Astronomical Calculations - Easter Sunday

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Easter Sunday is a moveable feast of the Church - it occurs on a different day each year. Western Churches use the Gregorian Calendar, while Eastern Churches use the Julian Calendar, meaning that Easter Sunday is most often different for the two religious groups. The calculations described here refer to the Easter Sunday defined by the Western Churches, occuring as early as March 22 or as late as April 25.

The date for Easter was set by the Council of Nicaea in 325 AD to be the first Sunday after the Paschal Full Moon (first full moon which falls after the Spring Equinox at the longitude of Venice). If the full moon falls on Monday through Saturday, Easter is the following Sunday. If the full moon falls on Sunday, Easter is not that day, but the following Sunday.

Specifying the longitude of Venice prevents any confusion that might result from the fact that at any particular instant in time, there are usually two different dates in use in the 24 or so time zones around the world. Full moon occurs at a point in its approximate 28-day orbit when it is most opposite the sun. At the instant it reaches this position, it is "full". The day it is on Earth when the moon is "full" depends on one's longitude or time zone. When the moon is "full", it may be 11:03 AM Sunday in Tokyo, 2:03 AM Sunday in London, 9:03 PM Saturday in New York, and 3:03 PM Saturday in Honolulu. To make Easter Sunday one specific Sunday for all locations, the location for the determination was specified to be the longitude of Venice.

Defined astronomically as described above, the date of Easter Sunday would depend on the physical motions of Earth, Moon, and Sun. Since these motions and positions are subject to minor changes and their predictions are subject to measurement errors, an integral arithmetic method of approximation was invented and adopted. To eliminate the uncertainties about Easter Sunday decisions, an "Ecclesiastical Moon" was adopted that approximates the motion of the real one, and Easter Sunday is now determined according the the "Ecclesiastical Moon". Algorithms based on integer mathematics, the "Ecclesiatical Moon", and the Gregorian Calendar can be used to calculate the date of Easter Sunday for Western Churches in any year.

The following algorithm is derived from a rule that appeared in Butcher's Ecclesiastical Calendar in 1876: (Examples are given for 2001)

Let A = the remainder after the four-digit year divided is by 19
(Example: 2001 divided by 19 = 105 with a remainder of 6. "A" would be the remainder 6).

Let B = the integer part of the four-digit year divided by 100
(Example: 2001 divided by 100 = 20 with a remainder of 1. "B" would be 20).

Let C = the remainder (if any) left over from the calculation of B
(Example: 2001 divided by 100 = 20 with a remainder of 1. "C" would be the remainder 1).

Let D = the integer part of B divided by 4
(Example: 20 divided by 4 = 5 with a remainder of 0. "D" would be 5).

Let E = the remainder (if any) left over from the calculation of D
(Example: 20 divided by 4 = 5 with a remainder of 0. "E" would be the remainder 0).

Let F = the integer part of (B + 8) divided by 25
(Example: (20 + 8) divided by 25 = 1 with a remainder of 3. "F" would be 1).

Let G = the integer part of (B - F + 1) divided by 3
(Example: (20 - 1 + 1) divided by 3 = 6 with a remainder of 2. "G" would be 6).

Let H = the remainder (if any) left over after dividing ((19 x A ) + B - D - G + 15) by 30
(Example: ((19 x 6) + 20 - 5 - 6 + 15) divided by 30 = 4 with a remainder of 18. "H" would be 18).

Let I = the integer part of C divided by 4
(Example: 1 divided by 4 = 0 with a remainder of 1. "I" would be 0).

Let J = the remainder (if any) left over from the calculation of I
(Example: "J" would be 1).

Let K = the remainder (if any) left over after dividing (32 + (2 x E) + (2 x I) - H - J) by 7
(Example: (32 + (2 x 0) + (2 x 0) - 18 - 1) divided by 7 = 1 with a remainder of 6. "K" would be 6).

Let L = the integer part of (A + (11 x H) + (22 x K)) divided by 451
(Example: (6 + (11 x 18) + (22 x 6)) divided by 451 = 0 with a remainder of 336. "L" would be 0).

Let M = the integer part of (H + K - (7 x L) + 114) divided by 31
(Example: (18 + 6 - (7 x 0) + 114) divided by 31 = 4 with a remainder of 14. "M" would be 4).

Let N = the 1 + remainder (if any) after the calculation of M
(Example: "N" would be 1+ 14, or 15).

Month of Easter = M
(Example: Easter month = 4, or April).

Date of Easter = N
(Example: Easter day = 15). Easter in 2001 occurs on April 15th.

Links for further reference on Easter Sunday

Butcher's Algorithm

Oudin's Algorithm

Carter's Method

The Orthodox Ecclesiastical Calendar

Easter Dating Method

Easter Algorithm

 

 

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Copyright 1998 by Leighton L. Paul
Last modified: January 27, 2000