A sidereal day is the time it takes the planet to do a full 360 degree revolution around its axis. It is influenced by nutation, tides and the gradual slowing of a planet’s rotation. When a distant star is in a particular location in the sky, the next sidereal day it is in the same location again.
A solar day is the time from a sun’s highest point in the sky to the next time the sun is at its highest point in the sky. The solar day can be longer or shorter than a sidereal day and is influenced by the orbit of the planet around the sun. The more elliptical the orbit is, the more the length of a solar day varies across the year. On Earth the solar days are approximately four minutes longer than the sidereal days, and in a year there is one more sidereal day than there are solar days. On Venus a solar day is shorter than a sidereal day because the planet has retrograde rotation.
Our current clocks use multiples of 12: 5 times 12 seconds in a minute, 5 times 12 minutes in an hour, 2 times 12 hours in a day. Twelve is easy to work with because it has many divisors. (Also the reason that there are 12 inches in a foot, 12 ounces in a troy pound and 12 pence in the old British shilling.) Therefore we should prefer the duodecimal system on other planets instead of the decimal system.
I propose a way of keeping daytime.
A planet with one sun
We humans like daylight very much, so we tend to count our days in solar days, or sols. Let’s define an hour as 1/24th of a day, and a minute as 1/60th of an hour, irrespective of how long a day on that planet actually is. Most planets are in a solar system with only one sun, so then it is logical to define a day on that planet as equal to a solar day, or sol (with some exceptions which I’ll talk about below). On any day for any position on the planet, on 12:00 (on a 24-hour clock) the sun is at its highest point in the sky for that sol.
This definition of hours and minutes means that the length of both hours and minutes depends on the planet you’re at. In fact, since a sol does not have a fixed length, the length of both hours and minutes not only varies from planet to planet, but also from day to day. This is also true on Earth. In practice this day to day difference in length is not big enough to matter, but over the course of a year the length of a sol on a planet with a highly elliptical orbit will fluctuate by several hundred seconds.
The second is a time unit that has now been defined independently of any planet’s rotation using the radiation of a cesium 133 atom. While it is nice that we generally assume that 60 seconds fit a minute, in our daily lives the smallest relevant time scale that matters are minutes. I propose to keep the defined length of a second constant (just like it is now) and just measure the length of a sol minute in those constant seconds.
This means that even on Earth an average minute is just a little bit more than 60 seconds long. An average Martian minute would be 61.6 seconds long.
There are of course also planets that have sols that are no where near an Earth day in length. For example Venus has a sol of approximately 224.7 Earth days. A human would not be able to wait that long before going to sleep, so such sols need to be adjusted to match our sense of time. I propose that this adjustment is made by dividing the sol into segments of 12 days. On Venus, a sol would be divided into 10 segments of 12 days, resulting in a day (thus 24 hours) with minutes of 58.4 seconds long. The number of segments is chosen such that the length of a day is closest to an Earth day (as that’s the time unit humans need for their biological clock).
For a planet that spins rapidly, we do the opposite: a day is divided into one or more segments of 12 sols.
A planet with no sun
There are stray planets flying around all the time. They were once flung out of a solar system and now don’t orbit any star. The only day that has any relevance to such planets is the sidereal day. So I propose to use the same time specification for sunless planets, substituting the sol by the sidereal day.
A planet with multiple suns
Although rare, there exist solar systems that have planets with multiple suns. In this case I propose we pick one (for example the brightest, closest, or the biggest) and use the solar day relative to that sun for measuring the length of a day. If the suns are sufficiently close together and far away, it doesn’t really matter which one you pick. If the suns are far apart, then daylight varies from day to day such that there is no simple way to define a day anyway.