THE DARIAN SYSTEM
1.0 The Darian Calendar for Mars
2.0 The Calendars of Jupiter
3.0 The Darian Calendar for Titan
Some time in the 21st century, there will be human settlements on Mars. Those pioneers will have left behind on Earth the familiar 24-hour day and the 365-day year, and they will be living and working according to the natural cycles of Mars. A work day will be 13 minutes longer than we're used to back here on Earth, but the work force on Mars will have an extra 26 minutes to show up the next morning. This is because Mars rotates a bit more slowly than Earth does. To devise a practical Martian clock, it's only necessary to take a terrestrial timepiece and slow it down sufficiently. The Martian clock therefore consists of the same units as we are used to on Earth -- 60 seconds per minute, 60 minutes per hour, and 24 hours per sol -- however, each of these Martian units of time are just slightly longer (2.7 percent) than their terrestrial counterparts. Most of the Martian clock applications on the World Wide Web use this system, although some authors, such as Bruce A. Mackenzie, have proposed Martian "metric" clocks based on powers of ten.
Figure 1-1: Length of Terrestrial and Martian Days
A much bigger difference is the length of the year on Earth and Mars. Because Mars is nearly 80 million kilometers further from the sun, it takes nearly twice as long for Mars to travel once around in its orbit. It will not seem at all odd to the Martians that ten-year-olds have the vote, or that the retirement age is 35. This difference between years on Earth and Mars will require a new calendar to mark the progress of the Martian year.
Mine is just one of several dozen calendars that have been devised for Mars. First described in a paper published in 1986, I chose to name it the Darian calendar for my son Darius. Hopefully, his generation will be the first to reach Mars.
While the Martian clock may be a "slam dunk", constructing a practical calendar for Mars is a bit more of a challenge. Astronomy tables give the length of the Martian year as 687 days. WARNING: these are Earth days, not Martian sols! The correct figure to use in expressing Martian time in consistently Martian units is 668.5907 sols per vernal equinox year (Note: earlier papers specified the 668.5921-sol tropical solar year). Now, just as Earth's Gregorian calendar uses a combination of common years of 365 days and leap years of 366 days to account for the 365.24219-day terrestrial tropical solar year, the same methodology can be applied on Mars to develop an accurate calendar. Of course, since the fractional amount of sols in a Martian vernal equinox year of 668.5907 solar days is different than in a terrestrial solar year, the sequence of common years and leap years will necessarily be different. In the Darian calendar, all even numbered years are 668 sols except for those divisible by ten. All other years are 669 sols, so that in ten calendar years there are 6,686 sols. In ten Martian tropical solar years there are 6,685.921 sols, the difference thus being -0.079 sols. A further correction is therefore needed every 100 years, and so every year divisible by 100 is 668 sols instead of 669. With this correction, there are 66,859 sols in 100 calendar years, while there are 66,859.21 sols in 100 tropical solar years. Finally, by making every year that is divisible by 500 a leap year, there are 334,296 sols in 500 calendar years, and the remaining error is only 0.05 sols. Theoretically, this error amounts to only one sol in 10,000 Martian years; however, the actual error will depend on the changes in Mars' orbital elements, rotational period, and the rate of the precession of the pole vector over this period of time.
To summarize, the intercalation formula is (Y-1)\2 + Y\10 - Y\100 + Y\500, where the backslash indicates integer division.
Figure 1-2: Length of Terrestrial and Martian Years
Since the Darian calendar year begins with the vernal equinox, the calendar should be based on the mean vernal equinox year rather than on the mean tropical year (for a discussion on the various astronomical years, see Michael Allison's "What is a 'Year' (on Earth or Mars)?"). At present, the mean vernal equinox year is 668.5907 sols; however, Allison estimates that the mean tropical year is increasing by 0.00042 sols per 1,000 Earth years, which times 1.88 (Earth years/Martian years) = 0.00079 sols per 1,000 Martian years. A simple intercalation formula of (Y-1)\2 + Y\10 - Y\100 + Y\1000 results in a mean calendar year of 668.5910 sols. This is a bit too long compared to the current mean vernal equinox year, so the calendar will slowly lose time at first. However, the rate at which the calendar loses time will slowly decrease until the mean vernal equinox year equals the mean calendar year. This will occur about the year 600 of the Telescopic period. The calendar will then begin to gain time at an ever-increasing rate as the vernal equinox year lengthens, eventually becoming off by more than one sol. Before this happens the intercalation formula will need to be changed, and as the year continues to lengthen, the intercalation formula will need to be changed again and again. The series of formulas in Table 1-1 would keep the date of the vernal equinox stable for 10,000 Martian years, as shown in Figure 1-3, assuming that the rate of increase in the vernal equinox year is constant. However, this is certainly not the case. Refinement of the intercalation series will need to await the determination of a value for the second order term for the variation of the Martian vernal equinox year. The table and figure are presented as an example of the accuracy that is achievable over long periods of time with simple formulas as our knowledge of Mars' solar orbit improves.
Range of Years
Mean Length of Calendar Year
|0-2000||(Y-1)\2 + Y\10 - Y\100 + Y\1000||668.5910 sols|
|2001-4800||(Y-1)\2 + Y\10 - Y\150||668.5933 sols|
|4801-6800||(Y-1)\2 + Y\10 - Y\200||668.5950 sols|
|6801-8400||(Y-1)\2 + Y\10 - Y\300||668.5967 sols|
|8401-10000||(Y-1)\2 + Y\10 - Y\600||668.5983 sols|
See Appendix 1 for a discussion of the precision of the 10-year intercalation cycle versus a 5-year cycle and the 4-year Gregorian cycle.
See Appendix 2 for a discussion of the variation in the length of various Martian astronomical years due to gravitational perturbations, and their affect on the dates of annual astronomical events.
Several of the calendars that have been devised for Mars stretch the months asymmetrically to reflect the changing angular velocity of Mars in its eccentric orbit around the sun. Such months would span equal arcs in Mars' orbit rather represent than equal spans of time. While this might appeal to the astronomical purist, it must be pointed out that, in all probability, as on Earth, comparatively few people on Mars will be concerned with astronomy. And let us be clear that we are discussing a civil timekeeping system, not a planetary ephemeris. No civil calendar will have the accuracy required for use by the space science community because of the need to insert leap sols. A timekeeping system is at least as much a societal construct as it is an astronomical one, and to be practical for the full spectrum of society, including those who can't program a VCR, a timekeeping system should be as simple as possible and as symmetrical as possible. As an analogy, here on Earth, few of us care, or even know, that on only four days of the year does the sun cross on the meridian at the precise moment that our clocks strike noon. We do not add or subtract minutes and seconds to our clocks throughout the year to adjust for the variable length of the solar day; rather, we set our timepieces according to the length of the mean solar day and let it go at that. Likewise, on Mars, we will find months whose lengths are nearly equal divisions of the tropical solar year to be far more useful than a system in which the longest month is nearly 50 percent longer than the shortest month. The stretched Gregorian calendar (Table 1-2) is a typical example of these wildly changing months. Imagine the difficulty of working out monthly budgets in such a variable system, or trying to remember how many sols each month contains. A mnemonic poem would be of epic length!
Since a Martian year is nearly twice as long as an Earth year, a logical approach to dividing the Martian year into smaller units is to give the calendar twice as many months. An alternative would be to maintain the division of the year by twelve and have months that are nearly twice as long as they are on Earth; however, a 24-month calendar year is more desirable for several reasons. The mean Earth month of 30.4368 days is already a familiar cycle the humans. Dividing 668.5907 by 24 results in a mean month of 27.8579 sols, or 28.6238 Earth days, so the difference between the mean Martian month and the mean Earth month would be only 6 percent. It will be much easier for humans to adjust to a slightly shorter month than to accept one that is nearly twice as long. Furthermore, although a 28-sol month has no astronomical basis on Mars, it will nevertheless be meaningful to human experience on Mars, since the statistical average of the human menstrual cycle is about 28 days. The purpose of a calendar is to mark the passage of time in human terms, so the more human factors that are designed into a calendar, the better.
In the Darian calendar, common years of 668 sols contain 20 months of 28 sols and four months of 27 sols. The 27-sol months occur at the end of each quarter. In leap years of 669 sols, the last month of the year (which also ends the fourth quarter, of course), instead of containing 27 sols, is a normal length of 28 sols. The leap sol is therefore the last sol of the year, rather than being stuck somewhere in the middle as it is on Earth's Gregorian calendar.
On the question of naming the 24 Martian months, the idea of using the names of the constellations of the zodiac naturally came to mind. Indeed, Robert Zubrin later adopted this idea for his own Martian calendar. These are the constellations through which the sun appears to pass as seen from Earth during the course of a year. This annual apparent path of the sun is called the ecliptic. Since Mars' orbit is inclined to Earth's by less than two degrees, as seen from Mars, the sun appears to pass through these same constellations along a very slightly different Martian ecliptic. There are only twelve such constellations, however, so two names must be used for each one. In the Darian calendar, twelve of the months bear the familiar Latin names of the zodiacal constellations. The names of the remaining twelve months are the Sanskrit names of these same constellations, and each appears in the calendar following its Latin counterpart. The nomenclature of the Darian calendar is thus a blend of Eastern and Western influences. Admittedly, the Sanskrit names are a bit difficult to remember, and to make it even worse, I recently changed the names of some of these months to reflect the predominant usage in the Hindu Solar calendar and in Vedic astrology.In early Roman religion Mars was the god of vegetation and fertility, and his festivals signified the return of life to the land (this was before Rome became an imperial power, and Mars the farm boy got drafted into the army). Back in that more pastoral era, Romulus chose to begin his calendar with the vernal equinox, and the first month of the year was named for Mars, the provider and protector of the Roman people. In this same vein, the Darian calendar is intended to symbolize the beginning of life on the planet named for Mars (or if ancient life once did flourish there, the return of life to Mars), and so the vernal equinox is chosen as the beginning of the Martian year. Furthermore, on Earth the vernal equinox is a standard astronomical reference point that marks the beginning of the astronomical year, and it seems reasonable to carry this idea to Mars. The present position of the Martian vernal equinox is on the western edge of the constellation of Sagittarius. The first month of the Darian calendar year is therefore named Sagittarius, and the rest follow in their appropriate order as listed in Figure 1-4.