Appendix/Ramblings/PatrickMotl - Revision history

Jet53man: /* Pick a Different Molecular-Weight Ratio */

2023-10-02T00:26:06Z

Pick a Different Molecular-Weight Ratio

← Older revision		Revision as of 20:26, 1 October 2023
Line 525:		Line 525:
	</td></tr></table>		</td></tr></table>
	Note:   <math>[\xi_i]_\mathrm{smooth} ~\rightarrow~ 0</math> when the argument inside the square brackets goes to zero, which occurs when <math>(\mu_e/\mu_c)\ln(\mu_e/\mu_c) ~\rightarrow ~ - 2.5\ln(5)</math>, that is, when <math>(\mu_e/\mu_c) \approx 0.299577998</math>. The implication is that ''all'' of the models along a sequence are stable against convection when <math>(\mu_e/\mu_c) < 0.299577998</math>.		Note:   <math>[\xi_i]_\mathrm{smooth} ~\rightarrow~ 0</math> when the argument inside the square brackets goes to zero, which occurs when <math>(\mu_e/\mu_c)\ln(\mu_e/\mu_c) ~\rightarrow ~ - 2.5\ln(5)</math>, that is, when <math>(\mu_e/\mu_c) \approx 0.299577998</math>. The implication is that ''all'' of the models along a sequence are stable against convection when <math>(\mu_e/\mu_c) < 0.299577998</math>.

			<div align="center">[[File:ConvectiveBoundaryLabeled.png\|400px\|Convective Boundary]]</div>

	====Use the Same Ratio of Specific Heats Throughout====		====Use the Same Ratio of Specific Heats Throughout====

Jet53man: /* Pick a Different Molecular-Weight Ratio */

2023-09-30T20:54:18Z

Pick a Different Molecular-Weight Ratio

← Older revision		Revision as of 16:54, 30 September 2023
Line 505:		Line 505:
	<td align="center">1/2</td>		<td align="center">1/2</td>
	<td align="right">2.86620</td>		<td align="right">2.86620</td>
			</tr>
			<tr>
			<td align="center">0.345</td>
			<td align="right">1.07128</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">1/3</td>		<td align="center">1/3</td>
	<td align="right">0.90754</td>		<td align="right">0.90754</td>
			</tr>
			<tr>
			<td align="center">0.31</td>
			<td align="right">0.4871479</td>
			</tr>
			<tr>
			<td align="center">0.299577998</td>
			<td align="right">0.00000000</td>
	</tr>		</tr>
	</table>		</table>
	</td></tr></table>		</td></tr></table>
			Note:   <math>[\xi_i]_\mathrm{smooth} ~\rightarrow~ 0</math> when the argument inside the square brackets goes to zero, which occurs when <math>(\mu_e/\mu_c)\ln(\mu_e/\mu_c) ~\rightarrow ~ - 2.5\ln(5)</math>, that is, when <math>(\mu_e/\mu_c) \approx 0.299577998</math>. The implication is that ''all'' of the models along a sequence are stable against convection when <math>(\mu_e/\mu_c) < 0.299577998</math>.

	====Use the Same Ratio of Specific Heats Throughout====		====Use the Same Ratio of Specific Heats Throughout====

Jet53man: /* Use the Same Ratio of Specific Heats Throughout */

2022-10-12T17:59:34Z

Use the Same Ratio of Specific Heats Throughout

← Older revision		Revision as of 13:59, 12 October 2022
Line 678:		Line 678:


	Note: ~~[http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)]~~ have examined radial oscillation modes in bipolytropic configurations that have a ''flipped'' set of indexes — that is, they studied equilibrium structures having <math>~(n_c, n_e) = (1, 5)</math> — assuming, as we have examined here, that oscillations in both the core and the envelope are governed by <math>~\gamma_g = 5/3</math>. The chapter of this H_Book in which we discuss the detailed analysis presented by ~~Murphy & Fiedler~~, we have inserted a short subsection titled, ''[[SSC/Stability/MurphyFiedler85#Aside_Regarding_Convectively_Unstable_Core\|Aside Regarding Convectively Unstable Core]],'' where we point out that the ''cores'' of the Murphy & Fiedler models should be convectively unstable whereas their ''envelopes'' should be stable against convection.		Note: {{ MF85bfull }} have examined radial oscillation modes in bipolytropic configurations that have a ''flipped'' set of indexes — that is, they studied equilibrium structures having <math>~(n_c, n_e) = (1, 5)</math> — assuming, as we have examined here, that oscillations in both the core and the envelope are governed by <math>~\gamma_g = 5/3</math>. The chapter of this H_Book in which we discuss the detailed analysis presented by {{ MF85b }}, we have inserted a short subsection titled, ''[[SSC/Stability/MurphyFiedler85#Aside_Regarding_Convectively_Unstable_Core\|Aside Regarding Convectively Unstable Core]],'' where we point out that the ''cores'' of the Murphy & Fiedler models should be convectively unstable whereas their ''envelopes'' should be stable against convection.

	====Behavior At Interface During Radial Oscillation====		====Behavior At Interface During Radial Oscillation====

Jet53man: /* Behavior At Interface During Radial Oscillation */

2022-06-02T22:52:24Z

Behavior At Interface During Radial Oscillation

Show changes

Jet53man: /* Behavior At Interface During Radial Oscillation */

2022-06-02T21:52:21Z

Behavior At Interface During Radial Oscillation

@@ Line 801: / Line 801: @@
    <td align="left">
 <math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{\gamma_e} \cdot \exp\biggl[ \frac{s_e}{\mu_e(\gamma_e-1)\mathfrak{R}} \biggr]</math>
    </td>
 </tr>

Jet53man: /* Can the Step Function be Flipped or Erased */

2022-06-02T18:51:59Z

Can the Step Function be Flipped or Erased

@@ Line 690: / Line 690: @@
    </td>
    <td align="center">
-<math>~=</math>
+<math>=</math>
    </td>
    <td align="left">
-<math>~
+<math>
 \ln \biggl\{
 \biggl(\frac{P}{\rho^{\gamma_g}}\biggr)^{\mu(\gamma_g-1)}
 \biggr\} \, .
 </math>
    </td>
 </tr>

Jet53man: /* Can the Step Function be Flipped or Erased */

2022-06-02T17:52:18Z

Can the Step Function be Flipped or Erased

← Older revision		Revision as of 13:52, 2 June 2022
Line 679:		Line 679:

	Note: [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] have examined radial oscillation modes in bipolytropic configurations that have a ''flipped'' set of indexes — that is, they studied equilibrium structures having <math>~(n_c, n_e) = (1, 5)</math> — assuming, as we have examined here, that oscillations in both the core and the envelope are governed by <math>~\gamma_g = 5/3</math>. The chapter of this H_Book in which we discuss the detailed analysis presented by Murphy & Fiedler, we have inserted a short subsection titled, ''[[SSC/Stability/MurphyFiedler85#Aside_Regarding_Convectively_Unstable_Core\|Aside Regarding Convectively Unstable Core]],'' where we point out that the ''cores'' of the Murphy & Fiedler models should be convectively unstable whereas their ''envelopes'' should be stable against convection.		Note: [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] have examined radial oscillation modes in bipolytropic configurations that have a ''flipped'' set of indexes — that is, they studied equilibrium structures having <math>~(n_c, n_e) = (1, 5)</math> — assuming, as we have examined here, that oscillations in both the core and the envelope are governed by <math>~\gamma_g = 5/3</math>. The chapter of this H_Book in which we discuss the detailed analysis presented by Murphy & Fiedler, we have inserted a short subsection titled, ''[[SSC/Stability/MurphyFiedler85#Aside_Regarding_Convectively_Unstable_Core\|Aside Regarding Convectively Unstable Core]],'' where we point out that the ''cores'' of the Murphy & Fiedler models should be convectively unstable whereas their ''envelopes'' should be stable against convection.

			====Behavior At Interface During Radial Oscillation====

			We have determined that the expression for the specific entropy, <math>s</math>, of any Lagrangian fluid element is,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>\frac{s}{\mathfrak{R}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\ln \biggl\{
			\biggl(\frac{P}{\rho^{\gamma_g}}\biggr)^{\mu(\gamma_g-1)}
			\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>

	===Eigenvectors of Marginally Unstable Models===		===Eigenvectors of Marginally Unstable Models===

Jet53man: /* October 07 - 08 (between Joel & Patrick) */

2021-10-19T18:04:53Z

October 07 - 08 (between Joel & Patrick)

← Older revision		Revision as of 14:04, 19 October 2021
Line 907:		Line 907:

	Please look at figure 6 and the lines of text that immediately precede and that immediately follow this figure on the following web page:		Please look at figure 6 and the lines of text that immediately precede and that immediately follow this figure on the following web page:
	https://~~www~~.~~vistrails.org~~/index.php~~/User:Tohline~~/Appendix/Ramblings/PatrickMotl		https://tohline.education/SelfGravitatingFluids/index.php/Appendix/Ramblings/PatrickMotl

	I put this material together some months ago, but had since forgotten about it. It contains quantitative results that will significantly strengthen any presentation that you make to show the results of your student's 3D simulations. I have carried out a numerical, linear-stability analysis of models along the pressure-truncated, n = 5 model sequence. Specifically, I have numerically determined the eigenfunction and the eigenfrequency (squared) of the fundamental mode of oscillation for 16 different models falling between \xi_i = 0.75 and \xi_i = 5.0. You should plot omega-squared versus \xi_i from this linear-stability analysis, then plot your oscillation frequencies on the same graph to see if they lie along the curve that you get from my analysis. Also, can you actually generate a plot of the radial eigenfunction of the vibration modes that develop in your simulations? You could directly compare these eigenfunction plots against my Figure-6 eigenfunction plots. [I'll be happy to explain this in more detail via a zoom session, if that would help.]		I put this material together some months ago, but had since forgotten about it. It contains quantitative results that will significantly strengthen any presentation that you make to show the results of your student's 3D simulations. I have carried out a numerical, linear-stability analysis of models along the pressure-truncated, n = 5 model sequence. Specifically, I have numerically determined the eigenfunction and the eigenfrequency (squared) of the fundamental mode of oscillation for 16 different models falling between \xi_i = 0.75 and \xi_i = 5.0. You should plot omega-squared versus \xi_i from this linear-stability analysis, then plot your oscillation frequencies on the same graph to see if they lie along the curve that you get from my analysis. Also, can you actually generate a plot of the radial eigenfunction of the vibration modes that develop in your simulations? You could directly compare these eigenfunction plots against my Figure-6 eigenfunction plots. [I'll be happy to explain this in more detail via a zoom session, if that would help.]

Jet53man: /* October 07 - 08 (between Joel & Patrick) */

2021-10-19T18:03:49Z

October 07 - 08 (between Joel & Patrick)

← Older revision		Revision as of 14:03, 19 October 2021
Line 932:		Line 932:

	I am quite sure that the answer is buried in the following chapter of my on-line book:		I am quite sure that the answer is buried in the following chapter of my on-line book:
	https://~~www~~.~~vistrails.org~~/index.php~~/User:Tohline~~/SSC/Perturbations#Summary_Set_of_Linearized_Equations		https://tohline.education/SelfGravitatingFluids/index.php/SSC/Perturbations#Summary_Set_of_Linearized_Equations
	For example, this specific link will take you to a subsection of the chapter that gives the relationship between p (fractional pressure variation), d (density), and x (lagrangian radial location) via the three key linearized differential equations. I am going to look for a better answer to your question, though.		For example, this specific link will take you to a subsection of the chapter that gives the relationship between p (fractional pressure variation), d (density), and x (lagrangian radial location) via the three key linearized differential equations. I am going to look for a better answer to your question, though.
	</td></tr></table>		</td></tr></table>


	==October 11==		==October 11==

Jet53man at 17:20, 19 October 2021

2021-10-19T17:20:24Z

Show changes