# Difficult units conversion

## 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?

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Chi Meson
Homework Helper
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
Homework Helper
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? )

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