To the rhythm of the sun: Sundial

In accordance with the laws of Kepler’s celestial mechanics, the speed of the Earth’s orbit varies from 29.291 to 30.287 km / s during the year, and the apparent speed of the Sun in the sky changes with it. It “accelerates” when the Earth is at perihelion (the point closest to the Sun, January 3), and slows down when our planet passes aphelion (at the maximum distance from the Sun). The maximum difference between the length of daylight hours and the day calculated by UTC can reach 7.9 s. During the year, the error accumulates

The annual schedule of such an error is a sinusoid with an amplitude of 7.66 minutes and a period of a year with an initial phase of January 3 (light line).

At the equinox moments, the projection of the ecliptic arc onto the equator is smaller than the arc itself, and at the solstice moments it is larger, therefore, at different times of the year, the sundial will have an error due to the inclination of the earth's axis. The correction can be represented as a sinusoid with an amplitude of 9.8 minutes with a period of six months (dashed line).

A mechanical watch is considered to be a rather complicated device. Indeed, random, at first glance, a heap of springs and gears anyone will be confused. Another thing is a solar watch: a simple stick casting a shadow on a flat disk. And yet, like many mechanisms of antiquity, in fact, a sundial is much more complicated than a mechanical one. Indeed, in addition to the dial and the gnomon, astronomical objects, the Earth and the Sun serve as their integral parts, the movement of which relative to each other obeys much more complex laws than the oscillations of the pendulum, and is not amenable to adjustment. A sundial is relatively simple to manufacture, however, in order to accurately calculate its design, deep knowledge in astronomy and trigonometry is required.

The biblical verse, cited as an epigraph, refers to a sundial built in Jerusalem under King Ahaz in the 8th century BC. One of the first sundials found in the burial of Nouth (Ireland) dates back to 5000 BC. Obelisks of Ancient Egypt and Babylon were used to determine the time of day by the length of the shadow. Perfection of a sundial was carried out by the greatest philosophers and mathematicians of Ancient Greece - Anaximander, Anaximenes, Eudoxus, Aristarchus. The ancient peoples did not divide the day into 24 equal parts. At 12 o’clock, they divided daylight hours, from dawn to sunset, so at different times of the year the length of the hour was different. In the ancient sundial - Skafis - time was determined by the length of the shadow cast by the gnomon on the surface of a spherical notch marked by complex curves. With the introduction of equal hours of day and night, time began to be determined not by the length of the shadow, but by its direction.

Earth model

The daily course of time is determined by the rotation of the Earth around its axis. To understand the basic principle of the work of a sundial, we will throw firewood into the bonfire of Giordano Bruno and imagine that objects of the celestial sphere revolve around our motionless planet. Looking at the night sky, we will see that over time all the stars change their position, moving in a circle around the North Star. Only she almost all night does not change her position. The fact is that at present the position of the North Star practically coincides with the North Pole of the world - a point on the celestial sphere into which the axis of rotation of the Earth is projected. Incidentally, over time, the position of the North Pole in the celestial sphere changes. So, 5000 years ago, the closest luminary to him was Tuban, the star of the constellation Dragon.

The gnomon of the simplest sundial should be directed to the North Pole of the world, in other words, be parallel to the earth's axis. The watch dial is perpendicular to the gnomon. In this case, the plane in which the Sun travels across the sky around our planet will also be perpendicular to the gnomon and parallel to the dial. And this means that the shadow from the gnomon will travel on the dial evenly during daylight hours, passing 15 degrees every hour. A sundial of this design is called equatorial, because its dial is parallel to the equator.

In order to correctly set the equatorial clock, it is necessary to take an even horizontal platform (determined using a level or plumb) as a guide and set the dial in such a plane that its angle to the horizontal equals the geographical latitude of the place. It forms a parallel to the equator if the gnomon of the clock is directed to the true north. For example, the giant stone gnomon of the sundial of the Jantar-Mantar observatory (Jaipur, India, the 18th century), rising 27 m above the ground, forms an angle of 26 ° 55 'with the earth's surface. The first sundials in Rome, brought by consul Valery Messala from Sicily, showed the time incorrectly, as they were calculated for a different latitude.

The North can be found at night by the North Star. Be careful: the direction of the gnomon cannot be determined by the compass, because the position of the North magnetic pole of the Earth and its geographical pole do not coincide. In addition, there are many magnetic anomalies on Earth: due to the metals contained in the rock in some parts of the planet, the compass error exceeds 15 degrees.

Equatorial watches have a characteristic feature. In the summer (between the days of the spring and autumn equinox) the gnomon casts a shadow on the upper side of the dial, and in the winter - on the lower. By summer, the sun rises higher and higher, and by the length of the shadow you can judge the time of year or even month. Therefore, it is easy to supplement the equatorial clock with a calendar by drawing concentric circles corresponding to the months (six on one side and six on the other) on the dial, and placing a ball or hole on the gnomon that can project a point onto the dial. Unlike a gnomon casting a shadow in the form of a line, any device projecting a point onto a dial is called a nodus.

Equatorial watches do not require complex calculations; positioning is more important for them. Of course, the shadows cast by a gnomon of any shape on a vertical, horizontal, spherical, any dial can also be associated with the time of day. Calculation of complex constructions of a sundial is first of all a trigonometric task.

Keeping up with the times

A simple sundial, correctly set at a certain point on the globe, shows the local solar time characteristic of a given geographical location and time of year. Today we all live according to Universal Coordinated Time (UTC), which is significantly different from local solar. The first difference is that the globe is divided into 24 time zones, within each of which the same time is taken for any longitude. Local solar time, on the contrary, has its own longitude. For example, the sundial of a resident of St. Petersburg will show noon later than the sundial of a resident of Moscow, while the wrist watches of both citizens are absolutely synchronized. So, to make the sundial show the “correct” time, you need to at least “bring” them in accordance with the time zone, shifting the time lines. The same shift should be made if daylight saving time is in effect in the region. Some sundials have two digital scales - for winter and summer time.

Hours, minutes, and seconds of standard time flow uniformly throughout the year, which cannot be said about the movement of the Sun across the horizon. The Earth’s orbit has the shape of an ellipse, in one of the foci of which the Sun is located. In accordance with the laws of celestial mechanics of Kepler, the speed of the Earth in orbit varies from 29.291 to 30.287 km / s during the year, and with it the apparent speed of the Sun in the sky also changes. It “accelerates” when the Earth is at perihelion (the point closest to the Sun, January 3), and slows down when our planet passes aphelion (at the maximum distance from the Sun). The maximum difference between the length of daylight hours and the day calculated by UTC can reach 7.9 s. During the year, the error accumulates. The annual schedule of such an error is a sinusoid with an amplitude of 7.66 minutes and a period of a year with an initial phase of January 3.

But that's not all. The influence of the annual motion of the Sun (that is, the motion of the Earth in its solar orbit) on the daily run of the star in the sky changes over time due to the inclination of the earth's axis by an angle of about 23.5 degrees. The annual movement of the Sun is most noticeably reflected in the diurnal, when the line of intersection of the planes of the equator and the ecliptic is directed tangentially to the Earth's orbit. Figuratively speaking, at this moment both the Earth observer and the Earth itself move relative to the Sun in approximately the same direction. This occurs during the summer or winter solstice. On the days of the equinox, on the contrary, the annual and daily movements of the Sun are directed at an angle to each other, so their mutual influence is minimal. In scientific terms, at the equinoxes, the projection of the ecliptic arc onto the equator is smaller than the arc itself, and at the solstice moments it is larger, therefore, at different times of the year, the sundial will have an error due to the inclination of the earth's axis. The correction can be represented as a sinusoid with an amplitude of 9.8 minutes with a period of six months.

The sum of annual deviations of solar time from standard time is expressed in the equation of time. It is customary to show it in the form of a graph of the dependence of the error on the calendar day. For example, according to the equation of time, we see that on February 12, the sundial is 14 minutes behind the wristwatch, and on November 3, they rush for 16.5 minutes.

One of the graphic expressions of the equation of time is the analemma, a line connecting all the positions of the Sun in the sky on different days of the year, but at one time of the day. The analemma shows not only the horizontal displacement of the Sun, expressing the change in its speed along the horizon, but also the vertical movement. Indeed, due to the inclination of the earth's axis in the summer, the Sun in the sky climbs much higher than in winter. There is an obvious correlation between these two biases. It is she who allows you to integrate the analemma in the design of the sundial in order to find out the exact standard time from it.

Formulas - for beauty

The simplest example of an analemic sundial is a spherical equatorial watch with a gnomon made in the form of the analemma itself. For example, in winter, the Sun rises low and casts a shadow on the dial from the thick part of the analemma. Its left edge shows the standard time with the put backward time from the sun. In the summer, when the Sun rises to the very top, the narrow part of the analemma works, casting a narrower shadow. An analemma can be expressed in the form of wavy time lines repeating the graph of the equation of time (for example, for a polar sundial), or in the form of a table on the dial. In almost all cases, the principle of correction is that the length of the shadow indicates the calendar day, and the direction indicates the time of day. The user can only compare these two values.

A gnomon casting a shadow on the dial is far from the only constructive solution for a sundial. The role of the gnomon can be played by a spherical mirror (something like a “crystal ball” in a disco), which at a certain point in time discards sunbeams into the corresponding parts of the dial. There are many unusual watches. A vivid example: a watch with a dial on a window in a room, which on a sunny day casts a shadow on the floor with the desired time value. At the foot of the 101-story Taipei 101 skyscraper in Taiwan, there is a circular park. Its paths and trees make up the dial of the sundial, which is clearly visible from the windows of a skyscraper. The gnomon of the clock is the building itself. In the sundial park of the Belgian city of Genk, you can find a digital sundial. Inside this complex device, the sundial undergoes numerous refractions and, having traveled through a system of mirrors, these or other points of the screen are highlighted. On a black screen, beautiful white numbers show the time in hours and minutes. At the end of the 18th - the beginning of the 19th centuries, sundials with midday battle came across in European parks. More precisely, not with battle, but with deafening firing. The watch was arranged in such a way that at noon the sun's rays fell on the lens, under which the blank cartridge was placed. The lens strengthened by the sun set fire to gunpowder, and a deafening shot rang out. During the year, the position of the lens and the cartridge was repeatedly regulated so that the midday volley would be heard at 12:00 local time.

Modern gnomonics is an interesting scientific hobby. Lovers of sundials have developed hundreds of different designs, and their number is constantly growing. The design and construction of watches is to a large extent astronomy, mathematics and geometry, and to the least extent is hand-held. You can be convinced of this by collecting two rather complex and accurate models from blanks placed on the pages of the magazine.

This watch is designed for Moscow. To do the same in your region, you can use one of the special computer programs for calculating the anemalmic dials, which are easy to find on the Internet. The development of such programs is a separate branch of the gnomonics as a hobby. They help novice sundial lovers not to bother with math and focus on experiments.

So do not be alarmed by the formulas given in the descriptions for individual watch models - they are here just for beauty!

The article was published in the journal Popular Mechanics (No. 3, March 2009).

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