In Planck, Stefan-Boltzmann, Kirchhoff and LTE one of our commenters asked a question about emissivity. The first part of that article is worth reading as a primer in the basics for this article. I don’t want to repeat all the basics, except to say that if a body is a “black body” it emits radiation according to a simple formula. This is the **maximum** that any body can emit. In practice, a body will emit less.

The ratio between actual and the black body is the emissivity. It has a value between 0 and 1.

The question that this article tries to help readers understand is the origin and use of the emissivity term in the Stefan-Boltzmann equation:

E = ε’σT^{4}

where E = total flux, ε’ = “effective emissivity” (a value between 0 and 1), σ is a constant and T = temperature in Kelvin (i.e., absolute temperature).

The term ε’ in the Stefan-Boltzmann equation is not really a constant. But it is often treated as a constant in articles that related to climate. Is this valid? Not valid? Why is it not a constant?

There is a constant material property called emissivity, but it is a function of wavelength. For example, if we found that the emissivity of a body at 10.15 μm was 0.55 then this would be the same regardless of whether the body was in Antarctica (around 233K = -40ºC), the tropics (around 303K = 30ºC) or at the temperature of the sun’s surface (5800K). How do we know this? From experimental work over more than a century.

Hopefully some graphs will illuminate the difference between emissivity the material property (that doesn’t change), and the “effective emissivity” (that does change) we find in the Stefan-Boltzmann equation. In each graph you can see:

- (top) the blackbody curve
- (middle) the emissivity of this fictional material as a function of wavelength
- (bottom) the actual emitted radiation due to the emissivity – and a calculation of the “effective emissivity”.

The calculation of “effective emissivity” = total actual emitted radiation / total blackbody emitted radiation (note 1).

At 288K – effective emissivity = 0.49:

At 300K – effective emissivity = 0.49:

At 400K – effective emissivity = 0.44:

At 500K – effective emissivity = 0.35:

At 5800K, that is solar surface temperature — effective emissivity = 0.00 (note the scale on the bottom graph is completely different from the scale of the top graph):

Hopefully this helps people trying to understand what emissivity really relates to in the Stefan Boltzmann equation. It is not a constant except in rare cases. But you can see that treating it as a constant over a range of temperatures is a reasonable approximation (depending on the accuracy you want), but change the temperature “too much” and your “effective emissivity” can change massively.

As always with approximations and useful formulas, you need to understand the basis behind them to know when you can and can’t use them.

Any questions, just ask in the comments.

**Note 1** – The flux was calculated for the wavelength range of 0.01 μm to 50μm. If you use the Stefan Boltzmann equation for 288K you will get E = 5.67×10^{-8} x 288^{4} = 390 W/m^{2}. The reason my graph has 376 W/m^{2} is because I don’t include the wavelength range from 50 to infinity. It doesn’t change the practical results you see.