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{{Short description|Days since the beginning of the Julian Period}}
{{redirect|Julian date|dates in the Julian calendar|Julian calendar|numbered day of year|Ordinal date|the comic book character Julian Gregory Day|Calendar Man|the artist and composer|Julian Day (artist)}}
{{Distinguish|Julian year (astronomy)}}
{{Use mdy dates|date=January 2023}}

The '''Julian day''' is a continuous count of days from the beginning of the Julian period; it is used primarily by [[astronomer]]s, and in [[software]] for easily calculating elapsed days between two events (e.g. food production date and sell by date).<ref>"Julian date" n.d.</ref>

The '''Julian period''' is a [[Chronology|chronological]] interval of 7980&nbsp;years, derived from three multi-year cycles: the [[Indiction]], [[Solar cycle (calendar)|Solar]], and [[Metonic cycle|Lunar]] cycles. The last year that was simultaneously the beginning of all three cycles was {{nowrap|4713 BC (−4712)}},<ref>''Astronomical Almanac for the year 2017'' p. B4, which states 2017 is year 6730 of the Julian Period.</ref> so that is year 1 of the current Julian period, making AD&nbsp;{{CURRENTYEAR}} year&nbsp;{{#expr: {{CURRENTYEAR}} + 4713 }} of that Period. The next Julian Period begins in the year AD&nbsp;3268. Historians used the period to identify Julian calendar years within which an event occurred when no such year was given in the historical record, or when the year given by previous historians was incorrect.<ref>Grafton 1975</ref>

The '''Julian day number''' ('''JDN''') shares the epoch of the Julian period, but counts days instead of years. Specifically, Julian day number 0 is assigned to the day starting at noon [[Universal Time]] on Monday, January 1, 4713&nbsp;BC, [[proleptic Julian calendar]] (November 24, 4714&nbsp;BC, in the [[proleptic Gregorian calendar]]).<ref>Dershowitz & Reingold 2008, 15.</ref><ref>Seidelman 2013, 15.</ref><ref>"Astronomical Almanac Online" 2016, Glossary, s.v. Julian date. Various timescales may be used with Julian date, such as [[Terrestrial Time]] (TT) or Universal Time (UT); in precise work the timescale should be specified.</ref>{{efn|Both of these dates are years of the [[Anno Domini]] or [[Common Era]] (which has no year 0 between 1 BC and AD 1). Astronomical calculations generally include a year 0, so these dates should be adjusted accordingly (i.e. the year 4713 BC becomes [[Astronomical year numbering|astronomical year number]] &minus;4712, etc.). In this article, dates before October 15, 1582, are in the (possibly proleptic) Julian calendar and dates on or after October 15, 1582, are in the Gregorian calendar, unless otherwise labelled.}} For example, the Julian day number for the day starting at 12:00&nbsp;[[Universal Time|UT]] (noon) on January 1, 2000, was {{val|2451545}}.<ref>McCarthy & Guinot 2013, 91–92</ref>

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.<ref name="IAU">"Resolution B1" 1997.</ref> For example, the Julian Date for 00:30:00.0&nbsp;UT January 1, 2013, is {{val|2456293.520833}}.<ref>US Naval Observatory 2005</ref> This article was loaded at {{#time:Y-m-d H:i:s|{{CURRENTTIMESTAMP}}}} ([[Coordinated Universal Time|UTC]]) – expressed as a Julian date this is {{#expr:{{#time:U/86400+2440587.5}}}}.

== 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,<ref name="usda">USDA {{Circa|1963}}.</ref> 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{{snd}}the date it was first established). Without an astronomical or historical context, a "Julian date" given as "36" most likely means the 36th&nbsp;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&nbsp;36 in the Julian calendar, or a duration of 36&nbsp;astronomical [[Julian year (astronomy)|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"'',<ref name="usda"/> but this could also mean that the calendars are of years in the Julian calendar system.

Historically, Julian dates were recorded relative to [[Greenwich Mean Time]] (GMT) (later, [[Ephemeris Time]]), but since 1997 the [[International Astronomical Union]] has recommended that Julian dates be specified in [[Terrestrial Time]].<ref>[https://www.iers.org/IERS/EN/Science/Recommendations/resolutionB1.html Resolution B1 on the use of Julian Dates] of the XXIIIrd International Astronomical Union General Assembly, Kyoto, Japan, 1997</ref> Seidelmann indicates that Julian dates may be used with [[International Atomic Time]] (TAI), [[Terrestrial Time]] (TT), [[Barycentric Coordinate Time]] (TCB), or [[Coordinated Universal Time]] (UTC) and that the scale should be indicated when the difference is significant.<ref>Seidelmann 2013, p. 15.</ref> 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 UTC, may need to be corrected for changes in time scales (e.g. [[leap second]]s).<ref name="IAU" />

== Variants ==<!-- This section is linked from [[Epoch (reference date)]] and Modified Julian Date -->
Because the starting point or [[Epoch (reference date)|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 unless otherwise specified. JD stands for Julian Date. 0h is 00:00 midnight, 12h is 12:00 noon, UT unless otherwise specified. ''Current value is at {{CURRENTDATE}} and may be cached. <small>[{{Purge|refresh}}]</small>''

{| class="wikitable"
|-
! Name
! [[Epoch (reference date)|Epoch]]
! Calculation
! <abbr title="{{datetime}}">Current value</abbr>
! Notes
|-
| Julian date
| {{nobr|12:00 January 1, 4713 BC}} {{nobr|[[proleptic Julian calendar]]}}
| JD
|align='right'| '''{{#expr: floor({{CURRENTJULIANDAY}}*1440)/1440 round 5 }}'''
|
|-
| Reduced JD
| 12:00 November 16, 1858
| JD − 2400000
|align='right'| '''{{#expr: floor(({{CURRENTJULIANDAY}}-2400000)*1440)/1440 round 5 }}'''
| <ref>Hopkins 2013, p. 257.</ref><ref>Pallé, Esteban 2014.</ref>
|-
| Modified JD
| 0:00 November 17, 1858
| JD − 2400000.5
|align='right'| '''{{#expr: floor(({{CURRENTJULIANDAY}}-2400000.5)*1440)/1440 round 5 }}'''
| Introduced by [[Smithsonian Astrophysical Observatory|SAO]] in 1957
|-
| Truncated JD
| 0:00 May 24, 1968
| floor (JD − 2440000.5)
|align='right'| '''{{#expr: floor( {{CURRENTJULIANDAY}} - 2440000.5 ) }}'''
| Introduced by [[NASA]] in 1979
|-
| Dublin JD
| 12:00 December 31, 1899
| JD − 2415020
|align='right'| '''{{#expr: floor(({{CURRENTJULIANDAY}} - 2415020)*1440)/1440 round 5 }}'''
| Introduced by the [[IAU]] in 1955
|-
| [[CNES|CNES JD]]
| 0:00 January 1, 1950
| JD − 2433282.5
|align='right'| '''{{#expr: floor(({{CURRENTJULIANDAY}} - 2433282.5)*1440)/1440 round 5 }}'''
| Introduced by the [[CNES]]<ref name="_TPtime">Theveny 2001.</ref>
|-
| [[CCSDS|CCSDS JD]]
| 0:00 January 1, 1958
| JD − 2436204.5
|align='right'| '''{{#expr: floor(({{CURRENTJULIANDAY}} - 2436204.5)*1440)/1440 round 5 }}'''
| Introduced by the [[CCSDS]]<ref name="_TPtime"/>
|-
| Modified JD2000 (MJD2000)
| 0:00 January 1, 2000
| JD - 2451544.5
|align='right'| '''{{#expr: floor(({{CURRENTJULIANDAY}}-2451544.5)*1440)/1440 round 5 }}'''
| Introduced by [[European Space Agency|ESA]]<ref>[https://eop-cfi.esa.int/Repo/PUBLIC/DOCUMENTATION/SYSTEM_SUPPORT_DOCS/Mission%20Convention%20Documents/MCD_conventions_v4_28.pdf ESA Earth Observation Mission CFI Software Conventions document]</ref>
|-
| [[Lilian date]]
| day&nbsp;1 = October 15, 1582{{efn|name=day0|This is an epoch starting with day 1 instead of 0. Conventions vary as to whether this is based on UT or local time.}}
| floor (JD − 2299159.5)
|align='right'| '''{{#expr: floor( {{CURRENTJULIANDAY}} - 2299159.5 ) }}'''
| Count of days of the [[Gregorian calendar]]
|-
| [[Rata Die]]
| day&nbsp;1 = January 1, 1{{efn|name=day0}} {{nobr|[[proleptic Gregorian calendar]]}}
| floor (JD − 1721424.5)
|align='right'| '''{{#expr: floor( {{CURRENTJULIANDAY}} - 1721424.5 ) }}'''
| Count of days of the [[Common Era]]
|-
| [[Timekeeping on Mars#Mars Sol Date|Mars Sol Date]]
| 12:00 December 29, 1873
| (JD − 2405522)/1.02749
|align='right'| '''{{#expr: floor(( {{CURRENTJULIANDAY}} - 2405522)*1440)/1440/1.02749125 round 5 }}'''
| Count of Martian days
|-
| [[Unix time]]
| 0:00 January 1, 1970
| (JD − 2440587.5) × 86400
|align='right'| '''{{#expr: ( {{CURRENTJULIANDAY}} - 2440587.5 ) * 86400 round 0 }}'''
| Count of seconds,<ref>''Astronomical almanac for the year 2001'', 2000, p. K2</ref> excluding leap seconds
|-
| JavaScript Date
| 0:00 January 1, 1970
| (JD − 2440587.5) × 86400000
|align='right'| '''{{#expr: ( {{CURRENTJULIANDAY}} - 2440587.5 ) * 86400000 round 0 }}'''
| Count of milliseconds,<ref>{{cite web | url=https://tc39.es/ecma262/multipage/numbers-and-dates.html#sec-time-values-and-time-range | title=ECMAScript® 2025 Language Specification }}</ref> excluding leap seconds
|-
| [[EXT4]] File Timestamps
| 0:00 January 1, 1970
| (JD − 2440587.5) × 86400000000000
|align='right'| '''{{#expr: ( {{CURRENTJULIANDAY}} - 2440587.5 ) * 86400000000000 round 0 }}'''
| Count of nanoseconds,<ref>{{cite web | url=https://www.kernel.org/doc/html/v4.19/filesystems/ext4/ondisk/index.html#inode-timestamps | title=2. Data Structures and Algorithms {{snd}} the Linux Kernel documentation }}</ref> excluding leap seconds
|-
| [[.NET Core|.NET]] DateTime
| 0:00 January 1, 1 {{nobr|[[proleptic Gregorian calendar]]}}
| (JD − 1721425.5) × 864000000000
|align='right'| '''{{#expr: trunc (( {{CURRENTJULIANDAY}} - 1721425.5 ) * 864000000000 round 0) }}'''
| Count of 100-nanosecond ticks, excluding ticks attributable to leap seconds<ref>{{cite web |url=https://docs.microsoft.com/en-us/dotnet/api/system.datetime.ticks |title = System.DateTime.Ticks documentation |publisher=[[Microsoft]] |access-date=January 14, 2022 |date=n.d. |quote=The value of this property represents the number of 100-nanosecond intervals that have elapsed since 12:00:00 midnight, January 1, 0001 in the Gregorian calendar,}}</ref>
|}

* The '''Modified Julian Date''' (MJD) was introduced by the Smithsonian Astrophysical Observatory in 1957 to record the orbit of [[Sputnik 1|Sputnik]] via an [[IBM 704]] (36-bit machine) and using only 18&nbsp;bits until August 7, 2576. MJD is the epoch of VAX/VMS and its successor [[OpenVMS#Timekeeping|OpenVMS]], using 63-bit date/time, which allows times to be stored up to July 31, 31086, 02:48:05.47.<ref>{{cite web|url=http://vms.tuwien.ac.at/info/humour/vms-base-time-origin.txt|title=38 Why Is Wednesday November 17, 1858 The Base Time For VAX/VMS? |date=June 6, 2007|website=[[Digital Equipment Corporation]]-Customer Support Center|location=Colorado Springs |url-status=dead|archive-url=https://web.archive.org/web/20070606064704/http://vms.tuwien.ac.at/info/humour/vms-base-time-origin.txt|archive-date=June 6, 2007}}</ref> The MJD has a starting point of midnight on November 17, 1858, and is computed by MJD&nbsp;=&nbsp;JD&nbsp;−&nbsp;2400000.5 <ref>Winkler n.d.</ref>
* The Truncated Julian Day (TJD) was introduced by [[NASA]]/[[Goddard Space Flight Center|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&nbsp;50000, or October 10, 1995, "which gives a long ambiguity period of 27.4&nbsp;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&nbsp;bits, allowing values up to 65535, which will occur in the year&nbsp;2147. There are five digits recorded after TJD&nbsp;9999.<ref>Chi 1979.</ref><ref>SPD Toolkit Time Notes 2014.</ref>
* 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]], [[Republic of Ireland|Ireland]], in 1955.<ref>Ransom {{Circa|1988}}</ref><!-- THE REST OF THESE ARE NOT RELATED TO JD -->
* The [[Lilian date|Lilian day number]] is a count of days of the Gregorian calendar and not defined relative to the Julian Date. It is an integer applied to a whole day; day&nbsp;1 was October 15, 1582, which was the day the Gregorian calendar went into effect. The original paper defining it makes no mention of the time zone, and no mention of time-of-day.<ref>Ohms 1986</ref> It was named for [[Aloysius Lilius]], the principal author of the Gregorian calendar.<ref>IBM 2004.</ref>
* [[Rata Die]] is a system used in [[Rexx]], [[Go (programming language)|Go]] and [[Python (programming language)|Python]].<ref>[https://docs.python.org/3/library/datetime.html#date-objects "datetime – Basic date and time types – date Objects"] (December 5, 2021). ''The Python Standard Library''.</ref> Some implementations or options use [[Universal Time]], others use local time. Day&nbsp;1 is January 1, 1, that is, the first day of the [[Christian Era|Christian]] or [[Common Era]] in the [[proleptic Gregorian calendar]].<ref>Dershowitz & Reingold 2008, 10, 351, 353, Appendix B.</ref> In Rexx, January 1 is Day&nbsp;0.<ref>[https://www-40.ibm.com/servers/resourcelink/svc0302a.nsf/pages/zVMV7R1sc246314/$file/dmsb1_v7r1.pdf#page=104 "Chapter 3. Functions – DATE – Base"] (September 29, 2022). ''z/VM: 7.1 REXX/VM Reference''</ref>

* 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&nbsp;minutes (498&nbsp;seconds), that being the time it takes light to reach [[Earth]] from the [[Sun]].{{efn|To illustrate the ambiguity that could arise from conflating Heliocentric time and Terrestrial time, consider the two separate astronomical measurements of an astronomical object from the Earth: Assume that three objects{{snd}}the Earth, the Sun, and the astronomical object targeted, that is whose distance is to be measured{{snd}}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&nbsp;light-seconds: For the first measurement, the Earth is roughly 500&nbsp;light seconds closer to the target than the Sun, and roughly 500&nbsp;light seconds further from the target astronomical object than the Sun for the second measure.

An error of about 1000&nbsp;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 ==
===Julian Period <span class="anchor" id="Julian period"></span>===
The ''Julian day number'' is based on the ''Julian Period'' proposed by [[Joseph Justus Scaliger|Joseph Scaliger]], a classical scholar, in 1583 (one year after the Gregorian calendar reform) as it is the product of three calendar cycles used with the Julian calendar:

{{block indent|1=28 ([[Solar cycle (calendar)|solar cycle]]) × 19 ([[Metonic cycle|lunar cycle]]) × 15 ([[indiction|indiction cycle]]) = 7980 years}}

Its epoch occurs when all three cycles (if they are continued backward far enough) were in their first year together. Years of the Julian Period are counted from this year, {{nowrap|4713 BC}}, as {{nowrap|year 1}}, which was chosen to be before any historical record.<ref>Richards 2013, pp. 591–592.</ref>

Scaliger corrected chronology by assigning each year a tricyclic "character", three numbers indicating that year's position in the 28-year solar cycle, the 19-year lunar cycle, and the 15-year indiction cycle. One or more of these numbers often appeared in the historical record alongside other pertinent facts without any mention of the Julian calendar year. The character of every year in the historical record was unique – it could only belong to one year in the 7980-year Julian Period. Scaliger determined that 1&nbsp;BC or year&nbsp;0 was Julian Period {{nowrap|(JP) 4713}}. He knew that 1&nbsp;BC or year&nbsp;0 had the character 9 of the solar cycle, 1 of the lunar cycle, and 3 of the indiction cycle. By inspecting a 532-year [[Paschal cycle]] with 19 solar cycles (each of 28&nbsp;years, each year numbered 1–28) and 28 lunar cycles (each of 19&nbsp;years, each year numbered 1–19), he determined that the first two numbers, 9 and 1, occurred at its year&nbsp;457. He then calculated via [[modulo operation|remainder division]] that he needed to add eight 532-year Paschal cycles totaling 4256&nbsp;years before the cycle containing 1&nbsp;BC or year&nbsp;0 in order for its year&nbsp;457 to be indiction 3. The sum {{nowrap|4256 + 457}} was thus JP&nbsp;4713.<ref>Grafton 1975, p. 184</ref>

A formula for determining the year of the Julian Period given its character involving three four-digit numbers was published by [[Jacques de Billy]] in 1665 in the ''[[Philosophical Transactions of the Royal Society]]'' (its first year).<ref>de Billy 1665</ref> [[John Herschel|John F. W. Herschel]] gave the same formula using slightly different wording in his 1849 ''Outlines of Astronomy''.<ref>Herschel 1849</ref>

{{blockquote|text=Multiply the ''Solar'' Cycle by 4845, and the ''Lunar'', by 4200, and that of the ''Indiction'', by 6916. Then divide the Sum of the products by 7980, which is the ''Julian Period'': The ''Remainder'' of the Division, without regard to the ''Quotient'', shall be the year enquired after.|author=Jacques de Billy}}

[[Carl Friedrich Gauss]] introduced the [[modulo operation]] in 1801, restating de Billy's formula as: 
{{block indent|1=Julian Period year = (6916''a'' + 4200''b'' + 4845''c'') MOD 15×19×28}}
where ''a'' is the year of the indiction cycle, ''b'' of the lunar cycle, and ''c'' of the solar cycle.<ref>Gauss 1966</ref><ref>Gauss 1801</ref>

[[John Collins (mathematician)|John Collins]] described the details of how these three numbers were calculated in 1666, using many trials.<ref>Collins 1666</ref> A summary of Collin's description is in a footnote.<ref>
{| class=wikitable
|+ Calculation of 4845, 4200, 6916 <br> by Collins
|-
| || align=center | ''Try'' 2+ until ||
|-
| align=center | {{sfrac|7980|28}} = 19×15 = 285 
| align=center | {{sfrac|285×''Try''|28}} = <br> {{nowrap|remainder 1}} 
| align=center | 285×17 = 19×15×17 = 4845
|-
| align=center | {{sfrac|7980|19}} = 28×15 =  420 
| align=center | {{sfrac|420×''Try''|19}} = <br> {{nowrap|remainder 1}} 
| align=center | 420×10 = 28×15×10 = 4200
|-
| align=center | {{sfrac|7980|15}} = 28×19 = 532 
| align=center | {{sfrac|532×''Try''|15}} = <br> {{nowrap|remainder 1}} 
| align=center | 532×13 = 28×19×13 = 6916
|}</ref>
Reese, Everett and Craun reduced the dividends in the ''Try'' column from 285, 420, 532 to 5, 2, 7 and changed remainder to modulo, but apparently still required many trials.<ref name="Reese, Everett and Craun 1981">Reese, Everett and Craun 1981</ref>

The specific cycles used by Scaliger to form his tricyclic Julian Period were, first, the indiction cycle with a first year of 313.{{efn|All years in this paragraph are those of the Anno Domini Era at the time of Easter}}<ref>Depuydt 1987</ref> Then he chose the dominant 19-year Alexandrian lunar cycle with a first year of 285, the [[Era of Martyrs]] and the Diocletian Era epoch,<ref>Neugebauer 2016, pp. 72–77, 109–114</ref> or a first year of 532 according to [[Dionysius Exiguus]].<ref name="Dionysius Exiguus 2003/525">Dionysius Exiguus 2003/525</ref> Finally, Scaliger chose the post-Bedan solar cycle with a first year of 776, when its first quadrennium of [[concurrent (Easter)|concurrent]]s, {{nowrap|1 2 3 4}}, began in sequence.{{efn|1=The concurrent of any Julian year is the weekday of its March{{nbsp}}24, numbered from Sunday=1.}}<ref>''De argumentis lunæ libellus'', col. 705</ref><ref>Blackburn and Holford-Strevens, p. 821</ref><ref>Mosshammer 2008, pp. 80–85</ref> Although not their intended use, the equations of de Billy or Gauss can be used to determined the first year of any 15-, 19-, and 28-year tricyclic period given any first years of their cycles. For those of the Julian Period, the result is AD&nbsp;3268, because both remainder and modulo usually return the lowest positive result. Thus 7980&nbsp;years must be subtracted from it to yield the first year of the present Julian Period, −4712 or 4713&nbsp;BC, when all three of its sub-cycles are in their first years.

Scaliger got the idea of using a tricyclic period from "the Greeks of Constantinople" as Herschel stated in his quotation below in [[#Julian day numbers|Julian day numbers]].<ref name="Herschel 1849, p. 634">Herschel 1849, p. 634</ref> Specifically, the monk and priest Georgios wrote in 638/39 that the Byzantine year 6149&nbsp;AM (640/41) had indiction 14, lunar cycle 12, and solar cycle 17, which places the first year of the [[Byzantine Era]] in 5509/08&nbsp;BC, the Byzantine Creation.<ref>Diekamp 44, 45, 50</ref> Dionysius Exiguus called the Byzantine lunar cycle his "lunar cycle" in argumentum 6, in contrast with the Alexandrian lunar cycle which he called his "nineteen-year cycle" in argumentum 5.<ref name="Dionysius Exiguus 2003/525"/>

Although many references say that the ''Julian'' in "Julian Period" refers to Scaliger's father, [[Julius Caesar Scaliger|Julius Scaliger]], at the beginning of Book V of his ''{{lang|la|Opus de Emendatione Temporum}}'' ("Work on the Emendation of Time") he states, "{{lang|la|Iulianam vocauimus: quia ad annum Iulianum accomodata}}",<ref>Scaliger 1629, p. 361</ref>{{refn|Scaliger used these words in his 1629 edition on p. 361 and in his 1598 edition on p. 339. In 1583 he used "{{lang|la|Iulianam vocauimus: quia ad annum Iulianum duntaxat accomodata est}}" on p. 198.}} which Reese, Everett and Craun translate as "We have termed it Julian because it fits the Julian year".<ref name="Reese, Everett and Craun 1981"/> Thus ''Julian'' refers to the [[Julian calendar]].

===Julian day numbers===
Julian days were first used by [[Christian Ludwig Ideler|Ludwig Ideler]] for the first days of the Nabonassar and Christian eras in his 1825 ''Handbuch der mathematischen und technischen Chronologie''.<ref>Ideler 1825, pp. 102–106</ref>{{refn|The Nabonassar day was elapsed with a typo – it was correctly printed later as 1448638. The Christian day (1721425) was current, not elapsed.}} [[John Herschel|John F. W. Herschel]] then developed them for astronomical use in his 1849 ''Outlines of Astronomy'', after acknowledging that Ideler was his guide.<ref>Herschel, 1849, p. 632 note</ref>

{{blockquote |text=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.<ref>Ideler 1825, p. 77</ref> 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 {{nowrap|4713 BC}}, and the noon of January 1 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.<ref name="Herschel 1849, p. 634">Herschel 1849, p. 634</ref>}}

At least one mathematical [[astronomer]] adopted Herschel's "days of the Julian period" immediately. [[Benjamin Peirce]] of [[Harvard University]] used over 2,800 Julian days in his ''Tables of the Moon'', begun in 1849 but not published until 1853, to calculate the lunar [[ephemeris|ephemerides]] in the new ''American Ephemeris and Nautical Almanac'' from 1855 to 1888. The days are specified for "Washington mean noon", with Greenwich defined as {{nowrap|18{{sup|h}} 51{{sup|m}} 48{{sup|s}}}} west of Washington (282°57′W, or Washington 77°3′W of Greenwich). A table with 197 Julian days ("Date in Mean Solar Days", one per century mostly) was included for the years –4713 to 2000 with no year&nbsp;0, thus "–" means BC, including decimal fractions for hours, minutes, and seconds.<ref>Peirce 1853</ref> The same table appears in ''Tables of Mercury'' by Joseph Winlock, without any other Julian days.<ref>Winlock 1864</ref>

The national ephemerides started to include a multi-year table of Julian days, under various names, for either every year or every leap year beginning with the French ''Connaissance des Temps'' in 1870 for 2,620&nbsp;years, increasing in 1899 to 3,000&nbsp;years.<ref>''Connaissance des Temps'' 1870, pp. 419–424; 1899, pp. 718–722</ref> The British ''Nautical Almanac'' began in 1879 with 2,000&nbsp;years.<ref>''Nautical Almanac and Astronomical Ephemeris'' 1879, p. 494</ref> The ''Berliner Astronomisches Jahrbuch'' began in 1899 with 2,000&nbsp;years.<ref>''Berliner Astronomisches Jahrbuch'' 1899, pp. 390–391</ref> The ''American Ephemeris'' was the last to add a multi-year table, in 1925 with 2,000&nbsp;years.<ref>''American Ephemeris'' 1925, pp. 746–749</ref> However, it was the first to include any mention of Julian days with one for the year of issue beginning in 1855, as well as later scattered sections with many days in the year of issue. It was also the first to use the name "Julian day number" in 1918. The ''Nautical Almanac'' began in 1866 to include a Julian day for every day in the year of issue. The ''Connaissance des Temps'' began in 1871 to include a Julian day for every day in the year of issue.

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, {{lang|fr|italic=yes|Traité de Mécanique Céleste}}, in 1823.<ref>Laplace 1823</ref> Other astronomers added fractions of the day to the Julian day number to create Julian Dates, which are typically used by astronomers to date [[astronomy|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 in 1860 by the English astronomer [[N. R. Pogson|Norman Pogson]], which he stated was at the suggestion of John Herschel.<ref>Pogson 1860</ref> They were popularized for variable stars by [[Edward Charles Pickering]], of the [[Harvard College Observatory]], in 1890.<ref>Furness 1915.</ref>

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 for his astronomical observations 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 clock]]s. Nevertheless, he double-dated most nighttime observations with both [[Egypt (Roman province)|Egyptian]] days beginning at sunrise and [[Babylonia]]n days beginning at sunset.<ref>Ptolemy {{Circa|150}}, p. 12</ref> Medieval Muslim astronomers used days beginning at sunset, so astronomical days beginning at noon did produce a single date for an entire night. Later medieval European astronomers used Roman days beginning at midnight so astronomical days beginning at noon also allow observations during an entire night to use a single date. When all astronomers decided to start their astronomical days at midnight to conform to the beginning of the civil day, on {{nowrap|January 1, 1925}}, it was decided to keep Julian days continuous with previous practice, beginning at noon.

During this period, usage of Julian day numbers as a neutral intermediary when converting a date in one calendar into a date in another calendar also occurred. An isolated use was by Ebenezer Burgess in his 1860 translation of the ''[[Surya Siddhanta]]'' wherein he stated that the beginning of the [[Kali Yuga]] era occurred at midnight at the meridian of [[Ujjain]] at the end of the 588,465th&nbsp;day and the beginning of the 588,466th&nbsp;day (civil reckoning) of the Julian Period, or between {{nowrap|February 17 and 18}} JP&nbsp;1612 or 3102&nbsp;BC.<ref>Burgess 1860</ref><ref>Burgess was furnished these Julian days by US Nautical Alamanac Office.</ref> Robert Schram was notable beginning with his 1882 ''Hilfstafeln für Chronologie''.<ref>Schram 1882</ref> Here he used about 5,370 "days of the Julian Period". He greatly expanded his usage of Julian days in his 1908 ''Kalendariographische und Chronologische Tafeln'' containing over 530,000 Julian days, one for the zeroth day of every month over thousands of years in many calendars. He included over 25,000 negative Julian days, given in a positive form by adding 10,000,000 to each. He called them "day of the Julian Period", "Julian day", or simply "day" in his discussion, but no name was used in the tables.<ref>Schram 1908</ref> Continuing this tradition, in his book "Mapping Time: The Calendar and Its History" British physics educator and programmer Edward Graham Richards uses Julian day numbers to convert dates from one calendar into another using algorithms rather than tables.<ref>Richards 1998, pp. 287–342</ref>

== Julian day number calculation ==<!-- This section is linked from [[Gregorian calendar]] -->
The Julian day number can be calculated using the following formulas ([[integer division]] rounding towards zero is used exclusively, that is, positive values are rounded down and negative values are rounded up):{{efn|Doggett in Seidenmann 1992, p. 603, indicates the algorithms are inspired by Fliegel & Van Flanderen 1968. That paper gives algorithms in [[Fortran]]. The Fortran computer language performs integer division by truncating, which is functionally equivalent to rounding toward zero.}}

The months January to December are numbered 1 to 12. For the year, [[astronomical year numbering]] is used, thus 1&nbsp;BC is 0, 2&nbsp;BC is −1, and 4713&nbsp;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|proleptic]]) Gregorian calendar dates after November 23, &minus;4713. '''Divisions are integer divisions towards zero'''; fractional parts are ignored.<ref>L. E. Doggett, Ch. 12, "Calendars", p. 604, in Seidelmann 1992. "These algorithms are valid for all Gregorian calendar dates corresponding to JD >= 0, i.e, dates after &minus;4713 November 23."</ref>

{{block indent|1=<math>\text{JDN}  =  \frac{1461 \cdot \left( \text{Y} + 4800 + \frac{\text{M}-14}{12} \right)}{4}  + \frac{367 \cdot \left( \text{M} - 2 - 12 \cdot \frac{\text{M}-14}{12} \right)}{12} - \frac{3 \cdot \frac{\text{Y} + 4900 + \frac{\text{M} - 14}{12}}{100}}{4}  + \text{D} - 32075</math>}}

===Converting Julian calendar date to Julian Day Number===
The algorithm<ref>L. E. Doggett, Ch. 12, "Calendars", p. 606, in Seidelmann 1992</ref> is valid for all (possibly [[Proleptic Julian calendar|proleptic]]) Julian calendar years ≥&nbsp;−4712, that is, for all JDN&nbsp;≥&nbsp;0. Divisions are integer divisions, fractional parts are ignored.

{{block indent|1=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&nbsp;UT one can use the following. Divisions are [[real number]]s.

{{block indent|1=<math>\begin{matrix}J\!D & = & J\!D\!N + \frac{\text{hour} - 12}{24} + \frac{\text{minute}}{1440} + \frac{\text{second}}{86400}\end{matrix}</math>}}

So, for example, January 1, 2000, at 18:00:00&nbsp;UT corresponds to ''JD''&nbsp;=&nbsp;2451545.25 and January 1, 2000, at 6:00:00&nbsp;UT corresponds to ''JD''&nbsp;=&nbsp;2451544.75.

=== Finding day of week given Julian day number ===
Because a Julian day starts at noon while a civil day starts at midnight, the Julian day number needs to be adjusted to find the day of week: for a point in time in a given Julian day after midnight UT and before 12:00&nbsp;UT, add 1 or use the JDN of the next afternoon.

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:
{{block indent|1='''W1''' = [[Modular arithmetic|mod]](''J'' + 1, 7)<ref>Richards 2013, pp. 592, 618.</ref>}}

{| class="wikitable"
|-style="text-align:center;"
!W1
| 0 || 1 || 2 || 3 || 4 || 5 || 6
|-
!Day of the week
|Sun||Mon||Tue||Wed||Thu||Fri||Sat
|}

If the moment in time is after midnight UT (and before 12:00&nbsp;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:
{{block indent|1='''W0''' = mod (''J'', 7) + 1}}

{| class="wikitable"
|- style="text-align:center;"
!W0
| 1 || 2 || 3 || 4 || 5 || 6 || 7
|-
!Day of the week
|Mon||Tue||Wed||Thu||Fri||Sat||Sun
|}

=== Julian or Gregorian calendar from Julian day number ===
This is an algorithm by Edward Graham 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.<ref>Richards 2013, 617–619</ref><ref>Richards 1998, 316</ref> All variables are integer values, and the notation "''a''&nbsp;div&nbsp;''b''" indicates [[integer division]], and "mod(''a'',''b'')" denotes the [[Modular arithmetic|modulus operator]].

{| class="wikitable"
|+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'' || &minus;38
|}

For Julian calendar:
# ''f&nbsp;''=&nbsp;'''J'''&nbsp;+&nbsp;''j''
For Gregorian calendar:
# ''f&nbsp;''=&nbsp;'''J'''&nbsp;+&nbsp;''j''&nbsp;+&nbsp;(((4 ×&nbsp;'''J'''&nbsp;+&nbsp;''B'')&nbsp;div&nbsp;146097)&nbsp;×&nbsp;3)&nbsp;div&nbsp;4&nbsp;+&nbsp;''C''
For Julian or Gregorian, continue:
{{ordered list|start=2
|1=''e'' = ''r'' × ''f'' + ''v''
|2=''g'' = mod(''e'', ''p'') div ''r''
|3=''h'' = ''u'' × ''g'' + ''w''
|4='''D''' = (mod(''h, s'')) div ''u'' + 1
|5='''M''' = mod(''h'' div ''s'' + ''m'', ''n'') + 1
|6='''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&nbsp;+&nbsp;4200m&nbsp;+&nbsp;4845s by 7980 and call the remainder r.
{{block indent|1=If r>4713, Y = (r − 4713) and is a year AD.}}
{{block indent|1=If r<4714, Y = (4714 − r) and is a year BC.}}

Example

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

{{block indent|1=(6916 × 8) = 55328; (4200 × 2) = 8400: (4845 × 8) = 38760.   55328 + 8400 + 38760 = 102488.}}
{{block indent|1=102488/7980 = 12 remainder 6728.}}
{{block indent|1=Y = (6728 − 4713) = AD 2015.<ref>Heath 1760, p. 160.</ref>}}

==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&nbsp;seconds (or in the unlikely event of a negative leap second, 86,399&nbsp;seconds). One authoritative source, the [[SOFA (astronomy)|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&nbsp;seconds, as required). SOFA refers to the result of such a calculation as "quasi-JD".<ref>"SOFA Time Scale and Calendar Tools" 2016, p. 20</ref>

== See also ==
{{Div col|colwidth=50em}}
* [[5th millennium BC]]
* [[Barycentric Julian Date]]
* [[Dual dating]]
* [[Decimal time]]
* [[Epoch (astronomy)]]
* [[Epoch (reference date)]]
* [[Era]]
* [[J2000]] – the epoch that starts on JD 2451545.0 (TT), the standard epoch used in astronomy since 1984
* [[Julian year (calendar)]]
* [[Lunation Number]] (similar concept)
* [[Ordinal date]]
* [[Time]]
* [[Time standard]]s
* [[Zeller's congruence]]
{{Div col end}}

==Notes==
{{notelist}}

== References ==
{{reflist|25em}}

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{{Refend}}

==External links==
*{{Cite web| title = Julian Date Converter| accessdate = 2023-08-30| url = https://aa.usno.navy.mil/data/JulianDate | publisher = US Naval Observatory}}

{{Time measurement and standards}}

{{DEFAULTSORT:Julian Day}}
[[Category:Calendar algorithms]]
[[Category:Calendaring standards]]
[[Category:Celestial mechanics]]
[[Category:Chronology]]
[[Category:Time in astronomy]]