Time and Dates (astropy.time)


The astropy.time package provides functionality for manipulating times and dates. Specific emphasis is placed on supporting time scales (e.g. UTC, TAI, UT1, TDB) and time representations (e.g. JD, MJD, ISO 8601) that are used in astronomy and required to calculate, e.g., sidereal times and barycentric corrections. It uses Cython to wrap the C language ERFA time and calendar routines, using a fast and memory efficient vectorization scheme.

All time manipulations and arithmetic operations are done internally using two 64-bit floats to represent time. Floating point algorithms from 1 are used so that the Time object maintains sub-nanosecond precision over times spanning the age of the universe.


Shewchuk, 1997, Discrete & Computational Geometry 18(3):305-363

Getting Started

The basic way to use astropy.time is to create a Time object by supplying one or more input time values as well as the time format and time scale of those values. The input time(s) can either be a single scalar like "2010-01-01 00:00:00" or a list or a numpy array of values as shown below. In general any output values have the same shape (scalar or array) as the input.

>>> import numpy as np
>>> from astropy.time import Time
>>> times = ['1999-01-01T00:00:00.123456789', '2010-01-01T00:00:00']
>>> t = Time(times, format='isot', scale='utc')
>>> t
<Time object: scale='utc' format='isot' value=['1999-01-01T00:00:00.123' '2010-01-01T00:00:00.000']>
>>> t[1]
<Time object: scale='utc' format='isot' value=2010-01-01T00:00:00.000>

The format argument specifies how to interpret the input values, e.g. ISO or JD or Unix time. The scale argument specifies the time scale for the values, e.g. UTC or TT or UT1. The scale argument is optional and defaults to UTC except for Time from epoch formats. We could have written the above as:

>>> t = Time(times, format='isot')

When the format of the input can be unambiguously determined then the format argument is not required, so we can simplify even further:

>>> t = Time(times)

Now let’s get the representation of these times in the JD and MJD formats by requesting the corresponding Time attributes:

>>> t.jd  
array([2451179.50000143, 2455197.5       ])
>>> t.mjd  
array([51179.00000143, 55197.        ])

The full power of output representation is available via the to_value method which also allows controlling the subformat, for example using numpy.longdouble as the output type for higher precision:

>>> t.to_value('mjd', 'long')  
array([51179.00000143, 55197.        ], dtype=float128)

The default representation can be changed by setting the format attribute:

>>> t.format = 'fits'
>>> t
<Time object: scale='utc' format='fits' value=['1999-01-01T00:00:00.123'
>>> t.format = 'isot'

We can also convert to a different time scale, for instance from UTC to TT. This uses the same attribute mechanism as above but now returns a new Time object:

>>> t2 = t.tt
>>> t2
<Time object: scale='tt' format='isot' value=['1999-01-01T00:01:04.307' '2010-01-01T00:01:06.184']>
>>> t2.jd  
array([2451179.5007443 , 2455197.50076602])

Note that both the ISO (ISOT) and JD representations of t2 are different than for t because they are expressed relative to the TT time scale. Of course, from the numbers or strings one could not tell; one format in which this information is kept is the fits format:

>>> print(t2.fits)
['1999-01-01T00:01:04.307' '2010-01-01T00:01:06.184']

One can set the time values in-place using the usual numpy array setting item syntax:

>>> t2 = t.tt.copy()  # Copy required if transformed Time will be modified
>>> t2[1] = '2014-12-25'
>>> print(t2)
['1999-01-01T00:01:04.307' '2014-12-25T00:00:00.000']

The Time object also has support for missing values, which is particularly useful for Table operations such as joining and stacking:

>>> t2[0] = np.ma.masked  # Declare that first time is missing or invalid
>>> print(t2)
[-- '2014-12-25T00:00:00.000']

Finally, some further examples of what is possible. For details, see the API documentation below.

>>> dt = t[1] - t[0]
>>> dt  
<TimeDelta object: scale='tai' format='jd' value=4018.00002172>

Here, note the conversion of the timescale to TAI. Time differences can only have scales in which one day is always equal to 86400 seconds.

>>> import numpy as np
>>> t[0] + dt * np.linspace(0.,1.,12)
<Time object: scale='utc' format='isot' value=['1999-01-01T00:00:00.123' '2000-01-01T06:32:43.930'
 '2000-12-31T13:05:27.737' '2001-12-31T19:38:11.544'
 '2003-01-01T02:10:55.351' '2004-01-01T08:43:39.158'
 '2004-12-31T15:16:22.965' '2005-12-31T21:49:06.772'
 '2007-01-01T04:21:49.579' '2008-01-01T10:54:33.386'
 '2008-12-31T17:27:17.193' '2010-01-01T00:00:00.000']>
>>> t.sidereal_time('apparent', 'greenwich')  
<Longitude [6.68050179, 6.70281947] hourangle>

Using astropy.time

Time object basics

In astropy.time a “time” is a single instant of time which is independent of the way the time is represented (the “format”) and the time “scale” which specifies the offset and scaling relation of the unit of time. There is no distinction made between a “date” and a “time” since both concepts (as loosely defined in common usage) are just different representations of a moment in time.

Time Format

The time format specifies how an instant of time is represented. The currently available formats are can be found in the Time.FORMATS dict and are listed in the table below. Each of these formats is implemented as a class that derives from the base TimeFormat class. This class structure can be easily adapted and extended by users for specialized time formats not supplied in astropy.time.



Example argument












datetime(2000, 1, 2, 12, 0, 0)












‘2000-01-01 00:00:00.000’



























{‘year’: 2010, ‘month’: 3, ‘day’: 1}





The TimeFITS format implements most of the FITS standard 2, including support for the LOCAL timescale. Note, though, that FITS supports some deprecated names for timescales; these are translated to the formal names upon initialization. Furthermore, any specific realization information, such as UT(NIST) is stored only as long as the time scale is not changed.


Rots et al. 2015, A&A 574:A36

Changing format

The default representation can be changed by setting the format attribute in place:

>>> t = Time('2000-01-02')
>>> t.format = 'jd'
>>> t
<Time object: scale='utc' format='jd' value=2451545.5>

Be aware that when changing format, the current output subformat (see section below) may not exist in the new format. In this case the subformat will not be preserved:

>>> t = Time('2000-01-02', format='fits', out_subfmt='longdate')
>>> t.value
>>> t.format = 'iso'
>>> t.out_subfmt
>>> t.format = 'fits'
>>> t.value

Many of the available time format classes support the concept of a subformat. This allows for variations on the basic theme of a format in both the input parsing / validation and the output

The table below illustrates available subformats for the string formats

iso, fits, yday formats:



Input / output



2001-01-02 03:04:05.678



2001-01-02 03:04






















Numerical formats such as mjd, jyear, or cxcsec all support the subformats: 'float', 'long', 'decimal', 'str', and 'bytes'. Here, 'long' uses numpy.longdouble for somewhat enhanced precision (with the enhancement depending on platform), and 'decimal' instances of decimal.Decimal for full precision. For the 'str' and 'bytes' sub-formats, the number of digits is also chosen such that time values are represented accurately.

When used on input, these formats allow creating a time using a single input value that accurately captures the value to the full available precision in Time. Conversely, the single value on output using Time to_value or TimeDelta to_value can have higher precision than the standard 64-bit float:

>>> tm = Time('51544.000000000000001', format='mjd')  # String input
>>> tm.mjd  # float64 output loses last digit but Decimal gets it
>>> tm.to_value('mjd', subfmt='decimal')  
>>> tm.to_value('mjd', subfmt='str')

The complete list of subformat options for the Time formats that have them is:




float, long, decimal, str, bytes


float, long, decimal, str, bytes


date_hms, date_hm, date


float, long, decimal, str, bytes


date_hms, date, longdate_hms, longdate


float, long, decimal, str, bytes


date_hms, date_hm, date


date_hms, date_hm, date


float, long, decimal, str, bytes


float, long, decimal, str, bytes


float, long, decimal, str, bytes


float, long, decimal, str, bytes


float, long, decimal, str, bytes


date_hms, date_hm, date

The complete list of subformat options for the TimeDelta formats that have them is:




float, long, decimal, str, bytes


float, long, decimal, str, bytes

Time from epoch formats

The formats cxcsec, gps, and unix are a little special in that they provide a floating point representation of the elapsed time in seconds since a particular reference date. These formats have a intrinsic time scale which is used to compute the elapsed seconds since the reference date.



Reference date



1998-01-01 00:00:00



1970-01-01 00:00:00



1980-01-06 00:00:19

Unlike the other formats which default to UTC, if no scale is provided when initializing a Time object then the above intrinsic scale is used. This is done for computational efficiency.

Time Scale

The time scale (or time standard) is “a specification for measuring time: either the rate at which time passes; or points in time; or both” 3. See also 4 and 5.

>>> Time.SCALES
('tai', 'tcb', 'tcg', 'tdb', 'tt', 'ut1', 'utc', 'local')




International Atomic Time (TAI)


Barycentric Coordinate Time (TCB)


Geocentric Coordinate Time (TCG)


Barycentric Dynamical Time (TDB)


Terrestrial Time (TT)


Universal Time (UT1)


Coordinated Universal Time (UTC)


Local Time Scale (LOCAL)


Wikipedia time standard article


SOFA Time Scale and Calendar Tools (PDF)




The local time scale is meant for free-running clocks or simulation times, i.e., to represent a time without properly defined scale. This means it cannot be converted to any other time scale, and arithmetic is possible only with Time instances with scale local and with TimeDelta instances with scale local or None.

The system of transformation between supported time scales (i.e., all but local) is shown in the figure below. Further details are provided in the Convert time scale section.


Scalar or Array

A Time object can hold either a single time value or an array of time values. The distinction is made entirely by the form of the input time(s). If a Time object holds a single value then any format outputs will be a single scalar value, and likewise for arrays. Like other arrays and lists, Time objects holding arrays are subscriptable, returning scalar or array objects as appropriate:

>>> from astropy.time import Time
>>> t = Time(100.0, format='mjd')
>>> t.jd
>>> t = Time([100.0, 200.0, 300.], format='mjd')
>>> t.jd  
array([2400100.5, 2400200.5, 2400300.5])
>>> t[:2]  
<Time object: scale='utc' format='mjd' value=[100. 200.]>
>>> t[2]
<Time object: scale='utc' format='mjd' value=300.0>
>>> t = Time(np.arange(50000., 50003.)[:, np.newaxis],
...          np.arange(0., 1., 0.5), format='mjd')
>>> t  
<Time object: scale='utc' format='mjd' value=[[50000.  50000.5]
 [50001.  50001.5]
 [50002.  50002.5]]>
>>> t[0]  
<Time object: scale='utc' format='mjd' value=[50000.  50000.5]>

Numpy method analogs

For Time instances holding arrays, many of the same methods and attributes that work on ndarray instances can be used. E.g., one can reshape Time instances and take specific parts using reshape(), ravel(), flatten(), T, transpose(), swapaxes(), diagonal(), squeeze(), take():

>>> t.reshape(2, 3)  
<Time object: scale='utc' format='mjd' value=[[50000.  50000.5 50001. ]
 [50001.5 50002.  50002.5]]>
>>> t.T  
<Time object: scale='utc' format='mjd' value=[[50000.  50001.  50002. ]
 [50000.5 50001.5 50002.5]]>

Note that similarly to the ndarray methods, all but flatten() try to use new views of the data, with the data copied only if that it is impossible (as discussed, e.g., in the documentation for numpy reshape()).

Some arithmetic methods are supported as well: min(), max(), ptp(), sort(), argmin(), argmax(), and argsort(). E.g.:

>> t.max()
<Time object: scale='utc' format='mjd' value=50002.5>
>> t.ptp(axis=0)  # doctest: +FLOAT_CMP
<TimeDelta object: scale='tai' format='jd' value=[2. 2.]>

Inferring input format

The Time class initializer will not accept ambiguous inputs, but it will make automatic inferences in cases where the inputs are unambiguous. This can apply when the times are supplied as datetime objects or strings. In the latter case it is not required to specify the format because the available string formats have no overlap. However, if the format is known in advance the string parsing will be faster if the format is provided.

>>> from datetime import datetime
>>> t = Time(datetime(2010, 1, 2, 1, 2, 3))
>>> t.format
>>> t = Time('2010-01-02 01:02:03')
>>> t.format

Internal representation

The Time object maintains an internal representation of time as a pair of double precision numbers expressing Julian days. The sum of the two numbers is the Julian Date for that time relative to the given time scale. Users requiring no better than microsecond precision over human time scales (~100 years) can safely ignore the internal representation details and skip this section.

This representation is driven by the underlying ERFA C-library implementation. The ERFA routines take care throughout to maintain overall precision of the double pair. The user is free to choose the way in which total JD is provided, though internally one part contains integer days and the other the fraction of the day, as this ensures optimal accuracy for all conversions. The internal JD pair is available via the jd1 and jd2 attributes:

>>> t = Time('2010-01-01 00:00:00', scale='utc')
>>> t.jd1, t.jd2
(2455198.0, -0.5)
>>> t2 = t.tai
>>> t2.jd1, t2.jd2  
(2455198., -0.49960648148148146)

Creating a Time object

The allowed Time arguments to create a time object are listed below:

valnumpy ndarray, list, str, or number

Data to initialize table.

val2numpy ndarray, list, str, or number; optional

Data to initialize table.

formatstr, optional

Format of input value(s)

scalestr, optional

Time scale of input value(s)

precisionint between 0 and 9 inclusive

Decimal precision when outputting seconds as floating point


Unix glob to select subformats for parsing input times


Unix glob to select subformat for outputting times

locationEarthLocation or tuple, optional

If a tuple, 3 Quantity items with length units for geocentric coordinates, or a longitude, latitude, and optional height for geodetic coordinates. Can be a single location, or one for each input time.


The val argument specifies the input time or times and can be a single string or number, or it can be a Python list or numpy array of strings or numbers. To initialize a Time object based on a specified time, it must be present. If val is absent (or None), the Time object will be created for the time corresponding to the instant the object is created.

In most situations one also needs to specify the time scale via the scale argument. The Time class will never guess the time scale, so a simple example would be:

>>> t1 = Time(50100.0, scale='tt', format='mjd')
>>> t2 = Time('2010-01-01 00:00:00', scale='utc')

It is possible to create a new Time object from one or more existing time objects. In this case the format and scale will be inferred from the first object unless explicitly specified.

>>> Time([t1, t2])  
<Time object: scale='tt' format='mjd' value=[50100. 55197.00076602]>


The val2 argument is available for specialized situations where extremely high precision is required. Recall that the internal representation of time within astropy.time is two double-precision numbers that when summed give the Julian date. If provided the val2 argument is used in combination with val to set the second the internal time values. The exact interpretation of val2 is determined by the input format class. As of this release all string-valued formats ignore val2 and all numeric inputs effectively add the two values in a way that maintains the highest precision. Example:

>>> t = Time(100.0, 0.000001, format='mjd', scale='tt')
>>> t.jd, t.jd1, t.jd2  
(2400100.500001, 2400101.0, -0.499999)


The format argument sets the time time format, and as mentioned it is required unless the format can be unambiguously determined from the input times.


The scale argument sets the time scale and is required except for time formats such as plot_date (TimePlotDate) and unix (TimeUnix). These formats represent the duration in SI seconds since a fixed instant in time which is independent of time scale.


The precision setting affects string formats when outputting a value that includes seconds. It must be an integer between 0 and 9. There is no effect when inputting time values from strings. The default precision is 3. Note that the limit of 9 digits is driven by the way that ERFA handles fractional seconds. In practice this should should not be an issue.

>>> t = Time('B1950.0', precision=3)
>>> t.byear_str
>>> t.precision = 0
>>> t.byear_str


The in_subfmt argument provides a mechanism to select one or more subformat values from the available subformats for input. Multiple allowed subformats can be selected using Unix-style wildcard characters, in particular * and ?, as documented in the Python fnmatch module.

The default value for in_subfmt is * which matches any available subformat. This allows for convenient input of values with unknown or heterogeneous subformat:

>>> Time(['2000:001', '2000:002:03:04', '2001:003:04:05:06.789'])
<Time object: scale='utc' format='yday'
 value=['2000:001:00:00:00.000' '2000:002:03:04:00.000' '2001:003:04:05:06.789']>

One can explicitly specify in_subfmt in order to strictly require a certain subformat:

>>> t = Time('2000:002:03:04', in_subfmt='date_hm')
>>> t = Time('2000:002', in_subfmt='date_hm')  
Traceback (most recent call last):
ValueError: Input values did not match any of the formats where the
format keyword is optional ['astropy_time', 'datetime',
'byear_str', 'iso', 'isot', 'jyear_str', 'yday']


The out_subfmt argument is similar to in_subfmt except that it applies to output formatting. In the case of multiple matching subformats the first matching subformat is used.

>>> Time('2000-01-01 02:03:04', out_subfmt='date').iso
>>> Time('2000-01-01 02:03:04', out_subfmt='date_hms').iso
'2000-01-01 02:03:04.000'
>>> Time('2000-01-01 02:03:04', out_subfmt='date*').iso
'2000-01-01 02:03:04.000'
>>> Time('50814.123456789012345', format='mjd', out_subfmt='str').mjd

See also the subformat section.


This optional parameter specifies the observer location, using an EarthLocation object or a tuple containing any form that can initialize one: either a tuple with geocentric coordinates (X, Y, Z), or a tuple with geodetic coordinates (longitude, latitude, height; with height defaulting to zero). They are used for time scales that are sensitive to observer location (currently, only TDB, which relies on the ERFA routine eraDtdb to determine the time offset between TDB and TT), as well as for sidereal time if no explicit longitude is given.

>>> t = Time('2001-03-22 00:01:44.732327132980', scale='utc',
...          location=('120d', '40d'))
>>> t.sidereal_time('apparent', 'greenwich')  
<Longitude 12. hourangle>
>>> t.sidereal_time('apparent')  
<Longitude 20. hourangle>


In future versions, we hope to add the possibility to add observatory objects and/or names.

Getting the Current Time

The current time can be determined as a Time object using the now class method:

>>> nt = Time.now()
>>> ut = Time(datetime.utcnow(), scale='utc')

The two should be very close to each other.

Using Time objects

The operations available with Time objects include:

  • Get and set time values(s) for an array-valued Time object:

  • Set missing (masked) values.

  • Get the representation of the time value(s) in a particular time format.

  • Get a new time object for the same time value(s) but referenced to a different time scale.

  • Calculate the sidereal time corresponding to the time value(s).

  • Do time arithmetic involving Time and/or TimeDelta objects.

Get and set values

For an existing Time object which is array-valued, one can use the usual numpy array item syntax to get either a single item or a subset of items. The returned value is a Time object with all the same attributes:

>>> t = Time(['2001:020', '2001:040', '2001:060', '2001:080'],
...          out_subfmt='date')
>>> print(t[1])
>>> print(t[1:])
['2001:040' '2001:060' '2001:080']
>>> print(t[[2, 0]])
['2001:060' '2001:020']

As of astropy version 3.1, one can also set values in-place for an array-valued Time object:

>>> t = Time(['2001:020', '2001:040', '2001:060', '2001:080'],
...          out_subfmt='date')
>>> t[1] = '2010:001'
>>> print(t)
['2001:020' '2010:001' '2001:060' '2001:080']
>>> t[[2, 0]] = '1990:123'
>>> print(t)
['1990:123' '2010:001' '1990:123' '2001:080']

The new value (on the right hand side) when setting can be one of three possibilities:

  • Scalar string value or array of string values where each value is in a valid time format that can be automatically parsed and used to create a Time object.

  • Value or array of values where each value has the same format as the Time object being set. For instance, a float or numpy array of floats for an object with format='unix'.

  • Time object with identical location (but scale and format need not be the same). The right side value will be transformed so the time scale matches.

Whenever any item is set then the internal cache (see Caching) is cleared along with the delta_tdb_tt and/or delta_ut1_utc transformation offsets, if they have been set.

If it is required that the Time object be immutable then set the writeable attribute to False. In this case attempting to set a value will raise a ValueError: Time object is read-only. See the section on Caching for an example.

Missing values

The Time and TimeDelta objects support functionality for marking values as missing or invalid (added in astropy 3.1). This is also known as masking, and is especially useful for Table operations such as joining and stacking. To set one or more items as missing, assign the special value numpy.ma.masked, for example:

>>> t = Time(['2001:020', '2001:040', '2001:060', '2001:080'],
...          out_subfmt='date')
>>> t[2] = np.ma.masked
>>> print(t)
['2001:020' '2001:040' -- '2001:080']


The operation of setting an array element to numpy.ma.masked (missing) overwrites the actual time data and therefore there is no way to recover the original value. In this sense the numpy.ma.masked value behaves just like any other valid Time value when setting. This is similar to how Pandas missing data works, but somewhat different from NumPy masked arrays which maintain a separate mask array and retain the underlying data. In the Time object the mask attribute is read-only and cannot be directly set.

Once one or more values in the object are masked, any operations will propagate those values as masked, and access to format attributes such as unix or value will return a MaskedArray object:

>>> t.unix  
masked_array(data = [979948800.0 981676800.0 -- 985132800.0],
             mask = [False False  True False],
       fill_value = 1e+20)

One can view the mask, but note that it is read-only and setting the mask is always done by setting the item to masked.

>>> t.mask
array([False, False,  True, False]...)
>>> t[:2] = np.ma.masked


The internal implementation of missing value support is provisional and may change in a subsequent release. This would impact information in the next section. However, the documented API for using missing values with Time and TimeDelta objects is stable.

Custom format classes and missing values

For advanced users who have written a custom time format via a TimeFormat subclass, it may be necessary to modify your class if you wish to support missing values. For applications that do not take advantage of missing values then no changes are required.

Missing values in a TimeFormat subclass object are marked by setting the corresponding entries of the jd2 attribute to be numpy.nan (but this is never done directly by the user). For most array operations and numpy functions the numpy.nan entries are propagated as expected and all is well. However, this is not always the case, and in particular the ERFA routines do not generally support numpy.nan values gracefully.

In cases where numpy.nan is not acceptable, format class methods should use the jd2_filled property instead of jd2. This replaces numpy.nan with 0.0. Since jd2 is always in the range -1 to +1, substituting 0.0 will allow functions to return “reasonable” values which will then be masked in any subsequent outputs. See the value property of the TimeDecimalYear format for any example.

Get representation

Instants of time can be represented in different ways, for instance as an ISO-format date string ('1999-07-23 04:31:00') or seconds since 1998.0 (49091460.0) or Modified Julian Date (51382.187451574).

The representation of a Time object in a particular format is available by getting the object attribute corresponding to the format name. The list of available format names is in the time format section.

>>> t = Time('2010-01-01 00:00:00', format='iso', scale='utc')
>>> t.jd        # JD representation of time in current scale (UTC)
>>> t.iso       # ISO representation of time in current scale (UTC)
'2010-01-01 00:00:00.000'
>>> t.unix      # seconds since 1970.0 (UTC)
>>> t.plot_date # Date value for plotting with matplotlib plot_date()
>>> t.datetime  # Representation as datetime.datetime object
datetime.datetime(2010, 1, 1, 0, 0)


>>> import matplotlib.pyplot as plt  
>>> jyear = np.linspace(2000, 2001, 20)  
>>> t = Time(jyear, format='jyear')  
>>> plt.plot_date(t.plot_date, jyear)  
>>> plt.gcf().autofmt_xdate()  # orient date labels at a slant  
>>> plt.draw()  

Convert time scale

A new Time object for the same time value(s) but referenced to a new time scale can be created getting the object attribute corresponding to the time scale name. The list of available time scale names is in the time scale section and in the figure below illustrating the network of time scale transformations.



>>> t = Time('2010-01-01 00:00:00', format='iso', scale='utc')
>>> t.tt        # TT scale
<Time object: scale='tt' format='iso' value=2010-01-01 00:01:06.184>
>>> t.tai
<Time object: scale='tai' format='iso' value=2010-01-01 00:00:34.000>

In this process the format and other object attributes like lon, lat, and precision are also propagated to the new object.

As noted in the Time object basics section, a Time object is immutable and the internal time values cannot be altered once the object is created. The process of changing the time scale therefore begins by making a copy of the original object and then converting the internal time values in the copy to the new time scale. The new Time object is returned by the attribute access.


The computations for transforming to different time scales or formats can be time-consuming for large arrays. In order to avoid repeated computations, each Time or TimeDelta instance caches such transformations internally:

>>> t = Time(np.arange(1e6), format='unix', scale='utc')

>>> time x = t.tt  
CPU times: user 263 ms, sys: 4.02 ms, total: 267 ms
Wall time: 267 ms

>>> time x = t.tt  
CPU times: user 28 µs, sys: 9 µs, total: 37 µs
Wall time: 32.9 µs

Actions such as changing the output precision or sub-format will clear the cache. In order to explicitly clear the internal cache do:

>>> del t.cache

>>> time x = t.tt  
CPU times: user 263 ms, sys: 4.02 ms, total: 267 ms
Wall time: 267 ms

Since astropy 3.1 these objects can be changed internally. In order to ensure consistency between the transformed (and cached) version and the original, the transformed object is set to be not writeable. For example:

>>> x = t.tt
>>> x[1] = '2000:001'
Traceback (most recent call last):
ValueError: Time object is read-only. Make a copy() or set "writeable" attribute to True.

If you require modifying the object then make a copy first, e.g. x = t.tt.copy().

Transformation offsets

Time scale transformations that cross one of the orange circles in the image above require an additional offset time value that is model or observation-dependent. See SOFA Time Scale and Calendar Tools for further details.

The two attributes delta_ut1_utc and delta_tdb_tt provide a way to set these offset times explicitly. These represent the time scale offsets UT1 - UTC and TDB - TT, respectively. As an example:

>>> t = Time('2010-01-01 00:00:00', format='iso', scale='utc')
>>> t.delta_ut1_utc = 0.334  # Explicitly set one part of the transformation
>>> t.ut1.iso    # ISO representation of time in UT1 scale
'2010-01-01 00:00:00.334'

For the UT1 to UTC offset, one has to interpolate the observed values provided by the International Earth Rotation and Reference Systems (IERS) Service. Astropy will automatically download and use values from the IERS which cover times spanning from 1973-Jan-01 through one year into the future. In addition the astropy package is bundled with a data table of values provided in Bulletin B, which cover the period from 1962 to shortly before an astropy release.

When the delta_ut1_utc attribute is not set explicitly then IERS values will be used (initiating a download of a few Mb file the first time). For details about how IERS values are used in astropy time and coordinates, and to understand how to control automatic downloads see IERS data access (astropy.utils.iers). The example below illustrates converting to the UT1 scale along with the auto-download feature:

>>> t = Time('2016:001')
>>> t.ut1  
Downloading https://maia.usno.navy.mil/ser7/finals2000A.all
|==================================================================| 3.0M/3.0M (100.00%)         6s
<Time object: scale='ut1' format='yday' value=2016:001:00:00:00.082>


The IERS_Auto class contains machinery to ensure that the IERS table is kept up to date by auto-downloading the latest version as needed. This means that the IERS table is assured of having the state-of-the-art definitive and predictive values for Earth rotation. As a user it is your responsibility to understand the accuracy of IERS predictions if your science depends on that. If you request UT1-UTC for times beyond the range of IERS table data then the nearest available values will be provided.

In the case of the TDB to TT offset, most users need only provide the lon and lat values when creating the Time object. If the delta_tdb_tt attribute is not explicitly set then the ERFA C-library routine eraDtdb will be used to compute the TDB to TT offset. Note that if lon and lat are not explicitly initialized, values of 0.0 degrees for both will be used.

The following code replicates an example in the SOFA Time Scale and Calendar Tools document. It does the transform from UTC to all supported time scales (TAI, TCB, TCG, TDB, TT, UT1, UTC). This requires an observer location (here, latitude and longitude).:

>>> import astropy.units as u
>>> t = Time('2006-01-15 21:24:37.5', format='iso', scale='utc',
...          location=(-155.933222*u.deg, 19.48125*u.deg))
>>> t.utc.iso
'2006-01-15 21:24:37.500'
>>> t.ut1.iso  
'2006-01-15 21:24:37.834'
>>> t.tai.iso
'2006-01-15 21:25:10.500'
>>> t.tt.iso
'2006-01-15 21:25:42.684'
>>> t.tcg.iso
'2006-01-15 21:25:43.323'
>>> t.tdb.iso
'2006-01-15 21:25:42.684'
>>> t.tcb.iso
'2006-01-15 21:25:56.894'


One can generate a unique hash key for scalar (0-dimensional) Time or TimeDelta objects. The key is based on a tuple of jd1, jd2, scale, and location (if present, None otherwise).

Note that two Time objects with a different scale can compare equal but still have different hash keys. This a practical consideration driven in by performance, but in most cases represents a desirable behavior.

Sidereal Time

Apparent or mean sidereal time can be calculated using sidereal_time(). The method returns a Longitude with units of hourangle, which by default is for the longitude corresponding to the location with which the Time object is initialized. Like the scale transformations, ERFA C-library routines are used under the hood, which support calculations following different IAU resolutions. Sample usage:

>>> t = Time('2006-01-15 21:24:37.5', scale='utc', location=('120d', '45d'))
>>> t.sidereal_time('mean')  
<Longitude 13.08952187 hourangle>
>>> t.sidereal_time('apparent')  
<Longitude 13.08950368 hourangle>
>>> t.sidereal_time('apparent', 'greenwich')  
<Longitude 5.08950368 hourangle>
>>> t.sidereal_time('apparent', '-90d')  
<Longitude 23.08950368 hourangle>
>>> t.sidereal_time('apparent', '-90d', 'IAU1994')  
<Longitude 23.08950365 hourangle>

Time Deltas

Simple time arithmetic is supported using the TimeDelta class. The following operations are available:

  • Create a TimeDelta explicitly by instantiating a class object

  • Create a TimeDelta by subtracting two Times

  • Add a TimeDelta to a Time object to get a new Time

  • Subtract a TimeDelta from a Time object to get a new Time

  • Add two TimeDelta objects to get a new TimeDelta

  • Negate a TimeDelta or take its absolute value

  • Multiply or divide a TimeDelta by a constant or array

  • Convert TimeDelta objects to and from time-like Quantities

The TimeDelta class is derived from the Time class and shares many of its properties. One difference is that the time scale has to be one for which one day is exactly 86400 seconds. Hence, the scale cannot be UTC.

The available time formats are:










Use of the TimeDelta object is easily illustrated in the few examples below:

>>> t1 = Time('2010-01-01 00:00:00')
>>> t2 = Time('2010-02-01 00:00:00')
>>> dt = t2 - t1  # Difference between two Times
>>> dt
<TimeDelta object: scale='tai' format='jd' value=31.0>
>>> dt.sec

>>> from astropy.time import TimeDelta
>>> dt2 = TimeDelta(50.0, format='sec')
>>> t3 = t2 + dt2  # Add a TimeDelta to a Time
>>> t3.iso
'2010-02-01 00:00:50.000'

>>> t2 - dt2  # Subtract a TimeDelta from a Time
<Time object: scale='utc' format='iso' value=2010-01-31 23:59:10.000>

>>> dt + dt2  
<TimeDelta object: scale='tai' format='jd' value=31.0005787037>

>>> import numpy as np
>>> t1 + dt * np.linspace(0, 1, 5)
<Time object: scale='utc' format='iso' value=['2010-01-01 00:00:00.000'
'2010-01-08 18:00:00.000' '2010-01-16 12:00:00.000' '2010-01-24 06:00:00.000'
'2010-02-01 00:00:00.000']>

The TimeDelta has a to_value method which supports controlling the type of the output representation by providing either a format name and optional subformat or a valid astropy unit:

>>> dt.to_value(u.hr)
>>> dt.to_value('jd', 'str')

Time Scales for Time Deltas

Above, one sees that the difference between two UTC times is a TimeDelta with a scale of TAI. This is because a UTC time difference cannot be uniquely defined unless one knows the two times that were differenced (because of leap seconds, a day does not always have 86400 seconds). For all other time scales, the TimeDelta inherits the scale of the first Time object:

>>> t1 = Time('2010-01-01 00:00:00', scale='tcg')
>>> t2 = Time('2011-01-01 00:00:00', scale='tcg')
>>> dt = t2 - t1
>>> dt
<TimeDelta object: scale='tcg' format='jd' value=365.0>

When TimeDelta objects are added or subtracted from Time objects, scales are converted appropriately, with the final scale being that of the Time object:

>>> t2 + dt
<Time object: scale='tcg' format='iso' value=2012-01-01 00:00:00.000>
>>> t2.tai
<Time object: scale='tai' format='iso' value=2010-12-31 23:59:27.068>
>>> t2.tai + dt
<Time object: scale='tai' format='iso' value=2011-12-31 23:59:27.046>

TimeDelta objects can be converted only to objects with compatible scales, i.e., scales for which it is not necessary to know the times that were differenced:

>>> dt.tt  
<TimeDelta object: scale='tt' format='jd' value=364.999999746>
>>> dt.tdb  
Traceback (most recent call last):
ScaleValueError: Cannot convert TimeDelta with scale 'tcg' to scale 'tdb'

TimeDelta objects can also have an undefined scale, in which case it is assumed that there scale matches that of the other Time or TimeDelta object (or is TAI in case of a UTC time):

>>> t2.tai + TimeDelta(365., format='jd', scale=None)
<Time object: scale='tai' format='iso' value=2011-12-31 23:59:27.068>

Barycentric and Heliocentric Light Travel Time Corrections

The arrival times of photons at an observatory are not particularly useful for accurate timing work, such as eclipse/transit timing of binaries or exoplanets. This is because the changing location of the observatory causes photons to arrive early or late. The solution is to calculate the time the photon would have arrived at a standard location; either the Solar system barycentre or the heliocentre.

Suppose you observed IP Peg from Greenwich and have a list of times in MJD form, in the UTC timescale. You then create appropriate Time and SkyCoord objects and calculate light travel times to the barycentre as follows:

>>> from astropy import time, coordinates as coord, units as u
>>> ip_peg = coord.SkyCoord("23:23:08.55", "+18:24:59.3",
...                         unit=(u.hourangle, u.deg), frame='icrs')
>>> greenwich = coord.EarthLocation.of_site('greenwich')  
>>> times = time.Time([56325.95833333, 56325.978254], format='mjd',
...                   scale='utc', location=greenwich)  
>>> ltt_bary = times.light_travel_time(ip_peg)  
>>> ltt_bary 
<TimeDelta object: scale='tdb' format='jd' value=[-0.0037715  -0.00377286]>

If you desire the light travel time to the heliocentre instead then use:

>>> ltt_helio = times.light_travel_time(ip_peg, 'heliocentric') 
>>> ltt_helio 
<TimeDelta object: scale='tdb' format='jd' value=[-0.00376576 -0.00376712]>

The method returns an TimeDelta object, which can be added to your times to give the arrival time of the photons at the barycentre or heliocentre. Here, one should be careful with the timescales used; for more detailed information about timescales, see Time Scale.

The heliocentre is not a fixed point, and therefore the gravity continually changes at the heliocentre. Thus, the use of a relativistic timescale like TDB is not particularly appropriate, and, historically, times corrected to the heliocentre are given in the UTC timescale:

>>> times_heliocentre = times.utc + ltt_helio  

Corrections to the barycentre are more precise than the heliocentre, because the barycenter is a fixed point where gravity is constant. For maximum accuracy you want to have your barycentric corrected times in a timescale that has always ticked at a uniform rate, and ideally one whose tick rate is related to the rate that a clock would tick at the barycentre. For this reason, barycentric corrected times normally use the TDB timescale:

>>> time_barycentre = times.tdb + ltt_bary  

By default, the light travel time is calculated using the position and velocity of Earth and the Sun from built-in ERFA routines, but one can also use more precise calculations using the JPL ephemerides (which are derived from dynamical models). An example using the JPL ephemerides is:

>>> ltt_bary_jpl = times.light_travel_time(ip_peg, ephemeris='jpl') 
>>> ltt_bary_jpl 
<TimeDelta object: scale='tdb' format='jd' value=[-0.0037715  -0.00377286]>
>>> (ltt_bary_jpl - ltt_bary).to(u.ms) 
<Quantity [-0.00132325, -0.00132861] ms>

The difference between the builtin ephemerides and the JPL ephemerides is normally of the order of 1/100th of a millisecond, so the builtin ephemerides should be suitable for most purposes. For more details about what ephemerides are available, including the requirements for using JPL ephemerides, see Solar System Ephemerides.

Interaction with Time-like Quantities

Where possible, Quantity objects with units of time are treated as TimeDelta objects with undefined scale (though necessarily with lower precision). They can also be used as input in constructing Time and TimeDelta objects, and TimeDelta objects can be converted to Quantity objects of arbitrary units of time. Usage is most easily illustrated by examples:

>>> import astropy.units as u
>>> Time(10.*u.yr, format='gps')   # time-valued quantities can be used for
...                                # for formats requiring a time offset
<Time object: scale='tai' format='gps' value=315576000.0>
>>> Time(10.*u.yr, 1.*u.s, format='gps')
<Time object: scale='tai' format='gps' value=315576001.0>
>>> Time(2000.*u.yr, format='jyear')
<Time object: scale='tt' format='jyear' value=2000.0>
>>> Time(2000.*u.yr, format='byear')
...                                # but not for Besselian year, which implies
...                                # a different time scale
Traceback (most recent call last):
ValueError: Input values did not match the format class byear:
ValueError: Cannot use Quantities for 'byear' format, as the interpretation would be ambiguous. Use float with Besselian year instead.

>>> TimeDelta(10.*u.yr)            # With a quantity, no format is required
<TimeDelta object: scale='None' format='jd' value=3652.5>

>>> dt = TimeDelta([10., 20., 30.], format='jd')
>>> dt.to(u.hr)                    # can convert TimeDelta to a quantity  
<Quantity [240., 480., 720.] h>
>>> dt > 400. * u.hr               # and compare to quantities with units of time
array([False,  True,  True]...)
>>> dt + 1.*u.hr                   # can also add/subtract such quantities  
<TimeDelta object: scale='None' format='jd' value=[10.04166667 20.04166667 30.04166667]>
>>> Time(50000., format='mjd', scale='utc') + 1.*u.hr  
<Time object: scale='utc' format='mjd' value=50000.0416667>
>>> dt * 10.*u.km/u.s              # for multiplication and division with a
...                                # Quantity, TimeDelta is converted  
<Quantity [100., 200., 300.] d km / s>
>>> dt * 10.*u.Unit(1)             # unless the Quantity is dimensionless  
<TimeDelta object: scale='None' format='jd' value=[100. 200. 300.]>

Writing a Custom Format

Some applications may need a custom Time format, and this capability is available by making a new subclass of the TimeFormat class. When such a subclass is defined in your code then the format class and corresponding name is automatically registered in the set of available time formats.

The key elements of a new format class are illustrated by examining the code for the jd format (which is one of the simplest):

class TimeJD(TimeFormat):
    Julian Date time format.
    name = 'jd'  # Unique format name

    def set_jds(self, val1, val2):
        Set the internal jd1 and jd2 values from the input val1, val2.
        The input values are expected to conform to this format, as
        validated by self._check_val_type(val1, val2) during __init__.
        self._check_scale(self._scale)  # Validate scale.
        self.jd1, self.jd2 = day_frac(val1, val2)

    def value(self):
        Return format ``value`` property from internal jd1, jd2
        return self.jd1 + self.jd2

As mentioned above, the _check_val_type(self, val1, val2) method may need to be overridden to validate the inputs as conforming to the format specification. By default this checks for valid float, float array, or Quantity inputs. In contrast the iso format class ensures the inputs meet the ISO format spec for strings.

One special case that is relatively common and easier to implement is a format that makes a small change to the date format. For instance one could insert T in the yday format with the following TimeYearDayTimeCustom class. Notice how the subfmts definition is modified slightly from the standard TimeISO class from which it inherits:

>>> from astropy.time import TimeISO
>>> class TimeYearDayTimeCustom(TimeISO):
...    """
...    Year, day-of-year and time as "<YYYY>-<DOY>T<HH>:<MM>:<SS.sss...>".
...    The day-of-year (DOY) goes from 001 to 365 (366 in leap years).
...    For example, 2000-001T00:00:00.000 is midnight on January 1, 2000.
...    The allowed subformats are:
...    - 'date_hms': date + hours, mins, secs (and optional fractional secs)
...    - 'date_hm': date + hours, mins
...    - 'date': date
...    """
...    name = 'yday_custom'  # Unique format name
...    subfmts = (('date_hms',
...                '%Y-%jT%H:%M:%S',
...                '{year:d}-{yday:03d}T{hour:02d}:{min:02d}:{sec:02d}'),
...               ('date_hm',
...                '%Y-%jT%H:%M',
...                '{year:d}-{yday:03d}T{hour:02d}:{min:02d}'),
...               ('date',
...                '%Y-%j',
...                '{year:d}-{yday:03d}'))

>>> t = Time('2000-01-01')
>>> t.yday_custom
>>> t2 = Time('2016-001T00:00:00')
>>> t2.iso
'2016-01-01 00:00:00.000'

Another special case that is relatively common is a format that represents the time since a particular epoch. The classic example is Unix time which is the number of seconds since 1970-01-01 00:00:00 UTC, not counting leap seconds. What if we wanted that value but do want to count leap seconds. This would be done by using the TAI scale instead of the UTC scale. In this case we inherit from the TimeFromEpoch class and define a few class attributes:

>>> from astropy.time.formats import erfa, TimeFromEpoch
>>> class TimeUnixLeap(TimeFromEpoch):
...    """
...    Seconds from 1970-01-01 00:00:00 TAI.  Similar to Unix time
...    but this includes leap seconds.
...    """
...    name = 'unix_leap'
...    unit = 1.0 / erfa.DAYSEC  # in days (1 day == 86400 seconds)
...    epoch_val = '1970-01-01 00:00:00'
...    epoch_val2 = None
...    epoch_scale = 'tai'  # Scale for epoch_val class attribute
...    epoch_format = 'iso'  # Format for epoch_val class attribute

>>> t = Time('2000-01-01')
>>> t.unix_leap
>>> t.unix_leap - t.unix

Going beyond this will probably require looking at the astropy code for more guidance, but if you get stuck the astropy developers are more than happy to help. If you write a format class that is widely useful then we might want to include it in the core!


When a Time object is constructed from a timezone-aware datetime, no timezone information is saved in the Time object. However, Time objects can be converted to timezone-aware datetime objects:

>>> from datetime import datetime
>>> from astropy.time import Time, TimezoneInfo
>>> import astropy.units as u
>>> utc_plus_one_hour = TimezoneInfo(utc_offset=1*u.hour)
>>> dt_aware = datetime(2000, 1, 1, 0, 0, 0, tzinfo=utc_plus_one_hour)
>>> t = Time(dt_aware)  # Loses timezone info, converts to UTC
>>> print(t)            # will return UTC
1999-12-31 23:00:00
>>> print(t.to_datetime(timezone=utc_plus_one_hour)) # to timezone-aware datetime
2000-01-01 00:00:00+01:00

Timezone database packages, like pytz for example, may be more convenient to use to create tzinfo objects used to specify timezones rather than the TimezoneInfo object.

Custom string formats with strftime and strptime

The Time object supports output string representation using the format specification language defined in the Python standard library for time.strftime. This can be done using the strftime method:

>>> from astropy.time import Time
>>> t = Time('2018-01-01T10:12:58')
>>> t.strftime('%H:%M:%S %d %b %Y')
'10:12:58 01 Jan 2018'

Conversely, to create a Time object from a custom date string that can be parsed with Python standard library time.strptime (using the same format language linked above), use the strptime class method:

>>> from astropy.time import Time
>>> t = Time.strptime('23:59:60 30 June 2015', '%H:%M:%S %d %B %Y')
>>> t
<Time object: scale='utc' format='isot' value=2015-06-30T23:59:60.000>


astropy.time Package



If the current ERFA leap second table is out of date, try to update it.


OperandTypeError(left, right[, op])


Time(val[, val2, format, scale, precision, …])

Represent and manipulate times and dates for astronomy.

TimeBesselianEpoch(val1, val2, scale, …[, …])

Besselian Epoch year as floating point value(s) like 1950.0

TimeBesselianEpochString(val1, val2, scale, …)

Besselian Epoch year as string value(s) like ‘B1950.0’

TimeCxcSec(val1, val2, scale, precision, …)

Chandra X-ray Center seconds from 1998-01-01 00:00:00 TT.

TimeDatetime(val1, val2, scale, precision, …)

Represent date as Python standard library datetime object

TimeDatetime64(val1, val2, scale, precision, …)

TimeDecimalYear(val1, val2, scale, …[, …])

Time as a decimal year, with integer values corresponding to midnight of the first day of each year.

TimeDelta(val[, val2, format, scale, copy])

Represent the time difference between two times.

TimeDeltaDatetime(val1, val2, scale, …[, …])

Time delta in datetime.timedelta

TimeDeltaFormat(val1, val2, scale, …[, …])

Base class for time delta representations

TimeDeltaJD(val1, val2, scale, precision, …)

Time delta in Julian days (86400 SI seconds)

TimeDeltaNumeric(val1, val2, scale, …[, …])

TimeDeltaSec(val1, val2, scale, precision, …)

Time delta in SI seconds

TimeEpochDate(val1, val2, scale, precision, …)

Base class for support floating point Besselian and Julian epoch dates

TimeEpochDateString(val1, val2, scale, …)

Base class to support string Besselian and Julian epoch dates such as ‘B1950.0’ or ‘J2000.0’ respectively.

TimeFITS(val1, val2, scale, precision, …)

FITS format: “[±Y]YYYY-MM-DD[THH:MM:SS[.sss]]”.

TimeFormat(val1, val2, scale, precision, …)

Base class for time representations.

TimeFromEpoch(val1, val2, scale, precision, …)

Base class for times that represent the interval from a particular epoch as a floating point multiple of a unit time interval (e.g.

TimeGPS(val1, val2, scale, precision, …[, …])

GPS time: seconds from 1980-01-06 00:00:00 UTC For example, 630720013.0 is midnight on January 1, 2000.

TimeISO(val1, val2, scale, precision, …[, …])

ISO 8601 compliant date-time format “YYYY-MM-DD HH:MM:SS.sss…”.

TimeISOT(val1, val2, scale, precision, …)

ISO 8601 compliant date-time format “YYYY-MM-DDTHH:MM:SS.sss…”.


Container for meta information like name, description, format.

TimeJD(val1, val2, scale, precision, …[, …])

Julian Date time format.

TimeJulianEpoch(val1, val2, scale, …[, …])

Julian Epoch year as floating point value(s) like 2000.0

TimeJulianEpochString(val1, val2, scale, …)

Julian Epoch year as string value(s) like ‘J2000.0’

TimeMJD(val1, val2, scale, precision, …[, …])

Modified Julian Date time format.

TimeNumeric(val1, val2, scale, precision, …)

TimePlotDate(val1, val2, scale, precision, …)

Matplotlib plot_date input: 1 + number of days from 0001-01-01 00:00:00 UTC

TimeString(val1, val2, scale, precision, …)

Base class for string-like time representations.

TimeUnique(val1, val2, scale, precision, …)

Base class for time formats that can uniquely create a time object without requiring an explicit format specifier.

TimeUnix(val1, val2, scale, precision, …)

Unix time: seconds from 1970-01-01 00:00:00 UTC.

TimeYMDHMS(val1, val2, scale, precision, …)

ymdhms: A Time format to represent Time as year, month, day, hour, minute, second (thus the name ymdhms).

TimeYearDayTime(val1, val2, scale, …[, …])

Year, day-of-year and time as “YYYY:DOY:HH:MM:SS.sss…”.

TimezoneInfo([utc_offset, dst, tzname])

Subclass of the tzinfo object, used in the to_datetime method to specify timezones.

Class Inheritance Diagram

Inheritance diagram of astropy.time.core.OperandTypeError, astropy.time.core.ScaleValueError, astropy.time.core.Time, astropy.time.formats.TimeBesselianEpoch, astropy.time.formats.TimeBesselianEpochString, astropy.time.formats.TimeCxcSec, astropy.time.formats.TimeDatetime, astropy.time.formats.TimeDatetime64, astropy.time.formats.TimeDecimalYear, astropy.time.core.TimeDelta, astropy.time.formats.TimeDeltaDatetime, astropy.time.formats.TimeDeltaFormat, astropy.time.formats.TimeDeltaJD, astropy.time.formats.TimeDeltaNumeric, astropy.time.formats.TimeDeltaSec, astropy.time.formats.TimeEpochDate, astropy.time.formats.TimeEpochDateString, astropy.time.formats.TimeFITS, astropy.time.formats.TimeFormat, astropy.time.formats.TimeFromEpoch, astropy.time.formats.TimeGPS, astropy.time.formats.TimeISO, astropy.time.formats.TimeISOT, astropy.time.core.TimeInfo, astropy.time.formats.TimeJD, astropy.time.formats.TimeJulianEpoch, astropy.time.formats.TimeJulianEpochString, astropy.time.formats.TimeMJD, astropy.time.formats.TimeNumeric, astropy.time.formats.TimePlotDate, astropy.time.formats.TimeString, astropy.time.formats.TimeUnique, astropy.time.formats.TimeUnix, astropy.time.formats.TimeYMDHMS, astropy.time.formats.TimeYearDayTime, astropy.time.formats.TimezoneInfo

Acknowledgments and Licenses

This package makes use of the ERFA Software ANSI C library. The copyright of the ERFA software belongs to the NumFOCUS Foundation. The library is made available under the terms of the “BSD-three clauses” license.

The ERFA library is derived, with permission, from the International Astronomical Union’s “Standards of Fundamental Astronomy” library, available from http://www.iausofa.org.