Cooling comes from the skyPDF
The cooling power of a surface exposed to sky comes from the off balance between emitted thermal radiation and radiation received from the sky – or, to be more precise, from the atmosphere. Radiation  corresponds to thermally excited atmosphere molecules. This thermal emission depends on temperature and concentration of molecules and their distance to the ground. It corresponds to an infra-red emission centred at 10 µm.
1. Kirchoff’s Law and Black Body
Thermal excitation of matter atoms and molecules make them vibrate, rotate and results in emission of electromagnetic radiation. Let us first consider a material with uniform temperature and composition called “black body” which absorbs and thus emits all radiations according to the Kirchoff’s law (Read: The thermal radiation of the black body).
2. Stefan-Boltzmann law and material emissivity
The total radiative power P emitted by a surface element at absolute temperature T can be obtained by integrating the black body spectral radiance over all wavelengths and over a half space above the surface. One obtains the well-known formula P = σT4 where P is in W.m-2 and T is in K; σ = 5.670 10-8 W.m-2. K-4 is the so-called Stefan–Boltzmann constant.
For materials that does not absorb or emit all radiations, called “grey body”, the Stefan-Boltzmann law above has to be corrected with what is called material emissivity (The thermal radiation of the black body). Emissivity has the same value as absorptivity because they correspond to the symmetrical process of light emission or absorption between two states of molecular excitations . For grey body with emissivity ε < 1 the Stefan-Boltzmann law above becomes P = εσT4 .
Note, however, that when dew starts to form, the substrate becomes wet by water. It is then the water emissivity which matters. The water emissivity is close to unity (0.98 in the atmospheric window, see below for its definition).
3. Atmosphere emissivity. Sky temperature
The atmosphere whose molecules absorb in specific spectral bands and emit radiation in the long wave part of the spectrum (3-100 µm) is a grey body. Oxygen and nitrogen, which compose the atmosphere by about 99% do not absorb or emit radiation in the far infrared, only the contribution from water vapour (about 0.2 % to 2 % by volume) and carbon dioxide (about 0.03 % in volume) matter. Radiation from water vapour is thus the more important. High absorption in this spectral region corresponds to a black body at about 300K temperature, except for a lower absorption between 7-14 µm. The latter is known as the atmospheric window and does not contain any water contribution. Only a peak due to stratospheric ozone is present, whose influence is relatively weak due to the stratosphere low temperature. Water vapour concentration usually decreases with altitude, which makes the atmospheric boundary layer  the place where most of IR radiations are emitted to the ground. The boundary layer ranges from tens of meters to a few km.
One can also define an apparent sky temperature Ts, which relates ground air temperature Ta and sky emissivity εs to give the same sky radiance on ground, as . One readily deduces the sky temperature .
4. Cooling by radiation deficit
The radiation deficit R is the difference between the radiative power of the condensing surface material emitted towards the sky and the power sent from the atmosphere when exposed to a clear sky radiation. With the material temperature Tc ≈ Ta , the air temperature :
Since and the materials cools down below the air temperature. R is thus the available cooling power for dew condensation. Because the sky emissivity is mostly dependent on the atmosphere water vapour content, the radiation deficit can be expressed as a function of the dew point temperature Td or relative humidity RH (defined in Focus Humid air and condensation). In Figure 1 are reported from Bliss (1961 ) the deficit as a function of Ta for several air relative humidity RH and Td. For typical nocturnal conditions where dew forms (Ta = 15°C and RH =85% – 90%, Td = 12.5 °C), R≈60 W/m2.
5. Angular dependence
The emissivity estimated above is the total sky emissivity where all angles and all wavelength contributions have been summed. There is, however, a marked dependence of the sky emissivity on zenithal angle  θ (Figure 2),.
εs is the total emissivity. The emissivity is the lowest for the atmosphere seen near the vertical (θ = 0 – 30°) corresponding to the smallest light path and the highest, close to unity, near the horizontal (θ = 75 – 90°). For efficient radiative cooling of a ground surface, all angles less than about 15° above the horizontal have then to be avoided. From the angle dependence of the sky emissivity, the influence of the tilt angle with horizontal of the cooling material can be evaluated . Unsurprisingly, the radiative deficit is maximum for the horizontal position and minimum for the vertical position.
Notes et references
Cover image. [Source : royalty free]
 RADIATION is the emission of energy by electromagnetic waves.
 STATES OF MOLECULAR EXCITATIONS: An excited state of an atom or a molecule is a quantum state with a higher energy than the minimum, ground state (that is, more energy than the absolute minimum). Excitations by e.g. the absorption of light (a photon) increase the energy level above a chosen starting point, the ground state or an already excited state. The return to a less excited state corresponds to the emission of a photon (light) whose wavelength depends on the difference in energy of the two states. For thermal excitation at room temperature, the range of energy corresponds to infra-red wavelengths
 THE ATMOSPHERIC BOUNDARY LAYER comprises the lowest part of the atmosphere extending from the ground). It is the place where ground and atmosphere exchange radiative, sensitive and latent heats (https://www.encyclopedie-environnement.org/physique/pression-temperature-et-chaleur/). It extends until where cumulus clouds form, which marks the commencement of the free atmosphere. In this layer many physical quantities (air flow velocity, temperature, humidity…) display rapid and turbulent fluctuations and vertical mixing is strong. The boundary layer thickness, h, can range from tens of meters to a few km and varies with time and can be expressed as where h is in km and Ta is the air temperature and Td the dew point temperature near the ground.
 BLISS R. A. (1961). Atmospheric radiation near the surface of the ground. Solar Energy 5 (3), 103–120
 BERGER, X., BATHIEBO, J. (2003). Directional spectral emissivities of clear skies. Renewable Energy 28 (12), 1925–1933.
 HOWELL, J.C., YIZHAQ, T., DRECHSLER, N., ZAMIR, Y., BEYSENS, D., SHAW, J.A. (2021). Generalized Nighttime Radiative Deficits. Journal of Hydrology 603 (B), 126971.