12
CHAPTER 12
GEOMETRY,
SPACE AND TIME
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In this chapter, we will look at astronomy and physics ideas that use the geometry concepts and their
applications we’ve learned in other chapters. Specifically, we develop tools for you to be able to esti-
mate the local time of day or time of year based on observing the sun, building on our work in Chapter 7
finding our latitude and longitude. Sundials of progressively increasing sophistication go back to prehis-
tory, and we will show you some basics and point to ideas to go beyond.
If you can handle some delayed gratification, we also describe a means of tracking the sun’s position for
a year at a set time of day. We’ll find that the sun (or, equivalently, its shadow) makes a sort of lopsided
figure-8 over the course of a year called the analemma. Like everything else scientific in the late medi-
eval period, early knowledge of it was shrouded in intrigue, but you can replicate it on a sunny patch
of driveway. The analemma captures a lot of information about Earth’s orbit around the sun over the
course of a year, and we will see how to interpret our observations.
THE EQUATION OF TIME
In Chapter 7, we talked about the Equation of Time, which isn’t really an “equation” the way we normally
think of one. It is more like a correction that we have to apply if we want to know the time (or our posi-
tion) based on where the sun is on a given day. One fallback for us modern folks, as we saw in Chapter
7, is to estimate from the graph in the Wikipedia
article “Equation of Time” or do an internet search
on “sundial correction.” However, now that we
know a little more geometry, can we be more intel-
ligent consumers of these sources? And, at the
same time, can we be more appreciative about how
observant some of those people who lived a thou-
sand or two years ago were?
First, let’s think about Kepler’s second law (Chapter
11), which says that, if we drew a line from a planet
going around the sun, the line would sweep out an
equal area of an orbit in equal time. One measure
of how much an ellipse is different from a circle
is its eccentricity, often called e (like many other
things in math, unfortunately). For an ellipse with
semimajor axis a and semiminor axis b, the eccen-
tricity is:
e= √
1-b/a
It turns out that the earth’s orbit has an eccentricity
of 0.01671. So it is not a lot different from a circle,
but different enough to be observable. If we square
3D Printable Models Used in
this Chapter
See Chapter 2 for directions on where and how to
download these models.
sun_dial_gnomon.scad
Makes the gnomon for a sundial at a user-input
latitude
Other supplies for this chapter
• Washable chalk
• Electrical tape
• Some paper to draw on
• Navigation compass (or compass phone app)
• Yardstick
• Tape
• A pencil
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both sides and rearrange a little,
(0.01671) = 1 - b/a and thus 0.99986 = b/a.
To put it another way, the shorter axis of our orbit is about 99.986% the size of
the longer one.
Isaac Newton figured out an equation called the vis-viva equation that will
let you figure out the speed at any point in an orbit. We won’t wade into the
algebra for that here, but it turns out that we get closest to the Sun in early
January of each year (called perihelion) at which time we are clicking along
our orbit at about 30.29km/s. We are farthest from the sun (aphelion) in early
July, and going about 29.29km/s (according to Wikipedia, “Earth’s Orbit.”). In
2023, the dates are January 4 and July 6 respectively. The website
https://www.timeanddate.com/ is a very good source for data like this.
The implication is that we spend a little less time hanging around the part of the
earth’s orbit around the December solstice than we do around the June solstice.
Note, though, that the way the tilt of the earth’s axis happens to line up with the
orbit is such that the time when we are closest to the sun does not line up with
when we will see the shortest and longest days of the year. Also, we are very
slightly closer to the sun while it is winter in the northern hemisphere.
The bottom line is that the effects of the tilt of the earth’s axis, plus the dif-
ferences in the earth’s speed around the sun at different times of the year,
sometimes add up and sometimes subtract from each other. They cancel
out four times a year. The dates vary plus or minus a day or so, because
the earth’s orbit is also not a round number of days, and there are several
effects (some of the biggest of which we will talk about in this chapter) that
happen out of sync with each other. That said, the dates when the Equation of
Time is zero are approximately April 15, June 13, September 1, and Decem-
ber 25. The rest of the time, the correction can be as much as 16 minutes one
way or the other; in the next section, we see what the variation looks like.
There are some other factors too, like the effects of other planets tugging
on the Earth-Moon-Sun system and a variety of other wobbles. To figure out
the Equation of Time in detail is a big job. For the most part, only people who
have to navigate spacecraft to where they are going and perform similar
tasks will need to figure out the details.
As a side note, read the definition of the sign of the correction carefully if
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you are reading the Equation of Time from a table. The conventions are not
consistent everywhere. Some people define a positive number as mean time
minus solar time, and others reverse the order. We describe the convention
where February is negative, where you would need to observe the sun at
12:16 on your clock to get a noon reading.
THE ANALEMMA
Other than intellectual curiosity, until recent times, there weren’t a lot of
reasons to need to know the time to very high resolution. That all changed
when people started to navigate across oceans. Thus, it is not a coinci-
dence that a lot of interest in better timekeeping and globe-spanning sea
travel came along about the same time (in the 1500s). We went over the
link between a good timepiece and being able to determine your position in
Chapter 7.
There was also a desire to know when the spring and fall equinoxes were
(the first day of spring and fall, respectively). People wanted to know when to
plant crops, or when to celebrate holidays. Thus some monks, and scientists
with patrons, started to do more thorough observations in the late 1500s and
early 1600s. A new calendar, the Gregorian calendar, was introduced in 1582
as a result of some of these observations. Yet it was still an era with inherent
contradictions for astronomy since it remained heretical in some countries to
say the earth went around the sun.
As scientists gathered more and more evidence that the earth went around
the sun, some observers in the late 1600s (like the Italian/French astrono-
mer Jean-Dominique Cassini) began to take careful observations of meridian
lines, which mapped the height of the sun above the horizon. Cassini worked
with a pinhole in a cathedral at San Petronio in Bologna, Italy. This pinhole
projected a disk of the sun on a measurement line on the cathedral floor.
This gave him good information on how the sun moved over a year along a
straight line, and other data that confirmed Kepler’s ideas about how planets
moved around the sun.
THE GEOMETRY OF THE ANALEMMA
As various instrumentation began to get better (including clocks that were
not dependent on the sun) people began to observe that if you marked the
position of a shadow at the same clock time each day, over the course of a
year it would not mark a straight up-and-down line. It would instead create a
lopsided figure-8. This is called the analemma (Figure 12-1).
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The analemma is essentially a graph on the
sky of the Equation of Time, drawn out by the
position of the sun every day at one fixed clock
time over the course of a year. Clock time
(sometimes called “mean time”) measures the
long-term average over the year of the length
of a day. On average, the sun will be at its
highest point in the sky near noon, corrected
for longitude (Chapter 7). However, sometimes
the sun will be a bit further east or west than
average at this time of day, and this creates the
east-west part of the analemma.
The vertical part of the analemma is the sea-
sonal elevation change, which we talked about
in Chapter 7, when the sun appears lower
in the sky in winter than it does in summer.
However, even if the earth’s orbit were per-
fectly circular, the analemma would be a
symmetrical figure eight, not a straight line.
The reason for this is a little subtle, and best
seen in animations. It comes about because
the earth’s tilt (which an astronomer would
call the earth’s obliquity) makes the actual path
of the sun in the sky take a little more or a little
less time than the imaginary average path of
the sun. For an excellent visualization look at
the “obliquity” animation at
http://analemma.com/obliquity.html or that
site’s associated phone app.
The difference between the top and bottom
of the analemma mostly comes from the fact
that the earth’s orbit is an ellipse, and so at
some parts of the orbit the earth will move a
little farther along its path than it “should” on
average (by Kepler’s second law). The sun will
seem to arrive a little later than it should (and
thus be a bit east). The sun is behind where it
should be on the left-hand side of Figure 12-1,
and ahead on the right side.
FIGURE 121: The analemma in the sky (North on top in the
northern hemisphere).
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