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User:Tohline/SSC/Stability/InstabilityOnsetOverview - Revision history
2024-03-28T18:40:01Z
Revision history for this page on the wiki
MediaWiki 1.36.2
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Stability/InstabilityOnsetOverview&diff=17379&oldid=prev
Tohline: /* Supporting Scratch Work */
2019-01-27T20:18:42Z
<p><span dir="auto"><span class="autocomment">Supporting Scratch Work</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:18, 27 January 2019</td>
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<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></table></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></table></div></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>[<font color="red">'''Note added on 1/4/2019:'''</font> The numerator of the first term on the RHS of this last expression used to be "n-3". This <del style="font-weight: bold; text-decoration: none;">type-setting </del>error has been fixed.]</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>[<font color="red">'''Note added on 1/4/2019:'''</font> The numerator of the first term on the RHS of this last expression used to be "n-3". This <ins style="font-weight: bold; text-decoration: none;">manuscript </ins>error has been fixed.]</div></td></tr>
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</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Stability/InstabilityOnsetOverview&diff=17378&oldid=prev
Tohline: /* Supporting Scratch Work */
2019-01-27T20:18:13Z
<p><span dir="auto"><span class="autocomment">Supporting Scratch Work</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:18, 27 January 2019</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l931">Line 931:</td>
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<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></table></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></table></div></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>[<font color="red">'''Note added on 1/4/2019:'''</font> The numerator of the first term on the RHS of this last expression used to be "n-3". <del style="font-weight: bold; text-decoration: none;">We have </del>fixed <del style="font-weight: bold; text-decoration: none;">this error</del>.]</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>[<font color="red">'''Note added on 1/4/2019:'''</font> The numerator of the first term on the RHS of this last expression used to be "n-3". <ins style="font-weight: bold; text-decoration: none;">This type-setting error has been </ins>fixed.]</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;"><div></div></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></div></div></td></tr>
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</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Stability/InstabilityOnsetOverview&diff=17275&oldid=prev
Tohline: /* Supporting Scratch Work */
2019-01-04T23:36:06Z
<p><span dir="auto"><span class="autocomment">Supporting Scratch Work</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:36, 4 January 2019</td>
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<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> <td align="left"></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> <td align="left"></div></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><math>~ \biggl(\frac{n<del style="font-weight: bold; text-decoration: none;">-3</del>}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr]_{\xi_1} \, .</math></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><math>~ \biggl(\frac{<ins style="font-weight: bold; text-decoration: none;">3-</ins>n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr]_{\xi_1} \, .</math></div></td></tr>
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<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></table></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></table></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;">[<font color="red">'''Note added on 1/4/2019:'''</font> The numerator of the first term on the RHS of this last expression used to be "n-3". We have fixed this error.]</ins></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;"><div></div></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></div></div></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><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"></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>We have already demonstrated that this is the boundary condition used by [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Schwarzschild_.281941.29|Schwarzschild (1941)]] in his examination of radial oscillations in n = 3 polytropes. Presumably this also matches up with the boundary condition employed by [[User:Tohline/SSC/Stability/Polytropes#Boundary_Conditions|HRW66]]. Notice that when n = 0, this final expression reduces to <math>~d\ln x/d\ln\xi = -3</math>, which is the surface boundary condition that we have been using to determine the eigenvectors of all pressure-truncated configurations.</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>We have already demonstrated that this is the boundary condition used by [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Schwarzschild_.281941.29|Schwarzschild (1941)]] in his examination of radial oscillations in n = 3 polytropes. Presumably this also matches up with the boundary condition employed by [[User:Tohline/SSC/Stability/Polytropes#Boundary_Conditions|HRW66]]. Notice that when n = 0, this final expression reduces to <math>~d\ln x/d\ln\xi = -3</math>, which is the surface boundary condition that we have been using to determine the eigenvectors of all pressure-truncated configurations.</div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Stability/InstabilityOnsetOverview&diff=13981&oldid=prev
Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */
2017-05-27T22:52:44Z
<p><span dir="auto"><span class="autocomment">Marginally Unstable Pressure-Truncated Gas Clouds</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:52, 27 May 2017</td>
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<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 colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;">{{LSU_HBook_header}}</ins></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>Additional details may be found [[User:Tohline/SSC/Stability/Isothermal#Radial_Oscillations_of_Pressure-Truncated_Isothermal_Spheres|here]].</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>Additional details may be found [[User:Tohline/SSC/Stability/Isothermal#Radial_Oscillations_of_Pressure-Truncated_Isothermal_Spheres|here]].</div></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><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"></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><del style="font-weight: bold; text-decoration: none;">{{LSU_HBook_header}}</del></div></td><td colspan="2"></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>==Isothermal==</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>==Isothermal==</div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Stability/InstabilityOnsetOverview&diff=13976&oldid=prev
Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */
2017-05-26T21:40:27Z
<p><span dir="auto"><span class="autocomment">Marginally Unstable Pressure-Truncated Gas Clouds</span></span></p>
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<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><tr><td align="center"></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><tr><td align="center"></div></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><font size="+2"><b>Abstract</b></font></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><font size="+2"><b>Abstract</b></font><ins style="font-weight: bold; text-decoration: none;"><p></p>Joel E. Tohline (April, 2017)</ins></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;"><div></td></tr></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></td></tr></div></td></tr>
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</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Stability/InstabilityOnsetOverview&diff=13794&oldid=prev
Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */
2017-04-28T04:15:47Z
<p><span dir="auto"><span class="autocomment">Marginally Unstable Pressure-Truncated Gas Clouds</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<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 04:15, 28 April 2017</td>
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<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>In a pair of papers that, to date, have not been widely cited, Yabushita ([http://adsabs.harvard.edu/abs/1974MNRAS.167...95Y 1974], [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y 1975]) proved that this is also precisely the configuration along the sequence of pressure-truncated isothermal spheres that marks the onset of dynamical instability. More specifically, drawing on traditional techniques for identifying the properties of radial pulsation modes in self-gravitating, isothermal fluids, Yabushita showed that an eigenvector prescribed analytically by the following eigenfrequency-eigenfunction pair<del style="font-weight: bold; text-decoration: none;">: </del></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>In a pair of papers that, to date, have not been widely cited, Yabushita ([http://adsabs.harvard.edu/abs/1974MNRAS.167...95Y 1974], [http://adsabs.harvard.edu/abs/1975MNRAS.172..441Y 1975]) proved that this is also precisely the configuration along the sequence of pressure-truncated isothermal spheres that marks the onset of dynamical instability. More specifically, drawing on traditional techniques for identifying the properties of radial pulsation modes in self-gravitating, isothermal fluids, Yabushita showed that an eigenvector prescribed analytically by the following eigenfrequency-eigenfunction pair<ins style="font-weight: bold; text-decoration: none;">, </ins></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;"><div><div align="center"></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><div align="center"></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;"><div><table border="0" cellpadding="5" align="center"></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><table border="0" cellpadding="5" align="center"></div></td></tr>
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<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>provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.</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>provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.</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:MassVsRadiusCombined02.png|left|200px|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the limiting mass (<del style="font-weight: bold; text-decoration: none;">or</del>, <del style="font-weight: bold; text-decoration: none;">as </del>the <del style="font-weight: bold; text-decoration: none;">case may be, </del>limiting pressure) occurs precisely where,</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:MassVsRadiusCombined02.png|left|200px|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the limiting mass (<ins style="font-weight: bold; text-decoration: none;">correspondingly</ins>, the limiting pressure <ins style="font-weight: bold; text-decoration: none;">in a P-V diagram</ins>) occurs precisely where,</div></td></tr>
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<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><table border="0" cellpadding="5" align="center"></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><table border="0" cellpadding="5" align="center"></div></td></tr>
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<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></div></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></div></div></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>We have discovered that, along each <del style="font-weight: bold; text-decoration: none;">polytropic </del>sequence, the transition from stable to dynamically unstable configurations occurs precisely at the location of this limiting mass configuration. <del style="font-weight: bold; text-decoration: none;">More specifically</del>, we have <del style="font-weight: bold; text-decoration: none;">shown </del>that, in each such transition model, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair: </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>We have discovered that, along each sequence, the transition from stable to dynamically unstable configurations occurs precisely at the location of this limiting mass configuration<ins style="font-weight: bold; text-decoration: none;">, if the adiabatic index governing the behavior of radial oscillations is tied to the polytropic index via the relation, <math>~\gamma = (n+1)/n</math></ins>. <ins style="font-weight: bold; text-decoration: none;">Specifically</ins>, we have <ins style="font-weight: bold; text-decoration: none;">found </ins>that, in each such transition model, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair: </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;"><div><div align="center"></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><div align="center"></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;"><div><table border="0" cellpadding="5" align="center"></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><table border="0" cellpadding="5" align="center"></div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Stability/InstabilityOnsetOverview&diff=13793&oldid=prev
Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */
2017-04-27T23:16:20Z
<p><span dir="auto"><span class="autocomment">Marginally Unstable Pressure-Truncated Gas Clouds</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<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 23:16, 27 April 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l36">Line 36:</td>
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<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>provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.</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>provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.</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:MassVsRadiusCombined02.png|left|<del style="font-weight: bold; text-decoration: none;">250px</del>|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the <del style="font-weight: bold; text-decoration: none;">"maximum </del>mass<del style="font-weight: bold; text-decoration: none;">" turning point </del>occurs precisely where,</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:MassVsRadiusCombined02.png|left|<ins style="font-weight: bold; text-decoration: none;">200px</ins>|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the <ins style="font-weight: bold; text-decoration: none;">limiting </ins>mass <ins style="font-weight: bold; text-decoration: none;">(or, as the case may be, limiting pressure) </ins>occurs precisely where,</div></td></tr>
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<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></div></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></div></div></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>We have discovered that, along each polytropic sequence, the transition from stable to dynamically unstable configurations occurs precisely at the location of this <del style="font-weight: bold; text-decoration: none;">"maximum </del>mass<del style="font-weight: bold; text-decoration: none;">" </del>configuration. <del style="font-weight: bold; text-decoration: none;">Specifically</del>, we <del style="font-weight: bold; text-decoration: none;">show </del>that, in each such transition model, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair: </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>We have discovered that, along each polytropic sequence, the transition from stable to dynamically unstable configurations occurs precisely at the location of this <ins style="font-weight: bold; text-decoration: none;">limiting </ins>mass configuration. <ins style="font-weight: bold; text-decoration: none;">More specifically</ins>, we <ins style="font-weight: bold; text-decoration: none;">have shown </ins>that, in each such transition model, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair: </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;"><div><div align="center"></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><div align="center"></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;"><div><table border="0" cellpadding="5" align="center"></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><table border="0" cellpadding="5" align="center"></div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Stability/InstabilityOnsetOverview&diff=13792&oldid=prev
Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */
2017-04-27T23:09:05Z
<p><span dir="auto"><span class="autocomment">Marginally Unstable Pressure-Truncated Gas Clouds</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<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 23:09, 27 April 2017</td>
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<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>provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.</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>provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.</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:MassVsRadiusCombined02.png|left|<del style="font-weight: bold; text-decoration: none;">200px</del>|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the "maximum mass" turning point occurs precisely where,</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:MassVsRadiusCombined02.png|left|<ins style="font-weight: bold; text-decoration: none;">250px</ins>|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the "maximum mass" turning point occurs precisely where,</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;"><div><div align="center"></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><div align="center"></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;"><div><table border="0" cellpadding="5" align="center"></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><table border="0" cellpadding="5" align="center"></div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Stability/InstabilityOnsetOverview&diff=13791&oldid=prev
Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */
2017-04-27T23:07:45Z
<p><span dir="auto"><span class="autocomment">Marginally Unstable Pressure-Truncated Gas Clouds</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:07, 27 April 2017</td>
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<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><tr><td align="center"></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><tr><td align="center"></div></td></tr>
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<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></div></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></div></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;">We have discovered that, along each polytropic sequence, the transition from stable to dynamically unstable configurations occurs precisely at the location of this "maximum mass" configuration. Specifically, we show that, in each such transition model, the eigenvector associated with the fundamental mode of radial oscillation is prescribed analytically by the following eigenfrequency-eigenfunction pair: </ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"><div align="center"></ins></div></td></tr>
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<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"><tr></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> <td align="right"></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"><math>~\sigma_c^2 = 0</math></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> </td></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> <td align="center"></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> </td></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> <td align="left"></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"><math>~x = \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, .</math></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> </td></ins></div></td></tr>
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<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></td></tr></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></td></tr></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;"><div></table></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></table></div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Stability/InstabilityOnsetOverview&diff=13790&oldid=prev
Tohline: /* Marginally Unstable Pressure-Truncated Gas Clouds */
2017-04-27T22:13:20Z
<p><span dir="auto"><span class="autocomment">Marginally Unstable Pressure-Truncated Gas Clouds</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-marker" />
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<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 22:13, 27 April 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l35">Line 35:</td>
<td colspan="2" class="diff-lineno">Line 35:</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>provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.</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>provides an exact definition of the fundamental mode of radial oscillation in the configuration that is associated with the Bonnor-Ebert limiting mass. Note that this expression for the eigenfunction has been given in terms of the dimensionless radial displacement, <math>~x \equiv \delta r/r_0</math>.</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:MassVsRadiusCombined02.png|left|200px|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. </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:MassVsRadiusCombined02.png|left|200px|Equilibrium sequences of Pressure-Truncated Polytropes]]Building upon Yabushita's investigation, we have analyzed the properties of radial oscillation modes in pressure-truncated ''polytropic'' configurations having indexes, <math>~3 \le n < \infty</math>. As is illustrated here, on the left, for any polytropic index within this stated range, at least one extremum in the mass can be identified along the corresponding mass-radius equilibrium sequence. <ins style="font-weight: bold; text-decoration: none;">Separately, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] have shown that, along each such sequence, the configuration associated with the "maximum mass" turning point occurs precisely where,</ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"><div align="center"></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"><table border="0" cellpadding="5" align="center"></ins></div></td></tr>
<tr><td colspan="2"></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> </div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"><tr></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> <td align="right"></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"><math>~</ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;">\biggl[ \frac{1}{\theta^{n+1}} \biggl(\frac{d\theta}{d\xi}\biggr)^2\biggr]_{\tilde\xi}</ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"></math></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> </td></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> <td align="center"></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"><math>~=</math></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> </td></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> <td align="left"></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"><math>~\frac{2}{n-3} \, .</math></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"> </td></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"></tr></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"></table></ins></div></td></tr>
<tr><td colspan="2"></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><ins style="font-weight: bold; text-decoration: none;"></div></ins></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;"><div></td></tr></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></td></tr></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;"><div></table></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></table></div></td></tr>
</table>
Tohline