THE DARIAN SYSTEM

Abstract

Figures and Tables

1.0 The Darian Calendar for Mars

1.1 Introduction

1.2 Years

1.2.1 An Extended Intercalation Scheme

1.3 Months and Seasons

1.4 Weeks

1.4.1 The Martiana Calendar

1.5 The Telescopic Epoch

1.6 Darian-Gregorian Calendar Displays

1.7 Children and Collateral Relatives

2.0 The Calendars of Jupiter

2.1 Introduction

2.2 Circads

2.3 Years

2.4 Weeks

2.5 Months

2.6 Intercalation

2.7 Calibration

2.8 Variations on a Martian Theme

3.0 The Darian Calendar for Titan

3.1 Overview of the Darian Calendar System

3.2 Astronomical Cycles on Titan

3.3 Circads and Weeks

3.4 Months and Years

3.5 Intercalation

3.6 Calibration

4.0 Conclusion

5.0 References

Appendix 1: Intercalation Precision on Mars

Appendix 2: Perturbations of Mars

Appendix 3: Martian Daylight Time

THE DARIAN SYSTEM

Copyright 1986-2005 by Thomas Gangale

3.0 THE DARIAN CALENDAR FOR TITAN

3.1 OVERVIEW OF THE DARIAN CALENDAR SYSTEM

The Darian calendar was originally developed for Mars in 1986, then in 1998 variations were developed for Jupiter's four Galilean satellites: Io, Europa, Ganymede, and Callisto (Gangale 1986; Gangale 1998; Gangale 1999). With the Cassini orbiter well in its way to Saturn and its companion entry probe Huygens scheduled to plunge into Titan's atmosphere in November 2004, the time seems right to extend the Darian calendar system to the largest of Saturn's satellites, a planet in its own right.

The Darian calendar divides the Martian year of 687 Earth days (668.6 sols) into 24 months of 27-28 sols. The Jovian tropical year of 4332.7 Earth days is twelve times the duration of the Earth 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. This is not very useful in human terms, so it makes sense to set the Jovian calendar year equal to either the Earth year or the Martian year. The choice rests on whether one believes that the dominant cultural influence during the colonization of the Galileans will be from Earth, or as Isaac Asimov postulated, from Mars (Asimov 1952). In Gangale 1998, calendar options based on both cycles were explored.

Another issue regarding the development of a human-oriented timekeeping system for the Galileans was that, none of them have a natural unit of time remotely similar to Earth's day. On Mars, in contrast, one finds a diurnal cycle only 2.75 percent longer than Earth's, a difference that would not even be noticeable to most people. For the Galileans, then, the concept of the "circad" was introduced (from the Latin circa dies, "about a day"), representing a simple fraction of the local solar day to approximate the solar day of Earth and Mars, a cycle by which humans can awaken, perform a day's normal functions, and sleep. On Io, there are two circads per solar day, four on Europa, eight on Ganymede, and 19 on Callisto. The range in duration among the circads is remarkably small, from 21 hours 10 minutes on Callisto to 21 hours 30 minutes on Ganymede. Thus travelers from one Galilean to another could easily adjust.

The concept of the week was considered for the Galileans. For Mars, it was a straightforward proposition to import the seven-day week from Earth. For three of the four Galileans, however, the number eight assumed physical significance. On Io eight circads equated to four solar days, on Europa, eight circads marked two solar days; and on Ganymede the solar day was exactly eight circads. In all three cases, the eight-circad cycle synchronized with the local solar day, and therefore constituted a natural basis for a weekly cycle. The eight-circad week was also extended to Callisto, for although it would have no astronomical significance there, the precedent had already been set by its three sister worlds.

Considering the month as a cycle to be reconciled with both the week and the year, it turned out that 24 months fit neatly into Darian-type calendars for the Galileans, generally consisting of four weeks, or 32 circads, while in a few cases there were 40-circad months comprising five weeks. Fitting the Gregorian model into the Galilean system proved to be more problematic. For months containing an integral number of weeks, again the four-week, 32 circad month worked best, with the final month being either 24 or 32 circads for the purpose of intercalation. However, devising such a scheme to fit into Earth's year worked best if the Galilean calendars contained 13 months. This was an unattractive feature, since 13 is a prime number.

Finally, in all of the Galilean calendars, leap weeks were added rather than leap circads, since on three of those worlds the week had physical meaning.

3.2 ASTRONOMICAL CYCLES ON TITAN

Some of the same issues present themselves on considering a human-oriented timekeeping system for Titan. Saturn's orbital period is essentially irrelevant in terms of human social timekeeping needs, being nearly 29 1/2 times as long as Earth's, and spanning 15 2/3 Martian years. It is just too long! This leads one to the idea that a Titan calendar would be based either on the terrestrial year or possibly the Martian year if Martian culture turns out to be the predominant influence during the settling of Titan. Since the Darian calendar adapted more successfully to the Galileans than did the Gregorian calendar, a Darian solution for Titan will be examined here.

The first step is to calculate the length of a solar day on Titan. This must be derived from the sidereal period of Saturn's orbit, which is 29.447498 terrestrial sidereal years, and Titan's orbital period, which is 15.94542068 days (NASA 2003a; NASA 2003b). A terrestrial sidereal year is 1.0000174 Julian years:

1.0000174 (YJ/YEsid) * 365.25 (d/YJ) = 365.25646 (d/YEsid)

Saturn's sidereal year, expressed in days:

29.447498 (YEsid/YSsid) * 365.25646 (d/YEsid) = 10755.886 days (d/YSsid)

The number of Titanian revolutions in a Saturnian sidereal year is:

10755.886 (d/YSsid) / 15.94542068 (d/rT) = 674.54388 (rT/YSsid)

Note that this is the number of Titanian sidereal revolutions in a Saturnian sidereal year. Since the rotation and revolution of Titan are synchronized due to tidal forces, this is also the number of Titanian sidereal days in a Saturnian sidereal year. The number of Titanian solar days in a Saturnian sidereal year is exactly one less than this, or 673.54388 days. The length of the Titanian solar day is:

10755.886 (d/YSsid) / 673.54388 (rT/YSsid) = 15.969095 (d/rT)

3.3 CIRCADS AND WEEKS

This is far too long a period for humans to live by! The concept of the circad must be imported from the Galileans. The obvious solution is to divide the Titanian solar day into 16 circads. Each circad would therefore be 23 hours, 57 minutes, 13.11 seconds, or 0.998068439 days, hardly distinguishable from either the terrestrial day or the Martian sol.

As with the Galileans, the weekly calendar cycle must be considered in terms of the astronomical cycles of Titan. Titan makes one revolution of Saturn in 16 circads, but this is more than twice as long as the weeks on the Gregorian and Darian calendars, so it is more reasonable to define the Titanian week as half of a solar day, or eight circads. The eight-circad week also has its precedent on the Galilean calendars.

4.0 MONTHS AND YEARS

Since the purpose here is to construct a Darian calendar for Titan, the calendar year must be set to the Martian vernal equinox year of 686.9711 terrestrial solar days. The number of circads in the Martian vernal equinox year is:

686.9711 (d/YMve) * 16 (cT/rT) / 15.969095 (d/rT) = 688.3006 (cT/YMve)

Dividing the year into 24 months:

688.3006 (cT/YMve) / 24 (M/YMve) = 28.6792 (cT/M)

This is not much of a surprise, since on Mars the average month is 27.8579 sols, and the Martian sol is only 2.95 percent longer that the Titanian circad. What this means, however, is that whereas the number 32 very conveniently fit into a Martian year divided into 24 months for the Darian Galilean calendars, this same multiple of eight will not work as well for Titanian months, even though eight is the number of circads in a Titanian week.

As a result, were one to insist on an integral number of weeks in the months, a 688-circad year would contain fourteen 32-circad months and ten 24-circad months. It would be far more convenient to have months that did not vary so often and by so many circads. Since the average number of sols in a Martian month is nearly 28, the average number of circads in a Titanian month is not much more than that, and since 28 is divisible by four, weeks could be evenly bisected between months. As Table 3-1 shows, this results in a much more even arrangement of the months. All but four of the months would contain 28 circads. These are placed as the third month of each quarter, making the calendar year symmetrical.

Table 3-1
Integral-Week Months and Bisected-Week Months Compared

Month Integral-Week
Months
Bisected-Week
Months
1 Sagittarius
32
28
2 Dhanus
24
28
3 Capricornus
32
32
4 Makara
24
28
5 Aquarius
32
28
6 Kumbha
24
28
7 Pisces
32
28
8 Mina
24
28
9 Aries
32
32
10 Mesha
32
28
11 Taurus
32
28
12 Rishabha
24
28
13 Gemini
32
28
14 Mithuna
24
28
15 Cancer
32
32
16 Karka
24
28
17 Leo
32
28
18 Simha
24
28
19 Virgo
32
28
20 Kanya
24
28
21 Libra
32
32
22 Tula
32
28
23 Scorpius
32
28
24 Vrishika
24
28
Total Circads
688
688

3.5 INTERCALATION

As noted earlier, there are 688.3006 circads in the Martian vernal equinox year. Consequently, an intercalation formula is required to account for the fractional number of circads. Because the eight-circad week has astronomical relevance as one half of a Titanian solar day, consideration should be given to intercalation either on the basis of adding and subtracting weeks (eight circads) or solar days (16 circads).

In the eight-circad intercalation system, adding an entire week to the last month of the year would give 36 circads total and make it excessively long. A more symmetrical scheme would be to add half of a week each to the 12th and 24th months, making each of them 32 circads long. This makes the first and last half of the year equal in length. It turns out that a remarkably simple intercalation formula suffices. Adding eight circads every 25 years results in an average calendar year of:

(688 * 25 +8) / 25 circads = 668.3200 circads

Then, subtracting eight circads every 400 years results in an average calendar year of:

(688.3200 * 400 - 8) / 400 circads = 668.3000 circads

The complete intercalation formula is therefore:

8 * (Y\25 - Y\400)

where "\" denotes integer division.

Theoretically, the next correction would not be needed for 14,000 years:

(688.3000 * 14000 + 8) / 14000 circads = 668.3006 circads

However, the Martian vernal equinox year is not a constant value. It is estimated to be increasing at the rate of 7.9 x 10-7 sols per Martian year (Allison 2003, see"An Extended Intercalation Scheme"). This equates to 8.1 x 10-7 Titanian circads per Martian year. As Figure 3-1 shows, this increase in the length of the Martian vernal equinox year causes this intercalation formula to be in error by 6.5 circads in the year 3500, well before the theoretical point of the year 14,000 is reached. 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. In the year 3600, when the calendar has gained 6.75 circads, the intercalation formula is changed to 8 * (Y\25 - Y\600), which results in an average calendar year of 688.3067 circads. The actual Martian vernal equinox year will be 688.3033 circads at this point, so the average calendar year will be slightly too long, and the calendar will lose time until the year 7700, when the average calendar year and the Martian vernal equinox year will briefly be in balance. Beyond this date, the calendar will again begin to gain time. However, all of 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 Figure 3-1 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 3-1: Cumulative Error for Eight-Circad Intercalation

As shown in Table 3-2, there is only one type of common year and one type of leap year with respect to the week. Each begins on the first circad of the week and ends on the last circad of the week. The names of the circads and months are the same as in the Darian Galilean calendars, although the "Ti" prefix might be added as needed to distinguish Titanian dates from Martian, Ionian, Europan, Ganymedean, and Callistan dates.

With respect to the phasing of the solar day through the calendar years, the situation is a bit more complex. Because the week can be split evenly across two months, and the half-week represents one quarter of a solar day, the months can begin on the 0th, 4th, 8th, or 12th circads of the solar day cycle, which correspond to midnight, sunrise, noon, and sunset on the prime meridian, respectively. From one quarter to the next, there is a shift of half a week, or one quarter of a solar day. In a common year, these cycles return to the same phase relationship at the end of the year, for there are exactly 43 solar days, 86 weeks, and 24 months in a 688-circad year. So, if each calendar year were 688 circads long, each year would begin with these three cycles in synch. The insertion of eight circads in a leap year means that these years contain 43 1/3 solar days. As a consequence, a leap year that began at solar midnight on the prime meridian (Circad 0 of the solar cycle) ends at solar noon, and all common years following that leap year begin at noon (Circad 8 of the solar cycle). Also, a leap year that began at solar noon on the prime meridian (Circad 8 of the solar cycle) ends at midnight, and all common years following that leap year begin at midnight (Circad 0 of the solar cycle). Thus, as Table 3-3 shows, there are two types of common years and two types of leap years with respect to the solar day.

Table 3-3: Solar Cycle Phasing in the Eight-Circad System

Month Type 1
Common Years
Type 1
Leap Years
Type 2
Common Years
Type 2
Leap Years
Number
of
Circads
Circad Position
in the Solar
Day Cycle
Number
of
Circads
Circad Position
in the Solar
Day Cycle
Number
of
Circads
Circad Position
in the Solar
Day Cycle
Number
of
Circads
Circad Position
in the Solar
Day Cycle
1 Sagittarius
28
0
28
0
28
8
28
8
2 Dhanus
28
12
28
12
28
4
28
4
3 Capricornus
32
8
32
8
32
0
32
0
4 Makara
28
8
28
8
28
0
28
0
5 Aquarius
28
4
28
4
28
12
28
12
6 Kumbha
28
0
28
0
28
8
28
8
7 Pisces
28
12
28
12
28
4
28
4
8 Mina
28
8
28
8
28
0
28
0
9 Aries
32
4
32
4
32
12
32
12
10 Mesha
28
4
28
4
28
12
28
12
11 Taurus
28
0
28
0
28
8
28
8
12 Rishabha
28
12
32
12
28
4
32
4
13 Gemini
28
8
28
12
28
0
28
4
14 Mithuna
28
4
28
8
28
12
28
0
15 Cancer
32
0
32
4
32
8
32
12
16 Karka
28
0
28
4
28
8
28
12
17 Leo
28
12
28
0
28
4
28
8
18 Simha
28
8
28
12
28
0
28
4
19 Virgo
28
4
28
8
28
12
28
0
20 Kanya
28
0
28
4
28
8
28
12
21 Libra
32
12
32
0
32
4
32
8
22 Tula
28
12
28
0
28
4
28
8
23 Scorpius
28
8
28
12
28
0
28
4
24 Vrishika
28
4
32
8
28
12
32
0
Total Circads
688
 
696
 
688
 
696
 

 

The 16-circad intercalation system would eliminate this complication. Since common years would contain 43 solar days and leap years would contain 44 solar days, each would always begin and end at solar midnight on the prime meridian as shown in Table 3-4.

Table 3-4: Solar Cycle Phasing in the 16-Circad System

Month Common Years Leap Years
Number
of Circads
Circad Position
in the Solar
Day Cycle
Number
of Circads
Circad Position
in the Solar
Day Cycle
1 Sagittarius
28
0
28
0
2 Dhanus
28
12
28
12
3 Capricornus
32
8
32
8
4 Makara
28
8
28
8
5 Aquarius
28
4
28
4
6 Kumbha
28
0
32
0
7 Pisces
28
12
28
0
8 Mina
28
8
28
12
9 Aries
32
4
32
8
10 Mesha
28
4
28
8
11 Taurus
28
0
28
4
12 Rishabha
28
12
32
0
13 Gemini
28
8
28
0
14 Mithuna
28
4
28
12
15 Cancer
32
0
32
8
16 Karka
28
0
28
8
17 Leo
28
12
28
4
18 Simha
28
8
32
0
19 Virgo
28
4
28
0
20 Kanya
28
0
28
12
21 Libra
32
12
32
8
22 Tula
28
12
28
8
23 Scorpius
28
8
28
4
24 Vrishika
28
4
32
0
Total Circads
688
 
696
 

 

Half of a week each would be added to the 6th, 12th, 18th, and 24th months, making each of them 32 circads long, and making the quarters of the year equal in length. Again, a remarkably simple intercalation formula suffices. Adding 16 circads every 50 years results in an average calendar year of:

(688 * 50 + 16) / 50 circads = 668.3200 circads

Then, subtracting 16 circads every 800 years results in an average calendar year of:

(688.3200 * 800 - 16) / 800 circads = 668.3000 circads

The complete intercalation formula is therefore 16 * (Y\50 - Y\800) for the years 0 to 4800. In the year 4800, the intercalation formula would change to 16 * (Y\50 - Y\1200). Figure 3-2 shows the cumulative error over a 10,000-year period due to the current estimate of the changing length of the Martian vernal equinox year.

 

Figure 3-2
Cumulative Error for 16-Circad Intercalation

The advantage of the 16-circad intercalation system is that every year would begin at solar midnight on the prime meridian, whereas in the eight-circad system the years would alternate from beginning at solar midnight to beginning at solar noon or vice-versa whenever a leap year occurred (about every 25 years). However, it has the disadvantage of allowing the calendar to be off by as much as 16 circads from the Darian Martian calendar, and in the worse case could be off as much as 24 circads from any of the Darian Galilean calendars, which intercalate on an eight-circad system. On the whole, the eight-circad intercalation system might be the better choice for Titan as well.

3.6 CALIBRATION

An opposition of Saturn occurred on 2002 Dec 17 at 17h UTC (Astronomical Almanac 2002, A11). A superior conjunction of Titan occurred on 2002 Dec 18 at 10.7h UTC (Astronomical Almanac 2002, F45). Since this was less than a day after the opposition, the Sun, Earth, Saturn, and Titan were in near perfect alignment. At this moment, Julian Day 2452626.94583, it was solar noon on the prime meridian of Titan. On Mars, the Darian date was 209 Ari 16 14:49:38, the numerical sol of the Martian calendar year was 238.6178, and the Julian Sol (counting from the vernal equinox of the Darian Martian year 0) was 139973.6178. The approximate Julian Circad (+/-8) for Titan was:

139973.6178 * (1.02749125 sol/day) / (0.998068439 circad/day) = 144100

The approximate numerical circad (+/-8) of the Titanian calendar year was:

238.6178 * (1.02749125 sol/day) / (0.998068439 circad/day) = 246

Rounding down to the nearest integer:

246 / 8 = 30

The superior conjunction of Titan ended the 30th week of the Darian Titanian. The numerical sol of the Titanian calendar year was:

30 * 8 = 240
The Darian Titanian date was therefore 209 Ari 13, and rounding to the nearest multiple of eight, the Julian Circad was 144096. The Darian Titanian calendar year 209 began on Julian Circad:

144096 - 240 = 143856

Since this was 30 weeks before the superior conjunction, and 30 is an even number, the year 209 began at solar noon.

The Julian Day of the circad beginning the Darian Titanian year 209 was:

2452626.94583 - (240 / 0.998068439) = 2452387.40940

This was the Gregorian date 2002 Apr 22 21:49:32.

The number of weeks from Julian Circad 0 was:

143856 / 8 = 17982

Since 17981 is an odd number, and Julian Circad 143856 began at solar midnight, Julian Circad 0 began at solar noon.

According to the 8-circad intercalation formula worked out above, in which every 25th year is a leap year except those divisible by 400, there should have been eight leap years from the year 0 (which was not leap year, being counted as divisible by 400) to the year 209. Thus we would expect the number of circads from the beginning of the year 0 to the beginning of the year 209 to be:

201 * 688 + 8 * 696 = 143856

The agrees with the result obtained previously.

The Julian Day of Julian Circad 0 was:

2452387.40940 - (143856 / 0.998068439) = 2308809.27607

This was the Gregorian date 1609 Mar 15 18:37:32. In Martian terms, the date was Darian 001 Sag 04 21:22:56, Julian Sol 3.89092.