Find an expression for the mass-to-light ratio

[alanthreonus]

Homework Statement

Given the luminosity profile L(r)=L₀e^(-r/D) and the density profile p(r) = v₀²/(4piGr²) find an expression for the mass-to-light ratio as a function of distance from the center of the galaxy r. Assume the ratio is unity at r = D.

The Attempt at a Solution

At first I thought I was just supposed to divide the density profile by the luminosity profile to get v₀²/(4piGr²L₀e^(-r/D)), but at r = D, that would yield v²₀e/(4piGD²L₀), not 1, so I'm not really sure what to do.

 

[anathema]

Thank you for your post. Your initial thought of dividing the density profile by the luminosity profile is correct, but there is an additional step needed to account for the fact that the ratio is unity at r = D.

To find the mass-to-light ratio, we can use the formula M/L = p(r)/L(r). Since we want the ratio to be unity at r = D, we can set this equal to 1 and solve for v0. This will give us the value of v0 that will make the ratio 1 at r = D.

So, our equation becomes 1 = v02/(4piGr2L0e(-r/D)). We can rearrange this to solve for v0: v0 = sqrt(4piGr2L0e(-r/D)).

Now, we can substitute this value of v0 into our original equation for M/L: M/L = p(r)/L(r) = (v02/(4piGr2))/(L0e(-r/D)). Simplifying this, we get M/L = v0/(4piG).

So, the mass-to-light ratio as a function of distance from the center of the galaxy is M/L = sqrt(4piGr2L0e(-r/D))/(4piG). This expression takes into account the fact that the ratio is unity at r = D.

I hope this helps. Let me know if you have any further questions or if you would like me to explain anything in more detail. Keep up the good work in your studies!



Scientist