A Semigray Planet Temperature Model

Barton Paul Levenson

Introduction

A semigray model of planet surface temperatures divides the electromagnetic spectrum into two broad bands, shortwave and longwave. If you place the division at about 4 microns wavelength, 99% of sunlight is shortwave and 99% of infrared from planetary surfaces or atmospheres is longwave.

An early attempt at such a model was the Milne-Eddington approximation (Milne 1916, Eddington 1922):

Here T_S is a planet's mean global annual surface temperature. In the SI we normally measure temperature in kelvins (K). Earth's T_S would then be 288 K, rather than 15°C or 59°F.

T_e is the equilibrium temperature, the temperature the planet would have from sunlight and reflection alone (for Earth T_e = 255 K). And τ is the longwave optical depth.

Optical depth is how much a medium resists the passage of light. At τ = 1, incoming light is reduced by a factor of e (Euler's number, 2.71828...), so only about 37% gets through. At τ = 2, it's reduced by e², so we're down to 14%. And so on.

From energy budget considerations, we know Earth's longwave optical depth is about 1.84 (Levenson 2021). Plug T_e = 255 K and τ = 1.84 into equation 1, and we get T_S = 317 K.

Clearly this is too high. The equation is imprecise because it doesn't account for factors which cool the surface at the expense of the atmosphere: atmospheric absorption of sunlight, convection, conduction, and evapotranspiration. An improved semigray method would account for these processes.

Venus, Earth, Mars

Let's fit a semigray model to three planets. I'll use Venus, Earth, and Mars, because they all get their greenhouse effect mainly from water and carbon dioxide. Here are some data to use later:

References: Levenson 2011, 2021; Lodders and Fegley 1998; Marov and Grinspoon 1998, Williams 2020, 2021.

The figures for Earth are for "preindustrial" Earth, before we started burning all those fossil fuels. From these data, we can derive some more:

Here σ is the Stefan-Boltzmann constant, 5.67037 x 10^-8 W m^-2 K^-4. The other new terms are defined here, along with their meanings:

In terms of electromagnetic energy, the flux in watts is the total power transferred from one object to another, or across a surface. The flux density is the flux per unit area. Flux density matters a lot when considering planet temperatures.

Energy Balance

Now we need some more equations. We want as much power (energy per unit time) coming into a planet's surface as we have going out, to have thermal equilibrium. If more comes in than goes out, the surface will heat up until it vaporizes. If more goes out than comes in, it will cool down to absolute zero. Averaged over a full year, planet surfaces are in thermal equilibrium. Ignoring climate change, that is.

Our energy balance equation is:

where

• F_net is the net (infrared) energy radiated up by the planet's surface,
• F_solar is the energy the surface absorbs from sunlight,
• F_green is the energy it absorbs from the atmosphere (the greenhouse effect), and
• F_conv is the energy it loses to convection (here also covering conduction and evapotranspiration).

We want parameterizations for each of the three terms on the right--equations that don't necessarily represent physical laws, but approximate the answers we want. Then we add up our right-hand-side flux density terms (note that F_conv is negative; it's energy going away from the surface) to find F_net. F_net gives us the surface temperature through the Stefan-Boltzmann law ("electromagnetic energy emitted by an object follows the fourth power of temperature"):

F_solar

We know the amount of solar energy hitting the top of the atmosphere. It's just S divided by 4, where S is the Solar constant. This is because Earth receives solar energy on its cross sectional area (π R²), but its total area is that of a sphere (4 π R²).

But not all of that gets to the ground. Some is reflected back out to space. Some is absorbed by gases and clouds along the way.

Reflection can be tricky. For Venus it takes place almost wholly at the cloud tops, very high in the atmosphere. Even for Earth, most reflection is by clouds. But Mars has almost no clouds, so most reflection takes place at the surface.

We define a new quantity--A_cloud, the fraction of incoming sunlight reflected away by the atmosphere:

Actually, not all the atmosphere's reflection to space is done by clouds--some is from Rayleigh scattering by air molecules. But the way we defined A_cloud takes care of both.

We now define F₀, the fraction of incoming sunlight not reflected away:

We now have two known quantities: F₀ coming down past the clouds, and F_SI (SI = Surface Illumination) making it all the way to the surface after absorption. F_SI is found from energy budgets which in turn come from instrument measurements. Figures for the three planets are 16.8 W m^-2 for Venus (Marov and Grinspoon 1998, p. 298), 188 W m^-2 for Earth (Stephens et al. 2012), and 125 W m^-2 for Mars (Read et al. 2014).

F₀ and F_SI together let us define the shortwave optical depth of the atmosphere:

Given what we have so far, the figures for the three planets are:

Our equation for F_solar is based on Beer's Law ("under certain circumstances, transmission through a medium is a negative exponential function of optical depth"):

Fsolar = (1 - Asurf) F0 exp(-τSW)

10

But for a good semigray method, we can't just have everything handed to us. Let's try to predict what τ_SW would be ahead of time, from some other value we have in hand.

On Earth, about 12% of incoming sunlight is absorbed by water vapor, 6% by ozone, 1% by carbon dioxide, and perhaps 4% by clouds (Chou 1990). Let's make a gross, crude, and not very accurate oversimplification and say that 2/3 of all absorption of sunlight is by water vapor and 1/3 by carbon dioxide. We can then fit the known values of τ_SW on Venus, Earth, and Mars to their greenhouse gas partial pressures. We also bring in a term for overall pressure in atmospheres:

P = Ps / 101,325

11

Why do we care about an artificial quantity like this? Because the spectroscopic absorption lines of greenhouse gases are affected by pressure broadening. Greenhouse gases are much more effective in a high pressure atmosphere; much less effective in one with low pressure.

We assume the partial optical depths are proportionate to a power of the partial pressure, times a power of the total pressure. Some good fits are:

This gives us:

F_green

For the greenhouse effect, we're going to deal with contributions to the longwave optical depth from both water vapor and carbon dioxide. In reality, Earth's greenhouse effect is, on average, 50% due to water vapor, 25% to clouds, 20% to carbon dioxide, and 5% to minor gases such as methane, ozone, nitrous oxide, and chlorofluorocarbons (Lacis et al. 2010). Again, we're going to grossly oversimplify and make it 79% water vapor and 21% carbon dioxide (clouds are, after all, water, if not water vapor).

We know from energy budget considerations that the longwave optical depth is about 1.84 for Earth and 0.27 for Mars (Levenson 2021). We don't know the value for Venus, but it should be somewhere around 100-200.

Chopping the Earth figure up into 1.454 from water vapor and 0.386 from CO₂, we can try to fit a curves like those of equations 12-14 to the figures for our three planets:

This gives us:

Manipulating the Milne-Eddington approximation (equation 1) gives us an expression for the flux density absorbed by the surface due to the greenhouse effect:

Here, α is the surface absorptivity, equivalent to the emissivity ε by Kirchhoff's Law ("in local thermal equilibrium, absorptivity equals emissivity at any given wavelength"). F is the absorbed climate flux density from equation 3, and τ is the longwave optical depth we just computed.

F_conv

We do not know the magnitude of the convective flux density on Venus. The atmosphere is very hot and it has a troposphere with a lapse rate, so we know convection is going on. It must be pretty fierce--in the hundreds or even thousands of watts per square meter--but the precise value has never been measured.

We do know the values for Earth and Mars, however: 102 W m^-2 and 5.5 W m^-2, respectively (Stephens et al. 2012, Lorenz and McKay 2003). The latter study suggests that F_conv should be a function of Fsolar and τ. A direct fit gives:

Results

Tying it all together, we can finally apply our energy balance equation and find the surface temperatures (equations 5 and 6):

Our results are pretty amazingly close considering how many liberties we took with the parameterizations. Let's line up the model values with the observed ones:

The r² value between the columns is 0.999984, significant at well beyond the 99.9% confidence level.

Caveat

This is extremely important. Do not take this model too seriously. It came out so close because we carefully fit everything so it would. It ignores real physical processes which are important to the atmospheres of all these planets. On Venus, a good fraction of the greenhouse effect actually comes from the scattering of infrared light by the sulfuric acid clouds. On Earth, clouds are an important greenhouse agent. On Mars, there is some scattering by the atmosphere and by the thin clouds that occasionally appear, so the planetary albedo and the surface albedo are not identical in real life.

But for a quick and dirty way to estimate planet surface temperatures, semigray models like this can be a good way to go.

References

Chou, M.-D. 1990. Parameterizations for the absorption of solar radiation by O₂ and CO₂ with application to climate studies. J. Clim. 3, 209-217.

Eddington, A.S. 1926. The Internal Constitution of the Stars. Cambridge Univ. Press, 407 pp.

Lacis, A.A., Schmidt, G.A., Rind, D., Ruedy, R.A. 2010. Atmospheric CO₂: Principal control knob governing Earth's temperature. Science 330, 356-359.

Levenson, B.P. 2011. Planet temperatures with surface cooling parameterized. Adv. Space Res. 47, 2044-2048.

Levenson, B.P. 2021. Habitable zones with an Earth climate history model. Planet. Space Sci. 206, 105318.

Lodders, K., Fegley Jr., B. 1998. The Planetary Scientist's Companion. Oxford, UK: Oxford Univ. Press.

Lorenz, R., McKay, C. 2003. A simple expression for vertical convective fluxes in planetary atmospheres. Icarus 165, 407-413.

Marov, M.Ya., Grinspoon, D.H. 1998. The Planet Venus. New Haven, CT: Yale Univ. Press.

Milne, A.E. 1922. Radiative equilibrium: The insolation of an atmosphere. Mon. Notices Roy. Astron. Soc. 24, 872-896.

Read, P.L., Chelvaniththilan, S., Irwin, P.G.J., Ruan, T., Tabataba-Vakili, F., Valeanu, A., Wang, Y., Montabone, L., Lewis, S.R. 2014. Atmospheric radiative and mechanical energy budgets for Mars, from GCMs and reanalyses. http://www-mars.lmd.jussieu.fr/oxford2014/abstracts/read_oxford2014.pdf, accessed 9/28/2020.

Stephens, G.L., Li, J., Wild, M., Clayson, C.A., Loeb, N., Kato, S., L'Ecuyer, T., Stackhouse Jr., P.W., Lebsock, M., Andrews, T. 2012. An update on Earth's energy balance in light of the latest global observations. Nature Geosci. 5, 691-696.

Williams, D.R. 2020. Mars Fact Sheet. https://nssdc.gsfc.nasa. gov/planetary/factsheet/marsfact.html, accessed 12/02/2021.

Williams, D.R. 2021. Venus Fact Sheet. https://nssdc.gsfc. nasa.gov/planetary/factsheet/venusfact.html, accessed 12/02/2021.