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.
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.
