Submitted by mossy2100 on Tue, 04/25/2017 - 00:14

Tropical and calendar mirs

The average length of a year in the Gregorian Calendar (the most commonly used calendar on Earth) is 365.2425 days. This is very close to the average period between northern hemisphere vernal equinoxes, which is 365.242374 days. This period is known as the tropical year, and it covers the full cycle of seasons from the start of the spring to the end of winter in the northern hemisphere.

In truth, this is just one option for determining the tropical year; any of the seasonal markers can be used (vernal equinox, summer solstice, autumnal equinox, or winter solstice), or an average of these durations. All of these values are slightly different, and change over time, due to gravitational interactions between the planets and the Sun. However, the northern vernal equinox tropical year is the most commonly used.

Astronomers measure Earth’s progress through the tropical year using longitude of the Sun (abbreviated “Ls”), which is the position of the Sun against the backdrop of fixed stars that it moves across. At the northern vernal equinox, Ls = 0°. During the year, as the Sun appears to move past the stars (from the perspective of Earth), Ls increases to 360°.

Following this scientific tradition, the Utopian Calendar mir begins at the northern hemisphere vernal equinox (within one sol). Thus, as Ls increases from 0° to 360°, the sols of the mir increase from 1 to 668 (or 669).

There is a poetic bonus gained by choosing to begin the mir with the northern spring. On Earth the northern spring begins in March, with the vernal equinox occurring on or close to March 20. In the early Roman calendar March was the first month of the year; thus, the year ended with winter and began with spring. March was named for the god Mars, who was originally the Roman god of springtime before he became the god of war. Therefore, it’s quite appropriate to begin the new mir at the beginning of spring, which is the god Mars’ season.

Short and long mirs

The average period between northern vernal equinoxes on Mars is 668.5907 sols, which is used as the basis for the calendar mir length in the Darian and Utopian Calendars.

However, we want calendar mirs to be a whole number of sols, just like the calendar year is a whole number of days in Terran calendars. This will make it possible to organise the sols of the mir into weeks and months. To ensure the average calendar mir length is 668.5907 sols, a pattern of short mirs of 668 sols and long mirs of 669 sols is defined, analogous to the common years of 365 days and leap years of 366 days in the Gregorian Calendar. However, the adjectives “common” and “leap” aren’t suitable for mirs because short mirs aren’t common, since they occur less often than long mirs.

The Utopian Calendar specifies a system of rules similar to those used by the Gregorian Calendar to determine when a calendar mir is short or long. These rules produce an average calendar mir length of 668.591 sols per mir, which is close enough to 668.5907 for practical purposes:

  1. A mir is short unless one of the following rules applies. (668 sols/mir)
  2. Odd-numbered mirs are long. (668 + 0.5 = 668.5 sols/mir)
  3. Mirs divisible by 10 (ending in 0) are also long. (668.5 + 0.1 = 668.6 sols/mir)
  4. Except, mirs divisible by 100 (ending in 00) are short. (668.6 - 0.01 = 668.59 sols/mir)
  5. Except, mirs divisible by 1000 (ending in 000) are long. (668.59 + 0.001 = 668.591 sols/mir)

There is a difference of 668.591 - 668.5907 = 0.0003 sols, or about 27 seconds, per mir. After 2000 mirs, the error will have accumulated to almost 15 hours. Then the intercalation rules are modified to resynchronise the calendar with Mars’ cycle. This is not only necessary to compensate for this slight difference between the length of the calendar mir and the northern vernal equinox mir, but also because the length of the northern vernal equinox mir is slowly increasing.

The Darian Calendar specifies intercalation rules for the first 10000 mirs, to achieve this resynchronisation:

Range of mirs Formula Mean length of calendar mir (sols)
0 to 2000 (M - 1)\2 + M\10 - M\100 + M\1000 668.5910
2001 to 4800 (M - 1)\2 + M\10 - M\150 668.5933
4801 to 6800 (M - 1)\2 + M\10 - M\200 668.5950
6801 to 8400 (M - 1)\2 + M\10 - M\300 668.5967
8401 to 10000 (M - 1)\2 + M\10 - M\600 668.5983

The resulting error is only about 1 sol after 12000 mirs.

The intercalary sol, i.e. the 669th sol of the mir in a long mir, is appended to the last month of the mir. The advantage of doing this is that the date will always map to the same sol number within a mir (1-669). This is not the case with the Gregorian Calendar, for example, because dates after 28 February (the 59th day of the year) map to different days of the year depending on whether or not it’s a leap year. For example, 1 March is the 60th day of the year in a common year, or the 61st in a leap year (29 February is the 60th). Perhaps this isn’t such a big deal, but it does make sense to append the extra sol at the end of the mir rather than inserting it in the middle somewhere.

This also reflects the early Roman calendar, when March was the first month of the year. January and February were appended to the original 10-month calendar around 713 BC, and February was the last month of the year up until about 450 BC. As the standard calendar year was only 355 days, it would fall out of sync with the seasons, so the Romans would periodically add extra days into February to resynchronise it. However, from 450 BC the calendar year started with January, making February the second month.

Incidentally, this is also why September, October, November, and December are now the 9th, 10th, 11th, and 12th months, instead of the 7th, 8th, 9th, and 10th months, which they originally were, as their Latin prefixes suggest.