THE DARIAN SYSTEM Copyright © 1986-2005 by Thomas Gangale 2.0 THE CALENDARS OF JUPITER 2.1 INTRODUCTION In the new millennium, humankind will establish permanent outposts on several planets in the Solar System. None of these prospective new residences for humankind has 24-hour days, nor do any of them have 365-day years. These are cycles that are peculiar to Earth, and as a product of billions of years of evolution on this planet, we are designed to operate by them. However, these Earthly cycles will have no physical meaning on other worlds. Inevitably, conflicts will arise between the natural cycles of the new worlds and those of human biology, and also between these new unearthly rhythms and the current timekeeping conventions of Earth. What will be the outcome of these conflicts? How will people on these other planets measure the passage of time? Will the system currently used on the old world suffice, or will the need arise for new systems of time measurement: new clocks and new calendars? The most immediate prospects for human colonization are of course the Moon and Mars. On Mars, the mean solar day (sol), being only 1.02749125 times as long than Earth's, is compatible with the human circadian rhythm. Also, the 25.19-degree inclination of Mars' axis causes the planet to undergo seasonal changes which will have a significant effect on human inhabitants. However, since a Martian year is 1.88 times as long as an Earth year, Martians seasons are much longer that those on Earth. Since the basic purpose of most Earth calendars is to reckon time in terms of the seasons, and no Earth calendar can perform this basic function for Mars, it is not at all surprising that many calendar systems have been devised for Mars. This very multiplicity of Martian calendars, most of which were developed without knowledge of other existing Martian calendars, eloquently bespeaks both the ease with which the natural cycles of Mars lend themselves to a new system of timekeeping that is compatible with human physiology and sociology, as well as the obvious need for such a new system. On the Moon, however, where the mean solar day is more than 29 times the duration of Earth's, daily human activity will probably continue to be dictated according to the 24-hour Earth day. Also, the length of the year on the Moon is obviously the same as it is on Earth. These facts would seem to militate against the adoption of a new calendar for lunar colonies, although it must be said that a lunisolar calendar, such as the Jewish or Chinese calendars, might be the optimal calendar for the Earth-Moon System as a whole, rather than the Gregorian calendar which now serves as Earth's civil calendar. Looking to other more or less Earth-like bodies in the Solar System, where astronomical cycles are compatible with human biological and social requirements they will naturally serve as the bases for units of time. Where they are completely unsuitable for regulating human activities, these alien astronomical cycles must be put aside in favor of artificial cycles that will almost certainly taken from the familiar chronometric conventions of Earth. Table 2-1 lists the orbital and rotational cycles for planet-sized bodies which may ultimately support human populations in the next century or two. I have ruled out Venus and the gas giants due to their extreme atmospheric pressures, and planets beyond Uranus because of their extreme distances from Earth. Of course, some of these candidate worlds are likely to be unsuitable for colonization for one reason or another, so that to include them in this discussion is to give them the benefit of considerable doubt. Table 2-1: Periods of Candidate Worlds (in Terrestrial Solar days) Planet Primary Year Revolution

About Primary Sidereal

Day Mean Solar

Day Mercury Sun 87.97079 87.97079 58.646225 175.93287 Moon Earth 365.24238 27.321661 27.321661 29.530680 Mars Sun 686.9710 686.9710 1.02595486 1.02749125 Io Jupiter 4332.8956 1.769138 1.769138 1.769860 Europa Jupiter 4332.8956 3.551181 3.551181 3.554094 Ganymede Jupiter 4332.8956 7.154553 7.154553 7.166386 Callisto Jupiter 4332.8956 16.689018 16.689018 16.753548 Titan Saturn 10765.9240 15.945421 15.945421 15.969072 On most of the worlds now to be considered, relating the local astronomical cycles to normal human experiences presents large problems. With the sole exception of Mars, each of these worlds is tidally coupled to its primary, the body about which it orbits. The large gravitational gradient exerted by the primary, in combination with uneven mass distribution in the secondary body, tends to lock the rotation of the secondary into the same period as that of its revolution about the primary. In other words, the heavier side of the secondary tends to sink toward the other body and come to rest in a position always facing it. The Moon is a familiar example of this; we always see the same side of the Moon from Earth. As can be seen in Table 2-1, this process has also been completed on the Galilean worlds and Titan. However, because of its highly elliptical orbit, Mercury has evolved a unique synchronism with the Sun; its period of sidereal rotation is exactly one-third of its period of revolution about the Sun, with the bizarre result that its solar day is twice as long as its solar year! The solar days on all of these tidally coupled worlds are far too long to be practical as days in the human sense. Even Io's solar day of 42 1/2 hours would be rather fatiguing for most people. As far as I know, no one has yet devised calendars for the four large moons of Jupiter. It is easy to see why. The prospect for human habitation of these world is far more remote than the colonization of Mars. Also, at first look the natural cycles of these worlds appear to be not only at odds with human biological and sociological rhythms, but at odds with each other as well. Even if there were a need for timekeeping systems base on local cycles, the implementation of four separate systems would seem to erect needless barriers to communication and commerce between four worlds so closely huddled together. However, it turns out that there is a certain harmony among the cycles of the Galileans that is not at all incompatible with the cycles of human individuals and human societies. 2.2 CIRCADS The basic natural unit of any proposed Galilean calendar must of course be the tidally coupled period of the particular world. Our first task is to divide each of these into periods more suitable to human circadian rhythms. These fractions of the local day I shall call "circads", from the Latin circa (around) and dies (day). Referring to Table 2-2, we see that on Io one has no choice but to divide the day into two circads of 21.24 hours. Similarly, on Europa, four circads of 21.32 hours yields the closest possible approximation to the terrestrial solar day. Table 2-2: Circads of the Galileans Planet Hours Per

Local Solar Day Circads Per

Local Solar Day Hours Per

Circad Io 42.47665 2 21.23833 Europa 85.29825 4 21.32456 Ganymede 171.99327 8 21.49916 Callisto 402.08515 19 21.16238 While dividing the Ganymedean day by seven would result in a circad of 25.13 hours, a period much closer to the terrestrial solar day, the circads of Io and Europa just derived have already established the precedent of a shorter period for governing human events on two of the four the Galilean worlds. It would be very desirable to have all four worlds operating according to circads of approximately the same duration, since one can easily envision a great deal of commerce between these worlds that are never separated by more than three million kilometers. Travelers between the Galilean planets could readily adjust to circads differing in duration by only a few minutes, but a variance of several hours between circads is sure to induce the ultimate in biorhythmic desynchronization; the "jet lag" we are familiar with on Earth would be minor by comparison. So alternatively, if one postulates eight circads per Ganymedean day, each period becomes 21.50 hours in duration, remarkably close to the circads of Io and Europa. Dividing the Callistan day by nineteen produces a circad of 21.16 hours, which is again very similar to the other three circads. The difference between the longest and the shortest of these circads is only twenty minutes! These fortuitous results give us some hope of devising for each world a calendar that bears a close family resemblance to the others. However, it remains to be seen how readily humans can adapt to circadian periods within these values. Before proceeding to develop the details of such calendars, however, a scheme for dividing the circads of each planet into smaller units should be addressed. In the absence of any obvious argument against it, it appears practical to simply transplant the 24-hour, 60-minute, 60-second system that we use on Earth. Keep in mind, though, that these Galilean units will be proportionally shorter in duration than their terrestrial counterparts. An alternative would be to divide the circad by powers of ten to create a "metric" time system for the Galileans. 2.3 YEARS Now let us consider the one cycle that all four Galilean worlds have in common: the orbital period of Jupiter. The Jovian year is twelve times the duration of the terrestrial year, which means that if calendar years were based on this cycle, colonists would receive their baccalaureates before the age of two, and six-year-olds would be pensioners. Such a calendar year is far removed from human traditions based on the annual cycle of Earth, and also it does not appear in any way to be a practical chronometric marker for the course of human events. Still another consideration is that the spin axes of the Galileans are inclined to Jupiter's orbit around the Sun by only three degrees, and thus these bodies undergo no significant seasonal changes in the course of a Jovian year. So let us put aside any further consideration of the Jovian year in our development of the Galilean calendars, and instead base Galilean calendar years the terrestrial year with which we are much more familiar. 2.4 WEEKS Every culture on Earth has had to invent a period of time whose length is shorter than a month but longer than a day. A society functions more efficiently if a given day out of every five or ten is set aside for bringing produce to market, social gatherings, recreation, religion, et cetera. There is no natural cycle of Earth, Moon, or Sun that satisfies this human sociological need, and across the face of the Earth and the span of history, civilizations have employed many methods for filling this void. In the Jovian System, however, there is a natural cycle corresponding to the week: the Ganymedean solar day, which is equivalent to 7.167 terrestrial solar days. Now we have divided this Ganymedean day into eight circads, so an eight-circad week is a "slam dunk" for Ganymede. Furthermore, there are exactly two Europan solar days per eight circads, and on Io four solar days per eight circads. On both of these planets, the eight-circad week would correspond to some simple multiple of the local solar day, and so as on Ganymede would have actual physical meaning. For example, at a specific location on Io the sun would always rise on the first, third, fifth, and seventh circads and set on the second, fourth, sixth, and eighth circads of the week. Likewise, at a given location on Europa the Sun would always rise on the second and sixth circads and set on the fourth and eighth circads. The phases of Jupiter would occur according to similarly simple rules, and, on Io, Europa, and Ganymede, eclipses of the Sun by Jupiter would occur on fixed circads of the week: four times per week on Io, twice a week on Europa, and once a week on Ganymede. One should bear in mind, however, that the seven-day week is so ingrained in our culture here on Earth that the two major attempts to change it, both of which were made by revolutionary governments seeking to purge society of all religious symbols, were complete failures. French revolutionaries attempted to establish a calendar containing ten-day weeks, but Napoleon Bonaparte abandoned it only thirteen years later. The early Soviet government promulgated a calendar incorporating five-day weeks, then replaced it three years later with a calendar of six-day weeks, which in turn had a life of only eight years. Meanwhile, the natural cycle of Callisto does not fall into the eight-circad scheme if we stick with circads of roughly 21.3 hours. The only way to fit Callisto into the octal circad framework is to divide its orbital period into sixteen parts, each of which would be 25.13 hours in duration. This is only a little more than an hour longer than an Earth day, but it would seem interminable to the citizens of the other three Galileans, being nearly four hours longer than their own circads. It may be more desirable to keep human circadian rhythms on the four Galileans compatible with each other than to juggle the length of the Callistan circad. Thus on Callisto, as on Earth, the week will be a purely sociological construct without astronomical relevance. It should be obvious that because the circads of each world are of slightly different lengths, the weeks on each Galilean will continually drift in and out of synchronization with each other. Because of this, it would be far too confusing to have a common set of eight names for the circads of the week. To eliminate any possible ambiguity we must have a distinct nomenclature for each planet. But the other side of the problem is this: can you imagine having to memorize 32 names for the days of the week, eight for each of the four Galileans? These names must be distinctive, yet similar enough and in some logical pattern, so that the scheme is not an onerous exercise in mnemonics. On Earth, the seven days of the week were originally named after the seven objects that could be seen moving through the sky with the naked eye: the Sun, the Moon, Mars, Mercury, Jupiter, Venus, and Saturn. In Latin these were Dies Solis, Dies Lunae, Dies Martis, Dies Mercurii, Dies Jovis, Dies Veneris, and Dies Saturni, which I have already used as the basis for naming the Martian sols of the week. I have used this same idea as a starting point for naming the circads on the calendars of Jupiter, with one named for Earth added to round out the eight-circad week. The root of these names is thus common to all four Galileans; they differ only in their prefixes, which are taken from the first letters of the names of the Galileans themselves. Table 2-3 and Figure 2-1 show the relationship between the circads of the week and the solar time of day on Io, Europa, and Ganymede. Table 2-3: Days of the Week Io Europa Ganymede Callisto Io Solis

(1st AM) Eu Solis

(1st Late Evening) Gan Solis

(After Midnight) Cal Solis Io Lunae

(1st PM) Eu Lunae

(1st Morning) Gan Lunae

(Before Dawn) Cal Lunae Io Terrae

(2nd AM) Eu Terrae

(1st Afternoon) Gan Terrae

(Early Morning) Cal Terrae Io Martis

(2nd PM) Eu Martis

(1st Early Evening) Gan Martis

(Late Morning) Cal Martis Io Mercurii

(3rd AM) Eu Mercurii

(2nd Late Evening) Gan Mercurii

(Early Afternoon) Cal Mercurii Io Jovis

(3rd PM) Eu Jovis

(2nd Morning) Gan Jovis

(Late Afternoon) Cal Jovis Io Veneris

(4th AM) Eu Veneris

(2nd Afternoon) Gan Veneris

(After Dusk) Cal Veneris Io Saturni

(4th PM) Eu Saturni

(2nd Early Evening) Gan Saturni

(Before Midnight) Cal Saturni Figure 2-1: Circads 2.5 MONTHS Since we are discussing a part of the Solar System that is nowhere near the Moon, we are not necessarily bound to adopt the month as a unit of time on any Galilean calendar, but since it is a very useful a unit of time with a long terrestrial tradition behind it, we should try to transplant it to the Jovian System if possible. Obviously, the calendars of Jupiter will have to consist not only of circads, weeks, and years, but also of some further intermediate unit as well. By multiplying the number of circads per local solar day by the number of local solar days per terrestrial year, we obtain the number of circads per year, as shown in the first column of Table 2-4. By dividing each of these numbers by twelve, the number of months in a terrestrial year, we obtain the results listed in the second column of Table 2-4. These seem like reasonable figures for the number of circads per month, being only slightly larger than the number of days per month we have on Earth, but how well would that other sociologically vital unit of time -- the week -- fit into such a month? Table 2-4: Dividing the Galilean Years (Earth-based) Planet Circads

Per Year Circads

Per 1/12 Year Circads

Per 1/13 Year Io 412.7358 34.39465 31.74890 Europa 411.0667 34.25556 31.62051 Ganymede 407.7284 33.97736 31.36372 Callisto 414.2170 34.51808 31.86285 In his reformation of the Roman calendar, Julius Caesar adopted the Egyptian four-year intercalation sequence, but he did not embrace the seven-day week that was also a feature of their calendar. Instead, he perpetuated the clumsy Roman method of dividing the months into Nones, Ides, and Calends, which consequently survived for nearly four more centuries. Meanwhile, the early Christians adopted the practice of the seven-day week from the Jews. Finally, in AD 321, Constantine I, the first Christian Roman emperor, incorporated the seven-day week into the Julian calendar. However, there was no attempt at this point to reconcile the varying lengths of the Roman months with the invariable seven-day week, so that the months and years were allowed to begin and end haphazardly on any day of the week. Ever since then, the lack of correlation between dates and days of the week has been a constant source of confusion. How many of us, without referring to a printed calendar, instantly know on what day of the week the 14th of next month will fall? We can avoid this unpleasantness in the calendars of Jupiter by making a judicious choice for the number of months in a calendar year, and arrive at a month that contains some number of circads that is a multiple of eight. We need not stick with the terrestrial convention of the twelve-month year if it turns out that some other arrangement is more convenient. In the second column of Table 2-4 we see that dividing the year into twelve parts produces months that are about 34 circads long. However, by dividing the year into thirteen months we find that such months would average just under 32 circads. A 32-circad month would contain precisely four weeks. Putting all thoughts of triskaidekaphobia aside, a thirteen-month year may seem unnatural to one who is used to the twelve-month Gregorian calendar, but every lunisolar calendar on Earth has an occasional thirteen-month year to make up for the fact that twelve lunations amount to only 355 days. In fact, the Roman calendar, on which the modern Gregorian calendar is based, was once a lunisolar calendar and had such a thirteenth month-like period -- called Mercedonius -- inserted in the month of February from time to time. Also, several proposed solar calendars, such as the International Fixed calendar, have thirteen months. The months on these Galilean calendars will always coincide to within a few circads. As with the names of the circads themselves, the names of the months can be common to all four Galileans with only variations in the prefixes to signify the specific planetary calendar. In fact, since this paper will later establish a reference point in time (epoch) such that New Year's Circad will always occur within a few terrestrial days of the beginning of the calendar year on Earth, for the sake of simplicity the names of the months on the Galilean calendars can be based on those in the Gregorian calendar, with the ancient Roman month of Mercedonius inserted between Februarius and Martius. Table 2-5: Months of the Year (Gregorian-based) Io Circads Europa Circads Ganymede Circads Callisto Circads 1 Io Januarius 32 Eu Januarius 32 Gan Januarius 32 Cal Januarius 32 2 Io Februarius 32 Eu Februarius 32 Gan Februarius 32 Cal Februarius 32 3 Io Mercedonius 32 Eu Mercedonius 32 Gan Mercedonius 32 Cal Mercedonius 32 4 Io Martius 32 Eu Martius 32 Gan Martius 32 Cal Martius 32 5 Io Aprilis 32 Eu Aprilis 32 Gan Aprilis 32 Cal Aprilis 32 6 Io Maius 32 Eu Maius 32 Gan Maius 32 Cal Maius 32 7 Io Junius 32 Eu Junius 32 Gan Junius 24 Cal Junius 32 8 Io Julius 32 Eu Julius 32 Gan Julius 32 Cal Julius 32 9 Io Augustus 32 Eu Augustus 32 Gan Augustus 32 Cal Augustus 32 10 Io September 32 Eu Septembris 32 Gan September 32 Cal September 32 11 Io October 32 Eu October 32 Gan October 32 Cal October 32 12 Io November 32 Eu November 32 Gan November 32 Cal November 32 13 Io December 24-32 Eu December 24-32 Gan December 24-32 Cal December 24-32 On Mars, both the Levitt and the Darian calendars are based on seven-sol weeks. The Levitt calendar consists of twelve months, all but four of which contain 56 sols in a 668-sol Martian year. These months comprise exactly eight seven-sol weeks, and thus each month always begins on Sunday and ends on Saturday. The remaining four months of the Levitt calendar have only 55 sols each, which of course end on Friday. Since the following sol, being the first sol of the next month, is a Sunday, the normally intervening Saturday is skipped over. Similarly, the Darian calendar is structured around a 24-month year, and in a 668-sol year all but four of these months are 28 sols long, or exactly four seven-sol weeks. As in the Levitt calendar, at the end of a short month the Darian calendar simply skips over the last sol of the week so that the next month can begin on the first sol of the week. On both the Levitt and Darian calendars, then, the last week of a short month is only six sols long, rather than seven. In constructing the calendars of Jupiter, we can follow a similar line of reasoning, but it will lead to a different conclusion due to a different set of circumstances. The standard month will contain 32 circads, but since thirteen such months would amount to a 416-circad year, whereas the actual values range from 407.728120 in the case of Ganymede to 414.216739 in the case of Callisto, some months must be shortened on all of these calendars. On Mars, where the week is an arbitrary unit of time, it was permissible to occasionally shorten the week to six sols from its standard length of seven sols, and correspondingly, a month could be short by a sol here and there. On three of the Galilean worlds, however, the eight-circad week contains some integral number of local solar days, and thus has a definite physical significance. On Callisto it makes no difference, but we could no more tamper with the length of the week on Ganymede than we could, on Earth for instance, have a quarter of a calendar day at the end of each year to account for the fractional number of days in a year. Since on Io two circads comprise a solar day, we could possibly shorten the week from eight to six circads, for the precedent of an occasional six-sol week has already been set on Mars. There is no precedent, however, for cutting a week in half, as would be necessary on Europa, where the solar day is four circads long. In the interest of maintaining as much commonality as possible among the calendars of Jupiter, the length of the eight-circad week should remain inviolate, and when a month is shortened, it must be shortened by a whole week. The result is a leap-week type of calendar. An example of a leap-week calendar for Earth is Bill Hollon's Fixed-Week calendar, while Josef Šurán designed a leap-week calendar for Mars. Reducing the length of the calendar year by one week shortens it to 408 circads from its full length of 416. According to Table 2-4, the solar years of three of the Galileans fall within this range, while the number of circads in a Ganymedean solar year is just short of 408. From this it can be seen that calendar years will range between 408 and 416 circads on Io, Europa, and Callisto. On Ganymede, however, the year will normally be 408 circads in duration, with an occasional 400-circad year inserted as necessary. On Io, Europa, and Callisto, 408-circad years, it will be the last month of the year that will be shortened to 24 circads. On Ganymede, the seventh month will always be 24 circads, and in the case of 400-circad years, the last month will also be reduced to 24 circads. Table 2-6 shows the layout of the Gregorian-based calendar year for all four Galilean moons. 2.6 INTERCALATION Although the calendar years on each of the Galilean worlds are approximately equal to the 365.242-day year of Earth, they each contain a somewhat different number of circads owing to the slightly different duration of the circads on each of these worlds. Because of this, each planet will have a different intercalation sequence. The basic ten-year intercalation scheme I have worked out for each planet is shown in Table 2-7. Further errors can be corrected by an extended intercalation scheme, shown in Table 2-8. As with other aspects of the calendars of Jupiter, the intercalation scheme has been designed for maximum commonality. Table 2-7: Ten-Year Intercalation Sequence (Gregorian-based) Years Ending In: Io Europa Ganymede Callisto 0 416 416 408 416 1 416 408 408 416 2 408 416 408 408 3 416 408 408 416 4 408 408 408 416 5 416 416 408 416 6 416 408 408 416 7 408 416 408 408 8 416 408 408 416 9 408 408 408 416 Circads Per 10 Calendar Years 4128 4112 4080 4144 Circads Per 10 Solar Years 4127.3575 4110.6667 4077.2838 4142.1701 Residual Error 0.6425 1.3333 2.7162 1.8299 Table 2-8: Extended Intercalation Scheme (Gregorian-based) Planet Extended Intercalation Formula

(\ denotes integer division) Io 8 * [-Y\5 - (Y-3)\5 + Y\50 - Y\100 + Y\200 - Y\400 + Y\800] Europa 8 * [(Y-1)\5 + (Y-3)\5 + Y\20 - Y\40] Ganymede 8 * (Y\10 - Y\20) Callisto 8 * (-Y\40 + Y\400) 2.7 CALIBRATION The final issue to resolve is the referencing of a date and time on the prime meridian on each of these worlds to an equivalent date and time on the Meridian of Greenwich on Earth. When Jupiter is in opposition, i.e., when Earth is aligned directly between Jupiter and the Sun, the moment at which each Galilean world is observed to pass in front of the central meridian of Jupiter -- inferior conjunction -- corresponds to midnight on that moon's prime meridian. The opposition of Jupiter on 2002 January 01 was used for calibrating each Jovian calendar. Data was obtained from NASA/Ames' Jupiter Ephemeris Generator 1.2. The times in Table 2-9 have been corrected for the light time between Earth and the Galileans. Table 2-9: Inferior Solar Conjunctions of Galileans

at 2002 January 01 Jupiter Opposition (all times UTC) Planet Inferior Conjunction

Nearest Opposition Io 2001 Dec 31 16:07:45 Europa 2002 Jan 02 17:12:57 Ganymede 2002 Jan 01 11:08:29 Callisto 2001 Dec 28 12:27:23 Since this Jupiter opposition occurred near the beginning of the Gregorian calendar year, setting these epochs to Galilean calendar dates is a trivial case. All of the dates and times in Table 2-9 correspond to 2002 January 01 00:00:00 in their respective timekeeping systems. 2.8 VARIATIONS ON A MARTIAN THEME The discussion above tailors Earth's Gregorian calendar to the cycles of the Galileans. However, Mars will be colonized before the Galileans, and Mars is generally closer to Jupiter than Earth is. Furthermore, the Jovian System may have natural resources that will become important to the Martian economy. Questions arises as to whether settlers of the Jovian System will come primarily from Earth or from Mars, and whether the Jovian economy will have stronger ties to Earth or to Mars (Isaac Asimov dealt with some of these issues in his short story, "The Martian Way"). If the Martian influence will predominate, it will make less sense to modify the Gregorian calendar for the Galilean worlds than to adapt Mars' Darian calendar for the Jovian System. Table 2-10 is a rephrasing of Table 2-4 in terms of the Martian vernal equinox year of 686.9711 days (or 668.5907 sols), which is the basis of the Darian calendar. It can be seen that dividing the Martian year by 24 (the number of months in the Darian calendar) once again results in Galilean months containing 32 circads, or exactly four eight-circad weeks. Thus a Darian-based system of Galilean calendars has a couple of advantages over the previously discussed Gregorian-based system. First of all, the number of months in the Darian calendar and its Galilean variants are the same, whereas in the Gregorian-based system, an thirteenth month needed to be added in order to have an integral number of weeks per month. Thus the Darian-based Galilean variants will generally be more in sync with their parent calendar. Secondly, a 24-month system is more attractive than a 13-month system because 24 is divisible by 2, 3, 4, 6, 8, and 12, while 13 is unfortunately a prime number. Table 2-10: Dividing the Galilean Years (Mars-based) Planet Circads

Per Year Circads

Per 1/24 Year Io 776.2997 32.34582 Europa 773.1603 32.21501 Ganymede 766.8814 31.95339 Callisto 779.0857 32.46190 Table 2-11 is the Darian-based analog of Table 2-5, showing the names of the Galilean months. Table 2-11: Months of the Year (Darian-based) Io Circads Europa Circads Ganymede Circads Callisto Circads 1 Io Sagittarius 32 Eu Sagittarius 32 Gan Sagittarius 32 Cal Sagittarius 32 2 Io Dhanus 32 Eu Dhanus 32 Gan Dhanus 32 Cal Dhanus 32 3 Io Capricornus 32 Eu Capricornus 32 Gan Capricornus 32 Cal Capricornus 32 4 Io Makara 32 Eu Makara 32 Gan Makara 32 Cal Makara 32 5 Io Aquarius 32 Eu Aquarius 32 Gan Aquarius 32 Cal Aquarius 32 6 Io Kumbha 32 Eu Kumbha 32 Gan Kumbha 32 Cal Kumbha 32 7 Io Pisces 32 Eu Pisces 32 Gan Pisces 32 Cal Pisces 32 8 Io Mina 32 Eu Mina 32 Gan Mina 32 Cal Mina 32 9 Io Aries 32 Eu Aries 32 Gan Aries 32 Cal Aries 32 10 Io Mesha 32 Eu Mesha 32 Gan Mesha 32 Cal Mesha 32 11 Io Taurus 32 Eu Taurus 32 Gan Taurus 32 Cal Taurus 32 12 Io Rishabha 40 Eu Rishabha 32 Gan Rishabha 32 Cal Rishabha 40 13 Io Gemini 32 Eu Gemini 32 Gan Gemini 32 Cal Gemini 32 14 Io Mithuna 32 Eu Mithuna 32 Gan Mithuna 32 Cal Mithuna 32 15 Io Cancer 32 Eu Cancer 32 Gan Cancer 32 Cal Cancer 32 16 Io Karka 32 Eu Karka 32 Gan Karka 32 Cal Karka 32 17 Io Leo 32 Eu Leo 32 Gan Leo 32 Cal Leo 32 18 Io Simha 32 Eu Simha 32 Gan Simha 32 Cal Simha 32 19 Io Virgo 32 Eu Virgo 32 Gan Virgo 32 Cal Virgo 32 20 Io Kanya 32 Eu Kanya 32 Gan Kanya 32 Cal Kanya 32 21 Io Libra 32 Eu Libra 32 Gan Libra 32 Cal Libra 32 22 Io Tula 32 Eu Tula 32 Gan Tula 32 Cal Tula 32 23 Io Scorpius 32 Eu Scorpius 32 Gan Scorpius 32 Cal Scorpius 32 24 Io Vrishika 32-40 Eu Vrishika 32-40 Gan Vrishika 24-32 Cal Vrishika 32-40

As with the Gregorian-based calendars, varying the length of the Darian-based calendars years are by multiples of eight circads. Ionian and Callistan calendar years are either 776 or 784 circads, Europan years either 768 or 776 circads and Ganymedean years either 760 or 768 circads. Twenty-four months of 32 circads each totals 768 circads. In 760-circad years the 24th month is shortened to three weeks while in 776-circad years, the 24th month is lengthened to five weeks, and in 784-circad years both the 12th and 24th months are lengthened to five weeks. Table 2-12, Table 2-13, Table 2-14, and Table 2-15 show the layout of the Darian-based calendar year for each Galilean moon. The basic ten-year intercalation scheme for each planet is shown in Table 2-16. Table 2-16: Ten-Year Intercalation Sequence (Darian-based) Years Ending In: Io Europa Ganymede Callisto 0 776 768 768 784 1 776 776 768 776 2 776 776 768 784 3 776 768 768 776 4 776 776 768 776 5 776 768 760 784 6 776 776 768 776 7 776 776 768 784 8 776 768 768 776 9 776 776 768 776 Circads Per 10 Calendar Years 7760 7728 7672 7792 Circads Per 10 Solar Years 7763.013 7731.618 7668.829 7790.872 Residual Error -3.013 -3.618 3.171 1.128

The Darian calendars are based on the Martian vernal equinox year, which is estimated to be increasing at the rate of 7.9 x 10-7 sols per Martian year (see"An Extended Intercalation Scheme"). This equates to 9.1 x 10-7 Europan and Ganymedean circads, and 9.1 x 10-7 Ionian and Callistan circads per Martian year. This steady increase in the Martian vernal equinox year causes any single intercalation formula to become inaccurate after a few thousand years. Thus a series of intercalation formulas must be applied to each moon's calendar. The series for each Galilean is given in Table 2-17. Table 2-17: Extended Intercalation Scheme (Darian-based) Planet Extended Intercalation Formula

(\ denotes integer division) Io Year 0-3200:

8 x (Y\20 - Y\40 + Y\80)



Year 3201-9600:

8 x (Y\20 - Y\80 + Y\1600)



Year 9601-10000:

8 x (Y\20 - Y\80 + Y\800) Europa Year 0-3200:

8 x [-Y\5 - (Y-3)\5 + Y\20 - Y\200]



Year 3201-8000:

8 x [-Y\5 - (Y-3)\5 + Y\20 - Y\200 + Y\1600]



Year 8001-10000:

8 x [-Y\5 - (Y-3)\5 + Y\20 - Y\200 + Y\800] Ganymede Year 0-3200:

8 x [-(Y-5)\10 - Y\20 + Y\80 - Y\400]



Year 3201-8400:

8 x [-(Y-5)\10 - Y\20 + Y\80 - Y\400 + Y\1200]



8401-10000:

8 x [-(Y-5)\10 - Y\20 + Y\80 - Y\400 + Y\800] Callisto Year 0-4800:

Y\5 + (Y-2)\5 - Y\40 + Y\80 - Y\400 + Y\1200



Year 4801-10000:

Y\5 + (Y-2)\5 - Y\40 + Y\80 - Y\400 + Y\1200

Since eight circads are alternately being added and subtracted in the intercalation formula, an excursion of eight circads or less from the actual occurrence of the Martian vernal equinox is tolerable. Figure 2-3, Figure 2-4, Figure 2-5, and Figure 2-6 show the accuracy of these intercalation formulas over 10,000 years. This assumes that the rate of increase in the vernal equinox year is constant, which 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 intercalation formulas and the figures 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. Figure 2-2: Cumulative Intercalation Error for Io Figure 2-3: Cumulative Intercalation Error for Europa Figure 2-4: Cumulative Intercalation Error for Ganymede Figure 2-5: Cumulative Intercalation Error for Callisto To calculate the Darian epoch for each Galilean, the corresponding date and time given in Table 2-9 was regressed an integral number of revolutions to arrive at the inferior conjunction (with respect to the Sun and Jupiter) occuring closest to the beginning of the Darian year on Mars, which is defined as the Martian vernal equinox of 1609 March 12 16:02:07 (000 Sagittarius 01 20:47:14 on Mars). The Darian Galilean epochs are given in Table 2-18. Table 2-18: Inferior Solar Conjunctions of Galileans

at 1609 March 12 Martian Vernal Equinox Planet Gregorian Date

(UTC) Darian Mars Date

(Airy Mean Time) Io 1609 Mar 13 05:29:26 000 Sag 02 09:52:57 Europa 1609 Mar 12 01:19:41 000 Sag 01 06:28:24 Ganymede 1609 Mar 11 09:52:12 -001 Vri 28 15:25:44 Callisto 1609 Mar 17 20:57:24 000 Sag 06 22:21:58