Editing SSC/Stability/Isothermal (section)

===From the Analysis of Taff &amp; van Horn (1974)===
The left-hand column of Composite Display 1, below, contains a pair of images that present properties of the eigenvectors that resulted from the {{ TVH74full }} analysis of radial oscillations in pressure-truncated isothermal spheres, assuming that the configurations remain isothermal &#8212; that is, adopting an adiabatic exponent of <math>~\Gamma_1 = 1</math> &#8212; during the oscillations.  In the image titled, "Table I", that we have extracted from their paper, the first column of numbers identifies values of nine adopted truncation radii in the range, <math>~2 \le x_0 \le 10</math>, while the second column lists the corresponding value of <math>~\lambda_0^2</math> that were determined via their numerical analysis.  For three values of the truncation radius &#8212; <math>~x_0 = 6, 7, \And 8</math> &#8212; the third column lists the values of <math>~\lambda_0^2</math> that had been previously reported by {{ Yabushita68full }}.

In the right-hand column of Composite Display 1, we have detailed some results from our own [[#Our_Numerical_Integration|numerical analysis]] of the same set of nine configurations that were studied by {{ TVH74 }}.  In the second column of our table that has the heading, "Fundamental Mode," we have listed the value of <math>~\mathfrak{F}</math> that was required in our analysis in order to generate an eigenfunction whose logarithmic derivative at each configuration's surface was precisely negative three (to five significant digits).  In order to facilitate quantitative comparison with the work of {{ TVH74 }}, the third column of our table lists for each model the corresponding value of <math>\lambda_0^2</math>; recalling that <math>\gamma = 1</math> and <math>\alpha = -1</math>, each of these values was determined via the relation,
<div align="center">
<math>~\lambda_0^2 = \frac{\gamma(\mathfrak{F}+2\alpha)}{6} = \frac{(\mathfrak{F}-2)}{6} \, .</math>
</div>
The agreement between our numerically determined fundamental-mode eigenvalues (highlighted by pink, rectangular boxes) and the ones reported by {{ TVH74 }} is excellent, across the board.  Notice that <math>\lambda_0^2</math> is negative for the models having <math>x_0 = 7, 8, 9, \And 10</math>, which indicates that these four models are dynamically unstable.


<div align="center" id="Yabushita68">
'''<font size="+1>Composite Display 1:</font>&nbsp; Select Eigenfrequencies for Pressure-Truncated Isothermal Spheres'''

<table border="1" cellpadding="5" width="75%">
<tr>
  <td align="center">
Data extracted from Tables I &amp; II (p. 429) of<br />{{ TVH74figure }}
  </td>
  <td align="center">
From [[#Our_Numerical_Integration|Our Analysis]]<p></p><math>~\alpha = -1</math>&nbsp;&nbsp;<math>~(\gamma=1)</math><p></p>
B.C.:&nbsp;&nbsp; <math>~\frac{d\ln x}{d\ln\xi} = -3.00000</math>
  </td>
</tr>

<tr>
  <td align="center" rowspan="1">
<!-- [[File:TaffAndVanHorn1974Table1.png|center|Taff &amp; Van Horn (1974)]] -->
Fundamental mode eigenvalues <math>\lambda_0^2</math> vs. <math>x_0</math> <math>(\Gamma_1=1)</math>
  </td>
  <td align="center">Fundamental Mode</td>
</tr>
<tr>
  <td align="center">
<table border="0" align="center" cellpadding="6">

<tr>
  <td align="center" rowspan="1"><math>x_0</math>
<br />&nbsp; <hr /></td>
  <td align="center"><math>\lambda_0^2</math> <br />from p. 429 of {{ TVH74hereafter }}<hr /></td>
  <td align="center"><math>\lambda_0^2</math> <br />from p. 116 of {{ Yabushita688hereafter }}<hr /></td>
</tr>

<tr>
  <td align="center"><math>2</math></td>
  <td align="center"><math>1.822~~</math></td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center"><math>3</math></td>
  <td align="center"><math>0.5860~</math></td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center"><math>4</math></td>
  <td align="center"><math>0.2161~</math></td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center"><math>5</math></td>
  <td align="center"><math>0.07607</math></td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center"><math>6</math></td>
  <td align="center"><math>0.01543</math></td>
  <td align="center"><math>+~0.01614</math></td>
</tr>

<tr>
  <td align="center"><math>7</math></td>
  <td align="center"><math>-~0.01316</math></td>
  <td align="center"><math>-~0.01274</math></td>
</tr>

<tr>
  <td align="center"><math>8</math></td>
  <td align="center"><math>-~0.02742</math></td>
  <td align="center"><math>-~0.02710</math></td>
</tr>

<tr>
  <td align="center"><math>9</math></td>
  <td align="center"><math>-~0.03477</math></td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center"><math>10</math></td>
  <td align="center"><math>-~0.03862</math></td>
  <td align="center">---</td>
</tr>
</table>
  </td>
  <td align="center">

<table border="0" align="center" cellpadding="6">
<tr>
  <td align="center"><math>~\xi_0</math></td>
  <td align="center"><math>~\mathfrak{F}</math></td>
  <td align="center"><math>~\lambda_0^2 = \frac{\gamma(\mathfrak{F}+2\alpha)}{6}</math></td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">12.92907</td>
  <td align="center" bgcolor="pink">+1.821512</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">5.51614</td>
  <td align="center" bgcolor="pink">+0.586023</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="center">3.29671</td>
  <td align="center" bgcolor="pink">+0.216118</td>
</tr>
<tr>
  <td align="center">5</td>
  <td align="center">2.456412</td>
  <td align="center" bgcolor="pink">+0.0760687</td>
</tr>
<tr>
  <td align="center">6</td>
  <td align="center">2.092651</td>
  <td align="center" bgcolor="pink">+0.0154418</td>
</tr>
<tr>
  <td align="center">7</td>
  <td align="center">1.921062</td>
  <td align="center" bgcolor="pink">-0.013156</td>
</tr>
<tr>
  <td align="center">8</td>
  <td align="center">1.8354928</td>
  <td align="center" bgcolor="pink">-0.027418</td>
</tr>
<tr>
  <td align="center">9</td>
  <td align="center">1.791388</td>
  <td align="center" bgcolor="pink">-0.034769</td>
</tr>
<tr>
  <td align="center">10</td>
  <td align="center">1.7683435</td>
  <td align="center" bgcolor="pink">-0.038609</td>
</tr>
</table>

  </td>
</tr>
<tr>
  <td align="center" rowspan="1">
Fundamental and First Harmonic Eigenfunctions and Eigenvalues <math>(\Gamma_1=1)</math>
  </td>
  <td align="center">First Harmonic</td>
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<tr>
  <td align="left" rowspan="1">
<!--[[File:TaffAndVanHorn1974Table2.png|center|Taff &amp; Van Horn (1974)]] -->
<table border="0" align="center" cellpadding="6">

<tr>
  <td align="center"><math>x_0</math><hr /></td>
  <td align="center"><math>\rho_c/<\rho></math><hr /></td>
  <td align="center"><math>\lambda_0^2</math><hr /></td>
  <td align="center"><math>\xi_0(0)</math><hr /></td>
  <td align="center"><math>x_\mathrm{pk}</math><hr /></td>
  <td align="center"><math>\xi_0(x_\mathrm{pk})</math><hr /></td>
  <td align="center"><math>\lambda_1^2</math><hr /></td>
  <td align="center"><math>\xi_1(0)</math><hr /></td>
  <td align="center"><math>x_{1,1}</math><hr /></td>
</tr>

<tr>
  <td align="center"><math>3</math></td>
  <td align="center"><math>1.934</math></td>
  <td align="center"><math>0.5860</math></td>
  <td align="center"><math>2.970</math></td>
  <td align="center">----</td>
  <td align="center">----</td>
  <td align="center" bgcolor="pink"><math>3.766</math></td>
  <td align="center"><math>-8.567</math></td>
  <td align="center"><math>2.07</math></td>
</tr>

<tr>
  <td align="center"><math>6</math></td>
  <td align="center"><math>5.018</math></td>
  <td align="center"><math>0.01543</math></td>
  <td align="center"><math>3.552</math></td>
  <td align="center">----</td>
  <td align="center">----</td>
  <td align="center" bgcolor="pink"><math>0.7178</math></td>
  <td align="center"><math>-5.561</math></td>
  <td align="center"><math>3.85</math></td>
</tr>

<tr>
  <td align="center"><math>9</math></td>
  <td align="center"><math>10.72</math></td>
  <td align="center"><math>-~0.03477</math></td>
  <td align="center"><math>5.311</math></td>
  <td align="center">----</td>
  <td align="center">----</td>
  <td align="center" bgcolor="pink"><math>0.2444</math></td>
  <td align="center"><math>-~4.685</math></td>
  <td align="center"><math>5.29</math></td>
</tr>
</table>

  </td>
  <td align="center">
<table border="0" align="center" cellpadding="6">
<tr>
  <td align="center"><math>~\xi_1</math></td>
  <td align="center"><math>~\mathfrak{F}</math></td>
  <td align="center"><math>~\lambda_1^2 = \frac{\gamma(\mathfrak{F}+2\alpha)}{6}</math></td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">56.87349</td>
  <td align="center" bgcolor="white">+9.14558</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">24.58903</td>
  <td align="center" bgcolor="pink">+3.76484</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="center">13.640525</td>
  <td align="center" bgcolor="white">+1.94009</td>
</tr>
<tr>
  <td align="center">5</td>
  <td align="center">8.798197</td>
  <td align="center" bgcolor="white">+1.13303</td>
</tr>
<tr>
  <td align="center">6</td>
  <td align="center">6.306545</td>
  <td align="center" bgcolor="pink">+0.71776</td>
</tr>
<tr>
  <td align="center">7</td>
  <td align="center">4.88991</td>
  <td align="center" bgcolor="white">+0.48165</td>
</tr>
<tr>
  <td align="center">8</td>
  <td align="center">4.024628</td>
  <td align="center" bgcolor="white">+0.33744</td>
</tr>
<tr>
  <td align="center">9</td>
  <td align="center">3.46662</td>
  <td align="center" bgcolor="pink">+0.24444</td>
</tr>
<tr>
  <td align="center">10</td>
  <td align="center">3.091117</td>
  <td align="center" bgcolor="white">+0.18185</td>
</tr>
</table>
  </td>
</tr>
</table>
</div>


For three of their models &#8212; the ones having <math>x_0 = 3, 6, \And ~9</math> &#8212; {{ TVH74 }} also determined an eigenvalue for the first harmonic mode of oscillation; see the seventh column of numbers, labeled <math>\lambda_1^2</math>, that appears in the digital image of their "Table II" that we have extracted from their paper and presented in Composite Display 1, above. Via our own [[#Our_Numerical_Integration|numerical analysis]], we have determined an eigenvalue for the first harmonic mode of oscillation in all nine of the models.  These results are presented in the right-hand panel of Composite Display 1, under the heading, "First Harmonic."  Again, the agreement between our numerically determined eigenvalues (see the three highlighted by pink, rectangular boxes) and the ones reported by {{ TVH74hereafter }} is excellent.

<span id="CompositeDisplay2">The right-hand column of Composite Display 2, below,</span> presents a pair of animations that display how our numerically derived displacement function, <math>x(\xi)</math>, varies with radius &#8212; from the center of the isothermal sphere, out to <math>\xi = 9</math> &#8212; for a variety of values of the square of the eigenfrequency, <math>\lambda_0^2</math>; each frame of both animations is tagged by the relevant value of <math>\lambda_0^2</math>.  The segment of the <math>x(\xi)</math> curve that has been drawn in blue identifies the ''eigenfunction'' that corresponds to the specified value of the eigenfrequency.  In each frame, the radial location at which the blue segment terminates simultaneously identifies the truncation radius of the relevant isothermal sphere, and the radius at which the boundary condition, <math>d\ln x/d\ln\xi = -3</math>, has been enforced.  In every frame, the <math>x(\xi)</math> function has been normalized such that the displacement amplitude is unity at the truncated configuration's surface.  


<div align="center" id="Yabushita68">
'''<font size="+1>Composite Display 2:</font>&nbsp; Select Eigenfunctions for Pressure-Truncated Isothermal Spheres'''

<table border="1" cellpadding="5" width="75%">
<tr>
  <td colspan="1" align="center">
Figure &amp; caption extracted from p. 430 of <br />{{ TVH74figure }}
</td>
  <td align="center">
From [[#Our_Numerical_Integration|Our Analysis]]<p></p><math>~\alpha = -1</math>&nbsp;&nbsp;<math>~(\gamma=1)</math><p></p>
B.C.:&nbsp;&nbsp; <math>~\frac{d\ln x}{d\ln\xi} = -3.00000</math>
  </td>
</tr>
<tr>
  <td align="left">
[[File:TaffAndVanHorn1974Fig1.png|600px|center|Taff &amp; Van Horn (1974)]]
  </td>
  <td align="left">
[[File:TaffVanHorn1974Harmonic.gif|450px|First Harmonic mode animation]]<p></p>
[[File:TaffVanHorn1974Fundamental.gif|450px|Fundamental mode animation]]
  </td>
</tr>
</table>
</div>

The blue curve segments in the bottom animation identify ''fundamental-mode'' eigenfunctions; most significantly, no radial nodes appear between the center and the surface of the truncated configuration.  The ones that terminate at <math>\xi_e = 3, 6, \And ~9</math> appear to be identical to the three fundamental-mode eigenfunctions with corresponding values of the truncation radius that appear in Figure 1a of {{ TVH74 }} &#8212; see the bottom half of the image that has been extracted from the {{ TVH74hereafter }} publication and presented here in the left-hand column of our Composite Display 2.  

The blue curve segments in the ''top'' animation identify "first harmonic" eigenfunctions; for each of the displayed eigenfunctions, one radial node exists between the center and the surface of the truncated configuration.  The ones that terminate at <math>\xi_e = 3, 6, \And ~9</math>, appear to be identical to the three first harmonic eigenfunctions with corresponding values of the truncation radius that appear in Figure 1b of {{ TVH74 }} &#8212; see the top half of the image that has been extracted from the {{ TVH74hereafter }} publication and presented here in the left-hand column of our Composite Display 2.