Difference between revisions of "User:Tohline/SSC/Synopsis StyleSheet"
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=Spherically Symmetric Configurations Synopsis= | =Spherically Symmetric Configurations Synopsis (Using Style Sheet)= | ||
{{LSU_HBook_header}} | {{LSU_HBook_header}} | ||
==Structure== | |||
===Tabular Overview=== | |||
{| class="wikitable" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12" | {| class="wikitable" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12" | ||
|+ style="height:30px;" | <font size="+1">'''Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion'''</font> — adiabatic index, <math>~\gamma</math> | |+ style="height:30px;" | <font size="+1">'''Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion'''</font> — adiabatic index, <math>~\gamma</math> | ||
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|- | |- | ||
! style="background-color:lightgreen;" colspan="2"|<font size="+1" | ! style="background-color:lightgreen;" colspan="2"|<b><font size="+1">Equilibrium Structure</font></b> | ||
|- | |- | ||
! style="text-align:center;" width="50%" |<b>Detailed Force Balance</b> | ! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">①</font></b> <b>Detailed Force Balance</b> | ||
! style="text-align:center;" |<b>Free-Energy | ! style="text-align:center; background-color:lightblue" |<b><font color="maroon" size="+1">③</font></b> <b>Free-Energy Identification of Equilibria</b> | ||
|- | |- | ||
! style="vertical-align:top; text-align:left;" |Given a barotropic equation of state, <math>~P(\rho)</math>, solve the equation of | ! style="vertical-align:top; text-align:left;" |Given a barotropic equation of state, <math>~P(\rho)</math>, solve the equation of | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~-a R^{-1} + | <math>~-a \biggl(\frac{R}{R_0}\biggr)^{-1} + b\biggl(\frac{R}{R_0}\biggr)^{3-3\gamma}+ c\biggl(\frac{R}{R_0}\biggr)^3 \, .</math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~\frac{ | <math>~R_0 ~\frac{\partial\mathfrak{G}}{\partial R}</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~a\biggl(\frac{R}{R_0}\biggr)^{-2} +(3-3\gamma)b\biggl(\frac{R}{R_0}\biggr)^{2-3\gamma} + 3c\biggl(\frac{R}{R_0}\biggr)^2</math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~\frac{ | <math>~\frac{R_0}{R}\biggl[ -W_\mathrm{grav} - 3(\gamma-1)U_\mathrm{int} + 3P_eV\biggr]</math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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</table> | </table> | ||
|- | |- | ||
! style="text-align:center;" |<b>Virial Equilibrium</b> | ! style="text-align:center; background-color:#ffff99;" |<b><font color="maroon" size="+1">②</font></b> <b>Virial Equilibrium</b> | ||
|- | |- | ||
! style="vertical-align:top; text-align:left;" | | ! style="vertical-align:top; text-align:left;" | | ||
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</tr> | </tr> | ||
</table> | </table> | ||
|} | |||
===Pointers to Relevant Chapters=== | |||
<!-- BACKGROUND MATERIAL --> | |||
<font size="+1" color="maroon"><b>⓪ </b></font> Background Material: | |||
{| class="wikitable" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5" | |||
|- | |||
! width="30px" style="text-align:right; vertical-align:top; "|· | |||
|[[User:Tohline/PGE#Principal_Governing_Equations|Principal Governing Equations]] (PGEs) in most general form being considered throughout this H_Book | |||
|- | |||
! width="30px" style="text-align:right; vertical-align:top; "|· | |||
|PGEs in a form that is relevant to a study of the ''Structure, Stability, & Dynamics'' of [[User:Tohline/SphericallySymmetricConfigurations/PGE|spherically symmetric systems]] | |||
|- | |||
! width="30px" style="text-align:right; vertical-align:top; "|· | |||
|[[User:Tohline/SR#Supplemental_Relations|Supplemental relations]] — see, especially, [[User:Tohline/SR#Barotropic_Structure|barotropic equations of state]] | |||
|} | |||
<!-- DETAILED FORCE BALANCE --> | |||
<font size="+1" color="maroon"><b>① </b></font> Detailed Force Balance: | |||
{| class="wikitable" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5" | |||
|- | |||
! width="30px" style="text-align:right; vertical-align:top; "|· | |||
|[[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Derivation of the equation of Hydrostatic Balance]], and a description of several standard strategies that are used to determine its solution — see, especially, what we refer to as [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Technique_1|Technique 1]] | |||
|} | |||
<!-- VIRIAL EQUILIBRIUM --> | |||
<font size="+1" color="maroon"><b>② </b></font> Virial Equilibrium: | |||
{| class="wikitable" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5" | |||
|- | |||
! width="30px" style="text-align:right; vertical-align:top; "|· | |||
|Formal derivation of the multi-dimensional, [[User:Tohline/VE#Second-Order_Tensor_Virial_Equations|2<sup>nd</sup>-order tensor virial equations]] | |||
|- | |||
! width="30px" style="text-align:right; vertical-align:top; "|· | |||
|[[User:Tohline/VE#Scalar_Virial_Theorem|Scalar Virial Theorem]], as appropriate for spherically symmetric configurations | |||
|- | |||
! width="30px" style="text-align:right; vertical-align:top; "|· | |||
|[[User:Tohline/VE#Generalization|Generalization]] of scalar virial theorem to include the bounding effects of a hot, tenuous external medium | |||
|} | |||
==Stability== | |||
===Isolated & Pressure-Truncated Configurations=== | |||
{| class="wikitable" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12" | |||
|- | |- | ||
! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Stability Analysis</b></font> | ! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Stability Analysis: Applicable to Isolated & Pressure-Truncated Configurations</b></font> | ||
|- | |- | ||
! style="text-align:center;" width="50%" |<b>Perturbation Theory</b> | ! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">④</font></b> <b>Perturbation Theory</b> | ||
! style="text-align:center;" |<b>Free-Energy Analysis</b> | ! style="text-align:center; background-color:lightblue;" |<b><font color="maroon" size="+1">⑦</font></b> <b>Free-Energy Analysis of Stability</b> | ||
|- | |- | ||
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</td> | </td> | ||
</tr> | </tr> | ||
<tr><td align="center" colspan="3"> | |||
[<b>[[User:Tohline/Appendix/References#P00|<font color="red">P00</font>]]</b>], Vol. II, §3.7.1, p. 174, Eq. (3.145) | |||
</td></tr> | |||
</table> | </table> | ||
</div> | </div> | ||
to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>. | to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>. | ||
! style="vertical-align:top; text-align:left;" rowspan=" | ! style="vertical-align:top; text-align:left;" rowspan="5"| | ||
The second derivative of the free-energy function is, | The second derivative of the free-energy function is, | ||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~\frac{ | <math>~R_0^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
- | -2a\biggl(\frac{R}{R_0}\biggr)^{-3} + (3-3\gamma)(2-3\gamma)b \biggl(\frac{R}{R_0}\biggr)^{1-3\gamma} + 6c\biggl(\frac{R}{R_0}\biggr) | ||
</math> | </math> | ||
</td> | </td> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~\frac{ | <math>~\biggl(\frac{R_0}{R} \biggr)^2\biggl[ | ||
2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V | 2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V | ||
\biggr] \, . | \biggr] \, . | ||
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<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~\Rightarrow~~~ R^2 \biggl[\frac{ | <math>~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</tr> | </tr> | ||
</table> | </table> | ||
Note the similarity with <b><font color="maroon" size="+1">⑥</font></b>. | |||
---- | |||
Alternatively, recalling that, | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | <tr> | ||
<td align=" | <td align="right"> | ||
< | <math>~3(\gamma - 1)U_\mathrm{int}</math> | ||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~2S_\mathrm{therm} \, , | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
< | </table> | ||
the conditions for virial equilibrium and stability, may be written respectively as, | |||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~ | <math>~3P_e V</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
<math>~=</math> | |||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ 2S_\mathrm{therm}+ W_\mathrm{grav} </math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~ | <math>~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~- \ | <math>~ | ||
2W_\mathrm{grav} - 2(2-3\gamma)S_\mathrm{therm} + 2 \biggl[ 2S_\mathrm{therm}+ W_\mathrm{grav} \biggr] | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
| |||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~\ | <math>~ | ||
4W_\mathrm{grav} + 6\gamma S_\mathrm{therm} \, . | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
|- | |||
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">⑤</font></b> <b>Variational Principle</b> | |||
< | |- | ||
< | ! style="vertical-align:top; text-align:left;" | | ||
Multiply the LAWE through by <math>~4\pi x dr</math>, and integrate over the volume of the configuration gives the, | |||
<div align="center"> | <div align="center"> | ||
<font color=" | <font color="#770000">'''Governing Variational Relation</font><br /> | ||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~ | <math>~0</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
\int_0^R 4\pi r^4 \gamma P \biggl(\frac{dx}{dr}\biggr)^2 dr | |||
- \int_0^R 4\pi (3\gamma - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
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</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
| |||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~- | <math>~ | ||
- 4\pi \biggr[r^4 \gamma Px \biggl(\frac{dx}{dr}\biggr) \biggr]_0^R | |||
- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, . | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
| |||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
\int_0^R x^2 \biggl(\frac{d\ln x}{d\ln r}\biggr)^2 \gamma 4\pi r^2P dr | |||
- \int_0^R (3\gamma - 4)x^2 \biggl( - \frac{GM_r}{r} \biggr) 4\pi \rho r^2 dr | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
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</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
| |||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~\frac{ | <math>~ | ||
+ \biggr[\gamma 4\pi r^3 Px^2 \biggl(-\frac{d\ln x}{d\ln r}\biggr) \biggr]_0^R | |||
- \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, . | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
</div> | |||
Now, by setting <math>~(d\ln x/d\ln r)_{r=R} = -3</math>, we can ensure that the pressure fluctuation is zero and, hence, <math>~P = P_e</math> at the surface, in which case this relation becomes, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~ | <math>~\omega^2</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
\frac{\gamma (\gamma -1) \int_0^R x^2 \bigl(\frac{d\ln x}{d\ln r}\bigr)^2 dU_\mathrm{int} | |||
- \int_0^R (3\gamma - 4)x^2 dW_\mathrm{grav} | |||
+ 3^2 \gamma x^2 P_eV}{ \int_0^R x^2 r^2 dM_r} | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
</div> | |||
|- | |||
< | ! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">⑥</font></b> <b>Approximation: Homologous Expansion/Contraction</b> | ||
</ | |- | ||
< | ! style="vertical-align:top; text-align:left;" | | ||
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~ | <math>~\omega^2 \int_0^R r^2 dM_r</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
<math>~ | <math>~\leq</math> | ||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~-\ | <math>~ | ||
(4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma P_eV \, . | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | |||
</div> | |||
|} | |||
===Bipolytropes=== | |||
{| class="wikitable" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12" | |||
|- | |||
! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Stability Analysis: Applicable to Bipolytropic Configurations</b></font> | |||
|- | |||
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">⑧</font></b> <b>Variational Principle</b> | |||
! style="text-align:center; background-color:lightblue;" |<b><font color="maroon" size="+1">⑩</font></b> <b>Free-Energy Analysis of Stability</b> | |||
|- | |||
! style="vertical-align:top; text-align:left;" | | |||
<div align="center"> | |||
<font color="#770000">'''Governing Variational Relation'''</font><br /> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^{R^*} (x r^*)^2 dM_r^* </math> | |||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~-\int_0^ | <math>~ | ||
\gamma_c (\gamma_c-1) \int_0^{r^*_\mathrm{core}} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int} | |||
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}} x^2 dW^*_\mathrm{grav} | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
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</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
| |||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~-\ | <math>~ | ||
+ ~\gamma_e (\gamma_e-1) \int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int} | |||
- (3\gamma_e - 4) \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav} | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
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</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
| | ||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~3(\ | <math>~ | ||
+ ~3^2(\gamma_c - \gamma_e) x_i^2 P_i^* V_\mathrm{core}^* \, . | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
</ | </div> | ||
! style="vertical-align:top; text-align:left;" rowspan="3"| | |||
As we have detailed in an [[User:Tohline/SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives|accompanying discussion]], the first derivative of the relevant free-energy expression is, | |||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~ | <math>~R ~\frac{\partial \mathfrak{G}}{\partial R}</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
\ | 2S_\mathrm{tot} + W_\mathrm{tot} | ||
\, , | |||
</math> | </math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
where, | |||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~\ | <math>~S_\mathrm{tot} \equiv S_\mathrm{core} + S_\mathrm{env}</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
and | |||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~W_\mathrm{tot} \equiv W_\mathrm{core} + W_\mathrm{env} \, ;</math> | ||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | |||
and the second derivative of that free-energy function is, | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math> | |||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
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</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~2\biggl[ | ||
W_\mathrm{tot} + (3\gamma_c - 2) S_\mathrm{core} + (3\gamma_e-2)S_\mathrm{env} | |||
\biggr] \, . | \biggr] \, . | ||
</math> | </math> | ||
Line 586: | Line 634: | ||
</tr> | </tr> | ||
</table> | </table> | ||
---- | |||
This stability criterion may be rewritten as, | |||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~ | <math>~\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
<math>~=</math> | <math>~=</math> | ||
</td> | </td> | ||
<td align="left"> | |||
<math>~ | <math>~ | ||
2[(3\gamma_c -4) S_\mathrm{core} | |||
+ (3\gamma_e -4) S_\mathrm{env} ] \, . | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | |||
Hence, in bipolytropes, the marginally unstable equilibrium configuration (second derivative of free-energy set to zero) will be identified by the model that exhibits the ratio, | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~ | <math>~\frac{S_\mathrm{core}}{S_\mathrm{env}}</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
Line 609: | Line 666: | ||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
\frac{(3\gamma_e - 4)}{(4 - 3\gamma_c)} \, . | |||
</math> | </math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | |||
See the [[User:Tohline/SSC/Stability/BiPolytropes#What_to_Expect_for_Equilibrium_Configurations|accompanying discussion]]. | |||
---- | |||
If — based for example on <b><font color="maroon" size="+1">⑦</font></b> — we make the reasonable assumption that, in equilibrium, the statements, | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~2S_\mathrm{core} = 3P_i V_\mathrm{core} - W_\mathrm{core}</math> | |||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
and | |||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~2S_\mathrm{env} = - 3P_i V_\mathrm{core} - W_\mathrm{env} \, ,</math> | ||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
hold separately, then we satisfy the virial equilibrium condition, namely, | |||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
Line 653: | Line 702: | ||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~2S_\mathrm{tot} + W_\mathrm{tot} \, ,</math> | ||
\ | |||
</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | |||
and the second derivative of the relevant free-energy function can be rewritten as, | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math> | |||
<math>~ | |||
</math> | |||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
Line 684: | Line 717: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
2(W_\mathrm{core} + W_\mathrm{env}) | |||
+ (3\gamma_c - 2) (3P_i V_\mathrm{core} - W_\mathrm{core}) | |||
</math> | </math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
Line 699: | Line 731: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
+ \ | + (3\gamma_e-2)(-3P_i V_\mathrm{core} - W_\mathrm{env}) | ||
</math> | </math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
| |||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
Line 719: | Line 744: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
\ | 3^2 P_i V_\mathrm{core}(\gamma_c - \gamma_e) | ||
- | + (4-3\gamma_c ) W_\mathrm{core} | ||
+ (4-3\gamma_e)W_\mathrm{env} \, . | |||
</math> | </math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
Note the similarity with <b><font color="maroon" size="+1">⑨</font></b> — temporarily, see [[User:Tohline/SSC/Stability/BiPolytropes#Revised_Free-Energy_Analysis|this discussion]]. | |||
< | |||
</ | |- | ||
< | ! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">⑨</font></b> <b>Approximation: Homologous Expansion/Contraction</b> | ||
Approximation: Homologous Expansion/Contraction | |||
</ | |||
|- | |||
! style="vertical-align:top; text-align:left;" | | |||
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives, | If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives, | ||
Line 747: | Line 765: | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
<math>~\ | <math>~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^R r^2 dM_r</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
<math>~\ | <math>~\leq</math> | ||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
(4- 3\ | (4- 3\gamma_c) W_\mathrm{core}+ (4- 3\gamma_e) W_\mathrm{env}+ 3^2 (\gamma_c - \gamma_e) P_i V_\mathrm{core} \, . | ||
</math> | </math> | ||
</td> | </td> | ||
Line 760: | Line 778: | ||
</table> | </table> | ||
</div> | </div> | ||
|} | |||
=See Also= | =See Also= |
Latest revision as of 23:02, 4 February 2019
Spherically Symmetric Configurations Synopsis (Using Style Sheet)
| Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
Structure
Tabular Overview
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Equilibrium Structure | ||||||||||||||||
① Detailed Force Balance | ③ Free-Energy Identification of Equilibria | |||||||||||||||
Given a barotropic equation of state, <math>~P(\rho)</math>, solve the equation of
for the radial density distribution, <math>~\rho(r)</math>. |
The Free-Energy is,
Therefore, also,
Equilibrium configurations exist at extrema of the free-energy function, that is, they are identified by setting <math>~d\mathfrak{G}/dR = 0</math>. Hence, equilibria are defined by the condition,
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② Virial Equilibrium | ||||||||||||||||
Multiply the hydrostatic-balance equation through by <math>~rdV</math> and integrate over the volume:
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Pointers to Relevant Chapters
⓪ Background Material:
· | Principal Governing Equations (PGEs) in most general form being considered throughout this H_Book |
---|---|
· | PGEs in a form that is relevant to a study of the Structure, Stability, & Dynamics of spherically symmetric systems |
· | Supplemental relations — see, especially, barotropic equations of state |
① Detailed Force Balance:
· | Derivation of the equation of Hydrostatic Balance, and a description of several standard strategies that are used to determine its solution — see, especially, what we refer to as Technique 1 |
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② Virial Equilibrium:
· | Formal derivation of the multi-dimensional, 2nd-order tensor virial equations |
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· | Scalar Virial Theorem, as appropriate for spherically symmetric configurations |
· | Generalization of scalar virial theorem to include the bounding effects of a hot, tenuous external medium |
Stability
Isolated & Pressure-Truncated Configurations
Stability Analysis: Applicable to Isolated & Pressure-Truncated Configurations | ||||||||||||||||||||||||||||||||||
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④ Perturbation Theory | ⑦ Free-Energy Analysis of Stability | |||||||||||||||||||||||||||||||||
Given the radial profile of the density and pressure in the equilibrium configuration, solve the eigenvalue problem defined by the, LAWE: Linear Adiabatic Wave (or Radial Pulsation) Equation
to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>. |
The second derivative of the free-energy function is,
Evaluating this second derivative for an equilibrium configuration — that is by calling upon the (virial) equilibrium condition to set the value of the internal energy — we have,
Note the similarity with ⑥.
the conditions for virial equilibrium and stability, may be written respectively as,
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⑤ Variational Principle | ||||||||||||||||||||||||||||||||||
Multiply the LAWE through by <math>~4\pi x dr</math>, and integrate over the volume of the configuration gives the, Governing Variational Relation
Now, by setting <math>~(d\ln x/d\ln r)_{r=R} = -3</math>, we can ensure that the pressure fluctuation is zero and, hence, <math>~P = P_e</math> at the surface, in which case this relation becomes,
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⑥ Approximation: Homologous Expansion/Contraction | ||||||||||||||||||||||||||||||||||
If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,
|
Bipolytropes
Stability Analysis: Applicable to Bipolytropic Configurations | ||||||||||||||||||||||||||||||||||||||||
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⑧ Variational Principle | ⑩ Free-Energy Analysis of Stability | |||||||||||||||||||||||||||||||||||||||
Governing Variational Relation
|
As we have detailed in an accompanying discussion, the first derivative of the relevant free-energy expression is,
where,
and the second derivative of that free-energy function is,
Hence, in bipolytropes, the marginally unstable equilibrium configuration (second derivative of free-energy set to zero) will be identified by the model that exhibits the ratio,
See the accompanying discussion. If — based for example on ⑦ — we make the reasonable assumption that, in equilibrium, the statements,
hold separately, then we satisfy the virial equilibrium condition, namely,
and the second derivative of the relevant free-energy function can be rewritten as,
Note the similarity with ⑨ — temporarily, see this discussion. | |||||||||||||||||||||||||||||||||||||||
⑨ Approximation: Homologous Expansion/Contraction | ||||||||||||||||||||||||||||||||||||||||
If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,
|
See Also
© 2014 - 2021 by Joel E. Tohline |