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What does the dynamical mass of a galaxy represent? Is it the mass of the gas in the galaxy or the total mass of the galaxy?

What can we infer from the rotation curves, is it dynamical mass or mass distribution?

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I found this reference for you. In the abstract, it is stated:

Rotation curves are the basic tool for deriving the distribution of mass in spiral galaxies.

Coming to answering your questions:

Is it the mass of the gas in the galaxy or the total mass of the galaxy?

The dynamical mass is the total mass in the galaxy is predicted using Newton's Inverse Square Law. It definitely doesn't represent the mass of gas in galaxy because the gas which you refer to is the visible mass from different spectrum of observations. If you tried to do so, then, it is similar to trying to measure the mass of Earth using a weighing machine.

What can we infer from the rotation curves, is it dynamical mass or mass distribution?

Actually we can't measure and pinpoint each and every mass present in a galaxy. So, using some mathematical modelling techniques and some basic assumptions and approximations, we compute the dynamical mass which represents the mass distribution of Galaxy. For inverse square law, we assume there is not much random acceleration among the masses.

Extras: We run into the trouble of Dark Matter because we assume Newton's Square Law holds in this situation.

But if you take MOND - Modified Newtonian Dynamics - the story is different. Here you don't run into trouble of dark matter as the inverse square law is modified with

$$F_N = m \mu \left ( \frac{a}{a_0} \right ) a$$

Here $F_N$ is the Newtonian force, $m$ is the object's (gravitational) mass, $a$ is its acceleration, $\mu(x)$ is an "interpolating function", and $a_0$ is a new fundamental constant which marks the transition between the Newtonian and deep-MOND regimes. You can read more about MOND here.

This modification actually fits nicely with the galaxy rotation curves and doesn't bring up unexplained mass in the dynamical mass distribution.

So, ultimately rotation curves help us understand dynamical mass which ought to represent Mass Distribution to the maximum extent.

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Some of the basics of galaxies rotation can be understood using basic mechanics (and a lot of assumptions that are probably not completely right but give a good approximation).

First, let's assume that the galaxy is a rotating, homogeneous plane disk (plane means that the radius $R$ is much bigger than the height $h$, which is mostly true for spiral galaxies, for example, Milky way has $R=100\,kpc$ and $h=0.6\,kpc$).

Let's focus now in a single star located at a distance $r$ from the center of the disk, and let's assume it's moving in a uniform circular motion. The velocity is then perpendicular to the position vector from the center to the star, and its modulus is

$$v = \sqrt{a\times r}$$

where $a$ is the centripetal acceleration. This acceleration is caused by gravity, and it can be shown that it's caused only the mass of the disk inside $r$: think of this as a Gauss law for magnetism, but applied to gravitation. As in magnetism, this "Gauss law for gravitation" allows us to change the mass of the disk inside $r$ for a point of the same mass located at the center. With this, the acceleration is

$$a = \frac{GM}{r^2} \Rightarrow M = \frac{v^2r}{G}$$

Where $M$ is the mass of the disk inside the distance $r$.

With this, you can infer the total mass in the disk measuring the velocity of stars at different distances from the center, and make a plot of distance $r$ vs. the mass of the disk $M$ inside that radius. This is called dynamical mass, since it was obtained from the dynamics of the system, and represents all the mass that contributes to gravity.

So far so good. The problem is that you can also calculate the mass of the galaxy using light: making some additional assumptions, you can relate the amount of light with the number of stars, and that number to the mass. If you do this using only the stars inside a certain radius, you end with the same plot as before, but now with the mass obtained from the amount of light, hence called visible mass.

As you may know, visible and dynamical masses do not match, and that's why the introduction of a new type of matter is needed, a matter that affects gravity but can't be seen, and has been called dark matter.

Going back to your question, dynamical mass represents the total mass affecting gravity, and from rotation curves you can obtain the mass enclosed inside a certain radius.

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    $\begingroup$ Your use of Newton's shell theorem for a mass distribution that is not spherically symmetric is incorrect. Propagating this error is responsible for half the crank literature on "solving" the dark matter problem. e.g. astronomy.stackexchange.com/questions/8172/… $\endgroup$
    – ProfRob
    Jul 25, 2020 at 13:38

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