Lagrange Points regarding a moon 1.5 times the mass of Earth's Moon and one 0.75 times the mass of Earth's Moon

Question

Working on a lunar orbit system. As place holder numbers, two moons are orbiting a planet equal to earth's mass. One moon, Selenara, is 1.5 times the mass of Earth's moon, and the other, Thoth, is .75 times the mass of Earth's moon. I want these two moons to have the same orbital period/lunar month--as in, it takes them each thirty days to complete their cycle.

Primary Question: How can I place the moons to make their orbits viable? I am interested in them preferably sharing the same lunar month, but if better options exist I would like to know.

Desired outcomes: I would prefer for both moons to be visible in the sky simultaneously at some points of the month, however I also wanted them to be in different points in their phases, which I suspect isn't possible.

If it is impossible, I want them at opposite orbits to each other(as in, while one is on the sun facing side of the planet, the other is on the night side).

Other Notes: Selenara, the bigger planet, is meant to support some life. Life was terraformed here at some point, and I need it to be viable. Any issues that orbital mechanics might cause should be avoided.

Answer 1

I want these two moons to have the same orbital period/lunar month--as in, it takes them each thirty days to complete their cycle.

With their specified masses, this is not possible. If Thoth was a lot smaller, at most a thirtieth of Selenara's mass, but preferably rather less, then Thoth could be in the L4 or L5 point of Selenara's orbit and would be stable there. As it is, their masses are too similar, and Lagrange stability will not apply. Their orbits would be chaotic and unpredictable; the likeliest outcome is that they collide and end up forming a single moon. The fragments thrown off in the collision would do a lot of damage to the planet's surface.

Putting Thoth at Selenara's L1, L2 or L3 points will not make a significant difference, and a collision is, again, the likeliest outcome. Those points are unstable no matter how tiny Thoth's mass is; satellites in Earth's L1 and L2 points need to use thrusters to stay there.

Making Thoth low enough mass for it to be stable in L4 or L5, while retaining its apparent size and visibility, requires that it be a balloon. Earth's moon's density is only 3.3g/cm3; cutting its mass enough requires a density of 0.1g/cm3 or less, and there's no solid natural material with such a low density. Of course, a pressurised balloon will be punctured by meteoroids very soon on an astronomical timescale.

Having the two moons in different elliptical orbits with the same 30-day period won't work either: they're too close together and will perturb each other, leading to a collision.

The nearest thing to a workable solution is to have Thoth in a circular orbit quite close to the planet, not far outside the Roche limit, and Selenara in a distant circular orbit, not far inside the planet's Hill Sphere. This is not stable either, but you'll probably get a few million years before they collide.

I would prefer for both moons to be visible in the sky simultaneously at some points of the month

You'll get that with this solution, but the time of the month when they're both visible will change from month to month.

I also wanted them to be in different points in their phases, which I suspect isn't possible.

They will be at different points in their phases most of the time, but what the phases are when they're both visible will change from day to day.

If it is impossible, I want them at opposite orbits to each other(as in, while one is on the sun facing side of the planet, the other is on the night side).

That's the really impossible one, if you're using anything like real astrophysics. To make that stable, you need a magical universe, or super-advanced science that can push large moons around without any kind of drive flame.

Answer 2

The only places I think you could put Thoth with stability is Selenara's L4 or L5 Lagrange Point, these are the attractive Lagrange points, they tend to accumulate mass in the form of Trojans. I'm pretty sure this would also lead to the phase inconsistency you're interested in as Thoth would be 60° ahead of, or behind, Selenara. Both bodies would, by default, be visible in the sky at certain times of the day and night, assuming one can see more than 60° of sky arc, since they share an orbital track. However, I'm pretty sure that as written Thoth is simply too massive to be a Trojan without creating orbital instability. You could play with the density, have Thoth look half the size but be a primarily icy body that has a very low mass for it's apparent size but that's a different question.