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https://www.vistrails.org//index.php?title=User:Tohline/Math/EQ_RadialPulsation02&diff=13452&oldid=prev
Tohline at 20:13, 8 March 2017
2017-03-08T20:13:22Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:13, 8 March 2017</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> <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> <td align="center" colspan="2"></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>where:&nbsp; &nbsp; <math>~Q(\xi) \equiv <del style="font-weight: bold; text-decoration: none;"> \biggl[ </del>- \frac{d\ln\theta}{d\ln\xi} <del style="font-weight: bold; text-decoration: none;">\biggr] </del>\, ,</math> &nbsp;&nbsp; <math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c} \, ,</math> &nbsp; &nbsp; and, &nbsp; &nbsp; <math>~\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</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>where:&nbsp; &nbsp; <math>~Q(\xi) \equiv - \frac{d\ln\theta}{d\ln\xi} \, ,</math> &nbsp;&nbsp; <math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c} \, ,</math> &nbsp; &nbsp; and, &nbsp; &nbsp; <math>~\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math></div></td></tr>
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Tohline
https://www.vistrails.org//index.php?title=User:Tohline/Math/EQ_RadialPulsation02&diff=13451&oldid=prev
Tohline at 20:11, 8 March 2017
2017-03-08T20:11:21Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:11, 8 March 2017</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><td align="center"></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><td align="center"></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>~0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 <del style="font-weight: bold; text-decoration: none;">+ </del>(n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>~0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 <ins style="font-weight: bold; text-decoration: none;">- </ins>(n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} </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>+ (n+1) \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_g } \biggr) \frac{\xi^2}{\theta} </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>+ (n+1) \biggl[ \biggl( \frac{\sigma_c^2}{6\gamma_g } \biggr) \frac{\xi^2}{\theta} </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>\alpha Q\biggr] \frac{x}{\xi^2} </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><ins style="font-weight: bold; text-decoration: none;">- </ins>\alpha Q\biggr] \frac{x}{\xi^2} </math></div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></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>
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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"></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"></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>where:&nbsp; &nbsp; <math>~Q(\xi) \equiv \frac{d\ln\theta}{d\ln\xi}</math><del style="font-weight: bold; text-decoration: none;">, </del>&nbsp;&nbsp; <math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c}</math> &nbsp; &nbsp; and, &nbsp; &nbsp; <math>~\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</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>where:&nbsp; &nbsp; <math>~Q(\xi) \equiv <ins style="font-weight: bold; text-decoration: none;">\biggl[ - </ins>\frac{d\ln\theta}{d\ln\xi} <ins style="font-weight: bold; text-decoration: none;">\biggr] \, ,</ins></math> &nbsp;&nbsp; <math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c} <ins style="font-weight: bold; text-decoration: none;">\, ,</ins></math> &nbsp; &nbsp; and, &nbsp; &nbsp; <math>~\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math></div></td></tr>
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Tohline
https://www.vistrails.org//index.php?title=User:Tohline/Math/EQ_RadialPulsation02&diff=13449&oldid=prev
Tohline at 20:06, 8 March 2017
2017-03-08T20:06:25Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:06, 8 March 2017</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><td align="center"></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><td align="center"></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>~0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 + (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} </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><math>~0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 + (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} </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>+ (n+1) \biggl[\frac{\sigma_c^2}{6\gamma_g } \<del style="font-weight: bold; text-decoration: none;">biggl( </del>\frac{\xi^2}{\theta}<del style="font-weight: bold; text-decoration: none;">\biggr) </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>+ (n+1) \biggl[ <ins style="font-weight: bold; text-decoration: none;">\biggl( </ins>\frac{\sigma_c^2}{6\gamma_g } \<ins style="font-weight: bold; text-decoration: none;">biggr) </ins>\frac{\xi^2}{\theta} </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>+ \alpha Q\biggr] \frac{x}{\xi^2} </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>+ \alpha Q\biggr] \frac{x}{\xi^2} </math></div></td></tr>
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Tohline
https://www.vistrails.org//index.php?title=User:Tohline/Math/EQ_RadialPulsation02&diff=13444&oldid=prev
Tohline: Created page with '<table border=0 cellpadding=2> <tr> <td align="right"> [[Image:LSU_Key.png|25px|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#Stability:__Radia…'
2017-03-08T16:26:48Z
<p>Created page with '<table border=0 cellpadding=2> <tr> <td align="right"> [[Image:LSU_Key.png|25px|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#Stability:__Radia…'</p>
<p><b>New page</b></p><div><table border=0 cellpadding=2><br />
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[[Image:LSU_Key.png|25px|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#Stability:__Radial_Pulsation]]<br />
</td> <br />
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<math>~0 = \frac{d^2x}{d\xi^2} + \biggl[ 4 + (n+1) Q \biggr] \frac{1}{\xi} \cdot \frac{dx}{d\xi} <br />
+ (n+1) \biggl[\frac{\sigma_c^2}{6\gamma_g } \biggl( \frac{\xi^2}{\theta}\biggr) <br />
+ \alpha Q\biggr] \frac{x}{\xi^2} </math><br />
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where:&nbsp; &nbsp; <math>~Q(\xi) \equiv \frac{d\ln\theta}{d\ln\xi}</math>, &nbsp;&nbsp; <math>~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi G\rho_c}</math> &nbsp; &nbsp; and, &nbsp; &nbsp; <math>~\alpha \equiv \biggl(3 - \frac{4}{\gamma_\mathrm{g}}\biggr)</math><br />
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Tohline