Editing SSC/Stability/BiPolytropes/HeadScratching (section)

==Selected Models==

Via a crude iterative technique, we have determined that the derivative, <math>dM^*_\mathrm{tot}/d\ell_i</math>, goes to zero when <math>\xi_i = 2.27276626</math> (to eight significant digits); this is therefore the minimum-mass model &#8212; identified by the light-blue diamond-shaped marker &#8212; along the <math>M^*_\mathrm{tot}(R^*)</math> sequence shown above in Figure 4. A few other properties of this model "<b>A</b>" are recorded in Table 2.  For example, <math>(M^*_\mathrm{tot}, R^*) = (38.97032951, 12.598233)</math>; and its position (also marked by a light-blue diamond) along the Figure 5 <math>(q, \nu) = (0.1246568, 0.0927131)</math>. The lower-left figure in Table 2 shows how <math>(r^*)</math> varies with enclosed mass-fraction for this minimum-mass model "<b>A</b>"; the core-envelope interface &#8212; where the blue and red segments of the plotted curve meet &#8212; is located at <math>(M_r/M_\mathrm{tot}, r^*) = (0.092713145, 1.5704549)</math>.

<table border="1" align="center" cellpadding="5">

<tr>
  <th align="center" colspan="8">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/Bipolytrope/Stability/qAndNuMax.xlsx --- worksheet = K-BK74 thru MinuPreparation]]Table 2<br />Bipolytrope with <math>(n_c, n_e) = (5, 1)</math><br />Selected Pairings along the <math>\mu_e/\mu_c = 0.25</math> Sequence</th>
</tr>

<tr>
  <td align="center">Pairing</td>
  <td align="center"><math>(\xi_i)_+</math></td>
  <td align="center"><math>\Lambda_i</math></td>
  <td align="center"><math>M^*_\mathrm{tot}</math></td>
  <td align="center"><math>\frac{dM^*_\mathrm{tot}}{d\ell_i}</math></td>
  <td align="center"><math>R^*</math></td>
  <td align="center"><math>q</math></td>
  <td align="center"><math>\nu</math></td>
</tr>

<tr>
  <td align="center">'''Example #1'''</td>
  <td align="center"><math>0.5</math></td>
  <td align="center"><math>4.715027199</math></td>
  <td align="center"><math>40.09338625</math></td>
  <td align="center"><math>-0.413955</math></td>
  <td align="center">--</td>
  <td align="center">--</td>
  <td align="center">--</td>
</tr>

<tr>
  <td align="center">'''A''' <font size="-1">(degenerate)</font></td>
  <td align="center"><math>2.27276626</math></td>
  <td align="center"><math>1.4535131</math></td>
  <td align="center"><math>38.97032951</math></td>
  <td align="center"><math>6.20\times 10^{-9}</math></td>
  <td align="center"><math>12.598233</math></td>
  <td align="center"><math>0.1246568</math></td>
  <td align="center"><math>0.0927131</math></td>
</tr>

<tr>
  <td align="center">'''B1'''</td>
  <td align="center"><math>2.0653386</math></td>
  <td align="center"><math>1.5156453</math></td>
  <td align="center"><math>39.00000000</math></td>
  <td align="center"><math>-0.491175</math></td>
  <td align="center"><math>11.31459</math></td>
  <td align="center"><math>0.1261314</math></td>
  <td align="center"><math>0.082829</math></td>
</tr>

<tr>
  <td align="center">'''B2'''</td>
  <td align="center"><math>2.4782510</math></td>
  <td align="center"><math>1.4088069</math></td>
  <td align="center"><math>39.00000000</math></td>
  <td align="center"><math>+0.500086</math></td>
  <td align="center"><math>13.987375</math></td>
  <td align="center"><math>0.1224277</math></td>
  <td align="center"><math>0.1013938</math></td>
</tr>

<tr>
  <td align="center">'''C1'''</td>
  <td align="center"><math>1.83343536</math></td>
  <td align="center"><math>1.612448</math></td>
  <td align="center"><math>39.10000000</math></td>
  <td align="center"><math>-0.990185582</math></td>
  <td align="center"><math>10.019034</math></td>
  <td align="center"><math>0.1264476</math></td>
  <td align="center"><math>0.0705448</math></td>
</tr>

<tr>
  <td align="center">'''C2'''</td>
  <td align="center"><math>2.70235958</math></td>
  <td align="center"><math>1.3746562</math></td>
  <td align="center"><math>39.10000000</math></td>
  <td align="center"><math>+1.042782519</math></td>
  <td align="center"><math>15.637446</math></td>
  <td align="center"><math>0.119412</math></td>
  <td align="center"><math>0.1095988</math></td>
</tr>

<tr>
  <td align="center" colspan="4">
[[File:MinMassProfile0.25.png|400px|Radius vs. Mass for Minimum-Mass Bipoltrope having &mu;-ratio = 0.250]]
  </td>
  <td align="center" colspan="4">
'''Eigenfunction Obtained Via B-KB74 Conjecture'''<br />
[[File:EigenfunctionCorrected.png|450px|Eigenfunction for Minimum-Mass Bipoltrope having &mu;-ratio = 0.250]]
  </td>
</tr>
</table>


In the context of our analysis of the stability of pressure-truncated n = 5 polytropes, we showed how the [[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_.3D_5_Polytropes|B-KB74 conjecture]] can be used to illustrate the approximate shape of the radial eigenfunction of the marginally unstable mode.  Proceeding along the lines of this independent discussion, here we have identified two equilibrium models &#8212; labeled "<b>B1</b>" and "<b>B2</b>" in Table 2 &#8212; that lie near to, but on either side of, the minimum-mass model along the equilibrium sequence and that have identical total masses: in this case, <math>M^*_\mathrm{tot} = 39.00000000</math> (identical, to nine significant digits).  Using the mass-fraction, <math>m_r \equiv M_r/M_\mathrm{tot}</math>, as the Lagrangian coordinate for both models, we subtracted the profile of model "<b>B1</b>" from the profile of model "<b>B2</b>" and divided this difference by the average profile, we obtained the approximate neutral-mode eigenfunction, <math>x(m_r)</math>, displayed in the lower-right figure of Table 2.

Things to note about this iteratively derived, approximate neutral-mode eigenfunction:
<ol>
  <li>The radial-displacement function, <math>x(m_r)</math>, has been normalized to unity at the surface.</li>
  <li>The location of the model "<b>A</b>" core-envelope interface <math>(m_r = \nu_{A} = 0.0927131)</math> has been marked by the vertical, red-dashed line segment.</li>
  <li>Throughout the core, <math>x</math> is very small; consistent with being zero throughout.</li>
  <li>Moving inward through the envelope, <math>x</math> appears to drop smoothly from "plus" one (at the surface) to approximately "minus" one (at the interface).</li>
  <li>Because <math>x</math> passes through zero one time inside the envelope, this cannot be the eigenfunction of the fundamental mode of radial oscillation; instead, it is likely associated with the 1st overtone, as discussed for example in connection with [[SSC/Stability/n3PolytropeLAWE#Fig1|Schwarzschild's modeling of radial eigenfunctions of n = 3 polytropes]].</li>
</ol>

With regard to the second itemized note, we should point out that, although models "<b>B1</b>" and "<b>B2</b>" have identical total masses, their core mass-fraction &#8212; that is, the location of the core-envelope interface as defined by the Lagrangian mass marker &#8212; is different: <math>\nu_{B1} = 0.082829</math> and <math>\nu_{B2} = 0.101394</math>.  As a result, the B-KB74 conjecture should not be expected to apply in the immediate vicinity of the core-envelope interface.