# Question re Galaxy Rotation Curve

1. Oct 17, 2015

### Buzz Bloom

The diagram below is from https://en.wikipedia.org/wiki/Galaxy_rotation_curve .

I would much appreciate a derivation explaining the shape of the "Expected from visible disk" curve in the diagram. Naively, based on Newtonian mechanics for the orbital velocity of a circular orbit,
V = √GM/R ∝ 1/R1/2 .​
Obviously, this is not the shape of the diagram curve. I suppose the diagram curve towards the right might be close to the above formula, but what is the derivation for the shape of the left part of the curve?

2. Oct 17, 2015

### Orodruin

Staff Emeritus
The formula you quote is only valid outside the mass distribution (assumed spherically symmetric). In a galaxy, the stars are orbiting inside the mass distribution itself (they are part of it!).

3. Oct 17, 2015

### Buzz Bloom

Hi @Orodruin:

Thanks for your post. I apologize for being vague in specifying the information I was seeking.

I interpret your quote as implying that projected onto the primary plane of a spiral galaxy, the "disk" is assumed to have a 2D radially symmetric distribution of baryonic mass, including star mass, dust, and gas, say ρb(R). Then the mass M(R) for r < R is
M(R) = ∫0R π r ρb(r) dr .​
The total mass of the baryonic matter in the galaxy would then be M(∞).

Q1: Is there some theory that produces a calculation of the function ρb(R) given M(∞)? If so, what is it, and what is this function ρb(R) that the theory produces?
I have tried to find the answer to this on the internet, but, if it is there, my research skills are inadequate to find it.

Given ρb(R), I know how to write down a complicated integral for the value of the gravitational acceleration Ab(R) of a test particle at radius R based on this distribution. I am guessing that to calculate the function of V(R) for the lower curve in the diagram, it would be first necessary to calculate the value of this integral Ab(R). Then V(r) = √Ab(R)/R.

Q2: Please cite a reference, if you know of one, that shows the calculation of Ab(R) given ρb(R).

Regards,
Buzz

4. Oct 18, 2015

### Buzz Bloom

CORRECTIONS

M(R) = ∫0R 2 π r ρb(r) dr

V(R) = √(Ab(R)/R)​

Here is the integral form of Ab(R).
Ab(R) = (G/2π) ∫0R ρb(r) ∫0 [(r-R cos θ)/(R2+r2-2 r R cos θ)3/2] dθ dr​

Last edited: Oct 18, 2015