Working out a binary planet setup with additional moons

[Yora]

I got this idea for a world that is an Earth-like moon orbiting a larger planet but also having other small moons visible in the sky. Since the world would be very Earth-like it would be assumed to be pretty big and I don't want the main planet to be super huge in the sky, so I am thinking of a setup like the Pluto-Charon-Hydra-Nyx system. Since we have an actual example we know that it can exist. But when you scale it up to a Super-Earth with a binary companion say 80% the mass of Earth, what other effects and phenomena would we expect to see?

The moon is tidally locked to Earth. Pluto and Charon are tidally locked to each other. Saturn and Titan aren't locked at all.

Ideally I'd want to have no tidal locking to have the main planet (let's designate it as A) move around in the sky of the world (designated B) with the other moons (C and D) moving around the whole thing.

If that doesn't work out with a Super-Earth, could it work with a small gas giant? Compared to Jupiter and the Earth, Neptune is not actually that big. Would smaller gas giants always be blue instead of brown, or is that just coincidence in the solar system?

[factotum]

I think the problem is having a moon which is 80% of the mass of the parent body. I would expect the parent to be tidally locked to the moon if the moon is locked to its parent in that situation, which would give it a fairly slow spin and thus a very, very long day. The only way to have other, smaller moons in such a system and make it anything like stable would be to have them in orbital resonance with the big one, I think.

From an orbital mechanics point of view it would make more sense for the habitable worlds to be two moons of a gas giant, IMHO. However, gas giants and large rocky planets tend to form at different places in the solar system, so getting two Earth-like worlds orbiting the same gas giant would be a bit of a stretch.

[Balyano]

Not sure if it would be much help but you could take a look at the Poul Anderson novel Murasaki.

Spoiler: From Wiki

Show

Poul Anderson, who had a degree in physics, worked out the physical framework for the anthology based on the characteristics of HD36395 as they were known in the early 1990s: one third of Earth Sun's mass, 82% of its diameter, spectral type M1 with a photosphere temperature of 3,400 K and a maximum emission in the near infrared. (The star is in fact very similar to Gliese 581, now known to have a planetary system.)

The twin terrestrial planets are separated by an average distance of only 156,000 km (about 40% of the Earth-Moon distance). They orbit around their center of mass in 91 hours in locked rotation, which minimizes the effects of the huge tidal forces which they exert on each other. This constellation orbits Murasaki within the habitable zone, at a distance of only 0.223 astronomical units (sidereal year, 66 Earth days) where the planets receive about the same amount of total irradiation Mars gets from the Sun; however, with a spectral power distribution shifted to much longer wavelengths. From this distance the star appears as a disk of 1° 40' diameter, almost three times as large as the Sun (or the Moon) when observed from Earth. Both planets have plate tectonics, causing most of their carbon dioxide being bound in their crusts, and have oxygen-nitrogen atmospheres of Earth-like composition.

Genji
Genji is a Super-Earth but only moderately so: it has 2.8 times the mass and 1.36 times the diameter of Earth, and 1.5 times Earth's gravity. The side of the planet that constantly faces its companion world Chujo (“Moonside”) is mostly land, the other hemisphere (“Starside”) is mostly ocean. The mean surface temperature is +20 °C, slightly warmer than Earth. Although humans in good condition can physically accommodate to the high gravity, the sea level air pressure of 3.1 bars which results from this gravity (as per the barometric formula) requires artificial decompression for safe breathing. Only at 5,800 meters (an altitude found on this planet only in the form of a few small highlands that are cold and arid) the atmospheric pressure drops to Earth-standard 1 bar.

Chujo
This is the smaller world, with 0.76 Earth masses at 94% of its diameter, and 85% of its surface gravity. The mean surface temperature is only +5 °C. Because most of this chilly planet's water is locked in permafrost and glaciers, the atmosphere (which has a sea-level pressure of 0.7 bar, equivalent to about 3,650 m altitude on Earth) is not only thin but also uncomfortably dry. From the hemisphere where the companion world Genji can be seen it hangs in the sky as a globe of 6° 20' diameter, and when full gives over 320 times the light of the full Moon as seen from Earth.

[Yora]

Oh yeah, Object A is going to be dead. Only object B is habitable. Should have made that clear.

I discovered that there is such a thing as gas dwarves. They have a rocky core with a huge Hyrogen-Helium atmosphere and a total mass of 1.7 to 4 Earths. That might actually be even cooler than a Super-Mars. And with the ratio of core to atmosphere being variable I think the two objects should be able to be pretty far apart while a low density A would still look pretty big in the sky of B.

To work out orbits I believe we can ignore diameter and density and only need to account for mass. If we take an object with 1 Earth mass (B) and one object with 3 Earth masses (A), it should be a simple equation to determine the minimum distance to prevent tidal locking, right?

[Trekkin]

Oh yeah, Object A is going to be dead. Only object B is habitable. Should have made that clear.

I discovered that there is such a thing as gas dwarves. They have a rocky core with a huge Hyrogen-Helium atmosphere and a total mass of 1.7 to 4 Earths. That might actually be even cooler than a Super-Mars. And with the ratio of core to atmosphere being variable I think the two objects should be able to be pretty far apart while a low density A would still look pretty big in the sky of B.

To work out orbits I believe we can ignore diameter and density and only need to account for mass. If we take an object with 1 Earth mass (B) and one object with 3 Earth masses (A), it should be a simple equation to determine the minimum distance to prevent tidal locking, right?

I went and built this in Universe Sandbox; it's stable over longish time scales, but the moons need to be really close to their parent bodies; Earth's Hill sphere gets distressingly close to the gas dwarf sometimes, so any other moon orbiting it with a period greater than about a week risks getting dragged off. (Or rapidly shot off into the Sun, as the case may be.) Earth stays in the Sun's habitable zone at all times, nothing's within anything else's Roche limit, and I can't see any other reason why this shouldn't be physically possible. Gas dwarf spins around the Sun, Earth spins around the gas dwarf, Earth and gas dwarf get one moon each.

As for the time to tidal locking: you also need the initial spin rate of the satellite, its semi-major axis, its moment of inertia (so yes, you do need the radius), its dissipation function, and its tidal Love number. Then it is a simple equation, yes; you may want to read B. Gladman; et al. (1996). "Synchronous Locking of Tidally Evolving Satellites". Icarus. 122: 166–192. and S.J. Peale, Rotation histories of the natural satellites, in J.A. Burns, ed. (1977). Planetary Satellites. Tucson: University of Arizona Press. pp. 87–112.).

[Yora]

I think that's a bit more work than I want to put into it to be able to say "There's a big planet in the sky instead of a moon, plus two little moons."

But thanks for checking it. I think I really should get Universe Sandbox 2 myself. My computer should be able to run it just fine.

[Trekkin]

I think that's a bit more work than I want to put into it to be able to say "There's a big planet in the sky instead of a moon, plus two little moons."

But thanks for checking it. I think I really should get Universe Sandbox 2 myself. My computer should be able to run it just fine.

I thought it might be, yes. Besides, does it really matter if the moons are tidally locked to the gas dwarf?

[Kalmageddon]

Only tangentially related, but did they fix the chance of life variables on Universe Sandbox? Last time I checked the only possible way to get something like a 0.1 chances of a planet having life was spawning earth, even a planet identical to earth wouldn't do.

[Yora]

I thought it might be, yes. Besides, does it really matter if the moons are tidally locked to the gas dwarf?

For little C and D it doesn't matter, and would make things needlessly complicated. But I don't want B to be locked to A, so that A keeps moving in the sky of B.

[factotum]

For little C and D it doesn't matter, and would make things needlessly complicated. But I don't want B to be locked to A, so that A keeps moving in the sky of B.

Didn't you also say that you didn't want A to appear huge in B's sky? Seems to me both problems have the same solution--have B orbiting a long distance from A. The further away it is the less likely tidal locking will occur, regardless of any other consideration. If you have the other moons closer in then they'll always be accompanying A in the sky, which may or may not be something you want?

[Yora]

That's what makes a gas dwarf such a nice option for A. With a relatively small core and big atmosphere it can be quite big in size while still having a relatively low mass. That should help with looking big when seen from B but far enough away to not tidally lock it.

Having C and D orbiting the binary pair instead of only A is probably a simpler setup than having them orbit only A. This arangement seems to work for Hydra and Nix.

[Trekkin]

That's what makes a gas dwarf such a nice option for A. With a relatively small core and big atmosphere it can be quite big in size while still having a relatively low mass. That should help with looking big when seen from B but far enough away to not tidally lock it.

Having C and D orbiting the binary pair instead of only A is probably a simpler setup than having them orbit only A. This arangement seems to work for Hydra and Nix.

It's simpler to set up as point masses, yes, but there are problems with tides to consider; the gas dwarf, being two orders of magnitude heavier than the Earth's moon, will produce significant tidal effects if B is not significantly farther away -- but they're both fairly light in terms of the size of their Hill spheres. I couldn't find an orbit of the A-B barycenter that won't eventually either fall off and orbit the Sun on its own or get dragged into the orbit of A or B alone.

[Yora]

So with small moons, Moon C would be orbiting Planet A and Moon D orbiting Planet B, for example? That also sounds pretty cool.

Any idea how long it would take for A to go through it's phases as seen from B? Or in other words, how long does it take A to make a full orbit around the center of gravity between A and B? I imagine it would be much faster than a month.
Though it would depend on the specific mass of A and B and their distance from the center of gravity, right? Just as a ballpark estimate, would we be talking weeks, days, or even hours?

Edit: I got US2 and tried it out myself. A 4 Earth-Mass Mini-Neptune and Earth as a binary pair at 1 AU of the sun seems to work very well. When I gave each of them a moon, Earth's moon flew off into space and shortly after Neptune's moon crashed into Earth. Not as easy as I thought it would be.

[Trekkin]

So with small moons, Moon C would be orbiting Planet A and Moon D orbiting Planet B, for example? That also sounds pretty cool.

Any idea how long it would take for A to go through it's phases as seen from B? Or in other words, how long does it take A to make a full orbit around the center of gravity between A and B? I imagine it would be much faster than a month.
Though it would depend on the specific mass of A and B and their distance from the center of gravity, right? Just as a ballpark estimate, would we be talking weeks, days, or even hours?

Edit: I got US2 and tried it out myself. A 4 Earth-Mass Mini-Neptune and Earth as a binary pair at 1 AU of the sun seems to work very well. When I gave each of them a moon, Earth's moon flew off into space and shortly after Neptune's moon crashed into Earth. Not as easy as I thought it would be.

Well, how long do you want it to take? These numbers are highly flexible and we haven't tried stabilizing with other planets yet.

Generally, I've found the most stable results where your Earth is orbiting the gas dwarf on a 30-60 day cycle, with the moons taking anywhere from a week to two weeks to orbit their primaries. I'll set it to Earth-centric motion when next I'm home, but in short, GasDwarf looks like it orbits every 60 days, GasDwarf A (Earth being Gasdwarf B) makes a pretty little spirograph as seen from Earth, and GasDwarf B' orbits every week; I should try setting B' to retrograde.