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User:Tohline/SSC/Virial/PolytropesEmbeddedOutline - Revision history
2024-03-28T13:27:57Z
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Tohline at 23:17, 10 July 2016
2016-07-10T23:17:14Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:17, 10 July 2016</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>[[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29|In an accompanying chapter]], we reproduce the discussion associated with our [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Second_Effort|"Second Effort", as referenced above]], but revise key sections to incorporate corrected expressions for the structural form factors.</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>[[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29|In an accompanying chapter]], we reproduce the discussion associated with our [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Second_Effort|"Second Effort", as referenced above]], but revise key sections to incorporate corrected expressions for the structural form factors.</div></td></tr>
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Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Virial/PolytropesEmbeddedOutline&diff=10607&oldid=prev
Tohline: Insert LSU_HBook_header
2015-09-14T18:56:04Z
<p>Insert LSU_HBook_header</p>
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Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Virial/PolytropesEmbeddedOutline&diff=10601&oldid=prev
Tohline: /* ASIDE: Isothermal Configurations */
2015-09-13T22:38:22Z
<p><span dir="auto"><span class="autocomment">ASIDE: Isothermal Configurations</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:38, 13 September 2015</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="center" colspan="2" rowspan="1" bgcolor="#D0FFFF"></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="center" colspan="2" rowspan="1" bgcolor="#D0FFFF"></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>[[File:<del style="font-weight: bold; text-decoration: none;">EnergyRadiusViewBottom5</del>.png|500px|Whitworth's (1981) Isothermal Free-Energy Surface]]</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:<ins style="font-weight: bold; text-decoration: none;">EnergyRadiusViewBottom6</ins>.png|500px|Whitworth's (1981) Isothermal Free-Energy Surface]]</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><!-- [[Image:AAAwaiting01.png|440px|Whitworth's (1981) Isothermal Free-Energy Surface]] --></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><!-- [[Image:AAAwaiting01.png|440px|Whitworth's (1981) Isothermal Free-Energy Surface]] --></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>Graphical depictions of the free-energy surface, <math>~\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressure-truncated, uniform-density isothermal configurations &#8212; see equation (10) of Whitworth (1981) or our restatement of this equation, [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#WhitworthFreeEnergyExpression|above]]. </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>Graphical depictions of the free-energy surface, <math>~\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressure-truncated, uniform-density isothermal configurations &#8212; see equation (10) of Whitworth (1981) or our restatement of this equation, [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#WhitworthFreeEnergyExpression|above]]. </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>*''Lower-right quadrant'': <sup>&dagger;</sup>A reproduction of the black &amp; white Figure 2 from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]; as displayed here, the aspect ratio of the figure has been modified slightly in order to facilitate comparison with the images displayed in other quadrants of this <del style="font-weight: bold; text-decoration: none;">image</del>. Continuous lines display the radial dependence of the free energy for a variety of different values of the external pressure, as labelled. Quoting directly from Whitworth's figure caption, "Filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state" at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</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>*''Lower-right quadrant'': <sup>&dagger;</sup>A reproduction of the black &amp; white Figure 2 from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]; as displayed here, the aspect ratio of the figure has been modified slightly in order to facilitate comparison with the images displayed in other quadrants of this <ins style="font-weight: bold; text-decoration: none;">figure</ins>. Continuous lines display the radial dependence of the free energy for a variety of different values of the external pressure, as labelled. Quoting directly from Whitworth's figure caption, "Filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state" at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</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;"><div>*''Upper-right quadrant'': The undulating free-energy surface is drawn in three dimensions and viewed from a vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy. The three coordinate axes are labeled and colored as follows: Radius (red), External Pressure (green), and Free Energy (blue). The properties of fifteen distinct equilibrium states are identified by the sequence of small colored spherical dots: Blue dots mark stable equilibria; white dots mark unstable equilibria; and the lone red dot identifies the critical neutral equilibrium state at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</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>*''Upper-right quadrant'': The undulating free-energy surface is drawn in three dimensions and viewed from a vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy. The three coordinate axes are labeled and colored as follows: Radius (red), External Pressure (green), and Free Energy (blue). The properties of fifteen distinct equilibrium states are identified by the sequence of small colored spherical dots: Blue dots mark stable equilibria; white dots mark unstable equilibria; and the lone red dot identifies the critical neutral equilibrium state at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</math>.</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>*''Upper-left quadrant'': The two-dimensional <del style="font-weight: bold; text-decoration: none;">"</del>projected<del style="font-weight: bold; text-decoration: none;">" </del>image that results from viewing the free-energy surface "from above," along a line of sight that is parallel to the free-energy <math>~(Z)</math> axis and looking directly down onto the radius-pressure <math>~(X-Y)</math> plane. From this vantage point, the sixteen small colored dots cleanly trace out the <math>~P_e(R_\mathrm{eq})</math> equilibrium sequence that is defined by the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#ScalarVirialTheorem|algebraic expression of the scalar virial theorem]]. </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>*''Upper-left quadrant'': The two-dimensional projected image that results from viewing the free-energy surface "from above," along a line of sight that is parallel to the free-energy <math>~(Z)</math> axis and looking directly down onto the radius-pressure <math>~(X-Y)</math> plane. From this vantage point, the sixteen small colored dots cleanly trace out the <math>~P_e(R_\mathrm{eq})</math> equilibrium sequence that is defined by the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#ScalarVirialTheorem|algebraic expression of the scalar virial theorem]]. </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>*''Lower-left quadrant'': The two-dimensional <del style="font-weight: bold; text-decoration: none;">"</del>projected<del style="font-weight: bold; text-decoration: none;">" </del>image that results from viewing the free-energy surface "from underneath," along a light of sight that is parallel to the external-pressure <math>~(Y)</math> axis and looking directly up at the radius-free-energy <math>~(X-Z)</math> plane. This image can be directly compared with Whitworth's Figure 2. Seven of the nine equilibrium configurations that are marked in Whitworth's diagram also appear among the fifteen equilibria that are identified (as small colored dots) in our projected image. For example, the red dot in our image corresponds to the marginally stable configuration that Whitworth marks with a cross; and the white dot that is <del style="font-weight: bold; text-decoration: none;">peaking </del>from behind the "Radius" axis &#8212; that is, the unstable equilibrium configuration that has a free-energy value of zero &#8212; corresponds to the open circle in Whitworth's plot that is labeled as <math>~P_e = 0.632</math> (see also the coordinate values given in our accompanying table).</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>*''Lower-left quadrant'': The two-dimensional projected image that results from viewing the free-energy surface "from underneath," along a light of sight that is parallel to the external-pressure <math>~(Y)</math> axis and looking directly up at the radius-free-energy <math>~(X-Z)</math> plane. This image can be directly compared with Whitworth's Figure 2. Seven of the nine equilibrium configurations that are marked in Whitworth's diagram also appear among the fifteen equilibria that are identified (as small colored dots) in our projected image. For example, the red dot in our image corresponds to the marginally stable configuration that Whitworth marks with a cross; and the white dot that is <ins style="font-weight: bold; text-decoration: none;">peeking </ins>from behind the "Radius" axis &#8212; that is, the unstable equilibrium configuration that has a free-energy value of zero &#8212; corresponds to the open circle in Whitworth's plot that is labeled as <math>~P_e = 0.632</math> (see also the coordinate values given in our accompanying table).</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 Identifying Properties of Virial Equilibria'' (see immediately below): Coordinates <math>~(X, Y, Z)</math> = <math>(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}, \mathcal{G})</math> and, hence, also the physical properties are provided for each of the sixteen equilibria that are marked by small colored spherical dots in our attending color plots of the free-energy surface. [Actually, two of the three attending color plots display only fifteen dots because the position of the (unstable) equilibrium configuration of highest energy falls outside the boundaries of the plot.] Seven of the nine equilibrium configurations that are identified in Whitworth's Figure 2 are among the sixteen equilibria that are identified here, including the critical neutral equilibrium state, which is highlighted in red. For completeness, the value of the corresponding normalization energy, <math>~\mathcal{G}_0(P_e)</math>, is also tabulated.</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 Identifying Properties of Virial Equilibria'' (see immediately below): Coordinates <math>~(X, Y, Z)</math> = <math>(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}, \mathcal{G})</math> and, hence, also the physical properties are provided for each of the sixteen equilibria that are marked by small colored spherical dots in our attending color plots of the free-energy surface. [Actually, two of the three attending color plots display only fifteen dots because the position of the (unstable) equilibrium configuration of highest energy falls outside the boundaries of the plot.] Seven of the nine equilibrium configurations that are identified in Whitworth's Figure 2 are among the sixteen equilibria that are identified here, including the critical neutral equilibrium state, which is highlighted in red. For completeness, the value of the corresponding normalization energy, <math>~\mathcal{G}_0(P_e)</math>, is also tabulated.</div></td></tr>
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</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Virial/PolytropesEmbeddedOutline&diff=10599&oldid=prev
Tohline: /* ASIDE: Isothermal Configurations */ Insert much cleaner animated gif
2015-09-13T22:19:58Z
<p><span dir="auto"><span class="autocomment">ASIDE: Isothermal Configurations: </span> Insert much cleaner animated gif</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:19, 13 September 2015</td>
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<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><table border="1" cellpadding="5" align="center" width="<del style="font-weight: bold; text-decoration: none;">600px</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><table border="1" cellpadding="5" align="center" width="<ins style="font-weight: bold; text-decoration: none;">500px</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><tr><td align="center" colspan="2"></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" colspan="2"></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>The black &amp; white line-drawing in lower-right quadrant of this composite image is &hellip;</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>The black &amp; white line-drawing in lower-right quadrant of this composite image is &hellip;</div></td></tr>
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<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><td align="center" colspan="1" rowspan="1"></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><td align="center" colspan="1" rowspan="1<ins style="font-weight: bold; text-decoration: none;">" bgcolor="#D0FFFF</ins>"></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>[[File:<del style="font-weight: bold; text-decoration: none;">EnergyRadiusViewCombined5</del>.png|<del style="font-weight: bold; text-decoration: none;">550px</del>|Whitworth's (1981) Isothermal Free-Energy Surface]]</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:<ins style="font-weight: bold; text-decoration: none;">EnergyRadiusViewTop5</ins>.png|<ins style="font-weight: bold; text-decoration: none;">250px</ins>|Whitworth's (1981) Isothermal Free-Energy Surface]]</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><!-- [[Image:AAAwaiting01.png|440px|Whitworth's (1981) Isothermal Free-Energy Surface]] --></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><!-- [[Image:AAAwaiting01.png|440px|Whitworth's (1981) Isothermal Free-Energy Surface]] --></div></td></tr>
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<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:<del style="font-weight: bold; text-decoration: none;">FEmovie01</del>.gif|<del style="font-weight: bold; text-decoration: none;">200px</del>|Animated Isothermal Free-Energy Surface]]</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:<ins style="font-weight: bold; text-decoration: none;">FEmovie02</ins>.gif|<ins style="font-weight: bold; text-decoration: none;">250px</ins>|Animated Isothermal Free-Energy Surface]]</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;">[[File:EnergyRadiusViewBottom5.png|500px|Whitworth's (1981) Isothermal Free-Energy Surface]]</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;"><!-- [[Image:AAAwaiting01.png|440px|Whitworth's (1981) Isothermal Free-Energy Surface]] --></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></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></tr></div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Virial/PolytropesEmbeddedOutline&diff=10595&oldid=prev
Tohline: Insert animated gif of Whitworth's free-energy surface
2015-09-12T23:56:30Z
<p>Insert animated gif of Whitworth's free-energy surface</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 23:56, 12 September 2015</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><div align="center" id="3DIsothermalSurface"></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" id="3DIsothermalSurface"></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="1" cellpadding="5" align="center" width="600px"></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="1" cellpadding="5" align="center" width="600px"></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><tr><td align="center"></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><tr><td align="center<ins style="font-weight: bold; text-decoration: none;">" colspan="2</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>The black &amp; white line-drawing in lower-right quadrant of this composite image is &hellip;</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>The black &amp; white line-drawing in lower-right quadrant of this composite image is &hellip;</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>Figure 2 extracted<sup>&dagger;</sup> from p. 973 of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]<p></p></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>Figure 2 extracted<sup>&dagger;</sup> from p. 973 of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]<p></p></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><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><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><td align="center" colspan="1" rowspan="1"></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="center" colspan="1" rowspan="1"></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>[[File:EnergyRadiusViewCombined5.png|550px|Whitworth's (1981) Isothermal Free-Energy Surface]] --<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>[[File:EnergyRadiusViewCombined5.png|550px|Whitworth's (1981) Isothermal Free-Energy Surface]]</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>[[Image:AAAwaiting01.png|440px|Whitworth's (1981) Isothermal Free-Energy Surface]]</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><ins style="font-weight: bold; text-decoration: none;"><!</ins>-- [[Image:AAAwaiting01.png|440px|Whitworth's (1981) <ins style="font-weight: bold; text-decoration: none;">Isothermal Free-Energy Surface]] --></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" colspan="1" rowspan="1"></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;">[[File:FEmovie01.gif|200px|Animated </ins>Isothermal Free-Energy Surface]]</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></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></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></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></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><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><tr></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> <td align="left" colspan="<del style="font-weight: bold; text-decoration: none;">1</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> <td align="left" colspan="<ins style="font-weight: bold; text-decoration: none;">2</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>Graphical depictions of the free-energy surface, <math>~\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressure-truncated, uniform-density isothermal configurations &#8212; see equation (10) of Whitworth (1981) or our restatement of this equation, [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#WhitworthFreeEnergyExpression|above]]. </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>Graphical depictions of the free-energy surface, <math>~\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressure-truncated, uniform-density isothermal configurations &#8212; see equation (10) of Whitworth (1981) or our restatement of this equation, [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#WhitworthFreeEnergyExpression|above]]. </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>*''Lower-right quadrant'': <sup>&dagger;</sup>A reproduction of the black &amp; white Figure 2 from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]; as displayed here, the aspect ratio of the figure has been modified slightly in order to facilitate comparison with the images displayed in other quadrants of this image. Continuous lines display the radial dependence of the free energy for a variety of different values of the external pressure, as labelled. Quoting directly from Whitworth's figure caption, "Filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state" at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</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>*''Lower-right quadrant'': <sup>&dagger;</sup>A reproduction of the black &amp; white Figure 2 from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]; as displayed here, the aspect ratio of the figure has been modified slightly in order to facilitate comparison with the images displayed in other quadrants of this image. Continuous lines display the radial dependence of the free energy for a variety of different values of the external pressure, as labelled. Quoting directly from Whitworth's figure caption, "Filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state" at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</math>. </div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Virial/PolytropesEmbeddedOutline&diff=10340&oldid=prev
Tohline: /* ASIDE: Isothermal Configurations */ More completely explain origin of Figure for copyright purposes
2015-08-07T18:46:36Z
<p><span dir="auto"><span class="autocomment">ASIDE: Isothermal Configurations: </span> More completely explain origin of Figure for copyright purposes</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 18:46, 7 August 2015</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><div align="center" id="3DIsothermalSurface"></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" id="3DIsothermalSurface"></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="1" cellpadding="5" align="center" width="600px"></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="1" cellpadding="5" align="center" width="600px"></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><tr></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><tr><<ins style="font-weight: bold; text-decoration: none;">td </ins>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><del style="font-weight: bold; text-decoration: none;"> </del><<del style="font-weight: bold; text-decoration: none;">th </del>align="center"></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><ins style="font-weight: bold; text-decoration: none;">The black &amp; white line-drawing in lower</ins>-<ins style="font-weight: bold; text-decoration: none;">right quadrant of this composite image is &hellip;</ins></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;">Virial Pressure</del>-<del style="font-weight: bold; text-decoration: none;">Radius Relation </del>for <del style="font-weight: bold; text-decoration: none;">Uniform</del>-<del style="font-weight: bold; text-decoration: none;">Density</del>, <del style="font-weight: bold; text-decoration: none;">Pressure</del>-<del style="font-weight: bold; text-decoration: none;">Truncated Isothermal Configurations</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><ins style="font-weight: bold; text-decoration: none;">Figure 2 extracted<sup>&dagger;</sup> from p. 973 of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]<p></p></ins></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></<del style="font-weight: bold; text-decoration: none;">th</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><ins style="font-weight: bold; text-decoration: none;">"''Global Gravitational Stability </ins>for <ins style="font-weight: bold; text-decoration: none;">One</ins>-<ins style="font-weight: bold; text-decoration: none;">dimensional Polytropes''"<p></p></ins></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></tr></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><ins style="font-weight: bold; text-decoration: none;">Monthly Notices of the Royal Astronomical Society</ins>, <ins style="font-weight: bold; text-decoration: none;">vol. 195, pp. 967</ins>-<ins style="font-weight: bold; text-decoration: none;">977 &copy; Royal Astronomical Society</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>></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><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><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><td align="center" colspan="1" rowspan="1"></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="center" colspan="1" rowspan="1"></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><!-- [[File:EnergyRadiusViewCombined5.png|550px|Whitworth's (1981) Isothermal Free-Energy Surface]] --></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><!-- [[File:EnergyRadiusViewCombined5.png|550px|Whitworth's (1981) Isothermal Free-Energy Surface]] --></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>[[Image:AAAwaiting01.png|<del style="font-weight: bold; text-decoration: none;">550px</del>|Whitworth's (1981) Isothermal Free-Energy Surface]]</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>[[Image:AAAwaiting01.png|<ins style="font-weight: bold; text-decoration: none;">440px</ins>|Whitworth's (1981) Isothermal Free-Energy Surface]]</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></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></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></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></tr></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l683">Line 683:</td>
<td colspan="2" class="diff-lineno">Line 684:</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 align="left" colspan="1"></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" colspan="1"></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>Graphical depictions of the free-energy surface, <math>~\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressure-truncated, uniform-density isothermal configurations &#8212; see equation (10) of Whitworth (1981) or our restatement of this equation, [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#WhitworthFreeEnergyExpression|above]]. </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>Graphical depictions of the free-energy surface, <math>~\mathcal{G}(R, P_e) = 2\mathcal{U}/3\mathcal{U}_\mathrm{rf}</math>, associated with pressure-truncated, uniform-density isothermal configurations &#8212; see equation (10) of Whitworth (1981) or our restatement of this equation, [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#WhitworthFreeEnergyExpression|above]]. </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>*''Lower-right quadrant'': A reproduction of the black &amp; white Figure 2 from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]. Continuous lines display the radial dependence of the free energy for a variety of different values of the external pressure, as labelled. Quoting directly from Whitworth's figure caption, "Filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state" at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</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>*''Lower-right quadrant'': <ins style="font-weight: bold; text-decoration: none;"><sup>&dagger;</sup></ins>A reproduction of the black &amp; white Figure 2 from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]<ins style="font-weight: bold; text-decoration: none;">; as displayed here, the aspect ratio of the figure has been modified slightly in order to facilitate comparison with the images displayed in other quadrants of this image</ins>. Continuous lines display the radial dependence of the free energy for a variety of different values of the external pressure, as labelled. Quoting directly from Whitworth's figure caption, "Filled circles mark stable equilibria; the open circles mark unstable equilibria; and the cross marks the critical neutral equilibrium state" at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</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;"><div>*''Upper-right quadrant'': The undulating free-energy surface is drawn in three dimensions and viewed from a vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy. The three coordinate axes are labeled and colored as follows: Radius (red), External Pressure (green), and Free Energy (blue). The properties of fifteen distinct equilibrium states are identified by the sequence of small colored spherical dots: Blue dots mark stable equilibria; white dots mark unstable equilibria; and the lone red dot identifies the critical neutral equilibrium state at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</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>*''Upper-right quadrant'': The undulating free-energy surface is drawn in three dimensions and viewed from a vantage point that illustrates its "valley of stability" and "ridge of instability;" the surface color correlates with the value of the free energy. The three coordinate axes are labeled and colored as follows: Radius (red), External Pressure (green), and Free Energy (blue). The properties of fifteen distinct equilibrium states are identified by the sequence of small colored spherical dots: Blue dots mark stable equilibria; white dots mark unstable equilibria; and the lone red dot identifies the critical neutral equilibrium state at <math>~(R_\mathrm{eq}/R_\mathrm{rf}, P_e/P_\mathrm{rf}) = (1.0, 1.0)</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;"><div>*''Upper-left quadrant'': The two-dimensional "projected" image that results from viewing the free-energy surface "from above," along a line of sight that is parallel to the free-energy <math>~(Z)</math> axis and looking directly down onto the radius-pressure <math>~(X-Y)</math> plane. From this vantage point, the sixteen small colored dots cleanly trace out the <math>~P_e(R_\mathrm{eq})</math> equilibrium sequence that is defined by the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#ScalarVirialTheorem|algebraic expression of the scalar virial theorem]]. </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>*''Upper-left quadrant'': The two-dimensional "projected" image that results from viewing the free-energy surface "from above," along a line of sight that is parallel to the free-energy <math>~(Z)</math> axis and looking directly down onto the radius-pressure <math>~(X-Y)</math> plane. From this vantage point, the sixteen small colored dots cleanly trace out the <math>~P_e(R_\mathrm{eq})</math> equilibrium sequence that is defined by the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#ScalarVirialTheorem|algebraic expression of the scalar virial theorem]]. </div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Virial/PolytropesEmbeddedOutline&diff=10339&oldid=prev
Tohline: /* Virial Equilibrium */
2015-08-07T18:26:22Z
<p><span dir="auto"><span class="autocomment">Virial Equilibrium</span></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 18:26, 7 August 2015</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l573">Line 573:</td>
<td colspan="2" class="diff-lineno">Line 573:</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>Recalling that <math>~\eta \leftrightarrow (n+1)/n</math>, it is clear that this <math>~P_e(R_\mathrm{eq})</math> relation exactly matches equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads:</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>Recalling that <math>~\eta \leftrightarrow (n+1)/n</math>, it is clear that this <math>~P_e(R_\mathrm{eq})</math> relation exactly matches equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads:</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;"></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" 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><table border="<del style="font-weight: bold; text-decoration: none;">2</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><table border="<ins style="font-weight: bold; text-decoration: none;">1</ins>" <ins style="font-weight: bold; text-decoration: none;">cellpadding="5" width="80%"></ins></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><tr><td></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><ins style="font-weight: bold; text-decoration: none;"><tr><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;">Equation and accompanying sentence extracted without modification from p. 970 of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]<p></p></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;">"''Global Gravitational Stability for One-dimensional Polytropes''"<p></p></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;">Monthly Notices of the Royal Astronomical Society, vol. 195, pp. 967-977 &copy; Royal Astronomical Society</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></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><tr></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;"> </ins><td <ins style="font-weight: bold; text-decoration: none;">align="center"</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><!-- [[File:Whitworth1981Eq5.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]] --></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><!-- [[File:Whitworth1981Eq5.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]] --></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>[[Image:AAAwaiting01.png|<del style="font-weight: bold; text-decoration: none;">500px</del>|center|Whitworth (1981<del style="font-weight: bold; text-decoration: none;">, MNRAS, 195, 967</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>[[Image:AAAwaiting01.png|<ins style="font-weight: bold; text-decoration: none;">400px</ins>|center|Whitworth (1981) <ins style="font-weight: bold; text-decoration: none;">Eq. 5</ins>]]</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></td></tr></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><ins style="font-weight: bold; text-decoration: none;"> </ins></td></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></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>
<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>====Stability====</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>====Stability====</div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Virial/PolytropesEmbeddedOutline&diff=10338&oldid=prev
Tohline: /* Coefficient Definitions */
2015-08-07T18:24:18Z
<p><span dir="auto"><span class="autocomment">Coefficient Definitions</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
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<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 18:24, 7 August 2015</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"></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>[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] <font color="red">'''Case P'''</font> analysis of pressure-truncated polytropic spheres produces the following governing free-energy function &#8212; referred to by Whitworth as the "global potential function":</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>[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] <font color="red">'''Case P'''</font> analysis of pressure-truncated polytropic spheres produces the following governing free-energy function &#8212; referred to by Whitworth as the "global potential function":</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;"></ins></div></td></tr>
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<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><table border="<del style="font-weight: bold; text-decoration: none;">2</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><table border="<ins style="font-weight: bold; text-decoration: none;">1</ins>" <ins style="font-weight: bold; text-decoration: none;">cellpadding="5" width="80%"></ins></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><tr><td></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><ins style="font-weight: bold; text-decoration: none;"><tr><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;">Equation extracted without modification from p. 973 of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]<p></p></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;">"''Global Gravitational Stability for One-dimensional Polytropes''"<p></p></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;">Monthly Notices of the Royal Astronomical Society, vol. 195, pp. 967-977 &copy; Royal Astronomical Society</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;"> </ins><td <ins style="font-weight: bold; text-decoration: none;">align="center"</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><!-- [[File:Whitworth1981Eq10.jpg|750px|center|Whitworth (1981, MNRAS, 195, 967)]] --></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><!-- [[File:Whitworth1981Eq10.jpg|750px|center|Whitworth (1981, MNRAS, 195, 967)]] --></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>[[Image:AAAwaiting01.png|<del style="font-weight: bold; text-decoration: none;">750px</del>|center|Whitworth (1981<del style="font-weight: bold; text-decoration: none;">, MNRAS, 195, 967</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>[[Image:AAAwaiting01.png|<ins style="font-weight: bold; text-decoration: none;">400px</ins>|center|Whitworth (1981) <ins style="font-weight: bold; text-decoration: none;">Eq. 5</ins>]]</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></td></tr></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><ins style="font-weight: bold; text-decoration: none;"> </ins></td></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 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;"></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>After setting <math>~\delta_{1\eta} = 0</math>, that is, by choosing to ignore isothermal systems, and after setting <math>~\eta = (n+1)/n</math>, that is, after rewriting his adiabatic exponent <math>~(\eta)</math> in terms of the corresponding polytropic index, Whitworth's free-energy expression becomes, </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>After setting <math>~\delta_{1\eta} = 0</math>, that is, by choosing to ignore isothermal systems, and after setting <math>~\eta = (n+1)/n</math>, that is, after rewriting his adiabatic exponent <math>~(\eta)</math> in terms of the corresponding polytropic index, Whitworth's free-energy expression becomes, </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" id="WhitworthFreeEnergyExpression"></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" id="WhitworthFreeEnergyExpression"></div></td></tr>
</table>
Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Virial/PolytropesEmbeddedOutline&diff=9854&oldid=prev
Tohline: /* Whitworth's (1981) Case P Analysis of Uniform-Density Configurations */ Comment out several images from Whitworth
2015-05-24T20:45:26Z
<p><span dir="auto"><span class="autocomment">Whitworth's (1981) Case P Analysis of Uniform-Density Configurations: </span> Comment out several images from Whitworth</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:45, 24 May 2015</td>
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<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:Whitworth1981Eq10.jpg|750px|center|Whitworth (1981, MNRAS, 195, 967)]]</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><ins style="font-weight: bold; text-decoration: none;"><!-- </ins>[[File:Whitworth1981Eq10.jpg<ins style="font-weight: bold; text-decoration: none;">|750px|center|Whitworth (1981, MNRAS, 195, 967)]] --></ins></div></td></tr>
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<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:Whitworth1981Eq5.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]]</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><ins style="font-weight: bold; text-decoration: none;"><!-- </ins>[[File:Whitworth1981Eq5.jpg<ins style="font-weight: bold; text-decoration: none;">|500px|center|Whitworth (1981, MNRAS, 195, 967)]] --></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 align="center" colspan="1" rowspan="1"></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="center" colspan="1" rowspan="1"></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>[[File:EnergyRadiusViewCombined5.png|550px|Whitworth's (1981) Isothermal Free-Energy Surface]]</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><ins style="font-weight: bold; text-decoration: none;"><!-- </ins>[[File:EnergyRadiusViewCombined5<ins style="font-weight: bold; text-decoration: none;">.png|550px|Whitworth's (1981) Isothermal Free-Energy Surface]] --></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;">[[Image:AAAwaiting01</ins>.png|550px|Whitworth's (1981) Isothermal Free-Energy Surface]]</div></td></tr>
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Tohline
https://www.vistrails.org//index.php?title=User:Tohline/SSC/Virial/PolytropesEmbeddedOutline&diff=9234&oldid=prev
Tohline: /* The Physics */ Insert text box to explain subscript M_limit
2015-02-23T20:44:56Z
<p><span dir="auto"><span class="autocomment">The Physics: </span> Insert text box to explain subscript M_limit</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:44, 23 February 2015</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>==The Physics==</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>==The Physics==</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>The above mathematical statements, ostensibly defining the free-energy function, the scalar virial theorem, and stability, cannot be interpreted in physical terms until the definitions of the various coefficients have been provided. In the discussion that follows, we will focus on sequences of equilibrium configurations that have a polytropic index <math>~n > 3</math> because, as has been foreshadowed in [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Overview|the above overview]], such sequences include both stable and unstable equilbria and are therefore of considerable interest in an astrophysical context. Isothermal sequences &#8212; corresponding to <math>~n = \infty</math> &#8212; are of particular astrophysical interest; however, we will devote a great deal of attention to <math>~n=5</math> configurations because their structures can be defined entirely in terms of analytic expressions. </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>The above mathematical statements, ostensibly defining the free-energy function, the scalar virial theorem, and stability, cannot be interpreted in physical terms until the definitions of the various coefficients have been provided. In the discussion that follows, we will focus on sequences of equilibrium configurations that have a polytropic index <math>~n > 3</math> because, as has been foreshadowed in [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Overview|the above overview]], such sequences include both stable and unstable equilbria and are therefore of considerable interest in an astrophysical context. Isothermal sequences &#8212; corresponding to <math>~n = \infty</math> &#8212; are of particular astrophysical interest; however, we will devote a great deal of attention to <math>~n=5</math> configurations because their structures can be defined entirely in terms of analytic expressions. </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;"><table align="center" border="1" width="65%" cellpadding="8"></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><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;">NOTE (copied from [[User:Tohline/SphericallySymmetricConfigurations/Virial#Volume_Integrals|here]]): The following considerations have led us to formally draw a distinction between <math>~M_\mathrm{limit}</math> and the "total" mass, <math>~M_\mathrm{tot}</math>, that we use for normalization. </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;"></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;"><font color="maroon"><b>Isolated Polytropes</b></font>: For [[User:Tohline/SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|isolated polytropes]], the limit of integration, <math>~R_\mathrm{limit}</math>, will be the natural edge of the configuration, where the pressure and mass-density drop to zero. In this case, <math>~M_\mathrm{limit}</math> quite naturally corresponds to the total mass of the configuration. </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;"></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;"><font color="maroon"><b>Pressure-Truncated Polytropes</b></font>: But, a [[User:Tohline/SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium|configuration embedded in an external medium]] of pressure, <math>~P_e</math>, will have a (pressure-truncated) surface whose radius, <math>~R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>~P_e</math>. In this case as well, one might choose to refer to <math>~M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>~M_\mathrm{limit}</math> from <math>~M_\mathrm{tot}</math>, continuing to rely on <math>~M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope. </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;"><font color="maroon"><b>BiPolytropes</b></font>: When discussing [[User:Tohline/SSC/BipolytropeGeneralization_Version2#Bipolytrope_Generalizatio|bipolytropes]], the limit of integration, <math>~R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope. In this case, <math>~M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.</ins></div></td></tr>
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Tohline