Collatz conjecture

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Julian day refers to a continuous count of days since the beginning of the Julian Period used primarily by astronomers.

The Julian Day Number (JDN) is the integer assigned to a whole solar day in the Julian day count starting from noon Greenwich Mean Time, with Julian day number 0 assigned to the day starting at noon on January 1, 4713 BC, proleptic Julian calendar (November 24, 4714 BC, in the proleptic Gregorian calendar). For example, the Julian day number for January 1, 2000, was 2,451,545.

The Julian Date (JD) of any instant is the Julian day number for the preceding noon plus the fraction of the day since that instant. 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 2,456,293.520833.

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 "May 12, 1629", this means that date in the Julian calendar (which is May 22, 1629, in Gregorian calendar— the date of the Treaty of Lübeck). Outside of an astronomical or historical context, if a given "Julian date" is "40", this most likely means the fortieth day of a given Gregorian year, namely February 9. But the potential for mistaking a "Julian date" of "40" to mean an astronomical Julian Day Number (or even to mean the year 40 Template:Scaps in the Julian calendar, or even to mean a duration of 40 astronomical Julian years) is justification for preferring the terms "ordinal date" or "day-of-year" instead. 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", in spite of the potential for misinterpreting this as meaning that the calendars are of years in the Julian calendar system.

The Julian Period is a chronological interval of 7980 years beginning 4713 BC. It has been used by historians since its introduction in 1583 to convert between different calendars. Template:Currentyear is year of the current Julian Period. The next Julian Period begins in the year 3268 AD.

Time scales

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 specified else wise.

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Julian Date	12h Jan 1, 4713 BC
Reduced JD	12h Nov 16, 1858	JD − 2400000
Modified JD	0h Nov 17, 1858	JD − 2400000.5		Introduced by SAO in 1957
Truncated JD	0h May 24, 1968	JD − 2440000.5		Introduced by NASA in 1979
Dublin JD	12h Dec 31, 1899	JD − 2415020		Introduced by the IAU in 1955
Chronological JD	0h Jan 1, 4713 BC	JD + 0.5 + tz	(UT)	Specific to time zone
Lilian date	Oct 15, 1582 (1)	floor (JD − 2299159.5)		Count of days of the Gregorian calendar
ANSI Date	Jan 1, 1601 (1)	floor (JD − 2305812.5)		Origin of COBOL integer dates
Rata Die	Jan 1, 1 (1)	floor (JD − 1721424.5)		Count of days of the Common Era (Gregorian)
Unix Time	0h Jan 1, 1970	(JD − 2440587.5) × 86400		Count of seconds
Mars Sol Date	12h Dec 29, 1873	(JD − 2405522)/1.02749		Count of Martian days

• 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 OpenVMS, using 63-bit date/time, postponing the next Y2K campaign to July 31, 31086, 02:48:05.47. MJD is defined relative to midnight, rather than noon.

• 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 Chronological Julian Date was recently proposed by Peter Meyer and has been used by some students of the calendar and in some scientific software packages. CJD is usually defined relative to local civil time, rather than UT, requiring a time zone (tz) offset to convert from JD. In addition, days start at midnight rather than noon. Users of CJD sometimes refer to Julian Date as Astronomical Julian Date to distinguish it.

• The 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 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. 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 a system (or more precisely a family of three systems) used in the book Calendrical Calculations. It uses the local timezone, and day 1 is January 1, 1, that is, the first day of the Christian or Common Era in the proleptic Gregorian calendar.

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 astronomic 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 measure. 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 in 1583, at the time of the Gregorian calendar reform, as 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 (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.

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, "Iulianum vocavimus: 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.

Calculation

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

The months (M) January to December are 1 to 12. For the year (Y) astronomical year numbering is used, thus 1 BC is 0, 2 BC is −1, and 4713 BC is −4712. D is the day of the month. JDN is the Julian Day Number, which pertains to the noon occurring in the corresponding calendar date.

Converting Julian or Gregorian calendar date to Julian Day Number

The algorithm is valid at least for all positive Julian Day Numbers. The meaning of the variables are explained by the Computer Science Department of the University of Texas at San Antonio.

It is worth noting that this algorithm does not follow the NASA or the US Naval Observatory - the convention in these systems being that the Gregorian Calendar did not exist before the date October 15, 1582 (Gregorian). This algorithm effectively back-dates the Gregorian calendar onto the Julian calendar for dates before the introduction of the Gregorian calendar. Thus any calculations made with this formula before October 15, 1582, will not agree with these previous ephemerides.

You must compute first the number of years (y) and months (m) since March 1 −4800 (March 1, 4801 BC):

$\begin{array}{rlrl}a& =\lfloor \frac{14-\text{month}}{12}\rfloor & & \text{(1 for January and February, 0 for other months)}\\ y& =\text{year}+4800-a\\ m& =\text{month}+12a-3& & \text{(0 for March, 11 for February)}\end{array}$

All years in the BC era must be converted to astronomical years, so that 1 BC is year 0, 2 BC is year −1, etc. Convert to a negative number, then increment toward zero.

Then, if starting from a Gregorian calendar date compute:

$JDN=\text{day}+\lfloor \frac{153m+2}{5}\rfloor +365y+\lfloor \frac{y}{4}\rfloor -\lfloor \frac{y}{100}\rfloor +\lfloor \frac{y}{400}\rfloor -32045$

Otherwise, if starting from a Julian calendar date compute:

$JDN=\text{day}+\lfloor \frac{153m+2}{5}\rfloor +365y+\lfloor \frac{y}{4}\rfloor -32083$

Note: (153m+2)/5 gives the number of days since March 1 and comes from the repetition of days in the month from March in groups of five:

Mar–Jul:	31 30 31 30 31
Aug–Dec:	31 30 31 30 31
Jan–Feb:	31 28

Finding Julian date given Julian day number and time of day

For the full Julian Date (divisions are real numbers):

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

So, for example, January 1, 2000, at 12:00:00 corresponds to JD = 2451545.0

Finding day of week given Julian day number

The US day of the week W1 can be determined from the Julian Day Number J with the expression:

W1 = mod(J + 1, 7)

W1	0	1	2	3	4	5	6
Day of the week	Sun	Mon	Tue	Wed	Thu	Fri	Sat

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

W0 = mod(J, 7)

W0	0	1	2	3	4	5	6
Day of the week	Mon	Tue	Wed	Thu	Fri	Sat	Sun

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 does not state which dates the algorithm is valid for. Reminder: all variables are integers, and the solidus (/) indicates integer division. The symbol * indicates multiplication 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

1. f = J + j + (((4 * J + B)/146097) * 3)/4 + C

2. e = r * f + v

3. g = mod(e, p)/r

4. h = u * g + w

5. D = (mod(h, s))/u + 1

6. M = mod(h/s + m, n) + 1

7. Y = e/p - y + (n + m - M)/n

D, M, and Y are the numbers of the day, month, and year respectively.

See also

• Julian year (astronomy)
• Julian year (calendar)
• Decimal time
• Epoch (reference date)
• Epoch (astronomy)
• Era
• Time
• Time standards
• Ordinal date
• Dual dating
• 5th millennium BC
• Lunation Number (similar concept)
• Zeller's congruence

References

• Astronomical almanac for the year 2001. (2000). U.S. Nautical Almanac Office and Her Majesty's Nautical Almanac Office.
• Astronomical Almanac Online. (2008). U.S. Nautical Almanac Office and Her Majesty's Nautical Almanac Office.
• <CS1063>...</CS1063>"CS 1063 Introduction to Programming: Explanation of Julian Day Number Calculation." (2011). Computer Science Department, University of Texas at San Antonio.
• Digital Equipment Corporation. Why is Wednesday, November 17, 1858, the base time for VAX/VMS? Modified Julian Day explanation
• Furness, C. E. (1915). An introduction to the study of variable stars. Boston: Houghton-Mifflin. Vassar Semi-Centennial Series.
• HORIZONS System. (April 4, 2013). NASA.
• Information Bulletin No. 81. (January 1998). International Astronomical Union.
• Julian Date Converter (March 20, 2013). US Naval Observatory. Retrieved September 16, 2013.
• Kempler, Steve. (2011). Day of Year Calendar. Goddard Earth Sciences Data and Information Services Center.
• McCarthy, D. & Guinot, B. (2013). Time. In S. E. Urban & P. K. Seidelmann, eds. Explanatory Supplement to the Astronomical Almanac, 3rd ed. (pp. 76–104). Mill Valley, Calif.: University Science Books. ISBN 978-1-89138-985-6
• Richards, E. G. (2013). Calendars. In S. E. Urban & P. K. Seidelmann, eds. Explanatory Supplement to the Astronomical Almanac, 3rd ed. (pp. 585–624). Mill Valley, Calif.: University Science Books. ISBN 978-1-89138-985-6
• Moyer, Gordon. (April 1981). "The Origin of the Julian Day System," Sky and Telescope 61 311−313.
• IBM. (2004). COBOL for AIX Programming Guide (1st ed. ver. 2.0).
• Noerdlinger, P. (April 1995 revised May 1996). Metadata Issues in the EOSDIS Science Data Processing Tools for Time Transformations and Geolocation. NASA Goddard Space Flight Center.
• Ohms, B. G. (1986). Computer processing of dates outside the twentieth century. IBM Systems Journal 25, 244–251.
• Ransom, D. H. Jr. (c. 1988) ASTROCLK Astronomical Clock and Celestial Tracking Program pages 69–143, "Dates and the Gregorian calendar" pages 106–111. Retrieved September 10, 2009.
• Reingold, E. M. & Dershowitz, N. (2008). Calendrical Calculations 3rd ed. Cambridge University Press.
• Seidelmann, P. Kenneth (ed.) (1992). Explanatory Supplement to the Astronomical Almanac pages 55 & 603–606. University Science Books, ISBN 0-935702-68-7.
• Strous, L. (2007) Astronomy Answers: Julian Day Number. Astronomical Institute / Utrecht University.
• US Naval Observatory. (2005, last updated July 2, 2011). Multiyear Interactive Computer Almanac 1800–2050 (ver. 2.2.2). Richmond VA: Willmann-Bell, ISBN 0-943396-84-0.

External links

• Julian day calculation by IMCCE at Paris Observatory ± Julian days with 16 significant digits (integer plus fraction)
• Julian Day and Civil Date calculator
• U.S. Naval Observatory Time Service article on Modified Julian Date
• U.S. Naval Observatory current MJD service
• Outlines of Astronomy by John Herschel, 1849 Table of Julian days for remarkable eras
• International Astronomical Union Resolution 1B: On the Use of Julian Dates
• Calendrica
• BASIC Programs from Sky & Telescope with CALJD.BAS and JDCAL.BAS, very small BASIC programs to convert Julian Day numbers. published in the May 1984 issue.