Derivation of Planetary Global Temperature Formula

Note: This post is highly mathematical. See the list of Quantities and Symbols[LINK] for further information on the symbols and notation used.

It is straightforward to compute the average planetary surface temperature by considering the balance of energy arriving and leaving at the top of the atmosphere (TOA). This is given by:

(1) $\begin{equation*} \TXH = \Sn - \OLR \end{equation*}$

where $\Sn$ is the net absorbed solar irradiance at TOA, $\OLR$ is the outgoing longwave (LW) radiation flux, and $\txh$ is the TOA excess heating flux, which is defined to the the imbalance between power arriving and leaving at TOA.

(Note: on a planet where there is a significant ongoing heating from a source other than the Sun―for example, a geothermal heat flux much larger than the 0.09 W/m² which Earth’s core provides―then that heat flux would also need to be included in the term $\Sn$ .)

Averaging this over the the surface of the planet (using $\ex{X}$ to denote the global average of a quantity $X$ ) leads to:

(2) $\begin{equation*} \ex{\TXH} = \ex{\Sn} - \ex{\OLR} \end{equation*}$

We can characterize the atmosphere as having a longwave effective absorption, $\LEA$ and a longwave effective transparency, $LTR$ , defined by:

(3) $\begin{equation*} \LTR = (1 - \LEA) = \frac{\OLR}{\SLR} \end{equation*}$

where $\SLR$ is the radiant flux of LW thermal radiation emitted by the surface. For Earth, the LW effective absorption is about $\LEA \approx$ 0.4, and the LW effective transparency is about $\LTR \approx$ 0.6. Note that in an atmosphere without any longwave-absorbing or scattering materials, the LW effective absorption. $\LEA$ , would be zero.

These definitions allow us to write the average outgoing LW radiation flux as:

(4) $\begin{equation*} \ex{\OLR} = \ex{\LTR \cdot \SLR} \end{equation*}$

Next, define the average of $X$ weighted relative to $W$ to be:

(5) $\begin{equation*} \exw{X}{W} = \frac{\ex{X \cdot W}}{\ex{W}} \end{equation*}$

This allows me to define the average of $\LTR$ , which I’ll denote $\avg{\LTR}$ , as a weighted average:

(6) $\begin{equation*} \avg{\LTR} = \exw{\LTR}{\SLR} = \frac{\ex{\LTR \cdot \SLR}}{\ex{\SLR}} \end{equation*}$

Then we can write $\ex{\OLR}$ as:

(7) $\begin{equation*} \ex{\OLR} = \avg{\LTR} \ex{\SLR} \end{equation*}$

However, I previously showed[LINK] that:

(8) $\begin{equation*} \ex{\SLR} = \sigma \avg{\emis} \ex{\TVEB} \ex{\tsurf}^4 \end{equation*}$

where $\avg{\emis} = \exw{\emis}{\tsurf^4}$ is the weighted average emissivity and $\ex{\TVEB}$ is the temperature variation emissivity boost factor, $\ex{\TVEB} \geq 1$ , and is given by:

(9) $\begin{equation*} \TVEB = \frac{\tsurf^4}{\ex{\tsurf}^4} \end{equation*}$

Substituting this expression for $\ex{\SLR}$ into our equation for $\ex{\OLR}$ yields:

(10) $\begin{equation*} \ex{\OLR} = {\avg{\LTR} \avg{\emis} \ex{\TVEB} \sigma} \ex{\tsurf}^4 \end{equation*}$

Bring this into the TOA energy balance equation and rearranging to solve for temperature, we find:

(11) $\begin{equation*} \ex{\tsurf} = \fourthrootr{\frac{\ex{\Sn} - \ex{\TXH}}{\avg{\LTR} \avg{\emis} \ex{\TVEB} \sigma}} \end{equation*}$

We can write this slightly differently if we define the LW baseline temperature boost factor, $\lbtb$ , and the temperature variation baseline temperature reduction factor, $\tvtr$ , as follows:

(12) $\begin{equation*} \LBTB = \frac{1}{\fourthrootr{\avg{\LTR}}} \end{equation*}$

(13) $\begin{equation*} \TVTR = \frac{1}{\fourthrootr{\ex{\TVEB}}} \end{equation*}$

I also define a temperature $T_{ip}$ as:

(14) $\begin{equation*} T_{ip} = \fourthrootr{ \frac{\ex{\Sn} - \ex{\TXH}}{ \avg{\emis} \sigma} } \end{equation*}$

This “idealized planet” temperature, $T_{ip}$ , is the temperature the planet would have if it was uniform in temperature and all surface longwave thermal emissions were able to reach space.

This allows our planetary temperature formula to be expressed as:

(15) $\begin{equation*} \ex{\tsurf} = \LBTB \cdot \TVTR \cdot T_{ip} \end{equation*}$

The above formula for $\ex{\tsurf}$ offers a simple but rigorous expression of the factors that determine the average surface temperature of a planet.

The average surface temperature is related to the idealized-planet temperature by two modifiers.

One modifier, the temperature variation temperature reduction factor, $\TVTR$ , reduces the average temperature as a consequence of the way that the temperature varies around the average. Temperature variations between the over the globe (and to a lesser extent over time) increase the value of $\TVTR$ . Upward variations in temperature boost emissions more than downward variations in temperature reduce emissions. This creates a net boost in emissions which means that a lower average temperature is sufficient for emissions to balance the rate at which the planet receives energy. If the surface temperature was uniform over the globe and over time, then $\TVTR$ would have its lowest possible value, $\TVTR=1$ .

The other modifier, the LW temperature boost factor, $\LBTB$ , increases the average temperature as a consequence of the atmosphere reducing the flux of longwave radiation that is emitted to space relative to what is emitted by the surface. For Earth, the LW temperature boost factor is about $\LBTB \approx$ 1.14, corresponding to a temperature boost of around 35℃/62℉. For an atmosphere transparent to longwave radiation, with no absorption, $\LBTB$ would be 1.

Note that in deriving this formula for temperature, there have been no approximations whatsoever.

The planetary global temperature formula we’ve just developed is exactly equivalent to the Stefan-Boltzmann law, expressed in terms of averaged quantities, and in terms of the emissions that leave Earth at TOA.

Note that if the global average of TOA excess heating, $\ex{\TXH}$ , is positive, then that means that there is a net energy imbalance, and the planet is accumulating thermal energy. Consequently, the temperature will tend to rise, increasing thermal emissions to space, until eventually $\ex{\TXH}=0$ , at which point equilibrium is reached and the temperature stabilizes. Typically, a planet is likely to be near equilibrium, in which case the the excess heating flux is small.

In equilibrium, we can take the TOA energy imbalance $\ex{\TXH}$ to be zero, in which case our the “ideal planet temperature”, $T_{ip}$ , simplifies to:

(16) $\begin{equation*} T_{ip} = \fourthrootr{ \frac{\ex{\Sn}}{ \avg{\emis} \sigma} } \end{equation*}$

To reiterate, this result is an exact consequence of the Stefan-Boltzmann law. There are no approximations involved. And only in this last equation was any assumption at all introduced―the assumption that is that the system has achieved equilibrium and there is no net energy imbalance at TOA. To avoid that assumption, using the preceding formula for temperature.

Some might be puzzled that there has been no mention of latent heat transport or convection, which are the primary methods by which Earth’s surface is cooled. Nor has there been any mention of atmospheric circulation, ocean circulation, thunderstorms, clouds, or any of the other complex phenomena that transport or redirect heat and create climate.

All of those factors are accounted for, implicitly but fully. Those factors influence, albeit possibly within certain limits, the empirical values that show up in the temperature formula: absorbed sunlight, temperature variations, and the effective longwave absorptance of the atmosphere.

Despite complexities like those being addressed only implicitly, the core laws of physics say that the formula for average planetary surface temperature is rigorously