What orbital setup allows for the 'most' solar eclipses?

Let’s start with the Earth-Moon system, but tweak it so that we can get the ‘most’ total solar eclipses possible. Obviously if we put the Moon into an orbit in the Earth’s orbital plane, we’d get a total solar eclipse somewhere between the Tropics every month. But can we do better?*

Can we stably put the Moon into a polar orbit such that we can sweep an eclipse from north to south or vice versa every month, instead? Maybe this gives us a greater swath of Earth from which an eclipse is visible. Bringing the Moon in as close as possible shortens the lunar month so we get eclipses more frequently, too. And polar tides would be kinda cool, right?

*Poorly specified, I know, but make of it what you will - in the plane of the ecliptic, a total eclipse won’t be seen north or south of the Tropics, so tilting it a little bit gets you total eclipses at higher latitudes but those are quicker. Maybe think of it as trying to minimize Earth’s total solar irradiation by blocking the Sun as much as possible.

Does your orbital setup need to be stable? Because if you make the Moon much bigger than the Earth, and then put the Earth in the L1 Lagrange point, you’d have a perpetual eclipse, covering the entire planet… but it’s an unstable arrangement.

That works, but I’d like to keep the Earth and Moon at their current sizes. I supose you could put the Moon at L1 for a permanent eclipse, but it would be pretty annular - covering something like only 5% of the Sun’s disk at that distance, I think?

I don’t mind nudging up against the Roche limit, though.

Are we allowed multiple moons?

Can you redefine “month”?

The polar orbit idea wouldn’t work, because (to a good approximation) the orientation of the plane of the moon’s orbit would remain fixed in space. But to get eclipses you need the plane of the moon’s orbit to always contain the Sun, which would require to orbital plane to slowly rotate once over the course of the year.

The fact that the Earth’s orbit isn’t exactly circular would mean that you’d actually need the plane of the orbit to rotate at different rates over the year — the Earth moves faster when it’s closer to the Sun, so the orbital plane would have to rotate faster at those times and slower at other times. I wouldn’t rule it out entirely, but it’s hard for me to see how the stars could align [heh] to make this possible.

I was thinking something along the lines of a Sun-synchronous orbit, where the near-polar orbit precesses once per year. The Moon probably wouldn’t survive such a low orbit, though. Nor would anything within 200 km of the coasts, for that matter.

So a lunar orbit in the ecliptic or just slightly tilted is probably the best bet that I can think of. Still, there are all those weird things like Molniya orbits and tundra orbits and horseshoe orbits and I don’t know what else, so maybe something better exists?

A lunar month is the time taken between full moons for an Earth observer. I suppose if you had a way to set up a permanent solar eclipse that would lose its meaning.

No, just the one.

Don’t you oppress us!

Interesting idea — I hadn’t heard of that those orbits!

If you’re worried about tides, you could also pare down the physical size of the moon. At an altitude of 800 km, you’d only need a moon 7–8 km in diameter to create a spot of totality on Earth’s surface. This would also drastically reduce the tidal effects relative to a full-sized Moon in a low orbit, though I’d have to do the math to figure out whether they’d be larger or smaller than what we have now.

The math on that is actually pretty simple. Tidal effects from a spherical body depend only on apparent angular size and density. So if your smaller moon is still just large enough to cover the disk of the Sun, and it’s made of the same material as the real Moon, it’ll raise the same amount of tides as the real Moon.

Locally, at least. The fact that the real Moon has tidal effects on the entire planet, while your micro-moon is only near a small portion of the planet, is probably relevant.

Yeah, I was concerned that you couldn’t use the usual approximations given that the micro-moon would be 800 km from one side of the Earth and 13,500 km from the other.