Question: SFH-measured IMFs

[keflavich]

In section 12, equations 12.1-12.3: https://github.com/Open-Astrophysics-Bookshelf/star_formation_notes/blob/707cb0783f7b29516622640c26885e10ea813c50/chapters/chapter12.tex#L86-L102

I don't see how 12.3 comes from the previous two. It looks like instead the equation should be, more simply:

\frac{dn}{dm} = \frac{dn_{\rm MS}}{dm} \left(\int_{-t_{\rm MS}(m)}^0 dt \, \dot{M}_*(t)\right)^{-1}

i.e., a direct rearrangement of 12.2.

Equation 12.3 seems to be a solution for dn_{form}/dm instead of dn/dm, but even then, the fraction seems flipped.

Am I just missing something here? The use of \propto hints that a term might have been dropped, but I can't see it. I'm prepping up notes for a class and can't quite make the text match the equations.

[keflavich]

(of course I'm happy to PR a fix if one is needed)

[markkrumholz]

Hi Adam,

You’re right, the fraction in equation 12.3 is flipped, and should be as you’ve noted. Equation 12.3 is indeed supposed to just be a re-arrangement of 12.2, and 12.1 is not needed to obtain it — 12.1 is there just as a conceptual way of understanding where 12.2 comes from. Equation 12.3 is written with a proportionality rather than an equality not because anything has been dropped, but because after making the lifetime correction one will have to re-normalise dn/dm to get back to a properly-normalised IMF.

Thanks for reporting this. I will check a fix into the notes now, and add this to my erratum list.

On 21 Nov 2019, at 1:32 am, Adam Ginsburg notifications@github.com wrote:

In section 12, equations 12.1-12.3: https://github.com/Open-Astrophysics-Bookshelf/star_formation_notes/blob/707cb0783f7b29516622640c26885e10ea813c50/chapters/chapter12.tex#L86-L102 https://github.com/Open-Astrophysics-Bookshelf/star_formation_notes/blob/707cb0783f7b29516622640c26885e10ea813c50/chapters/chapter12.tex#L86-L102 I don't see how 12.3 comes from the previous two. It looks like instead the equation should be, more simply:

\frac{dn}{dm} = \frac{dn{\rm MS}}{dm} \left(\int{-t{\rm MS}(m)}^0 dt \, \dot{M}*(t)\right)^{-1} i.e., a direct rearrangement of 12.2.

Equation 12.3 seems to be a solution for dn_{form}/dm instead of dn/dm, but even then, the fraction seems flipped.

Am I just missing something here? The use of \propto hints that a term might have been dropped, but I can't see it. I'm prepping up notes for a class and can't quite make the text match the equations.

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