https://selfgravitatingfluids.education/JETohline/index.php?action=history&feed=atom&title=Appendix%2FRamblings%2FNonlinarOscillation 2026-05-02T17:03:17Z Revision history for this page on the wiki MediaWiki 1.43.1 https://selfgravitatingfluids.education/JETohline/index.php?title=Appendix/Ramblings/NonlinarOscillation&diff=3224&oldid=prev 2025-12-03T01:01:07Z

<p><span class="autocomment">Radial Oscillations in Pressure-Truncated n = 5 Polytropes</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:01, 2 December 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l119">Line 119:</td> <td colspan="2" class="diff-lineno">Line 119:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where, &lt;math&gt;A_0&lt;/math&gt; is an arbitrary, overall scaling coefficient. This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the &quot;eigenfunction of the neutral mode.&quot;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where, &lt;math&gt;A_0&lt;/math&gt; is an arbitrary, overall scaling coefficient. This is what [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74] refer to as the &quot;eigenfunction of the neutral mode.&quot;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:Displacement5.gif|left|225px<del style="font-weight: bold; text-decoration: none;">|center</del>|Animated Comparison of Displacement Functions]]&lt;font color="maroon"&gt;Comparison:&lt;/font&gt; &amp;nbsp;By plotting one function on top of the other, in the righthand panel of [[#Fig2|Figure 2, below]] &amp;#8212; see, also, the animation displayed here on the left &amp;#8212; we illustrate that, as conjectured by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74], the nonlinear fractional displacement function, &lt;math&gt;x_\mathrm{BKB74} &lt;/math&gt;, does very accurately represent the exact eigenfunction of the neutral mode for small values of the parameter, &lt;math&gt;\mu&lt;/math&gt;.  The validity of this conjecture can be demonstrated even more quantitatively by expanding the function, &lt;math&gt;x_\mathrm{BKB74} &lt;/math&gt;, as a power series in &lt;math&gt;\mu&lt;/math&gt;. To lowest order, we find that,</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:Displacement5.gif|left|225px|Animated Comparison of Displacement Functions]]&lt;font color="maroon"&gt;Comparison:&lt;/font&gt; &amp;nbsp;By plotting one function on top of the other, in the righthand panel of [[#Fig2|Figure 2, below]] &amp;#8212; see, also, the animation displayed here on the left &amp;#8212; we illustrate that, as conjectured by [http://adsabs.harvard.edu/abs/1974A%26A....31..391B B-KB74], the nonlinear fractional displacement function, &lt;math&gt;x_\mathrm{BKB74} &lt;/math&gt;, does very accurately represent the exact eigenfunction of the neutral mode for small values of the parameter, &lt;math&gt;\mu&lt;/math&gt;.  The validity of this conjecture can be demonstrated even more quantitatively by expanding the function, &lt;math&gt;x_\mathrm{BKB74} &lt;/math&gt;, as a power series in &lt;math&gt;\mu&lt;/math&gt;. To lowest order, we find that,</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;div align=&quot;center&quot;&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;div align=&quot;center&quot;&gt;</div></td></tr> </table>

Joel2 https://selfgravitatingfluids.education/JETohline/index.php?title=Appendix/Ramblings/NonlinarOscillation&diff=805&oldid=prev 2024-01-03T23:47:15Z

<p>Created page with &quot;__FORCETOC__ &lt;!-- __NOTOC__ --&gt; =Radial Oscillations in Pressure-Truncated n = 5 Polytropes= &lt;table border=&quot;1&quot; width=&quot;90%&quot; align=&quot;center&quot; cellpadding=&quot;15&quot;&gt; &lt;tr&gt;&lt;td align=&quot;center&quot;&gt; &lt;table border=&quot;0&quot; cellpadding=&quot;8&quot;&gt; &lt;tr&gt;&lt;td align=&quot;center&quot;&gt; &lt;font size=&quot;+2&quot;&gt;&lt;b&gt;Summary&lt;/b&gt;&lt;/font&gt;&lt;br /&gt;Joel E. Tohline (May, 2017) &lt;/td&gt;&lt;/tr&gt; &lt;tr&gt;&lt;td align=&quot;center&quot;&gt; &lt;table border=&quot;0&quot; width=&quot;80%&quot; align=&quot;center&quot; cellpadding=&quot;0&quot;&gt; &lt;tr&gt;&lt;td align=&quot;left&quot;&gt; &lt;font color=&quot;darkgreen&quot;&gt;&quot;… a static c...&quot;</p> <a href="https://selfgravitatingfluids.education/JETohline/index.php?title=Appendix/Ramblings/NonlinarOscillation&amp;diff=805">Show changes</a>

Joel2