Why is Earth’s surface warmer than its radiation-to-space temperature? (science board)

We can work this out with measured sunlight, geometry and the laws of thermal radiation.

1. Start with conservation of energy (the first law of thermodynamics):

dE/dt = power absorbed - power emitted.

At equilibrium, the two powers balance. Earth intercepts sunlight over a disk of area pi*R^2, but its total surface area is 4*pi*R^2. With solar irradiance S approximately 1361 W/m^2 and planetary reflectivity A approximately 0.30:

P_absorbed = S*(1 - A)*pi*R^2
F_absorbed = P_absorbed/(4*pi*R^2)
F_absorbed = S*(1 - A)/4 approximately 238 W/m^2. [1]

The factor 4 comes from geometry, not an assumption that every place receives equal sunlight.

2. What temperature corresponds to that outgoing radiation?

The Stefan-Boltzmann law, obtained by integrating Planck's radiation law, gives blackbody emission:

F_emitted = sigma*T^4
sigma approximately 5.67 * 10^-8 W m^-2 K^-4

Temperature must be in kelvin. Therefore:

T_e = (S*(1 - A)/(4*sigma))^(1/4)
T_e approximately 255 K approximately -18 degrees C. [2]

This is the effective emission temperature: the blackbody temperature corresponding to Earth's radiation to space. The conventional rounded surface reference is about 288 K (15 degrees C), roughly 33 K warmer. A uniform blackbody surface at 288 K would emit sigma*(288 K)^4 approximately 390 W/m^2, considerably more than the 238 W/m^2 solar input. The surface therefore cannot be treated as directly radiating all its infrared energy to space. [3]

For a nonuniform surface with unit thermal-infrared emissivity radiating directly to space at equilibrium, with the same absorbed sunlight, sigma*mean(T^4) = 238 W/m^2. Because mean(T)^4 is less than or equal to mean(T^4) (Jensen's inequality), temperature variation cannot raise its mean above 255 K. This comparison holds today's reflectivity fixed; it is not an exact forecast for an Earth with its atmosphere removed, because clouds and reflectivity would also change. Solar reflectivity and infrared emissivity concern different wavelengths.

3. What changes the surface's energy budget?

Water vapour, CO2, other greenhouse gases and clouds absorb and emit thermal infrared radiation. Kirchhoff's law gives emissivity(lambda) = absorptivity(lambda) in local thermodynamic equilibrium: an absorber at a wavelength also emits at that wavelength. Its emission depends on its temperature through Planck's law. [2]

Some surface infrared escapes through atmospheric windows. Much of the infrared reaching space instead comes from colder atmospheric levels. The surface receives downward infrared as well as sunlight. In the steady-state approximation:

absorbed sunlight + downward infrared
= upward infrared + evaporation + sensible heat

The surface warming produced by the atmosphere's absorption and emission of infrared is, by definition, the atmospheric greenhouse effect. Energy still escapes to space; the atmosphere changes the temperatures needed to achieve balance. [1,3]

This obeys the second law. Radiation travels both ways, while net heat transfer remains from warmer to colder regions. For two ideal facing blackbodies:

F_net = sigma*(T_hot^4 - T_cold^4) > 0.

4. How do we identify the mechanism independently?

The temperature comparison alone does not identify the mechanism or a particular gas. Infrared spectroscopy does: satellite spectra show greenhouse-gas absorption signatures. AIRS observations have isolated the changing CO2 signal in its band near 15 micrometres and found agreement with radiative-transfer physics. [4]

Earth's internal heat is much too small to explain the comparison: an estimated 47 TW spread over Earth is about 0.09 W/m^2, versus roughly 238 W/m^2 of absorbed sunlight. [5]

The 33 K comparison concerns the natural greenhouse effect as a whole, not CO2 alone or a calculation of future warming from doubling CO2. The equilibrium calculation is a reference; Earth currently also stores energy when incoming power exceeds outgoing power.

Sources:
[1] NASA, Earth's energy budget:
https://science.nasa.gov/earth/earth-observatory/climate-and-earths-energy-budget/
[2] MIT atmospheric-radiation notes (Planck, Stefan-Boltzmann and Kirchhoff laws):
https://ocw.mit.edu/courses/12-815-atmospheric-radiation-fall-2006/resources/thermo/
[3] NASA GISS, measuring the greenhouse effect and the fixed-albedo comparison:
https://www.giss.nasa.gov/research/briefs/2010_schmidt_05/
[4] Teixeira et al. (2024), direct spectral observations from space:
https://acp.copernicus.org/articles/24/6375/2024/
[5] Davies & Davies (2010), Earth's internal heat flux:
https://se.copernicus.org/articles/1/5/2010/