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A Martian calendar.

By Rev. George D. Lardas.

OF ALL THE other planets in the Solar System, Mars is the most earthlike; it has a day-and-night cycle similar to the Earth’s, seasons, a (thin) atmosphere, and solid ground, with strong evidence of the hidden or one-time presence of water. As such, it has seemed for many, many years the most likely place in the Solar System apart from Earth where life might be found. Indeed, not long ago, only a century past, some believed it to be a possible habitation of intelligent civilization. As such the Red Planet has commanded the interest and attention of the public at large.

Clicking an image will launch a captioned slideshow. Charts and tabular material in this essay appear as linked .pdf files.

This interest is cyclical, following the periodic launch windows from Earth dictated by the relative positions of Earth and Mars, roughly two years apart, and the arrival times at Mars nine months later. The most recent spate of interest is spurred by the launch of NASA’s SUV-sized Mars Science Laboratory, the unsuccessful launch of the Russian Fobos-Grunt Probe and, earlier this month, the successful landing of the Curiosity Rover. At present there are three active probes on the Martian surface, three in orbit around it, and several expired probes both on the surface and in orbit.

 

A long-desired goal is human presence on Mars, an event not likely to occur in the next decade or so. The distances and conditions in interplanetary space are quite challenging. Once arrived, however, any explorers will be faced with the need to keep track of time – that is to say, to keep track of daylight and night time, and of the changing seasons of the Martian year. Also, even though there are no humans on Mars now, the day and night cycle and the seasons affect all probes on the surface of Mars, and hence there is need for a practical system of timekeeping. Such a system has a demonstrated appeal: some of the NASA ground support already use local Martian time for landed probes, and at least one JPL engineer had his entire family convert to Martian time when the Curiosity Rover landed1 on Monday, 6 August 2012.

At present, there is no official system, although a day count is kept for each lander, and a local time is noted. As more probes reach Mars, a more consistent and global system seems desirable. In this article we propose a calendar for Mars, building on an analysis by Allison and McEwen2 of the orbital and rotational elements of Mars.

This is not the first attempt at a Martian calendar. Previous proposals have been made by Levitt,3 Aitken4, Robinson5, Zubrin6, and Gangale7. We believe, however, that there is room for yet another proposal, and we think that this one has certain advantages not found in the older proposals.

Note that in this article we shall follow the astronomical convention of specifying a date as year, month, day, and if greater precision is desired, the time either as a (decimal) fraction of the day, or in hours, minutes and seconds. Thus the more significant units precede the less. If we specify day of the week, we shall insert it between the day and the time.

The Martian skies.

WHILE THERE ONLY have been robotic probes on the surface of Mars, we have a fairly good idea as to what a hypothetical observer on the surface may see. There would be much that is familiar – solid ground, rocks and soil, but very dry; a day and night cycle similar to our own; seasons (but nearly twice as long as our own); and weather. The outer planets, Jupiter and Saturn would appear somewhat as they do from the Earth. But there would be much that is unfamiliar – gravity one-third the Earth’s, a thin, unbreathable atmosphere, with a pressure similar to that of Earth’s stratosphere, mostly carbon dioxide; no ozone layer to protect the surface from the Sun’s damaging ultraviolet rays; a hazardous radiation environment (little protection from solar flare particles); temperatures cold enough to precipitate carbon dioxide from the atmosphere in the winter and at night as dry ice; weather that is mostly unchangeable from day to day as in an earthly desert (mostly clear skies); some part of the year when the planet is subject to global dust storms that obscure the entire surface; caramel colored skies due to the iron oxide dust prevalent in the atmosphere; the absence of a Red Planet, which happens to be under our feet, but with a double morning or evening star – Earth and Moon seen as two stars very close together. Venus and Mercury would be harder to see, as they would be closer to the Sun than as viewed from Earth.

The most notable difference – no Moon. Instead Mars has two small satellites, Phobos and Deimos. Phobos is a potato-shaped rock about 10 miles long. It is so close to Mars that it is not visible beyond latitudes greater than 70° from the equator, and orbits the planet in about 8 hours, lapping Mars three times while Mars rotates once. As a consequence, it rises in the west, reaches the meridian in about two and a half hours, and sets in the east five hours after rising (for a point on the equator). The next rising of Phobos is about thirteen hours after it sets. Phobos appears about one-third the diameter of the Sun at meridian, and appreciably smaller near the horizon because then it is farther away.

Deimos is a little over half the diameter of Phobos, and farther from Mars. It would have almost no visible disk, appearing nearly starlike, brighter than Venus appears from the Earth. It would never appear above the horizon at latitudes greater than about 83° north or south. Deimos orbits Mars in about 30 hours, a little longer than the Martian day. Consequently, it rises in the east, but very slowly, taking about five Martian days from rising to rising. It spends about two and a half days above the horizon and slightly more below (as seen from the equator), and takes more than a day to reach the meridian after rising.

The orbital periods and phases of the Martian moons are too short for convenient timekeeping, one being less than a Martian day, and one being somewhat greater. Therefore we will not take these into account in our discussion of the calendar. However, once a calendar is established, an almanac could be computed for the rising, setting, meridian passage, and phases of Phobos and Deimos. But that is a project for another day.

The Martian day.

TIMEKEEPING,  CONVENIENTLY, MAY be divided into horology and hemerology. Horology is concerned with time as kept by clocks, i.e. with time spans less than a day. Hemerology refers to groupings of days into larger units in a calendar. There exists a general agreement on the principles governing the former: a prime meridian has been defined, and consequently a global system of local time. As for the latter, several calendars have already been proposed for Mars since the 1950s, but we believe that each of these has important shortcomings. We therefore set forth a new offering, and a justification for its principles of construction. Nevertheless, we can determine the conversions between several proposals and this one.

For astronomical purposes, the mean solar day, the average time between noon and noon on a given body in the Solar System is called the sol. This can be as long as 176 Earth days for Mercury, or as short as 10 hours for Jupiter or Saturn. Some asteroids have a sol of a little over two hours.

The Earth’s sol, or day, is defined to be exactly 86,400 seconds, or 24 hours. In this article we will mainly follow the astronomical convention of numbering the hours from 0 through 23, counting from midnight. We notice that there would never be a time marked with a leading figure of 24, since 24:00:00 would be reduced to 00:00:00, the midnight commencing the next day. Additionally, all Earth times will be at the Greenwich prime meridian unless otherwise stated; this is also called Universal Time (UT). We will not concern ourselves with the differences between the various kinds of UT (UTC, ET), so roughly speaking, UT is Greenwich Mean Time (GMT), and we will use these terms synonymously. Any events given in any other time zone will be converted to UT.

As in other ways, Mars is the most Earthlike planet in that its sol is very close to the Earth’s. The Martian mean solar day is 24 hours, 39 minutes, and 35.244 seconds of Earth time, or 1.02749125 Earth day. A Martian prime meridian has also been defined as the meridian passing through the crater Airy-0. Longitudes are measured positive eastward, from 0 to 360 degrees (“west longitude” is not used).

A Martian clock would therefore divide the sol into 24 hours of 60 minutes of 60 seconds each. Following astronomical convention, the Martian clock would be a 24-hour clock starting with 0:00:00 at midnight, with 6 hours at mean dawn, 12 hours at noon, and 18 hours at mean sunset. If Mars ever hosts human settlers, then a more familiar system of reckoning time in AM and PM can be instituted. We should note that one hour on the Martian clock is not the same as one hour of an earthly clock; and likewise for the minute and the second. The corresponding Martian units are 1.02749125 longer than the Earth’s. Nominal time zones can be defined using every 15 degrees of longitude as central meridians, and the meridians exactly halfway in between (in odd multiples of 7.5 degrees) as time zone boundaries. The 180 degree meridian would then be the Martian date line. Any deviations from regularity (as we have on Earth) can be left to the needs of hypothetical future settlers.

One author (Kim Stanley Robinson) has proposed the use by human settlers of an earthly clock reckoning time in familiar Earth units from midnight until 12 PM, and leaving the remaining 39 minutes and 35.244 seconds unmarked. He calls this pause the Martian time slip. For our purposes we will not use this system due to the inconvenience of unmarked time intervals and the difficulty of synchronizing adjacent time zones.

The Martian week.

THE WEEK REGULATES many aspects of our lives: our work week, student class schedules, days of rest and worship. Our seven-day week is very ancient, and has been woven into the fabric of our civilization. Autonomous machinery, such as Mars probes, need never take the days of the week into account, since from a physical point of view they are arbitrary and conventional. For our purposes, however, we will include a seven-day week, since calendars are human devices and serve human needs. Some proposals for Martian calendars interrupt the sequence of the days of the week once a decade, once every other year, or more often. We, however, elect to keep the sequence of weekdays uninterrupted.

Because the Martian day is not synchronous with the Earth’s, a Martian week also cannot be synchronous with the earthly week. The Earth day, or 24 hours of Earth time, is 0.973244298 sol, or 23 hours 21 minutes and 28.307 seconds of Martian time. While the sol, the mean Martian solar day is 1.02749125 Earth day, or 24 hours, 39 minutes, and 35.244 seconds of Earth time.

Let us suppose, for instance, midnight beginning Sunday at Greenwich on Earth happens to coincide with midnight at Airy-0, and we decide to synchronize our Martian clock, along with the days of the week, at that point. When it is midnight again at Greenwich beginning Monday (12 PM), it is 11:21:28 PM on Sunday at Airy-0. A short while later it is midnight beginning Monday at Airy-0, but it is Monday 12:39:35 AM at Greenwich. One Martian day later, it is midnight beginning Tuesday at Airy-0, but 1:19:10 AM at Greenwich. Martian Wednesday begins at Greenwich Wednesday 1:58:46 AM. Ten days after our synchronization, we have Martian Wednesday beginning at Greenwich Wednesday 6:35:52 AM. Thirty-seven Martian days after our synchronization point we have Airy-0 Tuesday beginning at Greenwich 12:24:44 AM Wednesday, in other words, the Earth has lapped Mars by one full day. This happens at the rate of about 1 day in 37, or five and a half weeks. After 11 weeks, the Earth has now gained two days on Mars, in nine months the Earth has gained an entire week.

One way to keep the week on both planets synchronous is to drop a day of the week on Mars every time the Earth gains a day. Such a plan would require the omission of a day of the week every 36 or 37 days in an irregular pattern, which would be difficult to mesh with a calendar. It also has the drawback of interrupting the sequence of weekdays, and thus we run into a conflict with our resolve to keep the sequence uninterrupted.

The question then arises, how should we designate the seven days of the Martian week? If we take a given point in time, we have seven choices as to which day of the week we should assign to it. One way to make the choice is to define the weekday current at Greenwich noon on some given date of synchronization to be that of the sol current at Airy-0 at the same time. The matter then comes down to the choice of the date. For our purposes, we will choose the synchronization point to be Greenwich mean noon for Julian Date (JD) 0.0, which we will now explain.

For purposes of calculation and record, astronomers use a continuous count of days from Greenwich Mean Noon BC 4713 January 1, which is JD 0.0. This continuous day count, called the Julian Day (JD), ignores weeks, months and years, and simplifies the determination of the number of days between any two events without having to take into account the varying lengths of months and leap years, and has been in common astronomical use since it was proposed by Joseph Justus Scaliger in 1583 and adopted in 1890. This day count includes decimal fractions for greater precision, and has an integer value at Greenwich Mean Moon (12 h UT). The day of the week on BC 4713 January 1 was Monday.

We will therefore define our synchronization thus: the day current at Airy-0 at JD 0.0, i.e. Greenwich Mean Noon, BC 4713 January 1 in the Martian calendar shall be Monday. The days of the week will cycle uninterruptedly from that point on.

The synchronization point is a matter of convention, and actual settlers may choose a different synchronization point, say the day of the first human landing on Mars, being the day records are first kept on the surface of that planet. Also, different religious communities may choose different synchronization points, making it possible that the holy day may fall on days differing from those we observe on Earth.

If we wish observe neutrality in committing to specific weekday names, we might instead use the Latin word feria for the weekdays, and give them numerical names: Feria Prima, Feria Secunda, Feria Tertia, and so on to Feria Septima. We could also give Feria Prima and Feria Septima the special names as found in contemporary (Vatican) Latin, namely Dominicus and Sabbatum, respectively. For the sake of familiarity, however, we will use the usual weekday names in our Martian calendar.

The Martian Orbit.

THE FOLLOWING DIAGRAM compares the orbits of Earth and Mars, plotted roughly to scale:

 

The Martian orbit. Clicking the image will yield an annotated .pdf file.

Mars’ orbit is appreciably more elliptical than the Earth’s, with an eccentricity of 0.0934, as compared to Earth’s 0.0167. The closest point in the orbit to the Sun (perihelion) is therefore 1.38137 AU (206,649,000 km, or 128,406,000 mi), while the farthest point in the orbit from the Sun (aphelion) is 1.66599 AU (249,229,000 km, or 154,864,000 mi). The major axis of the orbit, which passes from the perihelion to the aphelion is called the line of apsides.

A planets position in orbit can be defined by the angle from the previous perihelion passage. This is called the anomaly. The mean anomaly tracks the fictitious position of the planet moving around the Sun at constant average speed, while the true anomaly tracks the actual position of the planet. An intermediate point in the orbit is the latus rectum, 90 degrees true anomaly from the line of apsides (see diagram). The distance from the Sun is actually less than the semi-major axis or mean distance, partly because the planet spends more time in the half of the orbit that includes the aphelion. The orbit is related to the overall position by comparison with Earth’s orbit, particularly ecliptic longitude, or the angle from Earth’s vernal equinox to the position of the planet in its orbit. In our calendar we will wish to track the apsides (perihelion and aphelion), the latera recta (outbound and inbound), and the point of zero ecliptic longitude.

Like the Earth’s, the equator of Mars is inclined with respect to its orbit. The obliquity of the equator to the orbit is 25°.1919, quite similar to Earth’s 23°.4393. Therefore, like Earth, Mars has seasons, and to about the same degree. On Earth, the predominant effect on the annual variation in temperature is the obliquity of the equator, with the eccentricity as only a minor factor in the seasons. On Mars, however, the 18% variation in distance from the Sun, combined with the inverse square law, yields a 37% variation in solar power between perihelion and aphelion. Compare this with the Earths annual variation of 6.7%. Thus both equatorial obliquity and orbital eccentricity affect the seasons on Mars. In fact, it happens that perihelion occurs fairly close to the Northern winter solstice (i.e. the Southern summer solstice). The effect of this is to mitigate the extremes of the northern seasons, and exaggerate the extremes of the southern seasons. As a consequence of this effect, the winter polar ice cap is much larger in the southern hemisphere than in the northern.

Orbital eccentricity also affects the length of seasons. The angle between the position of a planet at any solstice and the preceding or succeeding equinox is always 90 degrees. However, a planet travels faster in its orbit when closer to the Sun, and so the seasons around perihelion are shorter than those around aphelion, and so the seasons have varying lengths. For Earth, the number of days from the (northern) vernal equinox to the summer solstice is 92.75 days, the time from the summer solstice to the autumnal equinox is 93.65 days, from the autumnal equinox to the winter solstice is 89.85 days, and from the winter solstice to the vernal equinox is 88.99 days, a total of 365.24 days in Earth’s tropical year. Since Mars’ orbit is appreciably more elliptic than Earth’s, the variation in the lengths of the seasons is greater. The corresponding (northern) seasons are: Spring 193.30 sols, Summer 178.64 sols, Autumn 142.70 sols, and Winter 153.95 sols, a total of 668.59 sols in Mars’ tropical year. In our calendar we wish to track the equinoxes and solstices.

Other regular phenomena that we might wish to track in our calendar are the signs of the zodiac, and weather phenomena on Mars. For our purposes we will assign the signs of the zodiac to thirty degree segments beginning at Mars’ northern-hemisphere vernal equinox, the names being those of the actual constellation the Sun appears in at that time of the Martian year. The first sign is Sagittarius. Note, this is the exact reverse of the scheme proposed by Zubrin, where instead of the planetocentric constellations, he takes the heliocentric ones (i.e. the position of Mars as seen from the Sun).8

Criteria.

WE WILL NOW construct a Standard Calendar for Mars to which all others we have considered shall be compared. The desiderata are as follows:

  • The basic unit of time shall be the sol. There are approximately 668.6 sols in the Martian year. For calendar purposes the sol shall be called the day, and the day shall be divided into 24 hours of 60 minutes each, and each minute of 60 seconds. Thus the Martian day, hour, minute and second will be slightly longer than their Earth equivalents.
  • We will assume an Earth-based Martian Clock synchronized to the time at the Martian Prime Meridian, Airy-0. This clock will take into account the travel time of light signals between Earth and Mars, which varies as the distance between those two bodies from as little as 6 minutes 13 seconds to 22 minutes 11 seconds. We shall ignore the negligible (but measurable) relativistic effects due to the gravities of the Earth, Mars and the Sun, and their relative motions. A clock therefore runs imperceptibly slower on Earth than the same one on Mars, but we will ignore this effect.
  • We will emphasize the earthlike nature of Mars by dividing the year into twelve months, and by using the same names of the months as in our familiar Gregorian (and Julian) Calendar. This principle was followed previously by Levitt.9
  • We shall, however, make March the first month. This serves two purposes – first, March is from the Latin Martius meaning “of Mars,” an appropriate name for the first month in a calendar for Mars. Second, the numerical month names then correspond to their correct places in the year: September will indeed be the seventh month, October the eighth, and so on.
  • The months shall be of equal length, namely 56 days, or exactly 8 weeks. As a result, every month will begin on the same day of the week, and the same calendar can be used within any given year for all the months. Since 12 months of 56 days amount to 672 days, slightly longer than the true year, we shall make February the short month (naturally!). February shall have 52 or 53 days as needed by the intercalation (leap year) scheme.
  • As with the Earth, Mars’ tropical (seasonal) year is slightly longer than the sidereal year (i.e., with respect to the stars) due to the precession of its equinoxes (the wobbling of the polar axis like a top). Our calendar will track the tropical year.
  • We desire the equinoxes to occur in March and September, and the solstices in June and December. More specifically, we will place the northern hemisphere vernal equinox (southern hemisphere autumnal equinox) in March, the northern summer (southern winter) solstice in June, the northern autumnal (southern vernal) equinox in September, and the northern winter (southern summer) solstice in December.
  • Since the seasons are of unequal length, they will not begin around the same day of the month as the seasons do on Earth (March 21, June 22, September 22, and December 22). In order to keep the beginnings of the seasons in their proper months, the vernal equinox could begin anywhere between March 1 and March 18. This puts the autumnal equinox in the range of September 37 through September 56. If we set the vernal equinox on March 19, this puts the autumnal equinox on October 1, exceeding our desired range. If we set the autumnal equinox earlier than September 37, the vernal equinox will occur on the last day of February. We will therefore define the sol that contains the northern vernal equinox at the Prime Meridian (Airy-0) between midnight and midnight to be March 1. The other seasons then begin on June 26, September 37 and December 11 or 12.
  • The sequence of the days of the week shall not be interrupted. The week shall be synchronized such that the sol current at the Martian Prime Meridian (Airy-0) at epoch JD 0.0 (4713 BC January 1 Noon GMT) is Monday (the same as in the Julian Calendar).
  • The Martian sol and the seasons, and other important orbital points, and the correlation between Earth and Mars times will be based on the mean orbital calculations given by Alison and McEwen.10
  • Due to the ellipticity of the orbit, the mean tropical year, both on Earth and on Mars, differs slightly in time from one equinox to the same equinox or from one solstice to the same solstice. This calendar defines the northern vernal equinox to occur on March 1. Therefore we will base our calendar on the vernal equinox year of 668.5908 sols rather than on the mean tropical year of 668.5921 sols. To a high degree of accuracy this can be approximated by a Martian year of 668 with an extra day added 13 times in 22 years. The average length of the calendar year is then 668 13/22 (668.5909) Martian days (sols).
  • This results in a 22-year intercalation cycle of 668 for the first year, 669 for the second year, followed by four five-year cycles of 668, 669, 668, 669, 669 days. Since the number of sols in 22 Martian calendar years is not divisible by 7, the days of the week will repeat every 7 x 22 or 154 Martian years.
  • The leap year cycle shall be adjusted so that for the years immediately surrounding the J2000 epoch, March 1 will always contain the vernal equinox at Airy-0. The error is less than 1 sol in 10,000 Martian years.
  • We set the Martian year count so that Martian Year (MY) 1 is that which contains JD 0.0 (4713 BC January 1 Noon GMT).
  • We will set up a Julian Date-Martian (JDM) including fractional sols, commencing at Airy-0 Noon of the sol that contains the Julian Date epoch, JD 0.0. As we said before, this sol shall be Monday. This corresponds to MY 1 April 47, Airy-0 Noon (102 sols after March 1).
  • For ease of calculation of the 22-year cycle, we will set up a Martian Day (MD) count in sols commencing at midnight beginning the sol MY -14 March 1. This is the Julian Date-Martian augmented by 10130.5 sol. This is JDM augmented by 0.5 to bring the zero point from noon to the previous midnight, and by 102 sols to bring the zero point to March 1, and by 10028 to bring the zero point to the beginning of the first year of the 22-year cycle, as MY 1 is the 15th year of the cycle.
  • Our calendar is proleptic, meaning that we apply it to times before it is inaugurated. We justify this by the use by astronomers and historians of the Proleptic Gregorian and Julian Calendars to times BC before either calendar was established. Therefore we have extended the calculations of dates back to the beginning of the Julian Period, BC 4713. In order to do so we have given our figures a fictitious greater precision than is warranted by the astronomical facts with the intent that our figures should match the rotation of Mars and Earth for the years around the present time. We shall thus ignore the fact that the rotation of the Earth (and Mars) is slowing down, causing the calendar day to be more and more out of step with the constant rate of rotation assumed in these calculations the further we go back in time.

The Calendar Year.

WHEN ALL THE above conditions are taken into account we can calculate the events that take place during the Martian Calendar year. The results when the vernal equinox occurs at midnight commencing 1 March – along with a 22-year intercalation cycle chart and a perpetual calendar – may be seen by clicking the perpetual calendar image below.

Clicking the image above will produce a .pdf file containing these tables: Events of the Martian calendar year, the 22-Year intercalation cycle, and a perpetual calendar for Mars.

Calendar conversion.

IN WHAT FOLLOWS, the notation int (x) means to take the whole, or integer, part of x and discard the fractional part. The notation frac (x) means to take the fractional or decimal part of x, and discard the whole, or integer, part. The notation mod (x, y) means to divide x by y, and take the remainder, including any fractional part, and discard the quotient. Thus int (3.1415) = 3; frac (3.1415) = 0.1415; and mod (23, 7) = 2, mod (23.4, 7) = 2.4.

We will use the term Old Style (OS) as synonymous with the Julian Calendar, and New Style (NS) as synonymous with the Gregorian. If a date is given, but the style is not, we will bear in mind that the Julian Calendar was in effect until the Gregorian Reform on 1582 October 15 (NS) UT 0 h. In some countries the Julian Calendar remained in effect until the beginning of the 20th century (Russia, Greece, Serbia).

We also note the difference between historical and astronomical numbering of the years before Christ: Historians count the year before AD 1 as BC 1. This has the effect that leap years in the Julian calendar are those that have a remainder of 1 on division by 4, i.e. the years BC 1, 5, 9, 13… 97, 101, etc. Astronomers, however, use a different convention whereby the year AD 1 is preceded by the year 0, which in turn is preceded by the year –1. This has the effect that all leap years are evenly divisible by 4. Thus astronomical year –4712 is the calendar year BC 4713. We, however, will follow the historical convention, since dates before Christ are customarily given as years BC.

It is convenient to use the Julian Day, a continuous count of days unencumbered by years, months, weeks and days, as the medium of exchange between calendars, and so we will first determine how to convert a date in the Julian or Gregorian calendar into the Julian Day, and from the Julian Day into the Julian or Gregorian Calendar.

Conversion from the Julian or Gregorian Calendar to Julian Day.

There are several published methods of converting to and from Julian Day. We have adapted as most convenient, the one given in the article “Julian Day” in Wikipedia.11

We start with a year AD or BC, a month m (January = 1), a day d, and a time of day t in decimal fractions of a day since midnight UT. We convert the year AD or BC into the year of the Julian Period that commences at BC 4713 January 1 UT 12 h (Julian).

1)         JP = AD + 4713

2)         JP = 4714 – BC

For convenience in calculation, we wish to move the zero point of our day count to the beginning of the previous 400-year cycle of Gregorian leap years, namely to March 1 of BC 4801, and we wish to count the elapsed complete years y since that date.

3)         a = int ((14 – m) / 12)

4)         y = JP + 87 – a

The variable a determines whether our date is before March 1 (a = 1) or not (a = 0). We add 88 years to bring the zero point back to BC 4801, but decrement by one so as to count only the number of complete years since March 1 of that year. We then compute the number of completed months, n, since the March 1 previous to our date, and then we can compute the Julian Day.

5)         n = m + 12 a – 3

6)         JD = 365 y + int (y / 4) + int ((153 n + 2) / 5) + (d – 1) – 32082 + t – 0.5

The first term on the right-hand side of this equation counts complete years since BC 4801 March 1. The second term adds in the leap-year days. The third term is the number of days in the elapsed months since the previous March 1. It reflects the fact that the sequence of month lengths follow a pattern, starting from March that is repeated every 5 months: 31, 30, 31, 30, 31, a total of 153 days, which takes us to July. Then we begin again, 31, 30, 31, 30, 31, ending at December. The next cycle is cut short: 31, 28 or 29, ending with February, but since we never have more than 366 days left over counting from March 1, we never need be concerned with the remainder of that last cycle.

The variable d is the day of the month. This is decremented by 1, since we have our count start at 0 on the first day of the month. We decrement our count of days by the number of whole days between BC 4801 March 1 UT 0h and BC 4713 January 1 UT 0h, namely by 32082. The variable t is the (decimal) fraction of the day since 0 h UT. The final term, – 0.5, moves the commencement of the count from midnight beginning January 1 to the following noon. The constants could be combined, –1 – 32082 – 0.5 = ­–32083.5, but we wish to keep the underlying reasoning clear.

The above formulas are valid for all dates on or after BC 4800 March 1.0 and for the Julian Calendar (Old Style). If we wish to convert a Gregorian Calendar date, then we must add a correction, g:

7)         c = int (y / 100)

8)         g = 38 – c + int (c / 4)

9)         JD = 365 y + int (y / 4) + int ((153 n + 2) / 5) – 32083 + d + f – 0.5 + g

Conversion from Julian Day to the Julian or Gregorian calendar.

GIVEN THE JD we wish to find the year AD or BC, the month of the year m, the day of the month d, the time of day t, and the day of the week w. In what follows, we will give the formulas for conversion to both calendars, the Gregorian on the left and the Julian on the right wherever they differ.

We shift the zero point from BC 4713 January 1 UT 12 h to BC 4801 March 1 UT 0 h. We will compute integer days.

10)      h = int (JD + 32082.5)

11)      j = h + 38 [Gregorian]                                                                 j = h [Julian]

The constant 38 is the number of days of difference between the Julian and Gregorian calendar in the 48th century BC.

Compute g, the number of complete 400-year cycles since BC 4801 March 1 UT 0 h, but set this to 0 for the Julian Calendar. The variable dg is the number of days left over.

12)      g = int (j / 146097) [Gregorian]                                       g = 0 [Julian]

13)      dg = mod (j, 146097) [Gregorian]                                   dg = j [Julian]

Compute c, the number of completed centuries since the close of the previous 400-year cycle, but set this to 0 for the Julian Calendar. The variable dc is the number of days left over.

14)      c = int (3 (int (dg / 36524) + 1) / 4) c = 0

15)      dc = dg – 36524 c  [Gregorian]                                       dc = dg [Julian]

Compute b, the number of complete 4-year cycles since the close of the previous century. For the Julian Calendar we use the number of days since BC 4801 March 1 UT 0 h, i.e. dc = dg = j. The variable db is the number of days left over.

16)      b = int (dc / 1461)

17)      db = mod (dc, 1461)

Compute a, the number of complete 365-day common years since March 1 of the previous leap year. The variable da is the number of days left over. The complicated formula for a in (18) assures that it increments at the completion of each 365-day common year, but not on the last (365th) day of the last year of the 4-year cycle.

18)      a = int (3 (int (db / 365) + 1) / 4)

19)      da = db – 365 a

Compute y, the number of complete years (reckoned from March 1 through the last day of February) since BC 4801 March 1 and n, the number of complete months since the previous March.

20)      y = 400 g + 100 c + 4 b + a

21)      n = int ((5 da + 308) / 153) – 2

From the above we may now compute the year AD or BC, the month m, the day of the month d, the time of day, t, since GMT midnight, and the day of the week w.

22)      JP = y – 87 + int ((n + 2) / 12)

23)      AD = JP – 4713

24)      BC = 4714 – JP

25)      m = mod (n + 2, 12) + 1

26)      d = da – int (153 (n + 4) / 5) + 123

27)      t = frac (JD)

28)      w = mod (int (JD + 1.5), 7) + 1 = mod (j + 3, 7) + 1

Conversion from Julian Day to the Martian calendar.

ALLISON AND McEWAN define a Mars Sol-Date (MSD) as

29) MSD = MJD / 1.02749125 + 44696.0 – k

where k is a small adjustment, k < 0.001 sol, which we will ignore. Since the modified Julian Date, MJD = JD – 2400000.5, we can define a Julian Date-Mars, JDM = MSD + 2341160.5 such that JDM 0.0 is Airy-0 12h of the sol that contains JD 0.0. We find that

30) JDM = JD / 1.02749125 – 0.07140633

The first term on the right-hand side of the equation converts days to sols, the second term takes us from GMT Noon to Airy-0 Noon.

To conform to the requirement that March 1 should contain the northern vernal equinox between 0h and 24h at Airy-0 for the years around AD 2000, we seek the best fit to the data in Table A1, p. 231 ff in Allison and McEwen 12. If we commence the 22-year cycle with 0h at Airy-0 on the sol that contains the vernal equinox commencing orbit 3 in the Table (this corresponds to Earth calendar years 1879-1881), we find that the vernal equinox falls outside of March 1 only twice in the span of 134 orbits (252 Earth years), namely at the commencement of orbit 8, when the equinox is a few minutes past Airy-0 midnight beginning March 2, and at the commencement of orbit 69, when the equinox is a minute before Airy-0 midnight beginning March 1, i.e. just before midnight February 53 of orbit 68. These variations can be explained by irregularities in the Martian orbit caused by gravitation influence of the other planets, and we will consider them insignificant. Thus, a 22-year cycle begins on MSD 2149 or JDM 2343309.5.

Going back 160 22-year cycles before MDS 2149 takes us to the commencement of the 22-year cycle that contains JDM 0.0 (and also JD 0.0). Since 160 cycles is 160 x 14709 = 2353440 sols, we can define a Martian Day count (MD) as

31) MD = MSD + 2351291 = JDM + 10130.5

where 2351291 = 2353440 – 2149, and 10130.5 = 2353440 – 2343309.5.

From the MD, calculate the elapsed years, y, since MD 0,

32) y = int ((22 int (MD) + 21) / 14709)

We find that y for JDM 0.0 is 15. We define MY 1 as the Martian year containing JDM 0.0, so that

33) MY = y -14

Calculate x, the number of sols in the elapsed years, and z, the number of elapsed days since the beginning of the year,

34) x = int (14709 y / 22)

35) z = int (MD) – x

Since every month has 56 sols, we can calculate m, the number of the month, and d, the day of the month.

36) m = int (z / 56) + 1

37) d = mod (z, 56) + 1

Calculate the time of day, t, as the fractional part of MD, since MD begins at Airy-0 midnight, and the day of the week, w (Sunday = 1).

38) t = frac (MD)

39) w = mod (int (JDM + 1.5), 7) + 1 = mod ((int (MD) + 6) / 7) + 1

Conversion from the Martian Calendar to the Julian Day.

WE WISH TO find the Julian Day of a given Martian calendar date of MY, month m, day d, and time of day since midnight at Airy-0 (in decimal fractions) t. Calculate the MD,

40) MD = int (14709 (MY + 14) / 22) + 56 (m –1) + d – 1 + f

Deduct 10130.5 sols to bring the zero point of the count to JDM 0.0.

41) JDM = MD – 10130.5

Convert JDM to JD, the inverse of equation (29).

42) JD = 1.02749125 JDM + 0.07336938

From this use equations (10) through (28) to obtain the corresponding date in the Gregorian or Julian Calendar.

Conclusion.

AS A STUDENT of calendars and their history – and beginning in fact with his school days – this author has considered the problem of a calendar for Mars for many years. The present proposal is the fruit of much thought and development, and has afforded him many hours of diversion.

I do not propose to use this calendar to the exclusion of all others, but rather we offer this as a reasonable and practical proposal for timekeeping on Mars for astronomical and astronau­tical use. I believe that the regularity of construction, the uninterrupted sequence of weekdays, the use of the same calendar for every month of a year, the use of familiar and easy to remember names for time divisions, the maintenance of the vernal equinox on the same calendar day for a long span of years, the year count starting with a universally accepted astronomical reference point, and the result that it encompasses all observations of Mars both historical and current, all constitute virtues that strongly commend this proposal.

I leave aside for now the easily solved problem of correlating existing proposals for Martian calendars, especially for the Darian calendar of Gangale, and the Zodiacal calendar proposed by Zubrin. In fact, the study of historical terrestrial calendars has inspired this author to invent many alternative calendars for Mars. I leave all this for another project and another article.

Meanwhile, I have appended material here for the interested reader.

Appendices.

Appendices appear as .pdf files.

Appendix A gives a comparison between Earth and Mars.

Appendix B contains tables with correlation between the Gregorian and Martian Calendars for the few years surrounding the composition of this article.

Appendix C gives a chronology of historical and Mars-related events.


V Rev George Lardas is the Rector of St Nicholas Russian Orthodox Church in Stratford, Connecticut. He holds a BS in Astronomy and Physics and an MS in Aerospace Science, both from the University of Michigan, and has worked as an engineer for NASA and U.S. defense contractors, and remains a student of things astronomical.

Minor formatting edits were made to this article after publication. – Ed.

Published with “Mars” by Sir Robert Ball from the Fortnightly‘s archive.

NOTES:

  1. Huffington Post, 19 August 2012 at http://www.huffingtonpost.com/2012/08/19/david-oh-nasa_n_1807113.html, 19 August 2012.
  2. Allison, M., and McEwan, M, 2000. “A post-Pathfinder evaluation of areocentric solar coordinates with improved timing techniques for Mars seasonal/diurnal climate studies.” Planetary and Space Science, 48 (2000), pp. 215-235, Pergamon Press. Also at website: http://pubs.giss.nasa.gov/docs/2000/2000_Allison_McEwen.pdf, 1 January 2012.
  3. Levitt, I., Menzer, R. “Martian Calendar”, Sky and Telescope, May 1954.
  4. Cited by Richardson, R., 1954. Exploring Mars, pp. 96-97.
  5. Robinson, K., Mars Trilogy (fiction): Red Mars, 1993; Green Mars, 1995; Blue Mars, 1997; Spectra.
  6. Zubrin, R., 1997. The Case for Mars; Touchstone.
  7. Gangale, T., 1986-2005. Website, http://pweb.jps.net/~tgangale/mars/mst/darian.htm, 10 August 2012.
  8. Zubrin op. cit.
  9. Levitt op. cit.
  10. Alison and McEwen op. cit.
  11. Wikipedia article, “Julian Day”, http://en.wikipedia.org/wiki/Julian_day, 1 December 2011.
  12. Allison and McEwan op. cit.
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