Editing Apps/ImamuraHadleyCollaboration (section)

====Density Fluctuations in Tori with Small but Finite &beta;====
The eigenvector that describes density fluctuations in tori with small but finite <math>~\beta</math> is obtained by combining the [[#DensityPerturbation2|above expression]] for <math>~\rho^'/\rho_0</math> in terms of <math>~\delta W</math>, with the expressions for <math>~\delta W_{0,0,m}</math> and <math>~\sigma_{0,0,m}</math> derived by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] and [[#Tori_with_Small_but_Finite_.CE.B2|summarized above]].  Keeping in mind that the Blaes85 analysis targeted structures with uniform specific angular momentum, which means that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\Omega}{\Omega_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi^{-2}
= (1 - x\cos\theta)^{-2} \approx (1 + 2\eta\beta\cos\theta ) \, ,
</math>
  </td>
</tr>
</table>
</div>

we obtain,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[\frac{\rho^'}{\rho_0}\biggr]_{0,0,m} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{n}{n+1}\biggl[\biggl(\frac{\sigma }{\Omega_0 }\biggr)_{0,0,m} + m\cdot \frac{\Omega}{\Omega_0}\biggr] \frac{\delta W_{0,0,m}}{f} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{C}{(1-\eta^2)}\biggl(\frac{n}{n+1}\biggr)\biggl\{
\biggl[-m  - ~i~m\biggl( \frac{3}{2n+2}  \biggr)^{1/2}\beta\biggr] + m (1 + 2\eta\beta\cos\theta ) 
\biggr\} e^{[i(m\varphi + \sigma_{0,0,m} t)]}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~</math>
  </td>
  <td align="left">
<math>~
\times ~ \biggl\{
1 + \beta^2 m^2\biggl[
2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} ~\pm ~4i \biggl( \frac{3}{2n+2}  \biggr)^{1/2}\eta\cos\theta
\biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{C}{(1-\eta^2)}\biggl(\frac{n}{n+1}\biggr)e^{[i(m\varphi + \sigma_{0,0,m} t)]}  \biggl[
\mathrm{Re}(\Delta) + i~\mathrm{Im}(\Delta)
\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Re}(\Delta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
2\eta(\beta m)\cos\theta\biggl\{ 1 + (\beta m)^2\biggl[
2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~</math>
  </td>
  <td align="left">
<math>~\pm
(\beta m)\biggl[\frac{3}{2(n+1)}\biggr]^{1/2} \biggl\{4(\beta m)^2\biggl[\frac{3}{2(n+1)}\biggr]^{1/2} \eta\cos\theta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\eta(\beta m)\cos\theta \pm \mathcal{O}(\beta^3) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{Im}(\Delta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
-~(\beta m) \biggl[\frac{3}{2(n+1)}\biggr]^{1/2}\biggl\{ 1 + (\beta m)^2\biggl[
2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~</math>
  </td>
  <td align="left">
<math>~
\pm~2\eta (\beta m)\cos\theta \biggl\{ 4(\beta m)^2 \biggl[\frac{3}{2(n+1)}\biggr]^{1/2} \eta\cos\theta\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>-~(\beta m) \biggl[\frac{3}{2(n+1)}\biggr]^{1/2} \pm \mathcal{O}(\beta^3) \, .</math>
  </td>
</tr>
</table>
</div>

Therefore, at any instant in time, this density eigenfunction can be written in the [[#DensityEigenfunction|form discussed above]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{\rho^'}{\rho_0} \biggr]_{0,0,m}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~|g_{0,0,m}(\eta,\theta)|\exp\{i[\alpha_{0,0,m}(\eta,\theta)+m\varphi]\}  \, ,</math>
  </td>
</tr>
</table>
</div>
if we set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~|g_{0,0,m}(\eta,\theta)|</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{C}{(1-\eta^2)}\biggl(\frac{n}{n+1}\biggr)|\Delta| \, ,</math>
  </td>
</tr>
</table>
</div>

where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~|\Delta|^2</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl[ 2\eta(\beta m)\cos\theta\biggr]^2 + \biggl\{ -~(\beta m) \biggl[\frac{3}{2(n+1)}\biggr]^{1/2}  \biggr\}^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4(\beta m)^2 \biggl[ \eta^2\cos^2\theta + \frac{3}{8(n+1)} \biggr] \, ;
</math>
  </td>
</tr>
</table>
</div>
and if we simultaneously set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\alpha_{0,0,m}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan^{-1}\biggl[ \frac{\mathrm{Im}(\Delta)}{\mathrm{Re}(\Delta)} \biggr] 
\approx \tan^{-1}\biggl\{ \frac{-\sqrt{3/[2(n+1)]}}{2\eta\cos\theta} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

We note that, in the case of non-self-gravitating PP tori <math>~(\delta \Phi = 0)</math>, the amplitude of the "perturbed enthalpy" as defined by equation (38) of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley et al. (2014; Paper II)] is, to within a leading scale factor, just <math>~|\Delta|</math>.  Specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{W}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{P_0}{\rho_0}\biggl(\frac{n+1}{n}\biggr)  |g_{0,0,m}| + \cancelto{0}{\delta\Phi}
= \biggl(\frac{1-\eta^2}{n}\biggr)  |g_{0,0,m}|
= \biggl(\frac{C}{n+1}\biggr)  |\Delta| \, .
</math>
  </td>
</tr>
</table>
</div>


<div align="center" id="Table5">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="8">
<font size="+1">Table 5:&nbsp;&nbsp;Density Fluctuations in Tori with <math>~n = \tfrac{3}{2}</math> and Small but Finite</font> <math>~\beta</math>
<p></p>
<font size="+1">&hellip; &nbsp;Equatorial plane</font>&nbsp;&nbsp; &nbsp;&nbsp; <math>~~\Rightarrow~~~\cos\theta = \pm 1</math> 
</th></tr>
<tr>
  <td align="center" rowspan="2"><math>~\eta\cos\theta</math>
  <td align="center" rowspan="2"><math>~\frac{|\Delta|}{(\beta m)}</math>
  <td align="center" rowspan="2"><math>~\frac{|g_{0,0,m}|}{C(\beta m)}</math>
  <td align="center" colspan="3"><math>~\alpha_{0,0,m}+\varphi_\mathrm{shift}</math></td>
  <td align="center" colspan="2"><font size="+1"><math>~x\cos\theta</math>
<p></p>
(</font>assuming <math>~\beta = 0.12</math><font size="+1">)</font></td>
</tr>
<tr>
  <td align="center">Quadrant</td>
  <td align="center"><math>~\varphi_\mathrm{shift}=0</math></td>
  <td align="center"><math>~\varphi_\mathrm{shift}=\pi</math></td>
  <td align="center" rowspan="1">inner<p></p>
<math>~\cos\theta =-1</math>
  </td>
  <td align="center" rowspan="1">outer<p></p>
<math>~\cos\theta =+1</math>
  </td>
</tr>
<tr>
  <td align="center"><math>~-1.00</math></td>
  <td align="center"><math>~2.145</math></td>
  <td align="center"><math>~\infty</math></td>
  <td align="center">3<sup>rd</sup></td>
  <td align="center">---</td>
  <td align="center"><math>~+3.5111</math></td>
  <td align="center"><math>~- 0.1088</math></td>
  <td align="center"></td>
</tr>
<tr>
  <td align="center"><math>~-0.75</math></td>
  <td align="center"><math>~1.688</math></td>
  <td align="center"><math>~2.315</math></td>
  <td align="center">3<sup>rd</sup></td>
  <td align="center">---</td>
  <td align="center"><math>~+3.6183</math></td>
  <td align="center"><math>~- 0.0833</math></td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center"><math>~-0.50</math></td>
  <td align="center"><math>~1.265</math></td>
  <td align="center"><math>~1.012</math></td>
  <td align="center">3<sup>rd</sup></td>
  <td align="center">---</td>
  <td align="center"><math>~+3.8006</math></td>
  <td align="center"><math>~- 0.0569</math></td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center"><math>~-0.25</math></td>
  <td align="center"><math>~0.922</math></td>
  <td align="center"><math>~0.5901</math></td>
  <td align="center">3<sup>rd</sup></td>
  <td align="center">---</td>
  <td align="center"><math>~+4.1392</math></td>
  <td align="center"><math>~-0.0292</math></td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center"><math>~\mp 0.00</math></td>
  <td align="center"><math>~0.775</math></td>
  <td align="center"><math>~0.4648</math></td>
  <td align="center">---</td>
  <td align="center"><math>~-\tfrac{\pi}{2}</math></td>
  <td align="center"><math>~\tfrac{3\pi}{2}</math></td>
  <td align="center"><math>~0.0</math></td>
  <td align="center"><math>~0.0</math></td>
</tr>
<tr>
  <td align="center"><math>~+0.25</math></td>
  <td align="center"><math>~0.922</math></td>
  <td align="center"><math>~0.5901</math></td>
  <td align="center">4<sup>th</sup></td>
  <td align="center"><math>~-0.9976</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~0.0310</math></td>
</tr>
<tr>
  <td align="center"><math>~+0.50</math></td>
  <td align="center"><math>~1.265</math></td>
  <td align="center"><math>~1.012</math></td>
  <td align="center">4<sup>th</sup></td>
  <td align="center"><math>~-0.6591</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~0.0643</math></td>
</tr>
<tr>
  <td align="center"><math>~+0.75</math></td>
  <td align="center"><math>~1.688</math></td>
  <td align="center"><math>~2.315</math></td>
  <td align="center">4<sup>th</sup></td>
  <td align="center"><math>~-0.4767</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~0.1007</math></td>
</tr>
<tr>
  <td align="center"><math>~+1.00</math></td>
  <td align="center"><math>~2.145</math></td>
  <td align="center"><math>~\infty</math></td>
  <td align="center">4<sup>th</sup></td>
  <td align="center"><math>~-0.3695</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~0.1416</math></td>
</tr>
</table>
</div>


The second and third columns of [[#Table5|Table 5]] detail how, respectively, <math>~|\Delta|</math> and <math>~|g_{0,0,m}|</math> vary with location, <math>~-1 \le \eta\cos\theta \le +1</math>, in the equatorial plane <math>~(\cos\theta = \pm 1)</math> of a slim, <math>~n=\tfrac{3}{2}</math> PP torus.  Table 5 also contains an evaluation of the phase angle, <math>~\alpha_{0,0,m}</math>, across the equatorial plane of the same torus.  As indicated, in making this evaluation, care has been taken to place the phase angle in the proper quadrant of the equatorial plane.  Specifically &#8212; keeping in mind that, according to Blaes' analytic solution, the numerator of the arctangent argument, <math>~\mathrm{Im}(\Delta)</math>, is always negative &#8212; the phase angle should land in either the 3<sup>rd</sup> or 4<sup>th</sup> quadrant depending on whether the denominator is, respectively, negative <math>~(\eta\cos\theta < 0)</math> or positive <math>~(\eta\cos\theta > 0)</math>.  Because a standard evaluation of the arctangent function returns an angle that lies either in the 1<sup>st</sup> quadrant (positive argument) or the 4<sup>th</sup> quadrant (negative argument), we have added <math>~\varphi_\mathrm{shift} = \pi</math> to the value returned by the arctangent function in order to push the phase angle from the 1<sup>st</sup> to the 3<sup>rd</sup> quadrant wherever the denominator is negative  &#8212; that is to say, this phase shift has been implemented, throughout the range, <math>-1 \le \eta\cos\theta < 0</math>.