A Perihelion Poser
Today (January 4th) the Earth is at perihelion, ie its closest approach to the Sun. This may surprise folk in the Northern hemisphere who think that winter and summer are determined the Earth’s distance from the Sun…
Anyway, here’s an easy little question. The eccentricity of the Earth’s orbit is 0.017. Estimate the percentage difference in the flux of energy arriving at Earth from the Sun at the extremes of its orbit (ie at perihelion and aphelion). Is this difference likely to have any significant effect?
Answers through the comment box please..
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In the Dark 
January 4, 2014 at 5:12 pm
I presume you don’t want the answers expressed in astronomical magnitudes.
January 4, 2014 at 5:31 pm
I get a difference in insolation of 0.3%
Eccentricity e = \sqrt(1 – b^2 / a^2) where b and a are the semimajor and semiminor axes. Since insolation goes as the inverse square of distance the ratio of insolations is also b^2 / a^2. From there plug & chug to get b^2 / a^2 = 0.9997.
January 4, 2014 at 5:33 pm
Wait…the semiminor axis isn’t the perihelion distance, is it? Eh, close enough, it won’t be more than a percent.
January 4, 2014 at 7:37 pm
It’s easy, really. How does the perihelion distance relate to the semi-major axis a and the eccentricity e?
January 4, 2014 at 8:30 pm
Perihelion distance P = a – f/2 where f = a^2 – b^2, aphelion distance A = a + f/2. Eccentricity e = \sqrt(1 – b^2/a^2). Ratio of insolation R = P^2 / A^2 = ((2 – e)/(2 + e))^2.
e is small compared to 1 so take the first term of a geometric series, R \approx 1 – 2e = 0.966.
January 4, 2014 at 8:46 pm
More simply, P=a(1-e) and A=a(1+e)…you’re being too complicated with your eccentricity.
January 5, 2014 at 12:52 am
So we’ve established that the fractional difference in distance is 2e. The flux received goes like 1 / distance squared, of course. How much of a difference does that make to the surface temperature? In a crude model where the earth maintains it’s temperature by radiating away all the energy it receives as a black body, the temperature goes as the 1/4 power of the incoming flux. So the fractional change in temperature would be 1/4 times 4e, or e. 1.7% of 300 K is about 5 degrees Celsius.
Not totally negligible, but smaller than the seasonal variation.
This is an oversimplified model, of course, but I think it’s correct in order of magnitude. I think that the paleoclimate record does show measurable but small modulation in accordance with how perihelion lines up with the seasons.
January 5, 2014 at 1:42 am
Correct! And WMAP certainly saw this as an annual temperature variation of the optics. L2 also moves in and out with the Earth.
January 5, 2014 at 9:00 am
Yes, the point is that it’s not entirely negligible as is sometimes claimed but it is dominated by other effects.
Is this the reason why summer in the Southern hemisphere is hotter than summer in the North? Possibly, but there is also the difference in land mass and its effect on cooling etc.
January 5, 2014 at 12:09 pm
Well, there’s also the corollary that northern winters in western Europe are not quite as cold as they would be if it wasn’t for the eccentricity and the perihelion date (trifling complicating factors such as the Gulf Stream notwithstanding, of course ;-))
I shall allow that thought to provide succour to my extremities as I whizz down to the Hook of Holland on my bike this afternoon.
January 5, 2014 at 1:38 pm
Yes,it’s by no means a simple consequence of latitude. I still find it amazing that if you draw a latitude line across the Atlantic from London, it hits the coast of North America somewhere to the North of Newfoundland, where the climate is, um, very different…
January 5, 2014 at 5:11 pm
Or put another way, the whole of the United Kingdom lies north of the whole of the lower 48 states of the USA.
Sobering thought, particularly considering that one “low probability, high impact” consequence of climate change could be the shutting down of the thermohaline circulation in the Atlantic, leading to the cutting off of the the North Atlantic Drift, which brings a lot of heat to the UK and western Europe.
(The NAD is driven by density differences in the seawater and is a kind of over-turning of the sea; whereas the Gulf Stream is a wind-driven surface effect).
I recall hearing some interesting talks on this from Stefan Rahmstorf at the Potsdam Institut for Climate Research when I was working at the AIP in Potsdam: he has an informative webpage on the subject:
http://www.pik-potsdam.de/~stefan/thc_fact_sheet.html
(and much more interesting material on the oceans and climate change at his homepage).
January 6, 2014 at 3:17 pm
Here’s a picture of Torquay…