Holmes on the Relationship Between TSI and Temperature

I was just made aware today of a paper published in 2019 by  Robert Ian Holmes on the Relationship Between TSI and Temperature at 1-Bar Pressure. The paper claims to be able to "predict" planetary temperatures at 1-bar pressure on the basis of TSI values of rocky planets and moons with a surface pressure of 1-bar or higher. The logic is that if you calculate the relative TSI between two planets (rTSI) you can multiply ∜rTSI by the 1-bar temperature of one planet to get the 1-bar temperature of the other. We can summarize his math as:

T1 = ∜rTSI*T2

There are three rocky planets and moons that have as surface pressure of 1-bar or higher. Here they are with Holmes' values for their 1-bar temperatures: Venus (340K), Earth (288K), and Titan (85-90K). Holmes shows his calculations below.

This superficially looks convincing, but astute observers likely have questions, like 1) Are the empirical values accurate? 2) Holmes only did half the possible calculations with this data. What happens if you do the other three? And 3) Why did Holmes ignore albedo? Let's look at each in turn.

Empirical Values and Full Calculations

I checked on the data for TSI, surface pressure, effective temperature, and surface temperature on NASA's factsheets for Venus and Earth. I found the relevant info for Titan here and here. All these values were similar to those used by Holmes. I then checked on the 1-bar temperatures for Venus and Titan and found a NASA factsheet that put Venus' 1-bar temperature at 360K and Titans at 86K. I found a quora page that suggested 348K for Venus. Both of these estimates are larger than Holmes' value of 340K for Venus. I decided to stick with NASA sources, so to check Holmes I will use Tv = 360K, Te = 288K, and Tt = 86K.

With these data, I calculated 1-bar temps using Holmes' method, except for each planet I calculated what the 1-bar temp would be using data from both of the other two.

Using 1-bar Temperatures from NASA

As you can clearly see, with correct data, calculating 1-bar temperature for each planet from the other two showed that the resulting temperatures did not agree with each other or with the measured 1-bar temperature. They disagree by anywhere from 7K to 47K. If I use the values that Holmes published in his paper, they look closer, but clearly not close enough. Here's the same chart using Holmes values of 340K for Venus and 87.5K (midway between 85-90K) for Titan.

Using Holmes' 1-bar Temperatures

There are two glaring reasons why this is the case - Holmes completely ignored both albedo and the greenhouse effect. By ignoring albedo, Holmes essentially pretended that sunlight that is reflected by planets warms the planet just as much as the sunlight that gets absorbed. This is clearly absurd. His calculations should have been based on absorbed solar radiation (ASR) and not on TSI. And of course, he also ignored the effect of GHGs in these atmospheric systems. As a consequence, Holmes' calculations failed to predict 1-bar temperatures accurately, and they also don't agree with each other. Calculating 1-bar temps for Te from Tv should give the same result as calculating 1-bar temp or Te from Tt. They don't agree because they don't account for the relevant physics. But by using Tv = 340K, Holmes was able to find three of the six possible calculations that were close enough for him, and so he used those three calculations while ignoring the other three.

Calculating Temperature from ASR

At the same time, we should acknowledge that it should be possible to calculate the effective temperature (Teff) for each of these planets from the ASR ratio (rASR) and Teff for another planet. Let's rewrite Holmes' equation as one that works. We can start with a simple energy balance equation and solve for Teff:

ASR = OLR
TSI*(1-α)/4 = σT^4
T = ∜(TSI*(1-α)/σ4)

For simplicity, we can write this as:

T = ∜(ASR/σ)

So we can construct the following for two planets, generically named Planets 1 and 2

T1/T2 = ∜ASR1 / ∜ASR2

For convenience, we can rewrite ∜ASR1 / ∜ASR2 as ∜rASR and solve for T1:

T1 = ∜rASR*T2

This equation should work provided we use the effective temperature from T2 to calculate the effective temperature for T1. So I did those calculations. Here's what I came up with.

rASR "Predicts" Teff Very Well

So it's clear that since we can calculate Teff if we know TSI and albedo, we can also calculate Teff for Venus if we know rASR and Teff for Earth or Teff for Titan. This equation works. But now that we see that using ASR is valid, what happens if we use NASA's 1-bar temperatures and then do the calculations correctly using rASR? Here are the results.

rASR Doesn't Predict 1-bar Temps because of the GHE

As you can see, this doesn't work at all. And the reason why this doesn't work is because the greenhouse effect is different on each of these planets and moons. So what can we conclude?
  • Calculating 1-bar temperatures from rTSI only gave the superficial appearance of working because Holmes carefully selected which calculations he showed: 
    • He could only show a calculation of Te from Tv if he used 340K for Tv, but NASA has 360K for Tv. If he showed a calculation of Te from Tt, he would be off by 17K
    • He could only show a calculation of Tv from Te by using 340K for Tv. If he showed a calculation of Tv from Tt, he would be off by 22K.
    • He could only show a calculation of Tt from Te, and that was off by 5.5K. If he had calculated Tt from Tv = 360K, he would have been off by 13K.
  • It's absolutely wrong to ignore albedo. Reflected sunlight does not significantly affect a planet's temperature. Holmes' paper essentially pretended that solar energy reflected away from a planet warms the planet just liked absorbed solar energy.
  • Calculating the effective temperatures from rASR works just fine, and Holmes could have done this for himself to show why it's necessary to include albedo and use rASR instead of rTSI.
  • When looking at the discrepancies that result from doing the correct calculations from rASR, the influence of GHGs can be clearly seen.
    • Predicting Te from Tv produces a Te That is 117K too warm because the GHE on Venus is so much stronger than Earth's.
    • Predicting Tv from Te produces a Tv that 104K too cold because the GHE on Earth is so much weaker than Venus'.
    • Predicting Tt from Te produces a Tt that is 10K too warm, but predicting Tt from Tv produces a Tt that is 49K too warm, since the GHE on Titan is closer to Earth than to Venus.
Holmes failed to demonstrate that the GHE is irrelevant to planetary temperatures. He also failed to demonstrate that you can predict 1-bar temperatures on other planets with rTSI. In fact, if we do the math correctly, this shows the impact that the GHE actually has. The differences between actual Te and  predicted Te from Tv and Tt show why you can't ignore the GHE. Without accounting for the GHE from greenhouse gases, you simply can't calculate what Holmes thinks you can calculate.


References:

[1] Robert Ian Holmes. (2019). On the Apparent Relationship Between Total Solar Irradiance and the Atmospheric Temperature at 1 Bar on Three Terrestrial-type Bodies. Earth Sciences, 8(6), 346-351. https://doi.org/10.11648/j.earth.20190806.15


Comments

  1. Phil ClarkeJuly 22, 2025 at 1:18 PM

    Nice. Do we know where he got his '∜rTSI' from? Does it have a physical derivation or did he pull it out of a bodily orifice?

    ReplyDelete
    Replies
    1. Scott SimmonsJuly 22, 2025 at 1:57 PM

      Ha! I calculated rTSI from the TSI values I grabbed from NASA to verify he did the math properly, so presumably he got his TSI values from NASA as well. But the rest may well have come from various bodily orifices. :-)

      Delete

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