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    <h2 class="entry-title">
      <a href="https://keflavich.github.io/blog/hopkins-pdf-generalization.html" rel="bookmark"
         title="Permalink to Hopkins PDF Generalization">Hopkins PDF Generalization</a></h2>
 
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  <footer class="post-info">
    <abbr class="published" title="2013-06-04T15:00:00-06:00">
      2013/06/04
    </abbr>
    <address class="vcard author">
      By <a class="url fn" href="https://keflavich.github.io/blog/author/adam-keflavichgmailcom.html">Adam (keflavich@gmail.com)</a>
    </address>
  </footer><!-- /.post-info -->
  <div class="entry-content">
    <p><a class="reference external" href="http://adsabs.harvard.edu/abs/2013MNRAS.430.1880H">Hopkins 2013</a> presents a non-lognormal form of the probability distribution
function of density in turbulence, depending primarily on a parameter
<span class="formula"><i>T</i></span>.  I examine some of its properties here.  All of the equations shown
on this page are implemented in <a class="reference external" href="https://github.com/keflavich/turbulence_pdfs">https://github.com/keflavich/turbulence_pdfs</a>,
in particular the functions <tt class="docutils literal">moments_theoretical_hopkins</tt> and <tt class="docutils literal">moments</tt> in
<a class="reference external" href="https://github.com/keflavich/turbulence_pdfs/blob/master/hopkins_pdf.py">hopkins_pdf.py</a>.</p>
<p>The conservation equations, conservation of probability and mass</p>
<div class="formula">
<span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span><i>P</i><sub><i>V</i></sub>(ln<i>ρ</i>)<i>d</i>ln<i>ρ</i> ≡ 1
</div>
<div class="formula">
<span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span><i>ρ</i><i>P</i><sub><i>V</i></sub>(ln<i>ρ</i>)<i>d</i>ln<i>ρ</i> ≡ <i>ρ</i><sub>0</sub> = 1
</div>
<p><span class="formula"><i>ρ</i><sub>0</sub> = 1</span> is the definition used by <a class="reference external" href="http://adsabs.harvard.edu/abs/2013MNRAS.430.1880H">Hopkins 2013</a>.  Part of the goal
of this document is to generalize to <span class="formula"><i>ρ</i><sub>0</sub> &lt;  &gt; 1</span></p>
<p>The most important thing to do is determine the values of the moments of the
distribution, including the integral corrections for the Dirac Delta function
component of the PDF.</p>
<div class="section" id="the-pdf">
<h2>The PDF</h2>
<p>The full form of the Hopkins PDF is</p>
<div class="formula">
<i>P</i><sub><i>V</i></sub>(ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub>)<i>d</i>ln<i>ρ</i> = <span class="stretchy">[</span><i>I</i><sub>1</sub>(2<span class="sqrt"><span class="radical">√</span><span class="ignored">(</span><span class="root"><i>λ</i><i>u</i></span><span class="ignored">)</span></span>)<i>e</i><sup> − <i>λ</i> − <i>u</i></sup><span class="sqrt"><span class="radical">√</span><span class="ignored">(</span><span class="root"><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator"><i>u</i></span><span class="ignored">)</span></span></span><span class="ignored">)</span></span> + <i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<i>u</i>)<span class="stretchy">]</span><i>du</i>
</div>
<p>The mass-weighted PDF is</p>
<div class="formula">
<i>P</i><sub><i>M</i></sub>(ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub>)<i>d</i>ln<i>ρ</i> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>ρ</i><sub>0</sub></span><span class="ignored">)</span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(2<span class="sqrt"><span class="radical">√</span><span class="ignored">(</span><span class="root"><i>λ</i><i>u</i></span><span class="ignored">)</span></span>)<i>e</i><sup> − <i>λ</i> − <i>u</i></sup><span class="sqrt"><span class="radical">√</span><span class="ignored">(</span><span class="root"><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator"><i>u</i></span><span class="ignored">)</span></span></span><span class="ignored">)</span></span> + <i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<i>u</i>)<span class="stretchy">]</span><i>du</i>
</div>
<p>With definitions:</p>
<div class="formula">
<i>u</i> ≡ <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator">ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub></span><span class="ignored">)/(</span><span class="denominator"><i>T</i></span><span class="ignored">)</span></span>;(<i>u</i> ≥ 0)
</div>
<div class="formula">
<i>λ</i> ≡ <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>S</i><sub>ln<i>ρ</i>, <i>V</i></sub></span><span class="ignored">)/(</span><span class="denominator">2<i>T</i><sup>2</sup></span><span class="ignored">)</span></span>
</div>
<p>The ratio <span class="formula"><i>ρ</i> ⁄ <i>ρ</i><sub>0</sub></span> is needed to ensure self-consistency: the mass
PDF must also conserve probability, which means it must be scaled by
<span class="formula">1 ⁄ <i>ρ</i><sub>0</sub></span>.</p>
<p>These distributions are slightly different than those shown in <a class="reference external" href="http://adsabs.harvard.edu/abs/2013MNRAS.430.1880H">Hopkins 2013</a>.
The <span class="formula"><i>m</i> = 0</span> term in the summation version of the PDF yields a Dirac
<span class="formula"><i>δ</i></span> function in the integral form of the PDF.  To quote Phil:</p>
<pre class="literal-block">
If you check, the summation expression is exact, and does exactly integrate
to unity (and properly conserve mass) for any value of T or lambda (with
u&gt;0). But this is with the implicit understanding that the gamma-function
distribution with exponent &quot;m&quot; should be taken to go to a delta-function
about u=0 when m=0 (since otherwise it is ill defined; and physically this
is the &quot;no event/multiplication&quot; case, so that's the only allowed case for
the term in u).
</pre>
<!-- But note that both of these distributions can depend on :math:`\rho_0`, changing :math:`u` to be -->
<!--  -->
<!-- .. math:: u\equiv \frac{\lambda}{1+T} - \frac{\ln (\rho/\rho_0)}{T}  ;  (u \geq 0) -->
<div class="section" id="moment-equations">
<h3>Moment equations</h3>
<p>The important moments are the mean and variance of the volume and mass density
probability distribution and the volume and mass <em>log</em>-density probability
distribution.</p>
<p>The definitions used here are <span class="formula"><i>E</i>[<i>ρ</i>] ≡  &lt; <i>ρ</i> &gt; <sub><i>V</i></sub></span>, etc.  The
variance is computed as <span class="formula"><i>V</i>[<i>x</i>] = <i>E</i>[<i>x</i><sup>2</sup>] − <i>E</i>[<i>x</i>]<sup>2</sup></span>, which means we'll need to
compute the moments of various combinations of variables.</p>
<p>Also, <span class="formula"><i>S</i><sub>ln<i>ρ</i>, <i>V</i></sub> ≡ <i>V</i>[ln<i>ρ</i><sub><i>V</i></sub>]</span></p>
<div class="formula">
 &lt; <i>ρ</i> &gt; <sub><i>V</i></sub> ≡ <i>ρ</i><sub>0</sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span><i>ρ</i><i>P</i><sub><i>V</i></sub>(ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub>)<i>d</i>ln<i>ρ</i>
</div>
<div class="formula">
 &lt; ln<i>ρ</i> &gt; <sub><i>V</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span>ln<i>ρ</i><i>P</i><sub><i>V</i></sub>(ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub>)<i>d</i>ln<i>ρ</i>
</div>
<div class="formula">
 &lt; <i>ρ</i> &gt; <sub><i>M</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span><i>ρ</i>(<i>ρ</i><i>P</i><sub><i>V</i></sub>(ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub>))<i>d</i>ln<i>ρ</i>
</div>
<div class="formula">
 &lt; ln<i>ρ</i> &gt; <sub><i>M</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span>ln<i>ρ</i>(<i>ρ</i><i>P</i><sub><i>V</i></sub>(ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub>))<i>d</i>ln<i>ρ</i>
</div>
<div class="formula">
 &lt; <i>ρ</i><sup>3</sup> &gt; <sub><i>V</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span><i>ρ</i><sup>3</sup><i>P</i><sub><i>V</i></sub>(ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub>)<i>d</i>ln<i>ρ</i>
</div>
<div class="formula">
 &lt; ln<sup>2</sup><i>ρ</i> &gt; <sub><i>V</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span>(ln<i>ρ</i>)<sup>2</sup><i>P</i><sub><i>V</i></sub>(ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub>)<i>d</i>ln<i>ρ</i>
</div>
<div class="formula">
 &lt; ln<sup>2</sup><i>ρ</i> &gt; <sub><i>M</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span>(ln<i>ρ</i>)<sup>2</sup><i>ρ</i><i>P</i><sub><i>V</i></sub>(ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub>)<i>d</i>ln<i>ρ</i>
</div>
<p>In the moment equations, only the input to the PDF is scaled by a mean density
<span class="formula"><i>ρ</i><sub>0</sub></span>; the &quot;weighting factors&quot; for the expectation value are not
(i.e., we are measuring <span class="formula"><i>E</i>[<i>ρ</i>]</span>, not <span class="formula"><i>E</i>[<i>ρ</i> ⁄ <i>rho</i><sub>0</sub>]</span>).</p>
</div>
<div class="section" id="delta-terms">
<h3>Delta terms</h3>
<p>These can be integrated analytically.  In this section, I am just computing the
means; the variances are more complicated.</p>
<div class="formula">
 &lt; <i>ρ</i> &gt; <sub><i>V</i></sub> ≡ <i>ρ</i><sub>0</sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span><i>ρ</i><i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator">ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub></span><span class="ignored">)/(</span><span class="denominator"><i>T</i></span><span class="ignored">)</span></span>)<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>d</i>ln<i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>T</i></span><span class="ignored">)</span></span>
</div>
<div class="formula">
 &lt; ln<i>ρ</i> &gt; <sub><i>V</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span>ln<i>ρ</i><i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator">ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub></span><span class="ignored">)/(</span><span class="denominator"><i>T</i></span><span class="ignored">)</span></span>)<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>d</i>ln<i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>T</i></span><span class="ignored">)</span></span>
</div>
<div class="formula">
 &lt; <i>ρ</i> &gt; <sub><i>M</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span><i>ρ</i>(<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>ρ</i><sub>0</sub></span><span class="ignored">)</span></span><i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator">ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub></span><span class="ignored">)/(</span><span class="denominator"><i>T</i></span><span class="ignored">)</span></span>))<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>d</i>ln<i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>T</i></span><span class="ignored">)</span></span>
</div>
<div class="formula">
 &lt; ln<i>ρ</i> &gt; <sub><i>M</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span>ln<i>ρ</i>(<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>ρ</i><sub>0</sub></span><span class="ignored">)</span></span><i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator">ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub></span><span class="ignored">)/(</span><span class="denominator"><i>T</i></span><span class="ignored">)</span></span>))<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>d</i>ln<i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>T</i></span><span class="ignored">)</span></span>
</div>
<p>Substitution: <span class="formula"><i>v</i> = <span class="fraction"><span class="ignored">(</span><span class="numerator">ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub></span><span class="ignored">)/(</span><span class="denominator"><i>T</i></span><span class="ignored">)</span></span></span>,
<span class="formula"><i>dv</i> = <span class="fraction"><span class="ignored">(</span><span class="numerator">1</span><span class="ignored">)/(</span><span class="denominator"><i>T</i></span><span class="ignored">)</span></span><i>d</i>ln<i>ρ</i></span>, <span class="formula"><i>ρ</i> = <i>ρ</i><sub>0</sub><i>e</i><sup><i>v</i>*<i>T</i></sup></span>, <span class="formula">ln<i>ρ</i> = <i>v</i><i>T</i> + ln<i>ρ</i><sub>0</sub></span></p>
<div class="formula">
 &lt; <i>ρ</i> &gt; <sub><i>V</i><i>δ</i></sub> ≡ <i>ρ</i><sub>0</sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span><i>ρ</i><sub>0</sub><i>e</i><sup><i>vT</i></sup><i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <i>v</i>)<i>d</i><i>v</i>
</div>
<div class="formula">
 &lt; ln<i>ρ</i> &gt; <sub><i>V</i><i>δ</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span>(<i>vT</i> + ln<i>ρ</i><sub>0</sub>)<i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <i>v</i>)<i>d</i><i>v</i>
</div>
<div class="formula">
 &lt; <i>ρ</i> &gt; <sub><i>M</i><i>δ</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span><i>ρ</i><sub>0</sub><i>e</i><sup>2<i>vT</i></sup>(<i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <i>v</i>))<i>d</i><i>v</i>
</div>
<div class="formula">
 &lt; ln<i>ρ</i> &gt; <sub><i>M</i><i>δ</i></sub> ≡ <span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit"> − ∞</sub></span>(<i>vT</i> + ln<i>ρ</i><sub>0</sub>)<i>e</i><sup><i>vT</i></sup>(<i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <i>v</i>))<i>d</i><i>v</i>
</div>
<p>Solutions:</p>
<div class="formula">
 &lt; <i>ρ</i> &gt; <sub><i>V</i><i>δ</i></sub> = <i>ρ</i><sub>0</sub>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <i>λ</i><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span> = <i>ρ</i><sub>0</sub>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span> − <i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator">1</span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
 &lt; ln<i>ρ</i> &gt; <sub><i>V</i><i>δ</i></sub> = <i>e</i><sup> − <i>λ</i></sup><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + <i>e</i><sup> − <i>λ</i></sup>ln<i>ρ</i><sub>0</sub>
</div>
<div class="formula">
 &lt; <i>ρ</i> &gt; <sub><i>M</i><i>δ</i></sub> = <i>ρ</i><sub>0</sub>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">2<i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <i>λ</i><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span> = <i>ρ</i><sub>0</sub>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i> − 1</span><span class="ignored">)/(</span><span class="denominator"><i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
 &lt; ln<i>ρ</i> &gt; <sub><i>M</i><i>δ</i></sub> = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + ln<i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <i>λ</i><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
 = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + ln<i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"> − <i>λ</i></span><span class="ignored">)/(</span><span class="denominator"><i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<p>(for these next 3, I skipped intermediate steps)</p>
<div class="formula">
 &lt; <i>ρ</i><sup>3</sup> &gt; <sub><i>V</i><i>δ</i></sub> = <i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">3<i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <i>λ</i><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span> = <i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator">2<i>T</i> − 1</span><span class="ignored">)/(</span><span class="denominator"><i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
 &lt; ln<sup>2</sup><i>ρ</i> &gt; <sub><i>M</i><i>δ</i></sub> = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + ln<i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><sup>2</sup>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"> − <i>λ</i></span><span class="ignored">)/(</span><span class="denominator"><i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
 &lt; ln<sup>2</sup><i>ρ</i> &gt; <sub><i>V</i><i>δ</i></sub> = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + ln<i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><sup>2</sup><i>e</i><sup> − <i>λ</i></sup>
</div>
<p>Using <span class="formula"><i>ρ</i><sub>0</sub> = 1</span> as defined in <a class="reference external" href="http://adsabs.harvard.edu/abs/2013MNRAS.430.1880H">Hopkins 2013</a> simplifies all of these a great deal.</p>
</div>
<div class="section" id="pdf-integrals">
<h3>PDF Integrals</h3>
<p>These cannot be integrated analytically.</p>
<p>However, we can work from a few simple mathematica/sympy results:</p>
<div class="formula">
<span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit">0</sub></span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>L</i>)</sup><i>dx</i> = <i>e</i><sup><i>L</i></sup> − 1
</div>
<div class="formula">
<span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit">0</sub></span><i>x</i><sup>2</sup><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>L</i>)</sup><i>dx</i> = 4<i>L</i><sup>2</sup>*<i>e</i><sup><i>L</i></sup>
</div>
<div class="formula">
<span class="limits"><sup class="limit">∞</sup><span class="limit"><span class="bigoperator integral">∫</span></span><sub class="limit">0</sub></span><i>x</i><sup>4</sup><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>L</i>)</sup><i>dx</i> = 16<i>L</i><sup>3</sup>(<i>L</i> + 2)*<i>e</i><sup><i>L</i></sup>
</div>
<p>We use <span class="formula"><i>L</i></span> instead of <span class="formula"><i>λ</i></span> in these equations because it is often substituted in later equations.</p>
<div class="section" id="expectation-value-of-the-volume-weighted-density-e-rho">
<h4>Expectation Value of the Volume-Weighted Density <span class="formula"><i>E</i>[<i>ρ</i>]</span></h4>
<div class="formula">
<i>E</i>[<i>ρ</i>] ≡ <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><i>ρ</i><i>P</i><sub><i>v</i></sub>(ln<i>ρ</i> ⁄ <i>ρ</i><sub>0</sub>)<i>d</i>ln<i>ρ</i> = <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><i>ρ</i><span class="stretchy">[</span><i>I</i><sub>1</sub>(2<span class="sqrt"><span class="radical">√</span><span class="ignored">(</span><span class="root"><i>λ</i><i>u</i></span><span class="ignored">)</span></span>)<i>e</i><sup> − <i>λ</i> − <i>u</i></sup><span class="sqrt"><span class="radical">√</span><span class="ignored">(</span><span class="root"><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator"><i>u</i></span><span class="ignored">)</span></span></span><span class="ignored">)</span></span> + <i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<i>u</i>)<span class="stretchy">]</span><i>du</i>
</div>
<p>To get to the form of the above equations, we use the substitution</p>
<div class="formula">
<i>x</i> = 2<span class="sqrt"><span class="radical">√</span><span class="ignored">(</span><span class="root"><i>λ</i><i>u</i></span><span class="ignored">)</span></span>
</div>
<p>which gives us <span class="formula"><i>ρ</i></span> in terms of <span class="formula"><i>x</i></span>:</p>
<div class="formula">
<i>ρ</i> = <i>ρ</i><sub>0</sub>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><i>T</i><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<p>and leads to the rearrangement:</p>
<div class="formula">
<i>E</i>[<i>ρ</i>] = <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><i>ρ</i><sub>0</sub>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><i>T</i><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>) − <i>λ</i></sup><span class="stretchy">]</span><i>dx</i> + <i>ρ</i><sub>0</sub>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<p>where the rightmost term is kept from the first moment above.  The integral
term can straightforwardly be broken apart into equations of the form shown
above.</p>
<div class="formula">
<i>L</i> → <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span>
</div>
<div class="formula">
<i>E</i>[<i>ρ</i>] = <i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <i>λ</i> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>L</i>)</sup><span class="stretchy">]</span><i>dx</i> + exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
 = <i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <i>λ</i> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>(<i>e</i><sup><i>L</i></sup> − 1) + exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
 = <i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <i>λ</i> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>(<i>e</i><sup><i>λ</i> ⁄ 1 + <i>T</i></sup> − 1) + exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
 = <i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><i>e</i><sup> − <i>λ</i> ⁄ (1 + <i>T</i>)</sup>(<i>e</i><sup><i>λ</i> ⁄ 1 + <i>T</i></sup> − 1) + exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
 = <i>ρ</i><sub>0</sub>
</div>
<p>The same general approach can be followed for all expectation values, but we'll skip the detailed algebra.</p>
</div>
<div class="section" id="variance-of-the-volume-weighted-density-v-rho-s-ln-rho-v">
<h4>Variance of the Volume-Weighted Density <span class="formula"><i>V</i>[<i>ρ</i>] = <i>S</i><sub>ln<i>ρ</i>, <i>V</i></sub></span></h4>
<div class="formula">
<i>V</i>[<i>ρ</i>] = <i>E</i>[<i>ρ</i><sup>2</sup>] − <i>E</i>[<i>ρ</i>]<sup>2</sup> = <i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator">2<i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">1 + 3<i>T</i> + 2<i>T</i><sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span> − 1<span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<p>However, the &quot;correction factor&quot; is still important:</p>
<div class="formula">
<i>V</i><sub><i>δ</i></sub>[<i>ρ</i>] = <i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i> − 1</span><span class="ignored">)/(</span><span class="denominator"><i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span> − exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − 2<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
</div>
<div class="section" id="expectation-value-of-the-mass-weighted-density-e-m-rho">
<h4>Expectation Value of the Mass-Weighted Density <span class="formula"><i>E</i><sub><i>M</i></sub>[<i>ρ</i>]</span></h4>
<p>Start from halfway through <span class="formula"><i>E</i>[<i>ρ</i>]</span>, simply adding a factor of 2 in the exponent:</p>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[<i>ρ</i>] = <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><i>ρ</i><sub>0</sub>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>2<i>T</i><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>) − <i>λ</i></sup><span class="stretchy">]</span><i>dx</i> + <i>ρ</i><sub>0</sub>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i>(<i>T</i> − 1)</span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<p>Follow the same math, with <span class="formula"><i>L</i> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + 2<i>T</i></span><span class="ignored">)</span></span></span></p>
<div class="formula">
 = <i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <i>λ</i> + <span class="fraction"><span class="ignored">(</span><span class="numerator">2<i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>(<i>e</i><sup><i>L</i></sup> − 1) + exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i>(<i>T</i> − 1)</span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
 = <i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">(<i>T</i> − 1)<i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>(<i>e</i><sup><i>λ</i> ⁄ (1 + 2<i>T</i>)</sup> − 1) + exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[<i>ρ</i>] = <i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator">2<i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">1 + 3<i>T</i> + 2<i>T</i><sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span> − exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i> − 1</span><span class="ignored">)/(</span><span class="denominator"><i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span> + exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i> − 1</span><span class="ignored">)/(</span><span class="denominator"><i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<p>The right 2 terms cancel, yielding the value shown in Equation 7 of <a class="reference external" href="http://adsabs.harvard.edu/abs/2013MNRAS.430.1880H">Hopkins 2013</a>
scaled by <span class="formula"><i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span></span>.  However, the right-most term is the
correction factor from the Dirac Delta term needed to correct any
numerical computation of the mass-weighted density.</p>
<div class="formula">
<i>E</i><sub><i>δ</i>, <i>M</i></sub>[<i>ρ</i>] = exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i> − 1</span><span class="ignored">)/(</span><span class="denominator"><i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[<i>ρ</i>] = <i>ρ</i><sub>0</sub>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator">2<i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">1 + 3<i>T</i> + 2<i>T</i><sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
</div>
<div class="section" id="expectation-value-of-the-mass-weighted-density-squared-e-m-rho-2">
<h4>Expectation Value of the Mass-Weighted Density Squared <span class="formula"><i>E</i><sub><i>M</i></sub>[<i>ρ</i><sup>2</sup>]</span></h4>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[<i>ρ</i><sup>2</sup>] = <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><i>ρ</i><sup>2</sup><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>ρ</i><sub>0</sub></span><span class="ignored">)</span></span><i>e</i><sup> − <i>λ</i></sup><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><i>dx</i> + <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><i>ρ</i><sup>2</sup><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>ρ</i><sub>0</sub></span><span class="ignored">)</span></span><i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<i>u</i>)<i>du</i>
</div>
<div class="formula">
<i>E</i><sub><i>δ</i>, <i>M</i></sub>[<i>ρ</i><sup>2</sup>] = <i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator">2<i>T</i> − 1</span><span class="ignored">)/(</span><span class="denominator"><i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>left</i></sub>[<i>ρ</i><sup>2</sup>] = <i>e</i><sup> − <i>λ</i></sup><span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>3<i>T</i><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><i>dx</i>
</div>
<div class="formula">
 = <i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">(2<i>T</i> − 1)<i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − (3<i>T</i> + 1)<i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><i>dx</i>
</div>
<div class="formula">
 = <i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">(2<i>T</i> − 1)<i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">3<i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span> − 1<span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<div class="formula">
 = <i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">6<i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">3<i>T</i><sup>2</sup> + 4<i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span> − exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">(2<i>T</i> − 1)<i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[<i>ρ</i><sup>2</sup>] = <i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">6<i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">3<i>T</i><sup>2</sup> + 4<i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
</div>
<div class="section" id="variance-of-the-mass-weighted-density-v-m-rho-e-m-rho-2-e-m-rho-2">
<h4>Variance of the Mass-Weighted Density <span class="formula"><i>V</i><sub><i>M</i></sub>[<i>ρ</i>] = <i>E</i><sub><i>M</i></sub>[<i>ρ</i><sup>2</sup>] − <i>E</i><sub><i>M</i></sub>[<i>ρ</i>]<sup>2</sup></span></h4>
<p>Since correction factors are given for <span class="formula"><i>E</i><sub><i>M</i></sub>[<i>ρ</i><sup>2</sup>]</span> and
<span class="formula"><i>E</i><sub><i>M</i></sub>[<i>ρ</i>]</span>, they are not included separately here:</p>
<div class="formula">
<i>V</i><sub><i>M</i></sub>[<i>ρ</i>] = <i>E</i><sub><i>M</i></sub>[<i>ρ</i><sup>2</sup>] − <i>E</i><sub><i>M</i></sub>[<i>ρ</i>]<sup>2</sup> = <i>ρ</i><span class="scripts"><sup class="script">2</sup><sub class="script">0</sub></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">6<i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">3<i>T</i><sup>2</sup> + 4<i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span> − exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><i>λ</i><span class="fraction"><span class="ignored">(</span><span class="numerator">4<i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">1 + 3<i>T</i> + 2<i>T</i><sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
</div>
<div class="section" id="expectation-value-of-the-volume-weighted-log-density-e-ln-rho">
<h4>Expectation Value of the Volume-Weighted Log Density <span class="formula"><i>E</i>[ln<i>ρ</i>]</span></h4>
<div class="formula">
<i>E</i>[ln<i>ρ</i>] = <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span>ln<i>ρ</i><i>e</i><sup> − <i>λ</i></sup><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><i>dx</i> + <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span>ln<i>ρ</i><i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<i>u</i>)<i>du</i>
</div>
<div class="formula">
<i>E</i><sub><i>δ</i></sub>[ln<i>ρ</i>] = <i>e</i><sup> − <i>λ</i></sup><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + ln<i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>left</i></sub>[ln<i>ρ</i>] = <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>ln<i>ρ</i><sub>0</sub> + <i>T</i><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><i>e</i><sup> − <i>λ</i></sup><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><i>dx</i>
</div>
<div class="formula">
 = <i>e</i><sup> − <i>λ</i></sup><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><i>dx</i> − <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><i>dx</i><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<div class="formula">
 = <i>e</i><sup> − <i>λ</i></sup><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>(<i>e</i><sup><i>λ</i></sup> − 1) − <span class="fraction"><span class="ignored">(</span><span class="numerator">4<i>T</i><i>λ</i><sup>2</sup><i>e</i><sup><i>λ</i></sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<div class="formula">
 = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>(1 − <i>e</i><sup> − <i>λ</i></sup>) − <i>T</i><i>λ</i><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<div class="formula">
<i>E</i>[ln<i>ρ</i>] = ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <i>T</i><i>λ</i>
</div>
<div class="formula">
 = ln<i>ρ</i><sub>0</sub> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span>
</div>
</div>
<div class="section" id="expectation-value-of-the-mass-weighted-log-density-e-m-ln-rho">
<h4>Expectation Value of the Mass-Weighted Log Density <span class="formula"><i>E</i><sub><i>M</i></sub>[ln<i>ρ</i>]</span></h4>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[ln<i>ρ</i>] = <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span>ln<i>ρ</i><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>ρ</i><sub>0</sub></span><span class="ignored">)</span></span><i>e</i><sup> − <i>λ</i></sup><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><i>dx</i> + <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span>ln<i>ρ</i><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>ρ</i><sub>0</sub></span><span class="ignored">)</span></span><i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<i>u</i>)<i>du</i>
</div>
<div class="formula">
<i>E</i><sub><i>δ</i></sub>[ln<i>ρ</i>] = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + ln<i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"> − <i>λ</i></span><span class="ignored">)/(</span><span class="denominator"><i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>left</i></sub>[ln<i>ρ</i>] = <i>e</i><sup> − <i>λ</i></sup><span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>ln<i>ρ</i><sub>0</sub> + <i>T</i><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><i>T</i><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><i>dx</i>
</div>
<p>Again, separate into integrable terms:</p>
<div class="formula">
 = exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <i>λ</i><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − (1 + <i>T</i>)<i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span> + <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>Tx</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − (1 + <i>T</i>)<i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
<i>L</i> = <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span>
</div>
<div class="formula">
<i>E</i><sub><i>left</i></sub>[ln<i>ρ</i>] = exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <i>λ</i><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span> − 1<span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span> + <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">4<i>λ</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>left</i></sub>[ln<i>ρ</i>] = exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"> − <i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span> − 1<span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span> + <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">4<i>λ</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>left</i></sub>[ln<i>ρ</i>] = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>1 − exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"> − <i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span> − <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[ln<i>ρ</i>] = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span> − <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<div class="formula">
 = ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span>
</div>
</div>
<div class="section" id="expectation-value-of-the-mass-weighted-log-density-squared-e-m-ln-2-rho">
<h4>Expectation Value of the Mass-Weighted Log Density Squared <span class="formula"><i>E</i><sub><i>M</i></sub>[ln<sup>2</sup><i>ρ</i>]</span></h4>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[ln<sup>2</sup><i>ρ</i>] = <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span>(ln<i>ρ</i>)<sup>2</sup><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>ρ</i><sub>0</sub></span><span class="ignored">)</span></span><i>e</i><sup> − <i>λ</i></sup><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><i>dx</i> + <span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span>(ln<i>ρ</i>)<sup>2</sup><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>ρ</i></span><span class="ignored">)/(</span><span class="denominator"><i>ρ</i><sub>0</sub></span><span class="ignored">)</span></span><i>e</i><sup> − <i>λ</i></sup><i>δ</i>(<i>u</i>)<i>du</i>
</div>
<div class="formula">
<i>E</i><sub><i>δ</i></sub>[ln<sup>2</sup><i>ρ</i>] = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + ln<i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><sup>2</sup>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"> − <i>λ</i></span><span class="ignored">)/(</span><span class="denominator"><i>T</i> + 1</span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>left</i></sub>[ln<sup>2</sup><i>ρ</i>] = <i>e</i><sup> − <i>λ</i></sup><span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>ln<i>ρ</i><sub>0</sub> + <i>T</i><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><sup>2</sup>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><i>T</i><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><i>dx</i>
</div>
<p>This time it's just too ugly.  Define a new variable:</p>
<div class="formula">
<i>Q</i> = ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span>
</div>
<div class="formula">
<i>E</i><sub><i>left</i></sub>[ln<sup>2</sup><i>ρ</i>] = <i>e</i><sup> − <i>λ</i> ⁄ (1 + <i>T</i>)</sup><span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><i>Q</i> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><sup>2</sup>exp<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>Tx</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − <i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><i>dx</i>
</div>
<div class="formula">
<i>E</i><sub><i>left</i></sub>[ln<sup>2</sup><i>ρ</i>] = <i>e</i><sup> − <i>λ</i> ⁄ (1 + <i>T</i>)</sup><span class="limits"><span class="limit"><span class="bigoperator integral">∫</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><i>Q</i><sup>2</sup> − 2<i>Q</i><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>x</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>x</i><sup>4</sup></span><span class="ignored">)/(</span><span class="denominator">16<i>λ</i><sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="stretchy">[</span><i>I</i><sub>1</sub>(<i>x</i>)<i>e</i><sup> − (1 + <i>T</i>)<i>x</i><sup>2</sup> ⁄ (4<i>λ</i>)</sup><span class="stretchy">]</span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span><i>dx</i>
</div>
<div class="formula">
<i>E</i><sub><i>left</i></sub>[ln<sup>2</sup><i>ρ</i>] = <i>e</i><sup> − <i>λ</i> ⁄ (1 + <i>T</i>)</sup><span class="array"><span class="arrayrow"><span class="bracket align-left">⎡</span></span><span class="arrayrow"><span class="bracket align-left">⎣</span></span></span><i>Q</i><sup>2</sup>(<i>e</i><sup><i>λ</i> ⁄ (1 + <i>T</i>)</sup> − 1) − 2<i>Q</i><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i></span><span class="ignored">)/(</span><span class="denominator">4<i>λ</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator">4<i>λ</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span><i>e</i><sup><i>λ</i> ⁄ (1 + <i>T</i>)</sup><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">16<i>λ</i><sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>16<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><sup>3</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span>(<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + 2)<i>e</i><sup><i>λ</i> ⁄ (1 + <i>T</i>)</sup><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎤</span></span><span class="arrayrow"><span class="bracket align-right">⎦</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>left</i></sub>[ln<sup>2</sup><i>ρ</i>] = <i>Q</i><sup>2</sup>(1 − <i>e</i><sup> − <i>λ</i> ⁄ (1 + <i>T</i>)</sup>) − 2<i>Q</i><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + 2<span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<p>Add back the <span class="formula"><i>δ</i></span> termG</p>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[ln<sup>2</sup><i>ρ</i>] = <i>Q</i><sup>2</sup> − 2<i>Q</i><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + 2<span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<p>Re-expand to see if it's simplifiable....</p>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[ln<sup>2</sup><i>ρ</i>] = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><sup>2</sup> − 2<span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + 2<span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span>
</div>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[ln<sup>2</sup><i>ρ</i>] = ln<sup>2</sup><i>ρ</i><sub>0</sub> + 2ln<i>ρ</i><sub>0</sub><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span> − 2ln<i>ρ</i><sub>0</sub><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span> − 2<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span> + 2<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span>
</div>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[ln<sup>2</sup><i>ρ</i>] = ln<sup>2</sup><i>ρ</i><sub>0</sub> + 2ln<i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span> − <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i><sup>2</sup>(1 + <i>T</i>)</span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span> − 2<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span> + 2<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><sup>2</sup><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>4</sup></span><span class="ignored">)</span></span>
</div>
<div class="formula">
<i>E</i><sub><i>M</i></sub>[ln<sup>2</sup><i>ρ</i>] = ln<sup>2</sup><i>ρ</i><sub>0</sub> + 2ln<i>ρ</i><sub>0</sub><span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">1 + <i>T</i></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i><sup>2</sup>(1 + <i>T</i>)</span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span> − 2<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span> + 2<span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><sup>2</sup><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>4</sup></span><span class="ignored">)</span></span>
</div>
<div class="formula">
 = ln<sup>2</sup><i>ρ</i><sub>0</sub> + 2ln<i>ρ</i><sub>0</sub><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span> + <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><sup>2</sup> + <span class="fraction"><span class="ignored">(</span><span class="numerator">2<i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span>
</div>
<div class="formula">
 = <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><sup>2</sup> + <span class="fraction"><span class="ignored">(</span><span class="numerator">2<i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span>
</div>
<p>As expected, we recover the correct relation from <a class="reference external" href="http://adsabs.harvard.edu/abs/2013MNRAS.430.1880H">Hopkins 2013</a>:</p>
<div class="formula">
<i>S</i><sub>ln<i>ρ</i>, <i>M</i></sub> = <i>E</i><sub><i>M</i></sub>[ln<i>ρ</i><sup>2</sup>] − <i>E</i><sub><i>M</i></sub>[ln<i>ρ</i>]<sup>2</sup> = ln<sup>2</sup><i>ρ</i><sub>0</sub> + 2ln<i>ρ</i><sub>0</sub><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span> + <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span><span class="fraction"><span class="ignored">(</span><span class="numerator"><i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><sup>2</sup> + <span class="fraction"><span class="ignored">(</span><span class="numerator">2<i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span> − <span class="array"><span class="arrayrow"><span class="bracket align-left">⎛</span></span><span class="arrayrow"><span class="bracket align-left">⎝</span></span></span>ln<i>ρ</i><sub>0</sub> + <span class="fraction"><span class="ignored">(</span><span class="numerator"><i>T</i><sup>2</sup><i>λ</i></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>2</sup></span><span class="ignored">)</span></span><span class="array"><span class="arrayrow"><span class="bracket align-right">⎞</span></span><span class="arrayrow"><span class="bracket align-right">⎠</span></span></span><sup>2</sup>
</div>
<div class="formula">
 = <span class="fraction"><span class="ignored">(</span><span class="numerator">2<i>λ</i><i>T</i><sup>2</sup></span><span class="ignored">)/(</span><span class="denominator">(1 + <i>T</i>)<sup>3</sup></span><span class="ignored">)</span></span>
</div>
<p><em>independent</em> of <span class="formula"><i>ρ</i><sub>0</sub></span>.</p>
<div class="section" id="properties-of-the-hopkins-pdf">
<h5>Properties of the Hopkins PDF</h5>
<p>I began this investigation in order to find out whether a different parameter,
i.e. something related to the compressiveness of the turbulent driving, could
be responsible for the &quot;discrepancy&quot; between the Formaldehyde-derived density
and the volume-averaged density of some clouds.</p>
<p>I naively expected that, in compressive turbulence, more of the mass will be
concentrated at higher density, which should drive up the mass-weighted mean
density.</p>
<p><a class="reference external" href="http://adsabs.harvard.edu/abs/2013MNRAS.430.1880H">Hopkins 2013</a> showed that <span class="formula"><i>T</i> ∼ ℳ<sub><i>C</i></sub></span>.  If this is taken on
face value, without recognizing the relation between <span class="formula"><i>T</i></span> and
<span class="formula"><i>σ</i></span>, it means that for <em>fixed</em> <span class="formula"><i>σ</i></span>,
<span class="formula"> &lt; <i>ρ</i> &gt; <sub><i>M</i></sub> ∝ <i>e</i><sup> − <i>T</i><sup>2</sup></sup></span>.</p>
<p>This figure shows the relation of the various moments and <span class="formula"><i>T</i></span>.  The plots
show both the &quot;theoretical&quot; result (i.e., doing the integral by hand) and the
numerical result with 50,000 data points plus a correction for the
<span class="formula"><i>δ</i></span> terms; the agreement is essentially perfect excepting some
numerical noise (the only visible discrepancy is for a perfect lognormal at
high <span class="formula"><i>σ</i><sub><i>V</i></sub></span>):</p>
<img alt="" src="https://keflavich.github.io/blog/images/rho1_varyT_colorSigma.png" style="width: 800px;" />
<p>However, <a class="reference external" href="http://adsabs.harvard.edu/abs/2013MNRAS.430.1880H">Hopkins 2013</a> also found that simulations generally produced
<span class="formula"><i>T</i> ∼ 0.1<i>σ</i><span class="scripts"><sup class="script">2</sup><sub class="script"><i>s</i>, <i>M</i></sub></span></span> (though a relation of the form <span class="formula"><i>T</i> ∼ 0.25ln(1 + 0.25<i>σ</i><span class="scripts"><sup class="script">4</sup><sub class="script"><i>s</i>, <i>M</i></sub></span>)</span> is also a good fit).</p>
<p>If you use the simpler of these relations, the expected relationship (mean mass
increasing with increasing &quot;compressiveness&quot;) is recovered.  However, it comes
with an increasing <span class="formula"><i>σ</i><sub><i>s</i></sub></span>.  In these plots, half are labeled with
<span class="formula"><i>σ</i></span> and the others are labeled with <span class="formula"><i>T</i></span>, though the two are
equivalent.  The highest <span class="formula"><i>σ</i><sub><i>s</i></sub></span> observed in any of the simulations
shown in <a class="reference external" href="http://adsabs.harvard.edu/abs/2013MNRAS.430.1880H">Hopkins 2013</a> was about <span class="formula"><i>σ</i><sub><i>s</i></sub> = 4</span> (marked with a black
square below), so the maximum <span class="formula"><i>ρ</i><sub><i>M</i></sub> ⁄ <i>ρ</i><sub><i>V</i></sub> ∼ 10<sup>2</sup></span>.</p>
<img alt="" src="https://keflavich.github.io/blog/images/TSigma.png" style="width: 800px;" />
</div>
</div>
</div>
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