Orbiting supermassive black hole or galactic center of mass?

Question

One of the ways they measure the (supposed?) supermassive black hole at the galactic center of the milky way is to measure those tens of stars right at the galactic center that are orbiting what appears to be nothing, but what is said to be a supermassive black hole.

My question is why there is a distinction between all other stars in the galaxy orbiting (for instance, our own sun) and those few right near the galactic center orbiting. My point is, if you ask what our solar system orbits, an astronomer wouldn't say "the supermassive black hole at the center," he would say "the galactic center of mass." My assumption is that you couldn't even calculate the mass of the supermassive black hole by knowing our sun's mass and orbit - people would probably say we are too far out, or we are orbiting the galactic center of mass rather than the black hole, yadda yadda.

Questions:

1) Why the distinction? How do we know that the stars right at the galactic center aren't also orbiting the galactic center of mass vs. the supermassive black hole?

2) What would happen if you magically removed the supermassive black hole in an instant? Would the stars right at the center continue to orbit the center? If so, how is that different than what they were originally doing?

Note that I know that center of mass of the galaxy and the supermassive black hole are two different things. I also know that the center of mass of the galaxy is either superimposed on the supermassive black hole, or quite close to it. I just want to know how or why some objects feel the center of mass of the galaxy and respond to it, while others feel the pull of the supermassive black hole and respond to it. How far away do you have to be to be considered "orbiting the center of mass" rather than "orbiting the supermassive black hole?"

Answer 1

To a first approximation when we calculate how fast an object is orbiting around some mass distribution we can assume that the gravitational attraction it experiences is only that due to the mass interior to its orbit.

This approximation, known as the shell theorem should only be applied when the mass distribution is spherically symmetric, or most of the mass is concentrated well inside the orbit of the object.

The bulge of our Galaxy can be considered roughly spherically symmetric, and it appears that the mass distribution is strongly concentrated towards the centre. Hence we can say that the centripetal force on an orbiting object is provided by the gravitational force due to the mass interior to its (again, I am assuming a circular orbit for the sake of argument, but this doesn't greatly matter) orbit is $$ m\frac{v^{2}}{r} = Gm \frac{M(r)}{r^2},$$ where $m$ is the mass of the object (which cancels out) and $M(r)$ is the mass inside radius $r$. Thus the orbital speed $v$ is given by $\sqrt{GM(r)/r}$.

Thus by observing orbital speeds we can calculate roughly what mass exists interior to the orbit.

In the case of stars near the centre of our Galaxy, that turns out to be 4 million solar masses, packed into a space that is extremely small. The reason it is thought to be a black hole is not because of the mass, but because it is mass that is (i) densely packed and (ii) cannot be seen.

Even if it were a bunch of massive stars shrouded in dust, so that you could not see it in visible light, you can easily calculate that the heated dust should be kicking off huge amounts of infrared radiation; and that is not seen.

In the case of the Sun, well, the mass interior to the Earth's orbit is much larger than 4 million solar masses, it is many billions of solar masses. The contribution of the black hole at the centre is completely swamped by all the other mass in our Galaxy, so we are not directly influenced by the central black hole.

If you removed the central black hole, the stars that were near it would be orbiting way to fast for the new $M(r)$ that they experience. They would fly off and assume highly elliptical orbits with much larger semi-major axes, though I think they would still be part of the Galactic bulge.

Stars further out, including our Sun would have their orbits changed imperceptibly, because $M(r)$ will hardly have changed for them.

Answer 2

1) It is the mass internal to the orbit that determines the orbital speeds. These stars are so close to the center that the amount of bulge star mass contained within the orbit would be insignificant compared to the BH. 2) If the BH disappeared these stars would go into highly radial orbits out to much greater distances. That distance would be about where the mass of the bulge/nucleus stars and gas equals the mass of the BH.