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For the sake of the question, let's define a moon as a natural satellite at least half the size of our Moon. It needs to be spherical like our Moon, not like Phobos and Deimos on Mars. The imaginary solar system is identical to ours in every way (size and type of sun, number of planets, etc.). The planet in question is identical to Earth except for size.

The moons can be captured into orbit in any way you want, just so long as they stay there for a significant period.

If I wanted to have a habitable, Earth-like planet, what would be the minimum size for it to hold two spherical moons in orbit?

Bonus points if the planet can hold two moons and be half the size of Earth

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    $\begingroup$ What makes you think this is any different from the minimum size to hold a habitable atmosphere? $\endgroup$
    – Zeiss Ikon
    Jan 21 at 16:41
  • $\begingroup$ @ZeissIkon Mass plays a role in both, but I don't think there's a reason that the two would be the same - the minimum size to hold an atmosphere also depends on temperature and planetary radius (i.e. escape velocity). $\endgroup$
    – HDE 226868
    Jan 21 at 16:54
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    $\begingroup$ I heard people were saying there are bad things prepared for the life on the planet if there's no moon or more than one moon in orbit, since no moon will (allegedly) cause the planet to wobble constantly and have no predictable seasons, and more than one moon will cause too strong and too chaotic tides for life to survive. I think that both arguments are shaky, to put it lightly, and anthropocentric, but I felt like they should be mentioned. $\endgroup$ Jan 21 at 17:41
  • $\begingroup$ i mean depending on the types of moons and their masses you can have an earthlike habitable world with up to around ten moons with little actual impact. $\endgroup$
    – zackit
    Jan 25 at 17:09
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I think that a planet that can hold an Earth like atmosphere can keep two spherical moons with no big issues.

Let's break down your question in sub-problems.

For a body to become a sphere, it has to have sufficient self-gravity to pull it into a spherical shape. This depends on what the body is made of, for two reasons. The first reason is because the strength of self-gravity depends on the mass of the object rather than its size, implying that bodies made of denser materials become spherical at smaller radii.

The second reason is that some materials are easier to mould into a sphere than others, implying that less strong gravity is needed to push some materials into a spherical shape. The second reason tends to win out. Therefore, for bodies made mainly of rock, the minimum size to become a self-gravitating sphere is about 600km diameter; but, for bodies mainly made of ice, the minimum size is about 400km diameter.

  • Second: what is the minimum size of a planet to be Earth like?

I will take this as a planet capable of trapping water and oxygen in its atmosphere, while being at a distance from the star such that liquid water can exist.

Using the (probably) most quoted worldbuilding image ever, we get that it can be slightly bigger than Mars, or better with an escape velocity slightly higher than Mars, around 7 km/s.

enter image description here

  • Third: Can such a planet hold two moons?

It depends on how far they orbit the planet and if their Hill spheres avoid reciprocal gravitational interference between the two moons.

For a body the mass of Ceres (diameter 900 km) orbiting at 100 thousands km from a body the mass of Venus, the Hill sphere would be 3900 km. If you place the second moon with same mass at 400 thousands km from the main body, its Hill sphere would be 15600 km.

The Hill spheres of the two moons seems therefore to be far enough to not interfere with each other. If you play with the distances so that you have some orbital resonance between the moons you can be rather sure of their long term stability.

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  • $\begingroup$ Ok, this helps a lot, but I'm still a little confused. If the diameter of my fictional planet is about 7500 km (half the diameter of Earth) and has the same composition as Earth, does that mean it has half the mass? Assuming that, I typed in rough estimates (half the mass of our Moon and half the mass of Earth) into the Hill sphere calculator and got 111 m. Does that mean the two moon gravities would interact at 111 meters? $\endgroup$
    – Mandelbrot
    Jan 21 at 17:19
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    $\begingroup$ @Mandelbrot if it has half the diameter it has 1/8 of the Earth mass (volume scales like the cube of the radius). The calculator asks you the orbit distance, not the radius of the planets $\endgroup$
    – L.Dutch
    Jan 21 at 17:22
  • $\begingroup$ @L.Dutch - Reinstate Monica♦ My answer suggests that possibly you might want to redo your calculations using different values. $\endgroup$ Jan 21 at 23:17
  • $\begingroup$ Strictly speaking, the size needed for a liquid moon to be spherical is... small. Small enough that no reasonable person would call it a "moon". We're talking millimeters, maybe even less 🙂. At which point, surface tension, rather than gravity, is doing the rounding, though I'm pretty sure that as you scale up, a combination of the two will do the trick. $\endgroup$
    – Matthew
    Jan 22 at 12:45
  • $\begingroup$ @Matthew, a drop of liquid in the vacuum of space would quickly evaporate for a good part, unless its gravity was high enough to prevent it $\endgroup$
    – L.Dutch
    Jan 22 at 12:47
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What do you mean by "half the size of Earth" when you describe that as a desired goal?

The Planet Earth has a radius of 6,371 Kilometers and a diameter of 12,742 Kilometers. A planet with half the diameter of the Earth, or 6,371 kilometers, would have one eighth the volume. If that planet had the same average density as Earth it would have one eighth (0.125) the mass of Earth.

For a planet to have one half the mass of Earth and the same average density as the Earth, it would have to have half the volume of the Earth. Thus it would need to have approximately 0.7937 times the diameter of Earth, about 10,113.3254 Kilometers, to have a volume of about 0.499999006 that of Earth.

Compare those figures with the minimum masses for a planet to keep and/or to produce a breathable oxygen rich atmosphere which are given below.

Long, long ago, back in 1964, a book was published with a scientific discussion of what is necessary for a planet (or other world) to be habitable for humans.

Habitable Planets for Man, Stephen H. Dole, 1964, 2007. I don't know if the 2007 edition was updated with more recent scientific information.

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

There have been many more recent discussions of the habitability of other worlds using more recent and advanced science. But as far as I know most or all of those discussions are about habitability for life in general, not habitabilty for the more specific case of humans and life forms with similar requirements. On Earth, for example, many or maybe even most lifeforms flourish where humans would swiftly die.

On pages 53 to 58, Dole discusses how massive a world would have to be to retain a dense enough atmosphere of oxygen. On page 54 Dole concludes that a planet would have to have an escape velocity of 6.25 kilometers per second to retain an oxygen atmosphere for geological time periods. That corresponds to a planet with:

a mass of 0.195 Earth mass, a radius of 0.63 Earth, and a surface gravity of 0.49 g.

A radius of 0.63 Earth is a radius of 4,013.73 kilometers or a diameter of 8,027.46 kilometers.

Dole believed that a planet of that size could retain an oxygen rich atmosphere, but could not produce one. If Dole was correct, a planet of that size could only have an oxygen rich atmosphere if it was terraformed to have such an atmosphere by a highly advanced society.

Dole made two different calculations of the minimum mass that might be necessary for a world to not only retain an oxygen rich atmosphere but also to produce one. One was a mass of 0.25 Earth mass, and the other was a mass of 0.57 Earth mass. Dole considered those masses to be inaccurate, and settled on a mass of 0.4 Earth mass as the minimum mass required to produce an oxygen rich atmosphere.

This corresponds to a planet having a radius of 0.78 Earth radius and a surface gravity of 0.68 g.

A radius of 0.78 Earth radius is a radius of 4,969.38 kilometers and a diameter of 9,938.76 kilometers.

Mars has a mass of 0.107 Earth mass, a radius of 3,389.5 kilometers, and a diameter of 6,779 kilometers, so any world massive enough to retain and/or to produce an oxygen rich atmosphere should be significantly more massive and large than Mars.

Until and unless a science fiction writer finds a later and better set of calculations than Dole's they should not write about a planet with an oxygen rich atmosphere breathable for being similar to humans unless it has mass of at least 0.195 Earth and a diameter of at least 8,027.46 kilometers. And if they don't want the planet to have an artificial oxygen rich atmosphere created by a highly advanced civilization but have a naturally formed oxygen rich atmosphere instead, they should make their world have a mass of at least 0.4 Earth mass and a diameter of at least 9,938.76 kilometers.

And of course either minimum mass would be significantly greater than the mass of Mars, 1.822 or 3.738 times the mass of Mars. And also significantly less than the mass of Venus, 0.239 or 0.490 that of Venus. L Dutch - Reinstate Monica used a planet with the mass of Venus, 0.815 that of Earth, to calculate the Hill sphere of the planet in his answer.

I note that the size of your planet's Hill Sphere, will depend on the planet's mass, the distance to its star, and the mass of the star. I also note that a moon can have a stable orbit only within about 0.5 to 0.666 of the outer edge of the Hill Sphere.

https://en.wikipedia.org/wiki/Hill_sphere#True_region_of_stability[2]

I am not certain that two moons could have stable orbits around the least massive possible habitable planet at the distances indicated in L Dutch - Reinstate Monica's answer.

The Hill Sphere of Earth extends to about 1,500,000 kilometers, so the zone where moons can have stable orbits should extend to about 500,000 to 750,000 kilometers.

https://en.wikipedia.org/wiki/Hill_sphere#Formula_and_examples[3]

The example in L Dutch - Reinstate Monica's answer has two moons orbiting at about 100,000 and 400,000 kilometers, and both would be within the stable orbital zone of Earth. However, the question asks for a planet as small as possible, and L Dutch - Reinstate Monica mentioned a planet slightly larger than Mars earlier in his answer.

A planet significantly smaller than Earth would have a smaller Hill Sphere than Earth, and the moons would have to orbit closer. But if the planet is less massive, moons of a specific mass will have larger Hill spheres, perhaps interfering with one another.

Perhaps L Dutch - Reinstate Monica should recalculate his orbits for a planet massive enough to have an oxygen rich atmosphere orbiting a more massive and brighter star than the Sun at a greater distance than Earth orbits the Sun, to find a stable orbital configuration.

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