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Kinda ran out of characters for the title so had to improvise. Gravity decreases with the square of the distance due to the inverse-square law, and tides are inversely proportional to the cube of the distance since they're the difference between gravity.

So one would assume that if a gravitational phenomenon exists that's inversely proportional to the fourth power of the distance, it would be the difference between tides. If so, does it have a name? And if not, does any similar phenomenon that scales to 1/d^4 exist?

user267545
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    What would you class as a "similar phenomenon"? – ProfRob Jan 22 '24 at 19:47
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    I'm not sure this is right, but IIRC the force between magnetic dipoles is inverse 4th power of distance, and Weak nuclear force is inverse 5th power? This might be a better question for the Physics SE than Astronomy. I'd recommend dropping the first part of your title to just ask "Are any natural forces inversely proportional to the 4th power of distance?" or something like that. – Darth Pseudonym Jan 22 '24 at 22:22
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    @ProfRob Preferably a force related to gravity (inb4 “gravity is actually a result of spacetime distortions ”) but like Darth Pseudonym suggested, any force would be interesting. Off to Psychics SE – user267545 Jan 22 '24 at 22:37

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Yes, a very important one!

I'm pretty sure the force field due to a rotating body's $J_2$ force field due to it's equatorial oblateness varies as $r^{-4}$. See Wikipedial's Geopotential model; Largest terms

The potential field will be

$$u = J_2 \frac{1}{r^3}(3 \sin^2\theta-1)/2$$

and when moving away in a given direction, the magnitude of the force will then vary as $r^{-4}$.

Earth's $J_2$ term is pretty big. In low Earth orbit it's about a 1 part per thousand variation in Earth's gravitational field. It precesses the orbits of inclined satellites, so for example the plane of ISS' orbit precesses around the Earth (relative to the starts) about every two months, so that it alternates between constant daylight for several weeks then alternating day/night every 90 minutes for several weeks.

It has a small effect on the Moon as well.

This force also explains why Jupiter's and Saturn's moons (and Saturn's rings) are all so tightly constrained to the equatorial planes of those planets.

uhoh
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    Thank you! That explains why the cut-off distance where natural satellites stop having equatorial orbits lies so close to the planet. – user267545 Jan 23 '24 at 12:53
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    @user267545 that's a really interesting observation, I never though of that! – uhoh Jan 23 '24 at 13:31