Analemma

thumb|Afternoon analemma photo taken in 1998–99 in [[Murray Hill, New Jersey|Murray Hill, New Jersey, U.S., by Jack Fishburn. The Bell Laboratories building is in the foreground.]]

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 ecliptic). The diagram resembles a figure eight. Globes of the Earth often display an analemma as a two-dimensional figure of equation of time ("sun fast") 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 (disregarding daylight saving time and in as little cloud cover as possible).

Although the term analemma usually refers to Earth's solar analemma, it can be applied to other celestial bodies as well.

Description

thumb|Analemma on Earth. The red line shows the position on Earth where the Sun is directly overhead, taken at 24 hours intervals over the course of one year.

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 (or some multiple thereof); 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.

History

The book "Analemma" (Greek: Περὶ ἀναλήμματος) 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 (and probably in the world). The sundial is located at the church of Brou, and depicts the 8-shape analemma. Analemmas (in the modern sense of the term) have been used in conjunction with sundials since the 18th century to convert between apparent and mean solar time.

thumb|Analemma with date marks, printed on a globe, [[Globe Museum, Vienna, Austria]]

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

thumb|upright=1.5|Analemma plotted as seen at noon GMT from the [[Royal Observatory, Greenwich (latitude 51.48° north, longitude 0.0015° west)]]

thumb|The analemma cutout of this sundial causes the edge of the shadow to fall on the true time.

Owing to the tilt of Earth's axis (23.439°) 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.

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

[[File:Wreath_of_Analemmas.png|thumb|[https://play.google.com/store/apps/details?id=com.HWorks.AnalemmaMechanics]"Wreath of Analemmas". Analemmas calculated at 1-hour apart from each other for the geographic center of the contiguous United States. The gray part indicates it is nighttime.]]

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.

thumb|left| Analemma: Equation of time vs. declination of the Sun. Calculated for the year 2020 using the formulas from The [[Astronomical Almanac for the Year 2019.]]

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 ("Analemma: Equation of time...") is calculated using the algorithm presented in the reference

Times of sunrise and sunset

A similar geometrical method, based on the analemma, can be used to find the times of sunrise and sunset at any place on Earth (except within or near the Arctic Circle or Antarctic Circle), on any date.

The origin of the analemma, where the solar declination and the equation of time are both zero, rises and sets at 6 a.m. and 6 p.m. local mean time on every day of the year, irrespective of the observer's latitude. (This estimation does not take account of atmospheric refraction.) If the analemma is drawn in a diagram, tilted at the appropriate angle for an observer's latitude (as described above), and if a horizontal line is drawn to pass through the position of the Sun on the analemma on any given date (interpolating between the date markings as necessary), then at sunrise this line represents the horizon.

The origin appears to move along the celestial equator at a speed of 15° per hour, the speed of the Earth's rotation. The distance along the celestial equator from the point where it intersects the horizon to the position of the origin of the analemma at sunrise is the distance the origin moves between 6 a.m. and the time of sunrise on the given date. Measuring the length of this equatorial segment therefore gives the difference between 6 a.m. and the time of sunrise.

The measurement should, of course, be done on the diagram, but it should be expressed in terms of the angle that would be subtended at an observer on the ground by the corresponding distance in the analemma in the sky. It can be useful to compare it with the length of the analemma, which subtends 47°. Thus, for example, if the length of the equatorial segment on the diagram is 0.4 times the length of the analemma on the diagram, then the segment in the celestial analemma would subtend 0.4 × 47° = 18.8° at the observer on the ground. The angle, in degrees, should be divided by 15 to get the time difference in hours between sunrise and 6 a.m. The sign of the difference is clear from the diagram. If the horizon line at sunrise passes above the origin of the analemma, the Sun rises before 6 a.m., and vice versa.

The same technique can be used, mutatis mutandis, to estimate the time of sunset. The estimated times are in local mean time. Corrections must be applied to convert them to standard time or daylight saving time. These corrections will include a term that involves the observer's longitude, so both the latitude and longitude affect the final result.

Azimuths of sunrise and sunset

The azimuths (true compass bearings) of the points on the horizon where the Sun rises and sets can be easily estimated, using the same diagram as is used to find the times of sunrise and sunset, as described above.

The point where the horizon intersects the celestial equator represents due east or west. The point where the Sun is at sunrise or sunset represents the direction of sunrise or sunset. Simply measuring the distance along the horizon between these points, in angular terms (comparing it with the length of the analemma, as described above), gives the angle between due east or west and the direction of sunrise or sunset. Whether the sunrise or sunset is north or south of due east or west is clear from the diagram. The larger loop of the analemma is at its southern end.

Seen from other planets

thumb|An analemma as viewed from [[Mars ]]

On Earth, the analemma appears as a figure-eight, but on other Solar System bodies, it may be very different due to the interplay between the body's axial tilt, the non-circularity of its orbit (as characterized by its orbital eccentricity), and the angle between the northward equinox and the periapsis. The tilt tends to make the analemma into a figure-eight, since it causes the actual position of the Sun to run ahead of the mean solar time twice during the year (and therefore also to run behind solar time twice). The non-circularity of the orbit tends to make the analemma a figure-zero, with the position of the Sun running ahead of mean solar time once during the year due to Kepler's second law of planetary motion.

Notes

References

External links

Analemma Series from Sunrise to Sunset
Earth Science Photo of the Day (2005-01-22)
The Use of the Analemma from an inset from Bowles's New and Accurate Map of the World (1780)
Analemma.com—Dedicated to the analemma.
Calculate and Chart the Analemma—A web site offered by a Fairfax County Public Schools planetarium that describes the analemma and also offers a downloadable spreadsheet that allows the user to experiment with analemmas of varying shapes.
Analemma Sundial Applet—includes many reference charts.
Analemmas by Stephen Wolfram based on a program by Michael Trott, Wolfram Demonstrations Project.
The Making of a Tutulemma by Tunç Tezel
Making of a Solargraphy Analemma by Maciej Zapiór and Łukasz Fajfrowski
Equation-of-Time.info—A multipage website with many illustrations and videos dedicated to the Equation of Time, its components, its history, how it can be displayed in tables, curves, analemmas, etc., its use to correct sundials, astronomy, clocks, how it can be produced mechanically and much more : by Kevin Karney
Earth and Sun—An interactive blog post explaining the phenomenon
Astronomy Picture of the Day
2002-07-09: Analemma
2003-03-20: Sunrise Analemma
2004-06-21: Analemma over Ancient Nemea
2005-07-13: Analemma of the Moon
2006-12-23: Analemma over the Temple of Olympian Zeus
2006-12-30: Martian Analemma at Sagan Memorial Station (simulated)
2007-06-17: Analemma over Ukraine
2007-12-04: Analemma over New Jersey (film)
2008-12-21: Analemma over the Porch of Maidens
2009-12-20: Tutulemma: Solar Eclipse Analemma
2010-12-31: Analemma 2010
2012-09-20: Sunrise Analemma (with a little extra)
2013-10-14: High Noon Analemma Over Azerbaijan
2014-03-20: Solargraphy Analemma
2025-06-21:<includeonly></includeonly> Two Worlds, Two Analemmas