ParabolicDensity/Axisymmetric/Structure - Revision history

Joel2: /* Play With Vertical Pressure Gradient */

2024-11-16T19:36:46Z

Play With Vertical Pressure Gradient

← Older revision		Revision as of 15:36, 16 November 2024
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	<td align="left">		<td align="left">
	<math>		<math>
	(\mathrm{coef1}) \cdot \biggl[ \zeta^2 - (1-e^2)( 1 - \chi^2 \biggr]		(\mathrm{coef1}) \cdot \biggl[ \zeta^2 - (1-e^2)( 1 - \chi^2) \biggr]
	+ (\mathrm{coef2} )\cdot \biggl[ \zeta^4 - (1-e^2)^2( 1 - \chi^2)^2 \biggr]		+ (\mathrm{coef2} )\cdot \biggl[ \zeta^4 - (1-e^2)^2( 1 - \chi^2)^2 \biggr]
	+ ( \mathrm{coef3}) \cdot \biggl[ \zeta^6 - (1-e^2)^3( 1 - \chi^2)^3\biggr]		+ ( \mathrm{coef3}) \cdot \biggl[ \zeta^6 - (1-e^2)^3( 1 - \chi^2)^3\biggr]

Joel2: /* Play With Vertical Pressure Gradient */

2024-11-16T19:34:23Z

Play With Vertical Pressure Gradient

@@ Line 582: / Line 582: @@
    <td align="center">
 [[File:FerrersVerticalPressureD.png|center|500px|Ferrers Vertical Pressure ]]
    </td>
 </tr>

Joel2: /* Isobaric Surfaces */

2024-11-16T19:14:20Z

Isobaric Surfaces

← Older revision		Revision as of 15:14, 16 November 2024
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	</tr>		</tr>
	</table>		</table>
	For each of these five <math>(\chi,\zeta)</math> coordinate pairs, we have used our above derived expression for <math>P^_\mathrm{deduced}/P^_c</math> to calculate the "normalized pressure" at the relevant point inside the configuration. These results appear in the last column of the table; they also have been marked in the accompanying figure: dark green diamonds mark the points relevant to our choice of <math>\rho/\rho_c = 0.3</math> and purple diamonds mark the points relevant to our choice of <math>\rho/\rho_c = 0.6</math>. Notice that the normalized density is everywhere lower ~~that~~ <math>0.6</math> along the <math>\chi = 0.75</math> cut.		For each of these five <math>(\chi,\zeta)</math> coordinate pairs, we have used our above derived expression for <math>P^_\mathrm{deduced}/P^_c</math> to calculate the "normalized pressure" at the relevant point inside the configuration. These results appear in the last column of the table; they also have been marked in the accompanying figure: dark green diamonds mark the points relevant to our choice of <math>\rho/\rho_c = 0.3</math> and purple diamonds mark the points relevant to our choice of <math>\rho/\rho_c = 0.6</math>. Notice that the normalized density is everywhere lower than <math>0.6</math> along the <math>\chi = 0.75</math> cut, so the final row in the table has been marked "n/a" (not applicable).

	~~Note for later use~~ that,		The dark green diamond-shaped markers in the figure — along with the associated tabular data — show that at three separate points along the <math>\rho/\rho_c = 0.3</math> iso-density surface, the normalized pressure is ''nearly'' — but not exactly — the same; its value is approximately <math>0.057</math>. Similarly, the purple diamond-shaped markers show that at two separate points along the <math>\rho/\rho_c = 0.6</math> iso-density surface, the normalized pressure is nearly the same; in this case its value is approximately <math>0.28</math>. This seems to indicate that, throughout our configuration, the isobaric surfaces are almost — but not exactly — aligned with iso-density surfaces.

	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~<math> \~~frac{~~\~~partial C_\chi}{\partial\chi}~~</math><~~/td~~>
	<~~td align="center"~~><math>=</math><~~/td~~>
	<~~td align="left"~~>
	&~~hellip~~;
	~~</td>~~
	~~</tr>~~
	~~</table>~~

	====Now Play With Radial Pressure Gradient====		====Now Play With Radial Pressure Gradient====

Joel2: /* Isobaric Surfaces */

2024-11-13T19:28:38Z

Isobaric Surfaces

← Older revision		Revision as of 15:28, 13 November 2024
Line 676:		Line 676:
	</tr>		</tr>
	</table>		</table>
			For each of these five <math>(\chi,\zeta)</math> coordinate pairs, we have used our above derived expression for <math>P^_\mathrm{deduced}/P^_c</math> to calculate the "normalized pressure" at the relevant point inside the configuration. These results appear in the last column of the table; they also have been marked in the accompanying figure: dark green diamonds mark the points relevant to our choice of <math>\rho/\rho_c = 0.3</math> and purple diamonds mark the points relevant to our choice of <math>\rho/\rho_c = 0.6</math>. Notice that the normalized density is everywhere lower that <math>0.6</math> along the <math>\chi = 0.75</math> cut.

	Note for later use that,		Note for later use that,

Joel2: /* Isobaric Surfaces */

2024-11-13T19:07:06Z

Isobaric Surfaces

@@ Line 607: / Line 607: @@
 </table>
-In order to determine the relative stability of each configuration, it will be important to ascertain whether or not isobaric surfaces are also concentric spheroids.  (If they are, then we can say that each configuration obeys a [[SR#Barotropic_Structure|barotropic]] &#8212; but not necessarily a polytropic &#8212; equation of state; see, for example, the [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|accompanying relevant excerpt]] drawn from p. 466 of {{ Lebovitz67_XXXIV }}.)
+In order to determine the relative stability of each configuration, it will be important to ascertain whether or not isobaric surfaces are also concentric spheroids.  (If they are, then we can say that each configuration obeys a [[SR#Barotropic_Structure|barotropic]] &#8212; but not necessarily a polytropic &#8212; equation of state; see, for example, the [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|accompanying relevant excerpt]] drawn from p. 466 of {{ Lebovitz67_XXXIV }}.)  In an effort to make this determination for our <math>e = 0.6</math> spheroid, we first examine the iso-density surface for which <math>\rho/\rho_c = 0.3</math>.  Via the expression,
 <table border="1" align="center" cellpadding="5">
@@ Line 639: / Line 660: @@
 <tr>
    <td align="center" rowspan="3"><font color="purple">purple</font></td>
-   <td align="center" rowspan="3"><math>0.3</math></td>
+   <td align="center" rowspan="3"><math>0.6</math></td>
    <td align="center" rowspan="1"><math>0.00</math></td>
    <td align="center" rowspan="1"><math>0.50596</math></td>

Joel2: /* Isobaric Surfaces */

2024-11-13T18:32:58Z

Isobaric Surfaces

← Older revision		Revision as of 14:32, 13 November 2024
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	====Isobaric Surfaces====		====Isobaric Surfaces====

	By design, the mass within our oblate-spheroidal configuration is distributed in such a way that iso-density surfaces are concentric spheroids. As stated earlier, ~~mathematically~~ the relevant density distribution is,		By design, the mass within our oblate-spheroidal configuration is distributed in such a way that iso-density surfaces are concentric spheroids. As stated earlier, the relevant mathematically prescribed density distribution is,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>

	In order to determine the relative stability of each configuration, it will be important to ascertain whether or not isobaric surfaces are also concentric spheroids. (If they are, then we can say that each configuration obeys a [[SR#Barotropic_Structure\|barotropic]] — but not necessarily a polytropic — equation of state; see, for example, the [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|relevant excerpt]] drawn from p. 466 of {{ Lebovitz67_XXXIV }}.)		In order to determine the relative stability of each configuration, it will be important to ascertain whether or not isobaric surfaces are also concentric spheroids. (If they are, then we can say that each configuration obeys a [[SR#Barotropic_Structure\|barotropic]] — but not necessarily a polytropic — equation of state; see, for example, the [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|accompanying relevant excerpt]] drawn from p. 466 of {{ Lebovitz67_XXXIV }}.)

	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">

Joel2: /* Isobaric Surfaces */

2024-11-13T18:27:01Z

Isobaric Surfaces

← Older revision		Revision as of 14:27, 13 November 2024
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	<td align="center" rowspan="1"><math>0.00</math></td>		<td align="center" rowspan="1"><math>0.00</math></td>
	<td align="center" rowspan="1"><math>0.66933</math></td>		<td align="center" rowspan="1"><math>0.66933</math></td>
	<td align="center" rowspan="1"><math>0.~~057433~~</math></td>		<td align="center" rowspan="1"><math>0.060466</math></td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center" rowspan="1"><math>0.60</math></td>		<td align="center" rowspan="1"><math>0.60</math></td>
	<td align="center" rowspan="1"><math>0.</math></td>		<td align="center" rowspan="1"><math>0.46648</math></td>
	<td align="center" rowspan="1"><math>0.</math></td>		<td align="center" rowspan="1"><math>0.057433</math></td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center" rowspan="1"><math>0.75</math></td>		<td align="center" rowspan="1"><math>0.75</math></td>
	<td align="center" rowspan="1"><math>0.</math></td>		<td align="center" rowspan="1"><math>0.29665</math></td>
	<td align="center" rowspan="1"><math>0.</math></td>		<td align="center" rowspan="1"><math>0.055727</math></td>
	</tr>		</tr>
	~~</table>~~


	~~<table border="1" align="center" cellpadding="5">~~
	<tr>		<tr>
	<td align="center" rowspan="2">~~circular~~<br />~~marker~~<br />~~color~~</td>		<td align="center" rowspan="3"><font color="purple">purple</font></td>
	<td align="center" rowspan="2">~~chosen~~<br /><math>~~\chi~~</math></td>		<td align="center" rowspan="3"><math>0.3</math></td>
	<td align="center" ~~colspan~~="2">~~resulting …~~</td>		<td align="center" rowspan="1"><math>0.00</math></td>
			<td align="center" rowspan="1"><math>0.50596</math></td>
			<td align="center" rowspan="1"><math>0.292493</math></td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">~~surface~~ <math>~~\zeta~~</math></td>		<td align="center" rowspan="1"><math>0.60</math></td>
	<td align="center">~~mid-plane~~<br />~~pressure~~</td>		<td align="center" rowspan="1"><math>0.16000</math></td>
			<td align="center" rowspan="1"><math>0.280361</math></td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center"~~><font color~~="~~blue">blue</font></td>~~		<td align="center" rowspan="1"><math>0.75</math></td>
	~~<td align="center~~"><math>0.00</math></td>		<td align="center" rowspan="1">n/a</td>
	<td align="center"~~><math>0.8000</math></td>~~		<td align="center" rowspan="1">n/a</td>
	~~<td align~~="~~center"><math>~~1~~.00000</math></td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center~~">~~<font color="orange">orange<~~/~~font>~~</td>
	<td align="center"~~><math>0.60</math></td>~~
	~~<td align~~="~~center"><math>0.6400</math></td>~~
	~~<td align="center"><math>0.32667</math></td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center"><font color="gray">gray</font></td>~~
	~~<td align="center~~">~~<math>0.75<~~/~~math></td>~~
	~~<td align="center"><math>0.52915</math></td>~~
	~~<td align="center"><math>0.13085</math>~~</td>
	</tr>		</tr>
	</table>		</table>

Joel2: /* Isobaric Surfaces */

2024-11-13T18:18:31Z

Isobaric Surfaces

← Older revision		Revision as of 14:18, 13 November 2024
Line 608:		Line 608:

	In order to determine the relative stability of each configuration, it will be important to ascertain whether or not isobaric surfaces are also concentric spheroids. (If they are, then we can say that each configuration obeys a [[SR#Barotropic_Structure\|barotropic]] — but not necessarily a polytropic — equation of state; see, for example, the [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|relevant excerpt]] drawn from p. 466 of {{ Lebovitz67_XXXIV }}.)		In order to determine the relative stability of each configuration, it will be important to ascertain whether or not isobaric surfaces are also concentric spheroids. (If they are, then we can say that each configuration obeys a [[SR#Barotropic_Structure\|barotropic]] — but not necessarily a polytropic — equation of state; see, for example, the [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|relevant excerpt]] drawn from p. 466 of {{ Lebovitz67_XXXIV }}.)

			<table border="1" align="center" cellpadding="5">
			<tr>
			<td align="center" rowspan="2">diamond<br />marker<br />color</td>
			<td align="center" rowspan="2">chosen<br /><math>\rho/\rho_c</math></td>
			<td align="center" rowspan="2">chosen<br /><math>\chi</math></td>
			<td align="center" colspan="2">resulting …</td>
			</tr>
			<tr>
			<td align="center">    <math>\zeta</math>    </td>
			<td align="center">normalized<br />pressure</td>
			</tr>
			<tr>
			<td align="center" rowspan="3"><font color="darkgreen">green</font></td>
			<td align="center" rowspan="3"><math>0.3</math></td>
			<td align="center" rowspan="1"><math>0.00</math></td>
			<td align="center" rowspan="1"><math>0.66933</math></td>
			<td align="center" rowspan="1"><math>0.057433</math></td>
			</tr>
			<tr>
			<td align="center" rowspan="1"><math>0.60</math></td>
			<td align="center" rowspan="1"><math>0.</math></td>
			<td align="center" rowspan="1"><math>0.</math></td>
			</tr>
			<tr>
			<td align="center" rowspan="1"><math>0.75</math></td>
			<td align="center" rowspan="1"><math>0.</math></td>
			<td align="center" rowspan="1"><math>0.</math></td>
			</tr>
			</table>


	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">

Joel2: /* Isobaric Surfaces */

2024-11-13T17:49:36Z

Isobaric Surfaces

@@ Line 607: / Line 607: @@
 </table>
-In order to determine the relative stability of each configuration, it will be important to ascertain whether or not isobaric surfaces are also concentric spheroids.  If they are, then we can say that each configuration obeys a [[SR#Barotropic_Structure|barotropic]] &#8212; but not necessarily a polytropic &#8212; equation of state.
+In order to determine the relative stability of each configuration, it will be important to ascertain whether or not isobaric surfaces are also concentric spheroids.  (If they are, then we can say that each configuration obeys a [[SR#Barotropic_Structure|barotropic]] &#8212; but not necessarily a polytropic &#8212; equation of state; see, for example, the [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|relevant excerpt]] drawn from p. 466 of {{ Lebovitz67_XXXIV }}.)
 Note for later use that,

Joel2: /* Play With Vertical Pressure Gradient */

2024-11-13T17:42:24Z

Play With Vertical Pressure Gradient

← Older revision		Revision as of 13:42, 13 November 2024
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	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
	<tr>		<tr>
	<td align="center">circular<br />marker<br />color</td>		<td align="center" rowspan="2">circular<br />marker<br />color</td>
	<td align="center">chosen<br /><math>\chi</math></td>		<td align="center" rowspan="2">chosen<br /><math>\chi</math></td>
			<td align="center" colspan="2">resulting …</td>
			</tr>
			<tr>
			<td align="center">surface <math>\zeta</math></td>
	<td align="center">mid-plane<br />pressure</td>		<td align="center">mid-plane<br />pressure</td>
	~~<td align="center">surface <math>\zeta</math></td>~~
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">blue</td>		<td align="center"><font color="blue">blue</font></td>
	<td align="center"><math>0.00</math></td>		<td align="center"><math>0.00</math></td>
			<td align="center"><math>0.8000</math></td>
	<td align="center"><math>1.00000</math></td>		<td align="center"><math>1.00000</math></td>
	~~<td align="center"><math>0.8000</math></td>~~
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">orange</td>		<td align="center"><font color="orange">orange</font></td>
	<td align="center"><math>0.60</math></td>		<td align="center"><math>0.60</math></td>
			<td align="center"><math>0.6400</math></td>
	<td align="center"><math>0.32667</math></td>		<td align="center"><math>0.32667</math></td>
	~~<td align="center"><math>0.6400</math></td>~~
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">gray</td>		<td align="center"><font color="gray">gray</font></td>
	<td align="center"><math>0.75</math></td>		<td align="center"><math>0.75</math></td>
			<td align="center"><math>0.52915</math></td>
	<td align="center"><math>0.13085</math></td>		<td align="center"><math>0.13085</math></td>
	~~<td align="center"><math>0.52915</math></td>~~
	</tr>		</tr>
	</table>		</table>