The Martian Business Calendar

by Bruce Mills

Contents

  1. Introduction
  2. Terminology
  3. Martian Time Periods
  4. Possible Year Lengths
  5. The Calendar
  6. Month Names
    1. Numbered months
    2. Constellations
  7. Determination of Intercalary Weeks
    1. Future intercalation
    2. Gregorian-style rules
    3. Determining time of equinox or solstice
  8. Epoch
  9. Further Reading

Introduction

At present, Mars is a cold, dry, barren world that is not habitable. Despite these shortcomings, Mars is the second most habitable body in the Solar System after the Earth. Eventually, the time will come when Mars is colonised. When that time comes, the future Martians will require a calendar of their own.

When the Martian colonists choose a calendar, they will require a calendar that meets the needs of business and religion, as well as the day to day needs of the people in general. There have been many proposals for Martian calendars, but many of them have shortcomings that would make them unlikely to be adopted.

This document discusses a proposal for a Martian calendar called the Martian Business Calendar. The Martian Business Calendar has these features:

It is similar in structure to the Darian calendar, upon which it is based. The major differences between the two calendars are the lengths of the months, the number of intercalary days, and the method used to determine long years.

Terminology

The terminology used in this document is described here.

Leap years and common years are features of Earth-based calendars. Leap years receive their name from the two-day “leap” that occurs in dates after the intercalation is performed. Because the behaviour of a Martian calendar after intercalation will be different, alternative terminology for year types should be considered. The terms “long year” and “short year” are language-independent terms that are easier for new settlers to understand.

Martian Time Periods

A good estimate for the Martian mean solar day is 24 hours 39 minutes 35.24409 seconds. This is not the length of time it takes Mars to rotate on its axis, which is a few minutes shorter, but the mean time taken for successive passages of the Sun across the sky. Because this day is close to the length of an Earth day, it can be used as a basis for timekeeping.

Mars has two satellites, Phobos and Deimos. Both orbit Mars so quickly that their periods are not suitable for timekeeping. Therefore, if a month is to be employed, it will be an artificial month that has no relationship to the motion of Mars’ moons.

The Martian mean tropical year is presently about 668.5921 mean Martian solar days. The Martian Business Calendar is based on the length of the mean tropical year rather than the tropical year for a particular equinox or solstice because the mean tropical year would vary less over centuries. Mars has an eccentric orbit which causes the lengths of tropical years tied to a particular equinox or solstice to vary considerably. If a particular equinox or solstice was used as the basis for the tropical year, over timescales of centuries the variation in year length that would result would be quite pronounced, which would result in many adjustments to keep the year tied to the chosen equinox or solstice. Fixing the calendar to the mean tropical year would reduce the number of adjustments that would be needed to the calendar.

Possible Year Lengths

What periods would a year have if its average length was approximately 668.5921 days? Let's assume that a non-Gregorian long year rule is to be used with the year length approximated by suitable fractions instead of intricate mechanisms resembling the Gregorian calendar. The timing of intercalation in the Hebrew calendar is determined by fractional approximations, and a similar method is employed here.

If the year was constructed so that the year length varied in length by 1 day, then a quick analysis shows that the fractional part of the year length – 0.5921 – lies between the decimal expansions for the fractions 4/7 (0.571428) and 3/5 (0.6000). Indeed, when the length of the year is approximated by the fraction 45/76 (0.592105), the long year pattern that is seen will consist of periods of 3 long years in every 5, broken by the occasional period of 4 long years in 7. The cycles occur in the pattern: 4×5 1×7; 3×5 1×7; 4×5 1×7, for a total of 45 long years in 76 years. Note that 0.5921 × 76 = 44.9996.

The problem with year lengths of 668 and 669 days is that neither 668 or 669 are divisible by seven, so each year must begin on a different weekday. This makes the calendar have 14 different arrangements, like the Gregorian calendar. The problem with so many different arrangements is that a printed diary or calendar for the current year that remains unsold cannot be used again for an average of 14 years. On Earth, it is common practice to discard or recycle unused calendars, thus wasting resources. On Mars, resources may be in short supply, so such waste must be minimised.

A better arrangement can be made if the number of possible calendars was reduced, creating a perpetual calendar. Many proposed calendars for Mars do this by having only two arrangements and intercalate single days, with year lengths of 668 and 669 days. However, neither 668 nor 669 are divisible by seven, so the calendars must interrupt the cycle of the seven-day week. Interruptions to the seven-day week are unacceptable to Jews, Christians and Muslims because the seven-day week must remain uninterrupted for religious reasons.

There is an alternative that permits a perpetual calendar with only two arrangements without interrupting the seven-day week. If the calendar was arranged so that each year had a whole number of weeks, then the number of possible calendars would be reduced from 14 to two. Thus we consider a calendar which uses an intercalary week instead of an intercalary day.

For calendars that use an intercalary day, the fractional number of days that is used as a base for intercalation is simply the value after the decimal point when the length of the mean tropical year is expressed in days. For the Martian mean tropical year, that value is 0.5921.

For calendars that employ an intercalary week, to determine the length of the year, one must divide the length of the mean tropical year by seven before one can use the digits after the decimal point. Thus, 668.5921 / 7 = 95.5131571. This value can be approximated by 95+3976 (95.5131579), thus there would be 39 intercalary weeks in 76 years. 39/76 is equal to 12 + 176, so there would be about one intercalary week every two years with one extra intercalary week during a 76-year cycle.

Intercalary weeks may seem to be an odd way of managing a calendar, but intercalary periods longer than a day already exist in some calendars currently in use.

The acceptance of an intercalary period that is longer than a day is therefore a cultural matter and of no real consequence.

The Calendar

The calendar is very simple. There are 24 months in the year, and all months are arranged as follows.

Martian month
Week Mo Tu We Th Fr Sa Su
Varies 1 2 3 4 5 6 7
Varies 8 91011121314
Varies15161718192021
Varies22232425262728

Each week of the month also has a week number, in a similar manner to the standard weeks as specified in ISO 8601. This makes business planning easier, because one can determine the number of weeks between two dates by subtracting one week number from the other, rather than counting by hand. In addition to the month and day of the month, Martian dates can be specified by week number and day of the week because a particular date can always be converted to the same week number and day of the week. For example, the first day of the first month is always Monday in week 1.

The only exception to the 28-day month is the 24th month in short years. It has only three weeks, and is arranged as follows.

24th Martian month in short years
Week Mo Tu We Th Fr Sa Su
93 1 2 3 4 5 6 7
94 8 91011121314
9515161718192021

In short years, anniversaries of events that occured during an intercalary week (days 22 to 28 in the 24th month) would instead be observed on the corresponding day of the week during the last week of the year (days 15 to 21 of the 24th month).

Anniversaries of events that occurred in the last week of long years
  Week Mo Tu We Th Fr Sa Su
Last week of long years9622232425262728
maps to 
Last week of short years9515161718192021

This system of mapping dates is a little unusual, but makes sense when one considers that the days in the last week of the year can also be expressed as occurring a fixed number of days before the start of the following year. It is similar to the system of dates employed in the old Roman calendar. The old Roman calendar did not give each day a separate number as we do today, but instead gave three days of each month special names (Kalendae, Nōnae and Idūs), and then expressed the intervening dates as the number of days until the next named day. Kalendae were the first day of the month, Nōnae were the 5th or 7th day of the month, and Idūs were the 13th or 15th day of the month. For example, the date corresponding to our modern date of April 28 would be called “ANTE DIEM IV KALENDAS MAIVM” (A.D. IV KAL. MAI.), or “four days before the first of May”. The Romans counted four days not three because the Romans counted both days to determine the interval.

Wikipedia has more information about the Roman calendar.

Month Names

Two systems for the naming of the 24 months are presented here.

Numbered months

The names of the months are not defined, so each month only has a number. Giving each month a separate name is not a custom that occurs in all cultures. Instead, each month is referenced by number rather than by name, such as 1st Month, 2nd Month and so on. Such a system makes the month names easy to remember.

This system is not unusual. The names of the months in the Gregorian calendar as used in China, Japan and other similar places use a system with the number of the month combined with the character for “moon” (1, 2, 3 and so on, as represented by the Unicode characters ㋀ “” to ㋋ “”). It is also the origin of the Latin names for the months September (seventh month), October (eighth month), November (ninth month) and December (tenth month), as well as the old Latin names for the months of July (Quintilis, fifth month) and August (Sextilis, sixth month).

If the months remain unnamed and future settlers of Mars wish to give the months names, then the unnamed months gives them a blank canvas upon which they can paint their own visions.

Constellations

An alternative to the simple numbering of months is the naming of months after constellations. Others have done this with their own 24-month Martian calendars; one example is this one (Woods). I don’t agree fully with the two lists given because some of the constellations seem out of place from the strict seasonal progression as observed from the Martian equator. (Perhaps the creators were northern-hemisphere residents who incorrectly assumed that northern hemisphere seasonal conventions are laws of nature?) The twelve Zodiac constellations stand out as obvious candidates, and I have selected another 12 suitable constellations to complete a list of 24. The constellation month names with my version of this system would be:

  1. Sagittarius
  2. Aquila
  3. Capricornus
  4. Grus
  5. Aquarius
  6. Pegasus
  1. Pisces
  2. Cetus
  3. Aries
  4. Perseus
  5. Taurus
  6. Orion
  1. Gemini
  2. Canis Major
  3. Cancer
  4. Ursa Major
  5. Leo
  6. Corvus
  1. Virgo
  2. Boötes
  3. Libra
  4. Centaurus
  5. Scorpius
  6. Ophiuchus

Unabridged calendars can be found here with numbered months and here with constellation month names.

Determination of Intercalary Weeks

The number of long years every 76 years is 39. The intercalation of intercalary weeks is determined by using the following formula from here:

  1. (Year × Δ + ε ) mod C = R
  2. If R < Δ then the year has additional intercalary days

For the Martian Business Calendar, the values to use in the formula are:

Inserting the values into the formula, we get:

The zero in the above formula is an offset that controls where the cycle begins. Years are numbered from zero, so the offset is zero. It has been stated explicitly for reasons that will be explained shortly, but for general calendar calculations it can be omitted.

In a 76-year cycle, there will be periods of 39 years when the even-numbered years are long years, followed by periods of 37 years when the odd-numbered years are long years. During these half-cycles, the long years and short years alternate.

Future intercalation

The intercalation for the Martian Business Calendar is based on the assumption that the mean tropical year has a length of approximately 668.5921 Martian solar days. However, the Martian mean tropical year is not constant because Mars, like Earth, experiences tidal drag that gradually lengthens the rotation period of the planet. This lengthening of the mean solar day will cause the calendar to drift out of sync with the seasons. Eventually it will be necessary to adjust the intercalation.

With the intercalation formula shown above, all that is required to adjust the calendar to a new year length is to replace the three constants in the above formula (39, 0 and 76) with new constants that reflect the new length of the year. The offset, which in the above formula is zero, would be carefully chosen so that the overlap between the old rules and the new rules is as great as possible. Implementing the rules that applied in the past would be done simply by referencing the old values in a table and using the common algorithm.

Gregorian-style rules

Rules resembling the rules of the Gregorian calendar were rejected for the Martian Business Calendar because Gregorian-style intercalation rules are prone to errors of various kinds.

Gregorian-style intercalation rules produce larger errors.
Intercalating more regularly reduces errors.

The first class of errors are caused by the incorrect application of the intercalation rules. Examples of such errors in the Gregorian calendar are forgetting that century years are not leap years, or that century years divisible by 400 are leap years. Implementation errors of this kind were one of the problems identified with the Y2K bug. Such errors are compounded when it is necessary to make an adjustment to the intercalation rules. The accumulation of such rules over time would leave a calendar with a lot of complex scar tissue.

By contrast, the intercalation of the Martian Business Calendar would be computed using a single formula, and then the intercalation can be computed from tables. These tables would have the starting year and the three constants to use in the formula. When the intercalation changes, all that is necessary is to update the table with the new intercalation constants.

Another class of error is the difference between the astronomical date and the calendar date. When the intercalation is performed according to Gregorian-style rules, the error range of a Gregorian calendar is increased by almost a full intercalation period over the course of a complete intercalation cycle. With the Gregorian calendar, where one day is intercalated, the time of an equinox or solstice can vary by more than 2 days over a 400-year cycle.

If Gregorian-style intercalation rules were chosen, the intercalation could be performed by simply having alternating long years, with an additional long year every 76 years. This would increase the maximum error of the calendar to approximately eleven days. If the intercalation rules were more similar to those of the Gregorian calendar with rules that only come into effect at the end of a century, the errors would be larger and more irregular. Such errors are undesirable for an intercalary week calendar where the error in tracking the astronomical year already amounts to a week.

If the intercalation was based on a decimal approximation, the maximum error would be even larger and more irregular.

The formula chosen for the Martian Business Calendar performs the intercalation when it falls due, rather than postponing it as Gregorian-style rules tend to do. This minimises the error of the calendar.

Determining time of equinox or solstice

The intercalation rule has a side benefit that Gregorian-style rules lack. The approximate timing of an equinox or solstice can be determined by modifying the above formula as follows.

  1. Remainder = (Year × 39 + 0) mod 76
  2. (Remainder × 7/76) + date and time of earliest equinox or solstice = date and time of equinox or solstice

In the above formula, the fraction 7/76 is derived by dividing the number of intercalary days (7) by the length of the cycle (76). The formula cannot yield an exact value because the timing of an equinox or solstice is subject to variation, but it is good enough to calculate the date and time to within an hour. The result of the formula is in days, with the decimal part of the result being converted to a time.

Epoch

The epoch for this calendar is 1609 March 11. The exact epoch is the midnight on the Prime Meridian for Mars which occurs immediately before the Martian northward equinox that occurs on this date. This northward equinox is the last one prior to the invention of the telescope.

An alternative epoch is to set the epoch of the calendar to the same day as January 1, 1 AD in the proleptic Gregorian calendar. This would facilitate the conversion of dates.

Years are numbered from zero with the telescopic era epoch, but numbered from 1 in the Proleptic Gregorian epoch.

Further Reading