Dates and Time¶

Ancient and modern dates¶

Skyfield normally uses the modern Gregorian calendar, even for dates in history before the Gregorian calendar’s adoption in 1582. This “proleptic” use of Gregorian dates makes date calculations simple, is compatible with Python’s datetime, and is also the behavior of the United States Naval Observatory library on which many Skyfield routines were originally based.

But the Gregorian calendar is awkward for historians and students of ancient astronomy, because the calendar in actual use before 1582 was the old Julian calendar established by Julius Caesar’s calendar reform in 45 BC. The two calendars agree over the century between the leap day of AD 200 and the leap day of AD 300. But because the Julian calendar is not quite synchronized with the seasons, its dates run ahead of the Gregorian calendar before that century and run behind the Gregorian calendar after it.

If you would like Skyfield to switch to the Julian calendar for historical dates — both when interpreting the dates you input, and when producing calendar dates as output — simply give your Timescale object the Julian day on which you would like the calendar to switch.

from skyfield.api import GREGORIAN_START

ts.julian_calendar_cutoff = GREGORIAN_START

t = ts.tt_jd(range(2299159, 2299163))
for s in t.tt_strftime():
    print(s)

1582-10-03 12:00:00 TT
1582-10-04 12:00:00 TT
1582-10-15 12:00:00 TT
1582-10-16 12:00:00 TT

As you can see from these four successive days in history, Pope Gregory had the calendar jump directly from the old Julian calendar date 1582 October 4 to the new Gregorian calendar date 1582 October 15, bringing the date of Easter back into sync with the equinox. Skyfield provides two constants for popular cutoff dates:

• GREGORIAN_START — Julian day 2299161, on which the new Gregorian calendar went into effect in Rome.
• GREGORIAN_START_ENGLAND — Julian day 2361222, on which the new Gregorian calendar went into effect in England in 1752 (the reform having initially been rejected by the English bishops, “Seeing that the Bishop of Rome is Antichrist, therefore we may not communicate with him in any thing”).

You are free to choose your own cutoff Julian day if you are studying astronomy records from a country that adopted the Gregorian calendar on some other date. Russia, for example, did not adopt it until the twentieth century. The default value, that asks the timescale to always use Gregorian dates, is None:

ts.julian_calendar_cutoff = None

Note that even the Julian calendar becomes anachronistic before its adoption in 45 BC, so all dates generated by Skyfield are “proleptic” before that date. And, of course, the Julian calendar was local to the civilization that ringed the Mediterranean. If you are interested in relating astronomical events to more ancient Roman calendars, or the calendars of other civilizations, try searching for a third-party Python package that supports the calendar you are interested in.

Building and printing UTC¶

The utc parameter in the examples above specifies Coordinated Universal Time (UTC), the world clock known affectionately as “Greenwich Mean Time” which is the basis for all of the world’s timezones. If you are comfortable dealing directly with UTC instead of your local timezone, you can build and display dates without needing any other library besides Skyfield.

You can build a Time from a calendar date and UTC time using Timescale.utc. Provide only as many parameters as you want — year, month, day, hour, minute, and second — and Skyfield will fill in the rest by defaulting to January first and zero hours, minutes, and seconds.

Feel free to use fractional days, hours, and minutes. Here are several ways to specify the exact same time and date:

# Four ways to specify 2014 January 18 01:35:37.5

t1 = ts.utc(2014, 1, 18.06640625)
t2 = ts.utc(2014, 1, 18, 1.59375)
t3 = ts.utc(2014, 1, 18, 1, 35.625)
t4 = ts.utc(2014, 1, 18, 1, 35, 37.5)

assert t1 == t2 == t3 == t4    # True!

# Several ways to print a time as UTC.

print(tuple(t1.utc))
print(t1.utc_iso())
print(t1.utc_strftime())
print(t1.utc_strftime('On %Y %b %d at %H:%M:%S'))
print(t1.utc_jpl())

(2014, 1, 18, 1, 35, 37.5)
2014-01-18T01:35:38Z
2014-01-18 01:35:38 UTC
On 2014 Jan 18 at 01:35:38
A.D. 2014-Jan-18 01:35:37.5000 UTC

The 6 values returned by utc() can be accessed as the attributes year, month, day, hour, minute, and second.

print(t1.utc.year, '/', t1.utc.month, '/', t1.utc.day)
print(t1.utc.hour, ':', t1.utc.minute, ':', t1.utc.second)

2014 / 1 / 18
1 : 35 : 37.5

If you want to use the current time, Skyfield leverages the minimal support for UTC in the Python Standard Library to offer a now() function that reads your system clock and returns the current time as a Time object (assuming that your operating system clock is correct and configured with the correct time zone):

from skyfield.api import load

# Asking the current date and time

ts = load.timescale()
t = ts.now()
print(t.utc_jpl())

A.D. 2015-Oct-11 10:00:00.0000 UTC

UTC and your timezone¶

To move beyond UTC and work with other world timezones, you will need to install a time zone database for your version of Python.

• Every version of Python that Skyfield supports will work with the pytz package described in this section.
• Python 3.6 upgraded the Standard Library datetime type so that the contortions of pytz are no longer necessary, and instead recommends dateutil for working with timezones. Consult its documentation if you are interested in using it.
• Python 3.9 will offer a native zoneinfo module that for the first time brings timezone support into the Python Standard Library.

This documentation will focus on the first approach, which works universally across all Python versions. You can install the third-party pytz library by listing it in the dependencies of your package, adding it to your project’s requirements.txt file, or simply installing it manually:

pip install pytz

Once it is installed, building a Time from a local time is simple. Instantiate a normal Python datetime, pass it to the localize() method of your time zone, and pass the result to Skyfield:

from datetime import datetime
from pytz import timezone

eastern = timezone('US/Eastern')

# Converting US Eastern Time to a Skyfield Time.

d = datetime(2014, 1, 16, 1, 32, 9)
e = eastern.localize(d)
t = ts.from_datetime(e)

When Skyfield returns a Time at the end of a calculation, you can ask for either a UTC datetime or a datetime in your own timezone:

# UTC datetime

dt = t.utc_datetime()
print('UTC: ' + str(dt))

# Converting back to an Eastern Time datetime.

dt = t.astimezone(eastern)
print('EST: ' + str(dt))

UTC: 2014-01-16 06:32:09+00:00
EST: 2014-01-16 01:32:09-05:00

As we would expect, 1:32 AM in the Eastern time zone in January is 6:32 AM local time in Greenwich, England, five hours to the east across the Atlantic.

Note that Skyfield’s astimezone() method will detect that you are using a pytz timezone and automatically call its normalize() method for you — which makes sure that daylight savings time is handled correctly — to spare you from having to make the call yourself.

If you want a Time to hold an entire array of dates, as discussed below in Date arrays, then you can provide a list of datetime objects to the Timescale.from_datetimes() method. The UTC methods will then return whole lists of values.

UTC and leap seconds¶

The rate of Earth’s rotation is gradually slowing down. Since the UTC standard specifies a fixed length for the second, promises a day of 24 hours, and limits an hour to 60 minutes, the only way to stay within the rules while keeping UTC synchronized with the Earth is to occasionally add an extra leap second to one of the year’s minutes.

See The leap second table if you are interested in printing Skyfield’s full list of leap seconds.

The International Earth Rotation Service currently restricts itself to appending a leap second to the last minute of June or the last minute of December. When a leap second is inserted, its minute counts 61 seconds numbered 00–60 instead of staying within the usual range 00–59. One recent leap second was in June 2012:

# Display 5 seconds around a leap second

five_seconds = [58, 59, 60, 61, 62]
t = ts.utc(2012, 6, 30, 23, 59, five_seconds)

for string in t.utc_jpl():
    print(string)

A.D. 2012-Jun-30 23:59:58.0000 UTC
A.D. 2012-Jun-30 23:59:59.0000 UTC
A.D. 2012-Jun-30 23:59:60.0000 UTC
A.D. 2012-Jul-01 00:00:00.0000 UTC
A.D. 2012-Jul-01 00:00:01.0000 UTC

Note that Skyfield has no problem with a calendar tuple that has hours, minutes, or — as in this case — seconds that are out of range. When we provided a range of numbers 58 through 62 as seconds, Skyfield added exactly the number of seconds we specified to the end of June and let the value overflow cleanly into the beginning of July.

Keep two consequences in mind when using UTC in your calculations.

First, expect an occasional jump or discrepancy if you are striding forward through time using the UTC minute, hour, or day. For example, an hourly plot of planet’s position will show the planet moving slightly farther during an hour that was lengthened by a leap second than during other hours of the year. An Earth satellite’s velocity will seem higher when you reach the minute that includes 61 seconds. And so forth. Problems like these are the reason that the Time class only uses UTC for input and output, and insists on keeping time internally using the uniform time scales discussed below in Uniform time scales: TAI, TT, and TDB.

Second, leap seconds disqualify the Python datetime from use as a general way to represent time because in many versions of Python the datetime refuses to accept seconds greater than 59:

datetime(2012, 6, 30, 19, 59, 60)

Traceback (most recent call last):
  ...
ValueError: second must be in 0..59

That limitation is why Skyfield offers a second version of each method that returns a datetime. These fancier methods return a leap-second flag as an additional return value:

• Time.utc_datetime_and_leap_second()
• Time.astimezone_and_leap_second()

The leap-second return value is usually 0 but jumps to 1 when Skyfield is forced to represent a leap second as a datetime with the incorrect time 23:59:59.

# Asking for the leap_second flag to learn the whole story

dt, leap_second = t.astimezone_and_leap_second(eastern)

for dt_i, leap_second_i in zip(dt, leap_second):
    print('{0}  leap_second = {1}'.format(dt_i, leap_second_i))

2012-06-30 19:59:58-04:00  leap_second = 0
2012-06-30 19:59:59-04:00  leap_second = 0
2012-06-30 19:59:59-04:00  leap_second = 1
2012-06-30 20:00:00-04:00  leap_second = 0
2012-06-30 20:00:01-04:00  leap_second = 0

Using calendar tuples to represent UTC times is more elegant than using Python datetime objects because leap seconds can be represented accurately. If your application cannot avoid using datetime objects, then you will have to decide whether to simply ignore the leap_second value or to somehow output the leap second information.

Date arithmetic¶

Dates support a few simple math operations:

from datetime import timedelta

t - 10                   # 10 days earlier
t + 0.25                 # 6 hours later
t + timedelta(hours=12)  # 12 hours later

t2 - t1  # difference between times, in days

Raw numbers, like 10 and 0.25 above, specify days of Terrestrial Time — units of exactly 24 hours of 60 minutes of 60 SI seconds, measured in the Earth’s relativistic frame of reference. If you increment or decrement a date across a leap second, you will notice that the clock time returned by Skyfield’s UTC functions is one second earlier or later than you expect.

Date arrays¶

If you want to ask where a planet or satellite was across a whole series of times and dates, then Skyfield will work most efficiently if, instead of building many separate Time objects, you build a single Time object that holds the entire array of dates.

There are three techniques for building a Time array.

• Provide tai() or tt() or tdb() or ut1() with a Python list or NumPy array of numbers for one of the six components of the calendar date (year, month, day, hour, minute, or second).
• Provide tai_jd() or tt_jd() or tdb_jd() or ut1_jd() with a list or NumPy array of floating point numbers.
• Provide from_datetimes() with a Python list of datetime objects.

The first possibility is generally the one that is the most fun, because its lets you vary whichever time unit you want while holding the others constant. You are free to provide out-of-range values and leave it to Skyfield to work out the correct result. Here are some examples:

ts.utc(range(1900, 1950))     # Fifty years 1900–1949
ts.utc(1980, range(1, 25))    # 24 months of 1980 and 1981
ts.utc(2005, 5, [1, 11, 21])  # 1st, 11th, and 21st of May

# Negative values work too!  Here are the
# ten seconds crossing the 1974 leap second.
ts.utc(1975, 1, 1, 0, 0, range(-5, 5))

The resulting Time object will hold an array of times. As illustrated in the previous section (on leap seconds), you can use a Python for to print each time separately:

t = ts.utc(2020, 6, 16, 7, range(4))

for s in t.utc_strftime('%Y-%m-%d %H:%M'):
    print(s)

2020-06-16 07:00
2020-06-16 07:01
2020-06-16 07:02
2020-06-16 07:03

When you provide a time array as input to a Skyfield calculation, the output array will have an extra dimension that expands what would normally be a single result into as many results as you provided dates. We can compute the position of the Earth as an example:

# Single Earth position

planets = load('de421.bsp')
earth = planets['earth']

t = ts.utc(2014, 1, 1)
pos = earth.at(t).position.au
print(pos)

[-0.17461758  0.88567056  0.38384886]

# Whole array of Earth positions

days = [1, 2, 3, 4]
t = ts.utc(2014, 1, days)
pos = earth.at(t).position.au
print(pos)

[[-0.17461758 -0.19179872 -0.20891924 -0.22597338]
 [ 0.88567056  0.88265548  0.87936337  0.87579547]
 [ 0.38384886  0.38254134  0.38111391  0.37956709]]

Note the shape of the resulting NumPy array. If you unpack this array into three names, then you get three four-element arrays corresponding to the four dates. These four-element arrays are ready to be submitted to matplotlib and other scientific Python tools:

x, y, z = pos    # four values each
plot(x, y)       # example matplotlib call

If you instead slice along the second axis, then you can retrieve an individual position for a particular date — and the first position is exactly what was returned above when we computed the January 1st position by itself:

print(pos[:,0])

[-0.17461758  0.88567056  0.38384886]

You can combine a Python for loop with Python’s zip() builtin to print each time alongside the corresponding coordinates. There are two techniques, one of which is less efficient and the other more efficient.

# Less efficient: loop over `t`, forcing the creation of
# a separate `Time` object for each iteration of the loop.

for ti, xi, yi, zi in zip(t, x, y, z):
    print('{}  x = {:.2f} y = {:.2f} z = {:.2f}'.format(
        ti.utc_strftime('%Y-%m-%d'), xi, yi, zi,
    ))

2014-01-01  x = -0.17 y = 0.89 z = 0.38
2014-01-02  x = -0.19 y = 0.88 z = 0.38
2014-01-03  x = -0.21 y = 0.88 z = 0.38
2014-01-04  x = -0.23 y = 0.88 z = 0.38

# More efficient: loop over the output of a `Time` method,
# which returns an array of the same length as `t`.

t_strings = t.utc_strftime('%Y-%m-%d')

for tstr, xi, yi, zi in zip(t_strings, x, y, z):
    print('{}  x = {:.2f} y = {:.2f} z = {:.2f}'.format(
        tstr, xi, yi, zi,
    ))

2014-01-01  x = -0.17 y = 0.89 z = 0.38
2014-01-02  x = -0.19 y = 0.88 z = 0.38
2014-01-03  x = -0.21 y = 0.88 z = 0.38
2014-01-04  x = -0.23 y = 0.88 z = 0.38

Finally, converting an array Time back into a calendar tuple results in the year, month, day, hour, minute, and second each having the same dimension as the array itself:

print(t.utc)

[[2014. 2014. 2014. 2014.]
 [   1.    1.    1.    1.]
 [   1.    2.    3.    4.]
 [   0.    0.    0.    0.]
 [   0.    0.    0.    0.]
 [   0.    0.    0.    0.]]

Simply slice across the second dimension of the array to pull a particular calendar tuple out of the larger result:

print(t.utc[:,2])

[2014.    1.    3.    0.    0.    0.]

Slicing in the other direction, the rows can be fetched not only by index but also through the attribute names year, month, day, hour, minute, and second.

print(t.utc.year)
print(t.utc.month)
print(t.utc.day)
print(t.utc.hour)

[2014. 2014. 2014. 2014.]
[1. 1. 1. 1.]
[1. 2. 3. 4.]
[0. 0. 0. 0.]

Uniform time scales: TAI, TT, and TDB¶

Date arithmetic becomes very simple as we leave UTC behind and consider completely uniform time scales. Days are always 24 hours, hours always 60 minutes, and minutes always 60 seconds without any variation or exceptions. Such time scales are not appropriate for your morning alarm clock because they are never delayed or adjusted to stay in sync with the slowing rotation of the earth. But that is what makes them useful for astronomical calculation — because physics keeps up its dance, and the stars and planets move in their courses, whether humanity pauses to observe a UTC leap second or not.

Because they make every day the same length, uniform time scales can express dates as a simple floating-point count of days elapsed. To make all historical dates come out as positive numbers, astronomers traditionally assign each date a “Julian day” number that starts counting at 4713 BC January 1 in the old Julian calendar — the same date as 4714 BC November 24 in our Gregorian calendar. Following a tradition going back to the Greeks and Ptolemy, the count starts at noon, since the sun’s transit is an observable event but the moment of midnight is not.

So twelve noon was the moment of Julian date zero:

# When was Julian date zero?

bc_4714 = -4713
t = ts.tt(bc_4714, 11, 24, 12)
print(t.tt)

0.0

Did you notice how negative years work — that we expressed 4714 BC using the negative number -4713? People still counted by starting at one, not zero, when the scholar Dionysius Exiguus created the eras BC and AD in around the year AD 500. So his scheme has 1 BC followed immediately by AD 1 without a break. To avoid an off-by-one error, astronomers usually ignore BC and count backwards through a year zero and on into negative years. So negative year −n is what might otherwise be called either “n+1 BC” or “n+1 BCE” in a history textbook.

More than two million days have passed since 4714 BC, so modern dates tend to be rather large numbers:

# 2014 January 1 00:00 UTC expressed as Julian dates

t = ts.utc(2014, 1, 1)
print('TAI = %r' % t.tai)
print('TT  = %r' % t.tt)
print('TDB = %r' % t.tdb)

TAI = 2456658.5004050927
TT  = 2456658.5007775924
TDB = 2456658.500777592

What are these three different uniform time scales?

International Atomic Time (TAI) is maintained by the worldwide network of atomic clocks referenced by researchers with a need for very accurate time. The official leap second table is actually a table of offsets between TAI and UTC. At the end of June 2012, for example, the TAI−UTC offset was changed from 34.0 to 35.0 which is what generated the leap second in UTC.

Terrestrial Time (TT) differs from TAI only because astronomers were already maintaining a uniform time scale of their own before TAI was established, using a slightly different starting point for the day. For practical purposes, TT is simply TAI plus exactly 32.184 seconds. So it is now more than a minute ahead of UTC.

You can not only ask Skyfield for TT as a Julian date and a calendar date, but as a floating-point number of years of exactly 365.25 days each — a value which is often used as the time parameter in long-term astronomical formulae:

print('Julian year = {:.4f}'.format(t.J))

Julian year = 2014.0000

Finally, Barycentric Dynamical Time (TDB) runs at approximately the rate that an atomic clock would run if it were at rest with respect to the Solar System barycenter, and therefore unaffected by the Earth’s motion. The acceleration that Earth experiences in its orbit — sometimes speeding up, sometimes slowing down — varies the rate at which our atomic clocks run relative to an outside observer, as predicted by Einstein’s theory of General Relativity. So physical simulations of the Solar System use TDB as their clock. It is considered equivalent to the T_eph time scale traditionally used for Solar System and spacecraft simulations at the Jet Propulsion Laboratory.

UT1 and downloading IERS data¶

Finally, UT1 is the least uniform time scale of all because its clock cannot be housed in a laboratory, nor is its rate established by any human convention. It is, rather, the clock whose “hand” is the rotation of the Earth itself! The direction that the Earth is facing determines not only the coordinates of every city and observatory in the world, but also the local directions that each site will designate as their local “up”, “north”, and “east”.

It is hard to predict future values for UT1. The Earth is a young world with a still-molten iron core, a viscous mantle, and ice ages that move water weight into glaciers at the poles then release it back into the ocean. While we think we can predict, for example, Jupiter’s position thousands of years from now, predicting the fluid dynamics of the elastic rotating ellipsoid we call home is — at the moment — beyond us. We can only watch with sensitive instruments to see what the Earth does next.

Skyfield relies on the IERS, the International Earth Rotation Service, for accurate measurements of UT1 and for the schedule of leap seconds (discussed above) that keeps UTC from straying more than 0.9 seconds away from UT1.

Each new version of Skyfield carries recent IERS data in internal tables. This data will gradually fall out of date after each Skyfield release, however, with two consequences:

• The next time the IERS declares a new leap second that is not listed in Skyfield’s built-in tables, Skyfield’s UTC time will be off by 1 second for every date that falls after the leap second.
• As the Earth’s rotation speeds up or slows down in the coming years more than was predicted in Skyfield’s built-in UT1 tables, Skyfield’s idea of where the Earth is pointing will grow less accurate. This will affect both the position and direction of each GeographicPosition — whether used as an observer or a target — and will also affect Earth satellite positions.

You can avoid both of these problems by periodically downloading new data from the IERS. Simply specify that you don’t want Skyfield to use its builtin tables. In that case timescale() will instead download finals2000A.all from the IERS:

# Download and use the `finals.all` file.

ts = load.timescale(builtin=False)

[#################################] 100% finals2000A.all

As usual with data files, Skyfield will only download the file the first time you need it, then will keep using that same copy of the file that it finds on disk.

Note that the international agencies responsible for the file’s distribution sometimes have trouble keeping their servers up. For example, as I write this in May of 2022, the file cannot be fetched from ftp.iers.org because of an Outage of iers.org data servers reported on their website. At Skyfield issue #730 and Skyfield issue #732 you can find links to alternative data sources which various Skyfield users have been able to access in the meantime.

If your script will always have Internet access and you worry about the file falling out of date (and if you can trust the “modify time” file attribute on your filesystem), then you can have Skyfield download a new copy once the file on disk has grown too old (where “too old” for your application must be determined by comparing your accuracy needs with how quickly UT1 diverges without fresh IERS data; this example uses 30 days only as an illustration):

if load.days_old('finals2000A.all') > 30.0:
    load.download('finals2000A.all')

ts = load.timescale(builtin=False)

But, beware! For compatibility with versions of Skyfield ≤ 1.30, Skyfield will ignore finals2000A.all if the three old files deltat.data, deltat.preds, and Leap_Second.dat exist in the loader’s directory, in which case it will use them instead. This is to prevent users who specify builtins=False, but who downloaded the three necessary files long ago, from experiencing an unexpected download attempt. The hope is that all scripts which did not previously need Internet access will continue to run without it.

If you ever want to display or plot the behavior of UT1, astronomers use two common conventions for stating the difference between clock time and UT1. Skyfield supports them both.

print('{:+.4f}'.format(t.dut1))
print('{:+.4f}'.format(t.delta_t))

-0.0970
+67.2810

The two quantities are:

• DUT1 — The difference between UTC and UT1, which should always be less than 0.9 if the IERS succeeds at its mission. Note that there are two different reasons that this value changes: every day it changes a small amount because of the drift of UT1; and superimposed on this drift is a big jump of 1.0 seconds every time a leap second passes.
• ∆T — The difference between TT and UT1. This is a much more straightforward value than DUT1, without all the ugly discontinuities caused by leap seconds. Because TT is a uniform timescale, ∆T provides a pure and continuous record of how UT1 has changed over the decades that we have been measuring the Earth’s rotation to high precision.

Setting a Custom Value For ∆T¶

If you ever want to specify your own value for ∆T, then provide a delta_t keyword argument when creating your timescale:

load.timescale(delta_t=67.2810).utc((2014, 1, 1))

Values cached on the Time object¶

When you create a Time it goes ahead and computes its tt Terrestrial Time attribute starting from whatever time argument you provide. If you provide the utc parameter, for example, then the date first computes and sets tai and then computes and sets tt. Each of the other time attributes only gets computed once, the first time you access it.

The general rule is that attributes are only computed once, and can be accessed again and again for free, while methods never cache their results — think of the () parentheses after a method name as your reminder that “this will do a fresh computation every time.”

In addition to time scales, each Time object caches several other quantities that are often needed in astronomy. Skyfield only computes these attributes on-demand, the first time the user tries to access them or invokes a computation that needs their value:

gmst

Greenwich Mean Sidereal Time in hours, in the range 0.0 ≤ gmst < 24.0.

gast

Greenwich Apparent Sidereal Time in hours, in the range 0.0 ≤ gast < 24.0.

M, MT

This 3×3 matrix and its inverse perform the complete rotation between a vector in the ICRF and a vector in the dynamical reference system for this time and date.

C, CT

This 3×3 matrix and its inverse perform the complete rotation between a vector in the ICRF and a vector in the celestial intermediate reference system (CIRS) of this time and date.

You will typically never need to access these matrices yourself, as they are used automatically by the radec() method when you use its epoch= parameter to ask for a right ascension and declination in the dynamical reference system, and when you ask a GeographicPosition object for its position.