## Shadows, position and sundials

If you stick a vertical pole in the ground, its shadow will move as the day progresses. This is because the rotation of the earth makes the sun appear to travel across the sky. As the sun “moves”, the shadow that a fixed object casts moves with it. This suggests that you could make a clock: simply draw a line from the pole along the shadow at noon: mark it with a “12”; then draw another line along the shadow at 1 PM and put a “1” on it; the line at 11AM is marked with an “11”, etc. This gives a series of radial lines which should enable you to tell the hours on other days. Unfortunately this is not the case. If you lived at the North Pole it would work, but the problem is that when the sun travels through the sky for one hour, the shadow of a vertical pole moves through an angle *which is not the same* in every season; except at the north pole, the relation between time elapsed and the angle a pole’s shadow moves varies from day to day. You can not calibrate a vertical pole clock so as to tell time all year round! (More on this later.)

In Friday’s episode, the shadow pole is a basketball net post, presumably vertical. Two photographs are available of it, with local time stamps. Unfortunately, these just tell us the time it is *everywhere within a given time zone*. This is significant as we shall see.

What *do* the pictures tell us? If we know the height of the pole and the length of the shadow (which we do since a basketball rim is exactly 10 feet off the ground, and the shadow’s length is computed by counting bricks on the ground), we can use simple trigonometry to find the angle the sun makes with the ground, called its elevation.

Now the path of the sun through the sky depends only on the time of year (date) and your latitude. For example, in northern latitudes, the sun stays low on the horizon in the winter and higher in the summer. This is because the earth’s axis of rotation, which always points in the same direction (or very nearly) is tilted with respect to the plane of its orbit around the sun. So as the earth travels around the sun, the south-north axis may point toward or away from the sun. When it points toward the sun, the sun rises higher in the sky in the northern hemisphere, and so warms the Earth more. It is summer. You get theidea.

So let me repeat: The path of the sun through the sky depends only on the time of year and your latitude. We know the time of year and we would like to find the latitude, so we must determine the path. If we knew the sun’s *exact elevation* at a *particular time*, we could determine the path. This can be done by looking through all paths and finding the unique one with the sun at that position at that time; even better, there are mathematical ways of finding the path exactly, using what’s called *spherical trigonometry*. But we don’t know the exact time, just the time-zone time. Time zones are 1000 miles wide, approximately. So *here is why we need two pictures at a definite time interval apart*. Only one path will have the sun changing in altitude by exactly that amount in that time. Thus, two pictures taken a known time interval apart determine a unique path, which in turn gives a unique latitude.

Even better, once you know the path, you can predict at exactly what time the sun will be highest. Say this is at 12:20 local time. (Remember that the sun can’t be highest at noon *everywhere* within a time zone.) That means that you are 20 minutes west of noon in your time zone. There are tables which showexactly where (in terms of longitude) in a time zone noon occurs, so you can calculate your east-west coordinate — that is, your longitude — from this.

In summary, then, having two photos giving elevations of the sun at two local times enables you to find both your latitude and longitude. This can all be done using spherical trigonometry, a subject that started with the Greek mathematician Ptolemy nearly 2000 years ago, and was brought to fruition by the medieval Arab astronomer/mathematicians over 1000 years ago. Spherical trig applies the functions of trigonometry — sine, cosine, tangent — to studying circular arcs in 3-dimensional space. These arcs are either on the surface of the earth, or in the heavens directly above the earth. Such arcs are completely described by the angles which lines drawn from their endpoints make at the center of the earth; so, both the angles and the sides of a “spherical triangle” are actually given as angles. It’s a complex and beautiful theory, but I don’t have time to get into details here. In any case, it is exactly the math needed to describe the apparent paths of heavenly bodies through the sky, or ships moving across the seas.

How accurate are the calculations made on the show? Charlie claims that they are accurate to within 1/100 of a degree, which amounts to a little more than a kilometer. This is probably optimistic, since who knows if the basketball pole is exactly vertical, or how accurately we can count bricks in the photo to find the length of its shadow. Be that as it may, one might be able to find an aerial photo of a square kilometer block in California and have a computer do a pattern search with other features of the photos to pinpoint the exact location. Presumably that’s what happened.

By the way, the Arab mathematicians of about 600 years ago were able to fix the problem of the shadow pole not being accurate with the changing seasons. The idea was that the earth rotates around its axis, which is a line running between the North and South Poles. If you want equal hours to correspond to equal shadow angles, you need to have your shadow pole parallel to this axis. So, at least in the northern hemisphere, simply point the shadow pole in the direction of the North Star (Polaris). The angle that such a pole makes with the vertical turns out to be 90 degrees minus your latitude (so at the North Pole it is vertical). The Arabs produced very accurately calibrated sundials; the tilted “shadow pole” became a triangle with hypotenuse pointing north and angled so that it points to the North Star at night; it became known as a *gnomon*. They were able to calculate, using spherical trig, the exact calibrations so that it would tell accurate time all year long!