Monday, 8 December 2014

Astrophysics Corner, Part 14 – Solar Panels and Latitude: the Effect on the Sun’s Altitude and the Length of the Day

In an  earlier blog, I noted that solar panels on a planet are affected by both the latitude of the location that the panels are placed and the tilt of the planet’s axis of rotation, relative to its orbital plane.

  • The latitude of the location of the panels is obviously important.  Essentially, the solar flux that is experienced on the surface varies with the latitude of the site as the sine of the sun’s elevation above the ground.  The farther from the planet’s equator, the smaller that angle will be. 

  • That will be further compounded by the tilt of the planet’s axis, relative to the plane of its orbit around the star.  The greater the tilt, the more the elevation of its sun will vary throughout the year, so the solar panels output will be greater or lesser as the season’s progress.

Both of these effects can be modelled mathematically, with reasonable accuracy in an excel model. 

As noted above, one’s latitude affects both the sun’s maximum altitude in the sky (zenith) and the length of the day.  It is fairly intuitively obvious why latitude affects the sun’s altitude, as seen by an observer on the ground.  Near the equator, a person has to look high overhead to see the sun at mid-day, as his or her head is directly between the sun and the Earth, so the sun’s altitude will be somewhere near 90 degrees.  At the North Pole, the observer will be looking directly at the horizon to see the sun at mid-day, so the sun’s altitude will be somewhere near 0 degrees.

A simplified diagram is given below, as things would be on either of the equinoxes.  The people, of course, will be very small compared to the Earth (the diameter of the Earth would be more than 6 million times the observer’s height), and the distance to the sun will be many more Earth diameters than shown below (over 11,000 Earth diameters, in reality).  It’s hard to do those dimensions justice on a blog.

As we all know, things are more complicated than that, due to the tilt of the Earth’s axis.  This affects altitude of the sun at mid-day (through the course of the year, it will vary by twice the tilt of the Earth).  So, at the equator, the sun can vary between being 23.5 degrees to the person’s south, to 23.5 degrees to the person’s north.  At the north pole, the sun will be 23.5 degrees high in the mid-day sky at the summer solstice, and  23 degrees below the horizon at mid-day on the winter solstice.  At my latitude of 54 degrees, the sun can very from 13 degrees in the winter to 59 degrees in the summer.
When the sun is directly overhead (90 degrees from the horizon), its energy flux (watts per square meter) is focussed on a small spot, let’s say the top of your head, so you feel  dangerously hot.  When it is at a lesser angle, that energy is spread over a larger area, let’s say your head, torso and arms and legs, so you feel a lot cooler.  It’s the same with solar panels - they can generate a lot more power when the sun is high overhead than when it is low down towards the horizon.  You can see in the diagram below, that if you were shining light through a tube, the light would fall on a much smaller surface in the picture to the left than on the picture to the right, and thus would be more concentrated.  Mathematically, that  is accounted for by multiplying the flux of the sun by the sine of the angle of its altitude.
Also as we all know, the other major difference between the winter and summer is the length of the day.  That is a result of the tilt of the Earth’s rotation axis, relative to the plane of its orbit around the sun.  In the summer, the tilt favours the northern hemisphere, making the days long in addition to ensuring that the sun is high in the sky.  You can see how the regions in the far north will get 24 hours of sunlight, when the earth’s tilt is in the position shown, whereas the regions in the far south will not see the sun at all.  A similar situation prevails throughout the globe, with areas in the northern hemisphere generally getting longer days at this time of the year.  The effect varies with latitude.  The trigonometry is a lot trickier than in the case of the sun’s altitude above the horizon, though.

The graph below shows the sun’s highest mid-day altitude, by various latitudes on the Earth.  As you can see, at the equator, the sun goes from 90 degrees at the equinoxes to 23 degrees north and south of straight up, at the summer and winter (northern hemisphere) solstices.  At higher latitudes, the sun’s altitude throughout the year varies, correspondingly.  At latitude 66.5, the sun just touches the 0 latitude at the winter solstice - that is the arctic circle.  Above that point, the sun is actually below the horizon in mid-day during the depths of the winter - i.e. it never rises, so there are days with no sunlight at all.  It is interesting that even at fairly northerly attitudes, like 54 degrees (Edmonton, Canada which is only slightly north of London, England) the sun still gets quite high in the sky in mid-summer, nearly 60 degrees.

The next graph shows the length of the day at various latitudes, throughout the year.  We see that at the equator, the day is 12 hours long throughout the year.  That might be a bit monotonous.  At 30 degrees latitude (about the latitude of Shanghai or New Orleans), it varies roughly between 14 and 10 hours.  At 45 degrees (about the latitude of Toronto) it varies between about 15 hours and 9 hours .  At 54 degrees (about the latitude of Moscow) it varies between 17 and 7 hours.  At 66.5 degrees, we hit the arctic circle, so the day varies between 24 hours and 0 hours.  Interestingly, by just being half a degree south of that line, the day varies between 22 and 2 hours.  At the north pole, the day is either 24 hours long or 0 hours.

It should be noted that the sun is actually visible when it is below the horizon, due to the refraction of light through the Earth’s atmosphere, so the day is actually a bit longer, if we consider dawn and dusk.

An interesting result occurs if we combined these two graphs, as shown below.  In this case, I have multiplied the number of hours in the day by the sine of the angle of the midday sun, to account for the fact that that the total amount of solar flux is a combination of the two.  We can see that in higher latitudes, the solar irradiance is comparable to the tropics (because of the long hours of daylight), during the later spring and summer months.  Even at the fairly lofty latitude of 54 degrees, the energy of the sun is very similar to that of the tropics from early May until early August.  This explains why life, including people, migrated to high altitudes.  For a considerable time of the year, there is plenty of solar energy to sustain life.  For the rest of the year, life has had to store up this solar energy in the form of fat and/or migrate south.

It is probably just a matter of time before human technical ingenuity comes up with storage solutions that will allow us to  store the solar power we generate with our solar panels in the summer, to tide us through those short days of winter.

It is also interesting to note that the amount of solar energy that falls upon the earth in higher latitudes over the course of the year is a substantial fraction of that which falls on the tropics.    We will look more deeply into that in the next blog, where we will examine how the tilt of a planet’s axis affects the solar energy received at its surface. 

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