Julian day

The Julian day or Julian day number (JDN) is the (integer) number of days that have elapsed since Monday, January 1, 4713 BC in the proleptic Julian calendar . That day is counted as Julian day zero. Thus the multiples of 7 are Mondays. Negative values can also be used. Today (UTC) the value is round 0}}.

The Julian Date (JD) is the number of days (with decimal fraction of the day) that have elapsed since 12 noon Greenwich Mean Time (UT or TT) of that day. Rounding to the nearest integer gives the Julian day number. Currently the value is .

The Julian day number can be considered a very simple calendar, where its calendar date is just an integer. This is useful for reference, computations, and conversions. The Julian day system was introduced by astronomers to provide with a single system of dates that could be used when working with different calendars and to unify different historical chronologies. Apart from the choice of the zero point and name, this Julian day and Julian date are not related to the Julian calendar.

Julian Date

Historical Julian Dates were recorded relative to GMT or Ephemeris Time, but the International Astronomical Union now recommends that Julian Dates be specified in Terrestrial Time, and that when necessary to specify Julian Dates using a different time scale, that the time scale used be indicated when required, such as JD(UT1). The fraction of the day is found by converting the number of hours, minutes, and seconds after noon into the equivalent decimal fraction.

The term Julian date is also used to refer to:

Julian calendar dates
ordinal dates (day-of-year)

The use of Julian date to refer to the day-of-year (ordinal date) is usually considered to be incorrect.

Alternatives

The Heliocentric Julian Day (HJD) is the same as the Julian day, but adjusted to the frame of reference of the Sun, and thus can differ from the Julian day by as much as 8.3 minutes, that being the time it takes the Sun's light to reach Earth. The Julian day is sometimes referred to as the Geocentric Julian Day (GJD) in order to distinguish it from HJD.

The Chronological Julian Day (CJD) was introduced by Peter Meyer ^*. The starting point is set at midnight at the beginning of January 1, 4713 BC (proleptic Julian calendar) local time rather than noon UT. Chronographers found the Julian day concept useful, but they didn't like noon as the starting time. So CJD = JD + 0.5 (in the Greenwich time zone, anyway). Note that JD may use Universal Time (UT) or Terrestrial Time (TT), and so it is the same for all time zones and is independent of Summer Time or Daylight-Saving Time (DST). On the other hand, CJD is not, so it changes with different time zones and takes into account the different local DSTs. Users of CJD sometimes refer to the Julian day as astronomical Julian Day (AJD) to distinguish it from CJD.

Because the starting point is so long ago, numbers in the Julian day can be quite large and cumbersome. A more recent starting point is sometimes used, for instance by dropping the leading digits, in order to fit into limited computer memory with an adequate amount of precision.

The Modified Julian Day (MJD) is the number of days (with decimal fraction of the day) that have elapsed since midnight at the beginning of Wednesday November 17, 1858. In terms of the Julian day:

MJD = JD - 2,400,000.5

Currently the value is - 2400000.5 = - 2400000.5}}.

The day is found by rounding downward, currently giving - 2400001 round 0}}. This number changes at midnight UT or TT. It is 2,400,001 less than the Julian day number (the integer) of the same day. It is (of course) a multiple of 7 on Wednesdays.

The MJD was introduced by the Smithsonian Astrophysical Observatory in 1957 to record the orbit of Sputnik. MJD is the epoch of OpenVMS, needing only 18 bits until 2576-08-07.

The Reduced Julian Day (RJD) is also used by astronomers and counts days from the same day as MJD, but from noon UT or TT, and thus is defined as:

RJD = JD - 2400000

The Truncated Julian Day (TJD) was introduced by NASA for the space program. TJD began at midnight at the beginning of May 24, 1968. Since TJD exceeded four digits on October 10, 1995, some now count TJD from this date in order to maintain a four-digit number. NASA ^* treats it as a linear count:

TJD = JD - 2440000.5

but NIST treats it as cyclical:

TJD = (JD - 0.5) mod 10000

The Dublin Julian Day (DJD) is a count of days from noon UT of December 31, 1899, defined by the International Astronomical Union at their 1955 meeting in Dublin, Ireland, ^* as:

DJD = JD - 2415020

The Lilian day number defines day 1 as October 15, 1582, which was the first day of the Gregorian Calendar. It was named for Aloysius Lilius, the principal author of the Gregorian Calendar.

The ANSI Date defines January 1, 1601 as day 1, and is used as the origin of COBOL integer dates. This epoch is the beginning of the previous 400-year cycle of leap years in the Gregorian Calendar, which ended with the year 2000.

Rata Die is the epoch used in Calendrical Calculations by Edward M. Reingold and Nachum Dershowitz, where day 1 is January 1, 1, that is, the first day of the Christian or Common Era in the proleptic Gregorian Calendar. It's used as date('base') in REXX. Defined as:

RD = JD - 1721424.5

History

The Julian day number is based on the Julian Period proposed by Joseph Scaliger in 1583, at the time of the Gregorian calendar reform, but it is the multiple of three calendar cycles used with the Julian calendar:

15 (indiction cycle) × 19 (Metonic cycle) × 28 (Solar cycle) = 7980 years

Its epoch falls at the last time when all three cycles were in their first year together — Scaliger chose this because it pre-dated all historical dates.

Note: although many references say that the "Julian" in "Julian day" refers to Scaliger's father, Julius Scaliger, in the introduction to Book V of his Opus de Emendatione Tempore (Work on the Emendation of Time) he states: "Iulianum vocauimus: quia ad annum Iulianum dumtaxat accomodata est" which translates more or less as "We call this Julian merely because it is accommodated to the Julian year". This "Julian" refers to Julius Caesar, who introduced the Julian calendar in 46 BC.

In his book Outlines of Astronomy, first published in 1849, the astronomer John Herschel wrote:

The first year of the current Julian period, or that of which the number in each of the three subordinate cycles is 1, was the year 4713 B.C., and the noon of the 1st of January of that year, for the meridian of Alexandria, is the chronological epoch, to which all historical eras are most readily and intelligibly referred, by computing the number of integer days intervening between that epoch and the noon (for Alexandria) of the day, which is reckoned to be the first of the particular era in question. The meridian of Alexandria is chosen as that to which Ptolemy refers the commencement of the era of Nabonassar, the basis of all his calculations.

Astronomers adopted Herschel's Julian Days in the late ninteenth century, but used the meridian of Greenwich instead of Alexandria, after the former was made the Prime Meridian by international conference in 1884. This has now become the standard system of Julian days. Julian days are typically used by astronomers to date astronomical observations, thus eliminating the complications resulting from using standard calendar periods like eras, years, months, or weeks.

Julian days begin at noon because when Herschel recommended them, the astronomical day began at noon (it did so until 1925). The astronomical day had begun at noon ever since Ptolemy chose to begin the days in his astronomical periods at noon. He chose noon because the transit of the Sun across the observer's meridian occurs at the same apparent time every day of the year, unlike sunrise or sunset, which vary by several hours. Midnight was not even considered because it could not be accurately determined using water clocks. Nevertheless, he double dated most nighttime observations with both Egyptian days beginning at sunrise and Babylonian days beginning at sunset. Thus the astronomical day did not begin at noon to allow all observations of a single night to be in a single day.

Calculation

The Julian day number can be calculated using the following formulas:

The months January to December are 1 to 12. Astronomical year numbering is used, thus 1 BC is 0, 2 BC is −1, and 4713 BC is −4712. In all divisions (except for JD) the floor function is applied to the quotient (for dates since 1 March −4800 all quotients are non-negative, so we can also apply truncation).

$\begin{matrix}a & = & \left\lfloor\frac{14 - month}{12}\right\rfloor \\ \\y & = & year + 4800 - a \\ \\m & = & month + 12a - 3 \\\end{matrix}$

For a date in the Gregorian calendar (at noon):

$\begin{matrix}JDN & = & day + \left\lfloor\frac{153m + 2}{5}\right\rfloor + 365y + \left\lfloor\frac{y}{4}\right\rfloor - \left\lfloor\frac{y}{100}\right\rfloor + \left\lfloor\frac{y}{400}\right\rfloor - 32045\end{matrix}$

For a date in the Julian calendar (at noon):

$\begin{matrix}JDN & = & day + \left\lfloor\frac{153m + 2}{5}\right\rfloor + 365y + \left\lfloor\frac{y}{4}\right\rfloor - 32083\end{matrix}$

The constants used at the end of the Gregorian and Julian formulas are required to return exactly the same values of JDN between March 1, 200 and February 28, 300. The constants are the JDNs of February 29, −4800 in each calendar.

For the full Julian Date, not counting leap seconds (divisions are real numbers):

$\begin{matrix}JD & = & JDN + \frac{hour - 12}{24} + \frac{minute}{1440} + \frac{second}{86400}\end{matrix}$

So, for example, 1 January 2000 at midday corresponds to JD = 2451545.0

The day of the week can be determined from the Julian day number by calculating it modulo 7, where 0 means Monday.

JDN mod 7	0	1	2	3	4	5	6
Day of the week	Mon	Tue	Wed	Thu	Fri	Sat	Sun

Compare Template:YMD2MJD for an application of the Rata Die algorithm.

Gregorian calendar from Julian day number

Let J be the Julian day number from which we want to compute the date components.
With J, compute a relative Julian day number j from a Gregorian epoch starting on March 1 -4800 (i.e. March 1 4801 BC in the proleptic Gregorian Calendar), the beginning of the Gregorian quadricentennial 32,044 days before the epoch of the Julian Period.
With j, compute the number g of Gregorian quadricentennial cycles elapsed (there are exactly 146,097 days per cycle) since the epoch; subtract the days for this number of cycles, it leaves dg days since the beginning of the current cycle.
With dg, compute the number c (from 0 to 4) of Gregorian centennial cycles (there are exactly 36,524 days per Gregorian centennial cycle) elapsed since the beginning of the current Gregorian quadricentennial cycle, number reduced to a maximum of 3 (this reduction occurs for the last day of a leap centennial year where c would be 4 if it was not reduced); subtract the number of days for this number of Gregorian centennial cycles, it leaves dc days since the beginning of a Gregorian century.
With dc, compute the number b (from 0 to 24) of Julian quadrennial cycles (there are exactly 1,461 days in 4 years, except for the last cycle which may be incomplete by 1 day) since the beginning of the Gregorian century; subtract the number of days for this number of Julian cycles, it leaves db days in the Gregorian century.
With db, compute the number a (from 0 to 4) of Roman annual cycles (there are exactly 365 days per Roman annual cycle) since the beginning of the Julian quadrennial cycle, number reduced to a maximum of 3 (this reduction occurs for the leap day, if any, where a would be 4 if it was not reduced); subtract the number of days for this number of annual cycles, it leaves da days in the Julian year (that begins on March 1).
Convert the four components g, c, b, a into the number y of years since the epoch, by summing their values weighted by the number of years that each component represents (respectively 400 years, 100 years, 4 years, and 1 year).
With da, compute the number m (from 0 to 11) of months since March (there are exactly 153 days per 5-month cycle, however these 5-month cycles are offset by 2 months within the year, i.e. the cycles start in May, and so the year starts with a initial fixed number of days on March 1, the month can be computed from this cycle by a Euclidian division by 5); subtract the number of days for this number of months (using the formula above), it leaves d days past since the beginning of the month.
You can then deduce the Gregorian date (Y, M, D) by simple shifts from (y, m, d).

We can then develop these formulas into a single inlined formula per component, computed as above. All this computing requires only integers and so is not sensitive to rounding errors caused by floating point approximations (most decimal fractions have an inexact representation within the binary format used by floating point arithmetic used by most computer software, so using them would produce false results on some dates because of roundoff errors).

The formulas below (which use Euclidian division—integer division and modulo—without any negative number) are valid for the whole range of dates since −4800. The resulting date components are valid only in the Gregorian proleptic calendar using astronomical year numbering, which is based on the Gregorian calendar, but extended to cover dates before 1582, including the pre-Christian era. This calendar includes a zero year, which is 1 BC in the proleptic Gregorian calendar, as it is one year before 1 AD.

J = Julian day number

j = J + 32044

g = j div 146097

dg = j mod 146097

c = (dg div 36524 + 1) * 3 div 4

dc = dg - c * 36524

b = dc div 1461

db = dc mod 1461

a = (db div 365 + 1) * 3 div 4

da = db - a * 365

y = g * 400 + c * 100 + b * 4 + a

m = (da * 5 + 308) div 153 - 2

d = da - (m + 4) * 153 div 5 + 122

Y = y - 4800 + (m + 2) div 12

M = (m + 2) mod 12 + 1

D = d + 1

Footnotes

This equals November 24, 4714 BC in the proleptic Gregorian calendar.

References

Gordon Moyer, "The Origin of the Julian Day System," Sky and Telescope 61 (April 1981) 311-313.
Explanatory Supplement to the Astronomical Almanac, edited by P. Kenneth Seidelmann. University Science Books, 1992. ISBN 0935702687

External links

Article 'Julian Day Numbers' by Peter Meyer
U.S. Naval Observatory Julian Date Converter
Julian Day and Civil Date calculator
U.S. Naval Observatory Time Service article on Modified Julian Date
U.S. Naval Observatory current MJD service
MJD explanation (18 bits for dates)
Peter Baum's date algorithms cover Rata Die
Outlines of Astronomy by John Herschel
International Astronomical Union Resolution 1B: On the Use of Julian Dates
Calendrica
Another Julian Day calculator with conversions to many other calendars valid from 1 January 100 proleptic Gregorian calendar
Open-source date conversion software
Julian Date Convertor converts date and time (UT, also known as GMTor Z) to Julian Date and vice versa

Specific calendars | Celestial mechanics

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