Equation of time - Knowledge

8093: 344: 1310:(usually a hyperbola), since the circle of the Sun's motion together with the gnomon point define a cone. At the spring and autumnal equinoxes, the cone degenerates into a plane and the hyperbola into a line. With a different hyperbola for each day, hour marks can be put on each hyperbola which include any necessary corrections. Unfortunately, each hyperbola corresponds to two different days, one in each half of the year, and these two days will require different corrections. A convenient compromise is to draw the line for the "mean time" and add a curve showing the exact position of the shadow points at noon during the course of the year. This curve will take the form of a figure eight and is known as an 5924: 5502: 5919:{\displaystyle {\begin{aligned}e&=1.6709\times 10^{-2}-4.193\times 10^{-5}\left({\frac {D}{36\,525}}\right)-1.26\times 10^{-7}\left({\frac {D}{36525}}\right)^{2}\\\varepsilon &=23.4393-0.013\left({\frac {D}{36\,525}}\right)-2\times 10^{-7}\left({\frac {D}{36\,525}}\right)^{2}+5\times 10^{-7}\left({\frac {D}{36\,525}}\right)^{3}{\mbox{ degrees}}\\\lambda _{\mathrm {p} }&=282.938\,07+1.7195\left({\frac {D}{36\,525}}\right)+3.025\times 10^{-4}\left({\frac {D}{36\,525}}\right)^{2}{\mbox{ degrees}}\end{aligned}}} 614:, 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. 986:, 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. 30: 8087: 2268: 710: 22: 659:. 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. 766: 4672: 1294:

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.

4744:(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 using 1519:. 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. 621:. 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 a transit instrument was available, the sun's transit across the 363:(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 every 4343: 427: 644:. 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. 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. 446:", 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. 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 the 6531:. 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. 739:

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,

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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

731:

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

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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

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The following discussion describes a reasonably accurate (agreeing with almanac data to within 3 seconds over a wide range of years) algorithm for the equation of time that is well known to astronomers. It also shows how to obtain a simple approximate formula (accurate to within 1 minute

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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

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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

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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

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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

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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

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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

606:. 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. 1754:

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

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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, called

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Equation of time (red solid line) and its two main components plotted separately, the part due to the obliquity of the ecliptic (mauve dashed line) and the part due to the Sun's varying apparent speed along the ecliptic due to eccentricity of the Earth's orbit (dark blue dash & dot

530:. 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 774:

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

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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 due to presence of leap years, resetting itself every 4 years. The exact shape of the equation of time curve and the associated

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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 evaluate

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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.

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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 to

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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:

736:. 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. 685:

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:

5322: 625:(the moment the sun appears to be due south or north of the observer) 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 used 5454: 4123: 732:

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 by

1347:. 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. 5094: 6209: 1985:

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.

1125: 677:, 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". 3751: 662:

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 of

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of an hour (Book III, chapter 9). 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.

1306:(the shadow-casting object) is not an edge but a point (e.g., a hole in a plate), the shadow (or spot of light) will trace out a curve during the course of a day. If the shadow is cast on a plane surface, this curve will be a 5229:

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:

751:

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.

3384: 6635: 4981: 290: 6310: 924: 6501: 1857: 3085: 2967: 7720: 2601: 455:"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. 198: 485:) 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). 504:

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.

367:. 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's 1433: 7867:

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

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in the argument of the second term is written there in radians. Higher order approximations can also be written, but they necessarily have more terms. For example, the second order approximation in both

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The inclusion of small sources of additional variation in the apparent motion of the Sun, unknown in the 17th century but discovered from the 18th century onwards, including the effects of the Moon (See

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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 and

6435:) 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 cause 6221:, 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's 3854: 5507: 1716: 4629: 1476:= 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 693:

Slow intrinsic changes in the equation of time, occurring as a result of small long-term changes in the Earth's obliquity and eccentricity (affecting, for instance, the distance and dates of

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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

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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

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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,

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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

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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 "

4338:{\displaystyle \Delta t_{e^{2}y^{2}}=\Delta t_{ey}-{\frac {5}{4}}e^{2}\sin {2M}+4ey\sin {M}\cos \left(2M+2\lambda _{p}\right)-{\frac {1}{2}}y^{2}\sin \left(4M+4\lambda _{p}\right)} 2509: 6093: 6538:

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.

1866:, 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 410:

The equation of time vanishes only for a planet with zero axial tilt and zero orbital eccentricity. Two examples of planets with large equations of time are Mars and Uranus. On

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Sun and planets at local apparent noon (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).

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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. The

418:, 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. 2432: 1365:

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.

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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.

4987: 7991: 497:'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." 6124: 1018: 7914: 3673: 2186:, 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 1276:

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.

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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 planet

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The equation of time: above the axis a sundial will appear fast relative to a clock showing local mean time, and below the axis a sundial will appear slow

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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:

7724: 6570: 4890: 5210:, is less than 1 minute throughout the year; its largest value is 43.2 seconds and occurs on day 276 (3 October). The plot of 598:

for 1767: "Apparent Time is that deduced immediately from the Sun, whether from the Observation of his passing the Meridian, or from his observed

204: 6235: 833: 6968:, Studies in the History of Mathematics and Physical Sciences, vol. 1, New York / Heidelberg / Berlin: Springer-Verlag, pp. 984–986, 6445: 1785: 391:(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 4369:

must be allowed to vary with time. This creates additional calculational complications. Other approximations have been proposed, for example,

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is indistinguishable from the results of MICA, the largest absolute error between the two is 2.46 s on day 324 (20 November).

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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

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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 )

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The two above mentioned factors have different wavelengths, amplitudes and phases, so their combined contribution is an irregular wave. At

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The celestial sphere and the Sun's elliptical orbit as seen by a geocentric observer looking normal to the ecliptic showing the 6 angles (

970:, when the yearly movement of the Sun is parallel to the equator (causing amplification of perceived speed) and yields mainly a change in 2876:; the function that is returned by calculators and computer applications. Alternatively, this function can be expressed in terms of its 5496:

it is necessary to account for the secular change in the orbital parameters with time. The equations that describe this variation are:

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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 of

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The equation of time is obtained by substituting the result of the right ascension calculation into an equation of time formula. Here

7780: 1468: radians = 360° = 1 day = 24 hours. The difference, EOT, is measurable since GHA is an angle that can be measured and 8445: 7401: 4750:

and adding the actual periapsis date for a given year. However, high accuracy can be achieved by using the formulation in terms of

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This approximation has the potential for high accuracy, however, in order to achieve it over a wide range of years, the parameters

1379: 1314:. By comparing the analemma to the mean noon line, the amount of correction to be applied generally on that day can be determined. 799:

As a result, the eccentricity of the Earth's orbit contributes a periodic variation which is (in the first-order approximation) a

7900: 3794:, are not included, and in part because the goal is to obtain a simple analytical expression. Using two-term approximations for 1733:. In this formulation a measurement or calculation of EOT at a certain value of time depends on a measurement or calculation of 8671: 7813: 7008: 6981: 6933: 6908: 3998:{\displaystyle \Delta t_{ey}=-2e\sin {M}+y\sin \left(2M+2\lambda _{p}\right)=-7.659\sin {M}+9.863\sin \left(2M+3.5932\right)} 2448:

can be obtained from infinite (power or trigonometric) series, graphical, or numerical methods. Alternatively, note that for

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In terms of the equation of time, the inclination of the ecliptic results in the contribution of a sine wave variation with:

8414: 781: 6560:

in the above calculation is an accurate value for the Sun's ecliptic longitude (shifted by 90°), so the solar declination

4740:. This puts the periapsis on 4 January 2000 at 00:11:41 while the actual periapsis is, according to results from the 6421: 1674: 1361:

and Earth, simpler than but not as accurate as the former. Its accuracy can be improved by including small corrections.

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Another procedure for calculating the equation of time can be done as follows. Angles are in degrees; the conventional

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programmed to calculate the equation of time. The calculation can be numerical or analytical. The former are based on

8165: 7572: 7519: 7410: 7074: 6757: 4533: 2001: 7836: 5317:{\displaystyle \arctan _{\eta }x=\arctan x+\pi \operatorname {round} {\left({\frac {\eta -\arctan x}{\pi }}\right)}} 4475: 5449:{\displaystyle \Delta t(M)=M+\lambda _{p}-\arctan _{M+\lambda _{p}}\left(\cos {\varepsilon }\tan {\lambda }\right)} 1561:

of the apparent Sun (the angle between the apparent vernal equinox and the actual Sun in the plane of the equator).

784:, and its angular speed also varies, and thus the Sun appears to move faster (relative to the background stars) at 7233: 343: 7757: 1546: 1284:, and shifts the equinox backwards compared to the stars. But it can be ignored in the current discussion as our 946:

would still not be uniform. This is a consequence of the tilt of the Earth's rotational axis with respect to the

3150: 493:(equalization or adjustment), for Ptolemy's difference between the mean solar time and the apparent solar time. 8238: 4427: 2192:

or EOT) the small difference between them must be accounted for as well as the distinction between UT1 and UTC.

1917: 1668:, where the last term is the right ascension of the mean Sun. The equation is often written in these terms as 8419: 2300:) needed for the calculation of the equation of time. For the sake of clarity the drawings are not to scale. 718: 656: 2471: 526:(2nd century) is primarily concerned with the Sun's anomaly, and he tabulated the equation of time in his 33:

This graph shows how many minutes the clock is ahead (+) or behind (−) the apparent sun. See the section "

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is an approximation to EOT); eqeq is called the equation of the equinoxes and is due to the wobbling, or

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extreme values: beginning of February and August (negative) and beginning of May and November (positive).

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Maskelyne, Nevil (1764). "Some Remarks upon the Equation of Time, and the True Manner of Computing It".

1898:, and now called the Equation of Ephemeris Time (prior to the distinction that is now made between EOT, 8681: 8041: 8026: 7964: 7943: 7870: 6683: 5101: 2344: 573: 7691: 640:

The first tables to give the equation of time in an essentially correct way were published in 1665 by

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The resulting equation for years after 2000, written as a sum of two terms, given 1), 4) and 5), is:

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this approximation (or even just the first two terms) is a good one. Combining the approximation for

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as viewed from Earth. The equation of time values for each day of the year, compiled by astronomical

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is used in the medieval sense of "reconciliation of a difference". The two times that differ are the

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Term "a" represents the contribution of eccentricty, term "b" represents contribution of obliquity.

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The tiny increase of the mean solar day due to the slowing down of the Earth's rotation, by about 2

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General improvements in accuracy that came from refinements in astronomical measurement techniques,

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Doing calculations using Ptolemy's geocentric planetary models with a discussion of his E.T. graph

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in the year). 9 is the approximate number of days from the December solstice to 31 December.

5089:{\displaystyle b=9.863\sin \left(2(6.240\,040\,77+0.017\,201\,97(365.25(y-2000)+d))+3.5932\right)} 4092:

as given above, a calculator must be in radian mode to obtain correct values because the value of

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and continued iteration produces successively higher order terms of the power series expansion in

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subtracting 24 hours from the value of EOT in the small time interval after the discontinuity in

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Kort Onderwys aengaende het gebruyck der Horologien tot het vinden der Lenghten van Oost en West

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is used; in part because small corrections (of the order of 1 second), that would justify using

7624: 6204:{\displaystyle B=A+0.0167\cdot {\frac {360^{\circ }}{\pi }}\sin \left(\left(D-3\right)n\right)} 4707:

is that measured for the actual Sun at the epoch, 1 January 2000 at 12 noon UT1, and

3103: 1120:{\displaystyle b=9.863\sin \left(2(6.240\,040\,77+0.017\,201\,97(365(y-2000)+d))+3.5932\right)} 6900: 6893: 6415:. The subtraction gives the conventional sign to the equation of time. For any given value of 8608: 7091: 7064: 4657:

along with symbols locating the daily values at noon (at 10-day intervals) obtained from the

3746:{\displaystyle \alpha \approx \lambda -y\sin {2\lambda }+{\frac {1}{2}}y^{2}\sin {4\lambda }} 1873: 1357: 1289:

contributing waves had common zero points and the equation of time curve was symmetrical: in

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of the mean Sun (the angle from the mean vernal equinox to the mean Sun in the plane of the

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GMHA = Universal Time − Offset, the Greenwich Mean Hour Angle of the mean (fictitious) Sun.

1344: 741: 717:, with a circular scale showing the equation of time correction. This is now on display in 622: 399: 7784: 7285:"The Equation of Time: Why Sundial time Differs From Clock Time Depending On Time Of Year" 4381:

which uses the first order equation of the center but no other approximation to determine

2736:{\displaystyle \nu =2\arctan \left({\sqrt {\frac {1+e}{1-e}}}\tan {\tfrac {1}{2}}E\right)} 8: 8645: 8440: 8295: 8160: 8112: 6655: 6551: 6393:{\displaystyle C={\frac {A-\arctan {\frac {\tan B}{\cos 23.44^{\circ }}}}{180^{\circ }}}} 6010: 3606:{\displaystyle \alpha =\lambda -\arcsin \left(y\sin \left(\alpha +\lambda \right)\right)} 1443: 54: 7981: 7880: 7156:"Robert Hooke's 'universal joint' and its application to sundials and the sundial-clock" 6802: 6102:

is the date, counted in days starting at 1 on 1 January (i.e. the days part of the

78: 8696: 8620: 8394: 8318: 8248: 7672: 7590: 7539: 7200: 7051: 6814: 6788: 6689: 1959: 1285: 733: 641: 447: 318: 298: 3299:{\displaystyle \lambda \approx M+\lambda _{p}+2e\sin {M}+{\frac {5}{4}}e^{2}\sin {2M}} 2312:

seen from the Earth (the same as the Earth's orbit seen from the Sun). In this figure

1646:{\displaystyle \mathrm {EOT} =\mathrm {GAST} -\alpha -\mathrm {UT} +\mathrm {offset} } 816:

zero points: perihelion (at the beginning of January) and aphelion (beginning of July)

29: 8691: 8468: 8333: 8150: 7864: 7809: 7676: 7634: 7597: 7568: 7515: 7462: 7406: 7070: 7004: 6977: 6929: 6904: 6818: 6810: 6776: 6753: 6708: 3477:{\displaystyle \alpha =\arctan \left(\cos {\varepsilon }\tan {\lambda }+k\pi \right)} 2223: 1273: 1135: 963: 959: 943: 810: 664: 74: 7886: 7364: 4026:. The numerical values written here result from using the orbital parameter values, 8613: 8359: 8338: 8253: 8066: 7664: 7560: 7452: 7167: 7046: 7041: 7033: 6969: 6806: 6535: 2309: 2305: 2152: 955: 942:

Even if the Earth's orbit were circular, the perceived motion of the Sun along our

727: 652: 486: 438:

The United States Naval Observatory states "the Equation of Time is the difference

403: 97: 96:, a curve representing the angular offset of the Sun from its mean position on the 7111: 5179:, and results from MICA all for the year 2000 is shown in the figure. The plot of 2619:(much less than 1) two or three terms of the series give a good approximation for 618: 8344: 8323: 8155: 8127: 7892: 7745: 6961: 6679: 6408: 1558: 1447: 983: 971: 674: 634: 591: 494: 66: 1978:

is 1.3 s from 1960 to 2040. Therefore, over this restricted range of years

8640: 8399: 8389: 8328: 8285: 8117: 8107: 8076: 7976: 7948: 7457: 7438: 3379:{\displaystyle \alpha =\arctan \left(\cos {\varepsilon }\tan {\lambda }\right)} 2183: 1549:

and the meridian in the plane of the equator). This is a known function of UT.

1469: 1339:

is the local mean time for a meridian that often passes near the center of the

1139: 648: 611: 585: 477: 464: 58: 7668: 7069:(2nd ed.). Philadelphia, PA: American Philosophical Society. p. 19. 6973: 4871:= number of days from the epoch to noon on 1 January of the desired year 3118:

that characterises the Earth's orbit this gives a very good approximation for

993:

of the rotating Earth through the year, as seen from the Sun at solar midday.

77:. Apparent solar time can be obtained by measurement of the current position ( 8665: 8527: 8478: 8463: 8409: 8180: 8016: 8011: 7927: 7830: 7638: 7564: 7466: 7196: 6630:{\displaystyle \delta =-\arcsin \left(\sin 23.44^{\circ }\cdot \cos B\right)} 4976:{\displaystyle a=-7.659\sin(6.240\,040\,77+0.017\,201\,97(365.25(y-2000)+d))} 3201:

because the orbit is elliptical and not circular. From the approximation for

2877: 1542: 1326: 1307: 630: 364: 7858: 7738: 7284: 6860: 481: 472: 426: 285:{\displaystyle \Delta t_{ey}=-7.659\sin(D)+9.863\sin \left(2D+3.5932\right)} 8625: 8512: 8507: 8502: 8489: 8228: 7172: 7155: 6996: 6305:{\displaystyle B=A+1.914^{\circ }\cdot \sin \left(\left(D-3\right)n\right)} 6103: 3322:

from a relation for the right triangle on the celestial sphere shown above

2641: 2241: 2091: 1322: 1013:

This component of the EoT is represented by the aforementioned factor "b":

670: 6114:

move on its orbit at its average speed from the December solstice to date

4076: radians that correspond to the epoch 1 January 2000 at 12 noon 3823:

to be written as an explicit expression of two terms, which is designated

2437:

Although this equation cannot be solved exactly in closed form, values of

978:, when the Sun's apparent motion is more sloped and yields more change in 919:{\displaystyle a=-7.659\sin(6.240\,040\,77+0.017\,201\,97(365(y-2000)+d))} 8635: 8630: 8275: 8258: 8190: 8170: 8056: 7938: 7840: 7365:"How to find the exact time of solar noon, wherever you are in the world" 7255: 6652: – Horizontal angle from north or other reference cardinal direction 6547: 6496:{\displaystyle \mathrm {EOT} =720\left(C-\operatorname {nint} {C}\right)} 5191:

is seen to be close to the results produced by MICA, the absolute error,

1852:{\displaystyle E=\mathrm {GMST} -\alpha -\mathrm {UT} +\mathrm {offset} } 1464:

Here time and angle are quantities that are related by factors such as: 2

979: 776: 748: 347:

Clock with auxiliary dial displaying the equation of time. Piazza Dante,

101: 7705:(Supplement). Institute of Physics, Polish Academy of Sciences: S49–S67. 7615:

Whitman, Alan M (2007). "A Simple Expression for the Equation of Time".

6043:

is the Earth's mean angular orbital velocity in degrees per day, a.k.a.

1482:, an integer, is required in order to form the continuous quantity time 8650: 8603: 8587: 8473: 8379: 8200: 8061: 8031: 7969: 7845:(include rise/set/transit times of the Sun and other celestial objects) 7055: 6222: 2304:

All these angles are shown in the figure on the right, which shows the

1532: 1454: 1336: 1281: 990: 785: 709: 702: 694: 384: 46: 7309: 7205: 6833: 4671: 513:

The irregular daily movement of the Sun was known to the Babylonians.

8547: 8384: 8268: 8243: 8223: 8140: 8092: 8086: 8071: 4676: 3080:{\displaystyle \nu \approx M+2e\sin {M}+{\frac {5}{4}}e^{2}\sin {2M}} 2319: 2267: 2155:: the time interval between two successive passages of the periapsis; 2095: 1340: 1330: 800: 116: 109: 7484: 7037: 4784:(that depends on the year) from it to bring it into the range 0 to 2 4698:

at the chosen date and time. For the values given here, in radians,

4634: 2962:{\displaystyle \nu \approx E+e\sin {E}+{\frac {1}{4}}e^{2}\sin {2E}} 626: 8582: 8562: 8404: 8310: 8210: 8001: 7996: 4009:

This equation was first derived by Milne, who wrote it in terms of

1963: 1522:

According to the definitions of the angles on the celestial sphere

1311: 1269: 967: 951: 819:

extreme values: early April (negative) and early October (positive)

789: 560: 522: 452: 431: 388: 315:

represents the number of days since January 1 of the current year,

93: 21: 7024:

Kennedy, E. S. (1956). "A Survey of Islamic Astronomical Tables".

6793: 2213:

To complete the calculation three additional angles are required:

1333:

have to move in ways that are influenced by the equation of time.

823:

This component of the EoT is represented by aforementioned factor

8577: 8567: 8497: 8369: 8280: 8145: 6649: 4675:

Derivative of −Δt. The axis on the right shows the length of the

3106:; the expression written here is a second-order approximation in 2596:{\displaystyle E\approx M+e\sin {M}+{\frac {1}{2}}e^{2}\sin {2M}} 2098:

of the elliptical orbit to the mean Sun; its range is from 0 to 2

1318: 975: 599: 517: 501: 392: 193:{\displaystyle D=6.240\,040\,77+0.017\,201\,97(365.25(y-2000)+d)} 105: 82: 7601: 7482: 6858: 6831: 765: 8572: 8522: 8517: 7437:

Hughes, David W.; Yallop, B. D.; Hohenkerk, C. Y. (June 1989).

6942: 6727: 1882:

can be ignored unless one is interested in subsecond accuracy.

1303: 603: 470:, meaning "equation of days" or "difference of days". The word 415: 360: 356: 348: 7655:

Milne, R. M. (December 1921). "Note on the Equation of Time".

4788:. Likewise for years prior to 2000 one must add multiples of 2 8552: 8364: 8263: 8218: 6731:

Nautical Almanac and Astronomical Ephemeris for the year 1803

1428:{\displaystyle \mathrm {EOT} =\mathrm {GHA} -\mathrm {GMHA} } 6926:

Epitome of Copernican Astronomy & Harmonies of the World

4420:, the number of days past a specific date and time (epoch): 3531:

can be obtained from truncated Taylor series like those for

1472:, UT, is a scale for the measurement of time. The offset by 966:, along which "clock time" is measured, is a maximum at the 934: 8557: 8542: 8532: 7986: 7923: 7889:: Calculate your solar time, including the equation of time 7405:(4th ed.). Cambridge, UK: Cambridge University Press. 3512:

is in quadrant 4. For the values at which tan is infinite,

411: 70: 7992:

International Earth Rotation and Reference Systems Service

92:

The equation of time is the east or west component of the

8537: 4077: 1317:

The equation of time is used not only in connection with

372: 115:

The equation of time can be approximated by a sum of two

62: 6750:

The Sun in the Church: Cathedrals as Solar Observatories

633:, exploiting the relationship between sidereal time and 81:) of the Sun, as indicated (with limited accuracy) by a 7402:

Practical astronomy with your calculator or spreadsheet

6534:

Compared with published values, this calculation has a

6525:. On a computer, it can be programmed, for example, as 6217:

is the angle the Earth moves from the solstice to date

1321:

and similar devices, but also for many applications of

1138:

2000 these are the values (in minutes and seconds with

7876:

Equation of Time Longcase Clock by John Topping C.1720

5906: 5786: 2714: 1770:. The resulting EOT is a continuous function of time. 7883:

A page describing how to correct a clock to a sundial

7436: 6573: 6448: 6322: 6238: 6127: 6056: 5505: 5350: 5239: 5104: 4990: 4893: 4586: 4536: 4478: 4430: 4126: 3857: 3676: 3546: 3420: 3331: 3216: 3153: 3010: 2895: 2658: 2529: 2474: 2397: 2004: 1920: 1788: 1677: 1574: 1382: 1021: 836: 590:

A description of apparent and mean time was given by

321: 301: 207: 128: 6640:

which is accurate to within a fraction of a degree.

5480:

to the zero nearest value which is the desired one.

4713:

is the number of days past that epoch. At periapsis

4399:

which uses the second order equation of the center.

760: 2746:The correct branch of the multiple valued function 1373:The precise definition of the equation of time is: 7922: 7800: 7589: 7398: 7026:Transactions of the American Philosophical Society 6948: 6892: 6629: 6495: 6392: 6304: 6203: 6087: 5918: 5448: 5316: 5135: 5088: 4975: 4623: 4570: 4522: 4464: 4337: 3997: 3745: 3605: 3476: 3378: 3298: 3178: 3079: 2961: 2735: 2595: 2503: 2426: 2038: 1941: 1851: 1711:{\displaystyle \mathrm {EOT} =\alpha _{M}-\alpha } 1710: 1645: 1427: 1119: 918: 647:Another set of tables was published in 1672–73 by 489:uses the medieval term "equation", from the Latin 463:The phrase "equation of time" is derived from the 327: 307: 284: 192: 7444:Monthly Notices of the Royal Astronomical Society 7345: 5974:using a computer than any of its approximations. 5929:According to these relations, in 100 years ( 1911:the latter was known as the equation of time) is 1565:On substituting into the equation of time, it is 8663: 7394: 7392: 7390: 7388: 7386: 5225:For the choice of the appropriate branch of the 4624:{\displaystyle M=6.240\,040\,77+0.017\,201\,97D} 2369:by means of the following well-known procedure: 788:(currently around 3 January) and slower at 680: 7478: 7476: 6685:The Nautical Almanac and Astronomical Ephemeris 6001:or for any of its higher-order approximations. 4806:on 31 December at noon; the corresponding 4571:{\displaystyle =6.240\,040\,77+0.017\,201\,97D} 4080:. When evaluating the numerical expression for 2039:{\displaystyle \Delta t=M+\lambda _{p}-\alpha } 1438:The quantities occurring in this equation are: 45:describes the discrepancy between two kinds of 7731: 7399:Duffett-Smith, Peter; Zwart, Jonathan (2017). 7357: 4523:{\displaystyle =M_{D}+{\frac {2\pi }{t_{Y}}}D} 2514:This approximation can be improved, for small 1989: 1656:Like the formula for GHA above, one can write 617:The equation of time was used historically to 7908: 7559:. Cambridge, UK: Cambridge University Press. 7514:(3rd ed.). Bristol, England: A. Hilger. 7383: 6674: 6672: 6229:can be simplified by combining constants to: 3835:because it is a first order approximation in 3537:, it is more efficacious to use the equation 2174:, is the ecliptic longitude of the periapsis; 673:(1635–1703), who mathematically analyzed the 7764:(version 2.2.1), Richmond VA: Willmann-Bell. 7592:Handbook of Mathematical Tables and Formulas 7473: 7141:Vince, S. "A Complete System of Astronomy". 4881:(365 if the calculation is for a leap year). 929: 7714: 7712: 7278: 7276: 7180: 7003:. Princeton University Press. p. 171. 6966:A History of Ancient Mathematical Astronomy 6752:. Cambridge, MA: Harvard University Press. 2263:, the Sun's true longitude on the ecliptic. 1368: 728:Nautical Almanac and Astronomical Ephemeris 564:, values for the equation of time (Arabic 7915: 7901: 7806:TIME From Earth Rotation to Atomic Physics 7663:(155). Mathematical Association: 372–375. 7533: 7531: 7432: 7430: 7428: 7426: 7424: 7422: 7105: 7103: 6960: 6743: 6741: 6669: 6004: 3179:{\displaystyle \lambda =\nu +\lambda _{p}} 7628: 7623:. North American Sundial Society: 29–33. 7587: 7581: 7456: 7204: 7186: 7171: 7126: 7120: 7045: 6792: 6779:(2013). "The Sun's position in the sky". 6771: 6769: 6678: 5988:the article. This is not true either for 5888: 5841: 5818: 5768: 5715: 5668: 5574: 5038: 5034: 5024: 5020: 4936: 4932: 4922: 4918: 4614: 4610: 4600: 4596: 4561: 4557: 4547: 4543: 4465:{\displaystyle M={\frac {2\pi }{t_{Y}}}n} 4414:, the number of days past perihelion, or 1942:{\displaystyle \Delta t=\Lambda -\alpha } 1343:, and may possibly be further altered by 1272:slowly change over the centuries, due to 1069: 1065: 1055: 1051: 962:. The projection of this motion onto our 954:(the path the Sun appears to take in the 879: 875: 865: 861: 156: 152: 142: 138: 16:Apparent solar time minus mean solar time 8446:International Commission on Stratigraphy 7718: 7709: 7692:"Equation of Time: Problem in Astronomy" 7650: 7648: 7273: 7166:(2). Royal Society Publishing: 219–236. 7089: 7083: 6899:. New York: Dover Publications. p. 6747: 6728:British Commission on Longitude (1794). 5328:It produces a value that is as close to 4670: 4633: 2266: 933: 764: 708: 425: 342: 28: 20: 7721:"The Latitude and Longitude of the Sun" 7614: 7608: 7537: 7528: 7419: 7109: 7100: 7066:A survey of Islamic astronomical tables 7062: 7023: 6895:Sundials, Their Theory and Construction 6886: 6884: 6882: 6775: 6738: 6707:. New York: Macmillan. pp. 11–15. 2151: days is the length of time in an 430:Animation showing equation of time and 371:the equation of time was zero at 02:00 8664: 7762:Multiyear Interactive Computer Almanac 7689: 7683: 7541:An introduction to celestial mechanics 7483:Astronomical Applications Department. 7333: 7327: 7282: 7218: 7216: 6995: 6989: 6923: 6859:Astronomical Applications Department. 6832:Astronomical Applications Department. 6766: 6702: 6413:tilt of the Earth's axis ("obliquity") 4742:Multiyear Interactive Computer Almanac 4659:Multiyear Interactive Computer Almanac 3506:is in quadrants 2 or 3 and it is 2 if 2886:, the first three terms of which are: 2049:where the new angles that appear are: 579: 406:around the Sun, which is about 0.0167. 369:Multiyear Interactive Computer Almanac 7896: 7881:The equation of time correction-table 7859:The Equation of Time and the Analemma 7791: 7654: 7645: 7554: 7548: 7485:"Computing Approximate Sidereal Time" 7351: 7153: 7147: 7135: 6917: 6890: 6696: 4780:and one must subtract a multiple of 2 558:Based on Ptolemy's discussion in the 7751: 6879: 6541: 5488:The difference between the MICA and 4408:, can be written either in terms of 3757: 2504:{\displaystyle E\approx M+e\sin {M}} 1006:zero points: equinoxes and solstices 982:, leaving less for the component in 7796:. Richmond, Virginia: Willman-Bell. 7509: 7503: 7336:"Variation in the Equation of Time" 7213: 6088:{\displaystyle A=\left(D+9\right)n} 4826:and are reduced to the range 0 to 2 950:, or equivalently, the tilt of the 755: 13: 7778: 7283:Taylor, Kieran (4 November 2018). 7096:. New York: Collins & Brother. 6721: 6456: 6453: 6450: 5802: 5351: 5105: 4792:. For example, for the year 2010, 4160: 4127: 3858: 2649:from an elliptical orbit relation 2226:(note that this is different from 2005: 1930: 1921: 1845: 1842: 1839: 1836: 1833: 1830: 1822: 1819: 1805: 1802: 1799: 1796: 1685: 1682: 1679: 1639: 1636: 1633: 1630: 1627: 1624: 1616: 1613: 1599: 1596: 1593: 1590: 1582: 1579: 1576: 1421: 1418: 1415: 1412: 1404: 1401: 1398: 1390: 1387: 1384: 1129: 508: 208: 73:Sun with uniform motion along the 14: 8708: 8166:Discrete time and continuous time 7824: 7804:; Seidelmann, P. Kenneth (2009). 5483: 5136:{\displaystyle \Delta t_{ey}=a+b} 1442:EOT, the time difference between 782:Kepler's laws of planetary motion 761:Eccentricity of the Earth's orbit 8091: 8085: 7129:De Inaequalitate Dierum Solarium 5951:increases by about 0.5% (1.7°), 4830:by subtracting 10 and 11 times 2 3525:Although approximate values for 2631:, the better the approximation. 1545:(the angle between the apparent 1297: 568:) were standard for the tables ( 7758:United States Naval Observatory 7748:", "Naval Oceanography Portal". 7588:Burington, Richard S. (1965) . 7489:United States Naval Observatory 7334:Karney, Kevin (December 2005). 7302: 7248: 7017: 6954: 6865:United States Naval Observatory 6861:"Astronomical Almanac Glossary" 6838:United States Naval Observatory 5338:rounds to the nearest integer. 1892:to distinguish it from EOT and 1779:to distinguish it from EOT, is 1541:GAST is the Greenwich apparent 725:From 1767 to 1833, the British 8239:History of timekeeping devices 7794:Mathematical Astronomy Morsels 6949:McCarthy & Seidelmann 2009 6852: 6825: 5957:decreases by about 0.25%, and 5363: 5357: 5072: 5069: 5060: 5048: 5042: 5014: 4970: 4967: 4958: 4946: 4940: 4912: 1350: 1103: 1100: 1091: 1079: 1073: 1045: 913: 910: 901: 889: 883: 855: 245: 239: 187: 178: 166: 160: 1: 7771: 7739:Approximate Solar Coordinates 7596:(4th ed.). McGraw-Hill. 6658: – Global climate cycles 5220: 4802:on 1 January at noon to 3500:is in quadrant 1, it is 1 if 2753:to use is the one that makes 2427:{\displaystyle M=E-e\sin {E}} 1457:of the apparent (actual) Sun; 681:18th and early 19th centuries 651:, who later became the first 69:, which tracks a theoretical 34: 8672:Dynamics of the Solar System 7538:Moulton, Forest Ray (1914). 7110:Huygens, Christiaan (1665). 5995:considered as a function of 5160:. A comparison of plots of 1885:A third definition, denoted 1773:Another definition, denoted 1240:+16 min 25 s 1162:−14 min 15 s 807:amplitude: 7.66 minutes 719:Derby Museum and Art Gallery 566:taʿdīl al-ayyām bi layālayhā 57:, which directly tracks the 7: 8186:Gravitational time dilation 8022:Barycentric Coordinate Time 7853:Royal Greenwich Observatory 7719:Williams, David O. (2009). 7143:2nd edition, volume 1, 1814 7090:Olmstead, Dennison (1866). 6781:European Journal of Physics 6643: 6564:becomes readily available: 6411:". The value 23.44° is the 1990:Right ascension calculation 1214:−6 min 30 s 1188:+3 min 41 s 657:Royal Greenwich Observatory 421: 10: 8713: 8042:Geocentric Coordinate Time 8027:Barycentric Dynamical Time 7965:Coordinated Universal Time 7544:(2nd ed.). Macmillan. 7189:Philosophical Transactions 6811:10.1088/0143-0807/34/3/633 6703:Milham, Willis I. (1945). 6690:Commissioners of Longitude 6545: 5963:decreases by about 0.05%. 5334:as possible. The function 3389:Note that the quadrant of 2829:for which the argument of 713:A sundial made in 1812 by 583: 574:medieval Islamic astronomy 458: 338: 8596: 8487: 8454: 8428: 8309: 8209: 8196:Time-translation symmetry 8126: 8100: 8083: 8007:International Atomic Time 7957: 7934: 7669:10.1017/S0025557200232944 7131:. London: William Godbid. 7127:Flamsteed, John (1673) . 7093:A Compendium of Astronomy 6974:10.1007/978-3-642-61910-6 6924:Kepler, Johannes (1995). 6891:Waugh, Albert E. (1973). 3195:varies non-linearly with 3112:. For the small value of 2850:is the principal branch, 2759:a continuous function of 974:. It is a minimum at the 930:Obliquity of the ecliptic 119:(see explanation below): 7865:An article by Brian Tung 7657:The Mathematical Gazette 7565:10.1017/CBO9781139172189 7458:10.1093/mnras/238.4.1529 6748:Heilbron, J. L. (1999). 6734:. London, UK: C. Bucton. 6662: 2822:. At the specific value 1369:Mathematical description 104:, were widely listed in 8456:Astronomical chronology 8429:Archaeology and geology 8136:Absolute space and time 8052:IERS Reference Meridian 8047:International Date Line 7958:International standards 7808:. Weinheim: Wiley VCH. 7699:Acta Physica Polonica A 7229:Royal Museums Greenwich 7063:Kennedy, E. S. (1989). 7047:2027/mdp.39076006359272 6110:is the angle the Earth 6045:"the mean daily motion" 6005:Alternative calculation 4667:(day) for the year 2000 4116:consists of five terms 3395:is the same as that of 1291:Astronomical Algorithms 1000:amplitude: 9.87 minutes 375:on 16 April 2011. 7439:"The Equation of Time" 7224:"The Equation of Time" 7173:10.1098/rsnr.2006.0172 6834:"The Equation of Time" 6631: 6497: 6428:(sometimes written as 6394: 6306: 6225:. This expression for 6205: 6089: 5977:In all this note that 5920: 5450: 5341:Applying this yields: 5318: 5145:In plain text format: 5137: 5090: 4977: 4837:One can always write: 4680: 4668: 4625: 4572: 4524: 4466: 4339: 3999: 3747: 3607: 3478: 3380: 3300: 3180: 3104:equation of the center 3081: 2963: 2737: 2613:. For small values of 2597: 2520:, by iterating again: 2505: 2428: 2301: 2040: 1976:− (GMST − UT + offset) 1943: 1853: 1761:and before the one in 1712: 1647: 1429: 1121: 958:) with respect to the 939: 920: 771: 722: 544: time-degrees or 435: 352: 329: 309: 286: 194: 38: 26: 8415:Weekday determination 8301:Sundial markup schema 7831:NOAA Solar Calculator 7555:Hinch, E. J. (2002). 7154:Mills, Allan (2007). 6632: 6498: 6395: 6307: 6206: 6090: 5921: 5473:arranges here to set 5451: 5319: 5138: 5091: 4978: 4674: 4637: 4626: 4573: 4525: 4467: 4340: 4000: 3748: 3667:and iterating twice: 3608: 3479: 3381: 3301: 3181: 3144:from its definition: 3082: 2964: 2738: 2598: 2506: 2429: 2350:Now given a value of 2270: 2094:, the angle from the 2041: 1944: 1854: 1713: 1648: 1430: 1358:numerical integration 1122: 937: 921: 768: 712: 705:), Venus and Jupiter. 629:observations to give 429: 346: 330: 310: 287: 195: 32: 24: 8436:Chronological dating 8176:Theory of relativity 8037:Daylight saving time 7849:The equation of time 7837:"USNO data services" 7744:24 July 2011 at the 7557:Perturbation methods 7491:. United States Navy 7236:on 10 September 2015 6928:. Prometheus Books. 6867:. United States Navy 6840:. United States Navy 6705:Time and Timekeepers 6571: 6446: 6320: 6236: 6125: 6054: 5503: 5348: 5237: 5102: 4988: 4891: 4584: 4534: 4476: 4428: 4124: 3855: 3674: 3544: 3418: 3329: 3214: 3151: 3008: 2893: 2656: 2527: 2472: 2465:, and by iteration: 2395: 2358:, one can calculate 2002: 1918: 1786: 1731:= GAST − UT + offset 1675: 1572: 1380: 1345:daylight saving time 1019: 834: 715:Whitehurst & Son 319: 299: 205: 126: 8646:Time value of money 8441:Geologic time scale 8296:History of sundials 8161:Cosmological decade 8113:Greenwich Mean Time 7944:Orders of magnitude 7802:McCarthy, Dennis D. 7787:on 11 January 2004. 7690:Müller, M. (1995). 7510:Roy, A. E. (1988). 6803:2013EJPh...34..633J 6656:Milankovitch cycles 6552:Position of the Sun 6011:order of operations 4720:, so solving gives 4402:The time variable, 4056: radians, and 3407:to the range 0 to 2 3401:, therefore reduce 1739:at that time. Both 1453:GHA, the Greenwich 1444:apparent solar time 1325:. Machines such as 792:a half year later. 734:marine chronometers 580:Early modern period 440:apparent solar time 434:path over one year. 55:apparent solar time 8621:Mental chronometry 8249:Marine chronometer 8101:Obsolete standards 7861:, by Kieron Taylor 7792:Meeus, J. (1997). 7001:Ptolemy's Almagest 6777:Jenkins, Alejandro 6627: 6493: 6390: 6302: 6201: 6085: 5916: 5914: 5910: 5790: 5446: 5314: 5133: 5086: 4973: 4681: 4669: 4621: 4568: 4520: 4462: 4335: 3995: 3743: 3603: 3474: 3376: 3296: 3176: 3077: 2990:with this one for 2959: 2733: 2723: 2593: 2501: 2424: 2302: 2036: 1960:ecliptic longitude 1939: 1864:GMST = GAST − eqeq 1849: 1708: 1643: 1425: 1286:Gregorian calendar 1274:secular variations 1117: 948:plane of its orbit 940: 916: 772: 723: 642:Christiaan Huygens 572:) in the works of 436: 353: 325: 305: 282: 190: 39: 27: 8682:Time in astronomy 8659: 8658: 8469:Nuclear timescale 8151:Continuous signal 7851:described on the 7815:978-3-527-40780-4 7727:on 23 March 2012. 7160:Notes Rec. R. Soc 7010:978-0-691-00260-6 6983:978-0-387-06995-1 6935:978-1-57392-036-0 6910:978-0-486-22947-8 6542:Solar declination 6388: 6375: 6161: 6031:365.24 days, 5909: 5893: 5846: 5789: 5773: 5720: 5673: 5620: 5579: 5307: 4776:is greater than 2 4515: 4457: 4285: 4187: 3758:Final calculation 3717: 3310:Finally, knowing 3270: 3051: 2933: 2833:is infinite, use 2722: 2705: 2704: 2567: 2386:Kepler's equation 2224:eccentric anomaly 2209:at the periapsis. 1262: 1261: 964:celestial equator 960:celestial equator 944:celestial equator 665:Jeremiah Horrocks 328:{\displaystyle y} 308:{\displaystyle d} 75:celestial equator 8704: 8360:Dominical letter 8291:Equation of time 8254:Marine sandglass 8095: 8089: 8067:Terrestrial Time 7924:Time measurement 7917: 7910: 7903: 7894: 7893: 7887:Solar tempometer 7844: 7839:. Archived from 7819: 7797: 7788: 7783:. Archived from 7765: 7755: 7749: 7735: 7729: 7728: 7723:. Archived from 7716: 7707: 7706: 7696: 7687: 7681: 7680: 7652: 7643: 7642: 7632: 7612: 7606: 7605: 7595: 7585: 7579: 7578: 7552: 7546: 7545: 7535: 7526: 7525: 7507: 7501: 7500: 7498: 7496: 7480: 7471: 7470: 7460: 7451:(4): 1529–1535. 7434: 7417: 7416: 7396: 7381: 7380: 7378: 7376: 7369:Spot-On Sundials 7361: 7355: 7349: 7343: 7342: 7340: 7331: 7325: 7324: 7322: 7320: 7306: 7300: 7299: 7297: 7295: 7280: 7271: 7270: 7268: 7266: 7252: 7246: 7245: 7243: 7241: 7232:. Archived from 7220: 7211: 7210: 7208: 7184: 7178: 7177: 7175: 7151: 7145: 7139: 7133: 7132: 7124: 7118: 7117: 7107: 7098: 7097: 7087: 7081: 7080: 7059: 7049: 7021: 7015: 7014: 6993: 6987: 6986: 6962:Neugebauer, Otto 6958: 6952: 6946: 6940: 6939: 6921: 6915: 6914: 6898: 6888: 6877: 6876: 6874: 6872: 6856: 6850: 6849: 6847: 6845: 6829: 6823: 6822: 6796: 6773: 6764: 6763: 6745: 6736: 6735: 6725: 6719: 6718: 6700: 6694: 6693: 6680:Maskelyne, Nevil 6676: 6636: 6634: 6633: 6628: 6626: 6622: 6609: 6608: 6563: 6559: 6536:root mean square 6530: 6529: 6524: 6516: 6502: 6500: 6499: 6494: 6492: 6488: 6487: 6459: 6438: 6434: 6427: 6418: 6406: 6399: 6397: 6396: 6391: 6389: 6387: 6386: 6377: 6376: 6374: 6373: 6372: 6356: 6345: 6330: 6311: 6309: 6308: 6303: 6301: 6297: 6293: 6289: 6260: 6259: 6228: 6220: 6216: 6210: 6208: 6207: 6202: 6200: 6196: 6192: 6188: 6162: 6157: 6156: 6147: 6117: 6109: 6101: 6094: 6092: 6091: 6086: 6081: 6077: 6042: 6035: 6033: 6032: 6029: 6026: 6019: 6000: 5994: 5986: 5973: 5962: 5956: 5950: 5941: 5940: 5934: 5925: 5923: 5922: 5917: 5915: 5911: 5907: 5904: 5903: 5898: 5894: 5892: 5880: 5873: 5872: 5851: 5847: 5845: 5833: 5807: 5806: 5805: 5791: 5787: 5784: 5783: 5778: 5774: 5772: 5760: 5753: 5752: 5731: 5730: 5725: 5721: 5719: 5707: 5700: 5699: 5678: 5674: 5672: 5660: 5631: 5630: 5625: 5621: 5613: 5606: 5605: 5584: 5580: 5578: 5566: 5560: 5559: 5538: 5537: 5494: 5479: 5472: 5455: 5453: 5452: 5447: 5445: 5441: 5440: 5429: 5410: 5409: 5408: 5407: 5384: 5383: 5337: 5333: 5323: 5321: 5320: 5315: 5313: 5312: 5308: 5303: 5286: 5249: 5248: 5228: 5216: 5209: 5207: 5205: 5190: 5178: 5166: 5159: 5142: 5140: 5139: 5134: 5120: 5119: 5095: 5093: 5092: 5087: 5085: 5081: 4982: 4980: 4979: 4974: 4880: 4870: 4856: 4833: 4829: 4825: 4824: 4818: 4817: 4811: 4805: 4801: 4797: 4791: 4787: 4783: 4779: 4771: 4755: 4749: 4739: 4738: 4732: 4719: 4712: 4706: 4697: 4692:is the value of 4691: 4666: 4656: 4644: 4630: 4628: 4627: 4622: 4577: 4575: 4574: 4569: 4529: 4527: 4526: 4521: 4516: 4514: 4513: 4504: 4496: 4491: 4490: 4471: 4469: 4468: 4463: 4458: 4456: 4455: 4446: 4438: 4419: 4413: 4407: 4398: 4386: 4380: 4368: 4359: 4353: 4344: 4342: 4341: 4336: 4334: 4330: 4329: 4328: 4296: 4295: 4286: 4278: 4273: 4269: 4268: 4267: 4235: 4212: 4198: 4197: 4188: 4180: 4175: 4174: 4156: 4155: 4154: 4153: 4144: 4143: 4115: 4109: 4102: 4091: 4075: 4074: 4068: 4064: 4055: 4054: 4048: 4044: 4038: 4037: 4031: 4025: 4004: 4002: 4001: 3996: 3994: 3990: 3960: 3940: 3936: 3935: 3934: 3896: 3873: 3872: 3846: 3840: 3834: 3822: 3815: 3804: 3793: 3787: 3752: 3750: 3749: 3744: 3742: 3728: 3727: 3718: 3710: 3705: 3666: 3656: 3646:. Note that for 3645: 3641: 3639: 3638: 3635: 3632: 3612: 3610: 3609: 3604: 3602: 3598: 3597: 3593: 3536: 3530: 3521: 3511: 3505: 3499: 3493: 3483: 3481: 3480: 3475: 3473: 3469: 3459: 3448: 3410: 3406: 3400: 3394: 3385: 3383: 3382: 3377: 3375: 3371: 3370: 3359: 3321: 3315: 3305: 3303: 3302: 3297: 3295: 3281: 3280: 3271: 3263: 3258: 3238: 3237: 3206: 3200: 3194: 3185: 3183: 3182: 3177: 3175: 3174: 3143: 3137: 3128: 3117: 3111: 3101: 3086: 3084: 3083: 3078: 3076: 3062: 3061: 3052: 3044: 3039: 3000: 2989: 2978: 2968: 2966: 2965: 2960: 2958: 2944: 2943: 2934: 2926: 2921: 2885: 2875: 2874: 2872: 2871: 2868: 2865: 2858: 2849: 2842: 2832: 2828: 2821: 2809: 2801: 2790: 2782: 2769: 2758: 2752: 2742: 2740: 2739: 2734: 2732: 2728: 2724: 2715: 2706: 2703: 2692: 2681: 2680: 2648: 2640:, calculate the 2639: 2630: 2624: 2618: 2612: 2602: 2600: 2599: 2594: 2592: 2578: 2577: 2568: 2560: 2555: 2519: 2510: 2508: 2507: 2502: 2500: 2464: 2454: 2447: 2433: 2431: 2430: 2425: 2423: 2383: 2377: 2368: 2357: 2347:of the ellipse. 2342: 2341: 2340: 2317: 2310:elliptical orbit 2306:celestial sphere 2299: 2262: 2239: 2231: 2221: 2208: 2203:is the value of 2202: 2191: 2181: 2173: 2153:anomalistic year 2150: 2149: 2143: 2132: 2116: 2107: 2101: 2089: 2088: 2086: 2085: 2077: 2074: 2045: 2043: 2042: 2037: 2029: 2028: 1984: 1977: 1975: 1957: 1948: 1946: 1945: 1940: 1910: 1903: 1897: 1891: 1881: 1871: 1865: 1858: 1856: 1855: 1850: 1848: 1825: 1808: 1778: 1769: 1760: 1753: 1744: 1738: 1732: 1730: 1717: 1715: 1714: 1709: 1701: 1700: 1688: 1667: 1666: 1652: 1650: 1649: 1644: 1642: 1619: 1602: 1585: 1556: 1530: 1529: 1518: 1516: 1514: 1513: 1510: 1507: 1500: 1494: 1487: 1481: 1475: 1467: 1434: 1432: 1431: 1426: 1424: 1407: 1393: 1254: 1251:0 min 1228: 1225:0 min 1202: 1199:0 min 1176: 1173:0 min 1145: 1144: 1126: 1124: 1123: 1118: 1116: 1112: 1003:period: 1/2 year 956:celestial sphere 925: 923: 922: 917: 756:Major components 653:Astronomer Royal 596:Nautical Almanac 553: 552: 548: 543: 542: 538: 535: 487:Gerald J. Toomer 334: 332: 331: 326: 314: 312: 311: 306: 291: 289: 288: 283: 281: 277: 223: 222: 199: 197: 196: 191: 98:celestial sphere 43:equation of time 8712: 8711: 8707: 8706: 8705: 8703: 8702: 8701: 8687:Earth phenomena 8662: 8661: 8660: 8655: 8592: 8483: 8450: 8424: 8305: 8205: 8156:Coordinate time 8128:Time in physics 8122: 8096: 8090: 8081: 7953: 7930: 7921: 7843:on 29 May 2014. 7835: 7827: 7822: 7816: 7774: 7769: 7768: 7756: 7752: 7746:Wayback Machine 7736: 7732: 7717: 7710: 7694: 7688: 7684: 7653: 7646: 7630:10.1.1.558.1314 7613: 7609: 7586: 7582: 7575: 7553: 7549: 7536: 7529: 7522: 7508: 7504: 7494: 7492: 7481: 7474: 7435: 7420: 7413: 7397: 7384: 7374: 7372: 7363: 7362: 7358: 7350: 7346: 7338: 7332: 7328: 7318: 7316: 7308: 7307: 7303: 7293: 7291: 7281: 7274: 7264: 7262: 7254: 7253: 7249: 7239: 7237: 7222: 7221: 7214: 7185: 7181: 7152: 7148: 7140: 7136: 7125: 7121: 7108: 7101: 7088: 7084: 7077: 7060: 7038:10.2307/1005726 7022: 7018: 7011: 6994: 6990: 6984: 6959: 6955: 6947: 6943: 6936: 6922: 6918: 6911: 6889: 6880: 6870: 6868: 6857: 6853: 6843: 6841: 6830: 6826: 6774: 6767: 6760: 6746: 6739: 6726: 6722: 6715: 6701: 6697: 6677: 6670: 6665: 6646: 6604: 6600: 6593: 6589: 6572: 6569: 6568: 6561: 6557: 6554: 6546:Main articles: 6544: 6527: 6526: 6522: 6519:nearest integer 6508: 6507:The expression 6483: 6470: 6466: 6449: 6447: 6444: 6443: 6436: 6429: 6420: 6416: 6404: 6382: 6378: 6368: 6364: 6357: 6346: 6344: 6331: 6329: 6321: 6318: 6317: 6279: 6275: 6274: 6270: 6255: 6251: 6237: 6234: 6233: 6226: 6218: 6214: 6178: 6174: 6173: 6169: 6152: 6148: 6146: 6126: 6123: 6122: 6115: 6107: 6099: 6067: 6063: 6055: 6052: 6051: 6040: 6030: 6027: 6024: 6023: 6021: 6017: 6007: 5996: 5989: 5984: 5978: 5968: 5958: 5952: 5949: 5943: 5938: 5936: 5930: 5913: 5912: 5905: 5899: 5884: 5879: 5875: 5874: 5865: 5861: 5837: 5832: 5828: 5808: 5801: 5800: 5796: 5793: 5792: 5785: 5779: 5764: 5759: 5755: 5754: 5745: 5741: 5726: 5711: 5706: 5702: 5701: 5692: 5688: 5664: 5659: 5655: 5639: 5633: 5632: 5626: 5612: 5608: 5607: 5598: 5594: 5570: 5565: 5561: 5552: 5548: 5530: 5526: 5513: 5506: 5504: 5501: 5500: 5489: 5486: 5474: 5471: 5461: 5436: 5425: 5418: 5414: 5403: 5399: 5392: 5388: 5379: 5375: 5349: 5346: 5345: 5335: 5329: 5287: 5285: 5281: 5280: 5244: 5240: 5238: 5235: 5234: 5226: 5223: 5211: 5204: 5195: 5194: 5192: 5189: 5180: 5177: 5168: 5161: 5155: 5112: 5108: 5103: 5100: 5099: 5010: 5006: 4989: 4986: 4985: 4892: 4889: 4888: 4874: 4869: 4863: 4851: 4841: 4831: 4827: 4822: 4820: 4815: 4813: 4807: 4803: 4799: 4793: 4789: 4785: 4781: 4777: 4770: 4760: 4751: 4745: 4736: 4734: 4731: 4721: 4714: 4708: 4704: 4699: 4693: 4689: 4684: 4662: 4655: 4646: 4639: 4585: 4582: 4581: 4535: 4532: 4531: 4509: 4505: 4497: 4495: 4486: 4482: 4477: 4474: 4473: 4451: 4447: 4439: 4437: 4429: 4426: 4425: 4415: 4409: 4403: 4397: 4388: 4382: 4379: 4370: 4367: 4361: 4355: 4349: 4324: 4320: 4307: 4303: 4291: 4287: 4277: 4263: 4259: 4246: 4242: 4231: 4205: 4193: 4189: 4179: 4167: 4163: 4149: 4145: 4139: 4135: 4134: 4130: 4125: 4122: 4121: 4111: 4105: 4100: 4093: 4090: 4081: 4072: 4070: 4066: 4063: 4057: 4052: 4050: 4046: 4040: 4035: 4033: 4027: 4024: 4010: 3977: 3973: 3956: 3930: 3926: 3913: 3909: 3892: 3865: 3861: 3856: 3853: 3852: 3842: 3836: 3833: 3824: 3817: 3806: 3795: 3789: 3782: 3763: 3760: 3735: 3723: 3719: 3709: 3698: 3675: 3672: 3671: 3658: 3647: 3644: 3636: 3633: 3628: 3627: 3625: 3624: 3617: 3583: 3579: 3569: 3565: 3545: 3542: 3541: 3532: 3526: 3513: 3507: 3501: 3495: 3489: 3455: 3444: 3437: 3433: 3419: 3416: 3415: 3408: 3402: 3396: 3390: 3366: 3355: 3348: 3344: 3330: 3327: 3326: 3317: 3311: 3288: 3276: 3272: 3262: 3254: 3233: 3229: 3215: 3212: 3211: 3202: 3196: 3190: 3170: 3166: 3152: 3149: 3148: 3139: 3133: 3119: 3113: 3107: 3092: 3069: 3057: 3053: 3043: 3035: 3009: 3006: 3005: 2991: 2980: 2974: 2951: 2939: 2935: 2925: 2917: 2894: 2891: 2890: 2881: 2869: 2866: 2863: 2862: 2860: 2853: 2851: 2844: 2834: 2830: 2823: 2811: 2803: 2792: 2784: 2780: 2771: 2760: 2754: 2747: 2713: 2693: 2682: 2679: 2678: 2674: 2657: 2654: 2653: 2644: 2635: 2626: 2620: 2614: 2608: 2585: 2573: 2569: 2559: 2551: 2528: 2525: 2524: 2515: 2496: 2473: 2470: 2469: 2456: 2449: 2438: 2419: 2396: 2393: 2392: 2379: 2373: 2359: 2351: 2330: 2328: 2323: 2313: 2308:and the Sun's 2282: 2272: 2261: 2247: 2235: 2227: 2217: 2204: 2201: 2195: 2187: 2177: 2164: 2158: 2147: 2145: 2142: 2136: 2131: 2124: 2118: 2115: 2109: 2108:increases from 2103: 2099: 2084: 2078: 2075: 2072: 2061: 2060: 2058: 2053: 2024: 2020: 2003: 2000: 1999: 1992: 1979: 1971: 1970: 1969:The difference 1953: 1919: 1916: 1915: 1905: 1899: 1893: 1886: 1877: 1867: 1863: 1829: 1818: 1795: 1787: 1784: 1783: 1774: 1768: 1762: 1756: 1752: 1746: 1740: 1734: 1729: 1723: 1722: 1696: 1692: 1678: 1676: 1673: 1672: 1665: 1659: 1657: 1623: 1612: 1589: 1575: 1573: 1570: 1569: 1559:right ascension 1552: 1536: 1525: 1523: 1511: 1508: 1505: 1504: 1502: 1496: 1490: 1489: 1483: 1477: 1473: 1465: 1448:mean solar time 1411: 1397: 1383: 1381: 1378: 1377: 1371: 1353: 1300: 1252: 1226: 1200: 1174: 1132: 1130:Secular effects 1041: 1037: 1020: 1017: 1016: 984:right ascension 972:right ascension 932: 835: 832: 831: 763: 758: 683: 675:universal joint 635:mean solar time 612:equation clocks 592:Nevil Maskelyne 588: 582: 550: 546: 545: 540: 536: 533: 531: 511: 509:Early astronomy 495:Johannes Kepler 468:aequātiō diērum 461: 444:mean solar time 424: 341: 320: 317: 316: 300: 297: 296: 264: 260: 215: 211: 206: 203: 202: 127: 124: 123: 67:mean solar time 17: 12: 11: 5: 8710: 8700: 8699: 8694: 8689: 8684: 8679: 8674: 8657: 8656: 8654: 8653: 8648: 8643: 8641:Time metrology 8638: 8633: 8628: 8623: 8618: 8617: 8616: 8606: 8600: 8598: 8597:Related topics 8594: 8593: 8591: 8590: 8585: 8580: 8575: 8570: 8565: 8560: 8555: 8550: 8545: 8540: 8535: 8530: 8525: 8520: 8515: 8510: 8505: 8500: 8494: 8492: 8485: 8484: 8482: 8481: 8476: 8471: 8466: 8460: 8458: 8452: 8451: 8449: 8448: 8443: 8438: 8432: 8430: 8426: 8425: 8423: 8422: 8417: 8412: 8407: 8402: 8397: 8392: 8387: 8382: 8377: 8372: 8367: 8362: 8357: 8352: 8347: 8342: 8336: 8331: 8326: 8321: 8315: 8313: 8307: 8306: 8304: 8303: 8298: 8293: 8288: 8286:Dialing scales 8283: 8278: 8273: 8272: 8271: 8261: 8256: 8251: 8246: 8241: 8236: 8231: 8226: 8221: 8215: 8213: 8207: 8206: 8204: 8203: 8198: 8193: 8188: 8183: 8178: 8173: 8168: 8163: 8158: 8153: 8148: 8143: 8138: 8132: 8130: 8124: 8123: 8121: 8120: 8118:Prime meridian 8115: 8110: 8108:Ephemeris time 8104: 8102: 8098: 8097: 8084: 8082: 8080: 8079: 8077:180th meridian 8074: 8069: 8064: 8059: 8054: 8049: 8044: 8039: 8034: 8029: 8024: 8019: 8014: 8009: 8004: 7999: 7994: 7989: 7984: 7979: 7974: 7973: 7972: 7961: 7959: 7955: 7954: 7952: 7951: 7946: 7941: 7935: 7932: 7931: 7920: 7919: 7912: 7905: 7897: 7891: 7890: 7884: 7878: 7873: 7868: 7862: 7856: 7846: 7833: 7826: 7825:External links 7823: 7821: 7820: 7814: 7798: 7789: 7779:Helyar, A. G. 7775: 7773: 7770: 7767: 7766: 7750: 7730: 7708: 7682: 7644: 7607: 7580: 7573: 7547: 7527: 7520: 7512:Orbital motion 7502: 7472: 7418: 7411: 7382: 7356: 7344: 7326: 7301: 7272: 7256:"Eccentricity" 7247: 7212: 7179: 7146: 7134: 7119: 7116:. The Hague: . 7099: 7082: 7075: 7061:Reprinted in: 7016: 7009: 6988: 6982: 6953: 6941: 6934: 6916: 6909: 6878: 6851: 6824: 6787:(3): 633–652. 6765: 6758: 6737: 6720: 6714:978-0780800083 6713: 6695: 6667: 6666: 6664: 6661: 6660: 6659: 6653: 6645: 6642: 6638: 6637: 6625: 6621: 6618: 6615: 6612: 6607: 6603: 6599: 6596: 6592: 6588: 6585: 6582: 6579: 6576: 6543: 6540: 6505: 6504: 6491: 6486: 6482: 6479: 6476: 6473: 6469: 6465: 6462: 6458: 6455: 6452: 6401: 6400: 6385: 6381: 6371: 6367: 6363: 6360: 6355: 6352: 6349: 6343: 6340: 6337: 6334: 6328: 6325: 6314: 6313: 6300: 6296: 6292: 6288: 6285: 6282: 6278: 6273: 6269: 6266: 6263: 6258: 6254: 6250: 6247: 6244: 6241: 6212: 6211: 6199: 6195: 6191: 6187: 6184: 6181: 6177: 6172: 6168: 6165: 6160: 6155: 6151: 6145: 6142: 6139: 6136: 6133: 6130: 6096: 6095: 6084: 6080: 6076: 6073: 6070: 6066: 6062: 6059: 6037: 6036: 6006: 6003: 5982: 5947: 5927: 5926: 5902: 5897: 5891: 5887: 5883: 5878: 5871: 5868: 5864: 5860: 5857: 5854: 5850: 5844: 5840: 5836: 5831: 5827: 5824: 5821: 5817: 5814: 5811: 5809: 5804: 5799: 5795: 5794: 5782: 5777: 5771: 5767: 5763: 5758: 5751: 5748: 5744: 5740: 5737: 5734: 5729: 5724: 5718: 5714: 5710: 5705: 5698: 5695: 5691: 5687: 5684: 5681: 5677: 5671: 5667: 5663: 5658: 5654: 5651: 5648: 5645: 5642: 5640: 5638: 5635: 5634: 5629: 5624: 5619: 5616: 5611: 5604: 5601: 5597: 5593: 5590: 5587: 5583: 5577: 5573: 5569: 5564: 5558: 5555: 5551: 5547: 5544: 5541: 5536: 5533: 5529: 5525: 5522: 5519: 5516: 5514: 5512: 5509: 5508: 5485: 5484:Secular change 5482: 5469: 5460:The parameter 5458: 5457: 5444: 5439: 5435: 5432: 5428: 5424: 5421: 5417: 5413: 5406: 5402: 5398: 5395: 5391: 5387: 5382: 5378: 5374: 5371: 5368: 5365: 5362: 5359: 5356: 5353: 5326: 5325: 5311: 5306: 5302: 5299: 5296: 5293: 5290: 5284: 5279: 5276: 5273: 5270: 5267: 5264: 5261: 5258: 5255: 5252: 5247: 5243: 5222: 5219: 5200: 5185: 5173: 5132: 5129: 5126: 5123: 5118: 5115: 5111: 5107: 5084: 5080: 5077: 5074: 5071: 5068: 5065: 5062: 5059: 5056: 5053: 5050: 5047: 5044: 5041: 5037: 5033: 5030: 5027: 5023: 5019: 5016: 5013: 5009: 5005: 5002: 4999: 4996: 4993: 4972: 4969: 4966: 4963: 4960: 4957: 4954: 4951: 4948: 4945: 4942: 4939: 4935: 4931: 4928: 4925: 4921: 4917: 4914: 4911: 4908: 4905: 4902: 4899: 4896: 4883: 4882: 4872: 4867: 4849: 4834:respectively. 4768: 4729: 4702: 4687: 4651: 4632: 4631: 4620: 4617: 4613: 4609: 4606: 4603: 4599: 4595: 4592: 4589: 4578: 4567: 4564: 4560: 4556: 4553: 4550: 4546: 4542: 4539: 4519: 4512: 4508: 4503: 4500: 4494: 4489: 4485: 4481: 4461: 4454: 4450: 4445: 4442: 4436: 4433: 4393: 4375: 4365: 4346: 4345: 4333: 4327: 4323: 4319: 4316: 4313: 4310: 4306: 4302: 4299: 4294: 4290: 4284: 4281: 4276: 4272: 4266: 4262: 4258: 4255: 4252: 4249: 4245: 4241: 4238: 4234: 4230: 4227: 4224: 4221: 4218: 4215: 4211: 4208: 4204: 4201: 4196: 4192: 4186: 4183: 4178: 4173: 4170: 4166: 4162: 4159: 4152: 4148: 4142: 4138: 4133: 4129: 4098: 4086: 4061: 4022: 4007: 4006: 3993: 3989: 3986: 3983: 3980: 3976: 3972: 3969: 3966: 3963: 3959: 3955: 3952: 3949: 3946: 3943: 3939: 3933: 3929: 3925: 3922: 3919: 3916: 3912: 3908: 3905: 3902: 3899: 3895: 3891: 3888: 3885: 3882: 3879: 3876: 3871: 3868: 3864: 3860: 3829: 3780: 3759: 3756: 3755: 3754: 3741: 3738: 3734: 3731: 3726: 3722: 3716: 3713: 3708: 3704: 3701: 3697: 3694: 3691: 3688: 3685: 3682: 3679: 3642: 3622: 3614: 3613: 3601: 3596: 3592: 3589: 3586: 3582: 3578: 3575: 3572: 3568: 3564: 3561: 3558: 3555: 3552: 3549: 3486: 3485: 3472: 3468: 3465: 3462: 3458: 3454: 3451: 3447: 3443: 3440: 3436: 3432: 3429: 3426: 3423: 3387: 3386: 3374: 3369: 3365: 3362: 3358: 3354: 3351: 3347: 3343: 3340: 3337: 3334: 3308: 3307: 3294: 3291: 3287: 3284: 3279: 3275: 3269: 3266: 3261: 3257: 3253: 3250: 3247: 3244: 3241: 3236: 3232: 3228: 3225: 3222: 3219: 3187: 3186: 3173: 3169: 3165: 3162: 3159: 3156: 3132:Next, knowing 3102:is called the 3089: 3088: 3075: 3072: 3068: 3065: 3060: 3056: 3050: 3047: 3042: 3038: 3034: 3031: 3028: 3025: 3022: 3019: 3016: 3013: 2971: 2970: 2957: 2954: 2950: 2947: 2942: 2938: 2932: 2929: 2924: 2920: 2916: 2913: 2910: 2907: 2904: 2901: 2898: 2775: 2770:starting from 2744: 2743: 2731: 2727: 2721: 2718: 2712: 2709: 2702: 2699: 2696: 2691: 2688: 2685: 2677: 2673: 2670: 2667: 2664: 2661: 2634:Next, knowing 2625:; the smaller 2605: 2604: 2591: 2588: 2584: 2581: 2576: 2572: 2566: 2563: 2558: 2554: 2550: 2547: 2544: 2541: 2538: 2535: 2532: 2512: 2511: 2499: 2495: 2492: 2489: 2486: 2483: 2480: 2477: 2435: 2434: 2422: 2418: 2415: 2412: 2409: 2406: 2403: 2400: 2280: 2265: 2264: 2259: 2245: 2233: 2211: 2210: 2199: 2193: 2184:dynamical time 2175: 2162: 2156: 2140: 2134: 2129: 2122: 2113: 2082: 2070: 2047: 2046: 2035: 2032: 2027: 2023: 2019: 2016: 2013: 2010: 2007: 1991: 1988: 1950: 1949: 1938: 1935: 1932: 1929: 1926: 1923: 1860: 1859: 1847: 1844: 1841: 1838: 1835: 1832: 1828: 1824: 1821: 1817: 1814: 1811: 1807: 1804: 1801: 1798: 1794: 1791: 1766: 1750: 1727: 1719: 1718: 1707: 1704: 1699: 1695: 1691: 1687: 1684: 1681: 1663: 1658:GMHA = GAST − 1654: 1653: 1641: 1638: 1635: 1632: 1629: 1626: 1622: 1618: 1615: 1611: 1608: 1605: 1601: 1598: 1595: 1592: 1588: 1584: 1581: 1578: 1563: 1562: 1550: 1547:vernal equinox 1470:Universal Time 1462: 1461: 1458: 1451: 1436: 1435: 1423: 1420: 1417: 1414: 1410: 1406: 1403: 1400: 1396: 1392: 1389: 1386: 1370: 1367: 1352: 1349: 1327:solar trackers 1299: 1296: 1278: 1277: 1260: 1259: 1256: 1249: 1245: 1244: 1241: 1238: 1234: 1233: 1230: 1223: 1219: 1218: 1215: 1212: 1208: 1207: 1204: 1197: 1193: 1192: 1189: 1186: 1182: 1181: 1178: 1171: 1167: 1166: 1163: 1160: 1156: 1155: 1152: 1149: 1131: 1128: 1115: 1111: 1108: 1105: 1102: 1099: 1096: 1093: 1090: 1087: 1084: 1081: 1078: 1075: 1072: 1068: 1064: 1061: 1058: 1054: 1050: 1047: 1044: 1040: 1036: 1033: 1030: 1027: 1024: 1011: 1010: 1007: 1004: 1001: 931: 928: 927: 926: 915: 912: 909: 906: 903: 900: 897: 894: 891: 888: 885: 882: 878: 874: 871: 868: 864: 860: 857: 854: 851: 848: 845: 842: 839: 821: 820: 817: 814: 808: 762: 759: 757: 754: 707: 706: 698: 691: 682: 679: 649:John Flamsteed 586:Equation clock 581: 578: 510: 507: 478:Middle English 465:medieval Latin 460: 457: 423: 420: 408: 407: 396: 359: 33 340: 337: 324: 304: 293: 292: 280: 276: 273: 270: 267: 263: 259: 256: 253: 250: 247: 244: 241: 238: 235: 232: 229: 226: 221: 218: 214: 210: 200: 189: 186: 183: 180: 177: 174: 171: 168: 165: 162: 159: 155: 151: 148: 145: 141: 137: 134: 131: 59:diurnal motion 15: 9: 6: 4: 3: 2: 8709: 8698: 8695: 8693: 8690: 8688: 8685: 8683: 8680: 8678: 8675: 8673: 8670: 8669: 8667: 8652: 8649: 8647: 8644: 8642: 8639: 8637: 8634: 8632: 8629: 8627: 8624: 8622: 8619: 8615: 8612: 8611: 8610: 8607: 8605: 8602: 8601: 8599: 8595: 8589: 8586: 8584: 8581: 8579: 8576: 8574: 8571: 8569: 8566: 8564: 8561: 8559: 8556: 8554: 8551: 8549: 8546: 8544: 8541: 8539: 8536: 8534: 8531: 8529: 8526: 8524: 8521: 8519: 8516: 8514: 8511: 8509: 8506: 8504: 8501: 8499: 8496: 8495: 8493: 8491: 8490:units of time 8486: 8480: 8479:Sidereal time 8477: 8475: 8472: 8470: 8467: 8465: 8464:Galactic year 8462: 8461: 8459: 8457: 8453: 8447: 8444: 8442: 8439: 8437: 8434: 8433: 8431: 8427: 8421: 8420:Weekday names 8418: 8416: 8413: 8411: 8410:Tropical year 8408: 8406: 8403: 8401: 8398: 8396: 8393: 8391: 8388: 8386: 8383: 8381: 8378: 8376: 8375:Intercalation 8373: 8371: 8368: 8366: 8363: 8361: 8358: 8356: 8353: 8351: 8348: 8346: 8343: 8341:(lunar Hijri) 8340: 8337: 8335: 8332: 8330: 8327: 8325: 8322: 8320: 8317: 8316: 8314: 8312: 8308: 8302: 8299: 8297: 8294: 8292: 8289: 8287: 8284: 8282: 8279: 8277: 8274: 8270: 8267: 8266: 8265: 8262: 8260: 8257: 8255: 8252: 8250: 8247: 8245: 8242: 8240: 8237: 8235: 8232: 8230: 8227: 8225: 8222: 8220: 8217: 8216: 8214: 8212: 8208: 8202: 8199: 8197: 8194: 8192: 8189: 8187: 8184: 8182: 8181:Time dilation 8179: 8177: 8174: 8172: 8169: 8167: 8164: 8162: 8159: 8157: 8154: 8152: 8149: 8147: 8144: 8142: 8139: 8137: 8134: 8133: 8131: 8129: 8125: 8119: 8116: 8114: 8111: 8109: 8106: 8105: 8103: 8099: 8094: 8088: 8078: 8075: 8073: 8070: 8068: 8065: 8063: 8060: 8058: 8055: 8053: 8050: 8048: 8045: 8043: 8040: 8038: 8035: 8033: 8030: 8028: 8025: 8023: 8020: 8018: 8017:24-hour clock 8015: 8013: 8012:12-hour clock 8010: 8008: 8005: 8003: 8000: 7998: 7995: 7993: 7990: 7988: 7985: 7983: 7980: 7978: 7975: 7971: 7968: 7967: 7966: 7963: 7962: 7960: 7956: 7950: 7947: 7945: 7942: 7940: 7937: 7936: 7933: 7929: 7925: 7918: 7913: 7911: 7906: 7904: 7899: 7898: 7895: 7888: 7885: 7882: 7879: 7877: 7874: 7872: 7869: 7866: 7863: 7860: 7857: 7854: 7850: 7847: 7842: 7838: 7834: 7832: 7829: 7828: 7817: 7811: 7807: 7803: 7799: 7795: 7790: 7786: 7782: 7777: 7776: 7763: 7759: 7754: 7747: 7743: 7740: 7734: 7726: 7722: 7715: 7713: 7704: 7700: 7693: 7686: 7678: 7674: 7670: 7666: 7662: 7658: 7651: 7649: 7640: 7636: 7631: 7626: 7622: 7618: 7611: 7603: 7599: 7594: 7593: 7584: 7576: 7574:9781139172189 7570: 7566: 7562: 7558: 7551: 7543: 7542: 7534: 7532: 7523: 7521:0-85274-228-2 7517: 7513: 7506: 7490: 7486: 7479: 7477: 7468: 7464: 7459: 7454: 7450: 7446: 7445: 7440: 7433: 7431: 7429: 7427: 7425: 7423: 7414: 7412:9781108436076 7408: 7404: 7403: 7395: 7393: 7391: 7389: 7387: 7370: 7366: 7360: 7353: 7348: 7337: 7330: 7315: 7311: 7305: 7290: 7286: 7279: 7277: 7261: 7257: 7251: 7235: 7231: 7230: 7225: 7219: 7217: 7207: 7202: 7198: 7197:Royal Society 7194: 7190: 7183: 7174: 7169: 7165: 7161: 7157: 7150: 7144: 7138: 7130: 7123: 7115: 7114: 7106: 7104: 7095: 7094: 7086: 7078: 7076:9780871694621 7072: 7068: 7067: 7057: 7053: 7048: 7043: 7039: 7035: 7031: 7027: 7020: 7012: 7006: 7002: 6998: 6992: 6985: 6979: 6975: 6971: 6967: 6963: 6957: 6950: 6945: 6937: 6931: 6927: 6920: 6912: 6906: 6902: 6897: 6896: 6887: 6885: 6883: 6866: 6862: 6855: 6839: 6835: 6828: 6820: 6816: 6812: 6808: 6804: 6800: 6795: 6790: 6786: 6782: 6778: 6772: 6770: 6761: 6759:9780674005365 6755: 6751: 6744: 6742: 6733: 6732: 6724: 6716: 6710: 6706: 6699: 6691: 6687: 6686: 6681: 6675: 6673: 6668: 6657: 6654: 6651: 6648: 6647: 6641: 6623: 6619: 6616: 6613: 6610: 6605: 6601: 6597: 6594: 6590: 6586: 6583: 6580: 6577: 6574: 6567: 6566: 6565: 6556:The value of 6553: 6549: 6539: 6537: 6532: 6520: 6515: 6513: 6503: minutes 6489: 6484: 6480: 6477: 6474: 6471: 6467: 6463: 6460: 6442: 6441: 6440: 6433: 6426: 6423: 6414: 6410: 6383: 6379: 6369: 6365: 6361: 6358: 6353: 6350: 6347: 6341: 6338: 6335: 6332: 6326: 6323: 6316: 6315: 6298: 6294: 6290: 6286: 6283: 6280: 6276: 6271: 6267: 6264: 6261: 6256: 6252: 6248: 6245: 6242: 6239: 6232: 6231: 6230: 6224: 6197: 6193: 6189: 6185: 6182: 6179: 6175: 6170: 6166: 6163: 6158: 6153: 6149: 6143: 6140: 6137: 6134: 6131: 6128: 6121: 6120: 6119: 6113: 6105: 6082: 6078: 6074: 6071: 6068: 6064: 6060: 6057: 6050: 6049: 6048: 6046: 6016: 6015: 6014: 6012: 6002: 5999: 5993: 5985: 5975: 5972: 5964: 5961: 5955: 5946: 5935: = 5933: 5908: degrees 5900: 5895: 5889: 5885: 5881: 5876: 5869: 5866: 5862: 5858: 5855: 5852: 5848: 5842: 5838: 5834: 5829: 5825: 5822: 5819: 5815: 5812: 5810: 5797: 5788: degrees 5780: 5775: 5769: 5765: 5761: 5756: 5749: 5746: 5742: 5738: 5735: 5732: 5727: 5722: 5716: 5712: 5708: 5703: 5696: 5693: 5689: 5685: 5682: 5679: 5675: 5669: 5665: 5661: 5656: 5652: 5649: 5646: 5643: 5641: 5636: 5627: 5622: 5617: 5614: 5609: 5602: 5599: 5595: 5591: 5588: 5585: 5581: 5575: 5571: 5567: 5562: 5556: 5553: 5549: 5545: 5542: 5539: 5534: 5531: 5527: 5523: 5520: 5517: 5515: 5510: 5499: 5498: 5497: 5493: 5481: 5478: 5468: 5464: 5442: 5437: 5433: 5430: 5426: 5422: 5419: 5415: 5411: 5404: 5400: 5396: 5393: 5389: 5385: 5380: 5376: 5372: 5369: 5366: 5360: 5354: 5344: 5343: 5342: 5339: 5332: 5309: 5304: 5300: 5297: 5294: 5291: 5288: 5282: 5277: 5274: 5271: 5268: 5265: 5262: 5259: 5256: 5253: 5250: 5245: 5241: 5233: 5232: 5231: 5218: 5215: 5203: 5199: 5188: 5184: 5176: 5172: 5165: 5158: 5152: 5149: 5146: 5143: 5130: 5127: 5124: 5121: 5116: 5113: 5109: 5096: 5082: 5078: 5075: 5066: 5063: 5057: 5054: 5051: 5045: 5039: 5035: 5031: 5028: 5025: 5021: 5017: 5011: 5007: 5003: 5000: 4997: 4994: 4991: 4983: 4964: 4961: 4955: 4952: 4949: 4943: 4937: 4933: 4929: 4926: 4923: 4919: 4915: 4909: 4906: 4903: 4900: 4897: 4894: 4886: 4878: 4873: 4866: 4862: 4861: 4860: 4857: 4855: 4848: 4844: 4838: 4835: 4810: 4796: 4775: 4767: 4763: 4757: 4754: 4748: 4743: 4728: 4724: 4717: 4711: 4705: 4696: 4690: 4678: 4673: 4665: 4660: 4654: 4650: 4643: 4636: 4618: 4615: 4611: 4607: 4604: 4601: 4597: 4593: 4590: 4587: 4579: 4565: 4562: 4558: 4554: 4551: 4548: 4544: 4540: 4537: 4517: 4510: 4506: 4501: 4498: 4492: 4487: 4483: 4479: 4459: 4452: 4448: 4443: 4440: 4434: 4431: 4423: 4422: 4421: 4418: 4412: 4406: 4400: 4396: 4392: 4385: 4378: 4374: 4364: 4358: 4352: 4331: 4325: 4321: 4317: 4314: 4311: 4308: 4304: 4300: 4297: 4292: 4288: 4282: 4279: 4274: 4270: 4264: 4260: 4256: 4253: 4250: 4247: 4243: 4239: 4236: 4232: 4228: 4225: 4222: 4219: 4216: 4213: 4209: 4206: 4202: 4199: 4194: 4190: 4184: 4181: 4176: 4171: 4168: 4164: 4157: 4150: 4146: 4140: 4136: 4131: 4119: 4118: 4117: 4114: 4108: 4097: 4089: 4085: 4079: 4060: 4043: 4030: 4021: 4017: 4013: 3991: 3987: 3984: 3981: 3978: 3974: 3970: 3967: 3964: 3961: 3957: 3953: 3950: 3947: 3944: 3941: 3937: 3931: 3927: 3923: 3920: 3917: 3914: 3910: 3906: 3903: 3900: 3897: 3893: 3889: 3886: 3883: 3880: 3877: 3874: 3869: 3866: 3862: 3850: 3849: 3848: 3845: 3839: 3832: 3828: 3821: 3813: 3809: 3802: 3798: 3792: 3786: 3779: 3775: 3771: 3767: 3739: 3736: 3732: 3729: 3724: 3720: 3714: 3711: 3706: 3702: 3699: 3695: 3692: 3689: 3686: 3683: 3680: 3677: 3670: 3669: 3668: 3665: 3661: 3654: 3650: 3631: 3620: 3599: 3594: 3590: 3587: 3584: 3580: 3576: 3573: 3570: 3566: 3562: 3559: 3556: 3553: 3550: 3547: 3540: 3539: 3538: 3535: 3529: 3523: 3520: 3516: 3510: 3504: 3498: 3492: 3470: 3466: 3463: 3460: 3456: 3452: 3449: 3445: 3441: 3438: 3434: 3430: 3427: 3424: 3421: 3414: 3413: 3412: 3405: 3399: 3393: 3372: 3367: 3363: 3360: 3356: 3352: 3349: 3345: 3341: 3338: 3335: 3332: 3325: 3324: 3323: 3320: 3314: 3292: 3289: 3285: 3282: 3277: 3273: 3267: 3264: 3259: 3255: 3251: 3248: 3245: 3242: 3239: 3234: 3230: 3226: 3223: 3220: 3217: 3210: 3209: 3208: 3205: 3199: 3193: 3189:The value of 3171: 3167: 3163: 3160: 3157: 3154: 3147: 3146: 3145: 3142: 3136: 3130: 3126: 3122: 3116: 3110: 3105: 3099: 3095: 3091:The relation 3073: 3070: 3066: 3063: 3058: 3054: 3048: 3045: 3040: 3036: 3032: 3029: 3026: 3023: 3020: 3017: 3014: 3011: 3004: 3003: 3002: 2998: 2994: 2987: 2983: 2977: 2955: 2952: 2948: 2945: 2940: 2936: 2930: 2927: 2922: 2918: 2914: 2911: 2908: 2905: 2902: 2899: 2896: 2889: 2888: 2887: 2884: 2879: 2878:Taylor series 2857: 2848: 2841: 2837: 2826: 2819: 2815: 2807: 2800: 2796: 2788: 2778: 2774: 2767: 2763: 2757: 2751: 2729: 2725: 2719: 2716: 2710: 2707: 2700: 2697: 2694: 2689: 2686: 2683: 2675: 2671: 2668: 2665: 2662: 2659: 2652: 2651: 2650: 2647: 2643: 2638: 2632: 2629: 2623: 2617: 2611: 2589: 2586: 2582: 2579: 2574: 2570: 2564: 2561: 2556: 2552: 2548: 2545: 2542: 2539: 2536: 2533: 2530: 2523: 2522: 2521: 2518: 2497: 2493: 2490: 2487: 2484: 2481: 2478: 2475: 2468: 2467: 2466: 2463: 2459: 2452: 2445: 2441: 2420: 2416: 2413: 2410: 2407: 2404: 2401: 2398: 2391: 2390: 2389: 2387: 2382: 2376: 2372:First, given 2370: 2366: 2362: 2355: 2348: 2346: 2338: 2334: 2326: 2321: 2316: 2311: 2307: 2298: 2294: 2290: 2286: 2279: 2275: 2269: 2258: 2254: 2250: 2246: 2243: 2238: 2234: 2230: 2225: 2220: 2216: 2215: 2214: 2207: 2198: 2194: 2190: 2185: 2180: 2176: 2172: 2168: 2161: 2157: 2154: 2139: 2135: 2128: 2121: 2112: 2106: 2097: 2093: 2081: 2069: 2065: 2056: 2052: 2051: 2050: 2033: 2030: 2025: 2021: 2017: 2014: 2011: 2008: 1998: 1997: 1996: 1987: 1983: 1974: 1967: 1965: 1961: 1956: 1936: 1933: 1927: 1924: 1914: 1913: 1912: 1909: 1902: 1896: 1890: 1883: 1880: 1875: 1870: 1826: 1815: 1812: 1809: 1792: 1789: 1782: 1781: 1780: 1777: 1771: 1765: 1759: 1749: 1743: 1737: 1726: 1705: 1702: 1697: 1693: 1689: 1671: 1670: 1669: 1662: 1620: 1609: 1606: 1603: 1586: 1568: 1567: 1566: 1560: 1555: 1551: 1548: 1544: 1543:sidereal time 1540: 1539: 1538: 1534: 1528: 1524:GHA = GAST − 1520: 1499: 1493: 1486: 1480: 1471: 1459: 1456: 1452: 1449: 1445: 1441: 1440: 1439: 1408: 1394: 1376: 1375: 1374: 1366: 1362: 1359: 1348: 1346: 1342: 1338: 1334: 1332: 1328: 1324: 1320: 1315: 1313: 1309: 1308:conic section 1305: 1298:Practical use 1295: 1292: 1287: 1283: 1275: 1271: 1266: 1265: 1264: 1257: 1250: 1247: 1246: 1242: 1239: 1236: 1235: 1231: 1224: 1221: 1220: 1216: 1213: 1210: 1209: 1205: 1198: 1195: 1194: 1190: 1187: 1184: 1183: 1179: 1172: 1169: 1168: 1164: 1161: 1158: 1157: 1153: 1150: 1147: 1146: 1143: 1141: 1137: 1127: 1113: 1109: 1106: 1097: 1094: 1088: 1085: 1082: 1076: 1070: 1066: 1062: 1059: 1056: 1052: 1048: 1042: 1038: 1034: 1031: 1028: 1025: 1022: 1014: 1008: 1005: 1002: 999: 998: 997: 994: 992: 987: 985: 981: 977: 973: 969: 965: 961: 957: 953: 949: 945: 936: 907: 904: 898: 895: 892: 886: 880: 876: 872: 869: 866: 862: 858: 852: 849: 846: 843: 840: 837: 830: 829: 828: 826: 818: 815: 812: 809: 806: 805: 804: 802: 797: 793: 791: 787: 783: 778: 767: 753: 750: 745: 743: 737: 735: 730: 729: 720: 716: 711: 704: 699: 696: 692: 689: 688: 687: 678: 676: 672: 668: 666: 660: 658: 654: 650: 645: 643: 638: 636: 632: 631:sidereal time 628: 624: 620: 615: 613: 607: 605: 601: 597: 593: 587: 577: 575: 571: 567: 563: 562: 556: 529: 525: 524: 519: 514: 506: 503: 498: 496: 492: 488: 484: 483: 479: 475: 474: 469: 466: 456: 454: 449: 445: 441: 433: 428: 419: 417: 413: 405: 404:Earth's orbit 401: 397: 394: 390: 386: 382: 381: 380: 376: 374: 370: 366: 365:tropical year 362: 358: 350: 345: 336: 322: 302: 278: 274: 271: 268: 265: 261: 257: 254: 251: 248: 242: 236: 233: 230: 227: 224: 219: 216: 212: 201: 184: 181: 175: 172: 169: 163: 157: 153: 149: 146: 143: 139: 135: 132: 129: 122: 121: 120: 118: 113: 111: 107: 103: 102:observatories 99: 95: 90: 88: 84: 80: 76: 72: 68: 64: 60: 56: 52: 48: 44: 36: 31: 23: 19: 8626:Decimal time 8355:Astronomical 8290: 8234:Complication 8229:Atomic clock 7841:the original 7805: 7793: 7785:the original 7761: 7760:April 2010, 7753: 7733: 7725:the original 7702: 7698: 7685: 7660: 7656: 7620: 7616: 7610: 7591: 7583: 7556: 7550: 7540: 7511: 7505: 7493:. Retrieved 7488: 7448: 7442: 7400: 7373:. Retrieved 7368: 7359: 7347: 7329: 7317:. Retrieved 7313: 7304: 7292:. Retrieved 7288: 7263:. Retrieved 7259: 7250: 7238:. Retrieved 7234:the original 7227: 7192: 7188: 7182: 7163: 7159: 7149: 7137: 7128: 7122: 7112: 7092: 7085: 7065: 7029: 7025: 7019: 7000: 6997:Toomer, G.J. 6991: 6965: 6956: 6951:, p. 9. 6944: 6925: 6919: 6894: 6869:. Retrieved 6864: 6854: 6842:. Retrieved 6837: 6827: 6784: 6780: 6749: 6730: 6723: 6704: 6698: 6684: 6639: 6555: 6533: 6528:INT(C + 0.5) 6518: 6511: 6506: 6431: 6424: 6402: 6213: 6111: 6104:ordinal date 6097: 6044: 6038: 6008: 5997: 5991: 5980: 5976: 5970: 5965: 5959: 5953: 5944: 5931: 5928: 5491: 5487: 5476: 5466: 5462: 5459: 5340: 5330: 5327: 5224: 5213: 5201: 5197: 5193:Err = | 5186: 5182: 5174: 5170: 5163: 5156: 5153: 5150: 5147: 5144: 5097: 4984: 4887: 4884: 4876: 4864: 4858: 4853: 4846: 4842: 4839: 4836: 4808: 4798:varies from 4794: 4773: 4765: 4761: 4758: 4752: 4746: 4741: 4726: 4722: 4715: 4709: 4700: 4694: 4685: 4682: 4663: 4658: 4652: 4648: 4641: 4416: 4410: 4404: 4401: 4394: 4390: 4383: 4376: 4372: 4362: 4356: 4350: 4347: 4112: 4106: 4095: 4087: 4083: 4058: 4041: 4028: 4019: 4015: 4011: 4008: 3843: 3837: 3830: 3826: 3819: 3811: 3807: 3800: 3796: 3790: 3784: 3777: 3773: 3769: 3765: 3761: 3663: 3659: 3652: 3648: 3629: 3618: 3615: 3533: 3527: 3524: 3518: 3514: 3508: 3502: 3496: 3490: 3487: 3403: 3397: 3391: 3388: 3318: 3312: 3309: 3203: 3197: 3191: 3188: 3140: 3138:, calculate 3134: 3131: 3124: 3120: 3114: 3108: 3097: 3093: 3090: 2996: 2992: 2985: 2981: 2975: 2972: 2882: 2859:| < 2855: 2846: 2839: 2835: 2824: 2817: 2813: 2805: 2798: 2794: 2786: 2776: 2772: 2765: 2761: 2755: 2749: 2745: 2645: 2642:true anomaly 2636: 2633: 2627: 2621: 2615: 2609: 2606: 2516: 2513: 2461: 2457: 2450: 2443: 2439: 2436: 2380: 2378:, calculate 2374: 2371: 2364: 2360: 2353: 2349: 2345:eccentricity 2336: 2332: 2324: 2314: 2303: 2296: 2292: 2288: 2284: 2277: 2273: 2256: 2252: 2248: 2242:true anomaly 2240:, the Sun's 2236: 2228: 2222:, the Sun's 2218: 2212: 2205: 2196: 2188: 2178: 2170: 2166: 2159: 2137: 2126: 2119: 2110: 2104: 2092:mean anomaly 2079: 2067: 2063: 2054: 2048: 1993: 1981: 1972: 1968: 1954: 1951: 1907: 1900: 1894: 1888: 1884: 1878: 1868: 1861: 1775: 1772: 1763: 1757: 1747: 1741: 1735: 1724: 1720: 1660: 1655: 1564: 1553: 1526: 1521: 1497: 1491: 1484: 1478: 1463: 1437: 1372: 1363: 1354: 1335: 1323:solar energy 1316: 1301: 1290: 1279: 1263: 1258:25 December 1232:1 September 1165:11 February 1133: 1015: 1012: 995: 988: 941: 824: 822: 798: 794: 773: 746: 738: 726: 724: 684: 671:Robert Hooke 669: 661: 646: 639: 616: 608: 595: 589: 569: 565: 559: 557: 528:Handy Tables 527: 521: 516:Book III of 515: 512: 499: 490: 480: 471: 467: 462: 443: 439: 437: 409: 400:eccentricity 377: 368: 354: 294: 114: 91: 86: 50: 42: 40: 18: 8677:Timekeeping 8636:System time 8631:Metric time 8350:Solar Hijri 8276:Water clock 8259:Radio clock 8191:Time domain 8171:Proper time 8057:Leap second 7939:Chronometry 7310:"Obliquity" 7199:: 336–347. 6548:Declination 4812:values are 2783:. Thus for 1351:Calculation 1243:3 November 980:declination 742:summer time 655:of the new 110:ephemerides 49:. The word 8666:Categories 8651:Timekeeper 8604:Chronology 8588:Millennium 8474:Precession 8380:Julian day 8201:T-symmetry 8062:Solar time 8032:Civil time 7781:"Sun Data" 7772:References 7617:Compendium 7352:Meeus 1997 7319:29 January 7294:29 January 7265:29 January 7240:29 January 7032:(2): 141. 6688:. London: 6517:means the 6409:half-turns 6223:perihelion 5221:Continuity 5206:− MICA2000 4638:Curves of 3411:and write 3316:calculate 3001:produces: 2973:For small 2802:, and for 1533:hour angle 1455:Hour Angle 1337:Civil time 1331:heliostats 1282:precession 991:precession 813:: one year 786:perihelion 744:, if any. 703:barycentre 695:perihelion 619:set clocks 584:See also: 117:sine waves 79:hour angle 47:solar time 8697:Equations 8548:Fortnight 8395:Lunisolar 8385:Leap year 8319:Gregorian 8269:stopwatch 8244:Hourglass 8224:Astrarium 8141:Spacetime 8072:Time zone 7949:Metrology 7928:standards 7677:126276982 7639:1074-8059 7625:CiteSeerX 7467:0035-8711 6819:119282288 6794:1208.1043 6617:⁡ 6611:⋅ 6606:∘ 6598:⁡ 6587:⁡ 6581:− 6575:δ 6481:⁡ 6475:− 6384:∘ 6370:∘ 6362:⁡ 6351:⁡ 6342:⁡ 6336:− 6284:− 6268:⁡ 6262:⋅ 6257:∘ 6183:− 6167:⁡ 6159:π 6154:∘ 6144:⋅ 6013:applies. 5867:− 5859:× 5798:λ 5747:− 5739:× 5694:− 5686:× 5680:− 5650:− 5637:ε 5600:− 5592:× 5586:− 5554:− 5546:× 5540:− 5532:− 5524:× 5438:λ 5434:⁡ 5427:ε 5423:⁡ 5412:⁡ 5401:λ 5386:− 5377:λ 5352:Δ 5305:π 5298:⁡ 5292:− 5289:η 5278:⁡ 5272:π 5263:⁡ 5251:⁡ 5246:η 5106:Δ 5055:− 5004:⁡ 4953:− 4910:⁡ 4901:− 4677:solar day 4502:π 4444:π 4322:λ 4301:⁡ 4275:− 4261:λ 4240:⁡ 4229:⁡ 4203:⁡ 4177:− 4161:Δ 4128:Δ 3971:⁡ 3954:⁡ 3945:− 3928:λ 3907:⁡ 3890:⁡ 3878:− 3859:Δ 3740:λ 3733:⁡ 3703:λ 3696:⁡ 3687:− 3684:λ 3681:≈ 3678:α 3591:λ 3585:α 3577:⁡ 3563:⁡ 3557:− 3554:λ 3548:α 3467:π 3457:λ 3453:⁡ 3446:ε 3442:⁡ 3431:⁡ 3422:α 3368:λ 3364:⁡ 3357:ε 3353:⁡ 3342:⁡ 3333:α 3286:⁡ 3252:⁡ 3231:λ 3221:≈ 3218:λ 3168:λ 3161:ν 3155:λ 3067:⁡ 3033:⁡ 3015:≈ 3012:ν 2949:⁡ 2915:⁡ 2900:≈ 2897:ν 2816:= arctan 2797:= arctan 2711:⁡ 2698:− 2672:⁡ 2660:ν 2583:⁡ 2549:⁡ 2534:≈ 2494:⁡ 2479:≈ 2417:⁡ 2408:− 2320:obliquity 2096:periapsis 2090:, is the 2034:α 2031:− 2022:λ 2006:Δ 1937:α 1934:− 1931:Λ 1922:Δ 1816:− 1813:α 1810:− 1706:α 1703:− 1694:α 1610:− 1607:α 1604:− 1409:− 1341:time zone 1255:0 s 1229:0 s 1203:0 s 1180:15 April 1177:0 s 1086:− 1035:⁡ 976:equinoxes 968:solstices 896:− 853:⁡ 844:− 801:sine wave 777:culminate 385:obliquity 258:⁡ 237:⁡ 228:− 209:Δ 173:− 8692:Sundials 8609:Duration 8583:Saeculum 8563:Olympiad 8405:Solstice 8334:Holocene 8311:Calendar 8211:Horology 8002:ISO 8601 7997:ISO 31-1 7855:web site 7742:Archived 7602:63-23531 7495:1 August 7371:. London 7314:Analemma 7289:moonkmft 7260:Analemma 6999:(1998). 6964:(1975), 6871:1 August 6844:1 August 6682:(1767). 6644:See also 4067:282.9381 3494:is 0 if 2322:, while 1964:ecliptic 1874:nutation 1319:sundials 1312:analemma 1270:analemma 1237:maximum 1217:26 July 1211:minimum 1206:13 June 1185:maximum 1159:minimum 1142:dates): 952:ecliptic 790:aphelion 623:meridian 561:Almagest 523:Almagest 502:sundials 491:aequātiō 482:equation 473:aequātiō 453:mnemonic 432:analemma 422:Notation 389:ecliptic 106:almanacs 94:analemma 51:equation 37:" below. 35:Notation 8578:Century 8568:Lustrum 8498:Instant 8370:Equinox 8339:Islamic 8281:Sundial 8146:Chronon 7375:23 July 7056:1005726 6799:Bibcode 6650:Azimuth 6034:⁠ 6022:⁠ 5816:282.938 5647:23.4393 4859:where: 4047:23.4393 4005:minutes 3841:and in 3816:allows 3640:⁠ 3626:⁠ 2873:⁠ 2861:⁠ 2854:arctan 2845:arctan 2843:. Here 2812:arctan 2804:π < 2793:arctan 2748:arctan 2343:is the 2329:√ 2318:is the 2146:365.259 2087:⁠ 2059:⁠ 1958:is the 1557:is the 1537:where: 1515:⁠ 1503:⁠ 1302:If the 1191:14 May 627:stellar 604:Setting 594:in the 549:⁄ 539:⁄ 518:Ptolemy 459:History 402:of the 393:equator 387:of the 379:stars: 339:Concept 83:sundial 61:of the 8573:Decade 8528:Moment 8523:Minute 8518:Second 8488:Other 8345:Julian 8324:Hebrew 7970:offset 7812: 7675: 7637: 7627: 7600: 7571: 7518: 7465: 7409: 7206:105569 7203: 7073: 7054: 7007: 6980: 6932: 6907: 6817: 6756: 6711: 6584:arcsin 6422:arctan 6403:Here, 6339:arctan 6141:0.0167 6098:where 6039:where 5826:1.7195 5521:1.6709 5390:arctan 5295:arctan 5260:arctan 5242:arctan 5227:arctan 5208:| 5079:3.5932 5046:365.25 4944:365.25 4821:75.340 4814:69.078 4387:, and 4360:, and 3988:3.5932 3616:where 3560:arcsin 3488:where 3428:arctan 3339:arctan 2852:| 2789:< π 2669:arctan 1904:, and 1721:where 1304:gnomon 1110:3.5932 811:period 803:with: 697:), and 600:Rising 442:minus 416:Uranus 351:(1853) 349:Naples 295:where 275:3.5932 164:365.25 65:, and 8614:music 8553:Month 8513:Jiffy 8508:Shake 8503:Flick 8400:Solar 8390:Lunar 8365:Epact 8329:Hindu 8264:Watch 8219:Clock 7695:(PDF) 7673:S2CID 7339:(PDF) 7201:JSTOR 7052:JSTOR 6815:S2CID 6789:arXiv 6663:Notes 6602:23.44 6510:nint( 6366:23.44 6253:1.914 6112:would 5856:3.025 5653:0.013 5618:36525 5543:4.193 5336:round 5275:round 5032:0.017 5018:6.240 4998:9.863 4930:0.017 4916:6.240 4904:7.659 4879:≤ 364 4764:> 4759:When 4735:2.508 4683:Here 4608:0.017 4594:6.240 4555:0.017 4541:6.240 4530:days 4472:days 4071:4.938 4051:0.409 4034:0.016 3965:9.863 3948:7.659 3621:= tan 2384:from 2331:1 − ( 1952:here 1862:Here 1531:(see 1512:24 hr 1248:zero 1222:zero 1196:zero 1170:zero 1154:Date 1151:Value 1148:Point 1136:epoch 1063:0.017 1049:6.240 1029:9.863 873:0.017 859:6.240 847:7.659 770:line) 476:(and 395:; and 252:9.863 231:7.659 150:0.017 136:6.240 8558:Year 8543:Week 8533:Hour 7987:DUT1 7926:and 7810:ISBN 7635:ISSN 7598:LCCN 7569:ISBN 7516:ISBN 7497:2022 7463:ISSN 7407:ISBN 7377:2013 7321:2021 7296:2021 7267:2021 7242:2021 7071:ISBN 7005:ISBN 6978:ISBN 6930:ISBN 6905:ISBN 6873:2022 6846:2022 6754:ISBN 6709:ISBN 6550:and 6478:nint 6430:tan 6025:360° 5589:1.26 5058:2000 4956:2000 4875:0 ≤ 4840:5) 4823:4748 4819:and 4816:9468 4804:4017 4800:3653 4718:= 2π 4645:and 4580:4) 4110:and 4101:− 2π 4069:° = 4049:° = 3805:and 3772:) = 2810:use 2808:≤ 2π 2791:use 2785:0 ≤ 2356:≤ 2π 2352:0 ≤ 2148:6358 1745:and 1517:days 1446:and 1329:and 1089:2000 899:2000 448:sign 412:Mars 398:the 383:the 176:2000 108:and 87:Mean 71:mean 41:The 8538:Day 7665:doi 7561:doi 7453:doi 7449:238 7168:doi 7042:hdl 7034:doi 6970:doi 6901:205 6807:doi 6614:cos 6595:sin 6521:to 6464:720 6380:180 6359:cos 6348:tan 6265:sin 6164:sin 6150:360 5942:), 5939:525 5890:525 5843:525 5770:525 5717:525 5670:525 5576:525 5431:tan 5420:cos 5098:6) 5036:201 5022:040 5001:sin 4934:201 4920:040 4907:sin 4737:109 4661:vs 4612:201 4598:040 4559:201 4545:040 4424:3) 4298:sin 4237:cos 4226:sin 4200:sin 4120:2) 4078:UT1 4073:201 4053:093 4036:709 3968:sin 3951:sin 3904:sin 3887:sin 3851:1) 3730:sin 3693:sin 3655:= 0 3574:sin 3450:tan 3439:cos 3361:tan 3350:cos 3283:sin 3249:sin 3064:sin 3030:sin 2946:sin 2912:sin 2880:in 2831:tan 2827:= π 2820:+ π 2781:= 0 2708:tan 2580:sin 2546:sin 2491:sin 2453:= 0 2414:sin 2182:is 2117:to 2102:as 2062:2π( 1966:). 1077:365 1067:201 1053:040 1032:sin 887:365 877:201 863:040 850:sin 602:or 570:zij 520:'s 373:UT1 357:min 255:sin 234:sin 154:201 140:040 63:Sun 8668:: 7982:ΔT 7977:UT 7711:^ 7703:88 7701:. 7697:. 7671:. 7661:10 7659:. 7647:^ 7633:. 7621:14 7619:. 7567:. 7530:^ 7487:. 7475:^ 7461:. 7447:. 7441:. 7421:^ 7385:^ 7367:. 7312:. 7287:. 7275:^ 7258:. 7226:. 7215:^ 7195:. 7193:54 7191:. 7164:61 7162:. 7158:. 7102:^ 7050:. 7040:. 7030:46 7028:. 6976:, 6903:. 6881:^ 6863:. 6836:. 6813:. 6805:. 6797:. 6785:34 6783:. 6768:^ 6740:^ 6671:^ 6419:, 6118:. 6047:. 6020:= 5983:ey 5937:36 5886:36 5863:10 5839:36 5820:07 5766:36 5743:10 5713:36 5690:10 5666:36 5596:10 5572:36 5550:10 5528:10 5465:+ 5202:ey 5187:ey 5175:ey 5167:, 5040:97 5026:77 4938:97 4924:77 4852:+ 4845:= 4772:, 4756:. 4733:= 4725:= 4653:ey 4616:97 4602:77 4563:97 4549:77 4354:, 4088:ey 4065:= 4045:= 4039:, 4032:= 4018:+ 4014:= 3847:. 3831:ey 3783:− 3776:+ 3662:= 3657:, 3651:= 3522:. 3517:= 3207:: 3129:. 2838:= 2779:=0 2460:= 2455:, 2388:: 2327:= 2295:, 2291:, 2287:, 2283:, 2276:, 2255:+ 2251:= 2232:); 2169:− 2165:= 2144:= 2125:+ 2066:− 2057:= 1506:UT 1501:+ 1495:= 1488:: 1140:UT 1071:97 1057:77 881:97 867:77 827:: 749:ms 667:. 637:. 576:. 335:. 158:97 144:77 112:. 85:. 7916:e 7909:t 7902:v 7818:. 7737:" 7679:. 7667:: 7641:. 7604:. 7577:. 7563:: 7524:. 7499:. 7469:. 7455:: 7415:. 7379:. 7354:. 7341:. 7323:. 7298:. 7269:. 7244:. 7209:. 7176:. 7170:: 7079:. 7058:. 7044:: 7036:: 7013:. 6972:: 6938:. 6913:. 6875:. 6848:. 6821:. 6809:: 6801:: 6791:: 6762:. 6717:. 6692:. 6624:) 6620:B 6591:( 6578:= 6562:δ 6558:B 6523:C 6514:) 6512:C 6490:) 6485:C 6472:C 6468:( 6461:= 6457:T 6454:O 6451:E 6437:C 6432:x 6425:x 6417:x 6405:C 6354:B 6333:A 6327:= 6324:C 6312:. 6299:) 6295:n 6291:) 6287:3 6281:D 6277:( 6272:( 6249:+ 6246:A 6243:= 6240:B 6227:B 6219:D 6215:B 6198:) 6194:n 6190:) 6186:3 6180:D 6176:( 6171:( 6138:+ 6135:A 6132:= 6129:B 6116:D 6108:A 6100:D 6083:n 6079:) 6075:9 6072:+ 6069:D 6065:( 6061:= 6058:A 6041:n 6028:/ 6018:n 5998:M 5992:t 5990:Δ 5981:t 5979:Δ 5971:t 5969:Δ 5960:ε 5954:e 5948:p 5945:λ 5932:D 5901:2 5896:) 5882:D 5877:( 5870:4 5853:+ 5849:) 5835:D 5830:( 5823:+ 5813:= 5803:p 5781:3 5776:) 5762:D 5757:( 5750:7 5736:5 5733:+ 5728:2 5723:) 5709:D 5704:( 5697:7 5683:2 5676:) 5662:D 5657:( 5644:= 5628:2 5623:) 5615:D 5610:( 5603:7 5582:) 5568:D 5563:( 5557:5 5535:2 5518:= 5511:e 5492:t 5490:Δ 5477:t 5475:Δ 5470:p 5467:λ 5463:M 5456:. 5443:) 5416:( 5405:p 5397:+ 5394:M 5381:p 5373:+ 5370:M 5367:= 5364:) 5361:M 5358:( 5355:t 5331:η 5324:. 5310:) 5301:x 5283:( 5269:+ 5266:x 5257:= 5254:x 5214:t 5212:Δ 5198:t 5196:Δ 5183:t 5181:Δ 5171:t 5169:Δ 5164:t 5162:Δ 5157:d 5131:b 5128:+ 5125:a 5122:= 5117:y 5114:e 5110:t 5083:) 5076:+ 5073:) 5070:) 5067:d 5064:+ 5061:) 5052:y 5049:( 5043:( 5029:+ 5015:( 5012:2 5008:( 4995:= 4992:b 4971:) 4968:) 4965:d 4962:+ 4959:) 4950:y 4947:( 4941:( 4927:+ 4913:( 4898:= 4895:a 4877:d 4868:Y 4865:n 4854:d 4850:Y 4847:n 4843:D 4832:π 4828:π 4809:M 4795:D 4790:π 4786:π 4782:π 4778:π 4774:M 4769:p 4766:D 4762:D 4753:D 4747:n 4730:p 4727:D 4723:D 4716:M 4710:D 4703:D 4701:M 4695:M 4688:D 4686:M 4679:. 4664:d 4649:t 4647:Δ 4642:t 4640:Δ 4619:D 4605:+ 4591:= 4588:M 4566:D 4552:+ 4538:= 4518:D 4511:Y 4507:t 4499:2 4493:+ 4488:D 4484:M 4480:= 4460:n 4453:Y 4449:t 4441:2 4435:= 4432:M 4417:D 4411:n 4405:M 4395:e 4391:t 4389:Δ 4384:α 4377:e 4373:t 4371:Δ 4366:p 4363:λ 4357:ε 4351:e 4332:) 4326:p 4318:4 4315:+ 4312:M 4309:4 4305:( 4293:2 4289:y 4283:2 4280:1 4271:) 4265:p 4257:2 4254:+ 4251:M 4248:2 4244:( 4233:M 4223:y 4220:e 4217:4 4214:+ 4210:M 4207:2 4195:2 4191:e 4185:4 4182:5 4172:y 4169:e 4165:t 4158:= 4151:2 4147:y 4141:2 4137:e 4132:t 4113:y 4107:e 4099:p 4096:λ 4094:2 4084:t 4082:Δ 4062:p 4059:λ 4042:ε 4029:e 4023:p 4020:λ 4016:M 4012:λ 3992:) 3985:+ 3982:M 3979:2 3975:( 3962:+ 3958:M 3942:= 3938:) 3932:p 3924:2 3921:+ 3918:M 3915:2 3911:( 3901:y 3898:+ 3894:M 3884:e 3881:2 3875:= 3870:y 3867:e 3863:t 3844:y 3838:e 3827:t 3825:Δ 3820:t 3818:Δ 3814:) 3812:λ 3810:( 3808:α 3803:) 3801:M 3799:( 3797:λ 3791:E 3785:α 3781:p 3778:λ 3774:M 3770:M 3768:( 3766:t 3764:Δ 3753:. 3737:4 3725:2 3721:y 3715:2 3712:1 3707:+ 3700:2 3690:y 3664:λ 3660:α 3653:y 3649:ε 3643:) 3637:2 3634:/ 3630:ε 3623:( 3619:y 3600:) 3595:) 3588:+ 3581:( 3571:y 3567:( 3551:= 3534:ν 3528:α 3519:λ 3515:α 3509:λ 3503:λ 3497:λ 3491:k 3484:, 3471:) 3464:k 3461:+ 3435:( 3425:= 3409:π 3404:λ 3398:λ 3392:α 3373:) 3346:( 3336:= 3319:α 3313:λ 3306:. 3293:M 3290:2 3278:2 3274:e 3268:4 3265:5 3260:+ 3256:M 3246:e 3243:2 3240:+ 3235:p 3227:+ 3224:M 3204:ν 3198:M 3192:λ 3172:p 3164:+ 3158:= 3141:λ 3135:ν 3127:) 3125:M 3123:( 3121:ν 3115:e 3109:e 3100:) 3098:M 3096:( 3094:ν 3087:. 3074:M 3071:2 3059:2 3055:e 3049:4 3046:5 3041:+ 3037:M 3027:e 3024:2 3021:+ 3018:M 2999:) 2997:E 2995:( 2993:ν 2988:) 2986:M 2984:( 2982:E 2976:e 2969:. 2956:E 2953:2 2941:2 2937:e 2931:4 2928:1 2923:+ 2919:E 2909:e 2906:+ 2903:E 2883:e 2870:2 2867:/ 2864:π 2856:x 2847:x 2840:E 2836:ν 2825:E 2818:x 2814:x 2806:E 2799:x 2795:x 2787:E 2777:E 2773:ν 2768:) 2766:M 2764:( 2762:E 2756:ν 2750:x 2730:) 2726:E 2720:2 2717:1 2701:e 2695:1 2690:e 2687:+ 2684:1 2676:( 2666:2 2663:= 2646:ν 2637:E 2628:e 2622:E 2616:e 2610:e 2603:, 2590:M 2587:2 2575:2 2571:e 2565:2 2562:1 2557:+ 2553:M 2543:e 2540:+ 2537:M 2531:E 2517:e 2498:M 2488:e 2485:+ 2482:M 2476:E 2462:M 2458:E 2451:e 2446:) 2444:M 2442:( 2440:E 2421:E 2411:e 2405:E 2402:= 2399:M 2381:E 2375:M 2367:) 2365:M 2363:( 2361:α 2354:M 2339:) 2337:a 2335:/ 2333:b 2325:e 2315:ε 2297:E 2293:λ 2289:ν 2285:α 2281:p 2278:λ 2274:M 2260:p 2257:λ 2253:ν 2249:λ 2244:; 2237:ν 2229:M 2219:E 2206:t 2200:p 2197:t 2189:E 2179:t 2171:M 2167:Λ 2163:p 2160:λ 2141:Y 2138:t 2133:; 2130:Y 2127:t 2123:p 2120:t 2114:p 2111:t 2105:t 2100:π 2083:Y 2080:t 2076:/ 2073:) 2071:p 2068:t 2064:t 2055:M 2026:p 2018:+ 2015:M 2012:= 2009:t 1982:t 1980:Δ 1973:Λ 1955:Λ 1928:= 1925:t 1908:t 1906:Δ 1901:E 1895:E 1889:t 1887:Δ 1879:E 1869:E 1846:t 1843:e 1840:s 1837:f 1834:f 1831:o 1827:+ 1823:T 1820:U 1806:T 1803:S 1800:M 1797:G 1793:= 1790:E 1776:E 1767:M 1764:α 1758:α 1751:M 1748:α 1742:α 1736:α 1728:M 1725:α 1698:M 1690:= 1686:T 1683:O 1680:E 1664:M 1661:α 1640:t 1637:e 1634:s 1631:f 1628:f 1625:o 1621:+ 1617:T 1614:U 1600:T 1597:S 1594:A 1591:G 1587:= 1583:T 1580:O 1577:E 1554:α 1535:) 1527:α 1509:/ 1498:N 1492:t 1485:t 1479:N 1474:π 1466:π 1450:; 1422:A 1419:H 1416:M 1413:G 1405:A 1402:H 1399:G 1395:= 1391:T 1388:O 1385:E 1253:0 1227:0 1201:0 1175:0 1114:) 1107:+ 1104:) 1101:) 1098:d 1095:+ 1092:) 1083:y 1080:( 1074:( 1060:+ 1046:( 1043:2 1039:( 1026:= 1023:b 914:) 911:) 908:d 905:+ 902:) 893:y 890:( 884:( 870:+ 856:( 841:= 838:a 825:a 721:. 551:9 547:5 541:3 537:1 534:+ 532:8 361:s 323:y 303:d 279:) 272:+ 269:D 266:2 262:( 249:+ 246:) 243:D 240:( 225:= 220:y 217:e 213:t 188:) 185:d 182:+ 179:) 170:y 167:( 161:( 147:+ 133:= 130:D

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