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-

logical and oceanographic angular momentum budgets), UT1 gradually falls behind TAI.

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).

• calculate D0 = JD0 – 245 1545.0 (days, ending in 0.5).

• GST = 6.697374558 + 0.06570982441908D0 + 1.00273790935H + 0.000026T 2 (in hrs),

where T = D0/36525.

• reduce the answer to the interval [0, 24].

The term involving T is negligible for most purposes over periods of a century or less.

This algorithm has the virtue of separating the slow but steadily-growing contribution from

whole days in the D0 term from the rapidly-changing but periodic piece that occurs during

each 24 hr period (the H term). It is accurate to within 0.1 sec over 100 years. The term

proportional to D0 accounts for both the shorter sidereal day and the slow precession of the

equinox.

A simpler version of this calculation combines the daily and hourly terms:

• calculate D = JD – 245 1545.0 (days).

• GST = 18.697374558 + 24.06570982441908D (in hrs).

• reduce the answer to the interval [0, 24].

but is more susceptible to numerical rounding errors due to the very rapid and steady

increase in the D term.

Note that it is unnecessary to worry about leap seconds here, as UTC has already been

corrected to keep up with mean solar time; all we are doing is converting spin relative to

– 12 –

the Sun to spin relative to the stars. But for accurate work it is desirable first to convert

the time in UTC to UT1, and later to apply a small correction from mean to true sidereal

time to allow for nutation (or wobbling) of the Earth’s spin axis abut its mean direction.

These amount at most to 1–2 sec.

REFERENCES

Baillie, K., Colwell, J. E., Lissauer, J. J., Esposito, L. W., Sremcevic, M. 2011. Waves in

Cassini UVIS stellar occultations. 2. The C ring. Icarus 216, 292-308.

Borderies, N., Goldreich, P., Tremaine, S. 1986. Nonlinear density waves in planetary rings.

Icarus 68, 522-533.

This preprint was prepared with the AAS LATEX macros v5.2.