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Keeping the Earth Solar Month

People from Earth, living on Mars, would still have many sentimental ties to Earth, and they possibly would also have commercial dealings with Earth as well. While they would need a calendar that related to the days and the seasons on Mars, where they lived, they would have many emotional ties to Earth and its traditions. If the Martian calendar was built of months that matched months on Earth in length, they could celebrate many Earth holidays at about the same time as people on Earth.

As also previously noted, the Gregorian year is 365.2425 days in length, and so the average length of an Earth calendar month is 30.43685 Earth days, or 29.622485 Martian days. And the length of the Martian tropical year is 668.592018 Martian days. So, if one wished to maintain a long-term synchronicity between months on Mars and months on Earth, one could divide the Martian year into 22.570423 months. Approximately, that would lead to having a Martian year with 23 months for four years out of every seven, and with 22 months for the other three years.

If we do decide to synchronize the Martian month with the Earth month, on the basis that the Martian seasons will be of some importance, but a limited one, in Martian daily life, but Mars will remain in constant communication with the Earth, what would the Martian calendar look like as a result?

Twenty-two months of 29.622485 Martian days would be about 651.7 Martian days in length. So, let us fix that part of the Martian year at 652 days. This would amount to fourteen months of 30 days in length, and eight months of 29 days in length.

Seven years would include four intercalary months. If we kept rigidly to that simple scheme of intercalation, to remain synchronized with the Martian tropical year, these months would average 29.036 Martian days in length; to remain synchronized with the Earth calendar month they would average 29.088 Martian days in length.

In practice, it might be expected that the length of the month would be kept rigidly synchronized to the Earth month, and the length of the year would be kept rigidly synchronized to the Martian tropical year, and to do this, a more complicated scheme of intercalation than four extra months in every seven years would be used. For the time being, however, only a first approximation is presented here.

In any case, the above indicates that the intercalary month would usually be 29 days in length, only occasionally being 30 days long.

A cycle of seven years, four of which are 23 months long, and three of which are 22 months long, contains 158 months. So 42 Martian years are very nearly 79 Earth years.

To concretize my calendar proposal, here is a possible arrangement:

A cycle of seven years would consist of three regular years and four leap years, disposed as follows:

L R L R L R L

The two types of years would consist of the following months:

          REG LEA                 REG LEA
Aries      30  30    Libra         30  30
Germinal   29  29    Vendemaire    29  29
Taurus     30  30    Scorpio       30  30
Floreal    29  29    Brumaire      --  29
Gemini     30  30    Saggitarius   30  30
Prairial   30  30    Frimaire      30  30
Cancer     30  30    Capricorn     30  30
Messidor   --  29    Nivose        29  29
Leo        30  30    Aquarius      30  30
Thermidor  29  29    Pluviose      29  29
Virgo      30  30    Pisces        30  30
Fructidor  29  --    Ventose       29  29*

The calendar would begin with the Martian vernal equinox, which would, of course, have even less connection with Aries than the vernal equinox on Earth.

In the last year of the last cycle in a group of seven cycles of seven Martian years, Ventose would have 30 days instead of 29.

Since there are twelve months in the French Revolutionary Calendar, I have chosen, to treat them in a balanced way, to omit two of them in a regular year, and to include them both, but omit a different one, in a leap year. Also, I have chosen to make all the constellation months 30 day months, so in a sense I am treating all the FRC months as if they were intercalary, or at least poor cousins to the other twelve months.

Thus, the cycles would interrelate in a fashion like the following:

1) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
2) Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb
3) May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr
4) Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun
5) Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug
6) Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct

Year I
   ARI Ger TAU Flo GEM Pra CAN Mes LEO The VIR LIB
   Ven SCO Bru SAG Fri CAP Niv AQU Plu PIS Ven
Year II
                                               ARI
   Ger TAU Flo GEM Pra CAN LEO The VIR Fru LIB Ven
   SCO SAG Fri CAP Niv AQU Plu PIS Ven
Year III
                                       ARI Ger TAU
   Flo GEM Pra CAN Mes LEO The VIR LIB Ven SCO Bru
   SAG Fri CAP Niv AQU Plu PIS Ven
Year IV
                                   ARI Ger TAU Flo
   GEM Pra CAN LEO The VIR Fru LIB Ven SCO SAG Fri
   CAP Niv AQU Plu PIS Ven
Year V
                           ARI Ger TAU Flo GEM Pra
   CAN Mes LEO The VIR LIB Ven SCO Bru SAG Fri CAP
   Niv AQU Plu PIS Ven
Year VI
                       ARI Ger TAU Flo GEM Pra CAN
   LEO The VIR Fru LIB Ven SCO SAG Fri CAP Niv AQU
   Plu PIS Ven
Year VII
               ARI Ger TAU Flo GEM Pra CAN Mes LEO
   The VIR LIB Ven SCO Bru SAG Fri CAP Niv AQU Plu
   PIS Ven

Priority for the Twelvemonth

If the Earth month is the basic unit of the calendar, it may also be the case that the current twelvemonth is more important than the Martian year. In which case, it might well be appropriate to have the lengths of the months in the twelvemonth remain constant, except for a leap twelvemonth day.

Thus, one would have a calendar of Earth years, with an assignment of 23 or 22 months from that calendar to Martian years.

In that case, one would begin with the Gregorian Calendar Earth year of 365.2425 days of 86400 seconds each, and divide that by the Martian day of 88775.260726 seconds, to get the length of an Earth calendar year on Mars in days: 355.47011343 days.

So one has to omit ten days from the year, but almost half the years are leap years, instead of a quarter of the years.

This suggests an opportunity to keep the Martian week in a rough correspondence with the Earth week as well. If one were to apply the principle of an Earth calendar with a 364 day year, and occasional leap weeks, to produce a simplified Martian calendar. one could have a Martian calendar in which ten Tuesdays were omitted over the course of a year:

The normal year could look like this:

       January                February                March
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
  1  2  3  4  5  6  7             1  2  3  4                1  2  3
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    4  5  6  7  8  9 10
 15 16 -- 17 18 19 20   12 13 14 15 16 17 18   11 12 13 14 15 16 17
 21 22 23 24 25 26 27   19 20 -- 21 22 23 24   18 19 20 21 22 23 24
 28 29 30               25 26 27 28            25 26 -- 27 28 29 30

        April                   May                    June
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
  1  2  3  4  5  6  7          1  2  3  4  5                   1  2
  8  9 10 11 12 13 14    6  7 --  8  9 10 11    3  4  5  6  7  8  9
 15 16 17 18 19 20 21   12 13 14 15 16 17 18   10 11 -- 12 13 14 15
 22 23 24 25 26 27 28   19 20 21 22 23 24 25   16 17 18 19 20 21 22
 29 30                  26 27 28 29 30         23 24 25 26 27 28 29

         July                  August               September
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
  1  2  3  4  5  6  7             1  2  3  4                      1
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    2  3  4  5  6  7  8
 15 16 -- 17 18 19 20   12 13 14 15 16 17 18    9 10 11 12 13 14 15
 21 22 23 24 25 26 27   19 20 -- 21 22 23 24   16 17 18 19 20 21 22
 28 29 30               25 26 27 28 29 30      23 24 25 26 27 28 29

       October                November               December
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
  1  2 --  3  4  5  6             1  2  3  4                1  2  3
  7  8  9 10 11 12 13    5  6 --  7  8  9 10    4  5  6  7  8  9 10
 14 15 16 17 18 19 20   11 12 13 14 15 16 17   11 12 -- 13 14 15 16
 21 22 23 24 25 26 27   18 19 20 21 22 23 24   17 18 19 20 21 22 23
 28 29 30               25 26 27 28 29         24 25 26 27 28 29 30

Each year like this would be short by 1.47011343 Martian days compared to an Earth calendar year.

Now, the question comes: when a leap week is added, should it be a seven-day week, or a six-day week with an omitted Tuesday?

These two types of leap year would actually have to be mixed. In order to work out the ratio, what one has to do is look at a 364-day calendar for Earth with occasional leap weeks, to determine from Earth what the ratio of 52-week years to 53 week years is. Once that ratio is determined, allocating the left-over 1.47011343 Martian days in a year to the available 53rd weeks would determine how often seven days are added, and how often six days are added.

On a previous page, it was worked out that the average length of the year is 52.177457 weeks.

If we divide 1.47011343 Martian days by 0.177457 extra weeks per year, though, what we get is 8.284336 days per week.

Thus, instead of having two kinds of leap year, as the regular year is too short, we need to have some regular years of 355 Martian days instead of 354 Martian days.

Such a year might look like this, omitting nine Tuesdays instead of ten:

       January                February                March
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
  1  2  3  4  5  6  7             1  2  3  4                1  2  3
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    4  5  6  7  8  9 10
 15 16 -- 17 18 19 20   12 13 14 15 16 17 18   11 12 13 14 15 16 17
 21 22 23 24 25 26 27   19 20 -- 21 22 23 24   18 19 20 21 22 23 24
 28 29 30               25 26 27 28            25 26 -- 27 28 29 30

        April                   May                    June
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
  1  2  3  4  5  6  7          1  2  3  4  5                   1  2
  8  9 10 11 12 13 14    6  7 --  8  9 10 11    3  4  5  6  7  8  9
 15 16 17 18 19 20 21   12 13 14 15 16 17 18   10 11 12 13 14 15 16
 22 23 24 25 26 27 28   19 20 21 22 23 24 25   17 18 -- 19 20 21 22
 29 30                  26 27 28 29 30         23 24 25 26 27 28 29

         July                  August               September
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
  1  2  3  4  5  6  7             1  2  3  4                   1  2
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    3  4  5  6  7  8  9
 15 16 17 18 19 20 21   12 13 14 15 16 17 18   10 11 -- 12 13 14 15
 22 23 -- 24 25 26 27   19 20 21 22 23 24 25   16 17 18 19 20 21 22
 28 29 30               26 27 28 29 30         23 24 25 26 27 28 29

       October                November               December
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
  1  2  3  4  5  6  7             1  2  3  4                1  2  3
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    4  5  6  7  8  9 10
 15 16 17 18 19 20 21   12 13 14 15 16 17 18   11 12 -- 13 14 15 16
 22 23 -- 24 25 26 27   19 20 21 22 23 24 25   17 18 19 20 21 22 23
 28 29 30               26 27 28 29 30         24 25 26 27 28 29 30

And a 53-week year, then, would always omit ten Tuesdays, and never eleven Tuesdays, and it could look like this:

       January                February                March
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
  1  2  3  4  5  6  7             1  2  3  4                      1
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    2  3  4  5  6  7  8
 15 16 -- 17 18 19 20   12 13 14 15 16 17 18    9 10 11 12 13 14 15
 21 22 23 24 25 26 27   19 20 -- 21 22 23 24   16 17 18 19 20 21 22
 28 29 30               25 26 27 28 29 30      23 24 25 26 27 28 29
                                               30

        April                   May                    June
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
     1 --  2  3  4  5                1  2  3    1  2  3  4  5  6  7
  6  7  8  9 10 11 12    4  5 --  6  7  8  9    8  9 10 11 12 13 14
 13 14 15 16 17 18 19   10 11 12 13 14 15 16   15 16 -- 17 18 19 20
 20 21 22 23 24 25 26   17 18 19 20 21 22 23   21 22 23 24 25 26 27
 27 28 29 30            24 25 26 27 28 29 30   28 29 30

         July                  August               September
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
           1  2  3  4                      1          1  2  3  4  5
  5  6  7  8  9 10 11    2  3  4  5  6  7  8    6  7  8  9 10 11 12
 12 13 14 15 16 17 18    9 10 11 12 13 14 15   13 14 15 16 17 18 19
 19 20 -- 21 22 23 24   16 17 18 19 20 21 22   20 21 22 23 24 25 26
 25 26 27 28 29 30      23 24 -- 25 26 27 28   27 28 29 30
                        29 30

       October                November               December
 ====================   ====================   ====================
 SU M  TU W  TH F  SA   SU M  TU W  TH F  SA   SU M  TU W  TH F  SA
 --------------------   --------------------   --------------------
              1  2  3    1  2  3  4  5  6  7             1  2  3  4 
  4 --  5  6  7  8  9    8  9 -- 10 11 12 13    5  6  7  8  9 10 11
 10 11 12 13 14 15 16   14 15 16 17 18 19 20   12 13 -- 14 15 16 17
 17 18 19 20 21 22 23   21 22 23 24 25 26 27   18 19 20 21 22 23 24
 24 25 26 27 28 29 30   28 29 30               25 26 27 28 29 30 31

Then the basics of the calendar work out like this:

An Earth year is 355.47011343 Martian days.

The 53-week years are all 361 days long, and amount to the proportion 0.177457 of the years in the calendar.

So 355.47011343 = 0.177457 * 361 + 0.822543 * s, where s is the average length of a 52-week year. This value works out to 354.315853, and thus 0.315853 is the fraction of 52-week years that are 355 days long instead of 354 days long.

As noted previously, for determining the location of years that are 53 weeks long, one simple rule could be:

one leap year every five years                  52.2
omitting a leap year every 40 years             52.175
but putting the leap year back every 400 years  52.1775
and omitting it again every 20,000 years        52.17745

so the leap years would normally be those years ending in the digits 5 and 0.

If 31.5853% of 52-week years are 355 days long, this proportion of 52-week years, multiplied by 0.822543, gives the portion of all years that are 355 days long, rather than 361 or 354 days long. That fraction is 25.98% of all years.

The most obvious thing to do is to have 355 day long years correspond to leap years on Earth, delaying the 53-week year by one year when the two would coincide. Century years would always be leap years, unlike the case in the Gregorian calendar.

The extra 355-day year in each century could be the year ending in 02 or 98, or it could be the year ending in 50, although that would delay another 53-week year. That extra year would be omitted once every 5,000 years.

Alternatively, to distribute the 355 day years as uniformly as possible, in a normal century they could be distributed like this:

(blank) - normal 354 day year
      * - 355 day year
      X - possible 53-week, 361 day year (400 year schedule; replaced by 354 day year when not present)

   0 1 2 3 4 5 6 7 8 9
00 * X     * X     *
10 X   *     X *
20 * X     * X   *
30 X *       * X     *
40 X     *   X   *
50 * X     * X     *
60 X   *     X *
70 * X     * X   *
80 X *       * X     *
90 X     *   X   *

and in the exceptional case where the extra 355 day year is not present, every 50 centuries, the schedule would be the more regular:

   0 1 2 3 4 5 6 7 8 9
00 * X     * X     *
10 X   *     X *
20 * X     * X     *
30 X   *     X *
40 * X     * X     *
50 X   *     X *
60 * X     * X     *
70 X   *     X *
80 * X     * X     *
90 X   *     X *

Conclusions

The novel suggestions I advance here are the following, which I admit may well be rejected as impractical:

A Martian minute divided into 57 (or 92, or 43) units is not, I think, too high a price to pay to define the Martian time units very precisely in terms of the second. A Martian calendar with intercalary months, on the other hand, to tie the Martian month to the Earth month is an idea very unlikely to be adopted, but I simply present it to illustrate a possible application of the principles on which calendars are constructed.

Still, people from Earth, living on Mars, would still have many sentimental ties to Earth. While they would need a calendar that related to the days and the seasons on Mars, where they lived, they would have many emotional ties to Earth and its traditions. If the Martian calendar was built of months that matched months on Earth in length, they could celebrate many Earth holidays at the same time as people on Earth.


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