Mars Calendar Opus 7
From 1999 to about 2002, Richard Weidner's work was was located at http://wwwsisn.jpl.nasa.gov/ISSUE52/marsCal3txt.html. Other Martian time material written by Weidner was located at http://cicero.jpl.nasa.gov/~richard/Clock/index.html and http://cicero.jpl.nasa.gov/~richard/Calendar/Derivation/index.html. At the time of this writing, none of Weidner's original websites are extant, although we have reproduced one in its entirety at http://pweb.jps.net/~gangale3/weidner.htm. We found the following material on a website called The Christian Disciple Data Base. The material pertaining to Mars was interspersed with discussions of various Earth calendars, for no obvious reason and with no explanation given. No attribution to Weidner was given on this website, however it is our firm opinion that this material is entirely Weidner's work as it existed on the Jet Propulsion Laboratory's website. Because we found it interspersed with other, irrelevent material, we are hestitant to say that the following material represents the complete work that existed on Weidner's "Mars Calendar Opus 7" website. Rather, the material should be considered fragmentary, and the order of the fragments in this reconstruction may not reflect the order of the material on Weidner's original JPL website.
To the archaeologist's and archivist's eyes, the Weidneresque inclusions on a site professing to be a Christiancentered database represent various abrupt manuports or disturbances in the provenience (or provenance) of the other material that hangs together. That Weidner's websites are no longer extant presents us with an intriguing mystery, one that begs investigation by those interested in human measurement systems in extraterrestrial settings.
TG and MDR, June 29, 2003
A Mars Modified Tropical Year
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
Earthbased 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.
A Mars Tropical Year from DE405 with Mars Days
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.
A Cyclic 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).
A Keplerian Mars Tropical Year
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.
A Mars Tropical Year from DE405
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.
