# How would units of time be measured on a planet with a binary star system?

The days are long approximately 25 hours of daylight and 5 hours of darkness. There are two stars. The first star rises like the sun. The second star tracks the horizon. (I'm prepared to hand wave the physics on this.) They do not have the technology for clocks, although things like sundials would be possible. There is a magic system - think solar powered batteries that are only used by certain sections of the population. They do not have specific times when they eat. They are day long grazers, eating opportunistically. They tend to sleep when its dark and are up during daylight hours.

How would time be measured so I can pinpoint times in the day for a meeting for example?

• How can one sun "track the horizon"? If you walk to it, it'll be overhead at a certain point. Sep 14, 2020 at 9:37
• To further explain on @Trioxidane comment, there is one horizon per observer. Your second star cannot satisfy all of them
– L.Dutch
Sep 14, 2020 at 9:42
• Any way you like. There is no mathematically clear relationship between your premise and how people count time. Feel free to do anything really. If it makes sense to you, why not. Different people think differently about seasons on earth for example because it means something different depending on where you are Sep 14, 2020 at 9:50
• "How would units of time be measured...": Sundials are clocks. As for not having the technology for "clocks" in general, I am doubtful that there was ever a time when people had a need to measure time and did not have the technology to make water clocks or sand glasses. Sep 14, 2020 at 10:46
• @Trioxidane See my answer, with a tidally-locked planet the places with the sun overhead are there but simply uninhabitable... Sep 14, 2020 at 11:19

Time is remarkably arbitrary

Most people don't realize that the measurement of time (and therefore a LOT of our mathematics) is completely arbitrary. We've come up with a way to standardize the concept of a second, but the fact of a second was still completely arbitrary. In other words, we created a standard to fit the measurement, not a measurement to fit a standard.

The basics of time came from orbits, rotations, seasons, and culture. Culture's the easiest because your monarch can boldly declare anything he/she wants (e.g., the number of hours in a day). But let's focus on the astronomy first.

• Years are the measure of your planet's orbit around and among your stars. It's the "time" required to get from one arbitrarily defined point along the orbit to that same point. Behold! we have defined a year. It has nothing at all to do with the number of stars or their position in the sky — and that's why we're defining it first.

• Months on earth had more to do with the moon than the sun (a LOT more). There's a lot of icky history behind what I'm going to say. Icky history that I'm going to ignore. But simplistically, the time required to get from one New Moon to the next New Moon is a month. (One reason that it isn't quite that simple is that the lunar month does not conveniently divide evenly into a single solar year. So humans forced it.) You don't mention moons, so you may not have months at all.

*Days on earth are one rotation of the planet. When you say 25 hours of daylight and 5 hours of darkness, I'm assuming you mean Earth-hours. What you have is a period from sunrise-to-sunrise that's defined as a day. Your two stars complicate this a bit, but not much, because the planet is rotating.

OK, we've defined a year and a day ... and we might have a month (not that it's important). The next thing our primitive people care about is planting crops. We need seasons.

• We humans living in North America like four seasons. In reality, the world can really only depend on two: summer and winter. But those primitive people aren't fools. The seasons roughly correspond to Earth's apogee, perigee, and its two equinoxes. In a binary star system, that might be the most complicated aspect of defining seasons — because that planet might be bobbing and weaving all over the honking place. You'll need to decide some things here, like how elliptical your orbit is. The two stars orbit around one another. The center point is called a barycenter. Your planet is, simplistically, orbiting around that barycenter, too. therefore, you have an apogee, perigee, and two equinoxes — they might just be a bit more complex (a few "sub-seasons") due to the bob-and-weave thanks to the two stars. This has a LOT to do with how fast those two stars orbit the barycenter, the relative mass of the two stars, and the relative energy output of the two stars. You didn't provide that — so I'm going to assume the second star is much weaker than the first, and hopefully that means fourish standard seasons.

Now we have a year, broken down into four periods, which are further broken down into more periods (days).

• Weeks are incredibly arbitrary. See here. The Babylonians had 7-day weeks because they tracked seven celestial objects: the sun, the moon, Mercury, Venus, Mars, Jupiter and Saturn. The Egyptians had 10-day weeks. The Romans had 8-day weeks. Like I said, completely arbitrary. Go get your D&D 4-sided dice and roll three of them. That's the number of days in your week. (BTW, sun(day), moon(day), saturn(day) they all come from the original Babylonian astronomical reference.)

• Hours and seconds are just as arbitrary (see here). The Egyptians had a 12-hour night based on constellations, a 10-hour day, and two twilight hours. The Babylonians really liked base-60 numbers. It's that bad.

In the end, weeks, hours, minutes, and seconds were culturally defined on Earth. For the sake of realism, you want similar cultural influence in your time system.

And after a gazzilion years, people became so fed up with not knowing exactly how long a second was (mostly physicists...), that "the second has been defined as exactly "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom." And, just to make my point, "This length of a second was selected to correspond exactly to the length of the ephemeris second previously defined." In short, we crowbarred a highly precise measurement into our historical definition of a second because, IMO, it would have been way too hard to convince everyone to replace their wristwatches and desk calendars.

TL;DR

Use astronomical observations about your star system to determine a single year, a single day, and at least four seasons. If your planet has a moon, use it to define months, otherwise ignore months. Once you have these starting points, use the cultures of your world's people to define weeks, hours, minutes, and seconds, any-or-all as necessary, and arbitrarily.

Then, and only then, try to relate it back to terrestrial time. Otherwise what you'll have is Earth time and it won't feel natural.

• No moons. Well spotted. Sep 15, 2020 at 9:51
• I have very fast and extreme seasons. Going from hot to cold quickly. Sep 15, 2020 at 9:54
• Brilliant advice. I've also combined some of the other advice given and now have my time! Thanks Sep 15, 2020 at 9:59

## Pretty much the same as we do on Earth, possibly with much longer days.

When it comes to planets in a binary star system, there are two possible orbits. The circumbinary planets (P-type) orbit the center of mass of both stars (think "stars close to each other, the planet further away"), while the non-circumbinary planets (S-type) orbit only one of the two stars (think "planet close to one star, the other star further away").

Now we want to ignore the circumbinary planets, because for those there would simply be two suns close to each other and some very Earth-like days would occur. It might be pretty, but not very "special".

With an S-type planet, on the other hand, you get much more variability to play with. Typically, you'd have "normal" days from "your" star (A), while the more distant star B would run around similar to a very bright Moon. Still too Earth-like. But you want one of the stars on the horizon. And you can get it, without handwaving any physics, it will just need to be fixed on the horizon instead of tracking along!

Imagine your planet tidally locked to star A. The part (permanently) facing A would be a scorched desert wasteland, while the dark side would be a frigid frozen wasteland. But along the narrow-ish strip* where A is just high enough above the horizon to give you nearly enough light and warmth, life can flourish! Why nearly enough? Because if that was it, you would get your warm-ish sun on the horizon, permanent but not too bright days, a sunset till eternity. But don't forget star B! It's "orbiting" A roughly in the same plane as your planet. If it's bright and close enough** to be more than just a "brighter Moon", it will give you an extra boost energy, AND it will have proper sunrises and sunsets!

Now you have your horizon sun and your sky-climbing sun, and you can measure time using the latter. The length of your day will be given by the orbital period of the binary system. The day will be made a bit longer by the fact that your planet is orbiting A in the same direction as B (if we adopt A as our coordinate system origin), but given the difference in mass, this might not be by much.

* Actually, this strip can be possibly thousands of kilometres wide, ranging from just-about-livable Sahara-like to just-about-livable Greenland-like. It's also worth mentioning that the strip won't be perfectly circular, as the colder polar parts will be habitable with A higher on the horizon, while the warmer tropical parts will get more light from star B and thus the habitable zone will have A lower above the horizon.

** I don't have enough knowledge to tell if the "just right" combination of distance, mass and brightness of A, B and your planet is actually possible. Just right in this sense means that the combined light from A and B would cause the stuff I mentioned above without B's gravity disturbing the planet's orbit too much. Anyway, it wouldn't break my suspension of disbelief if this wasn't possible :)

NOTE: I don't think you get your 25+5 hour day, though! Light from A is weak but constant, while light from B would follow the same pattern as we have on Earth, just given by the difference in orbital planes rather than axis tilt. Due to the extra light from A you can count dawn&dusk as day time, but I would still not expect the common difference to be more than 20/10 with a 30-hour "day".

• Unfortunately it doesn't address the sun tracking the horizon. It would just be stationary at an area. People on the other side of the Earth wouldn't see this sun ever, so isn't tracking the horizon. And for all intent and purposes, you can move towards it, making it not track the horizon as well. Sep 14, 2020 at 11:30
• Yeah, having the sun on the horizon for everyone everywhere obviously can't be, so I tried to get as close as possible! Sep 14, 2020 at 11:40
• Would such a world be habitable only in its twilight zone? It is possible that all of the planet's air and water will freeze out on the dark side, making the entire planet uninhabitable. It is possible that the hyrodpshere and atmosphere will distribute heat over the planet rather evently, so it has the right temperatures everywhere. Both possibiities would eliminate the twilight zone being the only part of the planet where there is life, and thus elimnated the sun being on the horizon all the time. Sep 14, 2020 at 16:52
• If the planet is tidally locked to one star, as in BIOStheZerg's answer, that star would proably be a dim star the planet orbits very close to, and thus the planet should have a very short "year". as it orbits that star. The other star will appear to circle all around the sky in its synodic period, which will thus determne the day/night cycle of the planet. The farther star will have to be several times as far from the planet as the nearer star for orbital stability. And I think the farther away it is the shorter the synodic period will be. continued.. Sep 14, 2020 at 16:57
• Continued. So the farther star in the system should probably be several times as luminous as the nearer star, for it to have a major impact on the day/night cycle despite being several times as far away. But it cannot be too much more luminous, or else it and the entire system will be too young for the planet to have evolved intelligent life. Such a system will have to be designed carefully to work. Sep 14, 2020 at 17:00

This is a really long answer which elaborates on the answer by BIOStheZerg, pointing out some of the factors to be considered when designing a star system according the suggested answer of BIOStheZerg. I couldn't think of any other way to provide what AvengingEarth wants, so I thought that would fill in someof the details of BIOStheZerg's answer.

BIOStheZerg suggested that the planet should orbit in an S-type orbit, and thus orbit only one of the two stars, and the planet should be tidally locked to the star that it orbits, so that one side of the planet would always face the star and the other side would always face away from the star.

BIOStheZerg suggested that only the twilight zone of the planet would be habitable for the natives, and thus that the star would always be near the horizon of the planet from the twilight zone inhabited by the natives.

`A small change in the mass of a star will produce a much larger change in the luminosity of the star. Thus a planet will have orbit a less massive and dimmer star much deeper into the gravity well of that star, and the tidal braking of that star will slow down the planetary rotation much faster.

Thus the only tidally locked planets will be planets of low mass and very dim stars.

Astronomers for many years ruled out red dwarfs as potential abodes for life. Their small size (from 0.08 to 0.45 solar masses) means that their nuclear reactions proceed exceptionally slowly, and they emit very little light (from 3% of that produced by the Sun to as little as 0.01%). Any planet in orbit around a red dwarf would have to huddle very close to its parent star to attain Earth-like surface temperatures; from 0.3 AU (just inside the orbit of Mercury) for a star like Lacaille 8760, to as little as 0.032 AU for a star like Proxima Centauri[77] (such a world would have a year lasting just 6.3 days). At those distances, the star's gravity would cause tidal locking. One side of the planet would eternally face the star, while the other would always face away from it. The only ways in which potential life could avoid either an inferno or a deep freeze would be if the planet had an atmosphere thick enough to transfer the star's heat from the day side to the night side, or if there was a gas giant in the habitable zone, with a habitable moon, which would be locked to the planet instead of the star, allowing a more even distribution of radiation over the planet. It was long assumed that such a thick atmosphere would prevent sunlight from reaching the surface in the first place, preventing photosynthesis.

This pessimism has been tempered by research. Studies by Robert Haberle and Manoj Joshi of NASA's Ames Research Center in California have shown that a planet's atmosphere (assuming it included greenhouse gases CO2 and H2O) need only be 100 millibars (0.10 atm), for the star's heat to be effectively carried to the night side.[78] This is well within the levels required for photosynthesis, though water would still remain frozen on the dark side in some of their models. Martin Heath of Greenwich Community College, has shown that seawater, too, could be effectively circulated without freezing solid if the ocean basins were deep enough to allow free flow beneath the night side's ice cap. Further research—including a consideration of the amount of photosynthetically active radiation—suggested that tidally locked planets in red dwarf systems might at least be habitable for higher plants.[79]

https://en.wikipedia.org/wiki/Planetary_habitability#Size[1]

So on one extreme, a tidally locked planet might have so little heat circulation that all of the water and air might freeze out on the permanent night side, and there might not be any life on the planet.

And on the other extreme, there might be enough air and water circulation in a tidally locked planet that temperatures will be fairly good in all parts of the planet, and life might not be restricted to a narrow twilight zone.

So a situation where the natives are restricted to a rather narrow twilight zone where the star always appears low on the horizon, the desired situation, might be a rarity between dead tidally locked planets on one hand and tidally locked planets with decent temperatures in every part on the other hand, where the natives would not be restricted to a twilight zone and might be able to see the star high above the horizon in some places.

But since it has not been proved to be totally impossible yet we can accept that the planet might have such a situation.

As said earlier, a tidally locked planet would have its rotation rate slowed to match its orbital period around the star, so that one side always faced the star and the other side always faced away from the star, and the star would always be low on the horizon at the twilight zone. And it would have to be orbiting very close to a dime low mass star.

Assuming that the natives of the planet have environmental requirements similar to humans, the discussion of tidally locked planets in Habitable Planets for Man, Septen H. Dole, 1964. 2007, may be useful. Dole didn't believe that tidally locked planets could be habitable, so he ruled out any stars so dim that planets in their habitable zone would be tidally locked.

https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf[2]

Dole discusses the masses of stars capable of having habitable planets on pages 67 to 72.

Because more massive stars with larger habitable zones consume their nuclear "fuel" much faster than less massive stars, they spend less time in their main sequence stages before becoming red giants, and then white dwarf stars (or possibly neutron stars or even black holes). Since a planet should take billions of years to become habitable for humans, a star that is too massive will not spend enough time on the main sequence with a steady luminosity for its planets to become habitable for humans.

The only stars that conform with the requirement of stability for at least 3 billion years are main-sequence stars havng a mass less than about 1.4 solar masses - spectral type F2 and smaller - although the relationship between mass and time in the main sequence is probably not know with great accuracy and is subject to future revisions (see Figure 25).

Dole, believing that tidally locked planets would be uninhabitable for humans, unlike the assumptions made in the question, then discussed the stellar masses which would result in tidal locking of planets in their habitable zones, which Dole calls "ecospheres".

...habitable planets can exist in ecospheres only around stars having masses larger than about 0.72 solar mass. A "full" ecosphere can exist around primaries of stellar mass greater than about 0.88 solar mass, but the ecosphere is narrowed by the tidal braking effect for primaries of lesser mass until it disappers when the stellar mass reaches about 0.72 solar mass. The range in mass of stars which could have habitable planets is thus 0.72 ot 1.43 solar masses, corresponding to main-sequence stars of spectral types F2 through K. There is an extension of this range down to the larger class M stars (mass greater than 0.35 solar mass) for a special class of planets with large satellites. This will be discussed in the next section.

So Dole believed that for stars between 0.72 and 0.88 solar mass, part of the "ecosphere" or circumstellar habitable zone would be close enough to the star that planets orbiting there would e tidally locked to the star, while for stars lower than 0.72 stellar mass the entire "ecosphere" or circumstellar habitable zone woudl be close enough to the star that palnets within it would be tidally locked.

Since Dole believed that all tidally locked planets would be uninhabitable for humans, he disregard them and the dim stars they orbited.

But the question, and the answer of BIOStheZerg, require that the planet be both habitable and tidally locked, so that one star in the system always appears low on the horizon in the twilight zone of the planet. Since that has not been proved to be impossible, that is the set up I am using in my elaboration of BIOStheZerg's answer.

Astronomes have already discovered a number of planets orbiting within the habitable zones of their stars, and some orbit dim stars close enough they should be tidally locked. Wikipedia has a list of potentially habitable exoplanets orbiting within the habitable zones of their planets. The ones orbiting K1 class stars and less massive stars should be tidally locked to their stars.

https://en.wikipedia.org/wiki/List_of_potentially_habitable_exoplanets[3] The ones orbiting spectral type K and M stars include planets with orbital periods of hunderds of Earth days, including 289 days, 267 days, 259 days, 247 days, 198 days, 197 days, 177 days, 168 days, 147 days, 129 days, 122 days, 112 days, and 101 days.

Other exoplanets in the list, most of them, have orbital periods between 10 and 99 Earth days.

And some of them have orbital periods even shorter.

TRAPPIST-1f 9.2 days, TRAPPIST-1e 6.1 days, Teegarden b 4.91 days, and TRAPPIST-1d 4.05 days.

So these examples prove that a tidally locked planet in the habitable zone of its star could have an orbital period as short as 4.05 Earth days, and quite possibly as short as 4.000 Earth days. Considering how dim TRAPPIST-1 and Teegarden's star are, I don't feel it is safe to imagine a habitable planet having an orbital period much less than 4.00 Earth days long. Such tidally locked planets can also have orbital periods as long as 289 Earth days and probably longer.

The other star in the system.

To design a working star system, for AvengingEarth who asked theoriginal question, it will be necessary to decide whether the other star will provide both significant heat and light to the planet, or merely significant light.

The human eyes as a vast ability to adapt to a wide range of ilumination. Humans can see fairly well and function in a moonlit night. The dimmest moonlight, from a new moon, is magnitude -2.5, while the brightest moonlight, from a full moon, is magnitude -12.9, about 10,000 times brighter.

The Sun's apparant magnitude in broad daylight has an apparent magnitude of -23.0, which is about 400,000 times as bright as the full moon.

So it would be easy to make the other star in the system close emough to be many times as bright as the full moon on Earth, and thus make it easy for the native sto see whenit is above the horizon, while still much less bright than the Sun as seen from Earth.

A light source which is twice as fr away as an equally bright light source will appear a quarter as bright. A light source which is 3 times as far away will a ninth as bright. A light source 4 times as far away will appear 1/16 as bright, one 5 tiems as far away will appear 1/25 as bright, one 6 times as far away will appear /36 as bright, one 7 times as far awa will appear 1/49 as bright, one 8 times as far away will appear 1/64 as bright, one 9 times as far awy will appear 1/81 as bright, and one 10 times as far away will appear 1/100 as bright.

And if the light source is moved to 100 times its original distance, it will appear 1/10,000 as at the original distance.

So assuming that the two stars have equal luminosity, and the nearer star gives the planet the same amount of light as the Earth does, the farher star could be 100 times as far from the planet as the nearer star and still give the planet 1/10,000 as much light as the nearer star, which would be about 40 times the brightness of a full moon on Earth, and certainly enough light to see where someone is going.

So there would be no problem with designing a star system where the farther star gives the planet much less light than the nearer star does, but still enough light to see well and see where you are going. Certainly enough light for the natives of the planet to base their calendar on the rising and setting of the farther star.

But designing a star system where the farther star gives enough radiation to the planet to have a significant effect on the planet's temperature is a different story.

The planet Saturn is about 10 times as far from the Sun as Earth is, so the sunlight at Saturn is 1/100 times as bright as on Earth, which is still about 4,000 times as bright as the light of the full moon on Earth. You could see well enough in sunlight to walk around in your spacesuit on a moon of Saturn without using artifical light, and you would definitely notice how dark it got when the Sun set on that moon.

But the distance from the Sun makes a much more important difference in temperature. The average surface temperature on Titan, the big moon of Saturn, is 97 degrees K, or minus 179.5 degrees C, or minus 291.1 F.

Getting the farther star in the system to be luminous enough, and/or close enough to the planet, to contribute a signficant degree of heat to the planet will be much more difficult.

If a planet orbits a star in an S-type orbit in a binary star system, the other star sould be several tiems as far from the planet as the near star for the orbits to be stable over long periods of time.

In non circumbinary planets, if a planet's distance to its primary exceeds about one fifth of the closest approach of the other star, orbital stability is not guaranteed.

https://en.wikipedia.org/wiki/Habitability_of_binary_star_systems#Non-circumbinary_planet_(S-Type)[4]

So if the farther star's orbit periodically takes it less than five times as far from the nearer star as the planet orbits, the orbital stability is "not guaranteed".

If the two stars in the system ahave the same luminosity, the farther star can contribute no more than 1/25, or 4 percent, of the heat of the planet, and possibly a lot less.

The farther star would have to be at least 25 times as luminous as the nearer star in order for it to contribute equally more more to the temperature of the planet, and that is if the farther star is at is minimum possible distance.

Can a system with a habitable planet have one star that is at least 25 times as luminous as the other star? As quoted above, Dole said that a star had to have less than about 1.43 solar masses and be a spectral type F2 or less to have a habitable planet.

Alpha Corvi is spectral type F1 or F2, has a mass of 1.39 solar mass, and 4.91 times the luminosityof the Sun. I suspect that the most luminous star that could be old eneough to have a habitable planet would be somewhere between 4.5 and 5.0 times the luminosity of the Sun.

TRAPPIST-1 is a spectral type M8 star with 0.08 the mass of the Sun and a luminosity of about 0.00055 that of the Sun. Thus it is probably almost as dim a star as couldp possibly have habitable though tidally locked planet. That makes a difference of about 8,180 to 10,000 between the brightest and the dimmest possible stars with habitable planets.

So if the farther star is as much as 10,000 times as luminous as the nearer star, and is as little as 5 times as from from the planet as the nearer star, it could give the planet as much as 400 times the heat and light as the nearer but much dimmer star does.

But in order for the planet not to be overheated and too hot for life, it would have to be within the habitable zone of the farther star and it would also have to be far outside the habitable zone of the nearer star. And I don't know if even the dimmest star could be so dim that a planet far beyond its habitable zone would still be tidally locked to it.

Thus the farther star in the system could be about 90 to 100 times as far from the planet as the nearer star, and still provide equal amount of heat to the planet if it is about 8,180 to 10,000 times as luminous as the nearer star.

The farther star could provide up to 10 percent of the heat of the planet if it was 316 times as far away as the nearer star, if it was 10,000 times as luminous as the nearer star.

The farther star could provide up to 1 percent of the heat of the planet if it was 1,000 times as far as the nearer star, if it was 10,000 times as luminous as the nearer star.

But it would be a very rare binary star system where both stars were within the mass range suitable for having habitable planets and one was as much as 10,000 times as luminous as the other.

Clearly, in many, possibly most, binary systems where there is a habitable planet in an S-type orbit around one of the stars, the farther star will be a very important source of light on the habitable planet, appearing much brighter than the full moon does on Earth, being bright enough that its periods will become units in the time keeping methods of any native people, and yet the farther star will be a totally insignficant source of heat for the planet.

And there is the problem that the nearer star is supposed to always be at the horizon where the natives of the planet live. That can only be true in the twilight zone of the planet. So the natives have to stay in that twilightzone all the time, or almost all the time. So they should have no reason to go to the light side of the planet, facing the nearer star, where the nearer star will rise higher in the sky, no longer being at the horizon.

And they should have no reason to go to the side of the planet facing away from the nearer star. If they do go there, the nearer star will not be visiple at all, instead of being on the horizon. So the farther star should not heat up the fouter side of the planet enough to make it comfortably warm for the natives. And if the farther star provides enough light to the outer side of the planet, it may cause plants to grow there, and animals to feed on those plants, and thus there may be reasons for the natives to go to the outer side.

So the light the planet gets from the farther star should be enough to be significant to the natives to see by and consider it important, but it should probably not be enough for photosynthesis by plants on the side of the planet away from the nearer star, and thus there should be little life on that outer side of the planet and little incentive for the natives to go there.

So what type of star should be the nearer star that the tidally locked habitable planet orbits in an S-type orbit?

In my opinion, the nearer star should be a very, very dim class M star, like TRAPPIST-1, a star so dim that the tidally locked habitable planet orbits it with almost the shortest possbile orbital period.

Depending on the mass and luminosity of a star dim enough to have tidally locked planets in its habitable zone, the orbital periods of those tidally locked planets should range from about 4.0 Earth days to 289 Earth days to maybe 300 Earth days.

If the other star, the farther star, in the system is a much more luminous, and thus more massive star, the orbits of the two stars around each other can be discussed as though the dim nearer star and its tidally locked planet orbit around the more luminous and more massive farther star.

Thus the farther star, nearer star, and tidally locked habitable planet system can be considered to be analogous to the Sun-Earth-Moon system, where the Moon orbits the Earth which orbits the Sun. The tidally locked habitable planet would be analogous to the Moon, the nearer star would be analogous to the Earth, and the farther star would be analogous to the Sun.

Since the Moon is tidally locked to the Earth, the Earth never appears to move (much) from its position in the sky of a spot on the near side of the moon. But since the Moon orbits completely around the Earth in a month, the Moon makes a full circle in relation to the distance stars in a month, and also makes a full circle in relation to the Sun in a month.

To be precise, the Moon makes a full circle with respect to the background stars in one sidereal month, equal to one complete orbital period around tbe Earth. That orbital period and sidereal month is 27.321661 Earth days. One sidereal month of 27.321661 Earth days is also the length of a sidereal day on the Moon, when the Moon rotates a full 360 degrees with respect to the distant stars in interstellar space. Thus the moon rotates 13.176358 degrees per day with respect to the distant stars.

But during a sidereal month or sidereal lunar day of 27.321661 Earth days, the Earth also travels along its orbit around the Sun. If the Earth's orbit around the Sun was perfectly circular, the Earth would travel exactly 0.9856 degrees of its orbit each day, and thus exactly 26.928477 degrees along its orbit during a sidereal month. That means that the moon would have to rotate another 26.928477 degrees to be aligned with the Sun the same way it was at the begining of the month, which would take another 2.04369 Earth days. But during those Earth days the Earth would travel another couple of degrees, so the Moon would have to catch up a little more, and so on.

The synodic lunar month is the time it takes for the Earth, the Sun, and the Moon to become aligned the same way they were at the start of a synodic lumar month, the time it takes for the phases of the Moon, as seen from Earth, and the phases of the Earth, as seen from the Moon, to go through a complete cycle and return to their original appearance. It is 29.530589 Earth days long.

And I think that the synodic month is also the length of the syndoic day on the Moon, the length of time between two successive sunrises or sunsets at a specific location on the Moon.

And if the tidally locked habitable planet is analogous to the Moon, the synodic day of the tidally locked planet will be the time it takes the farther star to appear to circle the sky once, the time between successive sunrises or sunsets at a spot on the planet. And that will be a period of time which the natives of the planet will nclude in their time systems and calendars.

Note that the sidereal day of the Moon is 27.321661 Earth days, and the synodic day of the Moon, the time between successive sunrises, is 29.530589 Earth days, a somewhat longer period.

And it seems to me that it would be impossible to make the synodic day of the tidally locked planet, the period which would be part of the calendars of the natives, as short as the sidereal day of the tidally locked planet, which would equal one orbit of the tidally locked planet around the nearer star.

I think that the longer the orbital period of the nearer star and the planet around the farther star is compared to the orbital period of the planet around the nearer star, the less difference there will be between the sidereal day and the synodic day of the planet.

And the shorter the orbital period of the nearer star and the planet around the farther star is compared to the orbital period of the planet around the nearer star, the more difference there will be between the sidereal day and the synodic day of the planet.

So the synodic day of the planet could vary between being slighly longer than the sidereal day of the planet to being much longer than the sidereal day of the planet.

Adn how long could the sidereal day of a tidally locked but habitable planet be?

Stars dim enough to tidlaly lock any planets in their habitable zones vary greatly in mass and luminosity and the lengths of the orbital periods of planets in their habitable zones. The orbital periods of those tidally locked planets should range from about 4.0 Earth days to 289 Earth days to maybe 300 Earth days.

So if the orbital periods and thus sidereal days of tidally locked planets should range from about 4.0 Earth days to 289 Earth days to maybe 300 Earth days, and the synodic days of those tidally locked planets in respect to other stars in the system can vary between slightly longer than their sidereal days to several times as long as those sidereal days, the synodic days can be much longer than Earth years, as long as Earth seasons, as long as Earth months, as long as Earth weeks, or a few days long, depending on the length of orbital periods and sidereal days of those planets.

Thus I rather suspect that a really low mass and low luminosity star will be selected as the nearer star, so that the orbital period and sidereal day of the tidally locked planet will be as short as possible.

• Wow, what an answer! And I thought mine might be a bit too long! :) Anyway, although your answer is a bit hard to read, it's great that you actually went and put together the numbers! Sep 15, 2020 at 17:15
• By the way, when speaking about frigid frozen "dark side" of the planet, I didn't mean as cold as "frozen atmosphere", just enough to be uninhabitable, think Earth's very own Antarctica... After all, it will still get a reasonable amount of sunlight (and therefore energy) from the more distant star! Sep 15, 2020 at 17:18
• @BiostheZerg The temperature on the dark side of a tidally locked planet won't depend on what you or I mean, but on the physics of the situation. And some calculations indicate it will be not extremely cold. But without light from the nearer star itsn't have any plant life, unless light from the farther star is sufficient for the plants, so the dark side might be alost lifeless and there may be little rreason for the natives to go there. Sep 16, 2020 at 17:51
• Well, yeah, of course :) What I wanted to say is that the OP can create a world where situation is as I said and the "physics of the situation" will be realistic, or not so unrealistic as to break the suspension of disbelief. Such "dark side of the Earth" can be quite useful for the story, be it for an underground mining/rebel base or just to fly over with a jet liner in order to avoid a hostile neighbour country in the habitable zone! Sep 17, 2020 at 12:19

Probably in an extremely similar way we keep track of time now.

First, I'd briefly like to point out that, as mentioned by other commenters, the way you describe the motion of your stars ignores the fact that their position in the sky will vary based on your position on the planet. A way to describe a similar situation while maintaining internal consistency might be to say that one star tracks the equator while the other star has a perpendicular path along the sky which intersects both geographic poles. In this case, the stars will behave exactly as you described, assuming the observer is standing on one of the planet's poles.

With that established, there's no reason they wouldn't also divvy up their day into sections just like we did in order to keep track of the time of day. For reference, a quick wikipedia search shows that our hours originated from the Greeks and Romans divvying up the days and nights into 12 equal parts. I'd say this is likely due to the significance of the number in their culture. You can come up with a different cultural origin, or choose to have the segments be some other length of time, like maybe having there be six "Hours" in a day, with the origin of the hour being the length of one dark period. Or you could just have them go off of traditional hours and hand-wave the origin story of why there are 30 hours in a day.

As for how to mechanically tell what time of day it is at a given moment, that would likely mirror real life as well. you can tell what time of day it is by simply looking at either of the two stars and measuring how far along they are in their cycles. For simplicity, you may opt to have one star be a sort of designated "time keeper star" so everyone in this society sticks to the same standard.

Sundials can also work, but may have to work somewhat differently depending on how your stars work. if one star is bigger or brighter than the other (which is absolutely possible. For one example, this star system) then a sundial can function with little to no alteration; the brighter star will leave a darker shadow on the dial, and that shadow can be the one which is used to keep time (assuming the brighter star is the timekeeper star.)

Alternatively, if the stars are different colors, such as one being blue and one being red, then the dial can be made out of or surrounded by some material which filters out the light from only one of the stars. This material can be magical, or it could be as simple as colored glass. Either way if, for example, the material only allowed red light to pass through, then only the red star would cast a noticeable shadow on the dial.

If, however, both stars are the exact same size, brightness, and color, then the sundial would probably have to be altered to accommodate multiple light sources, and the way to do this vary, and may also depend on the geographical location of a given timepiece. One easy example may be an observatory building with openings in the ceiling through which the two stars are visible at certain landmark times of the day.