(c) 2026 by Barton Paul Levenson
Earth's surface is hotter than its radiative equilibrium temperature due to the greenhouse effect. This is also true for every other planet with a substantial atmosphere, and the moon Titan. Nonetheless, accepting this as a dry academic fact may not always be convincing.
But basic facts of radiation physics may be used to show that the greenhouse effect is not an option. Physical laws would be violated if it did not exist.
The first law of thermodynamics, also known as "1LOT" or the law of conservation of energy, might be considered the most basic law of modern physics. If conservation of energy does not hold, anything is possible, including perpetual motion machines.
A mathematical expression of 1LOT is:
dU = dQ + dW 1
where U is the total internal energy of a system, Q is thermal energy added to the system, and W is work done on the system. d means a small change in the amount. The change in internal energy is the sum of the change in thermal energy and the change in work done on the system.
The temperature of the system reflects its content of thermal energy. Assume the system is doing no work. If thermal energy is both coming into the system and leaving it, so that
dQ = dQin - dQout 2
Then,
If dQin > dQout, the system must heat up.
If dQin < dQout, the system must cool down.
All objects hotter than absolute zero radiate electromagnetic energy. Wien's Law describes the peak wavelength given off:
λpeak = c3 / T 3
Here c3 is the Wien's Law constant (sometimes written as b), and T is the absolute (i.e. Kelvin) temperature. For wavelengths in microns (micrometers, μ), c3 = 2897.771955 micron-kelvins.
If we divide "shortwave" radiation from "longwave" at a wavelength of 4 microns, 99% of sunlight is shortwave, while 99% of Earth's surface and atmospheric radiation is longwave. Using Wien's law, the peak wavelength of sunlight, at the sun's effective temperature of 5,777 K, is 0.50 microns, in the visible light region. The peak wavelength of Earth's radiation to space, at 255 K, is 11 microns, in the thermal infrared.
The Stefan-Boltzmann Law describes how much radiation an object at a given temperature emits:
F = ε σ T4 4
F is the "flux density" given off, in watts per square meter. ε is the emissivity, or radiative efficiency compared to a perfect radiator known as a black body. ε can be anywhere from 0 to 1. σ is the Stefan-Boltzmann constant, 5.670373 x 10-8 watts per square meter per kelvin to the fourth power. Finally, T, again, is the kelvin temperature.
The greenhouse gases, such as water vapor and carbon dioxide, preferentially absorb infrared light. So do clouds.
We now proceed to show that if energy is conserved, Wien's Law and the Stefan-Boltzmann law hold, and something in the atmosphere absorbs infrared light, there must a greenhouse effect.
We first consider the case of Earth with no greenhouse gases or clouds in its atmosphere (or no atmosphere!).
The solar constant at Earth's distance from the sun averages 1361.5 watts per square meter (Kopp and Lean 2011). From satellite observations, Earth's albedo (the fraction of sunlight reflected back out to space) is about 0.295 (cf Dewitt and Clerbaux 2017, Datseris and Stevens 2021). So the flux density of sunlight absorbed by the climate system is about
F = (S / 4) (1 - A) = 240 W m-2 5
With Earth's atmosphere transparent to infrared light, there is no greenhouse effect. All incoming sunlight is absorbed by the ground, and if sunlight is coming in at 240 W m-2, the ground is increasing its thermal energy. It must heat up.
But objects at any nonzero temperature radiate energy according to the Stefan-Boltzmann law. The hotter the ground, the more it radiates. At equilibrium, it must be radiating 240 W m-2 as well as absorbing that much.
We assume for the moment that the ground is a black body, a perfect emitter with ε = 1. (It actually isn't too far from that, about 0.98 (Levenson 2021), but we're keeping things simple here.) If so, the Stefan-Boltzmann law tells us its temperature is 255 K. This is the radiative equilibrium temperature of the planet.
255 K is cold--below the freezing point of water at 273 K. From sunlight alone, Earth's surface should be frozen over! This is what led Joseph Jean-Baptiste Fourier to conclude in 1824 that Earth's atmosphere must exert a greenhouse effect.
Now let us suppose the atmosphere absorbs 77% of the infrared light entering it. That's 184.8 W m-2 subtracted from the 240 W m-2 coming from the ground. So only 55.2 W m-2 is now getting out to space.
But with the air absorbing IR, it must heat up, just like the ground did. At equilibrium it will radiate as much as it takes in. Having both a top and a bottom, half the 184.8 W m-2 radiated out will go up to space--that's 92.4 W m-2--and the other half will go back down to the ground.
So the total getting out to space is now 55.2 + 92.4 = 147.6 W m-2. More is coming in (240 W m-2) than going out (147.6 W m-2). So the climate system must heat up.
Things will stabilize when as much is going out as is coming in. At equilibrium we have:
240 = (1 - 0.77) σ Tg4 + ½ (0.77) σ Tg4 6
where Tg is the temperature of the ground.
240 is the amount coming in from sunlight. The first term on the right of the equals sign is the infrared light reaching space from the ground. The second term is the amount reaching space from the atmosphere.
Working out the math, the ground temperature at equilibrium is 288 K. The IR-absorbing atmosphere has warmed Earth's surface by 33 K.
So if
then there must be a greenhouse effect. There's no way to avoid it. It's simple physics.
Dewitte, S., N. Clerbaux 2017. Measurement of the Earth radiation budget at the top of the atmosphere--A review. Remote Sensing 9, 1143-1152.
Datseris, G., B. Stevens 2021. Earth's albedo and its symmetry. AGU Advances 2, e2021AV000440.
Fourier, J.J.-B. 1824. Memoire sur les Temperatures du Globe Terrestre et des Espaces Planetaires. Annales de Chemie et de Physique 2d Ser. 27, 136-167.
Kopp, G., J.L. Lean 2011. A new, lower value of solar irradiance: Evidence and climate significance. Geophys. Res. Lett. 38, L01706.
Levenson, B.P. 2021. Habitable zones with an Earth climate history model. Planet. Space Sci. 206, 105318.
| Page created: | 03/02/2026 |
| Last modified: | 03/03/2026 |
| Author: | BPL |