<p>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<sup>-8</sup>  W/M<sup>2</sup>/K<sup>4</sup> 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.</p>

<p>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.</p>

CC BY-SA 3.0
2016-11-29T23:35:27.857Z
...which would be a value of....
AtmosphericPrisonEscape
CC BY-SA 3.0
2016-11-30T03:28:13.920Z
@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?
David Hammen
CC BY-SA 3.0
2016-11-30T12:03:02.027Z
For a complete answer, yes.
AtmosphericPrisonEscape
CC BY-SA 3.0
2016-12-07T05:42:55.153Z
@AtmosphericPrisonEscape -396 Fahrenheit.
Keith McClary
CC BY-SA 3.0
2016-12-07T10:58:54.913Z
@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.
David Hammen
CC BY-SA 4.0
2020-02-12T16:09:37.097Z
How can the answer be independent of properties of earth's atmosphere? If it isolates better the surface temperature should be higher, no?
emacs drives me nuts
CC BY-SA 4.0
2020-02-12T16:58:25.930Z
@emacsdrivesmenuts - What gases? The greenhouse gases (water, carbon dioxide, and methane, for example) would freeze out at much higher temperatures. Nitrogen and and then oxygen would freeze out at 63K and 54K, but neither of them are greenhouse gases. My assumption of an emissivity of one is more suspect, but even an unrealistically low emissivity value of 0.1 would have the surface temperature at about 63K. The oxygen and maybe the nitrogen wouldn't freeze out, but the greenhouse gases still would. An emissivity value of 0.2 or higher would result in the nitrogen and oxygen freezing out.
David Hammen
CC BY-SA 3.0
2016-11-29T20:57:59.753Z
2016-12-07T10:36:36.363Z
2016-12-07T10:36:36.363Z
David Hammen
<p>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. </p>

CC BY-SA 3.0
2016-11-29T14:11:59.893Z
@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.
stali
CC BY-SA 3.0
2016-12-08T09:30:06.870Z
Is that assuming with the current atmospheric contents (gases)? David Hammens answer assumes that it changes.
Jan Doggen
<geothermal-heat>
What is the current equilibrium surface temperature of Earth, i.e. without the sun?
Gordon Stanger
# What is the current equilibrium surface temperature of Earth, i.e. without the sun?

(Asked by: username_0 on 2016-11-27)

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.

**Tags:** geothermal-heat

> **Comments:**
> **username_1**: @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.  
> **username_2**: Is that assuming with the current atmospheric contents (gases)? username_3 Hammens answer assumes that it changes.  

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## 1 answers

### Answer by username_3 on 2016-11-29. Score: 10 | Accepted answer

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.

> **Comments:**
> **username_4**: ...which would be a value of....  
> **username_3**: @username_4 -- 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?  
> **username_4**: For a complete answer, yes.  
> **username_5**: @username_4 -396 Fahrenheit.  
> **username_3**: @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.  
> **username_6**: How can the answer be independent of properties of earth's atmosphere? If it isolates better the surface temperature should be higher, no?  
> **username_3**: @username_6 - What gases? The greenhouse gases (water, carbon dioxide, and methane, for example) would freeze out at much higher temperatures. Nitrogen and and then oxygen would freeze out at 63K and 54K, but neither of them are greenhouse gases. My assumption of an emissivity of one is more suspect, but even an unrealistically low emissivity value of 0.1 would have the surface temperature at about 63K. The oxygen and maybe the nitrogen wouldn't freeze out, but the greenhouse gases still would. An emissivity value of 0.2 or higher would result in the nitrogen and oxygen freezing out.  

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