In solar energy engineering, this effect is called the air mass, which defines the length of the sunlight’s path through the atmosphere (not to be confused with air mass in meteorology, which defines a volume of air that can be as large as thousands of square miles).
In Boston, the day in the winter can be as short as 9 hours and the day in the summer can be as long as 15 hours. So even if the projection effect favors the winter condition, the duration of the day doesn't. Our calculation must consider these two competing factors. After these considerations, the solar energy the window gains in 12 months is shown in Figure 3: It turns out that the window still gets more energy in the winter -- the projection effect wins big!
OK, let's now include the effect of the air mass in the calculation. Big surprise (at least to me when I first saw the results)!
As comparisons, Figures 3 and 4 also show the results for a west-facing and a north-facing window, both peaking in the summer (for different reasons that we will not elaborate here).
Given Figure 5, the fact that a south-facing window in Boston receives peak solar energy in the spring and fall becomes comprehensible now -- by the law of mathematics, the transition from the Equator to the North Pole must be smooth and the energy peak of any latitude in-between must be somewhere between winter (the season of peak energy in the Equator) and summer (the season of peak energy in the North Pole).
Interestingly enough, the peak energy at the North Pole is comparable to that at any other location in Figure 5. Considering that the Arctic has 24 hours of sun in the summer and the projection effect reaches maximum, this result is in fact a good demonstration of the air mass effect. Without the air mass, the North Pole would have gotten three times of solar energy as it does now, making the Arctic a tropical resort in the summer. Our planet would have been quite different.
All these analytic capabilities are freely available in our Energy3D software and all you need to do are some mouse clicks and some thinking. For the air mass calculation, you can choose to use the Homogeneous Sphere Model (default) or Kasten-Young Model. You can also turn the air mass off temporarily to evaluate its effect, just like what I showed in this article -- this is a piece of cake in Energy3D but is impossible to do in reality because you cannot turn the atmosphere off!