SSCpt1/Virial/FormFactors/Pt2 - Revision history

Joel2: /* First Detailed Example (n = 5) */

2024-07-19T22:16:37Z

First Detailed Example (n = 5)

← Older revision		Revision as of 18:16, 19 July 2024
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	==First Detailed Example (n = 5)==		==First Detailed Example (n = 5)==

	Here we complete these integrals to derive detailed expressions for the above subset of structural form factors in the case of spherically symmetric configurations that obey an <math>~n=5</math> polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should simplify the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases that cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures\|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#Free_Energy\|cores of bipolytropes]].		Here we complete these integrals to derive detailed expressions for the above subset of structural form factors in the case of spherically symmetric configurations that obey an <math>~n=5</math> polytropic equation of state. The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable. This should simplify the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases that cannot be derived analytically. The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures\|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/Analytic51#Free_Energy\|cores of bipolytropes]].

	===Foundation (n = 5)===		===Foundation (n = 5)===

Joel2: /* Mass1 (n = 5) */

2024-06-24T18:31:36Z

Mass1 (n = 5)

← Older revision		Revision as of 14:31, 24 June 2024
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	<font color="red">Normalized Mass:</font>		<font color="red">Normalized Mass:</font>
	<div align="center">		<div align="center">

			<math>
			\begin{align}
			M_r(r^\dagger) & = M_\mathrm{tot} \int_0^{r^\dagger} 3(r^\dagger)^2 \rho^\dagger dr^\dagger \, . \\
			\end{align}
			</math>

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	</div>		</div>
	Plugging in the profiles for <math>r^\dagger</math> and <math>\rho^\dagger</math>, and recognizing that,		Plugging in the profiles for <math>r^\dagger</math> and <math>\rho^\dagger</math>, and recognizing that,
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	gives, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], [[Image:OnlineIntegral01.png\|250px\|right\|Mathematica Integral]]		gives, with the help of [http://integrals.wolfram.com/index.jsp Mathematica's Online Integrator], [[Image:OnlineIntegral01.png\|250px\|right\|Mathematica Integral]]
	<div align="center">		<div align="center">
			<math>
			\begin{align}
			\frac{M_r(\xi)}{M_\mathrm{tot} } & = 3 \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{3/2} \biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2} \int_0^{\xi} \xi^2 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} d\xi \\
			& = 3 \biggl( \frac{1}{3} \biggr)^{3/2} \biggl[ \frac{\xi^3}{3}\biggl(1+\frac{\xi^2}{3}\biggr)^{-3/2} \biggr]_0^{\xi} \\
			& = \biggl[ \frac{\xi^2}{3}\biggl(1+\frac{\xi^2}{3}\biggr)^{-1}\biggr]^{3/2} \, . \\
			\end{align}
			</math>

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	</div>		</div>
	As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above. Notice that if we decide to truncate an <math>n=5</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> — as in the discussion that follows — the mass of this truncated configuration will be, simply,		As it should, this expression exactly matches the normalized <math>M_r</math> profile shown above. Notice that if we decide to truncate an <math>n = 5</math> polytrope at some radius, <math>\tilde\xi < \xi_1</math> — as in the discussion that follows — the mass of this truncated configuration will be, simply,
	<div align="center">		<div align="center">
			<math>
			\frac{M_\mathrm{limit}}{M_\mathrm{tot} } = \frac{M_r({\tilde\xi})}{M_\mathrm{tot} } = \biggl[ \frac{\tilde\xi^2}{3}\biggl(1+\frac{\tilde\xi^2}{3}\biggr)^{-1}\biggr]^{3/2} \, .
			</math>

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	</div>		</div>

Joel2: /* Foundation (n = 5) */

2024-06-24T18:07:53Z

Foundation (n = 5)

← Older revision		Revision as of 14:07, 24 June 2024
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	We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations\|our more general introductory discussion]]:		We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations\|our more general introductory discussion]]:

	<table border="1" align="center" cellpadding="5" width="80%">		<table border="1" align="center" cellpadding="5" width="70%">
	<tr><th align="center" colspan="2">		<tr><th align="center" colspan="1">
	Adopted Normalizations <math>(n=5; \gamma=6/5)</math>		Adopted Normalizations <math>(n=5; \gamma=6/5)</math>
	</th></tr>		</th></tr>
	<tr><td align="center" colspan="2">		<tr><td align="center" colspan="1">

	<math>		<math>
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	</td>		</td>
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	<tr><th align="left" colspan="2">		<tr><th align="left" colspan="1">
	Note that the following relations also hold:		Note that the following relations also hold:
	<div align="center">		<div align="center">
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	</div>		</div>

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	Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope\|accompanying discussion]] — in ''isolated'' <math>n=5</math> polytropes, the total mass is given by the expression,		Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope\|accompanying discussion]] — in ''isolated'' <math>n=5</math> polytropes, the total mass is given by the expression,
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	\rho_0^{1/5} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} M_\mathrm{tot}^{-1} \, .</math>		\rho_0^{1/5} = \biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2} M_\mathrm{tot}^{-1} \, .</math>
	</div>		</div>
	Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,		<span id="NormalizedProfiles">Employing this mapping</span> to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
	<~~span id~~="~~NormalizedProfiles~~">&~~nbsp;~~</~~span~~>
			<div align="center">
			<math>
			\begin{align}
			r^\dagger & \equiv \frac{r}{R_\mathrm{norm}} && = \biggl( \frac{\pi}{2\cdot 3^4} \biggr) \biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi = \biggl( \frac{\pi}{2^3\cdot 3^7} \biggr)^{1 / 2} \xi \\
			\rho^\dagger & \equiv \frac{\rho}{\rho_\mathrm{norm}} && = \biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, , \\
			P^\dagger &\equiv \frac{P}{P_\mathrm{norm}} && = \biggl( \frac{2\cdot 3^4}{\pi} \biggr)^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, , \\
			\frac{M_r}{M_\mathrm{tot}}& && = \biggl( \frac{\pi}{2\cdot 3^4} \biggr)^{1/2}
			\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]
			= \biggl[\frac{\xi^2}{3}\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-1}\biggr]^{3/2}
			\, .\\
			\end{align}
			</math>
			</div>

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	===Mass1 (n = 5)===		===Mass1 (n = 5)===

Joel2: /* Foundation (n = 5) */

2024-06-24T17:51:50Z

Foundation (n = 5)

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Joel2: /* Foundation (n = 5) */

2024-06-24T17:33:07Z

Foundation (n = 5)

← Older revision		Revision as of 13:33, 24 June 2024
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	the radial profile of various physical variables is as follows:		the radial profile of various physical variables is as follows:
	<div align="center">		<div align="center">
			<math>
			\begin{align}
			\frac{r}{[K^{1/2}/(G^{1/2}\rho_0^{2/5})]} & = \biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi \, , \\
			\frac{\rho}{\rho_0} & = \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, , \\
			\frac{P}{K\rho_0^{6/5}} & = \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} \, , \\
			\frac{M_r}{[K^{3/2}/(G^{3/2}\rho_0^{1/5})]} & = \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] \, . \\
			\end{align}
			</math>
			</div>

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	~~</div~~>
			-->

	Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope\|accompanying discussion]] — in ''isolated'' <math>n=5</math> polytropes, the total mass is given by the expression,		Notice that, in these expressions, the central density, <math>\rho_0</math>, has been used instead of <math>M_\mathrm{tot}</math> to normalize the relevant physical variables. We can switch from one normalization to the other by realizing that — see, again, our [[SSC/Structure/Polytropes#.3D_5_Polytrope\|accompanying discussion]] — in ''isolated'' <math>n=5</math> polytropes, the total mass is given by the expression,
	<div align="center">		<div align="center">

Joel2: /* Foundation (n = 5) */

2024-06-24T00:10:32Z

Foundation (n = 5)

← Older revision		Revision as of 20:10, 23 June 2024
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">

	----		----

	</td>		</td>
	</tr>		</tr>
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	</td>		</td>
	<td align="center"><math> \equiv </math> </td>		<td align="center"><math> \equiv </math> </td>
	<td align="left">		<td align="left"><math>P_\mathrm{norm} R_\mathrm{norm}^3 = \biggl( \frac{K^5}{G^3} \biggr)^{1/2} </math> </td>
	<math>P_\mathrm{norm} R_\mathrm{norm}^3 = \biggl( \frac{K^5}{G^3} \biggr)^{1/2} </math>
	</td>
	</tr>		</tr>

Joel2: /* Foundation (n = 5) */

2024-06-24T00:03:02Z

Foundation (n = 5)

← Older revision		Revision as of 20:03, 23 June 2024
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	===Foundation (n = 5)===		===Foundation (n = 5)===
	We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations\|our more general introductory discussion]]:		We use the following normalizations, as drawn from [[SSCpt1/Virial#Normalizations\|our more general introductory discussion]]:
	~~<div align="center">~~
	<table border="1" align="center" cellpadding="5" width="80%">		<table border="1" align="center" cellpadding="5" width="80%">
	<tr><th align="center" colspan="2">		<tr><th align="center" colspan="2">
	Adopted Normalizations <math>(n=5; ~\gamma=6/5)</math>		Adopted Normalizations <math>(n=5; \gamma=6/5)</math>
	</th></tr>		</th></tr>
	<tr><td align="center" colspan="2">		<tr><td align="center" colspan="2">
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	<math>E_\mathrm{norm}</math>		<math>E_\mathrm{norm}</math>
	</td>		</td>
	<td align="center">		<td align="center"><math> \equiv </math> </td>
	<math>\equiv</math>
	</td>
	<td align="left">		<td align="left">
	<math>P_\mathrm{norm} R_\mathrm{norm}^3 =		<math>P_\mathrm{norm} R_\mathrm{norm}^3 = \biggl( \frac{K^5}{G^3} \biggr)^{1/2} </math>
	\biggl( \frac{K^5}{G^3} \biggr)^{1/2} </math>
	</td>		</td>
	</tr>		</tr>
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	<math>\rho_\mathrm{norm}</math>		<math>\rho_\mathrm{norm}</math>
	</td>		</td>
	<td align="center">		<td align="center"><math> \equiv </math> </td>
	<math>\equiv</math>
	</td>
	<td align="left">		<td align="left">
	<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}		<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}		<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}} = \frac{4\pi}{3} \biggl( \frac{K^5}{G^3} \biggr)^{1/2} M_\mathrm{tot}^{-1} </math>
	= \frac{4\pi}{3} \biggl( \frac{K^5}{G^3} \biggr)^{1/2} M_\mathrm{tot}^{-1} </math>
	</td>		</td>
	</tr>		</tr>
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	</th></tr>		</th></tr>
	</table>		</table>
	~~</div>~~

	As is detailed in our [[SSC/Structure/BiPolytropes/Analytic51#Profile\|accompanying discussion of bipolytropes]] — see also our [[SSC/Structure/Polytropes#.3D_5_Polytrope\|discussion of the properties of ''isolated'' polytropes]] — in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{5}</math>, where,		As is detailed in our [[SSC/Structure/BiPolytropes/Analytic51#Profile\|accompanying discussion of bipolytropes]] — see also our [[SSC/Structure/Polytropes#.3D_5_Polytrope\|discussion of the properties of ''isolated'' polytropes]] — in terms of the dimensionless Lane-Emden coordinate, <math>\xi \equiv r/a_{5}</math>, where,
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	</div>		</div>
	Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,		Employing this mapping to switch to our "preferred" adopted normalizations, as defined in the above boxed-in table, the four radial profiles become,
	<~~div align="center"~~ id="NormalizedProfiles">		<span id="NormalizedProfiles"> </span>
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</td>		</td>
	<td align="center">		<td align="center">
	<math>=~~</math>~~		<math> = </math>
	~~</td>~~
	~~<td align="left">~~
	~~<math>\biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,~~</math>
	</td>		</td>
			<td align="left"><math>\biggl( \frac{2^3\cdot 3^6}{\pi} \biggr)^{3/2} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ,</math> </td>
	</tr>		</tr>

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	</td>		</td>
	<td align="center">		<td align="center">
	<math>=</math>		<math> = </math>
	</td>		</td>
	<td align="left">		<td align="left">
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	</tr>		</tr>
	</table>		</table>
	~~</div>~~

	===Mass1 (n = 5)===		===Mass1 (n = 5)===

Joel2 at 22:19, 16 January 2024

2024-01-16T22:19:43Z

Show changes

Joel2: Created page with "=Structural Form Factors (Pt 2)=

Part I: Synopsis
Part II: n = 5 Polytrope
Part III: n = 1 Polytrope..."
2024-01-16T22:09:36Z

Created page with "=Structural Form Factors (Pt 2)= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" bgcolor="lightblue" width="33%"><br />Part I: Synopsis </td> <td align="center" bgcolor="lightblue" width="33%"><br />Part II: n = 5 Polytrope </td> <td align="center" bgcolor="lightblue"><br />Part III: n = 1 Polytrope..."

New page
=Structural Form Factors (Pt 2)=
<table border="1" align="center" width="100%" colspan="8">
<tr>
<td align="center" bgcolor="lightblue" width="33%"><br />[[SSCpt1/Virial/FormFactors|Part I:  Synopsis]]
 
</td>
<td align="center" bgcolor="lightblue" width="33%"><br />[[SSCpt1/Virial/FormFactors/Pt2|Part II:  n = 5 Polytrope]]
 
</td>
<td align="center" bgcolor="lightblue"><br />[[SSCpt1/Virial/FormFactors/Pt3|Part III:  n = 1 Polytrope]]
 
</td>
</tr>
</table>

=See Also=

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