Hours, Dess, Centits and Mills
I would suspect that the Martians would find it useful to adjust their daily clock. I think they would find a 24 hour day convenient (because it allows the adaption of traditional clock designs, eight hours shifts, noon and midnight language, etc.). This means that the Martians need to fit 98.9 extra seconds into every hour. One way to do this would be to have a 1.02747 second long Martian second, but that would mess up SI units. A easier approach might be to use decihours (the favorite unit of time for lawyers and accountants) and centihours (rather than minutes) and millihours (rather than seconds), in customary local time measurement. Thus, a decihour would be 6 minutes and 9.89 seconds long, a centihour would be roughly 37 seconds long, and a millihour would be roughly 3.7 seconds long. This might be abbreviated to a "dess" (not conjugated to distinguish between the singular and plural, e.g. "one dess", "three dess", "a few dess"), a "centit" (in lieu of a minute) and a "mill" in popular conversation. A Martian Hour might simply be called an "hour" with an Earth hour called a "Standard Hour". The term "second" would always refer to the SI unit.
The time of day would be reported as, for example, "six point five eight o'clock". As in military time, there would be no AM and PM, only twenty four numbered hours. Thus, "fourteen point two one o'clock".
Sols and Weeks
I would expect that the people of Mars would adopt the natural Martian synodic day (midday to midday) (the 24 hour, 39 minute 35 second Sol), as people have been shown in sleep research to gravitate towards a little longer than 24 hour day when stripped of external references anyway, and would also probably adopt a seven day week, as it is familiar and has no connection to astronomical phenomena in any case. I suspect that the names of the days of the week would stay the same.
Months and Years
I would also think that a roughly menstrual based four week month (28.7 Earth days) called "months" with 24 months in a Martian year (in symmetry with the hours of the day, double the number of Earth months and each quite comparable in length to an Earth month), would be a useful figure (for things like accounting and billing purposes). Of course, this gives 672 sols in a year, rather than 669.65972 synodic days, so you would need two 27 sol months every year, and and another 27 sol month (a "skip day") in years divisible by three which would be called skip years (with an additional adjusting skip day every 144 years). One plausible way to space the short months would be to have the annual short months on the 8th and 16th months, with the skip month as the 24th month in a skip year. Every 144 years (a "grossery"), there would be an extra skip year, unless it fell on a skip year, in which case the extra skip year would be the following year. This would be one day off the true year every 3,126 years.
The symbolic time to start counting years would be the year the first human arrived on Mars. Years after the year zero of arrival would be AA (for after arrival). Years before this point in time would be BA (for before arrival). The symbolic month to start the year would be the first month of Spring at the equinox.
Choosing names for the 24 months of the year would be quite political. Since Mars has seasons, breaking them up into quarters of six for summer, autumn, winter and spring would make sense. To avoid confusion with Earth dates, names from other calanders might be used, for example the Hebrew, Chinese, Islamic and French Revolutionary calanders. An idea I like in particular is that a city name could be chosen from twenty-four sections of the Earth to name each month. For example, the months might be:
Summer: Paris (Par.), Cairo (Cai.), Moscow (Mos.), Tehran (Teh.), Mumbai (Mum.), and Kathmandu (Kat.).
Autumn: Bangkok (Ban.), Shanghai (Sha.), Tokoyo (Tok.), Brisbane (Bri.), Tarawa (Tar.), and Auckland (Auc.).
Winter: Nome (Nome), Oahu (Oahu), Juneau (Jun.), Seattle (Sea.), Denver (Den.), and Mexico (Mex.).
Spring: Boston (Bos.), Caracas (Car.), Rio (Rio), Canary (Can.), Dakar (Dak.), and London (Lon.).
Note that annual skip months would be months starting with "S".
Dates would be written in the European convention. For example, 5 Seattle 36 AA would be the fifth day of the month of Seattle in the year 36 after arrival. Of course, AA would usually be left unstated, while BA would normally be noted expressly. Also, months would usually be abbreviated. Thus, 25 Lon. 45 would be a more typical written date.
Holidays and Special Days
Arrival Day would probably be a major holiday, as would be New Year's Day, and indeed, probably the first day of each season.
As far as significant ages, ten Martian years old would be a convenient time for adulthood. This works out to be eighteen years and just under ten months in the Earth calander. Age thirty-five in Martian years (65.8 years) might be a marker for senior citizen status. Children would probably celebrate half-birthdays as well as birthdays. Adults would probably enjoy having birthdays half as often.
Legislative elections would probably be held once per Martian year, perhaps in the month of London with newly elected officials to take office the following month of Paris, as modern technology would permit fairly rapid vote counting.
Schools might be on-half quarters (three months each) called semesters (instead of conventional semesters), and might have a four day weekend semester, and a three week break at the end of each semester. Kids get bored in a full three month summer vacation which doesn't make sense in a modern economy anyway, but do need occassional lengthy breaks. The aggregate time off would be similar to the current system with its summer vacation, Spring Break, Christmas vacation and numerous holidays.
I would suspect that there would be a split between Orthodox religious people, who would keep track of Earth days for celebration of religious events (with the ultra-orthodox keeping even the Earth Sabbath) and ordinary religious people who would adapt most of their holidays (with the possible exception perhaps one holiday, like Easter, for Christians) to fit the local calendar.
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