Theequation of time describes the discrepancy between two kinds ofsolar time. The two times that differ are theapparent solar time, which directly tracks thediurnal motion of theSun, andmean solar time, which tracks a theoreticalmean Sun with uniform motion along thecelestial equator. Apparent solar time can be obtained by measurement of the current position (hour angle) of the Sun, as indicated (with limited accuracy) by asundial.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 would have a mean of zero.[1]
The equation of time is the east or west component of theanalemma, a curve representing the angular offset of the Sun from its mean position on thecelestial sphere as viewed from Earth. The equation of time values for each day of the year, compiled by astronomicalobservatories, were widely listed inalmanacs andephemerides.[2][3]: 14
The equation of time can be approximated by a sum of twosine waves:
where:
where represents the number of days since 1 January of the current year,.
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 11 February). The equation of time has zeros near 15 April, 13 June, 1 September, and 25 December. Ignoring very slow changes in the Earth's orbit and rotation, these events are repeated at the same times everytropical year. However, due to the non-integral number of days in a year, these dates can vary by a day or so from year to year. As an example of the inexactness of the dates, according to the U.S. Naval Observatory'sMultiyear Interactive Computer Almanac the equation of time was zero at 02:00UT1 on 16 April 2011.[4]: 277
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 equation of time vanishes only for a planet with zero axial tilt and zero orbital eccentricity.[5] Two examples of planets with large equations of time are Mars and Uranus. OnMars the difference between sundial time and clock time can be as much as 50 minutes, due to the considerably greater eccentricity of its orbit. The planetUranus, which has an extremely large axial tilt, has an equation of time that makes its days start and finish several hours earlier or later depending on where it is in its orbit.
The United States Naval Observatory states "the Equation of Time is the differenceapparent solar time minusmean solar time", i.e. if the sun is ahead of the clock the sign is positive, and if the clock is ahead of the sun the sign is negative.[6][7] The equation of time is shown in the upper graph above for a period of slightly more than a year. The lower graph (which covers exactly one calendar year) has the same absolute values but thesign is reversed as it shows how far the clock is ahead of the sun. Publications may use either format: in the English-speaking world, the former usage is the more common, but is not always followed. Anyone who makes use of a published table or graph should first check its sign usage. Often, there is a note or caption which explains it. Otherwise, the usage can be determined by knowing that, during the first three months of each year, the clock is ahead of the sundial. Themnemonic "NYSS" (pronounced "nice"), for "new year, sundial slow", can be useful. Some published tables avoid the ambiguity by not using signs, but by showing phrases such as "sundial fast" or "sundial slow" instead.[8]
The phrase "equation of time" is derived from themedieval Latinaequātiō diērum, meaning "equation of days" or "difference of days". The wordequation is used in the medieval sense of "reconciliation of a difference". The wordaequātiō (andMiddle Englishequation) was used in medieval astronomy to tabulate the difference between an observed value and the expected value (as in the equation of the centre, the equation of the equinoxes, the equation of the epicycle).Gerald J. Toomer uses the medieval term "equation", from the Latinaequātiō (equalization or adjustment), for Ptolemy's difference between the mean solar time and the apparent solar time.Johannes 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."[9]: 155
The difference between apparent solar time and mean time was recognized by astronomers since antiquity, but prior to the invention of accurate mechanical clocks in the mid-17th century,sundials were the only reliable timepieces, and apparent solar time was the generally accepted standard. Mean time did not supplant apparent time in national almanacs and ephemerides until the early 19th century.[10]
The irregular daily movement of the Sun was known to the Babylonians.[citation needed]
Book III ofPtolemy'sAlmagest (2nd century) is primarily concerned with the Sun's anomaly, and he tabulated the equation of time in hisHandy Tables.[11] 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 is8+1⁄3 time-degrees or5⁄9 of an hour (Book III, chapter 9).[12] However he did not consider the effect to be relevant for most calculations since it was negligible for the slow-moving luminaries and only applied it for the fastest-moving luminary, the Moon.
Based on Ptolemy's discussion in theAlmagest, values for the equation of time (Arabictaʿdīl al-ayyām bi layālayhā) were standard for the tables (zij) in the works ofmedieval Islamic astronomy.[13]
A description of apparent and mean time was given byNevil Maskelyne in theNautical Almanac for 1767: "Apparent Time is that deduced immediately from the Sun, whether from the Observation of his passing the Meridian, or from his observedRising orSetting. 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.[1]
The right time was originally considered to be that which was shown by a sundial. When good mechanical clocks were introduced, they agreed with sundials only near four dates each year, so the equation of time was used to "correct" their readings to obtain sundial time. Some clocks, calledequation clocks, included an internal mechanism to perform this "correction". Later, as clocks became the dominant good timepieces, uncorrected clock time, i.e., "mean time", became the accepted standard. The readings of sundials, when they were used, were then, and often still are, corrected with the equation of time, used in the reverse direction from previously, to obtain clock time. Many sundials, therefore, have tables or graphs of the equation of time engraved on them to allow the user to make this correction.[8]: 123
The equation of time was used historically toset clocks. Between the invention of accurate clocks in 1656 and the advent of commercial time distribution services around 1900, there were several common land-based ways to set clocks. A sundial was read and corrected with the table or graph of the equation of time.
If atransit instrument was available or accuracy was important, the sun's transit across themeridian (the moment the sun appears to be due south or north of the observer, known as itsculmination) was noted; the clock was then set to noon and offset by the number of minutes given by the equation of time for that date. A third method did not use the equation of time; instead, it usedstellar observations to givesidereal time, exploiting the relationship between sidereal time andmean solar time.[14]: 57–58 The more accurate methods were also precursors to finding the observer'slongitude in relation to aprime meridian, such as ingeodesy on land andcelestial navigation on the sea.
The first tables to give the equation of time in an essentially correct way were published in 1665 byChristiaan Huygens.[15] Huygens, following the tradition of Ptolemy and medieval astronomers in general, set his values for the equation of time so as to make all values positive throughout the year.[15] This meant that any clock being set to mean time by Huygens's tables was consistently about 15 minutes slow compared to today's mean time.
Another set of tables was published in 1672–73 byJohn Flamsteed, who later became the firstAstronomer Royal of the newRoyal Greenwich Observatory. These appear to have been the first essentially correct tables that gave today's meaning of Mean Time (previously, as noted above, the sign of the equation was always positive and it was set at zero when the apparent time of sunrise was earliest relative to the clock time of sunrise). 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.[16]
The equation of time, correctly based on the two major components of the Sun's irregularity of apparent motion, was not generally adopted until after Flamsteed's tables of 1672–73, published with the posthumous edition of the works ofJeremiah Horrocks.[17]: 49
Robert Hooke (1635–1703), who mathematically analyzed theuniversal 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".[18]: 219
The corrections in Flamsteed's tables of 1672–1673 and 1680 gave mean time computed essentially correctly and without need for further offset. But the numerical values in tables of the equation of time have somewhat changed since then, owing to three factors:
From 1767 to 1833, the BritishNautical Almanac and Astronomical Ephemeris tabulated the equation of time in the sense 'add or subtract (as directed) the number of minutes and seconds stated to or from the apparent time to obtain the mean time'. Times in the Almanac were in apparent solar time, because time aboard ship was most often determined by observing the Sun. This operation would be performed in the unusual case that the mean solar time of an observation was needed. In the issues since 1834, all times have been in mean solar time, because by then the time aboard ship was increasingly often determined bymarine chronometers. The instructions were consequently to add or subtract (as directed) the number of minutes stated to or from the mean time to obtain the apparent time. So now addition corresponded to the equation being positive and subtraction corresponded to it being negative.
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 33 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 due to one's distance from the local time zone meridian andsummer time, if any.
The tiny increase of the mean solar day due to the slowing down of the Earth's rotation, by about 2ms 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.
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 wouldculminate 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 not centered on the Sun, and its speed varies between 30.287 and 29.291 km/s, according toKepler's laws of planetary motion, and its angular speed also varies, and thus the Sun appears to move faster (relative to the background stars) atperihelion (currently around 3 January) and slower ataphelion a half year later.[20][21][22]
At these extreme points, this effect varies the apparent solar day by 7.9 s/day from its mean. Consequently, the smaller daily differences on other days in speed are cumulative until these points, reflecting how the planet accelerates and decelerates compared to the mean.
As a result, the eccentricity of the Earth's orbit contributes a periodic variation which is (in the first-order approximation) asine wave with:
This component of the EoT is represented by aforementioned factora:
Even if the Earth's orbit were circular, the perceived motion of the Sun along ourcelestial equator would still not be uniform.[5] This is a consequence of the tilt of the Earth's rotational axis with respect to theplane of its orbit, or equivalently, the tilt of theecliptic (the path the Sun appears to take in thecelestial sphere) with respect to thecelestial equator. The projection of this motion onto ourcelestial equator, along which "clock time" is measured, is a maximum at thesolstices, when the yearly movement of the Sun is parallel to the equator (causing amplification of perceived speed) and yields mainly a change inright ascension. It is a minimum at theequinoxes, when the Sun's apparent motion is more sloped and yields more change indeclination, leaving less for the component inright ascension, which is the only component that affects the duration of the solar day. A practical illustration of obliquity is that the daily shift of the shadow cast by the Sun in a sundial even on the equator is smaller close to the solstices and greater close to the equinoxes. If this effect operated alone, then days would be up to 24 hours and 20.3 seconds long (measured solar noon to solar noon) near the solstices, and as much as 20.3 seconds shorter than 24 hours near the equinoxes.[20][23][22]
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 apparentprecession 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 a sine wave variation with:
This component of the EoT is represented by the aforementioned factor "b":
(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. Atepoch 2000 these are the values (in minutes and seconds withUT dates):
| Point | Value | Date |
|---|---|---|
| minimum | −14 min 15 s | 11 February |
| zero | 0 min 0 s | 15 April |
| maximum | +3 min 41 s | 14 May |
| zero | 0 min 0 s | 13 June |
| minimum | −6 min 30 s | 26 July |
| zero | 0 min 0 s | 1 September |
| maximum | +16 min 25 s | 3 November |
| zero | 0 min 0 s | 25 December |
On shorter timescales (thousands of years) the shifts in the dates of equinox and perihelion will be more important. The former is caused byprecession, and shifts the equinox backwards compared to the stars. But it can be ignored in the current discussion as ourGregorian 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: inAstronomical 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.[25]
If thegnomon (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 aconic 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 ananalemma. 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 withsundials and similar devices, but also for many applications ofsolar energy. Machines such assolar trackers andheliostats 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 thetime zone, and may possibly be further altered bydaylight 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.[26]
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 onnumerical 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.[27]: 89 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.
The precise definition of the equation of time is:[28]: 1529
The quantities occurring in this equation are:
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 andUniversal Time, UT, is a scale for the measurement of time. The offset byπ = 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 numberN, an integer, is required in order to form the continuous quantity timet: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 sphereGHA = GAST −α (seehour angle)
where:
On substituting into the equation of time, it is
Like the formula for GHA above, one can writeGMHA = GAST −αM, where the last term is the right ascension of the mean Sun. The equation is often written in these terms as[4]: 275 [30]: 45
whereαM = GAST − UT + offset. In this formulation a measurement or calculation of EOT at a certain value of time depends on a measurement or calculation ofα at that time. Bothα andαM 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 inα and before the one inαM. The resulting EOT is a continuous function of time.
Another definition, denotedE to distinguish it from EOT, is
HereGMST = 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 (andE is an approximation to EOT); eqeq is called the equation of the equinoxes and is due to the wobbling, ornutation 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 andE can be ignored unless one is interested in subsecond accuracy.
A third definition, denotedΔt to distinguish it from EOT andE, and now called the Equation of Ephemeris Time[28]: 1532 (prior to the distinction that is now made between EOT,E, andΔt the latter was known as the equation of time) is
hereΛ is theecliptic longitude of the mean Sun (the angle from the mean vernal equinox to the mean Sun in the plane of theecliptic).
The differenceΛ − (GMST − UT + offset) is 1.3 s from 1960 to 2040. Therefore, over this restricted range of yearsΔt 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.
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:
where the new angles that appear are:
To complete the calculation three additional angles are required:
All these angles are shown in the figure on the right, which shows thecelestial sphere and the Sun'selliptical orbit seen from the Earth (the same as the Earth's orbit seen from the Sun). In this figureε is theobliquity, whilee =√1 − (b/a)2 is theeccentricity of the ellipse.
Now given a value of0 ≤M ≤ 2π, one can calculateα(M) by means of the following well-known procedure:[27]: 89
First, givenM, calculateE fromKepler's equation:[31]: 159
Although this equation cannot be solved exactly in closed form, values ofE(M) can be obtained from infinite (power or trigonometric) series, graphical, or numerical methods. Alternatively, note that fore = 0,E =M, and by iteration:[32]: 2
This approximation can be improved, for smalle, by iterating again:
and continued iteration produces successively higher order terms of the power series expansion ine. For small values ofe (much less than 1) two or three terms of the series give a good approximation forE; the smallere, the better the approximation.
Next, knowingE, calculate thetrue anomalyν from an elliptical orbit relation[31]: 165
The correct branch of the multiple valued functionarctanx to use is the one that makesν a continuous function ofE(M) starting fromνE=0 = 0. Thus for0 ≤E < π usearctanx = arctanx, and forπ <E ≤ 2π usearctanx = arctanx + π. At the specific valueE = π for which the argument oftan is infinite, useν =E. Herearctanx is the principal branch,|arctanx| <π/2; the function that is returned by calculators and computer applications. Alternatively, this function can be expressed in terms of itsTaylor series ine, the first three terms of which are:
For smalle this approximation (or even just the first two terms) is a good one. Combining the approximation forE(M) with this one forν(E) produces:
The relationν(M) is called theequation of the center; the expression written here is a second-order approximation ine. For the small value ofe that characterises the Earth's orbit this gives a very good approximation forν(M).
Next, knowingν, calculateλ from its definition:
The value ofλ varies non-linearly withM because the orbit is elliptical and not circular. From the approximation forν:
Finally, knowingλ calculateα from a relation for the right triangle on the celestial sphere shown above[33]: 22
Note that the quadrant ofα is the same as that ofλ, therefore reduceλ to the range 0 to 2π and write
wherek is 0 ifλ is in quadrant 1, it is 1 ifλ is in quadrants 2 or 3 and it is 2 ifλ is in quadrant 4. For the values at which tan is infinite,α =λ.
Although approximate values forα can be obtained from truncated Taylor series like those forν,[34]: 32 it is more efficacious to use the equation[35]: 374
wherey = tan2(ε/2). Note that forε =y = 0,α =λ and iterating twice:
The equation of time is obtained by substituting the result of the right ascension calculation into an equation of time formula. HereΔt(M) =M +λp −α[λ(M)] is used; in part because small corrections (of the order of 1 second), that would justify usingE, are not included, and in part because the goal is to obtain a simple analytical expression. Using two-term approximations forλ(M) andα(λ) allowsΔt to be written as an explicit expression of two terms, which is designatedΔtey because it is a first order approximation ine and iny.
This equation was first derived by Milne,[35]: 375 who wrote it in terms ofλ =M +λp. The numerical values written here result from using the orbital parameter values,e =0.016709,ε =23.4393° =0.409093 radians, andλp =282.9381° =4.938201 radians that correspond to the epoch 1 January 2000 at 12 noonUT1. When evaluating the numerical expression forΔtey as given above, a calculator must be in radian mode to obtain correct values because the value of2λp − 2π in the argument of the second term is written there in radians. Higher order approximations can also be written,[36]: Eqs (45) and (46) but they necessarily have more terms. For example, the second order approximation in bothe andy consists of five terms[28]: 1535
This approximation has the potential for high accuracy, however, in order to achieve it over a wide range of years, the parameterse,ε, andλp must be allowed to vary with time.[27]: 86 [28]: 1531,1535 This creates additional calculational complications. Other approximations have been proposed, for example,Δte[27]: 86 [37] which uses the first order equation of the center but no other approximation to determineα, andΔte2[38] which uses the second order equation of the center.
The time variable,M, can be written either in terms ofn, the number of days past perihelion, orD, the number of days past a specific date and time (epoch):
HereMD is the value ofM at the chosen date and time. For the values given here, in radians,MD is that measured for the actual Sun at the epoch, 1 January 2000 at 12 noon UT1, andD is the number of days past that epoch. At periapsisM = 2π, so solving givesD =Dp =2.508109. This puts the periapsis on 4 January 2000 at 00:11:41 while the actual periapsis is, according to results from theMultiyear Interactive Computer Almanac[39] (abbreviated as MICA), on 3 January 2000 at 05:17:30. This large discrepancy happens because the difference between the orbital radius at the two locations is only 1 part in a million; in other words, radius is a very weak function of time near periapsis. As a practical matter this means that one cannot get a highly accurate result for the equation of time by usingn and adding the actual periapsis date for a given year. However, high accuracy can be achieved by using the formulation in terms ofD.
WhenD >Dp,M is greater than 2π and one must subtract a multiple of 2π (that depends on the year) from it to bring it into the range 0 to 2π. Likewise for years prior to 2000 one must add multiples of 2π. For example, for the year 2010,D varies from3653 on 1 January at noon to4017 on 31 December at noon; the correspondingM values are69.0789468 and75.3404748 and are reduced to the range 0 to 2π by subtracting 10 and 11 times 2π respectively.
One can always write:
5)D =nY +d
where:
The resulting equation for years after 2000, written as a sum of two terms, given 1), 4) and 5), is:
6) [minutes]
In plain text format:
7) EoT = -7.659sin(6.24004077 + 0.01720197(365*(y-2000) + d)) + 9.863sin( 2 (6.24004077 + 0.01720197 (365*(y-2000) + d)) + 3.5932 ) [minutes]
Term "a" represents the contribution of eccentricity, term "b" represents contribution of obliquity.
The result of the computations is usually given as either a set of tabular values, or a graph of the equation of time as a function ofd. A comparison of plots ofΔt,Δtey, and results from MICA all for the year 2000 is shown in the figure. The plot ofΔtey is seen to be close to the results produced by MICA, the absolute error,Err = |Δtey − MICA2000|, is less than 1 minute throughout the year; its largest value is 43.2 seconds and occurs on day 276 (3 October). The plot ofΔt is indistinguishable from the results of MICA, the largest absolute error between the two is 2.46 s on day 324 (20 November).
For the choice of the appropriate branch of thearctan relation with respect to function continuity a modified version of the arctangent function is helpful. It brings in previous knowledge about the expected value by a parameter. The modified arctangent function is defined as:
It produces a value that is as close toη as possible. The functionround rounds to the nearest integer.
Applying this yields:
The parameterM +λp arranges here to setΔt to the zero nearest value which is the desired one.
The difference between the MICA andΔt results was checked every 5 years over the range from 1960 to 2040. In every instance the maximum absolute error was less than 3 s; the largest difference, 2.91 s, occurred on 22 May 1965 (day 141). However, in order to achieve this level of accuracy over this range of years it is necessary to account for the secular change in the orbital parameters with time. The equations that describe this variation are:[27]: 86 [28]: 1531,1535
According to these relations, in 100 years (D = 36525),λp increases by about 0.5% (1.7°),e decreases by about 0.25%, andε decreases by about 0.05%.
As a result, the number of calculations required for any of the higher-order approximations of the equation of time requires a computer to complete them, if one wants to achieve their inherent accuracy over a wide range of time. In this event it is no more difficult to evaluateΔt using a computer than any of its approximations.
In all this note thatΔtey as written above is easy to evaluate, even with a calculator, is accurate enough (better than 1 minute over the 80-year range) for correcting sundials, and has the nice physical explanation as the sum of two terms, one due to obliquity and the other to eccentricity that was used previously in the article. This is not true either forΔt considered as a function ofM or for any of its higher-order approximations.
Another procedure for calculating the equation of time can be done as follows.[37] Angles are in degrees; the conventionalorder of operations applies.
wheren is the Earth's mean angular orbital velocity in degrees per day, a.k.a."the mean daily motion".
whereD is the date, counted in days starting at 1 on 1 January (i.e. the days part of theordinal date in the year). 9 is the approximate number of days from the December solstice to 31 December.A is the angle the Earthwould move on its orbit at its average speed from the December solstice to dateD.
B is the angle the Earth moves from the solstice to dateD, including a first-order correction for the Earth's orbital eccentricity, 0.0167 . The number 3 is the approximate number of days from 31 December to the current date of the Earth'sperihelion. This expression forB can be simplified by combining constants to:
Here,C is the difference between the angle moved at mean speed, and at the angle at the corrected speed projected onto the equatorial plane, and divided by 180° to get the difference in "half-turns". The value 23.44° is thetilt of the Earth's axis ("obliquity"). The subtraction gives the conventional sign to the equation of time. For any given value ofx,arctanx (sometimes written astan−1x) has multiple values, differing from each other by integer numbers of half turns. The value generated by a calculator or computer may not be the appropriate one for this calculation. This may causeC to be wrong by an integer number of half-turns. The excess half-turns are removed in the next step of the calculation to give the equation of time:
The expressionnint(C) means thenearest integer toC. On a computer, it can be programmed, for example, asINT(C + 0.5). Its value is 0, 1, or 2 at different times of the year. Subtracting it leaves a small positive or negative fractional number of half turns, which is multiplied by 720, the number of minutes (12 hours) that the Earth takes to rotate one half turn relative to the Sun, to get the equation of time.
Compared with published values,[8] this calculation has aroot mean square error of only 3.7 s. The greatest error is 6.0 s. This is much more accurate than the approximation described above, but not as accurate as the elaborate calculation.
The value ofB in the above calculation is an accurate value for the Sun's ecliptic longitude (shifted by 90°), so the solar declinationδ becomes readily available:
which is accurate to within a fraction of a degree.
{{cite book}}:ISBN / Date incompatibility (help)
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