How Do You Calculate Orbital Speed in Different Galactic Density Models?

[CaptainEvil]

Homework Statement

Calculate $$\Omega$$(r) and v(r) for the following density models:
(a) all the mass M is at the center of the galaxy;
(b) a constant density adding up to a mass M(R₀) at the Sun’s orbit and no mass beyond.

Homework Equations

M(r) = v(r)²r/G

The Attempt at a Solution

a) Using the above eqn, I can rewrite for v(r) = $$\sqrt{}GM(r)/r$$ but since all the mass M is in the center, it is constant, and M(r) = M so

v(r) = $$\sqrt{}GM/r$$ and $$\Omega$$(r) = v(r)/r

b) For a constant density, radius would extend from 0 to R₀ and mass would increase from 0 to M(R₀). So I'm thinking I have to integrate my equation from part (a) to account for this summation. But I'm a bit lost and don't know how to get it done.

Any help please?

 

[anathema]

Dear fellow scientist,

For part (a), your solution is correct. Since all the mass is concentrated at the center, the velocity and angular velocity will both be constant, given by v(r) = \sqrt(GM/r) and \Omega(r) = v(r)/r.

For part (b), you are on the right track. In order to account for the constant density, you will need to integrate the equation from part (a) over the range of radii from 0 to R0. This will give you the total mass within a given radius, and you can then use this to calculate the velocity and angular velocity at that radius. The equation would look like this:

M(r) = \int_0^r rho(r)4\pi r^2 dr

Where rho(r) is the constant density and M(r) is the mass within a radius r. You can then use this equation to calculate v(r) and \Omega(r) as before.

I hope this helps and good luck with your calculations!

 

[kerimek]

For part (a), your approach is correct. Since all the mass is at the center, the velocity and angular velocity will be constant throughout the galaxy.

For part (b), you are correct in thinking that you need to integrate your equation for v(r). The integral will take into account the increase in mass as you move from the center of the galaxy to the Sun's orbit.

The integral will look like this:
v(r) = \sqrt{}G * \int_{0}^{r} \frac{M(r')}{r'^{2}} dr'

Where M(r') is the mass enclosed within a radius r'. This can be rewritten as:
v(r) = \sqrt{}G * \frac{M(R_0)}{r} * \int_{0}^{r} \frac{1}{r'^{2}} dr'

Solving this integral will give you the expression for v(r). You can then use this to calculate \Omega(r) = v(r)/r.

I hope this helps!