Without the sun to keep the Earth's temperature topped up to something comfortable, what would be the equilibrium surface temperature range. It would obviously be a balance between outgoing radiation and geothermal conduction to the surface. This question is prompted by Anixx's recent question on planetary cooling time without the sun.
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1$\begingroup$ Eventually, though, we'd run out of geothermal heating, since there's only a finite amount of heat in the Earth's core. So, the "equilibrium" temperature would ultimately be the same as "space temperature", right around 3K. Or do you mean we turn off the sun but then wrap the Earth is a perfect vacuum (or something) and prevent thermal radiation from the Earth so that we don't lose heat to outer space? $\endgroup$– user967Nov 27, 2016 at 22:07
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$\begingroup$ @BarryCarter Technically, earth generates half of its own heat (radiogenic part) and will keep doing so for a while. The "finite amount" is more true for primordial heat. $\endgroup$– staliNov 29, 2016 at 14:11
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$\begingroup$ Is that assuming with the current atmospheric contents (gases)? David Hammens answer assumes that it changes. $\endgroup$– Jan DoggenDec 8, 2016 at 9:30
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1 Answer
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Assuming a thermodynamic equilibrium between heat from below and heat escaping into outer space, and assuming an energy from below of 44 to 47 terawatts (the Earth's current internal heat budget), that means the surface temperature would be about 35 or 36 kelvins by the Stefan-Boltzmann law: $$\varepsilon \sigma A T^4 = \phi$$ where $\varepsilon$ is the surface's emissivity in the thermal range (which I assumed to be one), $\sigma$ is the Stefan-Boltzmann constant (5.670367×10-8 W/M2/K4 in SI units) , $A$ is the Earth's surface area, $T$ is the surface temperature, and $\phi$ is the energy supplied to the Earth's surface from below.
Note that atmospheric effects are a non-concern for this very low temperature. The Earth would have no atmosphere except perhaps some trace helium and hydrogen gas.
answered Nov 29, 2016 at 20:57
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1$\begingroup$ ...which would be a value of.... $\endgroup$ Nov 29, 2016 at 23:35
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$\begingroup$ @AtmosphericPrisonEscape -- Which would be a value of 35 or 36 kelvins. Do I need to spell out the Stefan-Boltzmann constant or the surface area of the Earth? $\endgroup$ Nov 30, 2016 at 3:28
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1$\begingroup$ For a complete answer, yes. $\endgroup$ Nov 30, 2016 at 12:03
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$\begingroup$ @AtmosphericPrisonEscape -396 Fahrenheit. $\endgroup$ Dec 7, 2016 at 5:42
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3$\begingroup$ @Fred - I undid your change from "35 or 36 kelvins" to "35 or 36 Kelvin". The kelvin as a unit of temperature is never capitalized. The same goes for newton (unit of force), watt (unit of work), hertz (unit of frequency), and many others. Regarding pluralization: The SI says that the plural of kelvin is kelvins, but CERN says it is kelvin. Since it's their unit (not CERN's), I'll go with the SI convention. $\endgroup$ Dec 7, 2016 at 10:58