In astronomy, an epoch is a moment in time used as a reference point
for some time-varying astronomical quantity, such as the celestial
coordinates or elliptical orbital elements of a celestial body,
because these are subject to perturbations and vary with time.
These time-varying astronomical quantities might include, for example,
the mean longitude or mean anomaly of a body, the node of its orbit
relative to a reference plane, the direction of the apogee or aphelion
of its orbit, or the size of the major axis of its orbit.
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
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. 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).
1 Epoch versus equinox
1.1 Date-references for coordinate systems
1.2 Epochs and periods of validity
2 Changing the standard equinox and epoch
3 Specifying an epoch or equinox
4 Besselian years
5 Julian years and J2000
6 Epoch of the day
7 See also
9 External links
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
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 Jan 1.5 TT (i.e. to 12h on the
Terrestrial Time scale on
2000 Jan 1), which occurred about 64 seconds sooner than noon
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).
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:
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. 2000
Jan 1.5 (12h on January 1) or JD 2451545.0.
Epochs and periods of validity
In the particular set of coordinates exampled above, much of the
time-dependence 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
IAU constellations 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, because of the lack of
information about their time-dependence and perturbations, be useful
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 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 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 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 most popular:
Julian days, e.g., JD 2433282.4235 for 1950 January 0.9235 TT
Besselian years (see below), e.g., 1950.0 or B1950.0 for 1950 January
Julian years, e.g., J2000.0 for 2000 January 1.5000 TT
All three of these are expressed in TT = Terrestrial Time.
Besselian years, used mostly for star positions, can be encountered in
older catalogs but are now becoming obsolete. The
summary, for example, defines the "catalog epoch" as J1991.25 (8.75
Julian years before 2000 January 1.5000 TT, e.g., 1991 April 2.5625
A Besselian year is named after the German mathematician and
Friedrich Bessel (1784–1846). Meeus 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 says that a "Besselian epoch" can be calculated from the
Julian date according to
B = 1900.0 + (
Julian date − 2415020.31352) / 365.242198781
This relationship is included in the SOFA software library.
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.
IAU constellation boundaries are defined in the equatorial
coordinate system relative to the equinox of B1875.0.
Henry Draper Catalog uses the equinox B1900.0.
The classical star atlas Tabulae Caelestes used B1925.0 as its
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 1 January 2000, and J1900 refers to the instant
of 12 noon on 0 January 1900, equal to 31 Dec 1899. 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 positive or negative and
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 2000
Jan 1 at 12h TT (about 64 seconds before noon UTC on the same
calendar day). (See also Julian year (astronomy).) Like the
Besselian epoch, an arbitrary Julian epoch is therefore related to the
Julian date by
J = 2000.0 + (
Julian date − 2451545.0)/365.25 .
IAU decided at their General Assembly of 1976 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
Different astronomers or groups of astronomers used to define
individually, but today standard epochs are generally defined by
international agreement 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, he/she then uses 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:
Gregorian date January 1, 2000 at 12:00 TT (Terrestrial Time).
Julian date 2451545.0 TT (Terrestrial Time).
January 1, 2000, 11:59:27.816 TAI (International Atomic Time).
January 1, 2000, 11:58:55.816 UTC (Coordinated Universal Time).
Epoch of the day
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 1 January 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. 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.
In 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. This practice was followed in the
Jewish and Islamic calendars and in Medieval Western Europe in
reckoning the dates of religious festivals.
Epoch (reference date)
International Celestial Reference System
International Celestial Reference Frame
^ Soop, E. M. (1994). Handbook of Geostationary Orbits. Springer.
^ M Chapront-Touzé (ed.), Jean le Rond d'Alembert, Oeuvres
Complètes: Ser.1, Vol.6, Paris (CNRS) (2002), p.xxx, n.50.
^ Examples of this usage are seen in: J L Simon et al., "Numerical
expressions for precession formulae and mean elements for the Moon and
Astronomy and Astrophysics 282 (1994), pp. 663-683.
^ Harvard Minor Planet Center, data for Pholus[permanent dead link]
^ See Explanation of Orbital Elements.
Hipparcos and Tycho Catalogues", ESA SP-1200, Vol. 1, page XV.
^ Meeus, J.: "Astronomical Algorithms", page 125. Willmann-Bell, 1991
^ Lieske, J.H.: "
Precession Matrix Based on
IAU (1976) System of
Astronomical Constants", page 282.
Astronomy & Astrophysics, 73,
^ a b "SOFA Libraries Issue 2007-08-10". 2007-08-18. Retrieved
^ See NASA Jet Propulsion Laboratory 'spice' toolkit documentation,
^ 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
Astronomy and Astrophysics. 128 (3): 263–267.
Bibcode:1983A&A...128..263A. ISSN 0004-6361.
^ Seidelmann, P. K., Ed. (1992). Explanatory Supplement to the
Astronomical Almanac. Sausalito, CA: University Science Books. p. 8.
^ Seidelmann, P. K., Ed. (1992). Explanatory Supplement to the
Astronomical Almanac. Sausalito, CA: University Science Books.
Glossary, s.v. Terrestrial Dynamical Time.
^ This article uses a 24-hour clock, so 11:59:27.816 is equivalent to
^ H. C. Wilson, "Change of astronomical time", Popular Astronomy, 33
^ Otto Neugebauer, A History of Ancient Mathematical Astronomy, (New
York: Springer, 1975), p. 1067. ISBN 0-387-06995-X
^ Otto Neugebauer, A History of Ancient Mathematical Astronomy, (New
York: Springer, 1975), pp. 1067-1069. ISBN 0-387-06995-X
^ Bede, The Reckoning of Time, 5, trans. Faith Wallis, (Liverpool:
Liverpool University Press, 2004), pp. 22-24. ISBN 0-85323-693-3
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.
What is Terrestrial Time? - U.S. Naval Observatory
International Celestial Reference System, or ICRS - U.S. Naval
IERS Conventions 2003 (defines ICRS and other related standards)
Elliptical / Highly elliptical
Inclined / Non-inclined
Orbit of the Moon
About other points
a Semi-major axis
b Semi-minor axis
Q, q Apsides
Ω Longitude of the ascending node
ω Argument of periapsis
ϖ Longitude of the periapsis
M Mean anomaly
ν, θ, f True anomaly
E Eccentric anomaly
L Mean longitude
l True longitude
T Orbital period
n Mean motion
v Orbital speed
Collision avoidance (spacecraft)
Low energy transfer
Transposition, docking, and extraction
Celestial coordinate system
Equatorial coordinate system
Interplanetary Transport Network
Kepler's laws of planetary motion
Orbital state vectors
Specific orbital energy
Specific relative angular momentum