Black Body Radiation and the Greenhouse Effect

A standard elementary argument for the occurrence of a greenhouse effect for the Earth assumes zero net-power, corresponding to an energy balance appropriate to a use of Stefan's radiation law.

The argument ignores coupling between the Sun and Earth. It also ignores the fact that Stafan's radiation law is a result valid only for in vacuo black body radiation, where all wavelengths are present.

The traditional view that a greenhouse effect for the Earth follows from Stefan's radiation law has been criticised, most recently by Gerlich and Tscheuscher [1].

Complex Problem

The total power density produced by two radiating bodies is a complex problem. Strictly, we should write the total power
\begin{displaymath} P=P_{1}+P_{2}+C(1,2) \end{displaymath} (1)

of two radiating bodies 1, 2 as the sum of the powers $P_{1}$, $P_{2}$ of the individual isolated bodies, plus an energy coupling $C(1,2)$ between 1 and 2.

I am not aware of a complete solution to the above equation. Even if the nature of the energy is assumed to be only electrodynamic then (1) still remains problematic due to arguments over the correct form of the coupling $C$.

Zero Net-Power

An elementary solution presented in textbooks involves putting $C$ equal to zero and making $P_{2}$ to be simply a function of $P_{1}$, which in turn is assumed to be the power density of a black body. No questions need be asked about the mechanisms involved in generating the black body radiation.

Sun & Earth

This situation is often claimed to represent the case of the Sun (body 1) and the Earth (body 2), and, as such, is an example of the net power radiated being the difference between the power emitted and the power absorbed. The net radiation power from Earth is assumed to be zero, enabling the equating of the power emitted by the Earth to the power it absorbs from the Sun.

The zero net-power assumption might seem to be obvious from the argument that the Earth is not a star and has therefore no internal source of radiation. However, this assumption ignores electromagnetic coupling through the magnetic field of the Earth. The mechanisms involved are not fully understood.

Stefan's Radiation Law

If we regard the Sun as a spherical black body of radius $R_{1}$, surface area $A=4\pi R_{1}^{2}$ and absolute temperature $T_{1}$ then its energy is generated at the rate of
\begin{displaymath} P_{1}=\sigma AT_{1}^{4}, \end{displaymath} (2)

where ${\sigma}$ is Stefan's constant. Eq. (2) becomes
\begin{displaymath} P_{1}=4{\pi\sigma}{\kern 1pt}R_{1}^{2}{\kern 1pt}T_{1}^{4} \end{displaymath} (3)

after substituting for the surface area $A$.

Eq. (2) is a form of Stefan's radiation law and pre-dates the work of Planck. It applies only for a black body, where all radiation wavelengths are present. Stefan's constant is derivable in terms of Planck's constant, the speed of light and Boltzmann's constant.

Temperature of the Earth

We can use (3) to obtain the power $P_{2}$ radiated by the Earth, and therefore the temperature $T_{2}$ of the Earth. The simplest way is to make some assumptions about the geometry.

We assume the Earth's orbit about the Sun to be a circle of radius ${r}$, which gives a power flux (power per square metre) from the Sun of $P_{1}/4\pi r^{2}$ at the surface of the Earth.

Although the Earth is taken to be a sphere of radius $R_{2}$, we also assume that the Earth presents itself as a disk of surface area $\pi R_{2}^{2}$ to this power flux, thereby collecting an amount of energy

\begin{displaymath} P_{2}=\left\{\frac{P_{1}}{4\pi r^{2}}\right\}\pi R_{2}^{2} \end{displaymath} (4)

per second. A substitution of $P_{1}$ from (3) gives
\begin{displaymath} P_{2}=\left\{\frac{4\pi R_{1}^{2}\sigma T_{1}^{4}}{4\pi r^{... ...eft\{\frac{R_{1}^{2}}{r^{2}}\right\}\pi R_{2}^{2}T_{1}^{4}. \end{displaymath} (5)

Again using Stefan's law,
\begin{displaymath} P_{2}=\sigma 4\pi R_{2}^{2}T_{2}^{4}. \end{displaymath} (6)

We now equate (5), (6) to obtain
\begin{displaymath} T_{2}=T_{1}\sqrt{\frac{R_{1}}{2r}} \end{displaymath} (7)

as the temperature of the Earth in terms of its distance from the Sun, and the Sun's radius and temperature.

Notice that the Earth is assumed to be a disk in deriving (5) but a sphere in (6). In equating (5), (6) we have effectively used the zero net-power argument mentioned above.

From Eq. (7), the temperature of the Earth is calculated to be about 280 degrees Kelvin. This figure is obtained by assuming an effective temperature of 5800 degrees Kelvin for the Sun's photosphere; a mean diameter of the Sun of 1.4 million Kilometres; and a mean distance of 150 million Kilometres of the Sun from the Earth.

Greenhouse Effect

Eq. (7) supports a greenhouse effect for the Earth, in the sense that the carbon dioxide envelope of the Earth is impermeable to radiation corresponding to its lower temperature $T_{2}$ but not to radiation from the Sun, appropriate to the higher temperature $T_{1}$. The support given by Eq. (7) does not necessarily mean that the conjecture of a greenhouse effect for the Earth is therefore proved.

Problems with Stefan's Law

There are major problems with a simple application of Stefan's law. For example, Stefan's law does not apply to filtered spectra and the integrated form (2) applies only for in vacuo radiation. Also, the power per unit area in Stefan's law refers to the power per unit perpendicular area. Simplified geometries, such as the one used above, do not take into account the orientation of emitting and absorbing surface elements.

Reference

[1] Gerhard Gerlich and Ralf D. Tscheuschner, Int. J. Mod. Phys. B, 23, 275 (2009).
DOI: 10.1142/S021797920904984X
arXiv eprint: arXiv:0707.1161v4 [physics.ao-ph]
Eprint can also be downloaded from here.

Footnotes

Stefan's constant

$5.7\times 10^{-8}$ Joule second^{-1} metre^{-2} degree-Kelvin^{-4}

Stefan's Radiation law

This is usually written in terms of radiation power per unit area. This may be achieved in Eq. (2) by moving the factor A to the left-hand side, as a denominator.

 

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