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Okay, so imagine that there is an Earth-like planet, with the following conditions:

  1. It's further from it's sun than the Earth is from the Sun (not much, but enough for weather to be a little colder than the Earth).
  2. It's more or less the same size as the Earth (same proportion of land and water, too).
  3. Since it's further from it's sun, it takes more time to complete an orbit that the Earth (for instance, 400 Earth-days, instead of 365).

Given those conditions, I would like to know which of the following would be more stable for this Earth-like planet to have a similar day/night and moon cycle to the Earth:

  • A moon that's the same size as the Earth's moon, only closer.

OR

  • A moon that's as far away as the Moon is to Earth, only that the moon of this planet is bigger (let's say, twice as big as our moon).

Sorry if it's kind of complicated. Please tell me if it's not understandable and I'll try to edit it so it's as clearly as possible! Thanks in advance :)

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Both would be stable. A single moon orbiting a planet is generally stable provided it's comfortably inside the true region of stability, in the Hill Sphere and provided that it's not so large that tidal forces become a stability issue or not so close that it dips inside the roche limit and might break apart.

The only way you might run into instability in your scenarios, is if you have the smaller moon very close, inside the Geosynchronous orbital distance, where the Moon would move ahead of the tidal bulge it creates and that would cause the Moon to be drawn towards the planet and in time, crash into it, similar to how Phobos is expected to crash into/break into a ring system around Mars.

As the planet moves away from the sun it's stable region expands. The geosynchronous orbit is unrelated to the sun, that's between the masses of the planet and moon only, but as long as your moon is orbiting between geosynchronous orbit and inside the stable region of the hill sphere, it should be fine. (unless you give the planet or moons a retrograde rotation, then you need to re-examine the long term tidal effects). For most orbits, the planet rotates ahead of the Moon so the tidal bulge pushes the Moon slowly outwards, but this effect gets weaker as the Moon gets further, so it's generally long term stable over billions of years, unless the planet is very close to the sun and it has a tiny hill-sphere.

Only two well known moons in our solar-system orbit ahead of their tidal bulges that leads to orbital decay (a handful of tiny captured moons do too, but those are so small and their tidal forces are largely irrelevant). Only Phobos and Neptune's wrong way moon Triton orbit ahead of their tidal bulges, resulting in them being pulled towards their planets, not pushed away. Triton is sufficiently distant that it's timing to crash into/break into a ring system around Neptune is billions of years off. All the other moons, for tidal reasons, should be moving away from their planets, mostly very very gradually.

The 4 Jovian moons, however, are more complicated as Jupiter has a gonzo magnetic field, a magnetosphere that's almost wind like and it's moons are tidally locked to each other and it's innermost moon is losing mass to Jupiter's magnetosphere - so . . . all bets are off when it comes to Jupiter's 4 inner moons.

2 or more large moons can sometimes interact with each other, creating some instability. One moon systems are, more often than not, long term stable.

(too long?)

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It looks like you are mixing things a little: If earth was farther away from the sun, it will mean nothing to the day/night cycle.

Day/night cycle only depends on the way the earth rotates on itself, not around the sun. So your question is the same for an planet with same orbit as ours as it is for a planet with an orbit a little larger.

To answer your question,

[for it] to have a similar day/night and moon cycle to the Earth:

Both are possible. Day/night cycle greatly depends on initial planet's spin. What changes here is the mass of the earth-like planet we are talking about, and this changes the speed at which the moon rotates around. the gravity of the moon will affect the planets rotation speed over very long time range.

  • first case, planet is heavier to have a moon the size our moon (I assume same mass as ours) rotating faster around - moon cycle is different, but you can have same day/night cycles as ours

  • second case, planet is lighter, the big moon rotates slower, but you can also have same day/night cycles

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A closer moon would bring higher gravitational pull and it orbits faster than our moon.

"A moon that's as far away as the Moon is to Earth, only that the moon of this planet is bigger (let's say, twice as big as our moon)."

In this case too the moon cycle won't be similar to ours, since again the size of object matters in gravitation.

but the rotation of That Earth like planet gives day/night cycles which you haven't touched of.

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