Motion of a star in a galaxy

[w3390]

Homework Statement

A star is moving in a circular orbit of radius r within a galaxy. What is it's orbital speed
v(r) as a function of $$\rho$$(r) and radius.

The galaxy is spherically symmetric with a mean density $$\rho(r)$$ and radius R.

Homework Equations

F = -Gm$$\int$$$$\frac{\rho(r')e_{r}}{r^{2}}$$

The Attempt at a Solution

My attempt to find the orbital speed was to equate the equation mentioned above to F = ma, which in this case would be F = mv^2/r. From here I solved for v. However, I am confused about what to do for the integral. As I have it now, my answer looks like:

v(r) = [-Gr$$\int$$$$\frac{\rho(r')e_r}{r^2}$$dV']^1/2

Is that correct?

Any help would be much appreciated.

 

[mark726]

Thank you for your question. Your approach to finding the orbital speed of the star is correct. However, there are a few things that need to be clarified.

Firstly, the integral in your equation should have limits of integration, which correspond to the radius of the star's orbit. This will give you the total force acting on the star due to the galaxy's mass within that radius. Secondly, the integral should be evaluated over the entire volume of the galaxy, not just at a single point. This is because the density of the galaxy may vary at different distances from the center.

With these changes, your equation would look like:

v(r) = [-Gm\int_{0}^{r}\int_{0}^{2\pi}\int_{0}^{\pi}\frac{\rho(r')e_r}{r^2}r'^2sin(\theta)d\theta d\phi dr']^1/2

Where m is the mass of the star, \theta and \phi are spherical coordinates, and e_r is the unit vector in the radial direction.

I hope this helps. Let me know if you have any further questions.