Analemma
In astronomy, an analemma is a diagram showing the position of the Sun in the sky as seen from a fixed location on Earth at the same mean solar time over the course of a year. The change of position is a result of the shifting of the angle in the sky of the path that the Sun takes in respect to the stars. The diagram resembles a figure eight. Globes of the Earth often display an analemma as a two-dimensional figure of equation of time vs. declination of the Sun.
The north–south component of the analemma results from the change in the Sun's declination due to the tilt of Earth's axis of rotation as it orbits around the Sun. The east–west component results from the nonuniform rate of change of the Sun's right ascension, governed by the combined effects of Earth's axial tilt and its orbital eccentricity.
An analemma can be photographed by keeping a camera at a fixed location and orientation and taking multiple exposures throughout the year, always at the same time of day.
Although the term analemma usually refers to Earth's solar analemma, it can be applied to other celestial bodies as well.
Description
An analemma can be traced by plotting the position of the Sun as viewed from a fixed position on Earth at the same clock time every day for an entire year, or by plotting a graph of the Sun's declination against the equation of time. The resulting curve resembles a long, slender figure-eight with one lobe much larger than the other. This curve is commonly printed on terrestrial globes, usually in the eastern Pacific Ocean, the only large tropical region with very little land. It is possible, though challenging, to photograph the analemma, by leaving the camera in a fixed position for an entire year and snapping images on 24-hour intervals ; see section below.The long axis of the figure—the line segment joining the northernmost point on the analemma to the southernmost—is bisected by the celestial equator, to which it is approximately perpendicular, and has a "length" of twice the obliquity of the ecliptic, i.e., about 47°. The component along this axis of the Sun's apparent motion is a result of the familiar seasonal variation of the declination of the Sun through the year. The "width" of the figure is due to the equation of time, and its angular extent is the difference between the greatest positive and negative deviations of local solar time from local mean time when this time-difference is related to angle at the rate of 15° per hour, i.e., 360° in 24 h. This width of the analemma is approximately 7.7°, so the length of the figure is more than six times its width. The difference in size of the lobes of the figure-eight form arises mainly from the fact that the perihelion and aphelion occur far from the equinoxes. Instead, they occur a couple of weeks after the solstices, which in turn causes a slight tilt of the figure eight and its minor lateral asymmetry.
There are three parameters that affect the size and shape of the analemma—obliquity, eccentricity, and the angle between the northward equinox and the periapsis. Viewed from an object with a perfectly circular orbit and no axial tilt, the Sun would always appear at the same point in the sky at the same time of day throughout the year and the analemma would be a dot. For an object with a circular orbit but significant axial tilt, the analemma would be a figure of eight with northern and southern lobes equal in size. For an object with an eccentric orbit but no axial tilt, the analemma would be a straight east–west line along the celestial equator.
The north–south component of the analemma shows the Sun's declination, its latitude on the celestial sphere, or the latitude on the Earth at which the Sun is directly overhead. The east–west component shows the equation of time, or the difference between solar time and local mean time. This can be interpreted as how much "ahead" or "behind" the Sun is compared to clock time. It also shows how far west or east the Sun is, compared with its mean position. The analemma can be considered as a graph in which the Sun's declination and the equation of time are plotted against each other. In many diagrams of the analemma, a third dimension, that of time, is also included, shown by marks that represent the position of the Sun at various, fairly closely spaced, dates throughout the year.
In diagrams, the analemma is drawn as it would be seen in the sky by an observer looking upward. If north is at the top, west is to the right. This corresponds with the sign of the equation of time, which is positive in the westward direction. The further west the Sun is, compared with its mean position, the more "fast" a sundial is, compared with a clock. If the analemma is a graph with positive declination plotted upward, positive equation of time is plotted to the right. This is the conventional orientation for graphs. When the analemma is marked on a geographical globe, west in the analemma is to the right, while the geographical features on the globe are shown with west to the left. To avoid this confusion, it has been suggested that analemmas on globes should be printed with west to the left, but this is not done, at least, not frequently. In practice, the analemma is so nearly symmetrical that the shapes of the mirror images are not easily distinguished, but if date markings are present, they go in opposite directions. The Sun moves eastward on the analemma near the solstices. This can be used to tell which way the analemma is printed. See the image above, .
An analemma that includes an image of a solar eclipse has been called a tutulemma, a term coined by photographers Cenk E. Tezel and Tunç Tezel based on the Turkish word for eclipse.
History
The book "Analemma" by Ptolemy deals with the means for plotting the celestial coordinates of the Sun or any other heavenly body for any geographical latitude at any time. The construction of sundials depends on such calculations.From this, analemma came to mean the graphical procedure of representing three-dimensional objects in two dimensions. In 1613, François d'Aguilon of Antwerp, began promoting orthographic projection as the name for this process. Today orthographic projection is what it is universally called.
In 1644, mathematician Jean-Louis Vaulezard described the first analemmatic sundial installed in France. The sundial is located at the church of Brou, and depicts the 8-shape analemma. Analemmas have been used in conjunction with sundials since the 18th century to convert between apparent and mean solar time.
Analemmas were often depicted on early globes. One such example is the terrestrial globe by George Woodward, made in 1846. In 1812, John Lathrop wrote:
As seen from Earth
Owing to the tilt of Earth's axis and the Earth's orbital eccentricity, the relative location of the Sun above the horizon is not constant from day to day when observed at the same clock time each day. If the time of observation is not 12:00 noon local mean time, then depending on one's geographical latitude, this loop will be inclined at different angles.The figure in this section is an example of an analemma as seen from the Earth's Northern Hemisphere. It is a plot of the position of the Sun at 12:00 noon at Royal Observatory, Greenwich, England during the year 2006. The horizontal axis is the azimuth angle in degrees. The vertical axis is the altitude in degrees above the horizon. The first day of each month is shown in black, and the solstices and equinoxes are shown in green. It can be seen that the equinoxes occur approximately at altitude, and the solstices occur approximately at altitudes where ε is the axial tilt of the Earth, 23.4°. The analemma is plotted with its width highly exaggerated, revealing a slight asymmetry.
The analemma is oriented with the smaller loop appearing north of the larger loop. At the North Pole, the analemma would be completely upright, and only the top half of it would be visible. Heading south, once south of the Arctic Circle, the entire analemma would become visible. If seen at noon, it continues to be upright, and rises higher from the horizon as the viewer moves south. At the equator it is directly overhead. Further south it moves toward the northern horizon, and is then seen with the larger loop at the top. If viewed at the analemma in the early morning or evening, it would start to tilt to one side as the viewer moves southward from the North Pole. At the equator the analemma would be completely horizontal. As the viewer continued south it would rotate so that the small loop was beneath the large loop in the sky. Crossing the Antarctic Circle the analemma, now nearly completely inverted, would start to disappear, until only 50%, part of the larger loop, was visible from the South Pole.
See equation of time for a more detailed description of the east–west characteristics of the analemma.
Photography
The first successful analemma photograph ever made was created in 1978–79 by photographer Dennis di Cicco over Watertown, Massachusetts. Without moving his camera, he made 44 exposures on a single frame of film, all taken at the same time of day at least a week apart. A foreground image and three long-exposure images were also included in the same frame, bringing the total number of exposures to 48.Calculated analemmas
While photographing analemmas may present technical and practical challenges, they can be calculated conveniently and presented in 3D plots for any given location on the surface of the Earth.The idea is based on the unit vector with its origin fixed at a chosen point on the surface of the Earth and its direction pointing to the center of the Sun all the time. If the position of the Sun is calculated, the solar zenith angle and solar azimuth angle at one-hour steps for an entire year, the head of the unit vector traces out 24 analemmas on the unit sphere centered on the chosen point. This unit sphere is equivalent to the celestial sphere. The figure on the right is the "wreath of analemmas" calculated for the geographic center of the contiguous United States.
As often seen on a globe, the analemma is also often plotted as a two-dimensional figure of equation of time vs. declination of the Sun. The adjacent figure is calculated using the algorithm presented in the reference that uses the formulas given in The Astronomical Almanac for the Year 2019.