What is the mass of the Galaxy's core?

[Lotto]

Homework Statement

Because of dark matter, our stars in the Galaxy with homogenous core of radius $r_1$ are moving with speed $v_0$ that is not dependent on distance from the Galaxy's centre. This is true for distances from the centre smaller than $r_2=7r_1$. Consider distribution of dark matter to be spherically symetric around the core. Determine mass of the Galaxy's core $M$.

Relevant Equations

$F_\mathrm g=G\frac{mM}{r^2}$

It is clear that the speed is constant because dark matter hasa gravitational effect on stars, so when a star is further from the core, gravitational force of it is smaller, but the net gravitational force of dark matter is bigger. So the net force acting on each star has to be the same. So there must be a centripetal force - the net force acting on a star. But centripetal force is dependent on a distance, so what to do? If I made a sum of all gravitational forces acting on a star, dependent on its distance from the centre of the Galaxy, then it would be trivial I guess. But that is according to me too hard to do. I had an idea to do it by using energies, but that seems to lead nowhere as well.

So how to start? I just want a small hint to be able to start because now I have no idea how to handle this problem.

P.S. It's a high school problem so the math shouldn't be that hard.

 

[gneill]

If the core of mass $M$ has a radius of $r_1$, and the speed of the closest stars is $v_0$, wouldn't that give you the mass of the core?

I presume that the mass of the core includes a mixture of regular as well as dark matter, but that the dark matter extends beyond the core and influences stars further out so that their velocities are kept constant. But you should be able to work out the core mass with the innermost stars?

 

[Lotto]

If the core of mass $M$ has a radius of $r_1$, and the speed of the closest stars is $v_0$, wouldn't that give you the mass of the core?

I presume that the mass of the core includes a mixture of regular as well as dark matter, but that the dark matter extends beyond the core and influences stars further out so that their velocities are kept constant. But you should be able to work out the core mass with the innermost stars?

First, I must edit the assignment, because the dark matter is distributed spherically symetrically around the Galaxy's centre, not the core. So your asscumption is correct.

But my problem is that when we take a star closest to the core, the net force caused by the dark matter is not zero, I mean the dark matter that is not part of the core. If the star was in the middle of the Galaxy, then the net force would by zero, but it si not in the middle. So one force is $G\frac{mM}{{r_1}^2}$ and there is also an another force. But how to calulate the other force?

Or are my thoughts wrong?

 

[gneill]

I find it doubtful that dark matter is not fully mixed with the core, and only surrounds it. But anyway, that's that's irrelevant; mass is mass. If you have a mass of dark matter mixed in with normal matter, you cannot distinguish the two. The mass of the core $M$ would not distinguish between dark and normal matter.

But my problem is that when we take a star closest to the core, the net force caused by the dark matter is not zero, I mean the dark matter that is not part of the core.

The dark matter is said to lie spherically around the core, as you have said. What's the gravity shell theorem say about parts of a body lying further from the center than the star's location?

 

[Lotto]

I find it doubtful that dark matter is not fully mixed with the core, and only surrounds it. But anyway, that's that's irrelevant; mass is mass. If you have a mass of dark matter mixed in with normal matter, you cannot distinguish the two. The mass of the core $M$ would not distinguish between dark and normal matter.

The dark matter is said to lie spherically around the core, as you have said. What's the gravity shell theorem say about parts of a body lying further from the center than the star's location?

Well, I would divide the space around the core into an infinite number of shells, so then the net force caused by this space acting on a star in a distance $r_1$ should be zero and the only force is $G\frac{mM}{{r_1}^2}$. And this force is equal to $m\frac{{v_0}^2}{r_1}$.

So the mass of the core should be then $M=\frac{{v_0}^2r_1}{G}$. It is correct?

 

[gneill]

Yes. That would be my answer.

 

[Lotto]

Well, I am sorry but I have just found out that the core might not be homogenous. In the assignment, it is stated:

Assume that a substantial part of the observed mass of our Galaxy is located in a spherical core with a radius of $r_1$, considerably smaller than the radius of the Galaxy, and that it is distributed spherically symmetrically.

Does it mean that the core is homogenous or that its density is a function of distance? Because then I couldn't use such a simple formula for the gravitational force.

So is there any difference when I say that "dark matter is distributed spherically symmetrically around Galaxy's centre" and is only "spherically symmetrically distibuted"? Is that "around the centre" important?

 

[nasu]

Any spherical symmetric distribution has a center. If it is not the center of the Galaxy, what it is? It must be a center.

But spherical symmetric does not mean homogeneous.

 

[gneill]

It doesn't matter. The core is stated to be spherically symmetric, so that it behave as a point mass to anything beyond it's boundary. It doesn't matter whether it's homogeneous (radially) or not, if it is given as spherically symmetric. It may have alternating layers of dark and normal matter, or not. Doesn't matter as far as gravity is concerned outside the sphere.

 

[Lotto]

It doesn't matter. The core is stated to be spherically symmetric, so that it behave as a point mass to anything beyond it's boundary. It doesn't matter whether it's homogeneous (radially) or not, if it is given as spherically symmetric. It may have alternating layers of dark and normal matter, or not. Doesn't matter as far as gravity is concerned outside the sphere.

OK, I thought that I can use $G\frac{mM}{r^2}$ only when the sphere is homogenous. So then it is correct.

 

[gneill]

OK, I thought that I can use $G\frac{mM}{r^2}$ only when the sphere is homogenous. So then it is correct.

Yes!