BPSDB
Following from Dan Pangburn: Dan commented on this, helpfully providing links to several pdfs showing where he derived his constant.
So I took a look at them.
Dan uses the First Law of Thermodynamics.
That’s a start: Energy(in) – Energy(out) = Energy(retained).
Let’s take a look at Energy(Out). A single term: X·T^{4}, where X is Dan’s constant.
Dan helpfully provides a link to Wikipedia’s A very simple model. This shows
(1a)S = 4εσT^{4}
where
 S is the solar constant – the incoming solar radiation per unit area—about 1367 W·m^{−2}
 a is the Earth’s average albedo, measured to be 0.3
 σ is the StefanBoltzmann constant—approximately 5.67×10^{−8} J·K^{−4}·m^{−2}·s^{−1}
 ε is the effective emissivity of earth, about 0.612
Dividing both sides by (1a)S, we get 1 = Y·T^{4} where Y = 4εσ/(1a)S.
Plugging in the terms into Y, we get
Y = (4 x 0.612 x 5.67×10−8)/(0.7 x 1367) K^{−4}
= 1.45·10^{10} K^{−4}
(or 1·10^{10} K^{−4} to 1 s.f. – we can’t justify more than one significant figure)
Y can’t really be a constant, though, since 1 = Y·T^{4}. If T increases then Y must decrease (and vice versa). Perhaps we should rewrite it as Y_{i}·T_{i}^{4} = 1. But for small ΔT Y will not change by much.
So far, so good.
Dan derives his constant in the same way, but then multiplies an additional term (the average sunspot count).
Quoting from his pdf on page 6:
The average sunspot number since 1700 is about 50, the energy radiated from the planet is about 342*0.7 = 239.4 (for the units used) and the earth’s effective emissivity is about 0.61 (http://en.wikipedia.org/wiki/Global_climate_model). Thus, as a place to start, X should be about 50/239.4 times the StephanBoltzmann constant times 0.61.
50/239.4*5.67E8 *0.61 = 7.2E9
Which then he “refines”, continuing:
With this plugged into the equation, a plausible graph is produced with a dramatic change observed to take place in about 1940. In EXCEL, 7.2E9 was placed in a cell and the cell (value for X) called by the equation which produced a graph. The graph was observed as the value for X was varied. X was adjusted until the net energy from 1700 to about 1940 exhibited a fairly level trend. This occurs when X is 6.519E9 (unbeknownst to me at the time, cell formatting rounded it to 6.52E9).If an average sunspot number of 6.52/7.2*50 = 45.28 had been used, no adjustment would have been needed.
This is, of course, nonsense.
But we will follow this for now to see where it goes.
If we now multiply Y by Dan’s sunspot average count, we get
45.28Y = 45.28 x 1.45·10^{10} K^{−4}
= 6.56·10^{9} K^{−4}.
This is pretty close to Dan’s value (the difference is probably due to slightly different values of the terms S & ε, which I had used from the model).
To all intents and purposes X = 45.28Y.
Now go back to Dan’s term X·T^{4}, the output energy.
Replacing X with 45.28Y, and remembering that Y·T^{4} = 1 (so that Y = T^{4}), we get
X·T^{4} = 45.28·T^{4}·T^{4}.
Gosh! The Temperature terms cancel, the StefanBoltzmann equation vanishes, and the energy we are left with is ….
45.28 Sunspots ….
Solar & Heliospheric Observatory
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:
How many brain cells did you have to kill to wade your way all the way through that?
My brain thanks you for your sacrifice.
—
I think you may be due for some kind of award.
It never occurred to me that anyone would actually read those things – let alone carefully enough to pinpoint the exact point where it all falls apart.
I like Dan’s post. It is nice to see someone who is working the problem on a physics level. My new work that somewhat mirrors his work is at nationalforestlawblog.com This month under Sun Hotz license plate was provided at the 3rd international Hurricane Conference at Rhodes Greece.
I think Dan has hit on something that should not be discarded.
Nice piece. I used a more simple math since most of my public is local business men.
Thank you for the opportunity to comment from time to time.
Paul Pierett
Pierett said:
Nice to see Pierett at last admitting that he is absolutely ignorant of science Only a fool, or a dyed in the wool AGW denier, would actually think (I know that is impossible for these types) that what Pangburn says is in any way related to real science and maths.
That’s hilarious! I glanced at Pierett’s comment and figured it was heuristically generated spam.
Following the forestry blog link, I found a post touting the fact that he comments at Huffington Post and Discover Magazine. I guess that makes me a “regular contributor to Scientific American.”
BUT THE R SQUARED!!!! CAN’T SOMEONE THINK OF THE R SQUARED!!!!!
:)
But even there Pangburn is full of fail. As I said in that other thread, R² is useful for correlating estimates of two different variables, not estimates of the same variable (temperature).
Of course, Pangburn will ignore all that, since in his own mind he’s authoring his very own Principia.
And a side note: Pierett says:
Sounds like there’s the expectation that “local businessmen” are mathematical idiots.
(But given the 2008 financial crisis, maybe that expectation isn’t too far off…)
– frank
The equation that you ended up with is pretty close. The difference that you introduced is intrinsically setting the yearly temperature always equal to the temperature that was used in Wikipedia. Since average global temperatures are actually fairly constant, a plot of the summation part of your equation should look a lot like that on page 14 of the pdf made public 4/11/10 at http://climaterealists.com/index.php?tid=145&linkbox=true (I checked, it does). I had considered doing it this way but decided to avoid the fixed temperature issue.
Another way to do it (less neuron damage) and get closer to the right answer is to write two equations in two unknowns and solve them simultaneously.
Average sunspot number = 45.28 = X*Ti^4
Energy radiated from the planet = 239.4 = 0.612 * SB * Ti^4
Then 239.4 = 0.612* SB *45.28/X
X = 45.28/239.4 * 0.612 * SB = 6.56E9
I thought the graph looked a bit better using 6.519E9.
Frank, R^2 is a measure of how close the calculated values are to the measured ones. It is explained on page 1 of the pdf made public 5/24/10.
Dan:
I stand corrected on that point. But you do know this, don’t you:
So your LOOK AT THE HIGH R SQUARED!!!! talking point still doesn’t hold.
– frank
Frank,
Yes.
How many regressors do you count (with d set to zero)? How would you describe the accuracy of the model? What other models are there to compare to?
Excuse me for piping in again, but the above is left dangling. . .
Could someone please offer a short bullet point summation of the mistakes in Dan’s formula and reasoning – I appreciate that a number of flaws have been highlighted here and there, but a list would be nice.
thank you
Peter M.
~ ~ ~ ~ ~ ~ ~
Also, if someone could shed light on how a Google of the title to Dan’s article gets wallpapered over seven pages of results, that would be interesting too.
I don’t mind admitting that you lost me somewhere in the middle of the first equation.
I do think that I understand the final point: energy isn’t measured in sunspots. If I have got the right end of the stick, somewhere inside I’m waiting for an allusion to babel fish, and the final and clinching proof of the nonexistence of god.
Thanks for what seems like grat blog :)
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