A Matter of Time: Part 2.

Mike Dziekan
VP Engineering, Connecticut Analytical Corporation

Part 1 appeared in the 30 June 2006 issue of The Citizen Scientist and is available here.

The Sundial

Most of us are familiar with that antique-looking, flat, horizontal metal plate that sits (mostly ignored) in many gardens. It was likely purchased as a decorative ornament where the purchaser intended to set it up as a sundial to indicate the time, only to be frustrated that the indicated time is way off. If you are even the slightest bit serious about purchasing a sundial, DO NOT buy one of those garden center sundials unless it has an adjustable gnomon that can be set for your exact latitude.

Not long ago the sundial was a very widely used indicator of time. If you can afford to get a sundial that can be adjusted for your latitude and longitude, then the sundial time will equal the clock time, with the exception of the approximately +/- 15 minute annual variation and daylight savings time (DST). There are many benefits and drawbacks of sundials. One very obvious drawback is contained in the name itself – sundial. The sun is an integral component of any sundial, and at night or during cloudy weather, a sundial no longer functions.

Recall from our previous discussion on time zones how the time difference is four minutes for every degree of longitude. Many people configure their sundial for DST with the logic being that one is more apt to be outdoors more often in the summer than in the winter!

Each sundial must have a gnomon angle that is equal to the exact latitude of your location to be accurate. If the seller happily agrees to sell you a garden variety sundial and states that it will be accurate without asking you where you live (your exact coordinates), then the seller intentionally deceptive or woefully ignorant of how sundials work! If the seller continues to argue that sundials aren't that accurate, and that the indicated time will be close enough, ask if 75% or 80% of the purchase price will be “close enough!” A properly adjusted and orientated sundial can be pretty accurate. But because of the popularity of garden center sundials, they have an undeserved reputation for inaccuracy.

Readers of The Citizen Scientist wise enough to own a copy of the Amateur Scientist CD Rom will be rewarded with a good deal of information relating to sundials, their construction and proper use. In fact, there is a good article by C. L. Stong, from March 1964, that explains how to construct an easy to make sundial that reads “clock time” directly.

Local time is what the sundial will indicate, and this is commonly referred to as “apparent time.” To correct your sundial to read “clock time,” a longitude correction is necessary. This is taken into consideration when you order a custom sundial, for you will need to provide your exact coordinates. The custom dial will include a gnomon set to the correct angle for your latitude, and the numbering will be adjusted to match your exact longitude. I have only been discussing one specific type of sundial, but there are many different types with different advantages to each. There is even a national organization called the North American Sundial Society (NASS).

I am a member of the North American Sundial Society. For a small membership fee you are privy to a wealth of very detailed information on “dialing.” The NASS has information ranging from the most basic concepts to very advanced theoretical analysis of the sun's motion and spherical trigonometry. I strongly urge members of the Society for Amateur Scientists to visit their website and download a sample copy of their quarterly Compendium . As far as I am concerned, the best source for “dialing” information is the NASS.

The Equation of Time (EOT)

Earlier I mentioned that a sundial can be off by about +/- 15 minutes. More precisely, it can be ahead by approximately 17 minutes and behind by 14 minutes during a calendar year. The equation of time (EOT) is used to calculate the amount of deviation in apparent time (sundial time) throughout the year. This provides mean solar time that assures that local noon does not vary throughout the year.

The Earth orbits the sun in a slight ellipse rather than a perfect circle. Kepler's laws of planetary motion, specifically his second law, provides that a body in an elliptical orbit around a star will sweep out equal areas in equal intervals of time.

As the Earth orbits near the Sun, the velocity of the orbit is greater. When the Earth is orbiting further away from the Sun, the orbital velocity is less. There are two key points here in the orbital path, perihelion and aphelion. Perihelion is the shortest distance in the orbital path between the star and the orbiting body, while the aphelion is the furthest distance. The orbital velocity at aphelion is slower than at perihelion. I should point out that the Earth's orbit around the sun is only slightly elliptical, and the image drawn in Fig. 1 is greatly exaggerated for illustrative purposes. The Earth orbit eccentricity in reality is only about 0.02, very nearly a circle.

Figure 1. An example of Kepler’s second law, equal areas in equal time.

Changes in the velocity of the Earth in its orbit, slight inclination of the Earth's orbital path and the precession about the Earth's axis will cause some minor but measurable differences in the observed local noon. This is the reason for the approximate +/- 15 minute difference in the time indicated by a properly adjusted sundial.

Figure 2. A plot of the EOT (Equation Of Time) due to the combination of the Earth's tilt and orbital velocity change at aphelion and perihelion.

I used Excel to create the plot in Fig. 2 of the equation of time to illustrate graphically why sundials indicate a difference in local noon throughout the year. As you look at Fig. 2, you will notice three distinct plots. The yellow dashed line is the error introduced by the tilt of the Earth on its axis. The red dashed line is the error introduced by the difference in orbital speed of the elliptical orbit of the Earth around the Sun – faster in the winter and slower in the summer. The combination of these two plots generates a third plot, the equation of time. The black plot is an indication of how much sundial time (apparent time) will differ from that of a clock.

The EOT corrected time is known as mean solar time and is usually applied to the sundial as an inscription or an attached plate that lists corrections in minutes throughout the year. Sometimes there is just a listing of numbers, and sometimes the graphical plot is also included. An example of such an EOT plaque for sundials is shown by the EOT chart image in Fig. 3..

Figure 3. A common example of an EOT plaque found on many sundials.

Since the sundial apparent time will be slower by 14 minutes in February, then the EOT chart indicates that we must add 14 minutes to correct for the error. If we had a custom made sundial that was specific to our latitude and longitude, then when our sundial indicates noon on 4 February, our clock would indicate fourteen minutes past noon. Therefore, by adding fourteen minutes to our sundial's apparent time we find the corrected time is 12:14 P.M. If we look at a sundial that was not corrected for its longitude, then if we correct the sundial time for the difference by using the EOT, we would be dismayed to find that it would probably not agree with our watch! Unless we correct for the longitude, our sundial can never give a direct reading of clock time.

I should point out that the EOT does not remain exactly the same from year to year. Each year the amounts of deviation will be slightly offset from that of the previous year.

Precession of the Equinoxes

The Earth not only rotates around its spin axis, but it also precesses . If you examine Fig. 4, you will notice that in addition to the rotational spin of the Earth, there is also an additional rotation. Just as a spinning top wobbles in a slow circular fashion as it spins, the Earth also wobbles. This is called gyroscopic precession. The dashed red line indicates the gyroscopic precession or wobble of the Earth. Although I have drawn a nice smooth circle, in reality this “circle” is more of a wobble motion, then a smooth circular motion.

Figure 4. Diagram of the Earth's axis in relation to the Earth's precessional motion.

The period of procession is approximately 25,800 years. This means that Polaris, the Northern Pole star, will no longer be aligned to the Earth's North Pole in several thousand years (see Fig. 5). In fact, several stars will take the place of Polaris as the Northern Pole star over the precession cycle. After one cycle of precession (25,800 years), we will return to alignment with Polaris as the Northern Pole star. If you have played with a spinning top, you may recall that the top wobbles in a slow circular fashion as it spins. This is called gyroscopic precession. As in the case of the top, the Earth also precesses. There is also a slight wobble, technically called nutation, caused by the Earth's varying gravitational attraction of the moon and sun on the Earth's equatorial bulge. The nutation period is only 18.6 years as opposed to the nearly 26,000 years for precession.

Figure 5. Diagram of the Earth's axis in relation to the ecliptic.

Figure 5 shows that the equinoxes do not occur on set dates, but at points of intersection between the ecliptic and the celestial equator. Precession of the equinoxes is the westward motion of the points of intersection along the ecliptic. The reason for this precession of the equinoxes is due to the Earth's own precession. Referring back to Fig. 4, as the Earth slowly precesses the points of intersection for the Equinoxes will slowly advance. The ecliptic lies in a plane of Earth's orbit around the Sun. The Celestial Equator lies in a plane of the Earth's equator. The intersection of these two planes determines the equinoxes. By looking at Fig. 5, one can more easily see how the precession of the Earth would cause these points of intersection to slowly move throughout the 25,800 year precession cycle.

The Analemma

Now we come to the analemma, the figure eight pattern printed on most globes like the one shown in Fig. 6. Technically, the analemma is a plot of the EOT on one axis and the Suns declination on the other. If you search the web for sundials, you will see various depictions and animations of the analemma.

Figure 6. Author Mike Dziekan holding a globe showing the analemma.

Figure 7 is a plot of the position of the Sun for one year with one point for each hour. This plot shows how the analemma varies over time in a pattern that would otherwise be hidden in the data. I color-coded the analemma in blue for the first half of the year and red for the second half. The plot also indicates the sun's position for the summer solstice, vernal and autumnal equinoxes and the winter solstice. You can also see from this plot of the sun's position that the sun actually moves east and west throughout the year. After the summer solstice, the sun rises more west each morning until the winter solstice, when it sets more eastward each evening. The two equinoxes' overlap each other in the plot and are difficult to distinguish.

Figure 7. This is a plot of the daily motion of the sun throughout the year. The analemmas are produced by plotting only the hourly position of the sun, while the solstices and equinoxes are plotted as continuous lines. The slight flattening of the top is produced because the arc is plotted as a flat structure instead of a curved one.

An actual photograph of the analemma made by photographer Anthony Ayiomamitis is shown in Fig. 8. Anthony photographed the sun about once a week throughout the year at the same time of day. Anthony's remarkable photograph shows that the analemma is literally written in the sky!

Figure 8. This remarkable photograph of the analemma was made by photographer Anthony Ayiomamitis. The image was made from his home in Athens by taking 47 separate images at the same time of day during one year (30 March 2003 to 24 March 2004 at 12:00:00 UT + 2). The ruins in the foreground are the Temple of Zeus. For more details about this image, see "Analemma photographed over the Temple of Zeus in Athens, Greece" by Anthony Ayiomamitis in The Citizen Scientist (30 June 2006). Also see Anthony's website. Copyright by Anthony Ayiomamitis. All rights reserved. Used with permission.

Now that you know that the sun will vary its apparent position in the sky throughout the year, we can address some typical Hollywood mistakes. If you have ever watched the movie “National Treasure”, then you may have noticed the scene where they had to wait for a specific time when the shadow of the top of the old Liberty bell tower in Philadelphia would cast a shadow on a specific brick in the wall. I am not saying that this would not work, but it would only work if you knew what specific day it was intended. Basically, the shadow of the old bell tower steeple would sketch out a shadow pattern of an analemma throughout the year, and, depending upon the distance between the two, it could be off by a meter (several feet)! It is also a good thing that it just happened to be sunny that day, but then again, it's Hollywood!

North American Sundial Society and the TV show NUMB3RS

No, it's not a typo. There is a television show called NUMB3RS, and it is about a brilliant mathematician who uses math to help solve cases for the FBI. A while back, the producers of the show contacted Fred Sawyer, president of the North American Sundial Society . They wanted to have an episode where several photographs were found in a digital camera at a crime scene. As Fred Sawyer explained, "They wanted a poor man's GPS.” The pictures indicated the exact time they were taken, and they depicted objects that had standard heights, including a basketball hoop that cast a shadow on a flat surface covered with bricks.

The bricks were all of uniform size, and the basketball hoop was of standard height. Because the surface was flat, the length of the shadow could be accurately measured from the photographs. Because each photograph was time stamped and several photos were taken at different times, the location of the photographs could be reasonably pinpointed. This is because the Sun can cast a shadow of a specific length and angle only at an exact time for any specific location. If you are a member of NASS, then you have access to the society's Compendium (volume 12, December 2005) in which Fred describes the entire correspondence related to this problem. It's a fascinating story.

If you are wondering why I related this story, it is all related to sundials. If you can determine the exact time from a sundial, then the opposite is true. If you know the exact height of an object, and the exact length and angle of its shadow, in addition to knowing the exact time and the Sun's position, then you can determine a unique location on Earth. I should mention that the only payment that Fred wanted for his assistance with the TV program was that two of the main characters would state that they are card carrying members of the North American Sundial Society. The small sundial that Charlie occasionally picked up and held during the program was donated to the show by Fred Sawyer.

In Part 3 Mike Dziekan will conclude his journey through the world of time with more about time and calendars. Editor.