Gravity from a Galaxy

Newton was a very serious man – you might even say an expert in gravity. First he invented gravity so that guys who juggle apples at parties wouldn’t have to keep climbing up to the ceiling to get them back down. Then he invented math to prove that the moon wouldn’t fall, too.

I’ve heard Newton worked hard to prove the following about his gravitational theorem: Suppose you have a spherical planet made out of apples, and you are standing on it. Each of the individual apples pulls on you. When you add up all those little tugs, you get the total force of gravity from the apple planet. The force is just the same as it would be if you were standing in the same spot, but all the apples were mashed up on top each other at the center of the planet.

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You are standing on a planet made of apples. Each apple pulls on you, and all their pulls combined are the gravitational force.

The fruit of Newton's labors.

When the apples are concentrated at the center, some get closer to you and some get further away. It just works out so the total force of gravity stays the same.

That’s convenient because it makes calculating the gravity of the apple planet easy. You don’t have to add up all the apples. You just need to know how far it is to the center of the planet and how many total apples there are. Or, you could measure the force and the size of the planet, and use that to calculate the number of apples.

But what about a galaxy instead of a planet? A galaxy isn’t a sphere. Since it spins, it’s more of a disc, like a wad a dough that gets twirled repeatedly in the air and flattens out into a pizza. (Caveat: I have never observed a pizza to form spiral arms).

A recent galaxy picture from APOD

The Earth orbits around the Milky Way, so if we want to do some physics with that, can we assume that all the mass of the Milky Way is located at the center? In other words, does the same theorem that holds for spherically-symmetric matter distributions (a planet) hold for cylindrically-symmetric ones (a galaxy)? Think about it a moment, do an integral if you like, then read on.

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Will the gravity from the disc galaxy also be the same as if all its mass were concentrated at the center?

In this post I won’t prove either theorem. Instead, I’ll show that if the theorem works for the sphere, then it does not work for the disc. To do this, imagine transforming a sphere into a disc. Start with the apple planet, then squash it down to two dimensions, keeping the radius the same, but moving all the apples into a disc shape. This is called projecting the sphere onto a plane.

Take a minute and form a mental image of a sloth having a seizure in slow motion.

The sphere gets projected onto the plane, making a disc. Imagine the apples are the same size, and there's the same number of them, and the center of the disc is filled. My drawing is not great.

Doing this, the force of gravity you feel must get stronger, because every apple that moved got at least a little bit closer to you. So the gravity from the disc is stronger than the gravity from the sphere. If we then collapsed all the apples to the point at the center, gravity would get weaker again. If the shell theorem works for spheres (and it does), it doesn’t work for discs.

One way of detecting dark matter is to look at the rate that stars orbit their galaxies and use that to infer how much matter there must be. If there’s lots of gravitating matter, but we don’t see that many stars, then we know the galaxy has dark matter somewhere. But when making that calculation, we’ll have to take into account the disc shape of the galaxy and its more-complicated gravity.

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