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)
to
Ts = Tgb * Ts / Tgb.
and finally to
Ts = Ts
TA-DA!
CONCLUSION
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,
w.
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.
w.
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Atmospheric compression creates heat due to that compression, by increasing the KE of that gas through the work done on it.. Jupiter, with its atmosphere of hydrogen and helium, radiates more heat than it receives. since there are no GHG’s in the Jovian atmosphere there must be another mechanism to create that extra heat and that must be gravity.
Gravity starts suns so why not also provide extra heat to a simple shallow atmosphere a few tens of Km deep.
They are using only eight datapoints and five tunable parameters with a specially-designed ad-hoc equation with no physical basis.
Presumably why the elevator speech is still “in the post” 🙂
W. I am surprised that you are surprised when you found the Boltzmann constant could be derived from eq 7.
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.
As they are working with gray body temperatures which follow the Boltzmann concept, it should not be a surprise.
It is a bit like being surprised that ohms law I = V/R is related to P = V^2/R etc etc and that substitutions can be made…
When dealing with natural laws, many relationships can be found between parameters of laws which bolster the acceptance of laws.
A proposed law that can not fit with all other known laws would have extreme difficulty in being accepted.
The concept of splitting your dataset in two to test your theory is fine when your dataset is large, but with 8 data points it is unrealistic.
They only have 8 data points.
tallbloke says:
January 24, 2012 at 1:33 am
Thanks, tallbloke. Here on this thread I’m focused on what they did in equations 7 and 8. There are many aspects to their work. I oppose the idea that it can all be settled in one thread. At present there are four threads. The original one. The answer by Ira Glickstein. The response to comments. And this thread. This thread is about equations 7 and 8. You have plenty of room to discuss the “evidence that their theory works” and the “similarity of their non linear regression” and the “clausius curve” on any one of those three threads. I have asked, and will continue to ask, that you stick to the topic here.
“Inserted the tautology”? I have shown that the underlying structure of equation 8 is Tgb * Ts / Tgb. I have not “inserted” anything.
You are correct. That’s how it is. However, the universe having given you very few data points doesn’t mean that you are justified in drawing conclusions from that tiny sample as though you had sufficient data.
Equation 7 contains parameters t1 thru t4, four parameters.
I discussed this above and acknowledged the possibility that it was not a parameter … including the fact that still leaves four parameters and eight data points. The oddity that confused me is that to get 29.3966 from your formula there you need to have an emissivity of 0.955, which seems strange. But no matter, four parameters is plenty to be too many.
That’s some excellent detective work, considering that in the head post I said:
But yes, you are correct.
Dang … that actually sounds about right in some strange way. Me and the sharks have been friends, well, not friends exactly but we have had a kind of wary mutual underwater respect for decades now. I’d sign up to be a shark jumper, sounds like fun.
As regards my charmed life, Napoleon is rumored to have said that he didn’t care if his generals were good, as long as they were lucky. And my aikido teacher said the best aikido is, when the fight is starting on third street, you’re walking down seventh street. I can only wish that everyone have that kind of good fortune.
My best to you,
w.
Jimmy_the_Dalek says:
……
Wayne,
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.
….
You are quite right that the 2 integal above give different results. But you have overlooked a small detail, that results in an incorrect conclusion in this context . The correct comparision would be
to compare
the integral of mu^0.25 on [0,1]
to
the integral of sin(theta)*cos(theta)^0.25 on [0, pi/2]
and if you do that you get the same result in both cases.
Why ? Well it goes like this. Think of theta and phi as the latitude and longitude of the Earth ( and assume Earth to be a sphere rather than a an ellipsoid ) you can then identify the position of every point on the surface with those two numbers.( elementary geography ok ?).
Now you may or may not know it but the vertical distance between two positions that have the same longitude, and f.x. a 1 degree diffrence in latitude is the same regardless of which longitude you are on , but when you have two points with the same latitude and a 1 degree difference in longitude the horisontal distances are different for different latitudes, on the equator the length of the 1 degree latitude arc is aproximately the same as length of a 1 degree longitude arc, but when yo go from there towards either one of the poles th horisontal degrees get smaller and end in a zero on the pole while length of the longitude degree is still the same as on the equator. And to calculate the length of 1 degree difference paralell with the equator at latitude theta you have to use sin(theta).
And the exact same thing happens when you use integation to calculate an area on the surface of the Earth and with positions determined by spherical coordinates, the sinus of the latitude sneaks uninvited into the equation as a scaling factor of for the paralells.
So there is nothing dubious about the mu = cos(theta) substituion , and its effect can be seen from the following
the derivative of mu w.r to theta is d(mu)/d(theta) = -sin(theta)
and from that we get d(theta)=(-1/sin(theta))d(mu)
insert mu for cos(theta),and the left hand in the equ. above for d(theta) into
the integral of [sin(theta)*cos(theta)^0.25 d(theta)] on [0, pi/2] as well
as adjust the interval limit to reflect the new variable mu to [cos(0),Cos(pi/2)]
the result becomes
The integal of sin(theta)*(mu^0.25)*(-1/sin(theta))d(mu) on [1,0]
the sin(theta) factor now cancels out and we are left with
The integral of (-1)*(mu^0.25) on [1,0]
switch the upper an lower integation limits to [0,1]
and change the sign on the integrand to reflect that switch
our final result now has become:
the integral of mu^0.25 on [0,1]
Q.E.D.
P:S. Jimmy , take a quick peek at the wikipedia entry for Spherical Coordinates if I have not managed to convince you that there is nothing sinister about tha<t substtution.
tallbloke says:
January 24, 2012 at 1:33 am
Upon further contemplation, I realized that this statement couldn’t be true, since the albedos are different for each planet. As a result, if it is a constant for all planets it is a tuned parameter.
w.
“that still leaves four parameters and eight data points”
It’s only six data points. If you look at the table in their original post, they didn’t try to fit the Moon or Mercury.
I get the feeling that there are a number who can see Willis’ limitations who are no longer coming here to post. There has been a lot of shouting recently and what I did in that environment was to opt out and go back to study, carefully, the original material.
I studied N&Z carefully only because the shouting drove me to do so… but presently I felt I had discovered a goldmine, a game-changer. The remarks of many others indicates that I am not the only person to have had that “aha!” experience. But from then on, much of my energies have been diverted into consolidating this, so that one can answer doubts just once, not tire oneself out with repeating oneself again and again to more and more individuals. So you won’t see me much. And it’s not just N&Z talking about pressure-induced atmospheric temperature, there is a tradition even older than that of Arrhenius and Callendar, that has recently produced a whole spate of work, practical experiments and data fitting theoretical maths and physics. Graeff, following Loschmidt. Now not just Jellbring and Gilbert but also Sorokhtin. And a growing number of climate skeptics, many of whom have diverted currently to Tallbloke’s threads.
The pressure figures of Huffman for Venus fit very precisely, they will not go away. These figures, Willis, are in addition to the single planetary fits used by N/Z. Then there are the experimental figures of Graeff that fit his theoretical calculations. Then there are those of Miskolczi, where data fit theory so precisely that it could not be accidental, again M overturns the ghg stuff. And I now start to wonder if the much-criticised Gerlich & Tscheuschner were on to something similar, and if they too will be vindicated.
It’s not easy coping with a whole paradigm shift. I am still reeling myself. There are still bits of the new paradigm I do not understand. And bits of the maths I still have to come to terms with – where N&Z might still be wrong. However, I prefer to stay with the work where the data clearly has excellent fit to the theory, than where there is more emotion and less data. Right now I’m working on developing a protected wiki environment that will be able to answer each major doubt issue for all comers. To me, this will help the science.
All the best Willis, I still think your earlier work on tropical thunderstorms and Darwin was superb.
Willis Eschenbach says:
January 24, 2012 at 3:21 am
tallbloke says:
January 24, 2012 at 1:33 am
Also, the constant 29.3966 in Eq. 8 is not a ‘tuned parameter’, but a result of combining 4 constants from the gray-body temperature in Eq. 2, i.e.
(2/5)*[(1 – 0.12)/(ϵ*σ)]^0.25 = 29.3966
Upon further contemplation, I realized that this statement couldn’t be true, since the albedos are different for each planet. As a result, if it is a constant for all planets it is a tuned parameter.
It’s the albedo for rocky planets without an atmosphere. Assumed to be the same for all the bodies tested. So, Moon: measured albedo 0.12 Earth with no atmosphere, about the same, etc. So, not tuned; measured from the Moon, and applied elsewhere.
Read Nikolov and Zellers work and their contributions to the threads on my website for details would be my advice to anyone who wants to address what they actually say. You’re not banned from reading my website Willis, just from trying to gishgallop their (or anyone elses) theory into the dust.
Willis,
Consider the following.
E=Mc^2
Therefore, c=sqrt(E/M)
therefore E=M * (sort(E/M))^2
therefore E=E
You can do it with any equation, and this is all you have done.
On the other hand, the fitting is definitely overdone, especially as 3 of the planets have P=0.
Also, your original objection has merit IMO – as they have omitted all properties of the atmosphere from their formulae, they have done the equivalent of postulating a totally transparent atmosphere with no chemical or optical properties – that cannot be right.
And whoever suggested that it is worth tracking down where that planetary data came from is correct – some of it must be spectroscopic in origin so the possibility of inadvertent circular argument is there.
Willis sez:
Eq. 7: Ts/Tgb = NTE(P)
Equation 7 contains parameters t1 thru t4, four parameters.
It doesn’t have to contain them because the parameters operating together in the form specified by N&Z are equivalent to (NTE)P and Ts/Tgb.
That’s why there’s two equals signs in there. One of them.between Ts/Tgb and NTE(P) and another one of those funny little parrallel line thingies (=) btween NTE(P) and the four parameters.
Sheesh.
Willis, I’m withholding detailed comment on the correctness of your ingenious analysis of equation (8), as it has distracted everybody from the equation (2) ‘elephant’ which is already glaring at them from within the original N & Z paper.
If equation (2) is correct, as I believe it is, and you do not yet understand it, then your above analysis (even if it is correct) is of secondary significance, compared to the paradigm shift which their equation (2) implies for GHE theory.
I have not seen a coherent criticism (by anyone!) of equation (2). Their new integration model is used to produce their fundamentally different “mean planetary gray body temperature” (Tgb) evaluation, which certainly features in (but eventually disappears from) your analysis.
Since Ned and Karl published Figure 1 in their “Reply to Comments” paper, I have found no error in the conception of that integration model or the resulting equation (2). So, show me where their equation (2) is incorrect (or where the simple geometry of its integration limits is in error) and I’ll reconsider whether your above analysis is worth analyzing in detail. The fact that your analysis effectively removes Tgb from the theory, as though it were irrelevant, is deeply troubling.
Could everybody have got the mean gray body temperature of an “airless, rotating, sun-lit sphere” so wrong in the past? I believe they could have, and await identification of any error in Ned & Karl’s equation (2) or in their brilliant new Tgb model.
Jorgekafkazar defines the ideal gas theory
——–
I am happy with your definition by I will respond with a derivation here: http://quantumfreak.com/derivation-of-pvnrt-the-equation-of-ideal-gas/
To summarize: the derivation is all about individual molecules, their kinetic energies and how that determines the impulse on the walls of the container, and hence the pressure. There is no need to consider collisions between molecules at all. It is not relevant.
You do need to consider molecular size at higher gas densities. Hence the old fashioned van der waals equation of state.
Willis
I think that the point you raise in this thread is valid.
However, that does not mean that N&Z are wrong but rather that the equation in question (and the ancillary fitting) does not necessarily carry the significance to which N&Z would have attribute to it.
When English is not the first language of an author, it is necessary to give wide latitude to the language being used. I appreciate that this can often be difficult since precision in language is often of utmost importance. It is unfortunate that they used the expression ‘miracle’. Clearly the choice of that expression to a native English speaker is wrong but this should not in itself be used as a vehicle of harsh criticism or judgment when dealing with someone whose first language is not English
“Why aren’t you all trying to disprove our MIRACLE equation”
That’s easy enough, really, there are no miracle equations.
Willis, your rebuttal almost makes me weep with joy. Well done! I hadn’t gotten around to N&Z as I’ve been distracted by Jelbring (see post coming soon that should finish that discussion once and for all) but I too was bothered by an empirical fit with no derivation or sound theoretical or physical basis (handwaving doesn’t count, especially not when the handwaving is based on sketchy ideas like “gravitational heating” in a steady-state planet.
Your quote of the Fermi-Dyson story reminds me of the good old days when I used to go to a lot of physics workshops where people presented the results of computations in condensed matter or nuclear physics based on diagrammatic perturbation theory. I don’t know if you are familiar with these fields, but they frequently involve trying to approximate a non-convergent or weakly convergent series with a few terms (usually terms selected because they come from the subset of all the possible terms that actually could be summed over, e.g. in some cases “ladder diagrams”, ignoring all the ones that they couldn’t or wouldn’t compute). I got so cynical while listening to talk after talk of this sort that I formulated the following:
The Fundamental Axiom of Diagrammatic Perturbation Theory
All the diagrammatic terms that I did not include in this computation do not significantly contribute.
I heard this axiom stated, in so many words, time after time after time, never with the slightest actual justification. Indeed, any justification offered was always a posteriori — they had a target result, or perhaps a few target results, that they wanted to explain. They had a computer code that would allow them to vary the degree of a non-convergent and/or incalculable series out to some order in a computable subset of all of the terms. They would then — intentionally or not — run the code a few dozen times with different numbers of included terms and — Surprise! Wow, look at that, if we include the first four regular terms and sum over all of the ladders, we get within 10% of the right numbers!
Of course, if one kept the first five or six terms, and summed over ladders and all of the other permutations of diagram types to systematically higher order — the kind of thing one would like to be able to do before claiming that a perturbative series result is actually valid — the answer would get worse. And the particular set of diagrams kept varied from computation to computation, presentation to presentation. Clearly another case of fitting the elephant, junk science skillfully hidden, although hey, theorists gotta eat.
I also spent my share of time trying to fit nonlinear functions that did have something of a theoretical basis in order to extract critical exponents, and that’s a damn difficult game. Nonlinear function fits with multiple, weakly covariant terms, often have many local optima, and one can sometimes get two or three distinct solutions all of which get worse if you perturb the solution parameters a bit locally (they are all gradient search optima) and which may be very nearly equivalently as good. This is well-known as the rough landscape problem — many optimization functions (and function fits are always optimization problems, although in e.g. linear regression there is frequently a unique best solution because of a smooth landscape) have parametric optimization surfaces that look like mountainous terrain — in N dimensions.
In that regard, their solution has a very suspicious aspect to it. It is the exponential of a sum of powers of the pressure. I have no idea how to enter equations into this interface yet (if you do know, or if Anthony knows — perhaps it would be a good idea to post a toplevel “Howto” document describing how, along with how to include figures and so on — can one embed figures in straight html?) so I’ll have to go with latex-style ascii equations, but:
N_TE = exp(t1*P^t2 + t3*P^t4) = exp(t1*P^t2) * exp(t3*P^t4)
Now, there is one other thing that Fermi was famous for — dimensional analysis and “Fermi estimates”. N_TE is a dimensionless number (ratio of two temperatures). The exponential function has a power series expansion (and hence must be dimensionless in physics. Its argument must be dimensionless in physics. This means that:
t1*P^t2
must be dimensionless! Well if it is, t1 must be some reference pressure, taken to the same power as the term it multiplies:
t1 = 1/P_1^t2 -> P_1^0.065 = 1/0.233 = 4.29
or:
P_1 = 5.407 x 10^9 = 5.4 x 10^4 atmospheres
(where I’m ignoring the difference between atmospheres and bar — really this is bar). Wow, the temperature on the surface of mercury depends on a pressure of 54 thousand atmospheres! I wonder where that number comes from?
For the uninitiated, the pressure in the ocean goes up by roughly one atmosphere for every ten meters of depth. 5.4 x 10^4 atm is thus the pressure at 5.4 x 10^5 meters, or 540 kilometers of depth in water! Alas, the Earth’s oceans are only a few tens of kilometers deep at their deepest points. That sort of pressure doesn’t exist on any of the planets in the list being fit. — the pressure on the surface of Venus is a mere 93 or so bar. You might find it — pretty far down — on one of the gas giants — or somewhere down inside the Sun.
This scale is, of course, absurd. There is no way that the temperature on tiny, nearly airless planets could meaningfully depend on a pressure of 54 kbar. So we have the first term:
exp(t1*P^t2) = exp((P/P_1)^t2) = exp( (P/54000)^0.065 )
where P 1/0.00154 = P_3^0.385
or
P_3 = 2.019 x 10^7 = 202 bar
Hmm, once again, I’m having a difficult time seeing where 200 bar could be a relevant pressure in any description of planetary climatology. It is over twice the size of the pressure at the base of the atmosphere on Venus, and yet is supposed to apply to a list of planetary bodies only two of which have atmospheres as high as 1 bar. Even Mars has almost negligible air pressure at the surface at 6 to 10 millibar.
Let’s write out N_TE properly then, shall we, in manifestly dimensionless form:
N_TE = exp( (P/54000)^0.065 ) * exp( (P/202)^0.385 )
Willis already dealt with the leading term in the T_s expansion — it is self-fullfiling prophecy. Let’s see what N_TE is for Earth:
N_TE(Earth) \approx exp(0.49) * exp(0.13) = 1.54 * 1.14 = 1.75
A nice number, order unity. What about the other planets? Well, for Venus this is clearly going to be a much larger number (but still not huge). For Mars it will be a much smaller number. The whole point of the big exponents is that it causes N_TE \approx 1 as soon as P << 1 bar. In fact, for small P — where even P = 1 bar is "small" in this game — we can power-series expand the exponential and observe that this is a linear fit to all of the planets but Earth and Venus, plus a quadratic and probably a cubic term that reach from Mars to Venus. The power series in P (again dimensionless) is certainly no less meaningless, and I’d predict that an ordinary 4th or 5th order power series expansion in P would work as well as this strange product of exponentials of powers of ratios of P and two utterly non-physical reference potentials.
I defy anyone to find any physics that would lead to the particular reference pressures 202 bar and 54 Kbar in the atmospheric science of Mercury, the Moon, Europa, Triton, Titan, or Mars. I double-dog-dare anyone to look for the mysterious meaning of 54 Kbar pressures in predicting the climatology of the Earth or Venus. Go on, show me how 1 g of gravity and atmospheric pressure that ranges from 1 bar to 0 can, for an ideal gas, make 54 Kbar an important, nay, critical pressure.
I’ll tell you exactly what this fit is. It is a double fit. One piece is negligible for very low pressures, where the other fits all the nearly airless planets on the list. That term, however, becomes comparable to the other right around “Earth” (1 bar) and fits only the range part between Mars and Venus! There isn’t the slightest bit of physics in either one.
rgb
Arrg. The interface (or my mouse) ate a half paragraph out of my previous reply. Somewhere between where I compute the first term and the second it should have said something like “Now let’s do the same thing for the second term, and evaluate its reference pressure.” plus a half line of algebra that seems to have disappeared but follows exactly the same solution approach as the first term.
Sorry,
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Anna v says
Please read a bit on how thermodynamical equations dovetail beautifully to statistical mechanics equations where the scatterings
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Thanks Anna. The ideal gas definition there matched mine pretty well, but I dont want to get hung up on definitions when it is clear what the derivation of the ideal gas law from first principles requires as it’s basic assumptions.
And I also misspoke about a “linear” fit for the small planets. I meant a fit with only one term such as N_TE = 1 + (P/P_0)^\alpha… for some entirely meaningless P_0 and \alpha. One might manage it with a linear fit indeed, or linear with a weak quadratic correction, though. I don’t care. Numerology isn’t physics, especially when the nonlinear curve fits turn out to depend on utterly nonphysical parameters.
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John Marshall says
Gravity starts suns so why not also provide extra heat to a simple shallow atmosphere a few tens of Km deep.
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The difference is that the initial heating of the sun or earth by gravity is only important during the sun or earth’s accretion phase.
That phase is largely over, though it is believed that gradual sinking of heavier material may be contributing a small amount to the internal energy budget of the sun, earth and Jupiter etc.
Nowadays the energy budget determining the surface temperature is determined largely by solar insolation.
Willis, you created the tautology yourself as others have pointed out. Should make you wonder about the rest of your analysis.
On the other issue I just happened to read a bridge column last night where the only way to defeat the contract was leading away from an ace. Normally considered a stupid play but in this case the bidding made it pretty clear that something unusual was needed.
Yes, using multiple parameters to fit an equation is normally wrong, but that does not make it wrong all the time. You are making the same mistake a poor bridge player would make in assuming certain general concepts are always true.
You need to understand what they have done more closely. I pointed out in another thread that you need to consider figures 5 and 6 and note what the curve fitting produces. All and all I think you are over-reacting in this case. I can understand it somewhat given the silly comments by N&Z attacking non-PHDs. It doesn’t help your case here which is pretty much waving your arms.
Look, it’s pretty clear that there are major problems in N&Z’s hypothesis. But, attacking what may turn out to be a reasonable correlation based on an “In general” set of logic leaves a bad taste in my mouth and is not up to your normal standards.
jimmi_the_dalek says:
January 24, 2012 at 3:42 am
On the other hand, the fitting is definitely overdone, especially as 3 of the planets have P=0.
Also, your original objection has merit IMO – as they have omitted all properties of the atmosphere from their formulae, they have done the equivalent of postulating a totally transparent atmosphere with no chemical or optical properties – that cannot be right.
Hi Jimmi,
One of the surprising things N&Z have discovered is that the particular composition doesn’t matter. The planets still fit the curve omitting the properties. Pressure clearly dominates the setting of surface temperature (along with insolation). However it is worth noting that even moons in the outer solar system have precipitable gases like methane, and these have radiative characteristics. So the question of transparent atmosphere’s with all the attendant radiative balance issues (incorrectly) raised in the Jelbring thread doesn’t arise.
Your elephant fitting skills are really impressive except that you kinda miss the point.
The N&Z claim is:
Ts/Tgs = f(Pn)
Now if you take values of Ts/Tgs and Pn and put them to the graph, you get a set of points which is interpolate-able by a smooth function. Yes, the function they used is rather arbitrary and four parameters is a lot but someone smarter may figure out a way how to use even less parameters to just fit these points. Just looking at the graph also shows that there are actually multiple subsets of our eight examples which are pretty likely to generate very similar fits: take one of (Moon, Mercury, Europa), one of (Triton, Mars), one of (Earth, Titan) and Venus and you’ll get pretty similar fit with each such combination.
That does neither prove nor disprove whether there is any physical reality behind that fit. My personal opinion is, what plays important role in real surface temperature compared to gray body temperature is thermal capacity of the atmosphere and wind speeds and directions – but if these two appear to be correlated to atmospheric pressure in a favorable way, it is possible this arbitrary fit is real.
Substituting (7) to (8) is nothing more than evaluation of how realistic the fit is, as you basically take the fit and get back to source data from it. The only thing you prove by reducing it to Ts = Ts is that you omitted all of the fit’s imperfections on the way.
So the sun still matters
1. Ts=Ts.
Of course it does. You’ve just proven by your own hand that their equations are properly balanced. If you could resolve them to Ts=1.5Ts theat would a be a problem.
2. E=IR and P=I^2*R. Using the precise same method that you have, I can resolve these to show that E=E, I=I, R=R and P=P. If I couldn’t, there would be a problem. That I can shows that the equations are properly balanced just as you’ve done by resolving Ts=Ts.
3. If SB Law did NOT show up as being integral to their equations, then there would be a problem. A major portion of their premise regards the proper application of SB Law, and they’ve produced equations that do precisely that, properly apply SB Law. That you can discover SB Law within their equations is no surprise. If you couldn’t, THAT would be a surprise.
4. I failed to understand your vitriol and agressiveness toward N&Z from day one. It hasn’t been objective, and your failed mathematical analysis in this thread is a prime example. Plenty of people have noticed it, and Lucy Skywalker’s admonishment upthread ought to give you pause. That said, perhaps the reason for your antagonism is contained in this thread where you once again draw attention to your own “thermostat hypothesis”. I’d ask you this Willis:
Is your cherished “thermostat hypothesis” so prescious to you that you would tear down the work of others to protect your own?