> Is each star within a galaxy assumed to be a point mass?

Here's what's cool, using Guass' Law for gravity you can derive the Shell theorem which says that for spherical objects, if you are outside of the mass of the object, then the gravity of the object is the same as if all of the mass is concentrated at a point right at the center. So, yes we get to treat them as a point mass, but that's (barely!) an approximation (I say barely because stars are not perfect circles, they are slightly bulged due to rotation. But this is such a minor effect, it wouldn't really cause an issue).

> Do we "simply" count the stars, and their mass, and get that as the mass of the galaxy, and then based on its size, work out if it should be capable of staying together? Or do we model the individual stars and their gravity?

To answer this, let's talk about something like a space shuttle orbiting the Earth. In Low Earth Orbit (LEO) a space shuttle travels about 17,500 mph (24,000 kph) to stay in orbit around the Earth. If it went too slow, it would fall into the Earth. If it went too fast, it would fly away from Earth and never return. The speeds are a function of the mass of the Earth. That is- if the Earth was heavier, the space shuttle would have to go faster to stay in LEO, and if it were lighter, it would have to go slower. Another cool thing is, the size of the object orbiting doesn't really impact it much at all (this is true as long as the object orbiting is much much smaller than the object being orbited). So, if we looked at some alien planet with a high powered telescope that had a satellite, and saw how fast the satellite was orbiting the planet, we could know the mass of that planet.

The stars in a galaxy are similar, but instead of orbiting a planet, they are orbiting around the galaxy. But all of the stars closer to the center of the galaxy are like the planet the stars are orbiting around. And if we take the link above on Shell's theorem even further, what's cool is if we model the galaxy as a rotating, flat disk (which it's not, and in the real simulations they do it more accurately, but honestly, modeling the galaxy as a disk is "good enough" for this), then it turns out that at any point on the disk, the gravity is completely determined by the amount of mass "closer to the center" of the disk then the point- that is, all the mass further away rotating doesn't matter. Why? Well, you can think of it like this: you're getting tugged both ways by mass further away from the center than you. The mass "behind you" is closer, and pulling you away, but there's more mass "in front of you", pulling you in. If you do all of the really complex math and add up all of those forces, it turns out they completely cancel. It's really cool.

So when we look at how much mass we can see between a star and the center of the galaxy, the star is orbiting at a speed that requires there be more mass than there actually is. Going to the shuttle example above, it's like if a shuttle was orbiting a planet that looked like Earth, but was moving at a speed like it was orbiting Jupiter, we'd say "wow, there is something weird in that planet, it has way more mass than it looks like." So, likewise, when we look at the galaxy we say "wow, there must be more mass in the galaxy than we can see, because the stars at every radius are just moving faster than we think they should be." That was the initial inspiration for Dark Matter- matter we can't see, but still creates gravity, giving more mass to the galaxy than we'd could otherwise see.

> If we do model the individual stars, do we model their cumulative gravity to infinity too?

Yeah, we can easily model their gravity out to infinity. The equations are simple for a computer to crunch through out to any distance.

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Thank you for taking you valuable time to explain this.

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Incredible answer... thank you

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I wonder, has it been completely ruled out that the extra mass necessary could be inside the black hole center? That is to say, that there could be a lot more mass in black holes than we have thought, and the stars closer to the center haven't yet fallen in because of [... some possible reason...]? Well certainly the intelligent people working on the problem have ruled this out, but I wonder how.

I agree with most of the stuff you have, but I don't think the shell theorem applies for flat disks in the way it applies for spheres. I tried my hand at working it out but wolframalpha doesn't like the integral I got, but when solving numerically the net force on an off-center mass in a hollow disk does not equal 0 like it would inside a hollow sphere.

Yes, for a couple reasons.The majority of mass in the galaxy does not orbit the central black hole, but orbits the all mass interior to its position in the galaxy. The strongest effects are from the stuff closest to it, as the force due to gravity decreases as a inverse square law. Adding mass to the central black hole would increase the gravitational forces felt by the stars that directly orbit the black hole, changing their orbital motion significantly, but the rest of the stars further away wouldn't feel these effects. You need some relativity spread out distribution of mass (dark matter) to explain why all the stars in the galaxy orbit faster than they "should", and not just the ones around the supermassive black hole.

Additionally, there are galaxies without supermassive black holes at their centers and we still see these "anomalies" in their orbital velocity, so you'd have to come up with some other explanation for these galaxies.

And finally, on a case by case basis, we have measurements of stellar orbits around SagA* constraining it's mass, and independent mass measurements from the event horizon telescope for SagA* and M87, so in these two cases we have even more evidence for it to not just be a bigger supermassive black hole.

Edit: Changed confusing pronouns

Thank you,

"... majority of mass in the galaxy does not orbit the central black hole but orbits the mass interior to it"

a little confused because, isn't orbiting the mass interior to the black hole the same as orbiting the black hole...?

Ah, your second point clears something up for me. I think I had read some time ago that some galaxies are found to actually lack dark matter, and by coincidence or not, also don't have black hole centers (were just start clusters?). But if galaxies exist without black holes and yet still have the velocity anomalies, then that's pretty definitive indeed.

Sorry, that's bad use of pronouns on my part. What I was trying to get across is that the effects of gravity a star, or gas cloud, or whatever feels is mostly not due to the central black hole. The star feels forces of gravity from all the mass interior to its position in the galaxy. For example, our Sun is 8 kpc away from the central black hole, SagA*. Even though SagA* is massive, we're just so ridiculously far away that the force of gravity exerted by SagA* on the Sun is tiny. Instead, what keeps the Sun orbiting is the force of gravity applied from stars and gas closer to the Sun. So increasing the mass of the central black hole would have almost no effect on the Sun's orbit, and the same goes for most the stars in the galaxy.

I see. Well explained, thanks.

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The calculations of mass and gravity were calculated By Edwin Hubble around 1932. He showed that The Milky Way had only 1/5th the mass needed to hold the galaxy together. Thus was Dark Matter, or it's theory, born. This doesn't even address the biggest problem with the orbital dynamics of galaxy sized systems.

Consider a solar system. Mercury travels faster in it's orbit than Venus. Venus moves faster than earth, and so on. Predictably, all the planets in our solar system move at a greater velocity than Neptune. Everybody knows this and understands why.

Now consider the galaxy and it's spiral shape. In order to maintain this shape the stars at the farthest outskirts have to be moving much faster that those closer in to the center. This configuration should force the galaxy to fly apart, but it doesn't. the question is, why don't the stars fly out into intergalatic space. Spiral galaxies shouldn't exist and nobody knows why.