# Why do grey body radiators not heat up?

So I'm doing so work on the earth's climate system, and modelling it as a grey body radiators, and I've come across the phrase: "$62\%$ of the outgoing energy is lost to space and the other $32\%$ is absorbed by earth's atmosphere".

Surely is energy is constantly being absorbed by the Earth it would just keep heating up for ever?

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I think we need a link to the document where you saw this statement so we can see the context. Also whoever made it can't add :-) – John Rennie May 3 '14 at 10:15
@JohnRennie Unless it's implicit that the remaining 6% goes to the Earth, haha! Although the sentence structure does suggest otherwise... – Danu May 3 '14 at 10:43
Just FYI, there is an earth science stack exchange (beta) these days as well – Nathaniel May 3 '14 at 17:24

Surely is energy is constantly being absorbed by the Earth it would just keep heating up for ever?

You are talking of grey body radiation, which means that the black body radiation is modified by the emissivity :

The SI unit for absolute temperature T is the kelvin. is the emissivity of the grey body; if it is a perfect blackbody, =1. In the still more general (and realistic) case, the emissivity depends on the wavelength,

The body absorbs less and emits less.

So a body at temperature T will absorb some energy E and its temperature will go up so it will radiate with the higher temperature. If the input radiance is much higher than the black/grey body radiation of the body irradiated, it will heat up until it combusts or melts or vaporizes. Fortunately the earth is far enough from the energy source to have reached an equilibrium , as we have seen by observations.The equilibrium can be disturbed by changes in the grey body constants, or by changes in the amount of energy impinging the earth. Then temperatures can go up, climate optima historically, or down, ice ages.

Vostok icecore measurements

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Yes, indeed, if the Earth were unable to get rid of all the incoming radiation, or if there were too much of it, the Earth's temperature might be like Mercury's or Venus'. Luckily for us, the Earth is in a "sweet spot" where the energy radiated from the Earth balances out all the energy coming in.

You can do simplified analysis yourself. Use 1365 W/m^2, which is the solar constant at the edge of the atmosphere, impinging on a 1m^2 blackbody plate in Earth's orbit which then radiates all of its energy into a 2.7K background from both sides of the plate. Solve for T in Planck's equation results in 331K, which is hot, but this simple equation ignores a bunch of stuff.

We know that ~30% of the sun's energy is directly reflected from the Earth: http://en.wikipedia.org/wiki/Albedo#Terrestrial_albedo so reducing the 1365 by 30% drops the temperature to 303K, which is 30C and close to the average temperature of the Earth.

Further refinements will get the number closer. See http://en.wikipedia.org/wiki/Earth%27s_energy_budget for more details

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Yes, unless all the accepted energy transformed into non-thermal form (like sunlight+CO$_2$+H$_2$O turning into sugar). It is generally assumed that in the long-term (more than one year?), the Earth radiates roughly as much energy as it receives from the Sun.

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