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

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,
*Y*_{E}, becomes

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

Davies *et al.* (1992) give Mars' rotation
as 350.8919830° d^{-1}. Mars' sidereal
day, d_{s}, then is 24h 37m 22.6636s, and
Mars' mean solar day, d_{M}, is (d_{s})^{2}/*Y*_{E}
= 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

*Y*_{E}
= 668.5936302 d_{M}.

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 *Y*_{E} by an integral
number of weeks gives 95 weeks (665 d_{M})
leaving a rest to *Y*_{E}

*r* = 3.5936302 d_{M}.

Assuming months of equal length, there are 19 months with 5 weeks
of 35 d_{M} in the standard calendar of 665
d_{M}.

A better arrangement (considered in the calendars derived below),
which is much more uniform, provides months with only 4 weeks
of 28 d_{M}. These, moreover, are closer to
the length of our terrestrial months (28 d_{M}
= 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 *Y*_{E}
can attain a negative value in addition to its positive value.
Completing this to 96 weeks = 672d_{M}, it
has 24 full months of 4 weeks and the rest to *Y*_{E}

*r'* = -3.4063698 d_{M}.

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 *Y*_{E}
(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 d_{M},
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 Ä of the calendar dates to correct dates
are given by the formula

Ä_{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. Ä_{1} is in Mars days.
An error of 1 d_{M} to mean dates occurs in
an interval of approximately 8350 yr_{M}.
It is a progressive error accumulating at multiples of 160 yr_{M}.
An average length of the calendar year in the latter interval
is 668.5937500 d_{M}.

(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 d_{M}. 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 d_{M}.
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 Ä of calendar dates to correct dates are

Ä_{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 yr_{M}.

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

Ä = *f(Y,p1,p2,p3)*

(3)

of *Y* and
parameters (periods) *p*_{1},
p_{2}. and *p*_{3}.
In the above cases *p*_{1}
= 2, *p*_{2} = 10, and
*p*_{3} = 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 *Y*_{E}
nears 0.5. This determines the basic frequency of leap or skip
years with *p*_{1} = 2,
another solution for *p*_{2}
and *p*_{3} could be 9
and 55, respectively. But (not to mention the inconvenience of
the rule) the resultant Ä diverge much more quickly.
An error of 1 d_{M}, would occur in an interval
of only about 1400 yr_{M}, (in a cycle of
990 yr_{M} an average calendar year would
be 668.5929293 d_{M}). If *p*_{2}
= 8 or 11, the divergence of Ä 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 *p*_{1}
( = 4) and *p*_{2} ( =
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
*p*_{1} and *p*_{2})
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 *Y*_{E}) 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 yr_{M}
@ 2 = *p*_{1}. This parameter
is the same in all calendars studied. Other parameters will differ.
*p*_{2} = 70, *p*_{3}
= 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

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

(4)

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

There are alternative leap rules with *p*_{2}
and *p*_{3}, respectively:
60 and 300; 50 and 150; 30 and 50. While magnitudes of Ä 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 d_{M} to mean dates accumulates
at an interval of about 3370 yr_{M}. (They
all have the same average calendar year of 668.5933333 d_{M})

(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 *Y*_{E} would be compensated
for by a skip week. The skip years would alternate on average
in the ratio 7/|*r'*| = 2.055 @ 2 = *p*_{1}.
Other periods would be as in (iii): *p*_{2}
= 70; *p*_{3} = 1100, and
gives Ä the formula

Ä_{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 *p*_{2}
and *p*_{3} 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 5000yr_{M};
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 d_{M} 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 Ä. 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.