Equation of time: Difference between revisions
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The equation of time was used historically to set clocks. Between the invention of accurate clocks in 1656 and the advent of commercial time distribution services around 1900, one of two common land-based ways to set clocks was by observing the passage of the sun across the local [[meridian (astronomy)|meridian]] at noon. The moment the sun passed overhead, the clock was set to noon, offset by the number of minutes given by the equation of time for that date. (The second method did not use the equation of time; instead, it used stellar observations to give [[sidereal time]], in combination with the relation between sidereal time and [[solar time]].)<ref>Olmstead 1866, pp. 57–58</ref> The equation of time values for each day of the year, compiled by astronomical [[Observatory|observatories]], were widely listed in [[almanac]]s and [[Ephemeris|ephemerides]].<ref>Milham 1945, pp. 11–15.</ref><ref>See for example, British Commission on Longitude 1794, p. 14.</ref> |
The equation of time was used historically to set clocks. Between the invention of accurate clocks in 1656 and the advent of commercial time distribution services around 1900, one of two common land-based ways to set clocks was by observing the passage of the sun across the local [[meridian (astronomy)|meridian]] at noon. The moment the sun passed overhead, the clock was set to noon, offset by the number of minutes given by the equation of time for that date. (The second method did not use the equation of time; instead, it used stellar observations to give [[sidereal time]], in combination with the relation between sidereal time and [[solar time]].)<ref>Olmstead 1866, pp. 57–58</ref> The equation of time values for each day of the year, compiled by astronomical [[Observatory|observatories]], were widely listed in [[almanac]]s and [[Ephemeris|ephemerides]].<ref>Milham 1945, pp. 11–15.</ref><ref>See for example, British Commission on Longitude 1794, p. 14.</ref> |
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Naturally, other [[planet]]s will have an equation of time too. On [[Mars]] the difference between sundial time and clock time can be as much as 50 minutes, |
Naturally, other [[planet]]s will have an equation of time too. On [[Mars]] the difference between sundial time and clock time can be as much as 50 minutes, because of the considerably greater eccentricity of its orbit. |
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==History== |
==History== |
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Until 1833, the equation of time was tabulated in the sense 'mean minus apparent solar time' in the British ''[[Nautical Almanac|Nautical Almanac and Astronomical Ephemeris]]'' published for the years 1767 onwards. Before the issue for 1834, all times in the almanac were in apparent solar time, because time aboard ship was most often determined by observing the Sun. In the unusual case that the mean solar time of an observation was needed, the extra step of adding the equation of time to ''apparent'' solar time was needed. In the Nautical Almanac issues for 1834 onwards, all times have been in mean solar time, because by then the time aboard ship was increasingly often determined by [[marine chronometer]]s. In the unusual case that the apparent solar time of an observation was needed, the extra step of applying the equation of time to ''mean'' solar time was needed, requiring all differences in the equation of time to have the opposite sign than before. |
Until 1833, the equation of time was tabulated in the sense 'mean minus apparent solar time' in the British ''[[Nautical Almanac|Nautical Almanac and Astronomical Ephemeris]]'' published for the years 1767 onwards. Before the issue for 1834, all times in the almanac were in apparent solar time, because time aboard ship was most often determined by observing the Sun. In the unusual case that the mean solar time of an observation was needed, the extra step of adding the equation of time to ''apparent'' solar time was needed. In the Nautical Almanac issues for 1834 onwards, all times have been in mean solar time, because by then the time aboard ship was increasingly often determined by [[marine chronometer]]s. In the unusual case that the apparent solar time of an observation was needed, the extra step of applying the equation of time to ''mean'' solar time was needed, requiring all differences in the equation of time to have the opposite sign than before. |
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As the apparent daily movement of the Sun is one revolution per day, that is 360° every 24 hours, and the Sun itself appears as a disc of about 0.5° in the sky, simple sundials can be read to a maximum accuracy of about one minute. Since the equation of time has a range of about 30 minutes, the difference between sundial time and clock time cannot be ignored. In addition to the equation of time, one also has to apply corrections |
As the apparent daily movement of the Sun is one revolution per day, that is 360° every 24 hours, and the Sun itself appears as a disc of about 0.5° in the sky, simple sundials can be read to a maximum accuracy of about one minute. Since the equation of time has a range of about 30 minutes, the difference between sundial time and clock time cannot be ignored. In addition to the equation of time, one also has to apply corrections because of one's distance from the local time zone meridian and [[Daylight saving time|summer time]], if any. |
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The tiny increase of the mean solar day itself |
The tiny increase of the mean solar day itself because of the slowing down of the Earth's rotation, by about 2 ms per day per century, which currently accumulates up to about 1 second every year, is not taken into account in traditional definitions of the equation of time, as it is imperceptible at the accuracy level of sundials. |
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==Explanations for the major components of the equation of time== |
==Explanations for the major components of the equation of time== |
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===Eccentricity of the Earth's orbit=== |
===Eccentricity of the Earth's orbit=== |
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{{Unreferenced section|date=August 2011}} |
{{Unreferenced section|date=August 2011}} |
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[[File:Zeitgleichung.png|thumb|250px|Graph showing the equation of time (red solid line) along with its two main components plotted separately, the part |
[[File:Zeitgleichung.png|thumb|250px|Graph showing the equation of time (red solid line) along with its two main components plotted separately, the part because of the obliquity of the ecliptic (mauve broken line) and the part caused by the Sun's varying apparent speed along the ecliptic because of the eccentricity & ellipticity of the Earth's orbit (dark dash-dotted line)]] |
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The Earth revolves around the Sun. As seen from Earth, the Sun appears to revolve once around the Earth through the background stars in one year. If the Earth orbited the Sun with a constant speed, in a circular orbit in a plane perpendicular to the Earth's axis, then the Sun would [[culmination|culminate]] every day at exactly the same time, and be a perfect time keeper (except for the very small effect of the slowing rotation of the Earth). But the orbit of the Earth is an ellipse, and its speed varies between 30.287 and 29.291 km/s, according to [[Kepler's laws of planetary motion]], and its angular speed also varies, and thus the Sun appears to move faster (relative to the background stars) at [[perihelion]] (currently around January 3) and slower at [[aphelion]] a half year later. At these extreme points, this effect increases (respectively, decreases) the real solar day by 7.9 seconds from its mean. This daily difference accumulates over a period. As a result, the eccentricity of the Earth's orbit contributes a sine wave variation with an amplitude of 7.66 minutes and a period of one year to the equation of time. The zero points are reached at perihelion (at the beginning of January) and aphelion (beginning of July) while the maximum values are in early April (negative) and early October (positive). |
The Earth revolves around the Sun. As seen from Earth, the Sun appears to revolve once around the Earth through the background stars in one year. If the Earth orbited the Sun with a constant speed, in a circular orbit in a plane perpendicular to the Earth's axis, then the Sun would [[culmination|culminate]] every day at exactly the same time, and be a perfect time keeper (except for the very small effect of the slowing rotation of the Earth). But the orbit of the Earth is an ellipse, and its speed varies between 30.287 and 29.291 km/s, according to [[Kepler's laws of planetary motion]], and its angular speed also varies, and thus the Sun appears to move faster (relative to the background stars) at [[perihelion]] (currently around January 3) and slower at [[aphelion]] a half year later. At these extreme points, this effect increases (respectively, decreases) the real solar day by 7.9 seconds from its mean. This daily difference accumulates over a period. As a result, the eccentricity of the Earth's orbit contributes a sine wave variation with an amplitude of 7.66 minutes and a period of one year to the equation of time. The zero points are reached at perihelion (at the beginning of January) and aphelion (beginning of July) while the maximum values are in early April (negative) and early October (positive). |
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===Obliquity of the ecliptic=== |
===Obliquity of the ecliptic=== |
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{{Unreferenced section|date=August 2011}} |
{{Unreferenced section|date=August 2011}} |
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[[Image:Middaysun.gif|thumb|300px|right|Sun and planets at solar midday (Ecliptic in red, Sun and Mercury in yellow, Venus in white, Mars in red, Jupiter in yellow with red spot, Saturn in white with rings).]] However, even if the Earth's orbit were circular, the motion of the Sun along the [[celestial equator]] would still not be uniform. This is a consequence of the tilt of the Earth's rotation with respect to its orbit, or equivalently, the tilt of the [[ecliptic]] (the path of the sun against the [[celestial sphere]]) with respect to the [[celestial equator]]. The projection of this motion onto the [[celestial equator]], along which "clock time" is measured, is a maximum at the solstices, when the yearly movement of the Sun is parallel to the equator and appears as a change in [[right ascension]], and is a minimum at the equinoxes, when the Sun moves in a sloping direction and appears mainly as a change in [[declination]], leaving less for the component in [[right ascension]], which is the only component that affects the duration of the solar day. As a consequence of that, the daily shift of the shadow cast by the Sun in a sundial, |
[[Image:Middaysun.gif|thumb|300px|right|Sun and planets at solar midday (Ecliptic in red, Sun and Mercury in yellow, Venus in white, Mars in red, Jupiter in yellow with red spot, Saturn in white with rings).]] However, even if the Earth's orbit were circular, the motion of the Sun along the [[celestial equator]] would still not be uniform. This is a consequence of the tilt of the Earth's rotation with respect to its orbit, or equivalently, the tilt of the [[ecliptic]] (the path of the sun against the [[celestial sphere]]) with respect to the [[celestial equator]]. The projection of this motion onto the [[celestial equator]], along which "clock time" is measured, is a maximum at the solstices, when the yearly movement of the Sun is parallel to the equator and appears as a change in [[right ascension]], and is a minimum at the equinoxes, when the Sun moves in a sloping direction and appears mainly as a change in [[declination]], leaving less for the component in [[right ascension]], which is the only component that affects the duration of the solar day. As a consequence of that, the daily shift of the shadow cast by the Sun in a sundial, because of obliquity, is smaller close to the [[equinox]]es and greater close to the [[solstice]]s. At the equinoxes, the Sun is seen slowing down by up to 20.3 seconds every day and at the solstices speeding up by the same amount. |
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In the figure on the right, we can see the monthly variation of the apparent slope of the plane of the ecliptic at solar midday as seen from Earth. This variation is due to the apparent [[precession]] of the rotating Earth through the year, as seen from the Sun at solar midday. |
In the figure on the right, we can see the monthly variation of the apparent slope of the plane of the ecliptic at solar midday as seen from Earth. This variation is due to the apparent [[precession]] of the rotating Earth through the year, as seen from the Sun at solar midday. |
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| 25 December |
| 25 December |
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E.T. = apparent − mean. Positive means: Sun runs fast and culminates earlier, or the sundial is ahead of mean time. A slight yearly variation occurs |
E.T. = apparent − mean. Positive means: Sun runs fast and culminates earlier, or the sundial is ahead of mean time. A slight yearly variation occurs because of the presence of leap years, resetting itself every 4 years. |
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The exact shape of the equation of time curve and the associated [[analemma]] slowly changes over the centuries |
The exact shape of the equation of time curve and the associated [[analemma]] slowly changes over the centuries because of secular variations in both eccentricity and obliquity. At this moment both are slowly decreasing, but they increase and decrease over a timescale of hundreds of thousands of years. If/when the Earth's orbital eccentricity (now about 0.0167 and slowly decreasing) reaches 0.047, the eccentricity effect may in some circumstances overshadow the obliquity effect, leaving the equation of time curve with only one maximum and minimum per year, as is the case on Mars.<ref>[http://www.giss.nasa.gov/research/briefs/allison_02/ Telling Time on Mars]</ref> |
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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 21 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 "maxima" of 15 min 39 sec and May and July of 4 min 58 sec. Before that time the February minimum was larger than the November maximum, and the May maximum larger than the July minimum. The secular change is evident when one compares a current graph of the equation of time (see below) with one from 2000 years ago, for example, one constructed from the data of Ptolemy. |
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 21 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 "maxima" of 15 min 39 sec and May and July of 4 min 58 sec. Before that time the February minimum was larger than the November maximum, and the May maximum larger than the July minimum. The secular change is evident when one compares a current graph of the equation of time (see below) with one from 2000 years ago, for example, one constructed from the data of Ptolemy. |
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* {{math|<var>λ</var><sub><var>p</var></sub>}} = {{math|Λ}} - ''M'' = 4.9412 = 283.11{{math|°}} is the ecliptic longitude of the periapsis written with its value on 1 Jan 2010 at 12 noon. |
* {{math|<var>λ</var><sub><var>p</var></sub>}} = {{math|Λ}} - ''M'' = 4.9412 = 283.11{{math|°}} is the ecliptic longitude of the periapsis written with its value on 1 Jan 2010 at 12 noon. |
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However, the displayed equation is approximate; it is not accurate over very long times because it ignores the distinction between dynamical time and mean solar time.<ref>Hughes p 1530</ref> In addition, an elliptical orbit formulation ignores small perturbations |
However, the displayed equation is approximate; it is not accurate over very long times because it ignores the distinction between dynamical time and mean solar time.<ref>Hughes p 1530</ref> In addition, an elliptical orbit formulation ignores small perturbations caused by the moon and other planets. Another complication is that the orbital parameter values change significantly over long times, for example {{math|<var>λ</var><sub><var>p</var></sub>}} increases by about 1.7 degrees per century. Consequently, calculating {{math|Δ''t''}} using the displayed equation with constant orbital parameters produces accurate results only for sufficiently short times (decades); when compared to more accurate calculations using the ''Multiyear Computer Interactive Almanac'' for each day in 2008 it disagrees by as much as 35.2 s.<ref>US Naval Observatory April 2010.</ref> It is possible to write an expression for the equation of time that is valid for centuries, but it is necessarily much more complex.<ref>Hughes p 1535</ref> |
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In order to calculate {{math|α}}, and hence {{math|Δ''t''}}, as a function of ''M'', three additional angles are required; they are |
In order to calculate {{math|α}}, and hence {{math|Δ''t''}}, as a function of ''M'', three additional angles are required; they are |
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This equation was first derived by Milne,<ref>Milne p 375</ref> who wrote it in terms of {{math|Λ}} = ''M'' + {{math|λ<sub>p</sub>}}. The numerical values written here result from using the orbital parameter values for ''e'', {{math|ε}}, and {{math|λ}}<sub>''p''</sub> given previously in this section. When evaluating the numerical expression for {{math|Δ}}''t''<sub>''a''</sub> as given above, a calculator must be in radian mode to obtain correct values. Note also that the date and time of periapsis (perihelion of the Earth orbit) varies from year to year; a table giving the connection can be found in [[perihelion]]. |
This equation was first derived by Milne,<ref>Milne p 375</ref> who wrote it in terms of {{math|Λ}} = ''M'' + {{math|λ<sub>p</sub>}}. The numerical values written here result from using the orbital parameter values for ''e'', {{math|ε}}, and {{math|λ}}<sub>''p''</sub> given previously in this section. When evaluating the numerical expression for {{math|Δ}}''t''<sub>''a''</sub> as given above, a calculator must be in radian mode to obtain correct values. Note also that the date and time of periapsis (perihelion of the Earth orbit) varies from year to year; a table giving the connection can be found in [[perihelion]]. |
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A comparative plot of the two calculations is shown in the figure on the right. The simpler calculation is seen to be close to the elaborate one, the absolute error, Err = |({{math|Δ}}''t''− {{math|Δ}}''t<sub>a</sub>'')|, is less than 45 seconds throughout the year; its largest value is 44.8 sec and occurs on day 273. More accurate approximations can be obtained by retaining higher order terms,<ref>Muller</ref> but they are necessarily more time consuming to evaluate. At some point it is simpler to just evaluate {{math|Δ''t''}}, but {{math|Δ}}''t<sub>a</sub>'' as written above is easy to evaluate, even with a calculator, and has a nice physical explanation as the sum of two terms, one |
A comparative plot of the two calculations is shown in the figure on the right. The simpler calculation is seen to be close to the elaborate one, the absolute error, Err = |({{math|Δ}}''t''− {{math|Δ}}''t<sub>a</sub>'')|, is less than 45 seconds throughout the year; its largest value is 44.8 sec and occurs on day 273. More accurate approximations can be obtained by retaining higher order terms,<ref>Muller</ref> but they are necessarily more time consuming to evaluate. At some point it is simpler to just evaluate {{math|Δ''t''}}, but {{math|Δ}}''t<sub>a</sub>'' as written above is easy to evaluate, even with a calculator, and has a nice physical explanation as the sum of two terms, one resulting from obliquity and the other from eccentricity. This is not true either for {{math|Δ''t''}} considered as a function of ''M'' or for higher order approximations of {{math|Δ}}''t<sub>a</sub>''. |
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{{clr}} |
{{clr}} |
Revision as of 02:23, 26 November 2011
The equation of time is the difference between apparent solar time and mean solar time. At any given instant, this difference will be the same for every observer. The equation of time can be found in tables (for example, The Astronomical Almanac) or estimated with formulas given below.
Apparent (or true) solar time can be obtained for example by measurement of the current position (hour angle) of the Sun, or indicated (with limited accuracy) by a sundial. Mean solar time, for the same place, would be the time indicated by a steady clock set so that over the year its differences from apparent solar time average to zero (with zero net gain or loss over the year).[1]
During a year the equation of time varies as shown on the graph; its change from one year to the next is slight. Apparent time, and the sundial, can be ahead (fast) by as much as 16 min 33 s (around 3 November), or behind (slow) by as much as 14 min 6 s (around 12 February). The equation of time has zeros near 15 April, 13 June, 1 September and 25 December.[2][3]
The graph of the equation of time is closely approximated by the sum of two sine curves, one with a period of a year and one with a period of half a year. The curves reflect two astronomical effects, each causing a different non-uniformity in the apparent daily motion of the Sun relative to the stars:
- the obliquity of the ecliptic (the plane of the Earth's annual orbital motion around the Sun), which is inclined by about 23.44 degrees relative to the plane of the Earth's equator; and
- the eccentricity of the Earth's orbit around the Sun, which is about 0.017.
The equation of time is also the east or west component of the analemma, a curve representing the angular offset of the Sun from its mean position on the celestial sphere as viewed from Earth.
The equation of time was used historically to set clocks. Between the invention of accurate clocks in 1656 and the advent of commercial time distribution services around 1900, one of two common land-based ways to set clocks was by observing the passage of the sun across the local meridian at noon. The moment the sun passed overhead, the clock was set to noon, offset by the number of minutes given by the equation of time for that date. (The second method did not use the equation of time; instead, it used stellar observations to give sidereal time, in combination with the relation between sidereal time and solar time.)[4] The equation of time values for each day of the year, compiled by astronomical observatories, were widely listed in almanacs and ephemerides.[5][6]
Naturally, other planets will have an equation of time too. On Mars the difference between sundial time and clock time can be as much as 50 minutes, because of the considerably greater eccentricity of its orbit.
History
Ancient history — Babylon and Egypt
The irregular daily movement of the Sun was known by the Babylonians, and Book III of Ptolemy's Almagest is primarily concerned with the Sun's anomaly. Ptolemy discusses the correction needed to convert the meridian crossing of the Sun to mean solar time and takes into consideration the nonuniform motion of the Sun along the ecliptic and the meridian correction for the Sun's ecliptic longitude. He states the maximum correction is 8 1/3 time-degrees or 5/9 of an hour (Book III, chapter 9).[7] However he did not consider the effect relevant for most calculations since it was negligible for the slow-moving luminaries and only applied it for the fastest-moving luminary, the Moon.
Medieval and Renaissance astronomy
Toomer uses the Medieval term equation, from the Latin term aequatio (equalization [adjustment]), for Ptolemy's difference between the mean solar time and the true solar time. Kepler's definition of the equation is "the difference between the number of degrees and minutes of the mean anomaly and the degrees and minutes of the corrected anomaly."[8]
Apparent time versus mean time
Until the invention of the pendulum and the development of reliable clocks during the 17th century, the equation of time as defined by Ptolemy remained a curiosity, of importance only to astronomers. However, when mechanical clocks started to take over timekeeping from sundials, which had served humanity for centuries, the difference between clock time and solar time became an issue for everyday life. Apparent solar time (or true or real solar time) is the time indicated by the Sun on a sundial (or measured by its transit over the local meridian), while mean solar time is the average as indicated by well-regulated clocks. The first tables for the equation of time which accounted for its annual variations in an essentially correct way were published in 1665 by Christiaan Huygens.[citation needed] Huygens set his values for the equation of time so as to make all values positive throughout the year.[9] This meant that a clock set by Huygens' tables would be consistently about 15 minutes slow on mean time.
Another set of tables was published in 1672/73 by John Flamsteed, who later became the first royal astronomer of the new Greenwich Observatory. These appear to have been the first essentially correct tables which also led to mean time without an offset. Flamsteed adopted the convention of tabulating and naming the correction in the sense that it was to be applied to the apparent time to give mean time.[10]
The equation of time, correctly based on the two major components of the Sun's irregularity of apparent motion, i.e. the effect of the obliquity of the ecliptic and the effect of the Earth's orbital eccentricity, was not generally adopted until after Flamsteed's tables of 1672/3, published with the posthumous edition of the works of Jeremiah Horrocks.[11]
Robert Hooke (1635-1703), who mathematically analyzed the universal joint, was the first to note that the geometry and mathematical description of the (non-secular) equation of time and the universal joint were identical, and proposed the use of a universal joint in the construction of a "mechanical sundial".[12]
Eighteenth and early nineteenth centuries
The corrections in Flamsteed's tables of 1672/3 and 1680 led to mean time computed essentially correctly and without an offset, i.e. in principle as we now know it. But the numerical values in tables of the equation of time have somewhat changed since then, owing to three kinds of factors:
- general improvements in accuracy that came from refinements in astronomical measurement techniques,
- slow intrinsic changes in the equation of time, occurring as a result of very slow long-term changes in the Earth's obliquity and eccentricity and the position of its perihelion (or, equivalently, of the Sun's perigee), and
- the inclusion of small sources of additional variation in the apparent motion of the Sun, unknown in the 17th century, but discovered from the eighteenth century onwards, including the effects of the Moon, Venus and Jupiter.[13]
Until 1833, the equation of time was tabulated in the sense 'mean minus apparent solar time' in the British Nautical Almanac and Astronomical Ephemeris published for the years 1767 onwards. Before the issue for 1834, all times in the almanac were in apparent solar time, because time aboard ship was most often determined by observing the Sun. In the unusual case that the mean solar time of an observation was needed, the extra step of adding the equation of time to apparent solar time was needed. In the Nautical Almanac issues for 1834 onwards, all times have been in mean solar time, because by then the time aboard ship was increasingly often determined by marine chronometers. In the unusual case that the apparent solar time of an observation was needed, the extra step of applying the equation of time to mean solar time was needed, requiring all differences in the equation of time to have the opposite sign than before.
As the apparent daily movement of the Sun is one revolution per day, that is 360° every 24 hours, and the Sun itself appears as a disc of about 0.5° in the sky, simple sundials can be read to a maximum accuracy of about one minute. Since the equation of time has a range of about 30 minutes, the difference between sundial time and clock time cannot be ignored. In addition to the equation of time, one also has to apply corrections because of one's distance from the local time zone meridian and summer time, if any.
The tiny increase of the mean solar day itself because of the slowing down of the Earth's rotation, by about 2 ms per day per century, which currently accumulates up to about 1 second every year, is not taken into account in traditional definitions of the equation of time, as it is imperceptible at the accuracy level of sundials.
Explanations for the major components of the equation of time
Eccentricity of the Earth's orbit
The Earth revolves around the Sun. As seen from Earth, the Sun appears to revolve once around the Earth through the background stars in one year. If the Earth orbited the Sun with a constant speed, in a circular orbit in a plane perpendicular to the Earth's axis, then the Sun would culminate every day at exactly the same time, and be a perfect time keeper (except for the very small effect of the slowing rotation of the Earth). But the orbit of the Earth is an ellipse, and its speed varies between 30.287 and 29.291 km/s, according to Kepler's laws of planetary motion, and its angular speed also varies, and thus the Sun appears to move faster (relative to the background stars) at perihelion (currently around January 3) and slower at aphelion a half year later. At these extreme points, this effect increases (respectively, decreases) the real solar day by 7.9 seconds from its mean. This daily difference accumulates over a period. As a result, the eccentricity of the Earth's orbit contributes a sine wave variation with an amplitude of 7.66 minutes and a period of one year to the equation of time. The zero points are reached at perihelion (at the beginning of January) and aphelion (beginning of July) while the maximum values are in early April (negative) and early October (positive).
Obliquity of the ecliptic
However, even if the Earth's orbit were circular, the motion of the Sun along the celestial equator would still not be uniform. This is a consequence of the tilt of the Earth's rotation with respect to its orbit, or equivalently, the tilt of the ecliptic (the path of the sun against the celestial sphere) with respect to the celestial equator. The projection of this motion onto the celestial equator, along which "clock time" is measured, is a maximum at the solstices, when the yearly movement of the Sun is parallel to the equator and appears as a change in right ascension, and is a minimum at the equinoxes, when the Sun moves in a sloping direction and appears mainly as a change in declination, leaving less for the component in right ascension, which is the only component that affects the duration of the solar day. As a consequence of that, the daily shift of the shadow cast by the Sun in a sundial, because of obliquity, is smaller close to the equinoxes and greater close to the solstices. At the equinoxes, the Sun is seen slowing down by up to 20.3 seconds every day and at the solstices speeding up by the same amount.
In the figure on the right, we can see the monthly variation of the apparent slope of the plane of the ecliptic at solar midday as seen from Earth. This variation is due to the apparent precession of the rotating Earth through the year, as seen from the Sun at solar midday.
In terms of the equation of time, the inclination of the ecliptic results in the contribution of another sine wave variation with an amplitude of 9.87 minutes and a period of a half year to the equation of time. The zero points of this sine wave are reached at the equinoxes and solstices, while the maxima are at the beginning of February and August (negative) and the beginning of May and November (positive).
Secular effects
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):
minimum | −14:15 | 11 February |
zero | 00:00 | 15 April |
maximum | +03:41 | 14 May |
zero | 00:00 | 13 June |
minimum | −06:30 | 26 July |
zero | 00:00 | 1 September |
maximum | +16:25 | 3 November |
zero | 00:00 | 25 December |
E.T. = apparent − mean. Positive means: Sun runs fast and culminates earlier, or the sundial is ahead of mean time. A slight yearly variation occurs because of the presence of leap years, resetting itself every 4 years.
The exact shape of the equation of time curve and the associated analemma slowly changes over the centuries because of secular variations in both eccentricity and obliquity. At this moment both are slowly decreasing, but they increase and decrease over a timescale of hundreds of thousands of years. If/when the Earth's orbital eccentricity (now about 0.0167 and slowly decreasing) reaches 0.047, the eccentricity effect may in some circumstances overshadow the obliquity effect, leaving the equation of time curve with only one maximum and minimum per year, as is the case on Mars.[14]
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 21 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 "maxima" of 15 min 39 sec and May and July of 4 min 58 sec. Before that time the February minimum was larger than the November maximum, and the May maximum larger than the July minimum. The secular change is evident when one compares a current graph of the equation of time (see below) with one from 2000 years ago, for example, 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 (usually) be the conic section of the hyperbola, since the circle of the Sun's motion together with the gnomon point define a cone. At the spring and fall 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.
Mathematical details
In terms of the right ascension of the Sun, α, and that of a mean Sun moving uniformly along the celestial equator, αM, the equation of time is defined as the difference,[15] Δt = αM - α. In this expression Δt is the time difference between apparent solar time (time measured by a sundial) and mean solar time (time measured by a mechanical clock). The left side of this equation is a time difference while the right side terms are angles; however, astronomers regard time and angle as quantities that are related by conversion factors such as; 2π radian = 360° = 1 day = 24 hour. The difference, Δt, is measureable because α can be measured and αM, by definition, is a linear function of mean solar time.
The equation of time can be calculated based on Newton's theory of celestial motion in which the earth and sun describe elliptical orbits about their common mass center. In doing this it is usual to write αM = 2πt/tY = Λ where
- t is dynamical time, the independent variable in the theory,
- tY is the length of time in a tropical year,
- Λ is the ecliptic longitude of a dynamical mean Sun; the angle from the vernal equinox to a fictitious Sun moving uniformly along the ecliptic and coinciding with the apparent sun at apoapsis and periapsis.
Substituting αM into the equation of time, it becomes[16]
The new angles appearing here are:
- M is the mean anomaly; the angle from the periapsis to the dynamical mean Sun,
- λp = Λ - M = 4.9412 = 283.11° is the ecliptic longitude of the periapsis written with its value on 1 Jan 2010 at 12 noon.
However, the displayed equation is approximate; it is not accurate over very long times because it ignores the distinction between dynamical time and mean solar time.[17] In addition, an elliptical orbit formulation ignores small perturbations caused by the moon and other planets. Another complication is that the orbital parameter values change significantly over long times, for example λp increases by about 1.7 degrees per century. Consequently, calculating Δt using the displayed equation with constant orbital parameters produces accurate results only for sufficiently short times (decades); when compared to more accurate calculations using the Multiyear Computer Interactive Almanac for each day in 2008 it disagrees by as much as 35.2 s.[18] It is possible to write an expression for the equation of time that is valid for centuries, but it is necessarily much more complex.[19]
In order to calculate α, and hence Δt, as a function of M, three additional angles are required; they are
- E the Sun's eccentric anomaly,
- ν the Sun's true anomaly,
- λ = ν + λp the Sun's true longitude on the ecliptic.
All these angles are shown in the figure on the right, which shows the celestial sphere and the Sun's elliptical orbit seen from the Earth (the same as the Earth's orbit seen from the Sun). In this figure ε = 0.40907 = 23.438° is the obliquity, while the eccentricity of the ellipse is e = [1 − (b/a)2]1/2 = 0.016705.
Now given a value of 0≤M≤2π, one can calculate α(M) by means of the following procedure:[20]
First, knowing M, calculate E from Kepler's equation[21]
A numerical value can be obtained from an infinite series, graphical, or numerical methods. Alternatively, note that for e = 0, E = M, and for small e, by iteration,[22] E ~ M + e sin M. This can be improved by iterating again, but for the small value of e that characterises the orbit this approximation is sufficient.
Next, knowing E, calculate the true anomaly ν from an elliptical orbit relation[23]
The correct branch of the multiple valued function tan−1x to use is the one that makes ν a continuous function of E(M) starting from ν(E=0) = 0. Thus for 0≤ E < π use tan−1x = Tan−1x, and for π < E ≤ 2π use tan−1x = Tan−1x + π. At the specific value E = π for which the argument of tan is infinite, use ν = E. Here Tan−1x is the principal branch, |Tan−1x| < π/2; the function that is returned by calculators and computer applications. Alternatively, note that for e = 0, ν = E and for small e, from a one term Taylor expansion, ν ~ E+e sin E ~ M +2 e sin M.
Next knowing ν calculate λ from its definition above
The value of λ varies non-linearly with M because the orbit is elliptical, from the approximation for ν, λ ~ M + λp + 2 e sin M.
Next, knowing λ calculate α from a relation for the right triangle on the celestial sphere shown above[24]
Like ν previously, here the correct branch of tan−1x to use makes α a continuous function of λ(M) starting from α(λ=0)=0. Thus for (2k-1)π/2 < λ < (2k+1)π/2, use tan−1x = Tan−1x + kπ, while for the values λ = (2k+1)π/2 at which the argument of tan is infinite use α = λ. Since λp ≤ λ ≤ λp+ 2π when M varies from 0 to 2π, the values of k that are needed, with λp = 4.9412, are 2, 3, and 4. Although an approximate value for α can be obtained from a one term Taylor expansion like that for ν,[25] it is more efficatious to use the equation[26] sin(α - λ) = - tan2(ε/2) sin(α + λ). Note that for ε = 0, α = λ and for small ε, by iteration, α ~ λ - tan2(ε/2) sin 2λ ~ M + λp + 2e sin M - tan2(ε/2) sin(2M+2λp).
Finally, Δt can be calculated using the starting value of M and the calculated α(M). The result is usually given as either a set of tabular values, or a graph of Δt as a function of the number of days past periapsis, n, where 0≤n≤ 365.242 (365.242 is the number of days in a tropical year); so that
Using the approximation for α(M), Δt can be written as a simple explicit expression, which is designated Δta because it is only an approximation.
This equation was first derived by Milne,[27] who wrote it in terms of Λ = M + λp. The numerical values written here result from using the orbital parameter values for e, ε, and λp given previously in this section. When evaluating the numerical expression for Δta as given above, a calculator must be in radian mode to obtain correct values. Note also that the date and time of periapsis (perihelion of the Earth orbit) varies from year to year; a table giving the connection can be found in perihelion.
A comparative plot of the two calculations is shown in the figure on the right. The simpler calculation is seen to be close to the elaborate one, the absolute error, Err = |(Δt− Δta)|, is less than 45 seconds throughout the year; its largest value is 44.8 sec and occurs on day 273. More accurate approximations can be obtained by retaining higher order terms,[28] but they are necessarily more time consuming to evaluate. At some point it is simpler to just evaluate Δt, but Δta as written above is easy to evaluate, even with a calculator, and has a nice physical explanation as the sum of two terms, one resulting from obliquity and the other from eccentricity. This is not true either for Δt considered as a function of M or for higher order approximations of Δta.
An alternative, quite simple and accurate, calculation of the equation of time can be done as follows.[29] (Angles are in degrees, so the calculator should be in degree mode, or the computer programmed appropriately.)
W is the earth's mean angular orbital velocity in degrees per day.
D is the date, in days starting at zero on January 1. 10 is the number of days from the December solstice to January 1. A is the angle the earth moves on its orbit, at mean velocity, from the December solstice to date D.
B is the angle the earth moves from the solstice to date D, including a first-order correction for the earth's orbital eccentricity. The number 0.0167 is the value of the eccentricity. The number 12 is the number of days from the solstice to the date of the earth's perihelion.
C is the difference between the angles moved at mean speed, and at the corrected speed projected onto the equatorial plane, and divided by 180 to get the difference into "half turns". (1 half turn = 180 degrees.) Arctan means arctangent or inverse tangent. It is sometimes written as tan−1, or in various computer programming languages as ATAN or ATN. The number 23.44 is the obliquity (tilt) of the earth's axis, in degrees. The subtraction is done in the direction that gives the conventional sign to the equation of time. Arctangent has multiple values, differing by half turns. The one that is produced by a calculator or computer may not be the right one for this calculation. The calculated value of C may therefore be wrong by an integer number of half turns. This uncertainty is resolved in the next step of the calculation.
This is the equation of time, in minutes. 720 is the number of minutes (12 hours) that the earth takes to rotate one half turn relative to the sun. The expression nint(C) means the nearest integer to C, which may be greater or less than C. The correct number of half turns must be close to zero so that when multiplied by 720 it makes the equation of time fairly small. Subtracting the nearest integer from C makes the number of half turns close to zero, which resolves the uncertainty from the previous step in the calculation and gives the correct value to the equation of time. The integer turns out to be 0, 1, or 2, at different times of the year. On a computer, nint(C) can be programmed, for example, as INT(C+0.5), where the INT function always rounds downward.
Compared with published values, the above calculation has a Root Mean Square error of only 3.7 seconds of time. The greatest error is 6.0 seconds.
See also
Footnotes
- ^ A description of apparent and mean time was given by Nevil Maskelyne in the Nautical Almanac for 1767: "Apparent Time is that deduced immediately from the Sun, whether from the Observation of his passing the Meridian, or from his observed Rising or Setting. This Time is different from that shewn by Clocks and Watches well regulated at Land, which is called equated or mean Time." (He went on to say that, at sea, the apparent time found from observation of the sun must be corrected by the equation of time, if the observer requires the mean time.)
- ^ As an example of the inexactness of the dates, according to the U.S. Naval Observatory's Multiyear Interactive Computer Almanac the equation of time will be 0 at 2:00 UT1 on 16 April 2011.
- ^ Heilbron 1999, p. 277.
- ^ Olmstead 1866, pp. 57–58
- ^ Milham 1945, pp. 11–15.
- ^ See for example, British Commission on Longitude 1794, p. 14.
- ^ Toomer 1998, p. 171.
- ^ Kepler 1995, p. 155.
- ^ Huygens 1665.
- ^ Flamsteed 1672
- ^ Vince 1814, p. 49.
- ^ Mills 2007, p. 219.
- ^ Maskelyne 1764, p. 163–169.
- ^ Telling Time on Mars
- ^ Heilbron p 275, Roy p 45
- ^ Duffett-Smith p 98, Meeus p 341
- ^ Hughes p 1530
- ^ US Naval Observatory April 2010.
- ^ Hughes p 1535
- ^ Duffet-Smith p 86
- ^ Moulton p 159
- ^ Hinch p 2
- ^ Moulton p 165
- ^ Burington p 22
- ^ Whitman p 32
- ^ Milne p 374
- ^ Milne p 375
- ^ Muller
- ^ http://www.green-life-innovators.org/tiki-index.php?page=Sunalign
References
- British Commission on Longitude (1794). Nautical Almanac and Astronomical Ephemeris for the year 1803. London, UK: C. Bucton.
- Burington R S 1949 Handbook of Mathematical Tables and Formulas (Sandusky, Ohio: Handbook Publishers)
- Duffett-Smith P 1988 Practical Astronomy with your Calculator Third Edition (Cambridge: Cambridge University Press)
- Flamsteed, John (1672 (for the imprint, and bound with other sections printed 1673)). De Inaequalitate Dierum Solarium. London: William Godbid.
{{cite book}}
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(help) - Heilbron J L 1999 The Sun in the Church, (Cambridge Mass: Harvard University Press|isbn=0-674-85433-0)
- Hinch E J 1991 Perturbation Methods, (Cambridge: Cambridge University Press)
- Hughes D W, et al. 1989, The Equation of Time, Monthly Notices of the Royal Astronomical Society 238 pp 1529–1535
- Huygens, Christiaan (1665). Kort Onderwys aengaende het gebruyck der Horologien tot het vinden der Lenghten van Oost en West. The Hague: [publisher unknown].
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- Kepler, Johannes (1995). Epitome of Copernican Astronomy & Harmonies of the World. Prometheus Books. ISBN 1-57392-036-3.
- Maskelyne, Nevil, "On the Equation of Time and the True Manner of Computing it", Philosophical Transactions, liv (1764), p.336 (as reprinted in an abridged edition, 1809, vol.12, at p.163–169)
- Meeus, J 1997 Mathematical Astronomy Morsels, (Richmond, Virginia: Willman-Bell)
- Milham, Willis I. (1945). Time and Timekeepers. New York: MacMillan. ISBN 0780800087. pp. 11–15
- Milne R M 1921, "Note on the Equation of Time", The Mathematical Gazette 10 (The Mathematical Association) pp 372–375.
- Mills, Allan (2007). "Robert Hooke's 'universal joint' and its application to sundials and the sundial-clock". Notes Rec. R. Soc. 61 (2). Royal Society Publishing: 219–236. doi:10.1098. Retrieved 2011-11-23.
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value (help) - Moulton F R 1970 An Introduction to Celestial Mechanics, Second Revised Edition, (New York: Dover).
- Muller M 1995, "Equation of Time - Problem in Astronomy", Acta Phys Pol A 88 Supplement, S-49.
- Olmstead, Dennison (1866). A Compendium of Astronomy. New York: Collins & Brother.
- United States Naval Observatory April 2010, Multiyear Computer Interactive Almanac (version 2.2.1), Richmond VA: Willmann-Bell.
- Roy A E 1978 Orbital Motion, (Adam Hilger|ISBN=0-85274-228-2)
- Toomer, G.J. (1998). Ptolemy's Almagest. Princeton University Press. p. 171. ISBN 0-691-00260-6.
- S Vince, "A Complete System of Astronomy", 2nd edition, volume 1, 1814.
- Whitman A M 2007, "A Simple Expression for the Equation of Time", Journal Of the North American Sundial Society 14 pp 29–33.
External links
- Graphical Visualisation of Equation of Time - Constantly updated
- NOAA Solar Calculator
- USNO rise/set/transit times of the Sun (and other celestial objects)
- The equation of time described on the Royal Greenwich Observatory website
- An analemma site with many illustrations
- The Equation of Time and the Analemma, by Kieron Taylor
- An article by Brian Tung containing a link to a C program using a more accurate formula than most (particularly at high inclinations and eccentricities). The program can calculate solar declination, Equation of Time, or Analemma.
- Doing calculations using Ptolemy's geocentric planetary models with a discussion of his E.T. graph
- Cosmology and the Equation of Time by Jack Forster. Watches that include the equation of time.
- Equation of Time Longcase Clock by John Topping C.1720
- Table giving the Equation of Time and the declination of the sun for every day of the year
- The equation of time correction-table A page describing how to correct a clock to a sundial.
- Solar tempometer - Calculate your solar time including the equation of time.