How can I convert the units of B to match the Kennicutt Schimdt law?

[matteo86bo]

Homework Statement

I need to infer the observed star formation efficiency of the Kennicutt Schimdt law starting from a volumetric SF law.
The KS law is a relationship between gas and stellar surface density that we can approximate as:

\frac{d\Sigma_*}{dt}=A\Sigma_{gas}^{1.5}

Where A is the efficiency and its values is roughly 2.5e-4 when \frac{d\Sigma_*}{dt} is measured in M_\odot kpc^{-2} yr^{-1} and \Sigma_{gas} in M_\odot pc^{-2}.

Now the problem is I want to derive this efficiency starting from this formula


\frac{d\rho_*}{dt}=B\frac{\rho_{gas}}{t_{ff}}
where {t_{ff}} is the free-falling time and is equal to
{t_{ff}}=\sqrt{\frac{3}{32\pi G \rho_{gas}}}

I need to compute B and then convert it in the same units of the Kennicutt Schimdt law.

2. The attempt at a solution

Since the Kennicutt law involes surface density I have multiplied both sides of the volumetric equation by a characteristic scale length \Delta x.

Therefore B should be equal to:

\sqrt{\frac{32\pi G}{3}} \Delta x

The things is I had to convert this number from cgs units to the units in which the Kennicutt law is given but I don't get the same order of magnitude. Can you help me?

 

[Chi Meson]

I gave this a shot, but it turns out I'm less able than you are at solving this. I would re-post this question in the Advanced forum if no one else here helps you soon.

 

[dynamicsolo]

Let's see if the units are making sense first:

For \frac{d\rho_*}{dt}=B\frac{\rho_{gas}}{t_{ff}} ,

we have M_\odot pc^{-3} yr^{-1} on the left and

B times M_\odot pc^{-3} divided by years on the right , so B should be dimensionless, no? (I'm taking it that the rho's are volume densities.)

So I don't think B = \sqrt{\frac{32\pi G}{3}} \Delta x can be right.(And A has units of \frac{pc }{ M_\odot^{1/2}\cdot yr} ,yes? )