# Julian day number

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The Julian day number (JDN) or Julian day is the integer number of days that have elapsed since the initial epoch defined as noon Universal Time (UT) Monday, January 1, 4713 BC in the proleptic Julian calendar [1]. That noon-to-noon day is counted as Julian day 0. Thus the multiples of 7 are Mondays. Negative values can also be used, although those predate all recorded history.

Now at 21:51, Friday February 17, 2017 (UTC) the JDN is 2457802. The remainder of this value divided by 7 is 4, an integer expression for the day of the week with 0 representing Monday.

The Julian date (JD) is a continuous count of days and fractions elapsed since the same initial epoch. Currently the JD is 2457802.4110532. The integral part (its floor gives the Julian day number. The fractional part gives the time of day since noon UT as a decimal fraction of one day or fractional day, with 0.5 representing midnight UT. Typically, a 64-bit floating point (double precision) variable can represent an epoch expressed as a Julian date to about 1 millisecond precision.

A Julian date of 2454115.05486 means that the date and Universal Time is Sunday 14 January 2007 at 13:18:59.9.

The decimal parts of a Julian date:
0.1 = 2.4 hours or 144 minutes or 8640 seconds
0.01 = 0.24 hours or 14.4 minutes or 864 seconds
0.001 = 0.024 hours or 1.44 minutes or 86.4 seconds
0.0001 = 0.0024 hours or 0.144 minutes or 8.64 seconds
0.00001 = 0.00024 hours or 0.0144 minutes or 0.864 seconds.

Almost 2.5 million Julian days have elapsed since the initial epoch. JDN 2,400,000 was November 16, 1858. JD 2,500,000.0 will occur on August 31, 2132 at noon UT.

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. It allows the time between any two dates in history to be computed by simple subtraction.

The Julian day system was introduced by astronomers to provide 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 directly related to the Julian Calendar, although it is possible to convert any date from one calendar to the other.

## Julian Date Edit

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:

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

## Alternatives Edit

• 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.

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 2457802.4110532 − 2400000.5 = 57801.9110532.

The day is found by rounding downward, currently giving 57801. This number changes at midnight UT or TT. It is 2,400,001 less than the Julian day number of the afternoon half of the same day (which is the same as the JD at noon). 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 via an IBM 704 (36-bit machine) and using only 18 bits until 2576-08-07. MJD is the epoch of OpenVMS, using 63 bit date/time postponing the next Y2K campaign to 31-JUL-31086 02:48:05.47.
• The Reduced Julian Day (RJD) is also used by astronomers and counts days from nearly the same day as the MJD, but lacks the additional offset of 12 hours that MJD has. It therefore starts from the previous noon UT or TT, on Tuesday November 16, 1858. It is defined as:
RJD = JD − 2400000
• The Truncated Julian Day (TJD) was introduced by NASA for the space program. TJD was zero at midnight UT at the beginning of May 24, 1968. It is defined by NASA as:
TJD = JD − 2440000.5

This was chosen so the number would resemble the MJD but be only four digits long. TJD exceeded four digits on October 10, 1995, and NASA now uses five-digit TJDs [1]. NIST, however, defines TJD cyclically so that it never exceeds four digits:

TJD = (JD − 0.5) mod 10000
• The Dublin Julian Day (DJD) is the number of days that has elapsed since the epoch of the solar and lunar ephemerides used from 1900 through 1983, Newcomb's Tables of the Sun and Ernest W. Brown's Tables of the Motion of the Moon (1919). This epoch was noon UT on 0 January 1900, which is the same as noon UT on December 31, 1899. The DJD was defined by the International Astronomical Union at their 1955 meeting in Dublin, Ireland.[2] as:
DJD = JD − 2415020
• The Lilian day number is a count of days of the Gregorian Calendar. It is an integer applied to a whole day; day 1 was October 15, 1582, which was the day the Gregorian calendar went into effect. It uses the local timezone, not UT. 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. It has three distinct forms. In the first form, it is a continously-increasing fractional number, taking integer values at midnight local time. It may be defined as ::RD = JD − 1721424.5 + Zoff
where Zoff is the offset of the timezone considered, in fractional days. In the second form, it is an integer that labels an entire day, midnight-to-midnight local time. This is the result of rounding the first form of RD downwards (towards negative infinity). Thus:
RD = floor(JD − 1721424.5 + Zoff)
In the third form, the RD is an integer labelling noon local time, and incapable of labelling any other time of day. This is defined as
RD = JD − 1721425 + Zoff
where the RD value must be an integer, thus constraining the choice of JD. This form of RD is used in the book for conversion between calendars that separate days on different boundaries. The book does not explicitly distinguish between these three forms, using the abbreviation "RD" for all of them.
Unix Time = (JD – 2440587.5) × 86400

## History Edit

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 Temporum (Work on the Emendation of Time) he states: "Iulianum vocauimus: quia ad annum Iulianum dumtaxat accomodata est" which translates more or less as "We have called it 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 nineteenth 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, or months.

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. This would seem to imply that his choice of noon was not, as is sometimes stated, made in order to allow all observations from a given night to be recorded with the same date.

## Calculation Edit

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 March 1, −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 Day of the week 0 1 2 3 4 5 6 Mon Tue Wed Thu Fri Sat Sun

### Gregorian calendar from Julian day number Edit

• 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 an 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 Edit

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

## References Edit

• 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 0-935702-68-7