The Mystery of Equation 8

I’ve been looking at the Nikolov and Zeller paper again. Among other things, they claim to be able to calculate the surface temperature Ts of eight different planets and moons from knowing nothing more than the solar irradiation So and the surface pressure Ps for each heavenly body. Dr. Zeller refers to this as their MIRACLE equation. He says:

Why aren’t you all trying to disprove our MIRACLE equation rather than banging your heads against walls trying to prove or disprove who knows what and exclaiming you have problems with this or that? The question is how can we possibly have done it – there is no question that our equations work – if you haven’t verified that it works, why haven’t you? […] Why aren’t you thinking: “hmmmm, N&Z have given us an equation that lo-and-behold when we plug in the measured pressures and calculate Tgb as they suggest, gives us a calculated Ts that also matches measured values! You can’t disprove the equation? So maybe we are cooking the data books somehow, but how?

This is supposed to be evidence that their theory is correct, and people keep telling me ‘but they’ve got real evidence, they can make predictions of planetary temperatures, check it out”. Plus it’s hard to ignore an invitation like Dr. Zellers, so I checked it out.

Figure 1. These are not the equations you are looking for.

They first postulate something called the “Near-surface Atmospheric Thermal Enhancement” or “ATE” effect that makes the earth warmer than it would be without an atmosphere.

The “ATE effect” is measured by something called Nte(Ps), which is defined and estimated in their paper as follows.

where Nte(Ps) is a measure of the “Near-surface Atmospheric Thermal Enhancement” effect.

Nte(Ps) is defined as the actual average surface air temperature of the planet Ts divided by the theoretical “graybody” temperature of the planet Tgb calculated from the total solar insolation So of the planet. Nte(Ps) is estimated using a fitted function of the surface pressure of the planet Ps.

Let me simplify things a bit. Symbolically, the right part of equation (7) can be written as

Nte(Ps) = e^(t1 * Ps ^ t2 + t3 * Ps ^ t4)       (7Sym)

where “e” is the base of natural logs and Ps is the surface pressure on the planet or moon. There are four tunable parameters (t1 through t4) that are “fitted” or tuned to the data. In other words, those values are repeatedly adjusted and tuned until the desired fit is obtained. This fitting can be easily done in Excel using the “Solve…” menu item. As you’d expect with four parameters and only eight datapoints, the fit is quite good, and their estimate is quite close to the actual value of Nte(Ps).

Amusingly, the result of equation (7) is then used in another fitted (tuned) equation, number (8). This is:

where So is total solar irradiation.

This is their piece de resistance, their MIRACLE equation, wherein they are saying the surface temperature of eight different planets and moons can be calculated from just two variables— Pr, the surface pressure, and So, the total Solar irradiation. This is what amazes the folks in the crowd so much that they write and tell me there is “evidence” that N&Z are right.

Obviously, there is another tuned parameter in equation (8), so we can rewrite this one symbolically as:

Ts = t5 * (Solar + adjustment ) ^ 1/4 * Nte(Ps).        (8Sym)

Let me pause a minute and point something out about equation (8). The total solar irradiation Solar ranges from over 9,000 W/m2 for Mercury down to 1.51 W/m2 for Triton. Look at equation 8. How will adding the adjustment = 0.0001325 to any of those values before taking the fourth root make the slightest bit of difference in the result? That’s just bizarre, that is.They say they put it in so that the formula will be accurate when there is no solar, so it will give the background radiation of 3 Kelvins. Who cares? Truly, it changes Ts by a maximum of a thousandth of a degree for Triton. So for the moment let me remove it, as it makes no practical difference and it’s just confusing things.

Back to the tale. Removing the adjustment and substituting equation 7 into equation 8 we get:

Ts = t5 * Solar^0.25 * e^(t1 * Ps ^ t2 + t3 * Ps ^ t4) (eqn 9)

This is amazing. These guys are seriously claiming that with only eight datapoints and no less than five tunable parameters , they can calculate the surface temperature of the eight planets knowing only their surface pressure and solar irradiation. And with that many knobs to turn, I am sure they can do that. I did it on my own spreadsheet using their figures. I get about the same values for t1 through t5. But that proves nothing at all.

I mean … I can only stand in awe at the sheer effrontery of that claim. They are using only eight datapoints and five tunable parameters with a specially-designed ad-hoc equation with no physical basis. And they don’t think that’s odd in the slightest.

I will return to this question of the number of parameters in a bit, because even though it’s gobsmacking what they’ve done there, it’s not the best part of the story. Here’s the sting in the tale. We can also substitute equation (7)  into equation (8) in a slightly different way, using the middle term in equation 7. This yields:

Ts = t5 * Solar^0.25 * Ts / Tgb  (eqn 10)

This means that if we start out by knowing the surface temperature Ts on the right side of the equation, we can then calculate Ts on the left side … shocking, I know, who would have guessed. Let’s check the rest of the math in equation (10) to see why that works out.

Upon inspection it can be seen that the first part of the right side of equation (10),

t5 * Solar^0.25

is an alternate form of the familiar Stefan-Boltzmann equation relating temperature and radiation. The S-B equation can be written as

T = (Solar / c1) ^ 0.25.

where T is temperature and c1 is a constant equal to the S-B constant times the emissivity. We can rewrite this as

T = 1/(c1^0.25) * Solar^0.25

Setting another constant c2 equal to 1 / (c1^0.25) gives me the Stefan-Boltzmann equation as:

T = c2 * Solar^0.25

But this is exactly the form of the first part of the right side of equation 10. More to the point, it is an approximation of the graybody temperature of the planet Tgb.

We can check this by observing that if emissivity is .9 then constant c1 is 5.103E-8, and c2 is therefore about 66. However, that value will be reduced by the rotation of the planet. Per the N&Z formula in their latest post, that gives a value of about 27.

Their fitted value is 25, not far from the actual value. So curiously, what it turns out they’ve done is to estimate the Stefan-Boltzmann constant by a bizarre curve fitting method. And they did a decent job of that. Actually, pretty impressive considering the number of steps and parameters involved.

But since  t5  * Solar^0.25  is an estimation of the graybody temperature of the planet Tgb, that means that Equation 10 reduces from

Ts = t5 * Solar^0.25 * Ts / Tgb (eqn 10)


Ts = Tgb * Ts / Tgb.

and finally to

Ts = Ts



Let me recap the underlying effect of what they have done. They are looking at eight planets and moons.

1. They have used an equation

e^(t1 * Ps ^ t2 + t3 * Ps ^ t4)

with four free parameters to yield an estimate of Ts/Tgb based on surface pressure. As one would expect given the fact that there are half as many free parameters as there are data points, and that they are given free choice to pick any form for their equation without limit, this presents no problem at all, and can be done with virtually any dataset.

2. They have used an equation

t5 * Solar^0.25

with one free parameter in order to put together an estimate of Tgb based on total planetary insolation. Since Tgb does depend inter alia on planetary insolation, again this presents no problem.

3. They have multiplied the two estimates together. Since the result is an estimate of Tgb times an estimate of Ts/Tgb, of course this has the effect of cancelling out Tgb.

4. They note that what remains is Ts, and they declare a MIRACLE.

Look, guys … predicting Ts when you start out with Ts? Not all that hard, and with five free parameters and a choice of any equation no matter how non-physically based, that is no MIRACLE of any kind, just another case of rampant curve fitting …

Finally, there is a famous story in science about this kind of pseudo-scientific use of parameters and equations, told by Freeman Dyson:

We began by calculating meson–proton scattering, using a theory of the strong forces known as pseudoscalar meson theory. By the spring of 1953, after heroic efforts, we had plotted theoretical graphs of meson–proton scattering. We joyfully observed that our calculated numbers agreed pretty well with Fermi’s measured numbers. So I made an appointment to meet with Fermi and show him our results. Proudly, I rode the Greyhound bus from Ithaca to Chicago with a package of our theoretical graphs to show to Fermi.

When I arrived in Fermi’s office, I handed the graphs to Fermi, but he hardly glanced at them. He invited me to sit down, and asked me in a friendly way about the health of my wife and our newborn baby son, now fifty years old. Then he delivered his verdict in a quiet, even voice. “There are two ways of doing calculations in theoretical physics”, he said. “One way, and this is the way I prefer, is to have a clear physical picture of the process that you are calculating. The other way is to have a precise and self-consistent mathematical formalism. You have neither.

I was slightly stunned, but ventured to ask him why he did not consider the pseudoscalar meson theory to be a selfconsistent mathematical formalism. He replied, “Quantum electrodynamics is a good theory because the forces are weak, and when the formalism is ambiguous we have a clear physical picture to guide us. With the pseudoscalar meson theory there is no physical picture, and the forces are so strong that nothing converges. To reach your calculated results, you had to introduce arbitrary cut-off procedures that are not based either on solid physics or on solid mathematics.”

In desperation I asked Fermi whether he was not impressed by the agreement between our calculated numbers and his measured numbers. He replied, “How many arbitrary parameters did you use for your calculations?”

I thought for a moment about our cut-off procedures and said, “Four.”

He said, “I remember my friend Johnny von Neumann used to say, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” With that, the conversation was over. I thanked Fermi for his time and trouble, and sadly took the next bus back to Ithaca to tell the bad news to the students.

The Nikolov and Zeller equation contains five parameters and only eight data points. I rest my case that it is not a MIRACLE that they can make the elephant wiggle his trunk, but an expected and trivial result of their faulty procedures.

My regards to everyone,


PS—There is, of course, a technical term for what they have done, as there are no new mistakes under the sun. It is called “overfitting”. As Wikipedia says, “Overfitting generally occurs when a model is excessively complex, such as having  too many parameters relative to the number of observations.” Five parameters is far, far too many relative to eight observations, that is a guaranteed overfit.

PPS—One problem with N&Z’s MIRACLE equation is that they have not statistically tested it in any way.

One way to see if their fit is even remotely valid is to leave out some of the datapoints and fit it again. Of course with only eight datapoints to start with, this is problematic … but in any case if the fitted parameters come out radically different when you do that, this casts a lot of doubt on your fit. I encourage N&Z to do this and report back on their results. I’d do it, but they don’t believe me, so what’s the point?

Aother way to check their fit is to divide the dataset in half, do the fit on one half, and then check the results on the other half. This is because fitted equations like they are using are known to perform very poorly “out of sample”, that is to say on data not used to  fit the parameters. Given only eight data points and four parameters for equation 7, of course this is again problematic, since if you divide the set in half you end up with as many parameters as data points … you’d think that might be a clue that the procedure is sketchy but what do I know, I was born yesterday. In any case I encourage N&Z to perform that test as well. My results from that test say that their fit is meaningless, but perhaps their test results will be different.

[UPDATE] One of the commenters below said:

Willis – go ahead – fit an elephant. Please!

Seriously N&Z are only demonstrating in algebra what has been observed in experiments, that heating a gas in a sealed container increases both pressure and temperature.

OK, here’s my shot at emulating the surface temperature using nothing but the data in the N&Z chart of planetary body properties:

Figure 1. Willis’s emulation of the surface temperature of the planetary bodies.

My equation contains one more variable and two less parameters than the N&Z equation. Remember their equation was:

Ts = 25.3966 * Solar^0.25 * e^(0.233001 * Pressure ^ 0.0651203 + 0.0015393 * Pressure ^ 0.385232)

My equation, on the other hand, is:

Ts = 0.8 * Tgb + 6.9 * Density + 0.2 * Gravity)

Note that I am absolutely not making any claim that temperature is determined by density and gravity. I am merely showing that fitting a few points with a few variables and a few parameters is not all that difficult. It also shows that one can get the answer without using surface pressure at all. Finally, it shows that neither my emulation nor N&Z’s emulation of the planetary temperatures are worth a bucket of warm spit …

[UPDATE 2] I figured that since I was doing miracles with the N&Z miracle equation, I shouldn’t stop there. I should see if I could beat them at their own game, and make a simpler miracle. Once again, their equation:

Ts = 25.3966 * Solar^0.25 * e^(0.233001 * Pressure ^ 0.0651203 + 0.0015393 * Pressure ^ 0.385232)

My simplified version of their equation looks like this:

Ts = 25.394 * Solar^0.25 * e^(0.092 * Pressure ^ 0.17)

Curiously, my simplified version actually has a slightly lower RMS error than the N&Z version, so I did indeed beat them at their own game. My equation is not only simpler, it is more accurate. They’re free to use my simplified miracle equation, no royalties necessary. Here are the fits:

Figure 2. A simpler version of the N&Z equation 8

Again, I make no claim that this improves things. The mere fact that I can do it with two less tuned parameters (three instead of five) than N&Z used does not suddenly mean that it is not overfitted.

Both the simplified and the complex version of the N&Z equations are nothing but curve fitting. This is proven by the fact that we already have three simple and very different equations that hindcast the planetary temperatures. That’s the beauty of a fitted equation, if you are clever you can fit a lot using only a little … but THAT DOESN’T MEAN THAT PRESSURE DETERMINES TEMPERATURE.

For example, I can do the same thing without using pressure at all, but using density instead. Here’s that equation:

Ts = 25.491 * Solar^0.25 * e^(0.603 * Density ^ 0.201)

And here’s the results:

Figure 3. An emulation of the planetary temperatures, using density instead of pressure.

Does this now mean that the planetary temperature is really controlled by density? Of course not, this whole thing is an exercise in curve fitting.



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Actually, there is a “miracle” here. Willis, you have the string: “too many parameters relative to the number of observations” with the HTML markup for bold displaying, rather than the bolding being applied. How you do dat in WordPress?
Nice summary BTW.
[Thanks, fixed. -w.]
[Reply: Use these symbols: “& g t ;” if you want this to appear: >. (first take out the spaces between the symbols). Use “lt” for the “less than” symbol: < ~dbs, mod.]

I truly wish I paid more attention in school. But from what I’m reading here, I can safely say my conclusion is: “That’s going to leave a mark.”

I agree. They actually fit their multi-parameter formula to only six planet bodies; two are said to have too little atmosphere. But one of those fitted is Europa. As they found in Wikipedia, it has a notional pressure of 10^-7 Pa. About a tenth of a milligram oxygen per sq m, estimated by spectrum. That is the whole “atmosphere”. And this, they claim, enables them to predict the “climate” to 3 figure accuracy using their miracle formula. Despite the fact that pressure has barely one significant figure accuracy, and the measure of “climate” little better.


Nice. And thanks for telling the whole Dyson/Fermi story. Had a chance to hear Dyson speak, oh, about 20 years ago. Good guy.
I do believe there is something useful to take from this exercise. I haven’t had time to cook up the math, but I am sure there is an important atmospheric effect. If you think about it, an atmosphere must cool hotspots on the surface of a planet and warm the cool spots. It will effectively prevent rapid re-radiation of incoming energy, and will lead to the accumulation of thermal energy in the surface. Sufficient atmosphere will delay the reradiation of energy back to space. When energy is retained until the next day, the surface will be warmer, and it will continue to get warmer until the surface is warm enough to reradiate over the entire day/night cycle an amount of energy equivalent to what it takes in during the day.
Here’s a question in the spirit of your post: How many characters are displayed as text adding up all pages available in WUWT?

Philip Finck

You were very polite to N&Z.


Darn Willis, you moved to a new thread. Here’s something to help your problem with the calculus of equations 4 & 5 and your attempt to word it correctly to your Mathematica model program:
A spherical integration of the per-point average temperature field: = ¼π ∫[0, 2π] ∫[0,1] T.i dμ dφ
or even clearer: = ½ * ½π ∫[0, 2π] ∫[0,1] T.i dμ dφ
As for μ = cos(θ.i), this will only need to be evaluated at two locations, 0 & π/2; first is when the sun is directly above and the second at the 360° terminator that is 90° from the first. So the ultimate results of these two evaluations are cos(0)=1 & cos(π/2)=0. = ½ * ½π * ∫[0,2π] ∫[0,1] root4( S.0 (1-α.0) μ / (εσ) ) dμ dφ = ½ * ½π * root4(S.0(1-α.0)/(εσ)) ∫[0,2π] ∫[0,1] μ^(1/4) dμ dφ
First integrate ∫ [0,1] μ^(1/4) dμ:
= 4/5 * ( cos(0)^(5/4) – cos(π/2)^(5/4) )
= 4/5 * ( 1^(5/4) – 0^(5/4) )
= 4/5 * 1^(5/4)
= 4/5
Giving: = 4/5 * ½ * ½π * root4(S.0(1-α.0)/(εσ)) ∫[0,2π] 1 dφ = 4/5 * ½ * ½π * root4(S.0(1-α.0)/(εσ)) ∫[0,2π] dφ
Next integrate ∫[0,2π] 1 dφ
2π*1 – 0*1 = 2π * 4/5 * ½ * ½π root4(S.0(1-α.0)/(εσ)) * 1
Simplifying: = 8π/20π root4(S.0(1-α.0)/(εσ)) = 2/5 root4(S.0(1-α.0)/(εσ))
or = 2/5 (S.0(1-α.0)/(εσ))^0.25
That is the final closed form equation given by Drs. Nikolov and Zeller.

Seems to me a planet with an atmosphere is actually a black body. By that I mean from the point where the atmopshere ends and becomes space where all radiation from the sun entering it is absorbed.
Clouds and rocks and ice are way inside the internals of a planet with an atmosphere so effectively there is no albedo effect preventing all solar radiation from entering and “heating” the planet’s atmosphere.
The ideal gas law holds good in my “it seems to me” opinion. As for so called “Green house” gases do they not simply help to spread the heat out more evenly/quickly with their radiative properties, such as an increase in CO2 in the Northern Hemisphere making the Arctic a bit warmer and the lower Northern lattitudes a bit cooler but the average mean remains the same?

Willis Eschenbach

Philip Finck says:
January 23, 2012 at 4:26 pm

You were very polite to N&Z.

Hey, I’m a reformed cowboy, and I’m working on reforming some more. Gotta say it’s a sight harder than it looks. However, at the urging of both my friends and my scientific opponents, I’ve been cutting back. In that spirit, I redacted some of my more memorable phrases before publication. And despite the loss to the colorful nature of my general reportage, I suspect that in this case they are right.

Steve O

I also have a miracle equation.
Take your age, and add 12… divide that number by 2, and subtract half your age…


Nicely done. The equations are subjected to steady heat to drive off the vapor, revealing a well-boiled bootstrap.


The right answer for the wrong reason is ALWAYS the wrong answer. (Used to teach cal. based Physics)

I suggest that to expect any reasonable statistical significance in a curve fit, you should have at least five times as many degrees of freedom (data points) as you have factors (parameters). You will always get a perfect fit when the degrees of freedom matches the number of factors. Years ago I heard a person proudly present a paper that did just that.


Sounds good to me. I will await the coming comments.

F. Ross

Willis of Ockham?


When I saw that MIRACLE quote in the other thread, I was (almost) sure it was a joke on somebody’s part. If it was actually from the real Zeller, then I think he just disqualified himself.
The problems with the integration occur before the steps you describe – it is the variable substitution mu = cos(theta) which is dubious. Simply try comparing
the integral of mu^0.25 on [0,1]
with the integral of cos(theta)^0.25 on [0, pi/2]
and see what you get.

@ Willis
> There is, of course, a technical term for what they have done,
> as there are no new mistakes under the sun. It is called “overfitting”.
I think you’re looking at this in the wrong way. You say ‘overfitting’, which suggests they are somehow dishonestly trying to ‘cook’ a formula to fit 8 examples.
I don’t think N&K (Ned&Karl) are dishonest. In fact, I think they are merely learning the relationships between pressure induced and radiative warming by trying to fit the set of parameters to a regression equation.
“Learning is compression” in the sense that they want to find the smallest set of parameters which fit the data. I.e Occam’s Razor: if two regressions, one with 5 parameters and another with 5000 parameters, both fit the data, which is better? Ans: keep it as simple as possible (but not too simple).
You’re also missing the main point:
“Pressure by itself is not a source of energy! Instead, it enhances (amplifies) the energy supplied by an external source such as the Sun through density-dependent rates of molecular collision. This relative enhancement only manifests as an actual energy in the presence of external heating. “
Look at my response to Tallbloke and GeoSmith …
… particular the derivation of the Ideal Gas Law.
Note that the temperature T of a system in equilibrium can be computed from the just kinetic energy of the moving gas particles and their mutual collisions (density, implying pressure). We don’t need to know the radiative aspects of the system to compute the temperature! What part of the Ideal Gas Law do you not understand here?
So, having computed the temperature T we can then ask the question: where did the kinetic energy come from? Probably from solar heat energy absorbed by the surface.
But the point is we don’t need to know where the energy came from. Temperature T is soley dependent upon the internal kinetic energy of the gas and its density.
No change in pressure required. Yes, a pressure “gradient” necessarily exists on all planets with atmospheres, but that is accidental in the sense that even the gradient itself is not required to understand that at any point x,y,z the temperature is solely a function of kinetic energy and transfer of momentum by collisions.
N&K further make the claims that show no pressure change is needed
“In the case of an isobaric process, where pressure is constant and independent of temperature such as the one operating at the Earth surface, it is the physical force of atmospheric pressure that can only fully explain the observed near-surface thermal enhancement (NTE). “
Yes, if you choose a long-enough time scale:
“the near-surface atmospheric dynamics can safely be assumed to be governed (over non-geological time scales) by nearly isobaric processes on average, i.e. operating under constant pressure. This isobaric nature of tropospheric thermodynamics implies that the average atmospheric volume varies in a fixed proportion to changes in the mean surface air temperature following the Charles/Gay-Lussac Law, i.e. Ts/V = const. “
Willis, please think about it some more before summarily rejecting it as nonsense.


“Figure 1. These are not the equations you are looking for.”
That’s hilarious!


And that is why I keep coming here.


This whole thing reminds me of a problem I have seen in other areas of science. Basically the scenario goes like this:
1) First make a really poor estimate of some physical quantity
2) Compare your estimate with reality, and you find there is a large difference!
3) Claim that the difference is not just the error in your calculations, but is actually due to a real hitherto unsuspected effect.
4) Announce you have found something new and important.
(and before anyone adds the obvious comment – yes, some parts of ‘standard’ AGW theory seems to suffer from that condition as well)


“With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” — John von Neumann


Yawn. It is sad, Willis, that you have to go to such great lengths to point out the obvious.
So the gist is N&Z have demonstrated that a fifth degree polynomial will fit exactly and perfectly 6 points (or something like that)?


jimmi_the_dalek says: “When I saw that MIRACLE quote in the other thread, I was (almost) sure it was a joke on somebody’s part. If it was actually from the real Zeller, then I think he just disqualified himself.”
Yes, “MIRACLE” does not engender confidence, does it? In fact, if I’d seen that first, I wouldn’t have read any further. And I suspect the statistical tests will be similarly discouraging, given the presence of exponentials in the “curve fit.”


Ts = (Ts/Tgb) * Tgb
crazy. Was this “peer reviewed”?


Very entertaining. And truly not all that complex of an analysis such that even a lowly chemist could follow along.
I suspect the AGW model maths follow along these lines also. The ole self fulfilling prophecy.

Joel Shore

I definitely am in agreement with what you say here. It is bizarre that they actually claim their fit as some sort of miracle!
The one comment I would make is I think that what your last part before the conclusion shows is simply that “t5” probably wasn’t a free parameter but was determined by them through their equation for T_sb. At least, that is what I assume that they did…although, who knows, maybe they tried to fit it! I wouldn’t put it past them.


If memory serves, when I reached essentially the same conclusion a couple of weeks ago, I found that even the “fitted” curve was not as good as it initially appeared if you put it on a log-log plot. I haven’t re-run the plot, but my recollection is that a couple of the bodies fall of the graph then.


Please don’t post the von neumann quote. I didn’t see it in the article until a bit later.
[Done. -w]

Equation 8 is not (I think) a fitted equation. It’s their equation 2, using (for all planets) their modified earth albedo, and emissivity close to 1. So they only(!) have four parameters to find some functional form fitting their six points.

Warren in Minnesota

Determining the temperature of an atmosphere using the Ideal Gas Law as mentioned by John Day or using the Boltzman Constant with volume and pressure gives the approximate temperatures for Venus, the Earth, and Mars.


The only pressure relationship that would make any sense to me would be that due to the increased partial pressure of greenhouse or infrared absorbing gases in the atmosphere. It would seem reasonable that the planetary surfaces would tend to become warmer as the atmosphere became more dense due to increased net IR absorption overhead. The effect would likely be logarithmic, like increasing concentration, which also increases the net mass of greenhouse gas above.
This effect would probably be complicated by the special attractive properties of polar water molecules if they were present.

JamesD said @ January 23, 2012 at 5:31 pm

Ts = (Ts/Tgb) * Tgb
crazy. Was this “peer reviewed”?

Yup! By Willis Eschenbach who seems to be suggesting the paper needs some revision 😉


One query that is related to Wilis analysis is that I suspect that most of the planetary temperatures have been determined spectroscopically. Which means that you will always end up with a SB distribution when fitting a curve, ie it is a circular argument. I aksed Nic about the temperature methods for the planets he used over at Spencer’s but he may have missed it.


jimmi_the_dalek says:
January 23, 2012 at 5:15 pm
“really poor estimate of some physical quantity”
Your are speaking of all of NASA’s data, right jimmi?


For those planets that have molten cores, the amount of heat flux at the planet’s surface due to its interior depends on the ratio of the planet’s surface area to its total volume. Would that have a significant effect on the surface temp for some of the smaller planets?

I’m amazed that most of you (including Willis) think that fitting a smooth (“natural”) curve with a few parameters is ‘overfitting’.
FYI, ‘overfitting’ happens when your regression produces an “unnatural” curves (i.e. full of ‘wiggles’).
Here’s a good example of overfitting:
Now look at the N&Z curve again and note how natural it appears. No wiggles. I.e. it matches the underlying physical process. This is a “good” regression, not overfitting.


Something someone is sure of is wrong.


My applogies, I meant Ned not Nic in my earlier post.


“Your are speaking of all of NASA’s data, right jimmi?”
No, but I am thinking of some of their computer models….

Willis Eschenbach

Nick Stokes says:
January 23, 2012 at 5:38 pm

Equation 8 is not (I think) a fitted equation. It’s their equation 2, using (for all planets) their modified earth albedo, and emissivity close to 1. So they only(!) have four parameters to find some functional form fitting their six points.

Interesting thought. I just looked to see. If so, they are using an emissivity of about 0.95, so that’s possible. What argues against it is that I doubt they realized that’s what they were doing. Also, to pull out all of the variables and constants (albedo, emissivity, S-V constant) into one single factor … what would be the reason to do that?
In any case, as you say … down to only(?) four parameters.

Carl Brannen

Looking back at the original article, my favorite was the claim that higher temperatures would increase the pressure at sea-level. I’ve assigned my students the problem of computing the sea-level pressure from the mass of the earth’s atmosphere and the acceleration of gravity. Temperature has absolutely zero to do with it.

Willis Eschenbach

John Day says:
January 23, 2012 at 5:02 pm

@ Willis

There is, of course, a technical term for what they have done, as there are no new mistakes under the sun. It is called “overfitting”.

I think you’re looking at this in the wrong way. You say ‘overfitting’, which suggests they are somehow dishonestly trying to ‘cook’ a formula to fit 8 examples.
I don’t think N&K (Ned&Karl) are dishonest. In fact, I think they are merely learning the relationships between pressure induced and radiative warming by trying to fit the set of parameters to a regression equation.

You misapprehend the term “overfitting”. It is a mathematical error, not an accusation of dishonesty or deliberate deception. Generally the only people who do it are those who don’t realize they’re doing it.
Let me be very clear. I don’t think N&Z are dishonest, that’s not my reading at all. I think they actually believe what they are claiming.
I got conned one time by a guy named Bill, who was so good at it you could hardly stay angry at him. I told a friend of mine the problem with Bill was that he could weave together a pack of lies, half truths, invented facts, actual truths, and artful omissions so well, he could make anybody believe it.
My friend disagreed. He said that was not giving Bill his full due. He said the real problem was that Bill could tell a story so damn well, and with such feeling and majestic artistry, he ended up believing it himself …

Willis Eschenbach

Terry says:
January 23, 2012 at 6:21 pm

One query that is related to Wilis analysis is that I suspect that most of the planetary temperatures have been determined spectroscopically. Which means that you will always end up with a SB distribution when fitting a curve, ie it is a circular argument. I aksed Nic about the temperature methods for the planets he used over at Spencer’s but he may have missed it.

Interesting, I hadn’t thought of that. So in addition to the circular argument I pointed out, where they use Ts to find Ts, we have one other.


This looks a lot like reducton for the “The Jello Equation”
“We have been able to simulate these unique and complex Jell-O ® dynamics using spatial deformations [Barr, 1986] and other hairy mathematics. From previous research with rendering systems we have learned that a good dose of gratuitous partial differential equations is needed to meet the paper quota for impressive formulas.”
Source: Ray Tracing J ELL-O ® Brand Gelatin

Werner Brozek

“John Day says:
January 23, 2012 at 5:02 pm
Temperature T is soley dependent upon the internal kinetic energy of the gas and its density.”
Is density even needed? Granted, temperature is a macroscopic property and not a microscopic property, but above a certain minimum density, is the following not true?
1/2mv2 = 3/2kT
P.S. From the article: “Pr, the surface pressure,” I assume you meant Ps since that is what was used in the equations.


4 parameters + choice of functional form. After all they could have put exponentials, logs, or anything in their equation. This provides a lot of a additional curve fitting power and reduces the need for parameters. A school friend of mine had to fit a wind turbine velocity power equation which had 10 points. He did it with a single well chosen function and two parameters.
Its insufficient to just look at the parameters its also necessary to consider the complexity of the functional form because this is also a source of “information”.

> You misapprehend the term “overfitting”. It is a mathematical error …
No Willis, you are misapprehending this term. Re-read my post above:
I think the term “overconfident” (i.e. “low entropy”) better describes an estimator which produces results which seem to be “too good” because there aren’t enough valid examples to sample. (Example, “label bias” in maximum-entropy classifiers)
That may be the case here, only 8 examples. But can you (or anybody) provide enough convincing counter-examples to disprove the N&Z claim?

@Werner Brozek
> Is density even needed?
Yes, because we need to compute the forces that the gas molecules exert on each other when then collide. (Force = rate of change of “motion” [momentum in Latin]). If there are no collisions, then “no change of motion”=”no pressure”=”no temperature”.
Why haven’t you addressed my comments on the Ideal Gas Law, the crux of this N&Z theory?

Bill Illis

We do need some new theories however. We should just start with the actual observations first (of all bodies) and try to work what is really happening in all cases.
GHGs are not the end-all be-all answer. They do not work sufficiently in any case and certainly not in all cases. Much more is going on.
Energy is coming in, and is being released and is being stored up and emission is delayed and all the molecules in the whole system are participating in this over every single nano-second. Exotic issues like pressure and gravitational potential also have to be included because this affects how the energy comes about and how it flows in and out etc.
There may be a long-term balance between the solar energy coming in and its re-emission at some layer in some atmosphere, but there are time differences and different things are going on above and below that layer. Hansen says CO2 and GHGs run the whole system. Hardly.


I don’t have time to look into the equation(s) in question but I agree with John Day. I was trying to make the same point by saying that ideal gas law (or any other, more accurate equation of state for real gases) is just a relation between state variables (pressure, density, temperature). That’s all. You don’t need a thermodynamic system, well defined system boundaries, energy inputs/outputs over system boundaries and so on. It’s just about state variables at any point (x,y,z).
“But the point is we don’t need to know where the energy came from. Temperature T is soley dependent upon the internal kinetic energy of the gas and its density.”


Terry“One query that is related to Wilis analysis is that I suspect that most of the planetary temperatures have been determined spectroscopically. Which means that you will always end up with a SB distribution when fitting a curve, ie it is a circular argument.”
Actually, you look at the emission spectra for the planet in question, and select the _highest_ possible SB curve to identify the surface temperature (coming through windows in the atmospheric absorption). Thermal emission has a very characteristic curve, and barring Martian death rays, that upper limit represents the hottest thing emitting. Notchs in the spectra dropping below that surface temp value show where there’s absorption from GHG’s or clouds or whatever. The integral of the entire spectrum, on the other hand, should match the energy coming in (insolation), which provides something of a check point.