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Is it possible for a planet's moon to always rise at the same time of the night? And in this way be used as a timekeeping device? What if there are two moons?

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It wouldn't qualify as a "moon" per se, but an object placed in the L2, L4, or L5 points of the planet/sun system would remain in a stationary position relative to the sun (L1 and L3 would also be stationary, but not visible). An object in L2 wouldn't be stable in a multi-body planetary system (it would drift away in a matter of months), which leaves L4 and L5.

What would an object in L4 look like? According to the Astronomy StackExchange, the limit on mass is about 10% that of the planet, so for an Earth-like planet, it would be about 10 times that of the Moon, roughly that of Mars. To an observer, it would have an angular diameter of 5 seconds of arc, slightly larger than that of Uranus, or of Mars at its furthest, and would be roughly second magnitude in brightness.

Would it be useful for timekeeping? For a pre-technological society, certainly. It's one of the brightest objects in the sky, and it rises or sets within a few minutes of the same time every night (give or take planetary axial tilt).

Could you have two of them? Yes, one in L4 and one in L5. You would probably need to make them smaller than the maximum, to keep them from interfering with each other's stability, but a pair of Moon-sized objects (third-magnitude or so in brightness) would be quite reasonable.

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    $\begingroup$ I should note that Mark originally first posted an answer involving Lagrangian points, but then deleted it, presumably because I pointed out what appeared to be an error. His new answer appears to be accurate and upvote-worthy. I say this to make it clear that the Lagrangian points were his original idea. I had been working on my answer during the same time as he had, which included the Lagrangian points - don't think I copied him or he copied me. So the point of this lengthy comment is A) Don't overlook this answer and B) We had the same idea at different times. $\endgroup$
    – HDE 226868
    Jan 11 '15 at 2:30
  • $\begingroup$ Forgive me if I am misunderstanding, it appears as if the two answers contradict each other. Mark seems to be saying a moon at L4 and / or L3 is possible, while the other answer indicates this is not stable? $\endgroup$
    – Jaz
    Jan 12 '15 at 21:12
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    $\begingroup$ L4 and L5 are stable, but are too distant to show up as a disk rather than a point. L2 and L1 are unstable but close enough to show up as a disk. L1 has the additional disadvantage of possibly being hidden by the glare of the sun. L3 is unstable, too distant, and behind the sun. $\endgroup$
    – Mark
    Jan 12 '15 at 22:26
  • $\begingroup$ Thank you for clarifying. L4 and L5 will do nicely, although appearing more like a star than a moon to the agrarian society. $\endgroup$
    – Jaz
    Jan 13 '15 at 11:20
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In order for this to be possible, the moon would have to be in the same place relative to the planet and the star throughout the day - in other words, if you drew a line from the center of the star through the center of the planet, then drew a line from the center of the planet through the center of the moon, you would have two lines that intersected at an angle that is forever constant.

For those who got past that first paragraph sentence, there are two places that would work: the Lagrangian points $L_1$ and $L_2$. The five Lagrangian points are laid out like this:

Lagrangian points

The "holes" and overall shape of space here is an analogy for gravity wells, so ignore that, but treat the green lines connecting the star, planet and points as rigid - in other words, as the planet revolves around the star, the green lines rotate with it, as do the Lagrangian points.

$L_1$ and $L_2$ will satisfy your scenario. However, $L_1$ should be discarded because putting the moon there would make it only appear in the daytime! So we'll take $L_2$.

The problem is that the only stable Lagrangian points are $L_4$ and $L_5$ notice how the others are near the metaphorical depressions of space. This indicates that if an object at them shifts, it will move away. For $L_4$ and $L_5$, the object will merely return.

To stay at $L_1$ or $L_2$, stationkeeping is required. So you'd have to attach thrusters to your moon to keep it in the same place! Alternatively, put it in a Lissajous orbit. You'll still need some thrusters, but it should be stable. Here's what the orbit will look like:

Lissajous orbit

This kind of orbit would be very tough - if not impossible - to put a very massive object in.

Two moons would make this job nearly impossible, because if they were near each other, they'd most likely perturb each other's orbits and ruin any stability. Your odds of getting this to work for one moon are slim to none; your odds for two moons are slightly worse.

Or you can use a little handwavium to solve all your technical difficulties!

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A moon whose orbital period is half a day would appear to observers on the ground to go around in a day, rising in the west.

Phobos is a more extreme example: its period is less than a third of a day, so it rises in the west twice a day.

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