The Moon's Orbit is WEIRD

We think of the moon as orbiting the earth, following a spiraling trajectory as the earth

itself orbits the sun. But this is wrong. Not only is the moon's orbit

NOT a spiral... there's an argument that the moon actually orbits the sun, not the earth!

Sure, from the earth's perspective, the moon appears to orbit the earth. But from the sun's

perspective (or from an observer far above the solar system), the moon orbits the earth

so slowly and the earth and moon are so far from the sun that the moon's trajectory never

actually makes any sort of a loop, and in fact, it never even curves outwards towards the earth

but always inwards towards the sun! The moon's trajectory is more like a

12-sided polygon with curved corners than it is a spiral or even a wiggly

line. Though the moon's trajectory is so close to being a circle it's kind of hard to see.

Here's a simpler example of an orbit where a planet's moon's trajectory only ever curves

towards its sun - it looks like this hypothetical moon is actually orbiting

the sun and just getting nudged in and out a little bit by the planet as it orbits.

This type of curve, where a circle rotates while attached to another rotating circle,

is called an "epitrochoid," though they're usually generated by rolling a circle around another

circle... but the curves are the same. It's a fun little bit of geometry: depending on how fast the

two circles rotate relative to each other, and how big they are relative to each other,

you get everything from spirals to wiggles to polygons with rounded corners. The faster a moon

orbits relative to its planet, or the larger the moon's orbit is relative to the planet's,

the more spirally the trajectory. The slower the moon moves relative to the planet,

or the smaller its orbit, the more wobbly or circular the trajectory. Looking at a whole

table of these trajectories, loops are in the upper right half here, and wobbles are

in this region. Orbits that never curve outwards - like the moon's - are in the remaining sliver.

In our case, the sun is about 400 times farther away than the moon, and the moon orbits the earth

around 13.5 times for each trip around the sun - once every 27 days. Looks pretty much like a

circle. Here's a zoomed-in view of what our moon and earth's orbits actually look like, exactly to

scale (though the earth and moon are not to scale) - it does look a lot like the earth and moon are

orbiting the sun independently but side by side, just nudging each other in and out as they go.

In order to start having a wobble, the moon would need to orbit about twice as fast as it does (but

at the same distance), or be twice as far from the earth (and still orbit once per month),

or the earth would need to be half as far from the sun (and still orbit once per year). In order to

start spiraling, the moon would need to orbit about 30 times faster at the same distance,

or be 30 times farther from the earth (and still orbit once per month). But in the real universe,

orbital speed and distance can't be adjusted independently: the farther out an object is,

the longer it takes to complete an orbit. It turns out the speed matters more,

so the moon would would have to come closer to earth where it could orbit faster in order

to have a wobbly orbit - one that even curves at all outwards away from the sun.

There's a good reason the moon's orbit, while affected by the earth, never even curves

outwards away from the sun: due to the distances between the earth, moon and sun, and their masses,

it turns out the sun pulls on the moon with almost twice the force that the earth pulls on the moon!

Technically speaking, the moon is outside of the Earth's "Chebotarev radius" where the pull from

the earth and sun are equivalent. That means the net force on the moon is ALWAYS towards the sun,

and that's why the moon's trajectory never curves outwards towards the earth - there's never a net

force that would pull it that way. The force on the moon is always pointing inwards, and just gets

stronger or weaker over time depending on whether the earth is pulling together with the sun,

or pulling against the sun. According to net forces, the moon is primarily orbiting

the sun with perturbations from the earth. Except, we shouldn't be using net forces,

because the earth and moon are moving in a circle around the sun, so we need to take into account

centrifugal effects. Those effects weaken the sun's relative pull enough that the earth's pull

on the moon is stronger. Technically speaking, we say the moon is inside the Earth's "Hill

radius," the zone where for practical purposes for objects orbiting along with the earth,

the earth's gravitation dominates the sun's. According to the hill radius,

the moon orbits the earth. Though technically speaking, the moon doesn't orbit the earth. The moon and the earth both actually orbit the center of mass

of the earth-moon system - that center of mass is inside the earth, so it's reasonable to say

the moon is orbiting the earth. But not by much! If the moon were exactly the same but just 40%

farther away, or in the same place but just 40% more massive, then the earth-moon center

of mass would be outside the earth, and the earth and moon might be better described as a

double planet rather than a planet and satellite! Though this would do basically nothing to change

the shape of the moon's trajectory around the sun. The point is: defining whether something is a

satellite vs independently orbiting the sun is actually a really tricky and possibly even

futile endeavor. The moon is certainly on the boundary between the two, with the balance of

evidence leaning on the side of "satellite." The best way to put it is that the earth and

moon and sun interact in a complex gravitational dance that defies simple classification - on the

one hand, it looks like the earth and moon orbit each other while together orbiting the sun,

and on the other, it looks like the earth and moon orbit the sun side by side,

pulling each other back and forth as they go. Both are true, from a certain point of view.

...Oh, and I definitely wouldn't rely too heavily on the shapes of trajectories to

think about orbits: trajectories can easily be used to mislead you! Like, you might imagine

that a point on earth's equator draws out loops as the earth spins along its orbit,

but because of how fast the earth is moving around the sun, points on the earth's surface

don't trace out loops but, like the moon, trace out something more like a wobbly circle.

Should we then say that each part of the earth independently orbits the sun? Of course not.

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