Version 1.3, 26 Aug 2018 Astronomical coordinates and timescales Philip D. Nicholson Department of Astronomy, Cornell University, Ithaca NY 14853 1. Equatorial and ecliptic coordinates Two spherical polar coordinate systems are commonly used to specify the geocentric positions on the sky of astronomical sources: equatorial coordinates for which the refer- ence plane is the Earth’s equatorial plane, and ecliptic coordinates for which the reference plane is the Earth’s orbital plane, or ecliptic. Equatorial coordinates right ascension α and declination δ, are analogous to terrestrial longitude and latitude, with 0 ◦ ≤ α ≤ 360 ◦ and -90 ◦ ≤ δ ≤ 90 ◦ . 1 Right ascension is measured counter-clockwise from a zero point at the ascending node of the ecliptic on the Earth’s equator, usually designated by the symbol γ and sometimes referred to as ‘the first point in Aries’, though it is actually now in the constellation Aquarius! As for terrestrial latitude, δ = +90 ◦ corresponds to the direction of the Earth’s (north) polar axis. Ecliptic longitude λ and latitude β are also analogous to terrestrial longitude and latitude, with λ being measured counter-clockwise from the same zero point γ (i.e., α = λ = 0 at the point γ .) Again, 0 ◦ ≤ λ ≤ 360 ◦ and -90 ◦ ≤ β ≤ 90 ◦ , but in this case, β = +90 ◦ corresponds to the direction of the (northern) normal to the Earth’s orbital plane. The definitions of both equatorial and ecliptic coordinates are illustrated in Fig. 1. In terms of the obliquity , the angle between the ecliptic and equatorial planes, the transformation from equatorial to ecliptic coordinates is given by: cos β cos λ = cos δ cos α (1) cos β sin λ = cos cos δ sin α + sin sin δ (2) sin β = - sin cos δ sin α + cos sin δ (3) where both the first and second equations are necessary to resolve the quadrant ambiguity in λ. (Note that β always lies within the range [-90 ◦ , +90 ◦ ], so that there is no quadrant ambiguity here.) The inverse transformation is given by: cos δ cos α = cos β cos λ (4) 1 For reasons which will become apparent below, right ascension is more often measured in hours, with 1 hr = 15 ◦ .
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Version 1.3, 26 Aug 2018
Astronomical coordinates and timescales
Philip D. Nicholson
Department of Astronomy, Cornell University, Ithaca NY 14853
1. Equatorial and ecliptic coordinates
Two spherical polar coordinate systems are commonly used to specify the geocentric
positions on the sky of astronomical sources: equatorial coordinates for which the refer-
ence plane is the Earth’s equatorial plane, and ecliptic coordinates for which the reference
plane is the Earth’s orbital plane, or ecliptic. Equatorial coordinates right ascension α
and declination δ, are analogous to terrestrial longitude and latitude, with 0◦ ≤ α ≤ 360◦
and −90◦ ≤ δ ≤ 90◦.1 Right ascension is measured counter-clockwise from a zero point at
the ascending node of the ecliptic on the Earth’s equator, usually designated by the symbol
γ and sometimes referred to as ‘the first point in Aries’, though it is actually now in the
constellation Aquarius! As for terrestrial latitude, δ = +90◦ corresponds to the direction of
the Earth’s (north) polar axis.
Ecliptic longitude λ and latitude β are also analogous to terrestrial longitude and
latitude, with λ being measured counter-clockwise from the same zero point γ (i.e., α = λ =
0 at the point γ.) Again, 0◦ ≤ λ ≤ 360◦ and −90◦ ≤ β ≤ 90◦, but in this case, β = +90◦
corresponds to the direction of the (northern) normal to the Earth’s orbital plane. The
definitions of both equatorial and ecliptic coordinates are illustrated in Fig. 1.
In terms of the obliquity ε, the angle between the ecliptic and equatorial planes, the
transformation from equatorial to ecliptic coordinates is given by:
cosβ cosλ = cos δ cosα (1)
cosβ sinλ = cos ε cos δ sinα+ sin ε sin δ (2)
sinβ = − sin ε cos δ sinα+ cos ε sin δ (3)
where both the first and second equations are necessary to resolve the quadrant ambiguity
in λ. (Note that β always lies within the range [−90◦,+90◦], so that there is no quadrant
ambiguity here.)
The inverse transformation is given by:
cos δ cosα = cosβ cosλ (4)
1For reasons which will become apparent below, right ascension is more often measured in hours, with
1 hr = 15◦.
– 2 –
Fig. 1.— (Left) The definitions of equatorial coordinates, right ascension (α) and declination
(δ) for an arbitrary point on the celestial sphere, X. PQ represents the prolongation of
the Earth’s polar axis and AB its equatorial plane, projected onto the celestial sphere.
(Right) The definitions of ecliptic longitude (λ) and latitude (β) for the same arbitrary
point on the celestial sphere. P again represents the Earth’s north pole while K is the
north ecliptic pole, both projected onto the celestial sphere. The obliquity ε is indicated, as
is the reference point γ at the ascending node of the ecliptic on the equator. From Green,
“Spherical Astronomy”.
– 3 –
cos δ sinα = cos ε cosβ sinλ− sin ε sinβ (5)
sin δ = sin ε cosβ sinλ+ cos ε sinβ (6)
where again the first and second equations are necessary to resolve the quadrant ambiguity
in α. Like β, δ always lies within the range [−90◦,+90◦] so it has no quadrant ambiguity.
Of course, it is always necessary to specify the epoch of the coordinate systems, e.g.,
J2000 for the standard epoch of 2000 Jan 1.5, as both the equatorial and ecliptic planes
slowly precess. As a consequence, the point γ moves slowly around the sky at a rate of
50 arcseconds (∼ 0.014◦) per year, in a clockwise direction. All astronomical catalogs
specify the epoch for which their positions are calculated. To obtain equatorial positions
more accurate than about 10 arcseconds, it is also necessary to apply an additional small
rotation due to the nutation (or wobbling) of the Earth’s spin axis about its mean direction.
2. Alt-azimuth coordinates
A third spherical polar coordinate system is used to specify the positions of objects
as seen by an observer located on the Earth’s surface. In this case the reference plane
is the observer’s local horizon. Altitude a is measured upwards from the horizon, with
−90◦ ≤ a ≤ 90◦, while azimuth A is measured clockwise around the horizon from a zero
point in the direction towards the North pole, towards the east, with 0◦ ≤ A ≤ 360◦.2
Two useful terms are the zenith, the point on the sky directly above the observer, and
the meridian, which is an imaginary great circle passing through both the zenith and
the direction to the Earth’s north pole. The meridian defines the local directions of north
and south. In astronomy, it is more common to use the zenith distance z, defined as
z = 90◦ − a, rather than the altitude. The geometry for northern and southern hemisphere
observers is illustrated in Fig. 2.
In terms of the observer’s east longitude ` and geographic latitude φ, the trans-
formation from equatorial to alt-azimuth coordinates is given by:
sin z sin A = − cos δ sin H (7)
sin z cos A = − sinφ cos δ cos H + cosφ sin δ (8)
cos z = cosφ cos δ cos H + sinφ sin δ (9)
where again both the first and second equations are necessary to resolve the quadrant
ambiguity in A. Objects with z < 90◦ are above the observer’s horizon, while those with
z > 90◦ are below the horizon and therefore invisible, either temporarily or permanently.
In the above expressions, the quantity H is known as the hour angle. It is an interme-
diate angle, usually given in the range [−180◦,+180◦] or [−12 hr, +12 hr], which depends
on both α and `, as well as on the time of observation t. If t is given in local sidereal time
2Some texts measure the azimuth in a counterclockwise direction, towards the west.
– 4 –
Fig. 2.— The definitions of altitude (a), zenith distance (z) and azimuth (A) for an arbitrary
point on the celestial sphere, X. PQ represents the Earth’s polar axis while Z is the
observer’s zenith, both projected onto the celestial sphere. The observer’s horizon is denoted
by the circle NWS, with N denoting the direction of north. The left diagram is for a
northern hemisphere observer and the right diagram for one in the southern hemisphere.
From Green, “Spherical Astronomy”.
– 5 –
(see below), then H = LST − α. If instead t is given in Greenwich sidereal time then we
have LST = GST + ` and
H = GST + `− α. (10)
(Note that one must be careful here to be consistent in using either degrees or hours for
all of the quantities in this equation. Recall that 15◦ = 1 hr.) H is simply the difference
in right ascension between the source (at α) and the observer’s meridian at the time of
observation, which is given by the local sidereal time. By convention, H is measured in the
direction opposite to α, so that it increases monotonically with time. It is zero when the
source is on the observer’s meridian. Greenwich sidereal time is also equal to the hour angle
of the reference point γ, as seen by an observer at Greenwich (i.e., at ` = 0◦). It increases
at the rate of 24 hr per sidereal day, which is approximately 23 hr 56 min 04 sec in civil
time.
Most small telescopes (e.g., the 12-inch refractor at Fuertes Observatory) have polar
and declination axes, so that they can point directly at stars using equatorial coordinates
without having to do any trig calculations. But larger optical (and most radio) telescopes
have vertical and horizontal axes, like a gun mount on a battleship, and must point using
alt-azimuth coordinates. All such telescopes are computer-controlled.
Another useful expression converts the observed azimuth and zenith distance of a source
to its declination:
sin δ = sinφ cos z + cosφ sin z cos A. (11)
This equation is important in celestial navigation, e.g., in obtaining an estimate of one’s
latitude from an observation of the sun or a standard star. For example, a ‘noon-sighting’ of
the sun (i.e., when it is on the meridian, where A = 0◦ or 180◦, yields sin δ� = sinφ cos z±cosφ sin z = sin(φ± z), so that we have φ = δ�∓ z. Consulting an Almanac for the current
value of δ� then yields the latitude φ.
Finally we note that Eqns (9) and (11) can both be derived from the cosine law of
spherical trigonometry:
cos a = cos b cos c+ sin b sin c cos A, (12)
where A is the angle opposite side a of a spherical triangle with sides a, b and c. This is
illustrated in Fig. 3.
3. Astronomical time scales
3.1. UT, ET and ATI
Several different systems of time are used in astronomy, depending on the context. The
subject can be confusing, especially when relativistic effects are considered, but the most
important systems in current use and their standard abbreviations — based on their names
in French — are as follows.
– 6 –
Fig. 3.— (Top) The relationship between equatorial coordinates (α, δ) and alt-azimuth
coordinates (z, A) for an arbitrary point on the celestial sphere, X. PQ represents the
prolongation of the Earth’s polar axis and NWS the observer’s horizon, projected onto the
celestial sphere. The observer’s latitude (φ) is indicated, as is the hour angle of the point X,
denoted by H. (Bottom) The spherical triangle PZX extracted from the upper diagram,
illustrating the application of the cosine law to derive expressions for sin δ and cos z. From
Green, “Spherical Astronomy”.
– 7 –
Universal Time (UT or UT1): Mean solar time, as measured at the longitude of
Greenwich, with 0 hr at midnight and 12 hr at mean noon. Usually given in hours, minutes
and seconds, UT was previously known as Greenwich Mean Time (GMT), often shortened
to “Zulu”. (A variant referred to as Greenwich Mean Astronomical Time (GMAT), was
measured from 0 hr at noon, but this was discontinued in 1925. This is sometimes encoun-
tered in the older literature, where it is sometimes simply called GMT.) UT differs from the
time measured by a sundial at Greenwich by up to ±15 min, due to the eccentricity of the
Earth’s orbit and the obliquity of the ecliptic. It is determined, in principle, by observations
of the Sun at Greenwich but in practice by observations of a network of standard stars at
various observatories around the world.
Coordinated Universal Time (UTC): Standard civil time, as measured at the
longitude of Greenwich, with 0 hr at midnight and 12 hr at mean noon. Also given in
hours, minutes and seconds, this is the time scale used for most civil and military purposes
and is broadcast by various national agencies, such as the USNO “talking clock” and the
WWV short-wave radio service. UTC was introduced in 1972 and is calculated from TAI,
defined below, but is maintained within ∼ 1 sec of UT1.
International Atomic Time (TAI): A timescale derived from a worldwide ensemble
of highly accurate atomic clocks. It was introduced in 1957 and is now the basis of UTC.
TAI is our closest practical approximation to an absolutely uniform timescale (neglecting
small relativistic effects due to the Earth’s orbital eccentricity). It’s rate was set equal to
that of ET (see below) and it was synchronized with UT1 on 1 Jan 1958.3
Because the Earth’s rotation rate changes both predictably (due to tides raised by the
Sun and Moon) and stochastically (due to unpredictable changes in the Earth’s meteoro-
In order to keep UTC within 0.9 sec of UT1, leap seconds are introduced periodically into
UTC, usually on 1 January or 1 July. So we have
UTC = TAI −∆AT (13)
where ∆AT is the sum of all the leap seconds added since January 1958. The necessary
pattern of leap seconds is unpredictable, and varies on decadal timescales due to geophysical
reasons that remain mysterious. Without them, 0 hr UTC would gradually drift away from
midnight at Greenwich. The residual difference, ∆UT = UT1 − UTC is distributed along
with broadcast time signals such as WWV.
Terrestrial Dynamical Time (TDT): Known as Ephemeris Time (and denoted ET)
prior to 1984, this is the theoretical time scale which underpins the planetary ephemerides.
It predates TAI, and was formally introduced in 1960 after the tidal variations in UT were
recognized, but its theoretical origins go back to Newcomb’s work on the motion of the
planets around 1900. Its rate was originally set to be equal to that of UT around 1870,
3Since atomic clocks are also used to define the SI second, this effectively guaranteed that the ET second
is essentially identical to the SI second, as far as could be measured at that time.
– 8 –
but the two time scales were actually synchronized c.1900. Originally, ET was measured
by comparing observations of the Sun, Moon and/or planets with Newcomb’s ephemerides,
but TDT is now also defined in terms of TAI, via the expression
TDT = TAI + 32.184 s (14)
where the additive constant was the best estimate of the accumulated difference ET − UTin January 1958. TDT was slightly revised in 2001 and renamed TT; the term TDT is no
longer used by the IAU or the IERS.
Delta T (∆T ): An important quantity which arises whenever one wishes to compare
a planetary or spacecraft position derived from an ephemeris with an observation recorded
in UT (which includes most spacecraft observations, whose internal clocks usually run on
some derivative of UTC) is the quantity:
∆T = TT − UTC = TDT − UTC = ET − UTC. (15)
Prior to 1984, ∆T was determined empirically, by observations of stars (for UTC) and
planets (for ET), but now it is simply given by:
∆T = 32.184 s + ∆AT. (16)
As of 2018, ∆T = 69.184 s, with an extra leap second being added every 2–3 years. Fig. 4
shows the observed variations in the Length of Day (LOD) over the past 40 years, based on
astronomical observations and, more recently, on the tracking of GPS satellites from fixed
ground stations. Note that an excess in the LOD of 1 msec results in a cumulative lag in
UT1 relative to TAI or TDT of 0.365 sec over 1 year, thus requiring that a leap second
be added to UTC every 3 years. In addition to the decades-long quasi-random variations
in the LOD, there is a small annual variation of ∼ 0.5 msec due to seasonal changes in
the atmosphere and ocean currents. If the LOD should ever drop below 86,400 sec then it
might be necessary to remove a second from UTC, but so far this has not happened.
Barycentric Dynamical Time (TDB): For many purposes, the above definitions
and quantities suffice, but if accuracies greater than a few msec are required then it is
necessary to correct for various relativistic effects. Chief among these is the variation in the
gravitational potential at the center of the Earth due to our planet’s eccentric orbit about
the sun. This amounts to a fractional clock-rate error of order ±eGM�/ac2 ' ±1.7×10−10,
where a and e are the semimajor axis and eccentricity of the Earth’s orbit. Relativistic time
dilation leads to a similar correction of order ±ev2⊕/c2, where v⊕ is the orbital velocity of
the Earth. The maximum accumulated correction due to both of these sources over a period
of 6 months is ±1.7 msec.4 The resulting time scale is known as Barycentric Dynamical
4The much larger constant effects of the mean gravitational potentials of the Sun and Earth, and of the
Earth’s mean orbital velocity of 30 km/s, are absorbed into the definitions of TAI and the SI second, which
are both defined at the surface of the Earth rather than at infinity.
– 9 –
Time (TDB), as it is referred to the center of mass of the solar system.5
This is the time scale used in JPL’s planetary ephemerides and for navigating interplan-
etary spacecraft. It was introduced in 1984, along with TDT, and is the first astronomical
timescale to explicitly include the effects of general relativity. TDB was slightly revised in
1991 and the term is no longer used by the IAU or the IERS. It has been replaced by the
‘coordinate times’ TCG and TCB.
3.2. Julian dates
For many purposes in astronomy, it is useful to have dates expressed in a simple, con-
tinuous calendar that avoids the foibles of most civil calendars, with their unequal months
and occasional leap days. The system in common use is that of the Julian date, which
is simply a continuous count of days starting at a time in the distant past, prior to any
astronomical records (so as to avoid problems with ‘day 0’). The zero date is 1 January
4713 BCE, at 12:00 GMT (chosen for now-obsolete religious reasons). Some recent reference
epochs often encountered in the literature are listed below.
B1900 = JD 241 5020.314 = 1900 Jan 0.814
B1950 = JD 243 3282.423 = 1950 Jan 0.923
J2000 = JD 245 1545.0 = 2000 Jan 1.500
A common use of Julian dates is to specify the epoch for binary star or exo-planet orbits.
Note that, in accordance with astronomical practice prior to c.1925, Julian dates are
reckoned to start at noon, Greenwich time, rather than at midnight. Hence the annoying
appearance of dates like Jan 1.5 = Jan 1, 12:00 GMT. Sometimes one encounters a Modifed
Julian Date (MJD), which is simply JD – 240 0000.5 and which ‘rolls over’ at 0:00 UTC.
Other variants include the Julian Ephemeris Date (JED), which is the Julian date in
ET (or TT) rather than UTC, and Barycentric Julian Date (BJD), which is the Julian
date for an Earth-based observation corrected for light travel time to the Solar System
barycenter (also useful for X-ray binary and exo-planet orbits, which can have very short
periods.)
3.3. Sidereal time
Already mentioned in Section 2 above, Earth-bound astronomers find it necessary to
keep track of the rotation of the Earth, on which their telescopes are fixed. This is done
5Note that TDB is not literally the time kept by an atomic clock at the solar system’s barycenter, which
is within the Sun, as that would be affected by the Sun’s deep gravitational well, but rather it is time as
kept by a clock orbiting at a fixed distance from the solar system barycenter, located on the surface of the
Earth at a distance of 1.0 AU from the Sun.
– 10 –
Fig. 4.— The variation in the Length of the Day (LOD) relative to 86,400 SI sec over the
past 40 years, as obtained from astronomical observations and tracking of GPS satellites.
The black curve is a smoothed average of the daily data, and reveals a variation with a
period of 1 year. From the USNO Time Service web site, and compiled by the International
Earth Rotation Service (IERS) in Paris.
– 11 –
using sidereal time, which is essentially time “measured by the stars”, rather than by the
Sun. The local sidereal time is simply the right ascension of a point on the observer’s
meridian. Equivalently, it is the local hour angle of the reference point γ, the zero of right
ascension. Sidereal time therefore increases by 24 hr over a period of 1 sidereal day, or
approximately 23 hr 56 min 04 sec in civil (i.e., solar) time. Over a period of 1 year, the
Earth spins on its axis by one extra rotation compared with the day-night cycle, so a sidereal
year is ∼ 366.25 sidereal days long and 24 hr/366 ' 1/15 hr = 4 min.
In practice, we only need to keep track of the sidereal time at one location (conveniently
chosen as Greenwich), as it is then easy to compute the local sidereal time at any other
place from the equation LST = GST + `, where ` is the longitude east of Greenwich (being
careful not to mix hours and degrees!).
Various formulae exist to calculate GST as a function of the date and UTC. We give
here a fairly simple but accurate prescription from the US Naval Observatory web site:
• calculate the Julian date, JD, corresponding to the desired UTC time.
• calculate the Julian date at the previous midnight, JD0 (ends in 0.5).
• calculate the time elapsed since midnight, H (hrs).