Astronomical Calendars

Year Definitions

The most common definition in the western world of the year is based on the revolution of the Earth around the Sun and is therefore called a `Solar Year'. However, there are several possibilites to define beginning and end of one revolution and thus also several kinds of solar years:

The years so defined differ in length because of the precession of Earth's rotation and the tumbling of the Earth orbit.
Julian year (365.25 days UT) and Gregorian year (365.2425 days UT) as defined in the calendars of the respective name are solar years as well.
Solar years have the disadvantage of not being easily observable. Many years of observations are required to fix them with any significant degree of accuracy. On the other hand, the phases of the Moon -- and the first visibility after the new moon in particular -- are very easy and quick to observe. Therefore, the first calendars defined a lunar year, usually consisting of 12 synodic months. A synodic month is the interval from one new moon to the next and lasts 29.5306 days UT. Since for practical reasons a month should contain an integer number of days, most calendars alternated betweens months of 29 and 30 days, respectively. A year made out of six months of each type has 354 days and is thus too short by 0.3672 days as compared with a true lunar year. Lunar calendars have to insert one leap day about every third year to keep in step with the moon phases. A pure lunar calendar is not synchronous with the seasons.
A luni-solar year is the attempt to combine the phases of the moon and the seasons into one calendar. This is possible if leap months are inserted. Several schemes were used in history. The best known solution was found by the Greek Meton in the year 432 BC but apparently was known to other cultures before. The Metonic cycle encompasses a total of 235 months of which 125 are full (i.e. they have 30 days) and 110 are `hollow' (having 29 days). The months are combined into 12 normal years with 12 months each and 7 leap years with 13 months each. The cycle covers 6940 days whereas 235 synodic months sum up to 6939.688 days and 19 tropical years to 6939.602 days. The difference in motion between Sun and Moon amounts to only 0.0866 days so that eclipses repeat in the Metonic cycle with high accuracy.

Julian Calendar

The Julian calendar is based on a solar year with originally 365 days. To account for the fact that the tropical year is longer than 365 days by about a quarter day, a leap day is inserted at the end of month of February in every fourth year. This simple leap year rule was already known in late Egypt. It was in fact an Alexandrian scholar named Sosigenes who advised Julius Cesar during the introduction of the calendar into the Roman empire in the year 46 BC. The calendar is named after Julius Cesar.
Julius Cesar had to start the introduction of his calendar with an anomalous leap year with 445 days for the year 46 BC to compensate for the inaccuracies of the Roman calendar used before. The following year 45 BC was a normal leap year with 366 days. After Cesar's death the new leap year rules were at first incorrectly applied and too many leap years occured. This was corrected under the government of Augustus and the Julian calendar was strictly obeyed since the year AD 8. For earlier years date estimates are uncertain by a few days since the sequence of leap years is not exactly known.
In astronomy and for historical purposes the Julian calendar is also applied to epochs earlier than the year 46 BC when this calendar was not yet defined and the people of that time could not know their date in it. To indicate this extension, the term proleptic Julian calendar is occasionally used (proleptic = brought forward).

Gregorian Calendar

The Julian year with its duration of 365.25 days was too long by 0.0078 days or 11 minutes 14 seconds with respect to the tropical year. Although this difference was not perceptible within a few years, it acculumated over the centuries. Astronomers first noticed that the true beginning of spring (when the Sun passes through the Vernal equinox) moved away from the nominal start of spring on March 21. This nominal date had been decreed by the Roman church in the connection with the Easter date. At the beginning of the 16th century the date in the Julian calendar already lagged 10 days behind the true position of Earth in its orbit and the Easter date began to lose its intended connection with the Jewish feast of Passover (that is tied to the true start of spring).
To solve this problem, pope Gregor XIII in AD 1582 ordered a calendar reform for the domain of the Catholic church. It consisted of three parts:

  1. Omission of 10 calendar days, the 4th of October 1582 was followed directly by the 15th of October 1582 in the new calendar. This brought the start of spring back to the 21th of March. The reckoning of week days was not changed.
  2. Introduction of a new leap year rule according to which no leap days occur in years that are divible by 100 but not by 400. This reduces the error in the year length and slows down the accumulation of this error. The leap day is inserted at the end of February as in the Julian calendar.
  3. Modification of the Easter rule to accomodate the new calendar.
The objectives and details of the new calendar were described in AD 1603 by Christoph Clavius in his book ``Explication Romani Calendarii a Gregorio XIII P.M. restitui''.
The leap year rule described under 2. is the basis for the Gregorian calendar still in use today. It results in a mean year length of 365.2425 days. The remaining difference with respect to the tropical year is small enough to require the insertion of an extra leap day only after 3333 years.
Although most sources date the conversion from Julian to Gregorian calendar for the pair of days October 4./15., 1582, this is in fact only true for countries where the Roman Catholic Church was influential. Other countries hesitated to adopt the new calendar, in some cases for very long times. Turkey, for instance, converted to the Gregorian calendar on January 1, 1927. Therefore, care must be taken in dating historical events to account for country-specific conversion dates. A fairly detailed list of conversion dates for many countries can be in the Explanatory Supplement (see the list of references).
The reckoning of our modern year count from the year of Christ's birth goes back to the roman abbot Dionysius Exiguus who at the year AD 525 endeavoured to set up tables for the computation of the Easter date. For reasons unknown to us, he equated the year 248 of the era of Diokletian with the year AD 532. (This assignment is considered dubious nowadays.) In this new reckoning, the year AD 1 is directly preceded by the year 1 BC, a year 0 does not exist in this system. In contrast, the astronomical reckoning indeed uses a year 0. For the purpose of distinction, astronomical reckoning drops the symbols AD and BC and uses a plus or minus sign before the year instead. The astronomical year +1 therefore corresponds to the year AD 1, the year 0 corresponds to 1 BC, and the year -1 to 2 BC.
The first century of Christian year reckoning began on January 1 of the year AD 1 and ended exactly a hundred years later on December 31, AD 100. Consequently, the second century had to begin on January 1, AD 101. A similar reasoning holds for the millennia. From that it follows that the next, the third millennium will not begin on Januar 1, AD 2000 -- as it is often assumed -- but on January 1, AD 2001. (That this misconception predictably leads to public debates whenever the occasion arises, is even noted in the Explanatory Supplement to the Astronomical Almanac, edition 1961, page 411., see Referencey)
The Gregorian calendar is regularly used in astronomy for dates later than October 14, 1582. For some applications, however, it is favourable to extrapolate it to epochs before this date (proleptic Gregorian calendar). On the other hand, even today some dates or time intervals are calculated according to the Julian calendar. These exceptions are labelled appropriately.

Easter Date

The Christian Easter feast was derived from the Jewish Passover which begins on the first full moon in spring. This day can obviously fall on a random day of the week. Easter, in contrast, begins on a Sunday by definition. At first, the Easter date was calculated very differently in the diverse Christian parishes. Only at the 1.council in Nicäa in the year AD 325 an agreement was achieved that Easter should begin on the first Sunday _after_ the first full moon in spring. The latter is the first full moon that occurs either on or after the day of the spring equinox.

However, with the decree of Nicäa the difficulties were not entirely removed because the precise determination of the first full moon in spring had its own problems. Finally, at the request of Pope John I, the roman abbot Dionysius Exiguus established in AD 525 the rule as previously used in Alexandria. According to this rule

  1. spring is defined to begin at March 21, 0 o'clock, and
  2. the moon is assumed to move at constant speed on a circular orbit.
Both assumptions are simplifications that lead to deviations from the true astronomical facts. The true beginning of spring happens some time between March 19, 8 o'clock and March 21, 20 o'clock UT. Consideration of the true lunar orbit leads to time differences of up +/- 0.7 days with respect to a circular orbit. Moreover, the Gregorian calendar reform forced the Easter date to fall into the time interval from March 22 to April 25 (both dates included). For these reasons, shifts between the factual Easter date and the date calculated from the astronomically correct spring full moon can occur which are called 'Easters paradoxes'. The last paradox happened in 1974 (Easter was celebrated on April 14 instead of April 7), the next one will be in the year 2000 (April 23 instead of March 26).
The Easter date is nowadays calculated from tables specifically constructed for that purpose or from the Easter formulae of Carl Friedrich Gauß. Both methods are valid for all years since AD 532. Simplified formulae for easier use, that explicitly assume either the Gregorian or the Julian calendar, are given by J.Meuus (see the list of literature).
Even today, the various Christian churches differ in the fixation of the Easter feast. The eastern churches, for example, stick to the beginning of spring on March 21 of the Julian calendar and calculate the true astronomical full moon for the meridian of Jerusalem.
(See here a list with the easter dates from 1901 to 2078.)

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© Dirk Husfeld --- 96/11/29 ---
Last modified:
C. Kronberg --- 97/07/17 ---