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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Equation of time | 5/9 | https://en.wikipedia.org/wiki/Equation_of_time | reference | science, encyclopedia | 2026-05-05T11:12:39.793727+00:00 | kb-cron |
EOT, the time difference between apparent solar time and mean solar time; GHA, the Greenwich Hour Angle of the apparent (actual) Sun; GMHA = Universal Time − Offset, the Greenwich Mean Hour Angle of the mean (fictitious) Sun. Here time and angle are quantities that are related by factors such as: 2π radians = 360° = 1 day = 24 hours. The difference, EOT, is measurable since GHA is an angle that can be measured and Universal Time, UT, is a scale for the measurement of time. The offset by π = 180° = 12 hours from UT is needed because UT is zero at mean midnight while GMHA = 0 at mean noon. Universal Time is discontinuous at mean midnight so another quantity day number N, an integer, is required in order to form the continuous quantity time t: t = N + UT/24 hr days. Both GHA and GMHA, like all physical angles, have a mathematical, but not a physical discontinuity at their respective (apparent and mean) noon. Despite the mathematical discontinuities of its components, EOT is defined as a continuous function by adding (or subtracting) 24 hours in the small time interval between the discontinuities in GHA and GMHA. According to the definitions of the angles on the celestial sphere GHA = GAST − α (see hour angle) where:
GAST is the Greenwich apparent sidereal time (the angle between the apparent vernal equinox and the meridian in the plane of the equator). This is a known function of UT. α is the right ascension of the apparent Sun (the angle between the apparent vernal equinox and the actual Sun in the plane of the equator). On substituting into the equation of time, it is
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{\displaystyle \mathrm {EOT} =\mathrm {GAST} -\alpha -\mathrm {UT} +\mathrm {offset} }
Like the formula for GHA above, one can write GMHA = GAST − αM, where the last term is the right ascension of the mean Sun. The equation is often written in these terms as
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{\displaystyle \mathrm {EOT} =\alpha _{M}-\alpha }
where αM = GAST − UT + offset. In this formulation a measurement or calculation of EOT at a certain value of time depends on a measurement or calculation of α at that time. Both α and αM vary from 0 to 24 hours during the course of a year. The former has a discontinuity at a time that depends on the value of UT, while the latter has its at a slightly later time. As a consequence, when calculated this way EOT has two, artificial, discontinuities. They can both be removed by subtracting 24 hours from the value of EOT in the small time interval after the discontinuity in α and before the one in αM. The resulting EOT is a continuous function of time. Another definition, denoted E to distinguish it from EOT, is
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{\displaystyle E=\mathrm {GMST} -\alpha -\mathrm {UT} +\mathrm {offset} }
Here GMST = GAST − eqeq, is the Greenwich mean sidereal time (the angle between the mean vernal equinox and the mean Sun in the plane of the equator). Therefore, GMST is an approximation to GAST (and E is an approximation to EOT); eqeq is called the equation of the equinoxes and is due to the wobbling, or nutation of the Earth's axis of rotation about its precessional motion. Since the amplitude of the nutational motion is only about 1.2 s (18″ of longitude) the difference between EOT and E can be ignored unless one is interested in subsecond accuracy. A third definition, denoted Δt to distinguish it from EOT and E, and now called the Equation of Ephemeris Time (prior to the distinction that is now made between EOT, E, and Δt the latter was known as the equation of time) is
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{\displaystyle \Delta t=\Lambda -\alpha }
here Λ is the ecliptic longitude of the mean Sun (the angle from the mean vernal equinox to the mean Sun in the plane of the ecliptic). The difference Λ − (GMST − UT + offset) is 1.3 s from 1960 to 2040. Therefore, over this restricted range of years Δt is an approximation to EOT whose error is in the range 0.1 to 2.5 s depending on the longitude correction in the equation of the equinoxes; for many purposes, for example correcting a sundial, this accuracy is more than good enough.
=== Right ascension calculation === The right ascension, and hence the equation of time, can be calculated from Newton's two-body theory of celestial motion, in which the bodies (Earth and Sun) describe elliptical orbits about their common mass center. Using this theory, the equation of time becomes:
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{\displaystyle \Delta t=M+\lambda _{p}-\alpha }
where the new angles that appear are:
M = 2π(t − tp)/tY, is the mean anomaly, the angle from the periapsis of the elliptical orbit to the mean Sun; its range is from 0 to 2π as t increases from tp to tp + tY; tY = 365.2596358 days is the length of time in an anomalistic year: the time interval between two successive passages of the periapsis; λp = Λ − M, is the ecliptic longitude of the periapsis; t is dynamical time, the independent variable in the theory. Here it is taken to be identical with the continuous time based on UT (see above), but in more precise calculations (of E or EOT) the small difference between them must be accounted for as well as the distinction between UT1 and UTC. tp is the value of t at the periapsis. To complete the calculation three additional angles are required: