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Encyclopedia > Equation of Time

The equation of time is the difference, over the course of a year, between time as read from a sundial and a clock. The sundial can be ahead (fast) by as much as 16 min 33 s (around November 3) or fall behind by as much as 14 min 6 s (around February 12). It is caused by irregularity of the motion of the Sun in the sky, due to a combination of the obliquity of the Earth's rotation axis and the eccentricity of its orbit. The equation of time is the east or west component of the analemma, a curve representing the angular offset of the Sun from its mean position on the celestial sphere as viewed from Earth. Wall sundial-a vertical direct south dial Wall sundial in Warsaws Old Town- a vertical south west decliner dial A sundial is a device that measures time by the position of the Sun. ... The massive clock on the Clock Tower of the Palace of Westminster, London (commonly known as Big Ben, although Big Ben is the bell inside - the picture is St Stephens Tower). ... A minute is a unit of time equal to 1/60th of an hour and to 60 seconds. ... Look up second in Wiktionary, the free dictionary. ... is the 307th day of the year (308th in leap years) in the Gregorian calendar. ... February 12 is the 43rd day of the year in the Gregorian calendar. ... “Sol” redirects here. ... In astronomy, Axial tilt is the inclination angle of a planets rotational axis in relation to a perpendicular to its orbital plane. ... This article is about Earth as a planet. ... In astrodynamics, under standard assumptions any orbit must be of conic section shape. ... Two bodies with a slight difference in mass orbiting around a common barycenter. ... The analemma photographed, looking east in the northern hemisphere. ...


Naturally, other planets will have an equation of time too. On Mars the difference between sundial time and clock time can be as much as 50 minutes, due to its orbit's considerably greater eccentricity. The eight planets and three dwarf planets of the Solar System. ... Mars is the fourth planet from the Sun in the solar system, named after the Roman god of war (the counterpart of the Greek Ares), on account of its blood red color as viewed in the night sky. ...

Contents

Apparent time versus mean time

The irregular daily movement of the Sun was known by the Babylonians, and Ptolemy has a whole chapter in the Almagest devoted to its calculation (Book III, chapter 9). However he did not consider the effect relevant for most calculations as the correction was negligible for the slow-moving luminaries. He only applied it for the fastest-moving luminary, the moon. A medieval artists rendition of Claudius Ptolemaeus Claudius Ptolemaeus (Greek: ; ca. ... Almagest is the Latin form of the Arabic name (al-kitabu-l-mijisti, i. ...


Until the invention of the pendulum and the development of reliable clocks towards the end of the 17th century, the equation of time as defined by Ptolemy remained a curiosity, not important to normal people except astronomers. Only when mechanical clocks started to take over timekeeping from sundials, which had served humanity for centuries, did the difference between clock time and solar time become an issue. Apparent solar time (or true or real solar time) is the time indicated by the Sun on a sundial, while mean solar time is the average as indicated by clocks. Solar time is based on the idea that when the sun reaches its highest point in the sky, it is noon. ... Solar time is based on the idea that, when the sun reaches its highest point in the sky, it is noon. ...


Until 1833, the equation of time was mean minus apparent solar time in the British Nautical Almanac and Astronomical Ephemeris. Earlier, all times in the almanac were in apparent solar time because time aboard ship was determined by observing the Sun. In the unusual case that the mean solar time of an observation was needed, the extra step of adding the equation of time to apparent solar time was needed. Since 1834, all times have been in mean solar time because by then the time aboard most ships was determined by marine chronometers. In the unusual case that the apparent solar time of an observation was needed, the extra step of adding the equation of time to mean solar time was needed, requiring all differences in the equation of time to have the opposite sign. A nautical almanac is a publication describing the positions and movements of celestial bodies, including the sun, moon, planets, and 57 stars chosen for their ease of identification and wide spacing. ... A marine chronometer is a timekeeper precise enough to be used as a portable time standard, used to determine longitude by means of celestial navigation. ...


As the daily movement of the Sun is one revolution per day, that is 360° every 24 hours or 1° every 4 minutes, and the Sun itself appears as a disc of about 0.5° in the sky, simple sundials can be read to a maximum accuracy of about one minute. Since the equation of time has a range of about 30 minutes, the difference between sundial time and clock time cannot be ignored. In addition to the equation of time, one also has to apply corrections due to one's distance from the local time zone meridian and summertime, if any. Though DST is common in Europe and North America, most of the worlds people do not use it. ...


The tiny increase of the mean solar day itself due to the slowing down of the Earth's rotation, by about 2 ms per day per century, which currently accumulates up to about 1 second every year, has nothing to do with the equation of time, and is completely irrelevant at the accuracy given by sundials.


Eccentricity of the Earth's orbit

The Earth revolves around the Sun. As such it appears that the Sun moves in one year around the Earth. If the Earth orbited the Sun with a constant speed in a plane perpendicular to its axis, then the apparent Sun would culminate every day at exactly 12 o'clock, and be a perfect time keeper (except for its slowing rotation). But the orbit of the Earth is an ellipse and its speed varies between 30.287 and 29.291 km/s, according to Kepler's laws of planetary motion, and as such the Sun seems to move faster at perihelion (currently around 3 January) and slower at aphelion a half year later. At these extreme instances this effect increases (respectively, decreases) the real solar day by 7.9 seconds. This accumulates every day. The final result is that the eccentricity of the Earth's orbit contributes a sine wave variation with an amplitude of 7.66 minutes and a period of one year to the equation of time. The zero points are reached at perihelion (at the beginning of January) and aphelion (beginning of July) while the maximum values are at the beginnings of April (negative) and October (positive). In astronomy, the culmination, at a given point, of a planet, star, constellation, etc. ... Johannes Keplers primary contributions to astronomy/astrophysics were his three laws of planetary motion. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... January 3 is the 3rd day of the year in the Gregorian calendar. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ...


Obliquity of the ecliptic

Sun and planets at solar midday (Ecliptic in red, Sun and Mercury in yellow, Venus in white, Mars in red, Jupiter in yellow with red spot, Saturn in white with rings).

However, even if the Earth's orbit were circular, the motion of the Sun along the celestial equator would still not be uniform. This is a consequence of the tilt of the Earth's rotation with respect to its orbit, or equivalently, the tilt of the ecliptic (the path of the sun against the celestial sphere) with respect to the celestial equator. The projection of this motion onto the celestial equator, along which "clock time" is measured, is a maximum at the solstices, when the yearly movement of the Sun is parallel to the equator and appears as a change in right ascension, and is a minimum at the equinoxes, when the Sun moves in a sloping direction and appears mainly as a change in declination, leaving less for the component in right ascension, which is the only component that affects the duration of the solar day. As a consequence of that, the daily shift of the shadow cast by the Sun in a sundial, due to obliquity, is smaller close to the equinoxes and greater close to the solstices. At the equinoxes, the Sun is seen slowing down by up to 20.3 seconds every day and at the solstices speeding up by the same amount. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The celestial equator is a great circle on the imaginary celestial sphere, which could be constructed by inflating the Earths equator until it intersects with said sphere. ... The plane of the ecliptic is well seen in this picture from the 1994 lunar prospecting Clementine spacecraft. ... The celestial sphere is divided by the celestial equator. ... The celestial equator is a great circle on the imaginary celestial sphere, which could be constructed by inflating the Earths equator until it intersects with said sphere. ... The celestial equator is a great circle on the imaginary celestial sphere, which could be constructed by inflating the Earths equator until it intersects with said sphere. ... Equatorial Coordinates Right ascension (abbrev. ... In astronomy, declination (abbrev. ... Equatorial Coordinates Right ascension (abbrev. ... Wall sundial-a vertical direct south dial Wall sundial in Warsaws Old Town- a vertical south west decliner dial A sundial is a device that measures time by the position of the Sun. ... In astronomy, an equinox is defined as the moment when the sun reaches one of two intersections between the ecliptic and the celestial equator. ... Solstice is an astronomical term regarding the position of the Sun in relation to the celestial equator. ... In astronomy, an equinox is defined as the moment when the sun reaches one of two intersections between the ecliptic and the celestial equator. ... Solstice is an astronomical term regarding the position of the Sun in relation to the celestial equator. ...


In the figure on the right, we can see the monthly variation of the apparent slope of the plane of the ecliptic at solar midday as seen from Earth. This variation is due to the apparent precession of the rotating Earth through the year, as seen from the Sun at solar midday. Precession of a gyroscope Precession refers to a change in the direction of the axis of a rotating object. ...


In terms of the equation of time, the inclination of the ecliptic results in the contribution of another sine wave variation with an amplitude of 9.87 minutes and a period of a half year to the equation of time. The zero points of this sine wave are reached at the equinoxes and solstices, while the maxima are at the beginning of February and August (negative) and the beginning of May and November (positive).


Secular effects

The two above mentioned factors have different wavelengths, amplitudes and phases, so their combined contribution is an irregular wave. At epoch 2000 these are the values: In astronomy, an epoch is a moment in time for which celestial coordinates or orbital elements are specified. ...

minimum −14:15 11 February
zero 00:00 15 April
maximum +03:41 14 May
zero 00:00 13 June
minimum −06:30 26 July
zero 00:00 1 September
maximum +16:25 3 November
zero 00:00 25 December

E.T. = apparent − mean. Positive means: Sun runs fast and culminates earlier, or the sundial is ahead of mean time. A slight yearly variation occurs due to presence of leap years, resetting itself every 4 years.


The exact shape of the equation of time curve and the associated analemma slowly changes over the centuries due to secular variations in both eccentricity and obliquity. At this moment both are slowly decreasing, but in reality they vary up and down over a timescale of hundreds of thousands of years. When the eccentricity, now 0.0167, reaches 0.047, the eccentricity effect may in some circumstances overshadow the obliquity effect, leaving the equation of time curve with only one maximum and minimum per year. The analemma photographed, looking east in the northern hemisphere. ...


On shorter timescales (thousands of years) the shifts in the dates of equinox and perihelion will be more important. The former is caused by precession, and shifts the equinox backwards compared to the stars. But it can be ignored in the current discussion as our Gregorian calendar is constructed in such a way as to keep the vernal equinox date at 21 March (at least at sufficient accuracy for our aim here). The shift of the perihelion is forwards, about 1.7 days every century. For example in 1246 the perihelion occurred on 22 December, the day of the solstice. At that time the two contributing waves had common zero points, and the resulting equation of time curve was symmetrical. Before that time the February minimum was larger than the November maximum, and the May maximum larger than the July minimum. The secular change is evident when one compares a current graph of the equation of time (see below) with one from about 2000 years ago, for example, one constructed from the data of Ptolemy. Precession of a gyroscope Precession refers to a change in the direction of the axis of a rotating object. ... The Gregorian calendar is the most widely used calendar in the world. ... March 21 is the 80th day of the year (81st in leap years) in the Gregorian calendar. ... December 22 is the 356th day of the year (357th in leap years) in the Gregorian calendar. ...


Practical use

If the gnomon (the shadow casting object) is not an edge but a point (e.g., a hole in a plate), the shadow (or spot of light) will trace out a curve during the course of a day. If the shadow is cast on a plane surface, this curve will (usually) be the conic section of the hyperbola, since the circle of the Sun's motion together with the gnomon point define a cone. At the spring and fall equinoxes, the cone degenerates into a plane and the hyperbola into a line. With a different hyperbola for each day, hour marks can be put on each hyperbola which include any necessary corrections. Unfortunately, each hyperbola corresponds to two different days, one in each half of the year, and these two days will require different corrections. A convenient compromise is to draw the line for the "mean time" and add a curve showing the exact position of the shadow points at noon during the course of the year. This curve will take the form of a figure eight and is known as an "analemma". By comparing the analemma to the mean noon line, the amount of correction to be applied generally on that day can be determined. The cantilever spar of this cable-stay bridge, the Sundial Bridge at Turtle Bay, forms the gnomon of a large garden sundial The gnomon is the part of a sundial that casts the shadow. ... Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ...


More details

The equation of time is the sum of two offset sine curves, with periods of one year and six months respectively. It can be approximated by

where is in minutes and

if sin and cos have arguments in degrees,

or This article describes the unit of angle. ...

if sin and cos have arguments in radians.

Here, is the so-called day number; i.e., N = 1 for January 1, N = 2 for January 2, and so on. Some common angles, measured in radians. ... is the 1st day of the year in the Gregorian calendar. ... is the 2nd day of the year in the Gregorian calendar. ...


The following is a graph of the current equation of time.

Notice that the appearance of this graph can be directly deduced from the time evolution of the projection into the celestial equator of the Earth's Analemma loop trajectory. File links The following pages link to this file: Equation of time Categories: Public domain images ... The analemma photographed, looking east in the northern hemisphere. ...


From one year to the next, the equation of time can vary by as much as 20 seconds, mainly due to leap years. [1].


The equation of time also has a phase shift of about one day in 24.23 years. The equation as read from a table of 1683 lags 13 days behind the one of 1998.


See also

Sidereal time is time measured by the apparent diurnal motion of the vernal equinox, which is very close to, but not identical with, the motion of stars. ... In horology terms, a complication in a mechanical timepiece is any feature beyond that of a simple hours, minutes, and seconds movement. ... An astrarium, also called a planetarium, is the mechanical representation of the cyclic nature of astronomical objects in one timepiece. ...

References

  • J. Meeus, Mathematical astronomy morsels, ISBN 0-943396-51-4

External links

  • Novel Visualisation of Equation of Time - Constantly updated
  • Table giving the Equation of Time and the declination of the sun for every day of the year
  • The equation of time described on the Royal Greenwich Observatory website
  • An analemma site with many illustrations
  • The Equation of Time and the Analemma, by Kieron Taylor
  • An article by Brian Tung containing a link to a C program using a more accurate formula than most (particularly at high inclinations and eccentricities). The program can calculate solar declination, Equation of Time, or Analemma.
  • Doing calculations using Ptolemy's ephemeres, such as his E.T. graph
  • A dynamic and unique Equation of Time visualisation.
  • Equation of Time function for Excel, CAD or other programs. The Sun API is free and extremely accurate. For Windows computers.
  • The equation of time correction-table A page describing how to correct a clock to a sundial.
  • An example of an Audemars Piguet mechanical wristwatch containing this concept as a complication, including a description of the implementation in horology and several videos/animations.
  • Two more examples of a mechanical wristwatch containing this complication, manufactured by Blancpain: Part 1 Part 2.
  • The Sundial Primer

  Results from FactBites:
 
Equation of time - Wikipedia, the free encyclopedia (1454 words)
This difference, known as the equation of time, results from an irregular movement of the Sun caused by a combination of the obliquity of the Earth's rotation axis and the eccentricity of its orbit.
The equation of time is visually illustrated by an analemma.
The Equation of Time and the Analemma, by Kieron Taylor
Equation of Time (1240 words)
Solar time, or true solar time, which was the time shown on the sundials of the past.
Equation of time is the difference between mean solar time and true solar time.
In “astronomical” watches and clocks, equation of time is often enhanced with sidereal time, the phases and age of the moon, a perpetual ca-lendar, the times of sunrise and sunset and the sun’s declination, the movement of the planets, the map of the skies, or the ephémérides (solstices, equinoxes, seasons).
  More results at FactBites »

 
 

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