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Clear-Sky Temperature (Berdahl-Martin)

Also known as effective sky temperature · sky emissivity · radiant sky temperature · Berdahl Martin correlation · clear sky emissivity from dew point · why frost forms on a clear night

Tsky=Tair[ 0.711+0.56(tdp100)+0.73(tdp100)2]1/4T_{sky} = T_{air}\left[\,0.711 + 0.56\left(\tfrac{t_{dp}}{100}\right) + 0.73\left(\tfrac{t_{dp}}{100}\right)^{2}\right]^{1/4}

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Constant used — built into this formula, no need to enter
T0=273.15 KT_0 = 273.15\ \text{K}Ice point (0 °C in kelvin) · exact
See also
Net Radiative Cooling to the Sky — What this sky temperature is FOR: it is the cold body a roof, a car or a leaf is radiating to.Surface Temperature Depression Under a Clear Sky — The end of the chain — how far below air temperature a surface actually settles once convection fights back.Dew Point (Magnus Approximation) — Where to get the dew point this correlation needs, if you have relative humidity instead.

Learning zone

Point an infrared thermometer at the sky on a clear night and it will read something absurd — twenty, thirty, forty degrees below the air temperature. It is not lying. The sky is not a surface, but it radiates, and the downward longwave it delivers is far less than a blackbody at air temperature would deliver. Wrap that deficit into an effective temperature and you have a single number you can put into a radiation calculation: T_sky, the temperature a blackbody would need to be to send you the same longwave flux. That number is why frost forms on a clear night and not a cloudy one, why a car windscreen frosts while the tarmac beside it does not, and why a bowl of water left out in a desert can freeze above freezing air temperature.

The mechanism is water vapour. Most of the downward longwave comes from the lowest few hundred metres of atmosphere, and most of that is emitted by water vapour, so the sky's apparent emissivity tracks how much vapour is present. Dew point is the practical proxy for that, and it is why this page asks for it. The correlation used here is Berdahl and Martin (1984): ε₀ = 0.711 + 0.56(t_dp/100) + 0.73(t_dp/100)², with the dew point in degrees Celsius, and T_sky = T_air × ε₀ to the power one quarter. Air at 20 °C with a dew point of 10 °C gives ε₀ = 0.7743 and a sky at 1.84 °C — eighteen degrees below the air.

This is an empirical fit, not physics, and it is one of several that disagree. Swinbank (1963) uses air temperature alone and ignores humidity entirely. Brunt uses vapour pressure. Idso and Jackson use the departure from 273 K. Applied to the same night, they can differ by several kelvin, and a several-kelvin error in T_sky is a ten-per-cent error in the radiative loss you compute from it. If you are comparing your figure to somebody else's, find out which correlation they used before you conclude anything. Berdahl and Martin's full published model also carries a diurnal correction and an elevation correction that this page leaves out; what you get here is the widely quoted reduced form.

Two boundaries. The fit was built on dew points roughly between −40 °C and +40 °C, and the page refuses to leave that band. And it is a CLEAR-sky formula, full stop. Cloud radiates at close to its own base temperature, so a solid overcast pushes the effective sky temperature back up toward the air temperature, radiative cooling nearly stops, and the frost does not come. That contrast — clear night, frost; cloudy night, none — is the single most useful thing on this page.

Clear-Sky Temperature (Berdahl-Martin)
Tsky=Tair[ 0.711+0.56(tdp100)+0.73(tdp100)2]1/4T_{sky} = T_{air}\left[\,0.711 + 0.56\left(\tfrac{t_{dp}}{100}\right) + 0.73\left(\tfrac{t_{dp}}{100}\right)^{2}\right]^{1/4}
TskyTairtdpclear sky, no cloud to radiate back
Where
  • TskyT_{sky}= Effective sky temperature (°C)
  • TairT_{air}= Air temperature (°C)
  • tdpt_{dp}= Dew-point temperature (°C)
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