Guest Post by Bob Wentworth

Ned Nikolov and Karl Zeller (N&Z) have written about their “discovery” of an interplanetary pressure-temperature relationship (e.g., in their 2017 paper). They offer a formula, developed via a curve-fitting process, which relates planetary Global Mean Annual near-surface Temperature (GMAT) to a function of solar irradiance and the average near-surface atmospheric pressure.

Their formula fits their version of the data quite well. They tried to fit their temperature data with formulas based on measures of greenhouse gases, and couldn’t achieve a convincing fit.

Based on a pressure formula fitting planetary temperatures, and greenhouse gas formulas not fitting, N&Z argue that they have discovered new physics, and that their formula establishes that:

“The ‘greenhouse effect’ is not a radiative phenomenon driven by the atmospheric infrared optical depth as presently believed, but a pressure-induced thermal enhancement analogous to adiabatic heating and independent of atmospheric composition”

There are plenty of reasons to be skeptical of this conclusion, at various levels: fundamental, procedural, and at the level of interpreting the significance of curve fitting.

At a fundamental level, there are simple thermodynamic arguments put forth by Willis Eschenbach in 2012 (and which I’ve spelled out in much more detail) which say that it’s impossible that the “pressure-induced thermal enhancement” hypothesis could be right.

At a procedural level, N&Z’s formula is built on top of their calculation of a planet’s “no atmosphere” temperature. For Earth, their calculated “no atmosphere” temperature is 90 K colder than the observed temperature. Yet, N&Z’s “no atmosphere” calculation doesn’t agree with anyone else’s (e.g., Smith 2008 or Spencer 2016). It appears the reason N&Z’s calculation (published under pseudonyms) doesn’t agree with others’ likely relates to an erroneous conclusion that the rotation rate of a planet doesn’t matter, and possibly also to their failure to take into account the heat storage capacity of the oceans. (Although oceans wouldn’t survive the absence of atmosphere, omitting their influence entirely and insisting that atmospheric effects alone must account for whatever effects oceans are responsible for seems logically suspect.) In any event, if N&Z’s “no atmosphere” temperature calculation is wrong, then that calls into question how meaningful any formula could be that purports to explain atmospheric warming relative to that wrong temperature.

However, today, I’d like to mainly focus on the significance (or lack of significance) of N&Z’s curve-fitting.

Essentially, N&Z argue that their formula fits the data so surprisingly and uniquely well that it must be more that a statistical coincidence—their formula must reflect real physics.

That argument hasn’t been convincing to me personally. After all, correlation doesn’t imply causation and there are plenty of examples of spurious correlations. But, I could argue all day long, at a philosophical level, about whether N&Z’s discovered correlation is spurious, and it likely wouldn’t convince anybody of anything.

If I think that N&Z’s formula only reveals a chance correlation, could I perhaps find another chance correlation? I set out to see if I could discover another formula that fits the data just as well as N&Z’s formula.

I succeeded. Once I had gotten to a point of being able to reproduce N&Z’s curve fitting, it took just a few hours of experimentation to discover another formula (or family of formulas) that fits N&Z’s data just as well.

The formula(s) I discovered depend on measures of the amount of greenhouse gases, not on total atmospheric pressure. The main difference from what N&Z tried is that I consider each greenhouse gas separately, rather than lumping them together as if their effects were indistinguishable.

Before I go into the details of the new formulas, let’s look at the results.

For each celestial body, the chart plots the ratio of the observed global mean annual temperature (GMAT) to N&Z’s calculated “no atmosphere” temperature for that body. For each body, I’ve plotted the actual observed temperature (using N&Z’s data), the predictions of N&Z’s pressure-based formula (which involved 4 tunable parameters, i.e., regression coefficients), and the predictions of three variants of my greenhouse gas formula. One variant, GH6, involves 6 tunable parameters, while the other two variants, GH4a and GH4b, each involve 4 tunable parameters.

It’s easier to assess the significance of the fits by looking at the residual errors (difference between observed and predicted values), as charted below.

I have normalized the residual errors relative to the uncertainty in the temperature data (as estimated by N&Z).

Note that I suspect N&Z have significantly underestimated some uncertainties. In their paper they repeatedly point out variability and lack of consensus in available data, and then proceed to offer a specific value for which they assign an uncertainty only modestly larger than the uncertainty associated with planets for which there is a strong consensus and vastly more data. So, I suggest taking the uncertainty values with a large “grain of salt.”

As the charts indicate, overall, all four models fit the data quite well. The NZ4 model is a little off for Titan, and the GH4a and GH4b models are a little off for Triton. The GH6 model matches the temperatures for all celestial bodies very well.

Note that the excellent fit of the GH6 model was not automatic, just because there were 6 tunable parameters. Closely related 6-parameter models, involving slightly different independent or dependent variables, were completely unable to fit the data and produced terrible fits.

I imagine that this experience, of slight variations in the model leading to terrible fits, is likely similar to the experience that led N&Z to believe that the fit of their model must be significant, must be due to more than chance.

Only, now we have two distinct formulas, depending on different variables, which offer comparably good fits to the data.

This strongly undermines N&Z’s argument that “such a good fit must mean it says something about the underlying physics.”

* * *

So, what’s this new formula of mine, and what motivated its form?

I wanted my formula to have at least some hint of a relationship to underlying physics. Different greenhouse gases absorb and re-radiate longwave radiation in different wavelength bands, with different strengths. Different gases are not the same, and it seems questionable to develop a model under the assumption that they are. So, I wanted to consider each primary greenhouse gas, CO₂, CH₄, and H₂O, separately.

Then there is the question of what metric to use to represent the amount of each gas. N&Z tried curve fitting using the total partial pressure or density of greenhouse gases near the surface. It made more sense to me to ask, “How much gas does longwave radiation need to pass through to make its way from the surface out to space?” So, the metric I use for the amount of gas x is the number of moles of gas x in a column of gas extending from the surface out to space, denoted Uₓ. (This is computed as Uₓ/Aᵣ = L⋅ρₓ/Mₓ, where Aᵣ=1 m² is a reference area, ρₓ and Mₓ are the near-surface density and molar mass of gas x, and L is the nominal scale height of the atmosphere, given by L = P/(g⋅ρ) where g is the surface gravity, P and ρ are the total atmospheric pressure and density at the surface, and g is the gravitational acceleration. Further details are available. All data was taken from N&Z.)

Another thing that we know about the underlying physics is that the radiative impact of greenhouse gases changes as their concentration increases. For a small amount of gas, we might expect the impact to vary linearly with the amount of gas. But, for higher concentrations, the impact of CO₂, for example, is said to be logarithmic in concentration. To reflect this, I assumed that the impact of gas x has the form aₓ⋅ln(1 + Uₓ/bₓ), where aₓ and bₓ are unknown parameters.

Altogether, the form I assumed for the ratio of overall temperature, T, to no-atmosphere temperature, Tₙₐ, is:

T/Tₙₐ = 1 + a꜀ₒ₂⋅ln(1 + U꜀ₒ₂/b꜀ₒ₂) + a꜀ₕ₄⋅ln(1 + U꜀ₕ₄/b꜀ₕ₄) + aₕ₂ₒ⋅ln(1 + Uₕ₂ₒ/bₕ₂ₒ)

(Note that I tried using (T/Tₙₐ)⁴ on the left, as might seem to make sense if we’re balancing energy flows. And I tried using greenhouse gas near-surface partial pressure or density. Each of these variations were terribly unsuccessful at fitting the data. Similarly, trying to introduce real albedo values also broke the fit.)

The models whose values were charted above corresponded to the following parameter values:

**GH6:**a꜀ₒ₂=2.47461964e-01, b꜀ₒ₂=3.46821712e+03, a꜀ₕ₄=2.52997123e-02, b꜀ₕ₄=1.49966410e-03, aₕ₂ₒ=1.81685678e-01, bₕ₂ₒ=7.97199109e+01**GH4a:**a꜀ₒ₂=2.47085039e-01, ꜀ₕ₄=1.16558785e-01, aₕ₂ₒ=1.99513528e+00, b꜀ₒ₂=b꜀ₕ₄=bₕ₂ₒ=3.42189402e+03**GH4b:**a꜀ₒ₂=a꜀ₕ₄=aₕ₂ₒ=2.47283033e-01, b꜀ₒ₂=3.44616690e+03, b꜀ₕ₄=3.36453603e+04, bₕ₂ₒ=1.67913332e+02

For model GH6, all six model parameters were allowed to vary independently. For model GH4a, all the bₓ parameters were assumed to be equal, and for model GH4b, all the aₓ parameters were assumed to be equal. Thus, model GH6 had 6 tunable parameters, but models GH4a and GH4b had only 4 parameters each. Thinking about the underlying physics, I would really want many parameters to describe each greenhouse gas. But, to prove my point, I wanted to show I could fit the data with as few parameters as N&Z had used.

* * *

What does all this mean?

Do I think my formula represents the “real physics” of atmospheric warming of planets?

No, not at all.

My formula, like N&Z’s formula, neglects albedo, which we know must have an effect on planetary temperature.

Both formulas assume the atmosphere accounts for the temperature difference between N&Z’s “no atmosphere” formula and what is observed. Yet, I’ve argued that it is almost certainly wrong to attribute that full temperature difference to atmospheric effects, when some of the effect is due to planetary rotation rate and the heat capacity of oceans.

Also, the temperatures of the celestial bodies involved vary from 39 K to 737 K. That means the wavelengths of thermal radiation on each body will be quite different, and will interact with different absorption bands of greenhouse gases. Without accounting for the impacts of absorption bands at different temperatures, it seems implausible that we could be accurately accounting for the real physics.

So, in terms of corresponding to underlying physics, I expect my formula is basically nonsense. But, it has at least as much correspondence to the underlying physics as is the case for N&Z’s formula based on atmospheric pressure. (I’ve omitted some relevant physics. N&Z omit relevant physics and, in addition, had to hypothesize new physics to justify their model. That hypothetical “new physics” is easily falsified.)

* * *

Let’s look at the logic of N&Z’s argument one more time.

They used curve fitting to find a model that “predicts” planetary temperature. The only model they were able to find that fit the data well depended on total atmospheric pressure, without regard to the presence or absence of greenhouse gases.

Because of the uniquely good fit of their empirical model, they argued that their model must correspond to actual physics.

Yet, based on the work I’ve presented here, we now know that the fit of N&Z’s model is *not* uniquely good. A model that relies only on amounts of greenhouse gases, without regard to total atmospheric pressure, fits the same data just as well.

N&Z’s pressure-causes-temperature model has no justification in terms of known physics (and is falsified by known physics).

If the pressure-causes-temperature model also does not offer a uniquely good empirical fit to the data, why should we believe that it signifies more than a chance, spurious correlation?

We shouldn’t.