What calendar would people use on Mars?
Short answer: For daily life they would use a new calendar based on the Martian day (roughly the same as Earth days) and Martian year (roughly two Earth years). However they would also need to monitor the Earth calendar in order to communicate between planets and to observe Earth dates such as annual celebrations.
Long answer: We need to consider the different units of calendar time (days, weeks, months and years) and decide which we would use on Mars, and how we'd define them.
The critical units of a calendar are days and years. Their lengths are determined by the planet we're on. They mark important natural cycles that we can't change - the length of time it takes the planet to rotate once on its axis (day) and the length of time it takes to orbit the Sun (year). If you're planning to live long-term on a different planet the only sensible option is to adopt the local days and years.
Days
Mars rotates once every 24 hours and 37 minutes, making its day 37 minutes longer than Earth's. This is conveniently close to an Earth day but not perfect. Human circadian rhythms are tied to a 24 hour cycle and it's not clear how a slight change would affect us. Although we can research the effects to some degree on Earth, Mars colonists will probably just live with a longer day and hope for the best.
If necessary, it would be possible to use a 24-hour day on Mars. Colonists could use Earth clocks, gradually going in and out of sync with the Martian day/night. This would be helped by the fact that they'll be spending a lot of time inside a habitat with no windows. However it would be preferable to be in sync with the planet's days if possible.
A day on Mars is called a "sol". When working on Mars robotic missions, engineers on Earth change their own daily schedule to synchronize with the Mars sol. This actually provides some small insight into what it would be like living with a longer day.
Years
The Martian year is 687 Earth days (668.6 sols). This annual cycle may be important when planning things like energy generation and consumption, but it doesn't really affect day-to-day life for humans. It also wouldn't affect crops brought from Earth as those crops would continue to grow in the same cycle they do on Earth, and we'd be growing them in greenhouses insulated from Martian weather.
We should have no problem working with the Martian year while simultaneously keeping an eye on passing Earth years. It's a bit like knowing how old your cat is in both cats years and human years. If you were born on Mars you'd grow up with two "ages" - your Mars age and your Earth age.
Months
On Earth, months are related to the length of time it takes the Moon to orbit Earth (27.3 days). Changing moonlight and other factors have an effect upon life on Earth, so the Moon's orbit has traditionally been an important part of human calendars. In modern times, however, these factors don't affect us much so we don't really consider the lunar cycle in our daily lives. Our calendar months are only approximations of the Moon's orbit - they're not in sync, they're just a hangover from the past.
However we do keep using months because they're a handy unit of time to fill the gap between and week and a year. It's possible that Martians will use their own version of a month - they'll be able to make it as long as they like.
Note: Mars does have two moons but their orbits are too short to be useful as monthly markers.
Weeks
Weeks are purely arbitrary. Our seven-day cycle is based on celestial objects that ancient people saw moving in the sky: The Sun (Sunday), Moon (Monday) and the five visible planets (Tuesday - Saturday). Unlike the day, year and month, there's no real connection between a week and the physical world around us. Martians would be free to base their week on seven sols, or perhaps they'd like to go decimal and have a ten-sol week.
Other Units
Theoretically, Martians could do away with weeks and months altogether and just use sols and years. However I think it would be useful to have at least one intermediate unit of time. Perhaps a Martian year could have 25 months of 50 sols with a 26th month of 37 sols. I don't care, it's up to them.
Trivia
- Because Mars has an eccentric orbit, its seasons are not of equal length as they are on Earth.
- Starting a new calendar provides a good chance to fix some issues. For example, the words "summer" and "winter" are specific to one hemisphere and can cause confusion, whereas the terms "northwarm" and "southwarm" are planet-wide and unambiguous. Using a single planet-wide timezone would also solve a lot of problems.
Summary
Days and years are set by the planet so people who live on Mars will eventually want to adopt Martian days and years. Weeks and months are completely flexible and may or may not be used at all. People living on Mars will be aware of dates in both their own calendar and Earth's calendar.
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