# Julian day

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Julian day is the continuous count of days since the beginning of the Julian Period and is used primarily by astronomers, and in software for easily calculating elapsed days between two events (e.g. food production date and sell by date).

The Julian Day Number (JDN) is the integer assigned to a whole solar day in the Julian day count starting from noon Universal time, with Julian day number 0 assigned to the day starting at noon on Monday, January 1, 4713 BC, proleptic Julian calendar (November 24, 4714 BC, in the proleptic Gregorian calendar), a date at which three multi-year cycles started (which are: Indiction, Solar, and Lunar cycles) and which preceded any dates in recorded history. For example, the Julian day number for the day starting at 12:00 UT on January 1, 2000, was 2 451 545.

The Julian date (JD) of any instant is the Julian day number plus the fraction of a day since the preceding noon in Universal Time. Julian dates are expressed as a Julian day number with a decimal fraction added. For example, the Julian Date for 00:30:00.0 UT January 1, 2013, is 2 456 293.520 833.

The Julian Period is a chronological interval of 7980 years; year 1 of the Julian Period was 4713 BC. It has been used by historians since its introduction in 1583 to convert between different calendars. The Julian calendar year 2019 is year 6732 of the current Julian Period. The next Julian Period begins in the year AD 3268.

## Terminology

The term Julian date may also refer, outside of astronomy, to the day-of-year number (more properly, the ordinal date) in the Gregorian calendar, especially in computer programming, the military and the food industry, or it may refer to dates in the Julian calendar. For example, if a given "Julian date" is "October 5, 1582", this means that date in the Julian calendar (which was October 15, 1582, in the Gregorian calendar—the date it was first established). Without an astronomical or historical context, a "Julian date" given as "36" most likely means the 36th day of a given Gregorian year, namely February 5. Other possible meanings of a "Julian date" of "36" include an astronomical Julian Day Number, or the year AD 36 in the Julian calendar, or a duration of 36 astronomical Julian years). This is why the terms "ordinal date" or "day-of-year" are preferred. In contexts where a "Julian date" means simply an ordinal date, calendars of a Gregorian year with formatting for ordinal dates are often called "Julian calendars", but this could also mean that the calendars are of years in the Julian calendar system.

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. Time intervals calculated from differences of Julian Dates specified in non-uniform time scales, such as Coordinated Universal Time (UTC), may need to be corrected for changes in time scales (e.g. leap seconds).

## Variants

Because the starting point or reference epoch 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. In the following table, times are given in 24-hour notation.

In the table below, Epoch refers to the point in time used to set the origin (usually zero, but (1) where explicitly indicated) of the alternative convention being discussed in that row. The date given is a Gregorian calendar date if it is October 15, 1582, or later, but a Julian calendar date if it is earlier. JD stands for Julian Date. 0h is 00:00 midnight, 12h is 12:00 noon, UT unless otherwise specified. Current value is as of 15:35, Friday, May 24, 2019 (UTC) and may be cached. (update)

Name Epoch Calculation Current value Notes
Julian Date 12h Jan 1, 4713 BC 2458628.14931
Reduced JD 12h Nov 16, 1858 JD − 2400000 58628.14931 
Modified JD 0h Nov 17, 1858 JD − 2400000.5 58627.64931 Introduced by SAO in 1957
Truncated JD 0h May 24, 1968 floor (JD − 2440000.5) 18627 Introduced by NASA in 1979
Dublin JD 12h Dec 31, 1899 JD − 2415020 43608.14931 Introduced by the IAU in 1955
CNES JD 0h Jan 1, 1950 JD − 2433282.5 25345.64931 Introduced by the CNES
CCSDS JD 0h Jan 1, 1958 JD − 2436204.5 22423.64931 Introduced by the CCSDS
Lilian date Oct 15, 1582 floor (JD − 2299159.5) 159468 Count of days of the Gregorian calendar
Rata Die Jan 1, 1 proleptic Gregorian calendar floor (JD − 1721424.5) 737203 Count of days of the Common Era
Mars Sol Date 12h Dec 29, 1873 (JD − 2405522)/1.02749 51685.25698 Count of Martian days
Unix Time 0h Jan 1, 1970 (JD − 2440587.5) × 86400 1558712131 Count of seconds
• The Modified Julian Date (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 August 7, 2576. MJD is the epoch of VAX/VMS and its successor OpenVMS, using 63-bit date/time, which allows times to be stored up to July 31, 31086, 02:48:05.47. The MJD has a starting point of midnight on November 17, 1858 and is computed by MJD = JD - 2400000.5 
• The Truncated Julian Day (TJD) was introduced by NASA/Goddard in 1979 as part of a parallel grouped binary time code (PB-5) "designed specifically, although not exclusively, for spacecraft applications." TJD was a 4-digit day count from MJD 40000, which was May 24, 1968, represented as a 14-bit binary number. Since this code was limited to four digits, TJD recycled to zero on MJD 50000, or October 10, 1995, "which gives a long ambiguity period of 27.4 years". (NASA codes PB-1—PB-4 used a 3-digit day-of-year count.) Only whole days are represented. Time of day is expressed by a count of seconds of a day, plus optional milliseconds, microseconds and nanoseconds in separate fields. Later PB-5J was introduced which increased the TJD field to 16 bits, allowing values up to 65535, which will occur in the year 2147. There are five digits recorded after TJD 9999.
• The Dublin Julian Date (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 January 0, 1900, which is the same as noon UT on December 31, 1899. The DJD was defined by the International Astronomical Union at their meeting in Dublin, Ireland, in 1955.

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 (498 seconds), that being the time it takes the Sun's light to reach Earth.

To illustrate the ambiguity that could arise, consider the two separate astronomical measurements of an astronomical object from the earth: Assume that three objects—the Earth, the Sun, and the astronomical object targeted, that is whose distance is to be measured—happen to be in a straight line for both measures. However, for the first measurement, the Earth is between the Sun and the targeted object, and for the second, the Earth is on the opposite side of the Sun from that object. Then, the two measurements would differ by about 1000 light-seconds: For the first measurement, the Earth is roughly 500 light seconds closer to the target than the Sun, and roughly 500 light seconds further from the target astronomical object than the Sun for the second measure.

An error of about 1000 light-seconds is over 1% of a light-day, which can be a significant error when measuring temporal phenomena for short period astronomical objects over long time intervals. To clarify this issue, the ordinary Julian day is sometimes referred to as the Geocentric Julian Day (GJD) in order to distinguish it from HJD.

## History

The Julian day number is based on the Julian Period proposed by Joseph Scaliger, a classical scholar, in 1583, at the time of the Gregorian calendar reform, as it is the least common 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 (if they are continued backward far enough) were in their first year together — Scaliger chose this because it preceded all historical dates. Years of the Julian Period are counted from this year, 4713 BC, which was chosen to be before any historical record.

Since it is now certain that every possible combination of the three cyclic numbers finds its place in the Julian Period, it is evident that the first year of the Christian era, which was the 10th year of a Solar Cycle, the 2nd of a Lunar Cycle, and the 4th of a Cycle of Indiction, finds its place within this artificial era, and must answer to that particular year of the period which is characterized by the same cyclic numbers. Hence, to refer the Christian era to the Julian Period is the same thing as to find out what year of that period it is which, when divided by 28 will leave a remainder 10, divided by 19 will leave a remainder 2, and divided by 15 will leave a remainder 4. The solution of this problem belongs to the higher mathematics, by which it is found that the year required is the 4714th of the period in question. Hence Jul. Per. 4714=A.D.1, and consequently Julian Period 4713=B.C.1.

In point of fact, finding the year is a very straightforward arithmetical procedure. See "Calculation" below.

Although many references say that the Julian in "Julian Period" 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, "Iulianam 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." Thus Julian refers to Julius Caesar, who introduced the Julian calendar in 46 BC.

Originally the Julian Period was used only to count years, and the Julian calendar was used to express historical dates within years. In his book Outlines of Astronomy, first published in 1849, the astronomer John Herschel added the counting of days elapsed from the beginning of the Julian Period:

The period thus arising of 7980 Julian years, is called the Julian period, and it has been found so useful, that the most competent authorities have not hesitated to declare that, through its employment, light and order were first introduced into chronology. We owe its invention or revival to Joseph Scaliger, who is said to have received it from the Greeks of Constantinople. 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 BC, 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 "days of the Julian period" in the late nineteenth century, but used the meridian of Greenwich instead of Alexandria, after the former was adopted as the Prime Meridian after the International Meridian Conference in Washington in 1884. This has now become the standard system of Julian days numbers.

The French mathematician and astronomer Pierre-Simon Laplace first expressed the time of day as a decimal fraction added to calendar dates in his book, Traité de Mécanique Céleste, in 1799. Other astronomers added fractions of the day to the Julian day number to create Julian Dates, which are typically used by astronomers to date astronomical observations, thus eliminating the complications resulting from using standard calendar periods like eras, years, or months. They were first introduced into variable star work by Edward Charles Pickering, of the Harvard College Observatory, in 1890.

Julian days begin at noon because when Herschel recommended them, the astronomical day began at noon. 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. When this practice ended in 1925, it was decided to keep Julian days continuous with previous practice.

## Julian day number calculation

The Julian day number can be calculated using the following formulas (integer division is used exclusively, that is, the remainders of all divisions are dropped):

The months January to December are numbered 1 to 12. For the year, astronomical year numbering is used, thus 1 BC is 0, 2 BC is −1, and 4713 BC is −4712. JDN is the Julian Day Number. Use the previous day of the month if trying to find the JDN of an instant before midday UT.

### Converting Gregorian calendar date to Julian Day Number

The algorithm is valid for all (possibly proleptic) Gregorian calendar dates after November 23, −4713.

JDN = (1461 × (Y + 4800 + (M − 14)/12))/4 +(367 × (M − 2 − 12 × ((M − 14)/12)))/12 − (3 × ((Y + 4900 + (M - 14)/12)/100))/4 + D − 32075

### Converting Julian calendar date to Julian Day Number

The algorithm is valid for all (possibly proleptic) Julian calendar years ≥ −4712, that is, for all JDN ≥ 0.

JDN = 367 × Y − (7 × (Y + 5001 + (M − 9)/7))/4 + (275 × M)/9 + D + 1729777

### Finding Julian date given Julian day number and time of day

For the full Julian Date of a moment after 12:00 UT one can use the following (divisions are real numbers):

${\begin{matrix}J\!D&=&J\!D\!N+{\frac {{\text{hour}}-12}{24}}+{\frac {\text{minute}}{1440}}+{\frac {\text{second}}{86400}}\end{matrix}}$ So, for example, January 1, 2000, at 18:00:00 UT corresponds to JD = 2451545.25

For a point in time in a given Julian day after midnight UT and before 12:00 UT, add 1 or use the JDN of the next afternoon.

### Finding day of week given Julian day number

The US day of the week W1 (for an afternoon or evening UT) can be determined from the Julian Day Number J with the expression:

W1 = mod(J + 1, 7)
 W1 Day of the week 0 1 2 3 4 5 6 Sun Mon Tue Wed Thu Fri Sat

If the moment in time is after midnight UT (and before 12:00 UT), then one is already in the next day of the week.

The ISO day of the week W0 can be determined from the Julian Day Number J with the expression:

W0 = mod (J, 7) + 1
 W0 Day of the week 1 2 3 4 5 6 7 Mon Tue Wed Thu Fri Sat Sun

### Julian or Gregorian calendar from Julian day number

This is an algorithm by Richards to convert a Julian Day Number, J, to a date in the Gregorian calendar (proleptic, when applicable). Richards states the algorithm is valid for Julian day numbers greater than or equal to 0. All variables are integer values, and the notation "a div b" indicates integer division, and "mod(a,b)" denotes the modulus operator.

Algorithm parameters for Gregorian calendar
variable value variable value
y 4716 v 3
j 1401 u 5
m 2 s 153
n 12 w 2
r 4 B 274277
p 1461 C −38

For Julian calendar:

1. f = J + j

For Gregorian calendar:

1. f = J + j + (((4 × J + B) div 146097) × 3) div 4 + C

For Julian or Gregorian, continue:

2. e = r × f + v
3. g = mod(e, p) div r
4. h = u × g + w
5. D = (mod(h, s)) div u + 1
6. M = mod(h div s + m, n) + 1
7. Y = (e div p) - y + (n + m - M) div n

D, M, and Y are the numbers of the day, month, and year respectively for the afternoon at the beginning of the given Julian day.

### Julian Period from indiction, Metonic and solar cycles

Let Y be the year BC or AD and i, m and s respectively its positions in the indiction, Metonic and solar cycles. Divide 6916i + 4200m + 4845s by 7980 and call the remainder r.

If r>4713, Y = (r − 4713) and is a year AD.
If r<4714, Y = (4714 − r) and is a year BC.

Example

i = 8, m = 2, s = 8. What is the year?

(6916 × 8) = 55328; (4200 × 2) = 8400: (4845 × 8) = 38760. 55328 + 8400 + 38760 = 102488.
102488/7980 = 12 remainder 6728.
Y = (6728 − 4713) = AD 2015.

## Julian date calculation

As stated above, the Julian date (JD) of any instant is the Julian day number for the preceding noon in Universal Time plus the fraction of the day since that instant. Ordinarily calculating the fractional portion of the JD is straightforward; the number of seconds that have elapsed in the day divided by the number of seconds in a day, 86,400. But if the UTC timescale is being used, a day containing a positive leap second contains 86,401 seconds (or in the unlikely event of a negative leap second, 86,399 seconds). One authoritative source, the Standards of Fundamental Astronomy (SOFA), deals with this issue by treating days containing a leap second as having a different length (86,401 or 86,399 seconds, as required). SOFA refers to the result of such a calculation as "quasi-JD".