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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Equation of time | 4/9 | https://en.wikipedia.org/wiki/Equation_of_time | reference | science, encyclopedia | 2026-05-05T11:12:39.793727+00:00 | kb-cron |
== Secular effects == (Note: the word “secular” in this context means “pertaining to the passage of time over long periods”, not “lacking religion”. See, for example, “secular trend”, which also uses the word in this sense.) The two above mentioned factors have different wavelengths, amplitudes and phases, so their combined contribution is an irregular wave. At epoch 2000 these are the values (in minutes and seconds with UT dates):
On shorter timescales (thousands of years) the shifts in the dates of equinox and perihelion will be more important. The former is caused by precession, and shifts the equinox backwards compared to the stars. But it can be ignored in the current discussion as our Gregorian calendar is constructed in such a way as to keep the vernal equinox date at 20 March (at least at sufficient accuracy for our aim here). The shift of the perihelion is forwards, about 1.7 days every century. In 1246 the perihelion occurred on 22 December, the day of the solstice, so the two contributing waves had common zero points and the equation of time curve was symmetrical: in Astronomical Algorithms Meeus gives February and November extrema of 15 m 39 s and May and July ones of 4 m 58 s. Before then the February minimum was larger than the November maximum, and the May maximum larger than the July minimum. In fact, in years before −1900 (1901 BCE) the May maximum was larger than the November maximum. In the year −2000 (2001 BCE) the May maximum was +12 minutes and a couple seconds while the November maximum was just less than 10 minutes. The secular change is evident when one compares a current graph of the equation of time (see below) with one from 2000 years ago, e.g., one constructed from the data of Ptolemy.
== Practical use == If the gnomon (the shadow-casting object) is not an edge but a point (e.g., a hole in a plate), the shadow (or spot of light) will trace out a curve during the course of a day. If the shadow is cast on a plane surface, this curve will be a conic section (usually a hyperbola), since the circle of the Sun's motion together with the gnomon point define a cone. At the spring and autumnal equinoxes, the cone degenerates into a plane and the hyperbola into a line. With a different hyperbola for each day, hour marks can be put on each hyperbola which include any necessary corrections. Unfortunately, each hyperbola corresponds to two different days, one in each half of the year, and these two days will require different corrections. A convenient compromise is to draw the line for the "mean time" and add a curve showing the exact position of the shadow points at noon during the course of the year. This curve will take the form of a figure eight and is known as an analemma. By comparing the analemma to the mean noon line, the amount of correction to be applied generally on that day can be determined. The equation of time is used not only in connection with sundials and similar devices, but also for many applications of solar energy. Machines such as solar trackers and heliostats have to move in ways that are influenced by the equation of time. Civil time is the local mean time for a meridian that often passes near the center of the time zone, and may possibly be further altered by daylight saving time. When the apparent solar time that corresponds to a given civil time is to be found, the difference in longitude between the site of interest and the time zone meridian, daylight saving time, and the equation of time must all be considered.
== Calculation == The equation of time is obtained from a published table, or a graph. For dates in the past such tables are produced from historical measurements, or by calculation; for future dates, of course, tables can only be calculated. In devices such as computer-controlled heliostats the computer is often programmed to calculate the equation of time. The calculation can be numerical or analytical. The former are based on numerical integration of the differential equations of motion, including all significant gravitational and relativistic effects. The results are accurate to better than 1 second and are the basis for modern almanac data. The latter are based on a solution that includes only the gravitational interaction between the Sun and Earth, simpler than but not as accurate as the former. Its accuracy can be improved by including small corrections. The following discussion describes a reasonably accurate (agreeing with almanac data to within 3 seconds over a wide range of years) algorithm for the equation of time that is well known to astronomers. It also shows how to obtain a simple approximate formula (accurate to within 1 minute over a large time interval), that can be easily evaluated with a calculator and provides the simple explanation of the phenomenon that was used previously in this article.
=== Mathematical description === The precise definition of the equation of time is:
E
O
T
=
G
H
A
−
G
M
H
A
{\displaystyle \mathrm {EOT} =\mathrm {GHA} -\mathrm {GMHA} }
The quantities occurring in this equation are: