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I've often wondered what exact effect the axial tilt of the earth has on the average amount of solar energy reaching the surface in a year, and how this varies by latitude. It doesn't take much to understand that zero axial tilt would result in a maximum annual average solar heat flux concentrated at the equator, as the sun remains directly over the equator every day of the year as shown below.
But exactly how does the axial tilt of the earth (or any planet for that matter) affect the distribution of heat flux across the latitudes? This is further complicated by the earth's orbit around the sun, as it results in the sun being positioned over a range of latitudes throughout the year as shown here.
We begin the process by finding the normal components of solar heat flux for each latitude and longitude. If the normal component is negative, it means that the sun is below the horizon, and it is therefore night; therefore the negative components can be ignored as there can be no negative solar heat flux on the planet's surface. For each latitude the daily average heat flux is the average heat flux for all longitudes. As the earth moves around the sun, the normal component changes for each latitude, and therefore so does the daily average heat flux.
The diagram here shows the heat flux distribution at mid-day in December, when the sun is directly over the Tropic of Capricorn, while North of the Arctic Circle, it is night 24 hours a day, and south of the Antarctic Circle it is daylight 24 hours a day. Although the poles are cold regions, you might wonder how much heat flux the poles actually receive when exposed to the sun (albeit at a shallow angle) 24 hours a day.
The data computed using the above approach was used to generate the plots below (using an axial tilt of 23.5° to represent Earth). Additionally, averaging the mean daily heat flux over a one-year period provides the annually heat flux distribution across latitudes.
Although the equatorial regions unsurprisingly receive the most heat flux when averaged over the entire year, the poles actually receive more daily heat flux during their summers than any other region on Earth! There is also a secondary local daily maximum heat flux at roughly 35° latitude during the summer, where moving either north or south results in a temporary decrease in the average daily heat flux. Winters on the other hand are a bit simpler, the further you travel toward the pole, the lower the daily average solar heat flux.
Now that we have the formulas in place, let's see what happens at other axial tilt angles. Here is an animation showing the same plots as above, but for a steadily changing axial tilt from 0° to 90°.
Attachments:
MATLAB script used to generate the plots above: Planetary_Heat_Flux.m
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