Autumnal Fading of the Insolation

The waning of summer has now been replaced by the rapid decline of the autumn as the sun passes the equinox. This is the time of year when the daily rate of change in sunlight -- the interval between sunrise and sunset -- is at its greatest. As we all know, as the days grow shorter and the sun gets lower in the sky the amount of energy we receive from the sun decreases. The why of it is pretty straight-forward although perhaps it is less clear just how great is the difference.

When I was a child and I first learned these facts I can clearly remember standing outside in the frigid winter air and looking at the sun hanging low in the south. It was cold but the sun seemed just as bright (or even brighter due to all the reflecting snow on the ground). I knew that the sun was no further away in the winter -- it's actually closer in the northern hemisphere winter -- and since I was facing it full on the energy I received should also be identical. This is in fact close to the truth, excepting the attenuation due to the longer path sunlight took through the atmosphere to reach my face.

The energy we receive from the sun by facing it, unfortunately, does not hold when you consider the wider expanse of the Earth's surface, both land and sea, which is what counts when it comes to weather and the seasons. At lower elevation angles, the energy is spread over a larger area, which directly lowers the flux. This is called solar insolation, and it is usually measured in watts per square meter, as power crossing a surface. Integrate by time and you get the energy, which matters when you have to take into account length of day.

This is all very qualitative and probably tells no one anything they don't already know. However I like to look into the details since it is intuitive that there are some interesting dynamics at play. For example, during the summer at higher latitudes the days are long, far longer than in the tropics, and the sun is high enough in the sky that the effect of sunlight spreading should be minimal. It should also be evident that at both equinoxes the total daily insolation should only depend on latitude, decreasing as one moves away from the equator, since daylight is 12 hours everywhere on the globe. In winter, the cooling effect due to latitude should be accentuated, and especially so in the arctic and antarctic which for a time has not daylight at all.

It is actually quite easy to do the calculation if you have a little bit of trigonometry at your fingertips. All you need to do is figure out when during the day the sun is in the sky and track its elevation between sunrise and sunset. If, like me, your trigonometry lessons are lost in the dim past you turn to the internet. If you do this you will quickly locate the equations for the sun's declination -- its angular displacement from the celestial equator -- and for the solar elevation angle. One then plugs these equations into a spreadsheet, enters a date and a latitude and spit out the numbers. Except that there are a couple of complications we need to deal with.

First, there is the matter of calculus since a proper calculation requires integrating the solar elevation angle over the time between sunrise and sunset. I took the easy way out and did a numerical integration with half-hour steps, which is a simple operation on a spreadsheet. The second is accounting for all the confounding factors if you want to achieve accuracy: the Earth's elliptical orbit, atmospheric scattering at various incidence angles, local terrain, and so forth. I opted for simplicity by discarding all these items and reduced the significant digits to two. The numerical integration adds its own inaccuracies but this is only a few percent.

With a tip of the hat to the prodigal spherical cow I proceeded to calculate. A summary of some interesting results is in the following table. Since watts and joules and all that stuff is not intuitive to most people I opted to normalize daily solar insolation where a value of 1.0 represents one hour of insolation with the sun standing still at the zenith. That is, directly overhead.

If you peruse the table you should notice some items of interest. One is that the loss of insolation in winter (it's at a minimum at the solstice, give or take a few days) is dramatic once you get to higher latitudes. I chose Iqaluit for my northernmost location since it's just south of the Arctic Circle. Edmonton gets only half the insolation of Ottawa, and Iqaluit gets only one-tenth of that, being only about 1.5% that Quito on the equator received on that same date.

The second interesting item is that at this same date Quito does not get its maximum insolation despite being on the equator. This is because the sun is in the northern sky by just enough to drop the insolation by about 10% from when it is at a maximum, both equinoxes. These are the two dates on which the sun crosses the zenith.

For me, the most striking result is the insolation at the summer solstice. All of these Canadian cities have a higher insolation than anywhere in the tropics. Even Iqaluit gets about the same amount as us southerners. This is of course due to the extra long summer days, when the sun barely gets below the horizon during the brief night, and compensates nicely for the lower solar elevation angle. More interesting is that the daily insolation gets even higher as you go north. This is because the daylight is 24 hours yet the sun is high enough in the sky to have half the intensity at the zenith even at the northern tip of Ellesmere Island, our most northerly point of land. But you have to time your visit to there just right because the warm sun doesn't last for long.

Only about 80 more days until we get our 2.0 zenith-hours of insolation here in Ottawa. I can hardly wait...until the day after.