmosphere (...) Even an Earth-sized body would have enough gravity to prevent the escape of the hydrogen and helium in its atmosphere.

Such worlds could maintain liquid water on their surfaces for maybe even billions of years without the need for geothermal flux, speciallly if they can get tidal heating from a moon.

I am sorry but I lack the knowledge to calculate the scenario for an H/He atmosphere enough for water to freeze.

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