Quantifying the Relative GHE for Various Planets

I decided to have some fun with a thought exercise to respond to claims I've been seeing about a paper published a few years ago by Robert Holmes. What I'm going to do below doesn't technically "predict" temperatures on other planets but it does show a simple model that explains why surface temperatures differ on planets in moons given both ASR and the GHE. In a previous post I debunked a silly paper from Holmes that claimed to be able to calculate the 1-bar temperatures of a planet knowing only the ratio of the TSI values for the two planets and the 1-bar T for the second planet. Holmes' used the following equation.

T1 = ∜rTSI*T2

I showed that this equation doesn't work because it ignores both the GHE and albedo. It gives the superficial appearance of working if you calculate 1-bar T of Earth from Venus and vice versa, since Venus has both a strong albedo and GHE. But even then it only "works" if you use 340 K for Venus' 1-bar T, and NASA currently reports the 1-bar T for Venus to be 360K. And this formula doesn't work for any of the possible calculations involving Titan at all, so it has at best a 67% failure rate (if you accept 340 K for Venus). One thing I thought was interesting, though, is that if you use ASR instead of TSI, the results were still wrong, but they were wrong in such a way that indicated the relative strength of the GHE on these planets. So I decided to play around to see if I quantify a factor that would show the strength of the GHE using surface temperatures of Venus, Titan, and Earth. In my last post I derived the following formula and showed that it actually works for predicting effective temperature (Teff) values (which is a rather trivial point).

T1 = ∜rASR*T2

This works for Teff, but if we use surface temperature (Ts), we will get the wrong answer because the strength of the GHE is different in each planet. For instance (see chart below for calculations),

1. Te = ∜rASR*Tv = 830 K, which is 2.88x the actual Te of 288 K.
2. Te = ∜rASR*Tt = 283 K, which is 0.98x the actual Te of 288 K.
3. Tv = ∜rASR*Te = 256 K, which is 0.35x the actual Tv of 737 K.
4. Tv = ∜rASR*Tt = 251 K, which is 0.34x the actual Tv of 737 K.
5. Tt = ∜rASR*Tv = 276 K, which is 2.93x the actual Tt of 94 K.
6. Tt = ∜rASR*Te = 96 K, which is 1.02x the actual Tt of 94 K.

The factors in bold above are not accidental. We can actually show exactly why they have these values.  These factors exist because on these planets not all the energy emitted from the surface (Fs) escapes to space, which is what is called a "greenhouse effect." We can quantify this as ε = ASR/Fs. So the factors above are actually the fourth root of ratio of the ε = ASR/Fs for the two planets. This is pretty easy to show. If we use Ts values, then we can use the following simple model:

ASR = εFs

Where ε is the fraction of the surface flux (Fs) that escapes to space. Here are the ε = ASR/Fs values for the three planets we're considering:

1. Earth: 240.2 Wm^-2 / 390.1 Wm^-2 = 0.62
2. Venus: 150 Wm^-2 / 16730 Wm^-2 = 0.009
3. Titan: 2.9 Wm^-2 / 4.4 Wm^-2 = 0.66

This means that on Earth, 62% of the energy emitted from the surface escapes to space, but on Venus only 0.9% of the energy emitted from the surface escapes to space, so the GHE is much stronger on Venus than it is on Earth. It turns out this is an excellent way to quantify the GHE on these planets. But we can do more here, since ASR = TSI*(1-α)/4 and Fs = σTs^4:

TSI*(1-α)/4 = εσTs^4

∜ε*Ts = ∜(ASR/σ)

We can now construct the following for two planets, generically named Planets 1 and 2:

(∜ε1*Ts1) / (∜ε2*Ts2) = ∜ASR1 / ∜ASR2

For simplicity, let's use ∜rASR for [∜ASR1 / ∜ASR2]  and ∜rε for [∜ε1/∜ε2]. We can now write:

∜rε*Ts1 = ∜rASR*Ts2

The ∜rε term is the factor by which all the above calculations are wrong. It's just the quarter root of the ratio of ε for two planets. I'm calling this rGHE = ∜rε for the purposes of this post. So that gives us this:

Ts1 = ∜rASR*Ts2/rGHE

Here the rGHE term is essentially a correction factor for what was "predicted" by Ts1 = ∜rASR*Ts2 in my last post. Using rGHE as a correction factor for the "predicted" Ts values above, we get the correct Ts value for each of the planets.

The rGHE term is mathematically identical to the ratio of the "predicted" Ts to actual Ts. But let's not make too much of this. These ε values are calculated using Fs = σTs^4, so this is not an argument that we can predict Ts on other planets using this method. You can't calculate Ts on any planet with rASR because you have to know rGHE, and that requires knowledge of Ts for both planets. What the above chart does show, though, is that we can quantify the relative strength of the GHE between two planets in a manner that consistently works, given our simple model. 

While this approach is certainly not revolutionary and admittedly based on a very simple model, it does not require any curve fitting with free parameters (like the Nikolov and Zeller model) or using a cherry-picked selection of calculations, that when combined with temperature values that are likely off by a bit, give you the superficial impression of a working model (Like Robert Holmes). This is just strait up calculations from a simple energy balance model that only works if you recognize that these planets and moons have a greenhouse effect in which more energy is radiated from the surface than escapes to space. And you can check out my calculations for yourself with other planets. This should work on any rocky planet or moon for which we have accurate data for Ts, TSI, and albedo.