Mars Calendar Opus 7

A Tropical Year is the time span between successive Vernal Equinoxes. An Equinox is the time when the Sun's apparent elevation in a planet's inertial equinoctial system is 0 or 180 degrees. A Modified Tropical Year is the time span between the Sun's apparent passage of the Prime Meridian on the days of the Vernal Equinoxes.

A planet's pole is inclined to the plane of the planets orbit around the Sun. The equatorial plane is thus at an angle to the planet's orbit plane. The angle between the position vector of the Sun and the equatorial plane is the elevation of the Sun in the planet's equatorial reference system. (We are not accounting for polar rotation or the effects of nutation and libration.) As the planet revolves in its orbit, the Sun occupies a different elevation in the sky. The position vector of the Sun relative to the center of the planet crosses the equatorial plane twice during a Tropical Year. These two equatorial plane crossings (0 and 180 degrees) are the Equinoxes.

In the Northern Hemisphere, the lowest elevation of the Sun occurs at the Winter Solstice. The highest elevation occurs at the Summer Solstice. The time when the elevation crosses the equator in its travel toward the North or Summer Solstice is the Vernal Equinox. The time it crosses the equator in its travel toward the South or Winter Solstice is the Autumnal Equinox.

The time span between successive Vernal Equinoxes changes every year. Thus, many ways have been adopted to fix the length for the purposes of a calendar. One method is to use the Mean Keplerian Model of the Earth's orbit. The Mean Keplerian Model does not give you a sufficiently accurate calendar for Mars. The deviation of the true orbit changes through time more than could be compensated with simple intercalary corrections of a calendar based on a Mean Keplerian Orbit Model.

A Mars Tropical Year can be derived from the relative position of the Sun with Respect to Mars as given in the JPL Planetary Ephemerides. Still, the numerical value of the tropical year is no more accurate than the ephemerides.

As of this date, the most accurate long term ephemerides is DE405. Having said that, it should also be added that there is no information on the accuracy of DE405. Neither JPL nor the IAU has a standard for describing the accuracy of the estimates of the position of the planets except in gross terms suitable for Earth-based observations (median seconds of arc).

The Mars Modified Tropical Year can be derived by first finding the Vernal Equinox and then finding the time of Prime Meridian passage closest to that time. Because the year is synchronized to the Prime Meridian passage, a constant day is also synchronized to the Prime Meridian and time passes with noon more or less approximating the time of Solar Zenith in the various time zones around Mars. The daily deviation will be discussed more when examining the Mars Day.

Converting the reference from Julian Days to Mars Days, the following plot shows the length of the Mars Tropical Year as a function of year with the year beginning at the time of the Solar Zenith at the Prime Meridian which is closest to the Vernal Equinox.

The Mean Modified Tropical Year is longer than 668 Mars Days in length. However the Mean Modified Tropical Year is not of interest. As you can see, the year varies widely over very short intervals.

Of more interest is a year definition that emulates the cyclic nature of the modified tropical year. Bear in mind that the calendar is based on an integral number of days with the use of intercalary days to track the true year. Such a system is specified by a Leap System.

The Mars Calendar Leap System isn't very complex. However, it is more complex than the Gregorian Leap System. In addition the cycle does not appear to finish and begin again in DE405. That is unfortunate, indicating a need for extension and revision before the end of the major cycle in 2190 CE Gregorian Calendar.

The years where the Sun crosses the Prime Meridian 669 times fall in a pattern. The pattern consists of an overall major sequence, composed of 14 minor sequences. The minor sequences fall in one of five types. The minor types are composed of sequences of five or seven years.

The smallest atomic cycle is a group of five years. The first,third, and fifth year are leap years. The next smallest atomic cycle is a group of seven years where the first, third, fifth, and seventh year are leap years. Thus a seven year cycle is composed of a five year cycle and ends with two years where the last year is a leap year.

The minor cycles are composed of four atomic cycles. Four of the minor cycles are composed of three five year cycles and one seven year cycle, in the permutations of that group (ie 7555,5755,5575,5557). The fifth minor cycle is composed of two seven year cycles and two five year cycles (7557). Thus the minor cycles are all 22 years except for the fifth cycle which is 24 years. The minor cycles are enumerated by 1: 7555, 2: 5755, 3:5575, 4: 5557, and 5: 7557.

The major cycle is 310 years long, consisting of the sequence of minor cycles, 5,4,4,3,4,3,3,2,3,2,2,2,2,1.

Finally, the leap years consist of 668 Mars Days and a leap day. A Mars Day is 88775.260726 seconds. The leap day is a Mars Day minus a leap second offset. The leap second offset is 18.591097 seconds. This definition tracks the Mars Modified Tropical Year within four seconds over the 310 years of DE405. No other leap system stays so closely to the ephemeris definition.

Bear in mind that the series does not repeat. There is no reason to believe that the series holds outside the span of 1609 to 2910 CE Gregorian Calendar. Be that as it may, exactly 10 major cycles will fit within the span of the Julian Calendar System and the start of the DE405 Ephemeris. This would arbitrarily imply that a starting Epoch for the Mars Calendar System (Opus 7) could be -4224/8/22 5:54:38.745113 CE Gregorian Proleptic Calendar. More accurately, that is Julian Date 178509.746282. That is the midnight of the day of the closest noon to the Vernal Equinox on Mars.

There is no reason to expect this number to hold with future versions of the calendar. Thus, the Mars Calendar Systems are marked with a version number (e.g. Opus 7).

The orbit of Earth is closely circular. It is possible to define a mean tropical year based on the Keplerian Model because the Equinoxes fall within a slowly changing position (angle) in the orbit. The slowly changing position is known as the Precession of the Equinoxes. Therefore there are many coincidences between the definition of Earth Calendars and the Keplerian orbit parameters of Earth. Perhaps the most evident coincidence is the definition of January 1st as an angle relative to the Keplerian orbit.

Comparison of the mean Keplerian Model with the planet positions from DE405 yields the opinion that the Keplerian Model is not sufficiently accurate to establish the Mars Tropical Year or the Calendar that would result.

The following graph shows the number of Julian days between perihelion passage for Mars. (Perihelion is the position where Mars comes closest to the Sun during its orbit.) The variation of 8 julian days over a approximately 20 orbits is difficult to compensate.

A more accurate year and day is required based on the Vernal Equinox instead of using mean Keplerian Values for an approximation to the Equinoctial Period.

JPL Ephemeris DE405 contains the major barycenters of the planets over approximately an 800 year interval centered on January 1, 2000. The Ephemeris is accessed by Julian Days. Calculating the times when the Sun crosses the equator on Mars, a series of Mars Tropical Years may be calculated. The following graph plots the length of the Mars Tropical Year from Julian Date 2305752 to 2524898.

The mean tropical year is just less than 687 Julian Days in length. However, you can tell the length of the year is definitely not polynomial in nature. The year length is apparently cyclic on a six year interval.

The duration of the year varies by less than one Julian day. Thus the Vernal Equinox period is much more stable than the interval between perihelion passages.

Still, a year based on an Earth reference does not tell you the solar cycle on Mars. A Mars Day reference is required.