I Lie Awake, Therefore I Calculate
I went on a calculating binge the other night; it was a sort of obsessive-compulsive insomnia thing. I got to thinking about global warming and climate physics and started to wonder how much of the temperature difference between the North Pole and somewhere on, or near the equator, like Nauru, was due to differences in the levels of solar input or whatever it's called between the two locations.
The one good thing about being kept awake by a mathematical or scientific problem, rather than, say, the theological question of whether Jesus died a virgin, is that with the help of a few reference books you can, with a little logical thinking and a few calculations arive at an answer. It may not be the correct answer but, if you're cunning enough to reach it through a chain of reasoning that seems plausible, you can at least get to sleep.
The first thing I needed to do, to bring the problem within a solvable scope, was to make some simplifying assumptions so I started by tossing away the Earth's axial tilt. This gets rid of all the seasonal variations in temperature; every day of the year is an equinox. While I was throwing out complications, I decided that the oceans and major land masses might as well go too. The oceans complicate the problem by feeding cloud formation and clouds, as we know, reflect sunlight so that it doesn't penetrate to the lower atmosphere. They also stuff things up in other ways, what with water being a greenhouse gas and all. It was much easier to work with a homogenous atmosphere, more or less completely transparent to solar radiation.
The first calculation I needed to do was to find the ratio between the distance from the sun to Nauru and the distance from the sun to one of the poles; north or south, it doesn't particularly matter. Memory, various reference works and other sources were consulted to produce the following figures:
Distance from the Earth to the Sun (d): 149,600,000km
Radius of the Earth (rE): 6,376.5 km
Radius of the Sun (rS): 110 times the radius of the Earth i.e. 701415 km
(That last calculation of the radius of the Sun is based on information in the Technical Appendix to The Copernican Revolution by Thomas S Kuhn).
From this information the ratio R between the distance from the surface of the sun to Nauru and the distance from the surface of the sun to one of the poles is fairly easy to calculate. I decided to skip the fart-arsing around with Pythagoras' theorem, and assume that the extra distance that sunlight has to travel to get to the North Pole, rather than Nauru, is the radius of the earth. If anything, this assumption inflates the final figure for the expected temperature difference (the trick with calculations of this kind is to make them just complicated enough that you finish with the feeling that you have dealt with the question thoroughly but not so complicated that you have to stay up all night doing the arithmetic. Remember, the point of this exercise was to clear my head of an annoying problem so that I could finally get a good night's sleep). The equation for calculating R is:
R = [d-(rE +rS)]/[d-rS]
Which works out to 0.999957176.
Just out of interest, at this point I decided to use the result to get a percentage difference between the distance from the sun to Nauru and the distance from the sun to the North (or South) pole. It's 0.0043%, which is chickenshit by any standard. Here I could have said to myself, there you go, whatever it is that makes the difference between Nauru being bloody hot and the North Pole being bloody cold, it's not because the North pole is further away from the sun than Nauru. I tried doing just that, but I still couldn't get to sleep.
The next stage is, given the ratio R, to calculate the ratio between solar input (or whatever it's called) at the equator and solar input at the poles. Memory helped out here, by providing the Inverse Square Law which relates the intensity of radiation to the distance from the source of the radiation. What it says, in essence, is that if you stand one metre away from an open fire, you'll be four times as warm as you would be standing two metres away, nine times as warm as you would be three metres away and so on. The ratio between the solar input at the pole and that at Nauru is, conveniently, the square of R, that is 0.999914353. The percentage difference between the two therefore works out to 0.009% ([1-0.999914353]*100). Which really is chickenshit.
To finish the job, we have to turn that 0.009% into an actual number of degrees Celsius. Actually we turn it into a number of degrees Kelvin, which is the same as a degree Celsius. To convert a temperature on the Celsius scale to a temperature on the Kelvin scale you simply add 273 degrees, which is a hell of a lot simpler than converting from Celsius to Fahrenheit or vice versa.
You might be wondering why convert to Kelvin in the first place. The answer is simple; there's a linear relationship between the temperature of any given mass of matter and the thermal energy stored in it. It takes as much energy (4.2 Joules) to raise the temperature of 1 gram of water from 0 degrees C to 1 degree C as it does to raise it from 1 degree C to 2 degree C. Trust me. I did high school physics and chemistry.
All I finally needed to do to convert that 0.009% into an actual number of degrees Kelvin (or Celsius) was to find out the average temperature at the equator, convert it to Kelvin, and multiply by 0.009% (or 0.00009 if you want to be picky about it). After finding a web-page that said the average surface temperature of the earth was 15 degrees C (288 degrees K) I decided the hell with it, that'll do, and calculated my final figure of 0.025 degrees. Then I went back to bed.
While I was drifting off to sleep, for some reason I thought of PT Barnum's famous saying "There's a sucker born every minute." It got me thinking about how many suckers would be born in a year (525600, or 527040 in leap years) and what percentage of the annual birthrate that would have made up in Barnum's time. If you knew that you could calculate the current sucker production rate and, if you tossed in some assumptions about life expectancy, you could arrive at an estimate of the current world population of suckers and I'm really too tired for this right now, it will just have to wait until later.