What is the formula for how far a planet has to be from its sun to have a moon?

How do you know that? Last I read, there was not even any particularly prevalent theory about why Mercury and Venus have no moons.

So unless there’s something new, I don’t think it’s true that they there is any reason they could not have moons given their distance from the sun. They just happen not to (and/or it may be something to do with the presumed increased traffic closer to the sun having been more likely to disrupt moons closer in). So, no formula.

Your questions about the planets have been popping up for a few days now, and I have rigorously held my tongue. But this one forces me to ask why it is YOU believe that Venus and mercury must be moonless? Do you think they’re too close to the sun to capture or form moons, or too close to hold onto any satelites they formerly held?

Well I read that their proximity to the Sun does not allow them to have moons… That must have been false information.

Do you think it possible for
us to plant satelites around those planets in stable orbits?

I don’t know. Maybe. But if Mercury had a moon, I would make it very small. Venus’s moon would be just large enough to be spherical.

I looked it up and the reason why Mercury does not have a moon is because it doesn’t have enough gravity.

Venus is perfectly capable of having a moon. But sadly, it doesn’t.

Why do you think Mercury lacks the gravity to have a small moon? It doesn’t. Mercury is quite larger than Pluto, for example, which has five moons.

maybe he means Mercury would lose its moon to the tug of the sun.

Why is it sad that Venus does not have a moon?
Venus probably does not care, and has enough going on to keep it busy.

It would have been cool for Venus to have a moon – then it would be even more like a twin of Earth.

To answer the OP’s question, it might be that gravity of the Sun is stronger as you get closer, so a wandering body might be more likely captured by the Sun than by Mercury or Venus. The origins of each planet’s moons are all different – some might have been wanderers, some might have been remnant debris. The remnant debris would not be as influenced by the Sun since the pieces would already be in the same orbit as the respective planet. And our Moon might have been a wayward protoplanet, which is an entirely different formation process.

There probably is an inner limit to the distance from the sun that a planet might approach and maintain its moon in a stable orbit. The limit would probably vary considerably with the masses of the bodies, orbital distances of the moons, etc. It makes sense that anything approaching the orbit of Mercury is in fact well on its way to the sun and accelerating at speeds that would make capture by feeble Mercury extremely unlikely.

Venus and especially Mercury are so close to the sun that any moon orbiting these planets at a distance will be in an unstable orbit and will be captured by the sun. Too close to the planet and the moon will be destroyed by tidal forces. There is only a narrow band of stability which makes it unlikely a moon will survive in orbit for any length of time. The mathematics are probably what describes the three body problem

@flutherother this is exactly what I read about.

Ok, so there are two calculations that describe that:

The Roche Limit (the distance from a planet within which a moon would be torn apart by tidal forces).

The Hill sphere (the region where a planet can hold a moon in orbit without it being pulled out of orbit by the star they both orbit).

These do limit the likelihood of moons for Venus and (especially) Mercury, because they are close to each other, or perhaps overlapping for Mercury depending on how big the proposed moon would be.