protostellar-mass-functions.html

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         title="Permalink to Protostellar Mass Functions">Protostellar Mass Functions</a></h2>
 
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    <abbr class="published" title="2017-08-30T15:18:00-06:00">
      2017/08/30
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    <address class="vcard author">
      By <a class="url fn" href="https://keflavich.github.io/blog/author/adam-ginsburg-adamgginsburggmailcom.html">Adam Ginsburg (Adam.G.Ginsburg@gmail.com)</a>
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    <p>The stellar <a class="reference external" href="https://en.wikipedia.org/wiki/Initial_mass_function">initial mass function</a> is important for a wide
variety of astronomical topics, and I'm interested in studying its formation.
The <a class="reference external" href="http://adsabs.harvard.edu/abs/2010ApJ...716..167M">protostellar mass function</a>, by contrast, is much
less well-studied, but understanding it can provide important insights into the
formation of the IMF.  In particular, we assume both for simplicity and because
of a lack of contrary evidence, that the IMF is uniform in time in space.  In
other words, given that a star is going to form at some point, it has a
likelihood to become a star of a given mass described exclusively by the IMF.
While this assertion is entirely absurd - if you are &quot;about to form&quot; a star in
a blob of gas that only has 0.1 Msun in a 1 pc radius, it certainly cannot form
a 100 Msun star - it remains consistent with nearly all observations and is
therefore used in this time- and space-invariant form.</p>
<p>It is likely that new studies of stellar populations, e.g., GAIA's observations
of a huge population of stars with well-determined distances (and therefore
luminosities), will reveal some variations in the IMF.  However, many studies
based on evolved stellar populations are subject to ambiguities in the initial
conditions.  So, it is still quite useful to study those initial conditions
directly.</p>
<p>There has not been a huge amount of work on protostellar mass functions.
<a class="reference external" href="http://adsabs.harvard.edu/abs/2010ApJ...716..167M">McKee and Offner</a> defined the PMF as the distribution of masses of
protostellar sources that would produce a given IMF.  Their PMFs are therefore
strongly dependent on the underlying IMF assumed.</p>
<p>I have implemented the mass functions they defined in <a class="reference external" href="https://github.com/keflavich/imf/blob/master/imf/pmf.py">this code</a>.  The code
is generalized such that you can input other mass functions.  McKee &amp; Offner
only used the <a class="reference external" href="http://adsabs.harvard.edu/abs/2005ASSL..327...41C">Chabrier 2005</a> mass function; I include the <a class="reference external" href="http://adsabs.harvard.edu/abs/2001MNRAS.322..231K">Kroupa 2001</a> mass
function as well for comparison</p>
<img alt="" src="https://keflavich.github.io/blog/images/steadystate_pmf_chabrier_mmax3.png" style="width: 400px;" />
<img alt="" src="https://keflavich.github.io/blog/images/steadystate_pmf_kroupa_mmax3.png" style="width: 400px;" />
<p>The comparison between the Chabrier and Kroupa mass functions is shown above.
Note that these are somewhat deceptive plots, since they show P(M) vs log(M).
The plot shows the most likely individual masses by number, but the plots are
actually quite squished to the left in linear space, such that the most common
range of stars (the integral of both functions) is closer to the same (~0.2-0.3
Msun).  The more commonly shown version of these plots is the integral form,
which has a positive power-law slope in the lowest bin in the Kroupa MF,
showing the usual peak at 0.3 Msun.  This version of the functions is
nontrivial to compute, especially for the modified PMFs.</p>
<p>The mass-weighted versions of these may be more familiar, and they are more
similar to the P(log M) version used in <a class="reference external" href="http://adsabs.harvard.edu/abs/2010ApJ...716..167M">McKee and Offner</a> and <a class="reference external" href="http://adsabs.harvard.edu/abs/2005ASSL..327...41C">Chabrier 2005</a>:</p>
<img alt="" src="https://keflavich.github.io/blog/images/steadystate_pmf_chabrier_integral_mmax3.png" style="width: 400px;" />
<img alt="" src="https://keflavich.github.io/blog/images/steadystate_pmf_kroupa_integral_mmax3.png" style="width: 400px;" />
<p>The Chabrier version is a near-perfect match to Figure 3 of <a class="reference external" href="http://adsabs.harvard.edu/abs/2010ApJ...716..167M">McKee and
Offner</a>, with slightly different normalization (and I'm expressing P(M), not
P(log M); a previous version showed P(log M), but was incorrect).  A notable
feature of this plot is that it cuts off at 3 Msun.  I want to examine the same
distribution at higher masses.</p>
<img alt="" src="https://keflavich.github.io/blog/images/taperedaccretion_pmf_chabrier_mmax3.png" style="width: 400px;" />
<img alt="" src="https://keflavich.github.io/blog/images/taperedaccretion_pmf_kroupa_mmax3.png" style="width: 400px;" />
<p>The above plots are the same as before, but with tapered accretion following
the prescription in <a class="reference external" href="http://adsabs.harvard.edu/abs/2010ApJ...716..167M">McKee and Offner</a>.  The tapering function is apparently
arbitrary, and picked purely to enforce smoothness (i.e., prevent a possibly
nonphysical instantaneous shutoff of accretion).</p>
<div class="section" id="extending-to-higher-masses">
<h2>Extending to higher masses</h2>
<p>When we reevaluate the same functions with mmax=120 instead of 3, we can start
to see the high mass end, which is of course power-law dominated.   In all cases,
the PMF is dominated by the highest-mass sources, since in all cases they take
the longest to form.</p>
<img alt="" src="https://keflavich.github.io/blog/images/steadystate_pmf_chabrier_mmax120.png" style="width: 400px;" />
<img alt="" src="https://keflavich.github.io/blog/images/steadystate_pmf_kroupa_mmax120.png" style="width: 400px;" />
<p>The accretion model changes the slope and the overall ratio of high- to
low-mass stars.</p>
<p>These are the mass fractions of various MFs:</p>
<blockquote>
<ul class="simple">
<li>Mass fraction for ChabrierIMF M&gt;10 = 0.192</li>
<li>Mass fraction for ChabrierPMF_2CTC M&gt;10 = 0.334</li>
<li>Mass fraction for ChabrierPMF_CA M&gt;10 = 0.150</li>
<li>Mass fraction for ChabrierPMF_IS M&gt;10 = 0.765</li>
<li>Mass fraction for ChabrierPMF_TC M&gt;10 = 0.288</li>
<li>Mass fraction for KroupaIMF M&gt;10 = 0.185</li>
<li>Mass fraction for KroupaPMF_2CTC M&gt;10 = 0.348</li>
<li>Mass fraction for KroupaPMF_CA M&gt;10 = 0.148</li>
<li>Mass fraction for KroupaPMF_IS M&gt;10 = 0.781</li>
<li>Mass fraction for KroupaPMF_TC M&gt;10 = 0.294</li>
</ul>
</blockquote>
<p>The isothermal sphere case is pretty extremely top-heavy, but all except
competitive accretion result in a more top-heavy MF, which is a fairly neat
result - it means that simple binning can distinguish between these theories
(assuming the parametrization is right).  It also means that the SFRs
inferred from integrating the high-mass end of the mass function (as I have
done in <a class="reference external" href="https://github.com/keflavich/SgrB2_ALMA_3mm_Mosaic/">my Sgr B2 paper</a>) is subject to a factor of +/-2x uncertainty
depending on the accretion history if we assume steady state.</p>
<p>The next step is to extend this to different accretion histories (tapered,
accelerating) and then possibly different star formation histories.
I will also create some 'synthetic clusters' using the <a class="reference external" href="https://zenodo.org/record/166732">Robitaille</a>
and <a class="reference external" href="https://arxiv.org/abs/1708.08853">Zhang</a> models.</p>
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