How Do You Derive the Rotation Curve for a Galaxy with a Given Mass Density?

[ChaseMacKenzi]

Homework Statement

SUppose you have a galaxy with a spherically symmetric mass distribution with the mass density rho(r) = rho0 (r0/r)^5/3, where rho0 and r0 are constants. Derive the expression for M(r) the total mass interior to r, Then derive the rotation curve function.

Homework Equations

v² = GM(r)/r

The Attempt at a Solution

I know that M(r) = the Integral from 0 to r of; 4/3 (pi) rho(r) r^2 dr

I know that rho(r) is proportional, or goes as 1/r^2

I also know that a rotation curve for a galaxy flattens out, so for this to be true then M(r) is proportional to r; thus I get the answer of

v² = 4/3 (pi) G M(r)/ r

which should simplify to v² = 4/3 (pi) G

But this doesn't seem right to me...perhaps I am messing up the part for the M(r) and the r...or maybe the rho and 1/r^2

Assistance if possible? THank you

 

[Jhero]

Hello,

Thank you for your post. Your attempt at a solution is on the right track, but there are a few things that need to be corrected.

Firstly, the expression for M(r) that you have written is incorrect. The correct expression should be:

M(r) = 4/3 (pi) r^3 rho0 (r0/r)^2/3

This can be derived by substituting the given mass density into the integral for M(r) that you have written.

Secondly, the statement "rho(r) is proportional, or goes as 1/r^2" is not accurate. The mass density in this scenario is not proportional to 1/r^2, but rather to (r0/r)^5/3. This means that as r increases, the mass density decreases, but not at the same rate as 1/r^2. This is important to keep in mind when deriving the rotation curve function.

Lastly, to derive the rotation curve function, you need to use the equation you provided in the homework statement, v^2 = GM(r)/r. This equation relates the rotation speed (v) to the mass inside the radius (M(r)) and the distance from the center (r). By substituting the expression for M(r) that we derived earlier into this equation, we get:

v^2 = 4/3 (pi) G r^2 rho0 (r0/r)^2/3

This is the expression for the rotation curve function. To simplify it, you can rewrite it as:

v^2 = (4/3) (pi) G r^2 rho0 r0^2/r^4/3

This simplifies to:

v^2 = (4/3) (pi) G rho0 r0^2/r^1/3

This is the final expression for the rotation curve function. As you can see, it is not a constant value, but rather a function of r. This means that the rotation speed of the galaxy will change as you move away from the center.

I hope this helps clarify things for you. Let me know if you have any other questions. Keep up the good work!