Planet. Space Sci., Vol. 45, No. 6. pp. 705-708, 1997
©1997 Elsevier Science Ltd
All rights reserved. Printed in Great Britain
0032-0633;97 $17.00+0.00

PH: S0032-0633(97)00033-0

A Calendar for Mars

Josef Šurán

Geodetical Observatory Pecny, Ondrejov, Czech Republic
Received 22 October 1996; accepted 16 January 1997

Abstract

In a generalized approach to calendar construction for Earth, two types of perpetual calendars (with dates fixed to the days of a week) were studied for Mars: with leap and skip days; and with leap and skip weeks. Calendars with skip days or weeks (omitted days or weeks) are preferable, because the frequency of skip years is appreciably lower than that of leap years. Unlike our terrestrial (Gregorian) calendar with a 2-parametric leap rule (periods of 4 and 400 years), a Mars calendar of comparable accuracy requires a 3-parametric rule with three periods. The rules derived possess this accuracy and represent an optimum solution. With the skip week calendar, which appears to be the best compromise for a calendar for Mars, an error of I day would occur (theoretically) in an interval > 100,000 Martian years. (However, unknown secular changes in the length of the Martian year, an inaccuracy in the adopted value of its length, and possible non-uniform rotation of Mars, may affect the calendar accuracy over such long intervals of time.) A common year would have 672 Martian days distributed into 24 months of 28 days (or 4 weeks of 7 days each). In skip years a week at the end of the twelfth month would be omitted. The above most regular arrangement of months (corresponding to 12 bi-months) and a 7 day Martian week, also offer the possibility of conveniently adapting terrestrial month and day names to the calendar of Mars. The month names could be, e.g. Januarione, Januaryide; Februarione, Februaryide, etc., and those for days, e.g. Mondim, Tuesdim, etc. ©1997 Elsevier Science Ltd


1. Introduction

During the next century a manned mission may be sent to Mars and later a permanent station may be installed there. In the still more distant future, people may try to colonize this planet which is, besides Earth, the only one in our solar system offering some chance in this respect. With a view to this possibility a question arises of a suitable calendar for people who may stay there for years, or even generations.

Though Mars exhibits some Earth-like features regarding its period of rotation, inclination of rotational axis and distinct seasons, our Earth calendar cannot be readily adapted. A different calendar is needed to conform with new parameters. The calendar in question should be as practical and accurate as calendars on Earth.

Every calendar represents natural, as a rule, astronomical cycles. As these are not formed by integral multiples of days, there is always a variety of possible solutions. The calendar chosen is therefore necessarily only a certain compromise balancing the criteria of simplicity, uniformity and accuracy.

In our era there have been efforts to replace our present Gregorian calendar with a calendar of "perpetual" form. This innovative approach suggests a standard calendar (of 364 days) constant for every year, adjusted to correct length by additional (extra-calendrical) days-the 365th and the 366th in a leap year. These (also called blank days) would not be part of the week.

Another possibility is to introduce a leap week after a term of years (to replace extra-calendrical days). In perpetual calendars the dates are fixed to the days of the week, and they would be very convenient for use. Only this form of calendar will be considered in this paper.

A perpetual calendar assumes employment of a week. As the rotation of Mars is only about 37m longer than that of the Earth, it is reasonable to employ this chronological cycle on Mars, also thereby replicating a little piece of life's rhythm from Earth. A Martian month, another possible chronological unit of measure, would be too long to be easily remembered, and a subcycle is therefore needed. (A suitable length of Martian months will be given later, when we determine the length of a Mars year of seasons.)


2. The Mars year of seasons and the Martian month

The best value of Mars' sidereal year is presently givenby the theory VSOP82 by Bretagnon (1982). It equals 686.979851902d = 686d 23h 30m 59.20s of mean solar time. To obtain the Mars year of seasons (in analogy to the terrestrial tropical year) we have to take into account the precession of the equinoxes of Mars. Taking the value of the total precession in longitude of 7.296" ▒ 0.021" yr-1 determined by Hilton (1991) for the solid-core Mars model (including planetary perturbations and effects of Mars' satellites Phobos and Deimos), the precessional value for Mars' sidereal year is 13.7227" ▒eek 0.039". The latter value yields a correction of -10m 28.48s to Mars' sidereal year; then Mars' year of seasons, YE, becomes

686 d 23h 20m 30.72s = 686.972577778d.

Davies et al. (1992) give Mars' rotation as 350.8919830° d-1. Mars' sidereal day, ds, then is 24h 37m 22.6636s, and Mars' mean solar day, dM, is (ds)2/YE = 2m 12.3831s longer, and equals

24h 39m 35.0467s = 1.027488966d.

(Fractions of seconds of time may have only computational accuracy, limited mainly by our knowledge of the precessional motion of Mars and of the rate of its rotation. They are given for numerical consistency of all constants.) Time refers to Earth time. Accordingly, Mars' year of seasons is

YE = 668.5936302 dM.

The latter value in Martian mean solar days, which could be accurate to a few seconds, will be utilized in further calculations.

A perpetual calendar requires an integral number of weeks forming a standard calendar, constant from year to year. The closest approximation of YE by an integral number of weeks gives 95 weeks (665 dM) leaving a rest to YE

r = 3.5936302 dM.

Assuming months of equal length, there are 19 months with 5 weeks of 35 dM in the standard calendar of 665 dM.

A better arrangement (considered in the calendars derived below), which is much more uniform, provides months with only 4 weeks of 28 dM. These, moreover, are closer to the length of our terrestrial months (28 dM = 28.8 d). In this arrangement there are 24 months ( 12 bi-months) in the standard year. with 23 full months of 4 weeks, and 1 month deficient (incomplete) of only 3 weeks. There is still another possibility--if the rest of the standard year to YE can attain a negative value in addition to its positive value. Completing this to 96 weeks = 672dM, it has 24 full months of 4 weeks and the rest to YE

r' = -3.4063698 dM.

This generalization leads to new kinds of calendars: with skip days or skip weeks. If |r'| < |r|, they have better chronological properties than calendars with leap days or leap weeks. The chronological unit, adjusting the length of -the standard year to YE (the day or week), with the character of the rest value adjusted (r or r'), thus generates four kinds of possible Martian calendars. They will be treated in sequence.


3. Leap day and skip day perpetual calendar

(i) Assuming a standard calendar of 95 weeks, the three days from r would be blank and the total number of days in a common year would be 668. The remaining fraction of a day, r - 3 = 0. 5936302 dM, would have to be compensated for by intercalated leap days. The blank days could be added either all at the end of the deficient month (of 3 weeks), or (perhaps more properly) placed individually between quarter-years (at the beginning of the first, second and fourth). The deficient month could be at the end of the second quarter-year and the leap day would follow its end. Thus the year would end with a full month. The following leap rule was devised:

Leap years would be even years, save multiples of 160, with additional odd leap years after each multiple of 10 (i.e. 11, 21, 31, etc.).

Deviations ─ of the calendar dates to correct dates are given by the formula

1 = (Y/2)INT+[(Y- 1)/10]INT-(Y/160)INT- Y(r-3)

(1)

where Y designates Mars years from an arbitrary origin and INT means rounding-off to an integer. ─1 is in Mars days. An error of 1 dM to mean dates occurs in an interval of approximately 8350 yrM. It is a progressive error accumulating at multiples of 160 yrM. An average length of the calendar year in the latter interval is 668.5937500 dM.

(ii) If in a standard calendar of 95 weeks as above in (i) there were four blank days (spaced through the beginning of each quarter-year), a common year would have 669 dM. The fourth blank day at the beginning of the third quarter-year would then be skipped in skip years, to compensate for the remaining fraction of a day r-4 = - 0.4063698 dM. This is smaller in absolute value than in the first proposed system (i) above, and consequently, skip years will be less frequent than leap years. The relevant skip rule would be:

Skip years would be even years, save multiples of 10, with the exception of multiples of 160 (i.e. 160, 320, 480, etc.) which would also be skip years.

Deviations ─ of calendar dates to correct dates are

2 = -(Y/2)INT + ( Y/10)INT - Y/160)INT - Y(r - 4).

(2)

The accuracy of the skip rule. as well as the value of an average calendar year, are the same as for leap days. However, the number of skip years is considerably lower than that of leap years, with a difference of 937 over a period of 5000 yrM.

Deviations ─ of the studied calendars can be generally expressed as an algebraic function

─ = f(Y,p1,p2,p3)

(3)

of Y and parameters (periods) p1, p2. and p3. In the above cases p1 = 2, p2 = 10, and p3 = 160. Should there be a better solution with different parameters in (1) or (2)? A more detailed analysis of the problem shows that this is not possible. The remaining fraction of a day to YE nears 0.5. This determines the basic frequency of leap or skip years with p1 = 2, another solution for p2 and p3 could be 9 and 55, respectively. But (not to mention the inconvenience of the rule) the resultant ─ diverge much more quickly. An error of 1 dM, would occur in an interval of only about 1400 yrM, (in a cycle of 990 yrM an average calendar year would be 668.5929293 dM). If p2 = 8 or 11, the divergence of ─ is so great that the solutions are meaningless for a calendar.

If we compare our terrestrial (Gregorian) calendar, deviations are a function of only two parameters, of p1 ( = 4) and p2 ( = 400). This is due to the fact that in this calendar a fraction of only about 0.25d, adjusted by leap days, is nearly twice as small as that for Mars. The first two terms in for Mars (with p1 and p2) provide merely an accuracy comparable to that of the Julian calendar, and a third term is needed to approximate the length of the Martian year over periods of thousands of years.


4. Leap week and skip week perpetual calendar

(iii) Let us suppose that a standard calendar would consist of 95 weeks, with one deficient month of only 3 weeks. Its length would be 365 d, and the rest of the days (r to YE) would be compensated for by a leap week. It would be inserted at the end of the deficient month, on average in 7/r = 1.95 yrM @ 2 = p1. This parameter is the same in all calendars studied. Other parameters will differ. p2 = 70, p3 = 1100 and the corresponding leap rule is:

Leap years would be even years, save multiples of 1100, with additional odd leap years after each multiple of 70 (i.e. 71, 141, 21 1. etc.).

Deviations of calendar dates to correct dates give the formula

3 = 7{(Y/2)INT+ [(Y-1)/70]INT-(Y/1100)INT} - Yr.

(4)

The rule is extremely accurate. An error of 1 dM to mean dates, accumulating at multiples of the cycle of 7700 yrM, arises (theoretically) in an interval of approximately 161,300 yrM, and an average calendar year in the cycle is 668.5936364 dM. Deviations ─ varying about zero attain a maximum absolute value of approximately 10 dM (resulting from an addition of 7 d, inserted in leap years, and of r).

There are alternative leap rules with p2 and p3, respectively: 60 and 300; 50 and 150; 30 and 50. While magnitudes of ─ approach closely those of equation (4), these leap rules are all considerably less accurate than the first. Moreover, this generates a more uniform pattern of intercalations. With alternative leap rules, an error of 1 dM to mean dates accumulates at an interval of about 3370 yrM. (They all have the same average calendar year of 668.5933333 dM)

(iv) If a standard calendar had 96 weeks, there would be 24 full months of 28 days, totalling 672 days. The surplus of days r' over YE would be compensated for by a skip week. The skip years would alternate on average in the ratio 7/|r'| = 2.055 @ 2 = p1. Other periods would be as in (iii): p2 = 70; p3 = 1100, and gives ─ the formula

4= -7{(Y/2)INT-(Y/70)INT+[(Y-1)/1100]INT}-Yr'

(5)

which corresponds to the following skip rule:

Skip years would be even years, save multiples of 70, with additional odd skip years after each multiple of 1100 (i.e. 1101, 2201, 3301. etc.).

In this case too, there are three alternative skip rules, and all skip rules have the same p2 and p3 as in (iii). Accuracy, length of an average calendar year, and magnitude of are likewise the same. The first skip rule, which is extremely accurate, is most suitable. There are also fewer skip years than there are leap years. Even if this difference is not as great as in leap and skip day calendars. it is still appreciable. There are 134 fewer skip years over a period of 5000yrM; and skip calendars are generally much more uniform.


5. Conclusions

Four kinds of possible perpetual calendars for Mars were studied with leap days and skip days, and with leap weeks and skip weeks. Unlike our terrestrial (Gregorian) calendar with its 2-parametric leap rule (of 4 and 400 yr periods), a comparable Martian calendar requires a 3-parametric rule (with three periods) to approximate the Mars year of seasons over a span of thousands of years. This is because the residual fraction of a day in the Martian year is significantly larger than (about twice) that of the Earth's calendar.

Of the calendars mentioned, those of the skip type are more suitable. They have fewer skip years (with an omitted day or week) than leap years. This is again due (as in the above-mentioned case) to (somewhat) smaller fractional values corrected for in this type of calendar. As a consequence, calendars of the leap type have more leap years than normal years, and skip type calendars are more uniform both in an overall structure and in detail.

As the period of rotation of Mars is not very different from that of the Earth, a week of 7 Martian days may also be used in Mars' calendar. However, extra-calendrical days, used in leap and skip day calendars, interrupt its flow between years and quarter-years. This chronological inconsistency does not appear with leap and skip week calendars. A Martian month's most suitable length is 28 dM which renders a standard calendar year of 24 months (of 12 bi-months). This system gives rise to their most regular arrangement. Parameters (periods) for the leap and skip type calendars are the same for the same kind of calendars (with days or weeks adjusted to Mars' year), and this is their neatly interesting property. The final leap and skip rules derived have optimum parameters.

Calendars with leap and skip week have maximum deviations to correct dates up to 10days in absolute value (owing to intercalation or omission of a week). This apparent disadvantage is balanced by an uninterrupted continuity of weeks. If we consider that Mars' seasons (due to its significant orbital eccentricity of 0.093) vary in length up to (nearly) 27 days from their average value, this difference does not appear to be very large. Nevertheless. actual dates of the seasonal points for a particular year can be easily determined by using corrections from the relevant formula for ─. A continuity of weeks has another beneficial feature. If Martian Day Numbers, analogous to terrestrial Julian dates, were also recorded on Mars from a certain epoch (for observation purposes, etc.), their simple division by 7 would determine the day of the week and thus check the date.

The skip week calendar appears to be most suitable for Mars. With the skip rule devised there is virtually no cumulative error in dates arising over a period of several tens of thousands of Martian years. Note: an actual interval of validity of the skip rule (and of other rules derived) may need to be adjusted to accommodate secular changes in the length of the Mars year and possible non-uniformity of Mars' rotation. Also, an error of 1s in the adopted value of a Mars year of seasons would cause an appreciable error in regard to intervals of ten-thousands of years. The skip week calendar has the most uniform standard year, consisting of 24 equal months, two equal half-years and four equal quarter-years. It represents the best compromise for uniformity, accuracy and simplicity.

The 12 bi-month system with a 7-day Martian week makes it sensible to introduce Earth's (Gregorian) month and day names, with pertinent adaptation to Mars' calendar. A possible naming of Martian months, because Mars has two "moons", could be: January of Phobos, January of Deimos, etc., abbreviated to January P, January D, etc. Or, they could be January A, January B, etc.; eventually, the second month, or both, could have different endings (e.g. -ione, -yide; i.e. Januarione, Januaryide; Februarione, Februaryide; Marchone, Marchide; Aprilone, Aprilide; Mayone, Mayide; Junione, Junyide; Julione, Julyide; Augustone, Augustide; Septemberone, Septemberide; Octoberone, Octoberide; Novemberone, Novemberide; Decemberone, Decemberide). The Martian days could, likewise, have an ending different from Earth's names, e.g. -im (or -am, or -ah). Their names could be: Mondim, Tuesdim, etc. (or Mondam, Tuesdam, etc.; or Mondah, Tuesdah, etc.), distinguishing them thus from the days on Earth. A Martian year could start with the beginning of winter in the Mars' northern hemisphere. All this would introduce into the Martian calendar an element of kinship with Earth's calendar, making it easier to remember. It might also remind human beings in that hostile new world that their real home was Earth.


References

Bretagnon, P. (1982) Astron. Astrophys. 114, 278-288.

Davies, M. E. et al. (1992) Report of the IAU/IAG/COSPAR, Working Group on Cartographic Coordinates and Rotational Elements of the Planets and Satellites: 1991. Celest. Mech. Dyn. Astron. 53, 377-397.

Hilton, J. L. (1991) The motion of Mars' pole. I. Rigid body precession and nutation. Astron. J. 102(4) (October), 1510-1527.