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For example, if the Earth - by some unknown force - stopped spinning, would the orbit of the Moon be effected at all? Its not a perfect example because Earth's gravity is not uniform - it has mountains and tectonic anomalies - so what if, instead of Earth, it was a perfect spheroid with perfectly uniform gravity?

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The most prominent effect of spin is that it deforms planets into oblate spheroids, they get wider at the equator.

The gravity of perfectly round planets can be treated as an equivalent point mass, due to the shell theorem. But the shell theorem is not valid for any shape that deviates from a sphere.

These deviations from a perfect point mass gravity are the spherical harmonics. The by far largest and most important of these is the J2 term, caused by the oblateness.

So what effect does J2 have on orbits?

It causes the orbital plane of a satellite to rotate. The ascending and descending nodes (the points where the orbit crosses the equatorial plane) will slowly move around (Nodal precession). This is sometimes useful, for instance for making a satellite that stays in sunlight all year round.

The oblateness of the Earth has some effect on the constantly oscillating orbit of the Moon. But this is more complicated due to the significant influence the gravity of the Sun has over the Moon.

It's also important to note that the spin itself does not directly cause these effects, only the shape does. If an object is small enough that it stays rigid (asteroids, spacecraft), the spin will not be strong enough to deform the object into an oblate spheroid (hydrostatic equilibrium).

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    $\begingroup$ J2 also causes objects orbiting close to equatorial to orbit a bit faster than objects with the same semi major axis length but orbiting closer to polar. Compared to $GM/r^2$, oblateness increases gravitational acceleration over the equator but reduces it over the poles. $\endgroup$ Jan 25, 2023 at 14:24
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    $\begingroup$ In addition to the oblateness effect (which is Newtonian), there's also gravitational frame dragging, which is a general relativistic effect. For the Earth, frame dragging is tiny (minuscule) compared to oblateness. $\endgroup$ Jan 25, 2023 at 14:28
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    $\begingroup$ Also, the tidal interaction is slowly transferring angular momentum from the Earth's spin to the Moon's orbit. $\endgroup$
    – PM 2Ring
    Jan 25, 2023 at 17:19
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Yes, over time, through tidal interactions.

Think of the Moon orbiting Earth. As it does so its tidal gravity creates a bulge on Earth, but Earth rotates this bulge out from under the Moon due to Earth rotating with more angular velocity than the Moon's orbit. Then:

  1. The Moon will try to pull the tidal bulge back under it, tending to slow Earth's rotation.

  2. Equally, the displaced bulge will pull the Moon forward, accelerating it beyond the equilibrium orbital velocity. This extra acceleration pushes the Moon outwards. We have even measured this outward displacement.

In effect, the excess angular momentum feeds into the Moon's orbital energy and, given enough time, will continue to do so until Earth's rotation and the Moon's orbit come into synchronization (tidal locking). In practice the required energy transfer is too great and the energy transfer rate too small for this to go to completion before changes in the Sun destroy the Earth-Moon completely. But the faster-acting Pluto-Charon system has evolved into this equilibrium, giving the double dwarf planet one of its most distinctive characteristics.

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