Analemma

Analemma was also a book by Ptolemy.

The analemma photographed, looking east in the northern hemisphere. The dates of the sun's position are shown.

Analemma (pronounced anna-lemma) is a term in astronomy used to describe the plot of the positions of the Sun on the celestial sphere at the same time of day (at approximately 24 hour intervals) and from the same location on Earth on successive days through the calendar year. This appears as a figure eight. If the course of the Earth around the sun were a precise circle and if the Earth's polar axis stood perpendicular to the plane of its orbit, 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. If the orbit were circular, but the polar axis tilted as it is, the northern and southern lobes of the figure of eight would be equal in size; if the polar axis were not tilted but the orbit had its current eccentricity then the analemma would be a straight east-to-west line. It is difficult but possible to construct an analemma by taking a photograph of the sun each day of the year at the same time and place and with a camera positioned in exactly the same way each time.

Explanation

The analemma for Earth

Plotting the analemma with the width exaggerated shows that it is slightly asymmetrical due to the misalignment of apsides and solstices.

Due to the earth's tilt on its axis (23.45°) 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 time on each day. Depending on one's geographical latitude, this loop will be inclined at different angles.

The vertical coordinate of a point corresponding to a date corresponds to the declination of the sun on that date, while the horizontal coordinate indicates whether the sun is "fast" or "slow" compared to mean time as measured by clocks.

The deviation between solar time and mean time is known as the equation of time and is due to the variation in the length of the synodic day, which is due to variation in the distance between the earth and the sun. An analemma is a visual representation or illustration of the equation of time. (See aphelion, perihelion.)

Plotting the analemma with the width exaggerated shows that it is slightly asymmetrical due to the misalignment of apsides and solstices.

Analemma on other planets

The analemma for Mars

On Earth, the analemma shows a "8"-shape figure, but on Mars it has a teardrop shape. Other planets also have analemmas with various shapes [link]. The variation is due to the interplay between the tilt of each planet's axis and the elliptical shape of the planetary orbit.

Seven of the solar system's planets (including Earth) exhibit the right orbital characteristics for the Sun to form an analemma when viewed from their surface (or atmospheric cloud tops) throughout the planet's solar year. The shapes differ however:

Ellipse - Jupiter
Figure 8 - Earth, Uranus, Neptune, Pluto
Teardrop - Mars, Saturn

The two exceptions are Mercury and Venus. In Mercury's case, the apparent solar day lasts exactly two revolutions, so its "analemma" degenerates to a single point. In Venus' case, the apparent solar day (if the Sun could be seen from the surface) lasts a little more than half a revolution: the two periods are not really commensurate with each other, although the ratio 77:40 comes close (i.e. Venus completes 40 revolutions in about 77 Venusian solar days). An analemma of sorts can be accumulated over the Venusian years, but it is no longer an analemma as per the usual definition.

External links

[Astronomy Picture of the Day, 2002 9 July: Analemma]
[Astronomy Picture of the Day, 2003 20 March: Sunrise Analemma]
[Astronomy Picture of the Day, 2004 21 June: Analemma over Ancient Nemea]
[Astronomy Picture of the Day, 2005 13 July: Analemma of the Moon]
[Analemma Series from Sunrise to Sunset]
[Analemma explanation by John Holtz]
[Earth Science Photo of the Day, Jan 22, 2005]
[The Equation of Time and the Analemma, by Kieron Taylor]
[An article by Brian Tung, 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. It provides more depth than most analemma sources. PDF format. 1,433 Kb.
[Analemma.com] is dedicated to the analemma.
[Calculate and Chart the Analemma] is 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.

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