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I do know that the atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention (greenhouse effect),and reducing temperature extremes between day and night (the diurnal temperature variation). I wonder what temperature would earth reach if there was no atmosphere?.

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My recollection is an average sufrace temp of 255 K based on radiative equilibrium. I'll make an answer if I can back that number and how it is determined. –  casey May 7 at 3:59
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The question is a little ambiguous. Are you asking about the temperature difference between the earth with surface as it is now, but without an atmosphere? Or about the earth with no atmosphere and with a bare surface? It makes a significant difference to the answer. If you're just interested in how much difference the greenhouse effect gives, then it's best just to ask that explicitly - and the answer then, as casey says, is a surface temp of 255 K, and a temp diff of about +33 K –  EnergyNumbers May 7 at 7:28
    
@EnergyNumbers What I meant to ask was usual temperature one can read on thermometer. –  Praveen Kadambari May 7 at 8:01
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OK. Of the earth exactly as the surface is now, just in the absence of the atmosphere (that is, what it would look like the instant the atmosphere vanished)? Or of the earth without the atmosphere, and if the Earth's surface was like the moon's: no water, no snow, no forests, etc (that is, what the surface would come to look like after centuries of no atmosphere)? –  EnergyNumbers May 7 at 10:06
    
Assuming no change to the Sun, and after the atmosphere vanishes we've allowed enough time for all water to disappear, all ice to sublimate, all dead flora and fauna to crumble to dust, etc.? Then I would expect temperatures to be very similar to the Moon's, modified by a slightly higher albedo (the Moon is quite dark, and there could be a bright coating of salt where the oceans used to be) (-, but might radiate less at night) and a little bit more heat flow from the Earth's interior (+). A lot depends on the history of the planet -- did it ever have oceans, etc. –  Phil Perry May 7 at 16:28

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up vote 16 down vote accepted

According to Wikipedia an approximate average surface temperature for a bare earth is 274.5 K. This scenario is quite reasonable in my opinion as stripping the atmosphere without changing much else would (on a geological timescale) rather quickly result in a bare earth without ice caps or vegetation, causing circumstances quite close to those on the moon.

This is estimated by comparing the black body radiation of the Earth and Moon, which is then corrected for albedo (fraction of incoming radiation that is reflected) and emissivity (ability of a material to emit radiation), which are properties of a material. Since the Earth and moon are both at the same distance to the sun and made up from the same material on average, measurements of the albedo and emissivity of the moon can than be used as estimations of these properties for the Earth.

The black body radiation of the sun is calculated with the Stefan-Boltzmann law:

$$P_{\text{S,emit}} = 4\pi R_S^2 \sigma T_S^4$$

$P_{\text{S,emit}}$ is the emitted energy by the sun, $R_S$ is the radius of the sun, and $T_S$ is the temperature of the sun. The fraction of this energy recieved by the Earth is then proportional to the circular surface area facing the sun and the energy density at the distance $D$ between the Earth and the sun.

$$P_{\text{SE}} = P_{\text{S,emit}}\left(\frac{\pi R_E^2}{4\pi D^2}\right)$$

$R_E$ is the Earth's radius. Using albedo $\alpha$ the absorbed energy can be calculated:

$$P_{\text{E,abs}} = (1-\alpha)P_{\text{SE}}$$

Applying the Stefan-Boltzman law to the Earth, corrected for the emissivity $\overline{\epsilon}$, the emitted energy is then:

$$P_{\text{E,emit}} = \overline{\epsilon} 4\pi R_E^2 \sigma T_E^4$$

Assuming energy equilibrium $P_{\text{E,abs}} = P_{\text{E,emit}}$ we can now calculate $T_E$:

$$\begin{aligned} \frac{(1-\alpha)4\pi R_S^2 \sigma T_S^4\pi R_E^2}{4\pi D^2} & = \overline{\epsilon}4\pi R_E^2 \sigma T_E^4 \\ T_E^4 & = \frac{(1-\alpha)4\pi R_S^2 \sigma T_S^4\pi R_E^2}{\overline{\epsilon}4\pi D^2 4\pi R_E^2 \sigma} \\ T_E^4 & = \frac{(1-\alpha) R_S^2 T_S^4}{ 4\overline{\epsilon}D^2 } \\ T_E & = \left( \frac{(1-\alpha) R_S^2 T_S^4}{4 \overline{\epsilon}D^2 }\right)^{\frac{1}{4}} \\ T_E & = T_S \left( \frac{(1-\alpha) R_S^2}{4 \overline{\epsilon} D^2 }\right)^{\frac{1}{4}} \\ T_E & = T_S \sqrt{ \frac{ R_S \sqrt{\frac{1-\alpha}{\overline{\epsilon}}} }{2 D } } \end{aligned}$$

Finally we only need to insert the correct values:

  • $R_S = 6.96\times 10^8$ m
  • $T_S = 5778$ K
  • $D = 1.496\times 10^{11}$ m
  • $\alpha = 0.1054$ (assuming value of the moon)
  • $\overline{\epsilon} = 0.95$ (assuming value of the moon)

This gives us a temperature of 274.5 K.

Note that there are many factors that can cause local and temporal variations. For example, incoming radiation varies with latitude and season, and if the removal of the atmosphere would be caused by a dying sun that grows to engulf the earth temperatures would be much higher than this. All in all, to account for all those factors a very large model must be made that can analyse the influence of each factor, including the decrease in temperature of a dying sun etc., but that would be nearly impossible to build if only for the resources it would take to do so.

Since one of the most contested factors is the albedo after the atmosphere is removed the following graph shows how the average surface temperature changes with albedo. At an albedo of zero all incoming solar radiation is absorbed, while at 1 all radiation is reflected. Note that the temperature of 0K is an effect of the assumed equilibrium between incoming and emitted radiation, which will not hold at that point. As said above, the albedo for a bare earth will be approximately 0.1, while current values on average range from 0.3-0.4, largely contributed to by clouds. An average for the albedo of the Earth in its current vegetated state, but without clouds I haven't been able to find.

Black body temperature of earth with albedo

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According to your wikipedia link, the temp is 254-255 K. But you've reported it as saying 274.5 K. Any idea where that discrepancy comes from? –  EnergyNumbers May 7 at 7:04
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@EnergyNumbers Please read the wikipedia article more carefully. The 254K is the BLACK BODY temperature, the value of 274K I provide here is corrected for albedo and emissivity of a bare earth without atmosphere and thus a more realistic value than the plain black body radiation. –  hugovdberg May 7 at 7:08
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@EnergyNumbers fixed a couple of copy paste errors, and it doesn't say 274.5K but 1.36C instead (a couple of lines below the 254K), but I prefered to use Kelvin throughout my answer. –  hugovdberg May 7 at 7:13
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But isn't the question about earth in the absence of the atmosphere, rather than a bare earth? Now, I agree there's an ambiguity there in the question because of that. Do you think it might be worth explicitly dealing with that ambiguity? –  EnergyNumbers May 7 at 7:26
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@EnergyNumbers I think an earth without atmosphere would quickly turn into a bare earth, or never had plants over it. Besides, large parts of the albedo are caused by clouds and ice coverage, removing the atmosphere would at the very least remove the clouds, so still decrease the albedo, and cause a higher temperature, although that is probably hard to qualify. –  hugovdberg May 7 at 7:30

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