There is a formula you can use to figure out the maximum elevation of the
sun in-between these four times of the year:
Max elevation = 90° - latitude + solar declination
That means that to find latitude, we need to find the sun’s maximum ele-
vation above the horizon for the day. We also need to calculate the solar
declination for that day, which depends on how many days have passed since
the last winter solstice. We’ll show how to calculate that in a later section in
this chapter. We can find the sun’s elevation by measuring it, as we’ll show in
the next section.
Incidentally, the earth’s axial tilt varies slowly over time, and is currently
decreasing. Over many millions of years, it has varied from about 22° to
24.5°. Unless you are doing precision navigation, you can safely ignore this
variation over a human lifetime. However, astronomers often think about the
long term.
ELEVATION OF THE SUN AT LOCAL NOON
Let’s measure the elevation of the sun, the angle between it and the horizon
when the sun is at its maximum height for the day. First we need something
that will cast a shadow that is easy to measure. The fancy name for this is
a gnomon. A toothpick in modeling clay or something else pointy held ver-
tically will work, or we have a 3D printable one that will be a little easier to
use. If you are using our 3D printed gnomon, first create it from the model in
gnomon.scad
Measuring elevation is not as simple as going outside when the clock says
noon. Depending on where you are in the time zone, the sun might be at its
highest point as much as an hour before or after noon on the clock. In some
extreme cases, like the far west of Alaska, it might be even more if a state is
kept in one time zone for convenience.
For that reason, you should start doing the following measurements shortly
after 11 AM if you are currently observing standard time, and at about noon if
you are someplace using daylight savings (summer) time. You might have to
do this for as much as two hours. If you are using daylight savings (summer)
time, start measuring at noon on your clock and you might have to go as far
as 2 PM Daylight Savings time.
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If you know you live near a time zone boundary, you might be able to think
through how that works out and make the following process shorter. If you
want to assume nothing, however, here is what you should do. The follow-
ing directions are for standard time; shift them all an hour later for daylight
savings time.
Take a piece of paper and your 3D printed or toothpick-and-ruler gnomon
outside. Every five minutes, starting at 11:00, do the following:
• Put the gnomon on a sturdy (and level) surface that will be in the sun
for at least two hours.
• Arrange it so that the shadow of the vertical (shorter) part
falls on the horizontal part. Read off the solar angle directly
(Figure 7-16).
• If you are using a toothpick in clay, first place the toothpick right
over the zero marking on the ruler. Then pack around it with a bit of
modeling clay to make it stay upright. Be sure that the toothpick is
vertical and making a right angle with the ruler, as shown in Figure
7-17. You want to use something pointy so that the shadow of the tip
is crisp and easy to read. If you can figure out how to stand it up, a
pencil with its tip up would work too.
• Measure the shadow. You may need to rotate the gnomon to have
the shadow fall squarely on it. Be sure you are measuring from the
center of the toothpick or pencil, not from the edge of the clay!
• Re-do this measurement every five minutes until the shadow
reaches a minimum height and starts getting longer again. (If it is
already getting longer when you start, try again the next day, but
start earlier.)
FIGURE 716: 3D printed gnomon
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• Write down as accurately as you can the time when you saw the short-
est shadow. You’ll need that time to calculate your longitude later in this
chapter.
In Chapter 6, we learned about sine, cosine, and tangent. The tangent (oppo-
site side divided by adjacent side) of the angle of the sun above the horizon
is the height of the vertical part of the gnomon (the toothpick) divided by the
length of the shadow. For example, if the gnomon is a 6.5cm tall toothpick
stuck into a bit of modeling clay so it stands on its own (rather than being
stuck in the ground), and the shadow is 6.8cm long, the angle whose tangent
is 6.5/6.8 = 0.96 is 43°.
Once we get this angle, we have one of the key numbers we need for our
calculation of latitude: the maximum elevation angle of the sun today. Write
down the angle you got from your observations. Remember that we are using
this equation:
Max elevation = 90° - latitude + solar declination
Now we just need solar declination to calculate latitude. For our example we
got 43° for solar elevation, and we would have
43° = 90° - latitude + solar declination.
FIGURE 717: Using a ruler, toothpick, and clay as a gnomon
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SOLAR DECLINATION
The solar declination is the angle between a line connecting the centers of
the earth and sun, and the earth’s equator. It adds or subtracts as much as
23.44° to the elevation angle over the course of the year. To do it right, we
would need to use a complicated formula that takes account of the earth’s
orbit not being perfectly circular, in addition to the tilt of the earth’s axis. For-
tunately, though, we can get within a degree or two with this approximation:
Solar declination (in degrees) = - 23.44° * cos ((360°/365) * (days
since the first day of winter)).
There are 360° in a circle, and it takes 365 days to go around the sun in a
year. That’s where the 360°/365 comes in since that is how many degrees the
earth goes around its orbit per day. We multiply that times how many days it
has been since the sun was at its most extreme low point, when it is 23.44°
below where it would be if the earth’s axis pointed straight up and down. This
day is called the winter solstice in the northern hemisphere, and we also think
of it as the first day of winter. In other words,
(360/365) * days since the winter solstice (which is 10 + days since Jan 1)
equals the fraction of the almost-circle of earth’s orbit we have gone around
since the first day of northern winter. If we plot declination versus days since
the first day of winter, we get the graph in Figure 7-18.
FIGURE 718: Solar declination in degrees versus days since the winter solstice
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Let’s say we want to know the solar declination on
October 22, 2023. That turns out to be day 295 in
2023, which is not a leap year. So the solar decli-
nation on October 22, 2023, is:
Solar declination (in degrees) =
- 23.44° * cos ((360/365) *(295+10))
= - 23.44° * cos(300.8°)
= - 23.44° * 0.51237 = - 12.0°.
Since we need to subtract this number of degrees
off the altitude of the sun on the first day of winter
(our starting point), it is a negative number.
Therefore the maximum sun elevation on October
22, 2023 anywhere in the world will equal:
90° - latitude - 12°, or 78° - latitude.
Notice that this means at latitudes north of 78°,
the sun will not be above the horizon on October
21 at all. This line of latitude runs roughly through
the middle of Greenland. You can play with this
and a globe or online maps to see what part of the
world will start missing the sun altogether on a
particular day.
On the first day of winter, we subtract off 23.44°
from the sun’s highest point above the horizon.
About 182 days later, we add 23.44°. It’s tricky
to think about this in the Southern Hemisphere,
where the negative latitude gives us sun angles
of more than 90°. This means the sun will be in the
north during most or all of the day.
Suppose we wanted to be a little more accurate
and take account of the fact that the earth’s
orbit isn’t quite 365 days. It is more like 365 and
a quarter days, which is why we have leap years
every four years.
Also, the winter solstice isn’t exactly at local noon
in any given place. In 2022, it is at 1:48 PM Pacific
Standard Time. To correct for that, we can use
fractional days. If we are measuring around noon
Pacific Standard Time (close to 1 PM Pacific Day-
light Time, in October 2023, when local noon is),
that’s about 1 hour and 48 minutes less than a full
day from than the solstice was, or 0.075 of a day
less. We could use both these corrections to get:
Solar declination (in degrees) =
-23.44° * cos((360 / 365.25) * (295 - 0.075 + 10)) =
-11.9°
This difference is probably less than the accuracy
we will get in measuring our elevation angle, but
good to know that our approximation is close. We
will call it -12° in our example.
Solar Declination Example
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