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## Dan’s constant 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·T4, where X is Dan’s constant.

Dan helpfully provides a link to Wikipedia’s A very simple model. This shows

(1-a)S = 4εσT4

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 Stefan-Boltzmann 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 (1-a)S, we get 1 = Y·T4 where Y = 4εσ/(1-a)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·T4. If T increases then Y must decrease (and vice versa). Perhaps we should rewrite it as Yi·Ti4 = 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 Stephan-Boltzmann constant times 0.61.

50/239.4*5.67E-8 *0.61 = 7.2E-9

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, 7.2E-9 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 6.519E-9 (unbeknownst to me at the time, cell formatting rounded it to 6.52E-9).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·T4, the output energy.

Replacing X with 45.28Y, and remembering that Y·T4 = 1 (so that Y = T-4), we get

X·T4 = 45.28·T-4·T4.

Gosh! The Temperature terms cancel, the Stefan-Boltzmann equation vanishes, and the energy we are left with is ….

45.28 Sunspots ….

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## Dan Pangburn

Dan Pengburn has been commented several times, so I think that he deserves a post of his own. 🙂 anom(Y) = calculated temperature anomaly in year Y
N(i) = average daily Brussels International sunspot number in year i
Y = number of years that have passed since 1700 (or any other year where the net summation is approximately zero such as 1856, 1902, 1910, 1938, or 1943)
T(i) = agt (average global temperature) of year i in °K,
ESST(c,Y) = ESST (Effective Sea Surface Temperature) in year Y calculated using an ESST range (magnitude) of c
CO2(Y) = ppmv CO2 in year Y
CO2start = ppmv CO2 in 1880

Dang, his equation is just too big to fit the image.
However we could simplify his equation and tidy it up a bit.

In the first summation, N(i) will always be positive, since N(i) >= 0 (you cannot have negative sunspots).

It is also dimensionless. For dimensional analysis Dan has to change this to degrees, but doesn’t say how it does.

In the second term, 6.52×10-9T4 is quoted.
Dan did not state how this was derived, or cited. Is it based on science, or has Dan just made it up?

It is vaguely reminiscent of the Stefan Boltzmann law, but he is certainly not using the Stefan–Boltzmann constant (which is more than 10 times larger than Dans). Besides which the the Stefan Boltzmann equation is in units of Watts per square metre. We need to derive the anomaly in degrees Kelvin, so somehow Dan needs to define his constant.

Using Dan’s value at 288K, term 2 results in ~ 45. So (according to Dan) we should be warming whenever the “average daily Brussels International sunspot number” is more than 45 then we should result in warming (e.g. 2000), and a lower number should mean cooling (e.g. 2007).

Check any of your favourite datasets and see if how many years agree with Dan’s figures. I don’t know how many that you might find in agreement with Dan’s, since I gave up looking after seeing 2000 and 2007.

But there are more terms – perhaps we need to look at them to see if this makes more sense further along.

(Or maybe not. :))

ESST(c,Y) = ESST (Effective Sea Surface Temperature) in year Y calculated using an ESST range (magnitude) of c

I haven’t got a clue what this means, since Dan does not say how he derives it. But I don’t think it means the Sea Surface Temperature. Perhaps he meant the anomaly?

The last term does makes sense – that CO2 will warm logarithmically.

Then we have Dan has four “coefficients”, a, b, c and d.

Usually are coefficients are constants without dimensions (e.g. π). I know that some engineering terms use coefficients with units, but if they are they quoted in units. Since Dan doesn’t explain the units for his terms the equation , there is a real problem with a, b & d when looking at dimensional analysis (I can’t comment on c since I don’t understand the term). Possibly he meant that a and d were in units of K and b was in K-1 it might make more sense – but he didn’t say this.

But wait – Dan earlier stated that the coefficients are “to be determined” (i.e. not known).

They are not coefficients or even constants – he selects his terms according to the year (and even offers different versions for the same year).
His “coefficients” are variables! He even it states that the “coefficients” are adjusted to get the “best fit” of R2.
If I’m reading this correctly, then there is no supporting science of his coefficients. His “coefficients” are nothing more than pattern matching.

Since the coefficients were determined using all available data, some reviewers asserted that the equation may have no predictive ability in spite of it being formulated from relevant physical
phenomena and a known law of thermodynamics.

(My emphasis)

Of course I would expect Dan as an engineer would understand “a known law of thermodynamics”. In fact I would expect him to know at least three of them.
Which one has he selected? It would help.

Dan has however predicted the temperature for the next 25 years or so (and, surprisingly enough, we see that it will be cooling).
He is assuming that the sunspot variability over the next years is the same as the pattern between 1915 to 1941 – which is fair enough, since that he knows that it is a guess.
If sunspots do resemble then Dan predicts a cooling of about of between 0.2 K and 0.4 K (depending on his variable “coefficients”, despite that he has no idea what almost all his terms are unknown).

The beauty of it is that his own graph shows substantial warming between 1915 to 1941. 🙂

Shot in the foot? I think so.

Finally, Dan “shows” that the temperature has been declining between 2005 and 2011 (despite that the 2011 isn’t yet known yet).
He draws a straight line between 2005 and 2011(using UAH).

This is just sloppy. If Dan knows how to calculate R2 then he is perfectly capable of working out an OLS trend.

Over to you, Dan. 🙂

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# Comment Policy

Discussion on this blog is to be guided by:

It is worth knowing and abiding by whether you comment on this blog or not.

Comments that are not relevant to the post that they appear under or the evolving discussion will simply be moved or deleted, as will links to Denier spam known to be scientific gibberish
• The “Mostly” Open Thread” is for general climate discussion that is not relevant to a particular post. Spam and abuse rules still apply;
• The “Challenging the Core Science” Comment Thread is for comments that purport to challenge the core science of anthropogenic climate change.
• The “Spam” Comment Thread is for comments posted by people who think that they can ignore site policy.