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


Copyright 1986-2005 by Thomas Gangale


Having addressed the accuracy of the Darian calendar's intercalation formulae, let's now take a look at its precision. It must be said up front that precision was not a criterion in developing the intercalation scheme. Rather, the scheme emphasizes simplicity and accuracy; an algebraic expression containing only four terms ((Y-1)\2 + Y\10 - Y\100 + Y\1000) results in an accuracy of one sol over several thousand years. Referring to Figure 8, one can see that in the year 200, the vernal equinox occurs on the numerical date 1.12673, which corresponds to early on the evening (03:02:29 to be precise) of the first sol of the calendar year, 200 Sagittarius 01. The year 200, being a centennial year, contains 668 sols, whereas the vernal equinox year is 668.5907 sols. This has the effect of causing the next vernal equinox (beginning the year 201) to occur a fraction of a sol later in the next calendar year, at 16:49:08 on 201 Sagittarius 01. The year 201, being odd-numbered, is a leap year, so the next vernal equinox (beginning the year 202) occurs earlier on the next Sagittarius 01, at 07:31:51. The year 202, being even-numbered, contains 668 sols, so the vernal equinox beginning the year 203 occurs at 21:28:18 on Sagittarius 01. The problem is that one can only either push the vernal equinox back relative to the calendar 0.5906 sols or advance it by 0.4094 (1 - 0.5906) sols. The combined effect in a two-year period is for the equinox to slip further into the calendar year by 0.1812 (0.5906 - 0.4094) sols. As a result, in the year 205 the equinox occurs at 01:38:03 on Sagittarius 02. The equinox occurs on Sagittarius 02 in the years 207 and 209 as well. At this point, three successive leap years bring the vernal equinox to the morning of Sagittarius 01 in the year 212, then the decade-long cycle begins again. In the course of a decade, the combined fluctuation in the time of the vernal equinox comes to 1.1342105 sols. This makes it impossible to keep the vernal equinox on the same date. Moreover, the result of the decennial cycle is to advance the equinox by -0.094 sols (4 * 0.5906 - 6 * 0.4094), causing it to occur progressively earlier in the calendar year. In fact, in the year 242, the vernal equinox occurs a few hours before the end of the calendar year, at 21:32:02 on 241 Vrishika 28. The year 249 is the last time in the third century that the vernal equinox occurs on Sagittarius 02, but as the century advances, the vernal equinox occurs on Vrishika 28 with increasing frequency. However, for most of the third century (75 years), the vernal equinox occurs on the first sol of the calendar year, Sagittarius 01. The vernal equinox occurs on Sagittarius 02 ten times, and on Vrishika 28 fifteen times.

Figure A1-1: Precision of the Darian Calendar in the 3rd Century

A slightly more complicated intercalation scheme can reduce the wandering of the vernal equinox across the calendar, but not by much. In the Darian intercalation scheme, three successive leap years bracket the decennial years. Using a five-year cycle, in which there are never more than two successive leap years, one can decrease the total travel of the vernal equinox relative to the calendar date. This scheme can be algebraically expressed as Y\5 + (Y-1)\5 + (Y-3)\5, i.e., the leap years are the 0th, 1st, and 3rd, compared with the Darian calendar's cycle of odd-numbered years and decennial years, (Y-1)\2 + Y\10. The total fluctuation of the vernal equinox in the quintennial intercalation scheme during the course of the century is 1.7104 sols, which is not a lot better than the total travel of 1.9789 sols to which the Darian calendar restricts the vernal equinox. This slightly better precision does not guarantee better performance in any given century however. Referring to Figure 9, it happens that the vernal equinox occurs on the first day of the calendar year only 71 times (the equinox occurs twice on Sagittarius 02 and 27 times on Vrishika 28), actually fewer times than using the Darian decennial intercalation formula.

Figure A1-2: Precision of a Five-Year Intercalation Cycle in the 3rd Century

In comparison, the vernal equinox shifts 1.4568 days in a century on the Gregorian calendar. In the 21st century, for instance (Figure 10), the vernal equinox occurs as early as 19 March and as late as 21 March.

Figure A1-3: Precision of the Gregorian Calendar in the 21st Century

Both the decennial and quintennial Martian schemes need to subtract a leap day every century. Comparing Figures 11 and 12 shows that over a 500-year period, both of the schemes allow the vernal equinox to occur on the three different calendar dates previously mentioned.

Figure A1-4: Precision of the Darian Calendar, Years 0 to 500

Figure A1-5: Precision of a Five-Year Intercalation Cycle, Years 0 to 500

The Gregorian calendar also eliminates one leap day every one hundred years, but then adds the subtracted centennial leap day ever four hundred years. As a result, the 20th century began with the vernal equinox occurring as late as 22 March, while the 21st century will end with the vernal equinox occurring as early as 19 March (Figure 13). During this period, the vernal equinox takes place on four distinct dates, with a total travel of 2.2168 days.

Figure A1-6: Precision of the Gregorian Calendar, Years 1900 to 2400

An intercalation scheme developed by Richard Weidner that guarantees the occurrence of the vernal equinox on the first sol of the calendar year is far more complex, and is probably too unwieldy to gain wide acceptance for civil use. While it is important that the seasons not drift unchecked in relation to the calendar, variations of a sol one way or the other is of little consequence to most people. Thos who need greater accuracy than that provided by a civil calendar would refer to ephemerides in any case.