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Difficult units conversion

  1. Aug 18, 2011 #1
    1. The problem statement, all variables and given/known data

    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:

    [itex]\frac{d\Sigma_*}{dt}=A\Sigma_{gas}^{1.5}[/itex]

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

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


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

    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 [itex]\Delta x[/itex].

    Therefore B should be equal to:

    [itex]\sqrt{\frac{32\pi G}{3}} \Delta x[/itex]

    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?
     
  2. jcsd
  3. Aug 19, 2011 #2

    Chi Meson

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    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.
     
  4. Aug 19, 2011 #3

    dynamicsolo

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

    For [itex]\frac{d\rho_*}{dt}=B\frac{\rho_{gas}}{t_{ff}}[/itex] ,

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

    B times [itex]M_\odot pc^{-3}[/itex] 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 [itex]B = \sqrt{\frac{32\pi G}{3}} \Delta x[/itex] can be right.


    (And A has units of [itex] \frac{pc }{ M_\odot^{1/2}\cdot yr} [/itex] ,yes? )
     
    Last edited: Aug 19, 2011
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