analemma

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Diagram of an analemma looking east in the northern hemisphere. The dates of the Sun's position are shown. This analemma is calculated, not photographed.

In astronomy, an analemma ( /ˌænəˈlɛmə/; from Greek ἀνάλημμα "pedestal of a sundial") is a curve representing the angular offset of a celestial body (usually the Sun) from its mean position on the celestial sphere as viewed from another celestial body relative to the viewing body's celestial equator. The term is commonly applied nowadays to the figure traced in the sky when the position of the Sun is plotted at the same time each day over a calendar year from a particular location on Earth.

Description

Analemma printed on a globe. The Globe Museum, Vienna

Since the Earth's mean solar day is almost exactly 24 hours, 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. The resulting curve resembles a lemniscate of Bernoulli. This curve is commonly printed on 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.

There are three parameters that affect the size and shape of the analemma: obliquity, eccentricity, and the angle between the apse line and the line of solstices. 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 is the declination, or the latitude at which the Sun is directly overhead. The east-west component is the equation of time, or the difference between solar time and local mean time. This can be interpreted as how "fast" or "slow" the Sun is compared to clock time.

An analemma that includes an image of a total solar eclipse is called a tutulemma — a portmanteau coined by photographers based on the Turkish word for eclipse.^[1]

Earth's analemma

Analemma for Earth as seen from the northern hemisphere with altitude and azimuth to the same scale.

Analemma for Earth plotted as seen from the Royal Observatory, Greenwich (latitude 51.4791° north, longitude 0°).

Owing to the tilt of Earth's axis (23.439°) and its elliptical orbit around the Sun, 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. Depending on one's geographical latitude, this loop will be inclined at different angles.

The figure on the left is an example of an Earth analemma as seen from the northern hemisphere. It is a plot of the position of the Sun at 12:00 noon at Royal Observatory, Greenwich, England (latitude 51.4791°N, longitude 0°) during the year 2006. The horizontal axis is the azimuth angle in degrees (180° is facing south). 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 at altitude φ = 90° − 51.4791° = 38.5209°, and the solstices occur at altitudes φ ± ε where ε is the axial tilt of the earth, 23.439°. The analemma is plotted with its width highly exaggerated, which permits noting that it is slightly asymmetrical (due to the two-week misalignment between the apsides of the Earth's orbit and its solstices).

The analemma is oriented with the smaller loop appearing north of the larger loop.

See equation of time for a more detailed description of the east-west characteristics of the analemma.

Photography of the Earth's analemma

The first analemma photograph ever made was created in 1978 /1979 by photographer Dennis di Cicco over New England in the USA. He made 44 exposures on a single frame of film all taken from the same location with a solar filter. Since then most photographers take a single foreground image and superimpose multiple sun images on top of it.^[2]

Other analemmas

Analemma for Mars

On Earth, the analemma appears as a figure eight, but on other solar system bodies it may be very different^[3] because of the interplay between the tilt of each body's axis and the elliptical shape of its orbit.

In the following list, "day" and "year" refer to the synodic day and sidereal year of the particular body:

• Mercury: Because orbital resonance makes the day exactly two years long, the method of plotting the Sun's position at the same time each day would yield only a single point. However, the equation of time can still be calculated for any time of the year, so an analemma can be graphed with this information. The resulting curve is a nearly straight east-west line.
• Venus: There are slightly less than two days per year, so it would take several years to accumulate a complete analemma by the usual method. The resulting curve is an ellipse.
• Mars: teardrop
• Jupiter: ellipse
• Saturn: technically a figure 8, but the northern loop is so small that it more closely resembles a teardrop
• Uranus: figure 8
• Neptune: figure 8
• Pluto: figure 8

See also

• Analemmatic sundial
• Armillary sphere
• De architectura
• On the Dioptra

References

External links

Wikimedia Commons has media related to: Analemmatic sundials

• Analemma Series from Sunrise to Sunset
• Earth Science Photo of the Day (2005-01-22)
• The Equation of Time and the Analemma — by Kieron Taylor
• The Use of the Analemma — from an inset from Bowles's New and Accurate Map of the World (1780)
• Figure-Eight in the Sky — contains link to a C program using a more accurate formula than most (particularly at high inclinations and eccentricities)
• The Analemma for Latitudinally-Challenged People — explains rising and setting analemmas as viewed from different latitudes and provides more depth on analemma.
• 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 lots of reference charts.
• Analemmas — by Stephen Wolfram based on a program by Michael Trott, Wolfram Demonstrations Project.
• Analemma in Verse by Tad Dunne

• 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