The main use of astronomical quantities specified in this way is to calculate other relevant parameters of motion, in order to predict future positions and velocities. The applied tools of the disciplines of celestial mechanics or its subfield orbital mechanics (for predicting orbital paths and positions for bodies in motion under the gravitational effects of other bodies) can be used to generate an ephemeris, a table of values giving the positions and velocities of astronomical objects in the sky at a given time or times.
Astronomical quantities can be specified in any of several ways, for example, as a polynomial function of the time interval, with an epoch as a temporal point of origin (this is a common current way of using an epoch). Alternatively, the time-varying astronomical quantity can be expressed as a constant, equal to the measure that it had at the epoch, leaving its variation over time to be specified in some other way—for example, by a table, as was common during the 17th and 18th centuries.
The word epoch was often used in a different way in older astronomical literature, e.g. during the 18th century, in connection with astronomical tables. At that time, it was customary to denote as "epochs", not the standard date and time of origin for time-varying astronomical quantities, but rather the values at that date and time of those time-varying quantities themselves.[2] In accordance with that alternative historical usage, an expression such as 'correcting the epochs' would refer to the adjustment, usually by a small amount, of the values of the tabulated astronomical quantities applicable to a fixed standard date and time of reference (and not, as might be expected from current usage, to a change from one date and time of reference to a different date and time).
Epoch versus equinox
Astronomical data are often specified not only in their relation to an epoch or date of reference but also in their relations to other conditions of reference, such as coordinate systems specified by "equinox", or "equinox and equator", or "equinox and ecliptic" – when these are needed for fully specifying astronomical data of the considered type.
Date-references for coordinate systems
When the data are dependent for their values on a particular coordinate system, the date of that coordinate system needs to be specified directly or indirectly.
Celestial coordinate systems most commonly used in astronomy are equatorial coordinates and ecliptic coordinates. These are defined relative to the (moving) vernal equinox position, which itself is determined by the orientations of the Earth's rotation axis and orbit around the Sun. Their orientations vary (though slowly, e.g. due to precession), and there is an infinity of such coordinate systems possible. Thus the coordinate systems most used in astronomy need their own date-reference because the coordinate systems of that type are themselves in motion, e.g. by the precession of the equinoxes, nowadays often resolved into precessional components, separate precessions of the equator and of the ecliptic.
The epoch of the coordinate system need not be the same, and often in practice is not the same, as the epoch for the data themselves.
The difference between reference to an epoch alone, and a reference to a certain equinox with equator or ecliptic, is therefore that the reference to the epoch contributes to specifying the date of the values of astronomical variables themselves; while the reference to an equinox along with equator/ecliptic, of a certain date, addresses the identification of, or changes in, the coordinate system in terms of which those astronomical variables are expressed. (Sometimes the word 'equinox' may be used alone, e.g. where it is obvious from the context to users of the data in which form the considered astronomical variables are expressed, in equatorial form or ecliptic form.)
The equinox with equator/ecliptic of a given date defines which coordinate system is used. Most standard coordinates in use today refer to 2000 TT (i.e. to 12h (noon) on the Terrestrial Time scale on January 1, 2000, see below), which occurred about 64 seconds sooner than noon UT1 on the same date (see ΔT). Before about 1984, coordinate systems dated to 1950 or 1900 were commonly used.
There is a special meaning of the expression "equinox (and ecliptic/equator) of date". When coordinates are expressed as polynomials in time relative to a reference frame defined in this way, that means the values obtained for the coordinates in respect of any interval t after the stated epoch, are in terms of the coordinate system of the same date as the obtained values themselves, i.e. the date of the coordinate system is equal to (epoch + t).[a]
It can be seen that the date of the coordinate system need not be the same as the epoch of the astronomical quantities themselves. But in that case (apart from the "equinox of date" case described above), two dates will be associated with the data: one date is the epoch for the time-dependent expressions giving the values, and the other date is that of the coordinate system in which the values are expressed.
For example, orbital elements, especially osculating elements for minor planets, are routinely given with reference to two dates: first, relative to a recent epoch for all of the elements: but some of the data are dependent on a chosen coordinate system, and then it is usual to specify the coordinate system of a standard epoch which often is not the same as the epoch of the data. An example is as follows: For minor planet (5145) Pholus, orbital elements have been given including the following data:[3]
Epoch 2010 Jan. 4.0 TT . . . = JDT 2455200.5 M 72.00071 . . . . . . . .(2000.0) n. 0.01076162 .. . . . Peri . 354.75938 a 20.3181594 . . . . . Node . 119.42656 e. 0.5715321 . . . . . Incl .. 24.66109
where the epoch is expressed in terms of Terrestrial Time, with an equivalent Julian date. Four of the elements are independent of any particular coordinate system: M is mean anomaly (deg), n: mean daily motion (deg/d), a: size of semi-major axis (AU), e: eccentricity (dimensionless). But the argument of perihelion, longitude of the ascending node and the inclination are all coordinate-dependent, and are specified relative to the reference frame of the equinox and ecliptic of another date "2000.0", otherwise known as J2000, i.e. January 1.5, 2000 (12h on January 1) or JD 2451545.0.[4]
Epochs and periods of validity
In the particular set of coordinates exampled above, much of the elements has been omitted as unknown or undetermined; for example, the element n allows an approximate time-dependence of the element M to be calculated, but the other elements and n itself are treated as constant, which represents a temporary approximation (see Osculating elements).
Thus a particular coordinate system (equinox and equator/ecliptic of a particular date, such as J2000.0) could be used forever, but a set of osculating elements for a particular epoch may only be (approximately) valid for a rather limited time, because osculating elements such as those exampled above do not show the effect of future perturbations which will change the values of the elements.
Nevertheless, the period of validity is a different matter in principle and not the result of the use of an epoch to express the data. In other cases, e.g. the case of a complete analytical theory of the motion of some astronomical body, all of the elements will usually be given in the form of polynomials in interval of time from the epoch, and they will also be accompanied by trigonometrical terms of periodical perturbations specified appropriately. In that case, their period of validity may stretch over several centuries or even millennia on either side of the stated epoch.
Some data and some epochs have a long period of use for other reasons. For example, the boundaries of the IAUconstellations are specified relative to an equinox from near the beginning of the year 1875. This is a matter of convention, but the convention is defined in terms of the equator and ecliptic as they were in 1875. To find out in which constellation a particular comet stands today, the current position of that comet must be expressed in the coordinate system of 1875 (equinox/equator of 1875). Thus that coordinate system can still be used today, even though most comet predictions made originally for 1875 (epoch = 1875) would no longer be useful today, because of the lack of information about their time-dependence and perturbations.
Changing the standard equinox and epoch
To calculate the visibility of a celestial object for an observer at a specific time and place on the Earth, the coordinates of the object are needed relative to a coordinate system of the current date. If coordinates relative to some other date are used, then that will cause errors in the results. The magnitude of those errors increases with the time difference between the date and time of observation and the date of the coordinate system used, because of the precession of the equinoxes. If the time difference is small, then fairly easy and small corrections for the precession may well suffice. If the time difference gets large, then fuller and more accurate corrections must be applied. For this reason, a star position read from a star atlas or catalog based on a sufficiently old equinox and equator cannot be used without corrections if reasonable accuracy is required.
Additionally, stars move relative to each other through space. Apparent motion across the sky relative to other stars is called proper motion. Most stars have very small proper motions, but a few have proper motions that accumulate to noticeable distances after a few tens of years. So, some stellar positions read from a star atlas or catalog for a sufficiently old epoch require proper motion corrections as well, for reasonable accuracy.
Due to precession and proper motion, star data become less useful as the age of the observations and their epoch, and the equinox and equator to which they are referred, get older. After a while, it is easier or better to switch to newer data, generally referred to as a newer epoch and equinox/equator, than to keep applying corrections to the older data.
Specifying an epoch or equinox
Epochs and equinoxes are moments in time, so they can be specified in the same way as moments that indicate things other than epochs and equinoxes. The following standard ways of specifying epochs and equinoxes seem the most popular:
Julian days, e.g., JD 2433282.4235 for January 0.9235, 1950 TT
Besselian years (see below), e.g., 1950.0 or B1950.0 for January 0.9235, 1950 TT
Julian years, e.g., J2000.0 for January 1.5, 2000 TT
Besselian years, used mostly for star positions, can be encountered in older catalogs but are now becoming obsolete. The Hipparcos catalog summary,[5] for example, defines the "catalog epoch" as "J1991.25" (8.75 Julian years before January 1.5, 2000 TT, e.g., April 2.5625, 1991 TT).
Besselian years
A Besselian year is named after the German mathematician and astronomer Friedrich Bessel (1784–1846). Meeus 1991, p.125 defines the beginning of a Besselian year to be the moment at which the mean longitude of the Sun, including the effect of aberration and measured from the mean equinox of the date, is exactly 280 degrees. This moment falls near the beginning of the corresponding Gregorian year. The definition depended on a particular theory of the orbit of the Earth around the Sun, that of Newcomb (1895), which is now obsolete; for that reason among others, the use of Besselian years has also become or is becoming obsolete.
Lieske 1979, p.282 says that a "Besselian epoch" can be calculated from the Julian date according to
B = 1900.0 + (Julian date − 2415020.31352) / 365.242198781
Lieske's definition is not exactly consistent with the earlier definition in terms of the mean longitude of the Sun. When using Besselian years, specify which definition is being used.
To distinguish between calendar years and Besselian years, it became customary to add ".0" to the Besselian years. Since the switch to Julian years in the mid-1980s, it has become customary to prefix "B" to Besselian years. So, "1950" is the calendar year 1950, and "1950.0" = "B1950.0" is the beginning of Besselian year 1950.
The IAU constellation boundaries are defined in the equatorial coordinate system relative to the equinox of B1875.0.
The classical star atlas Tabulae Caelestes used B1925.0 as its equinox.
According to Meeus, and also according to the formula given above,
B1900.0 = JDE 2415020.3135 = 1900 January 0.8135 TT
B1950.0 = JDE 2433282.4235 = 1950 January 0.9235 TT
Julian years and J2000
A Julian year is an interval with the length of a mean year in the Julian calendar, i.e. 365.25 days. This interval measure does not itself define any epoch: the Gregorian calendar is in general use for dating. But, standard conventional epochs which are not Besselian epochs have been often designated nowadays with a prefix "J", and the calendar date to which they refer is widely known, although not always the same date in the year: thus "J2000" refers to the instant of 12 noon (midday) on January 1, 2000, and J1900 refers to the instant of 12 noon on January 0, 1900, equal to December 31, 1899.[6] It is also usual now to specify on what time scale the time of day is expressed in that epoch-designation, e.g. often Terrestrial Time.
In addition, an epoch optionally prefixed by "J" and designated as a year with decimals (2000 + x), where x is either positive or negative and is quoted to 1 or 2 decimal places, has come to mean a date that is an interval of x Julian years of 365.25 days away from the epoch J2000 = JD 2451545.0 (TT), still corresponding (in spite of the use of the prefix "J" or word "Julian") to the Gregorian calendar date of January 1, 2000, at 12h TT (about 64 seconds before noon UTC on the same calendar day).[7] (See also Julian year (astronomy).) Like the Besselian epoch, an arbitrary Julian epoch is therefore related to the Julian date by
J = 2000 + (Julian date − 2451545.0) ÷ 365.25
The IAU decided at their General Assembly of 1976[8] that the new standard equinox of J2000.0 should be used starting in 1984. Before that, the equinox of B1950.0 seems to have been the standard.[citation needed]
Different astronomers or groups of astronomers used to define individually, but today standard epochs are generally defined by international agreements through the IAU, so astronomers worldwide can collaborate more effectively. It is inefficient and error-prone if data or observations of one group have to be translated in non-standard ways so that other groups could compare the data with information from other sources. An example of how this works: if a star's position is measured by someone today, they then use a standard transformation to obtain the position expressed in terms of the standard reference frame of J2000, and it is often then this J2000 position which is shared with others.
On the other hand, there has also been an astronomical tradition of retaining observations in just the form in which they were made, so that others can later correct the reductions to standard if that proves desirable, as has sometimes occurred.
The currently used standard epoch "J2000" is defined by international agreement to be equivalent to:
Over shorter timescales, there are a variety of practices for defining when each day begins. In ordinary usage, the civil day is reckoned by the midnight epoch, that is, the civil day begins at midnight. But in older astronomical usage, it was usual, until January 1, 1925, to reckon by a noon epoch, 12 hours after the start of the civil day of the same denomination, so that the day began when the mean sun crossed the meridian at noon.[11] This is still reflected in the definition of J2000, which started at noon, Terrestrial Time.
In traditional cultures and in antiquity other epochs were used. In ancient Egypt, days were reckoned from sunrise to sunrise, following a morning epoch. This may be related to the fact that the Egyptians regulated their year by the heliacal rising of the star Sirius, a phenomenon which occurs in the morning just before dawn.[12]
In some cultures following a lunar or lunisolar calendar, in which the beginning of the month is determined by the appearance of the New Moon in the evening, the beginning of the day was reckoned from sunset to sunset, following an evening epoch, e.g. the Jewish and Islamic calendars[13] and in Medieval Western Europe in reckoning the dates of religious festivals,[14] while in others a morning epoch was followed, e.g. the Hindu and Buddhist calendars.
In astronomy, declination is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. The declination angle is measured north (positive) or south (negative) of the celestial equator, along the hour circle passing through the point in question.
The ecliptic or ecliptic plane is the orbital plane of Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic against the background of stars. The ecliptic is an important reference plane and is the basis of the ecliptic coordinate system.
Right ascension is the angular distance of a particular point measured eastward along the celestial equator from the Sun at the March equinox to the point in question above the Earth. When paired with declination, these astronomical coordinates specify the location of a point on the celestial sphere in the equatorial coordinate system.
A year is the time taken for astronomical objects to complete one orbit. For example, a year on Earth is the time taken for Earth to revolve around the Sun. Generally, a year is taken to mean a calendar year, but the word is also used for periods loosely associated with the calendar or astronomical year, such as the seasonal year, the fiscal year, the academic year, etc. The term can also be used in reference to any long period or cycle, such as the Great Year.
The zodiac is a belt-shaped region of the sky that extends approximately 8° north and south of the ecliptic, the apparent path of the Sun across the celestial sphere over the course of the year. Also within this zodiac belt appear the Moon and the brightest planets, along their orbital planes. The zodiac is divided along the ecliptic into 12 equal parts ("signs"), each occupying 30° of celestial longitude. These signs roughly correspond to the astronomical constellations with the following modern names: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, and Pisces.
The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fundamental plane consisting of the projection of Earth's equator onto the celestial sphere, a primary direction towards the March equinox, and a right-handed convention.
In astronomy, the ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets and many small Solar System bodies have orbits with only slight inclinations to the ecliptic, using it as the fundamental plane is convenient. The system's origin can be the center of either the Sun or Earth, its primary direction is towards the March equinox, and it has a right-hand convention. It may be implemented in spherical or rectangular coordinates.
Sidereal time is a system of timekeeping used especially by astronomers. Using sidereal time and the celestial coordinate system, it is easy to locate the positions of celestial objects in the night sky. Sidereal time is a "time scale that is based on Earth's rate of rotation measured relative to the fixed stars".
A sidereal year, also called a sidereal orbital period, is the time that Earth or another planetary body takes to orbit the Sun once with respect to the fixed stars.
In astronomy, axial precession is a gravity-induced, slow, and continuous change in the orientation of an astronomical body's rotational axis. In the absence of precession, the astronomical body's orbit would show axial parallelism. In particular, axial precession can refer to the gradual shift in the orientation of Earth's axis of rotation in a cycle of approximately 26,000 years. This is similar to the precession of a spinning top, with the axis tracing out a pair of cones joined at their apices. The term "precession" typically refers only to this largest part of the motion; other changes in the alignment of Earth's axis—nutation and polar motion—are much smaller in magnitude.
The first point of Aries, also known as the cusp of Aries, is the location of the March equinox, used as a reference point in celestial coordinate systems. In diagrams using such coordinate systems, it is often indicated with the symbol ♈︎. Named for the constellation of Aries, it is one of the two points on the celestial sphere at which the celestial equator crosses the ecliptic, the other being the first point of Libra, located exactly 180° from it. Due to precession of the equinoxes since the positions were originally named in antiquity, the position of the Sun when at the March equinox is now in Pisces; when it is at the September equinox, it is in Virgo.
In astronomy, a Julian year is a unit of measurement of time defined as exactly 365.25 days of 86400 SI seconds each. The length of the Julian year is the average length of the year in the Julian calendar that was used in Western societies until the adoption of the Gregorian Calendar, and from which the unit is named. Nevertheless, because astronomical Julian years are measuring duration rather than designating dates, this Julian year does not correspond to years in the Julian calendar or any other calendar. Nor does it correspond to the many other ways of defining a year.
Spherical astronomy, or positional astronomy, is a branch of observational astronomy used to locate astronomical objects on the celestial sphere, as seen at a particular date, time, and location on Earth. It relies on the mathematical methods of spherical trigonometry and the measurements of astrometry.
The Moon orbits Earth in the prograde direction and completes one revolution relative to the Vernal Equinox and the stars in about 27.32 days and one revolution relative to the Sun in about 29.53 days. Earth and the Moon orbit about their barycentre, which lies about 4,670 km from Earth's centre, forming a satellite system called the Earth–Moon system. On average, the distance to the Moon is about 384,400 km (238,900 mi) from Earth's centre, which corresponds to about 60 Earth radii or 1.282 light-seconds.
The semi-analytic planetary theory VSOP is a mathematical model describing long-term changes in the orbits of the planets Mercury to Neptune. The earliest modern scientific model considered only the gravitational attraction between the Sun and each planet, with the resulting orbits being unvarying Keplerian ellipses. In reality, all the planets exert slight forces on each other, causing slow changes in the shape and orientation of these ellipses. Increasingly complex analytical models have been made of these deviations, as well as efficient and accurate numerical approximation methods.
In astronomy, an equinox is either of two places on the celestial sphere at which the ecliptic intersects the celestial equator. Although there are two such intersections, the equinox associated with the Sun's ascending node is used as the conventional origin of celestial coordinate systems and referred to simply as "the equinox". In contrast to the common usage of spring/vernal and autumnal equinoxes, the celestial coordinate system equinox is a direction in space rather than a moment in time.
Earth-centered inertial (ECI) coordinate frames have their origins at the center of mass of Earth and are fixed with respect to the stars. "I" in "ECI" stands for inertial, in contrast to the "Earth-centered – Earth-fixed" (ECEF) frames, which remains fixed with respect to Earth's surface in its rotation, and then rotates with respect to stars.
A tropical year or solar year is the time that the Sun takes to return to the same position in the sky – as viewed from the Earth or another celestial body of the Solar System – thus completing a full cycle of astronomical seasons. For example, it is the time from vernal equinox to the next vernal equinox, or from summer solstice to the next summer solstice. It is the type of year used by tropical solar calendars.
In lunar calendars, a lunar month is the time between two successive syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month.
Astronomical nutation is a phenomenon which causes the orientation of the axis of rotation of a spinning astronomical object to vary over time. It is caused by the gravitational forces of other nearby bodies acting upon the spinning object. Although they are caused by the same effect operating over different timescales, astronomers usually make a distinction between precession, which is a steady long-term change in the axis of rotation, and nutation, which is the combined effect of similar shorter-term variations.
↑ SOFA Time Scale and Calendar Tools(PDF) (Report). C programming language. International Astronomical Union. October 9, 2017. (Document revision 1.5.)
↑ Bede, The Reckoning of Time, 5, trans. Faith Wallis, (Liverpool: Liverpool University Press, 2004), pp. 22–24. ISBN0-85323-693-3
Sources
Aoki, S.; Soma, M.; Kinoshita, H.; Inoue, K. (December 1983). "Conversion matrix of epoch B 1950.0 FK 4-based positions of stars to epoch J 2000.0 positions in accordance with the new IAU resolutions". Astronomy and Astrophysics. 128 (3): 263–267. Bibcode:1983A&A...128..263A. ISSN0004-6361.
Lieske, J.H. (1979). "Precession Matrix Based on IAU (1976) System of Astronomical Constants". Astronomy & Astrophysics. 73 (3): 282–284. Bibcode:1979A&A....73..282L.
Simon, J. L.; Bretagnon, P.; Chapront, J.; Chapront-Touze, M.; Francou, G.; Laskar, J (1994). "Numerical expressions for precession formulae and mean elements for the Moon and the planets". Astronomy and Astrophysics. 282: 663–683. Bibcode:1994A&A...282..663S.
Wilson, H. C. (1925). "Change of astronomical time". Popular Astronomy. 33: 1–2. Bibcode:1925PA.....33....1W.
Further reading
Standish, E. M. Jr. (November 1982). "Conversion of positions and proper motions from B1950.0 to the IAU system at J2000.0". Astronomy and Astrophysics. 115 (1): 20–22. Bibcode:1982A&A...115...20S.
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