User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain - Revision history

Tohline: /* Plotting Concise Mass-Radius Relation */

2019-02-10T22:08:01Z

Plotting Concise Mass-Radius Relation

← Older revision		Revision as of 22:08, 10 February 2019
Line 1,852:		Line 1,852:


	<span="TabulatedValues">First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.</span>		<span id="TabulatedValues">First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.</span>
	<div align="center">		<div align="center">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">

Tohline: /* Plotting Concise Mass-Radius Relation */

2019-02-10T22:06:23Z

Plotting Concise Mass-Radius Relation

← Older revision		Revision as of 22:06, 10 February 2019
Line 1,852:		Line 1,852:


	First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.		<span="TabulatedValues">First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.</span>
	<div align="center">		<div align="center">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">

Tohline: /* Deriving Concise Virial Theorem Mass-Radius Relation */

2019-02-09T20:02:37Z

Deriving Concise Virial Theorem Mass-Radius Relation

← Older revision		Revision as of 20:02, 9 February 2019
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	</div>		</div>

	as well as the modified length- and mass-normalizations, <math>~R_\mathrm{mod}</math> and <math>~M_\mathrm{mod}</math>, such that,		<span id="modNormalizations">as well as the modified length- and mass-normalizations,</span> <math>~R_\mathrm{mod}</math> and <math>~M_\mathrm{mod}</math>, such that,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Plotting Concise Mass-Radius Relation */

2017-04-05T22:04:59Z

Plotting Concise Mass-Radius Relation

← Older revision		Revision as of 22:04, 5 April 2017
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~R_\mathrm{SWS}\biggr\|_\mathrm{isothermal} \equiv \biggl( \frac{c_s^2}{G P_e} \biggr)^{1 / 2} \, .</math>		<math>~R_\mathrm{SWS}\biggr\|_\mathrm{isothermal} \equiv \biggl( \frac{c_s^4}{G P_e} \biggr)^{1 / 2} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Plotting Concise Mass-Radius Relation */

2017-04-05T21:53:47Z

Plotting Concise Mass-Radius Relation

← Older revision		Revision as of 21:53, 5 April 2017
Line 1,843:		Line 1,843:
	<math>~\frac{R}{R_\mathrm{SWS}} = \biggl\{ \biggl[ \biggl(\frac{3}{4\pi}\biggr)\frac{M}{M_\mathrm{SWS}}\biggr]^{4/3}		<math>~\frac{R}{R_\mathrm{SWS}} = \biggl\{ \biggl[ \biggl(\frac{3}{4\pi}\biggr)\frac{M}{M_\mathrm{SWS}}\biggr]^{4/3}
	- \biggl(\frac{1}{5\pi}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggr\}^{1/4} </math>         for,     <math>~0 \le \frac{M}{M_\mathrm{SWS}} \le \biggl(\frac{3^4\cdot 5^3}{2^8\pi}\biggr)^{1/2} \, .</math>		- \biggl(\frac{1}{5\pi}\biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggr\}^{1/4} </math>         for,     <math>~0 \le \frac{M}{M_\mathrm{SWS}} \le \biggl(\frac{3^4\cdot 5^3}{2^8\pi}\biggr)^{1/2} \, .</math>
			</div>
			* <span id="Isothermal">''Isothermal''</span> (explained [[#IsothermalExplained\|immediately below]])
			<div align="center">
			<math>~\frac{M}{M_\mathrm{SWS}} = \frac{5}{2} \biggl( \frac{R}{R_\mathrm{SWS}} \biggr) \biggl\{
			1 \pm \biggl[ 1 - \frac{16\pi}{15}\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^2 \biggr]^{1 / 2} \biggr\} </math>
			for,     <math>~0 \le \frac{R}{R_\mathrm{SWS}} \le \biggl(\frac{3\cdot 5}{2^4\pi}\biggr)^{1/2} \, .</math>
	</div>		</div>


	First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.		First, we'll create a table of the normalized coordinate values that satisfy this nonlinear expression.
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	</table>		</table>
	</div>		</div>

			<span id="IsothermalExplained">From a</span> [[User:Tohline/SSC/Virial/Isothermal#Bonnor.27s_.281956.29_Equivalent_Relation\|free-energy analysis of isothermal spheres]], we have demonstrated that, when the structural form factors are all set to unity, the statement of virial equilibrium is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl( \frac{R}{R_\mathrm{SWS}} \biggr)^4
			- \frac{3}{4\pi} \biggr( \frac{M}{M_\mathrm{SWS}} \biggr) \biggl( \frac{R}{R_\mathrm{SWS}} \biggr)
			+ \frac{3}{20\pi} \biggl( \frac{M}{M_\mathrm{SWS}} \biggr)^2 \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			where, in order to be consistent with the above polytropic normalizations, we have adopted the ''isothermal'' normalizations,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~M_\mathrm{SWS}\biggr\|_\mathrm{isothermal} \equiv \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1 / 2} \, ,</math>
			</td>
			<td align="center">
			and
			</td>
			<td align="left">
			<math>~R_\mathrm{SWS}\biggr\|_\mathrm{isothermal} \equiv \biggl( \frac{c_s^2}{G P_e} \biggr)^{1 / 2} \, .</math>
			</td>
			</tr>
			</table>
			</div>
			This is a quadratic equation that can be readily solved to provide an analytic expression for the ''isothermal'' mass-radius relation; the relevant expression has already been [[#Isothermal\|provided, above]].

	====Confirmation====		====Confirmation====

Tohline: /* Plotting Concise Mass-Radius Relation */ Added columns of data for n = 2

2017-04-05T00:41:59Z

Plotting Concise Mass-Radius Relation: Added columns of data for n = 2

← Older revision		Revision as of 00:41, 5 April 2017
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	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
	<tr>		<tr>
			<td align="center" colspan="1"><math>~n =2</math></td>
	<td align="center" colspan="1"><math>~n =2.8</math></td>		<td align="center" colspan="1"><math>~n =2.8</math></td>
	<td align="center" colspan="1"><math>~n =3.5</math></td>		<td align="center" colspan="1"><math>~n =3.5</math></td>
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	</tr>		</tr>
	<tr>		<tr>
			<td align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
			<td align="center"> </td>
			<td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
			</tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr>
			<td align="right">0.3800</td>
			<td align="center"> </td>
			<td align="left">0.26562</td>
			</tr>
			<tr>
			<td align="right">0.4500</td>
			<td align="center"> </td>
			<td align="left">0.477153</td>
			</tr>
			<tr>
			<td align="right">0.5000</td>
			<td align="center"> </td>
			<td align="left">0.70919</td>
			</tr>
			<tr>
			<td align="right">0.5500</td>
			<td align="center"> </td>
			<td align="left">1.063602</td>
			</tr>
			<tr>
			<td align="right">0.5800</td>
			<td align="center"> </td>
			<td align="left">1.39755</td>
			</tr>
			<tr>
			<td align="right">0.5950</td>
			<td align="center"> </td>
			<td align="left">1.64662</td>
			</tr>
			<tr>
			<td align="right">0.6050</td>
			<td align="center"> </td>
			<td align="left">1.893915</td>
			</tr>
			<tr>
			<td align="right">0.6120</td>
			<td align="center"> </td>
			<td align="left">2.22372</td>
			</tr>
			<tr>
			<td align="right"><font color="darkgreen">0.6131721</font></td>
			<td align="center"> </td>
			<td align="left"><font color="darkgreen">2.433375</font></td>
			</tr>
			<tr>
			<td align="right">0.6120</td>
			<td align="center"> </td>
			<td align="left">2.64923</td>
			</tr>
			<tr>
			<td align="right">0.6050</td>
			<td align="center"> </td>
			<td align="left">3.01688</td>
			</tr>
			<tr>
			<td align="right">0.5950</td>
			<td align="center"> </td>
			<td align="left">3.32037</td>
			</tr>
			<tr>
			<td align="right">0.5800</td>
			<td align="center"> </td>
			<td align="left">3.658702</td>
			</tr>
			<tr>
			<td align="right">0.5500</td>
			<td align="center"> </td>
			<td align="left">4.19097</td>
			</tr>
			<tr>
			<td align="right">0.5000</td>
			<td align="center"> </td>
			<td align="left">4.94599</td>
			</tr>
			<tr>
			<td align="right">0.4700</td>
			<td align="center"> </td>
			<td align="left">5.38791</td>
			</tr>
			<tr>
			<td align="right">0.4500</td>
			<td align="center"> </td>
			<td align="left">5.69164</td>
			</tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			</table>
			</td>
	<td align="center">		<td align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Plotting Concise Mass-Radius Relation */ Finished entering tabular data for n=2.8 virial-equilibrium sequence

2017-04-05T00:31:33Z

Plotting Concise Mass-Radius Relation: Finished entering tabular data for n=2.8 virial-equilibrium sequence

← Older revision		Revision as of 00:31, 5 April 2017
Line 1,849:		Line 1,849:
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
	<tr>		<tr>
			<td align="center" colspan="1"><math>~n =2.8</math></td>
	<td align="center" colspan="1"><math>~n =3.5</math></td>		<td align="center" colspan="1"><math>~n =3.5</math></td>
	<td align="center" colspan="1"><math>~n = 4</math></td>		<td align="center" colspan="1"><math>~n = 4</math></td>
Line 1,854:		Line 1,855:
	</tr>		</tr>
	<tr>		<tr>
			<td align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
			<td align="center"> </td>
			<td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
			</tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr>
			<td align="right">0.3800</td>
			<td align="center"> </td>
			<td align="left">0.266134</td>
			</tr>
			<tr>
			<td align="right">0.4500</td>
			<td align="center"> </td>
			<td align="left">0.47971</td>
			</tr>
			<tr>
			<td align="right">0.5000</td>
			<td align="center"> </td>
			<td align="left">0.71765</td>
			</tr>
			<tr>
			<td align="right">0.5250</td>
			<td align="center"> </td>
			<td align="left">0.831825</td>
			</tr>
			<tr>
			<td align="right">0.5600</td>
			<td align="center"> </td>
			<td align="left">1.20977</td>
			</tr>
			<tr>
			<td align="right">0.5750</td>
			<td align="center"> </td>
			<td align="left">1.427183</td>
			</tr>
			<tr>
			<td align="right">0.5850</td>
			<td align="center"> </td>
			<td align="left">1.653232</td>
			</tr>
			<tr>
			<td align="right">0.5900</td>
			<td align="center"> </td>
			<td align="left">1.89304</td>
			</tr>
			<tr>
			<td align="right"><font color="darkgreen">0.5904492</font></td>
			<td align="center"> </td>
			<td align="left"><font color="darkgreen">1.989927</font></td>
			</tr>
			<tr>
			<td align="right">0.5900</td>
			<td align="center"> </td>
			<td align="left">2.086584</td>
			</tr>
			<tr>
			<td align="right">0.5850</td>
			<td align="center"> </td>
			<td align="left">2.32394</td>
			</tr>
			<tr>
			<td align="right">0.5750</td>
			<td align="center"> </td>
			<td align="left">2.54527</td>
			</tr>
			<tr>
			<td align="right">0.5600</td>
			<td align="center"> </td>
			<td align="left">2.75612</td>
			</tr>
			<tr>
			<td align="right">0.5250</td>
			<td align="center"> </td>
			<td align="left">3.07134</td>
			</tr>
			<tr>
			<td align="right">0.4500</td>
			<td align="center"> </td>
			<td align="left">3.460304</td>
			</tr>
			<tr>
			<td align="right">0.3500</td>
			<td align="center"> </td>
			<td align="left">3.75881</td>
			</tr>
			<tr>
			<td align="right">0.2500</td>
			<td align="center"> </td>
			<td align="left">3.97835</td>
			</tr>
			<tr>
			<td align="right">0.2000</td>
			<td align="center"> </td>
			<td align="left">4.09302</td>
			</tr>
			<tr>
			<td align="right">0.1500</td>
			<td align="center"> </td>
			<td align="left">4.232786</td>
			</tr>
			<tr>
			<td align="right">0.1000</td>
			<td align="center"> </td>
			<td align="left">4.430303</td>
			</tr>
			<tr>
			<td align="right">0.0700</td>
			<td align="center"> </td>
			<td align="left">4.60984</td>
			</tr>
			<tr>
			<td align="right">0.0400</td>
			<td align="center"> </td>
			<td align="left">4.9057</td>
			</tr>
			<tr>
			<td align="right">0.0150</td>
			<td align="center"> </td>
			<td align="left">5.47056</td>
			</tr>
			</table>
			</td>
	<td align="center">		<td align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 1,900:		Line 2,028:
	</tr>		</tr>
	<tr>		<tr>
	<td align="right">0.58</td>		<td align="right">0.5800</td>
	<td align="center"> </td>		<td align="center"> </td>
	<td align="left">1.745165</td>		<td align="left">1.745165</td>
Line 1,910:		Line 2,038:
	</tr>		</tr>
	<tr>		<tr>
	<td align="right">0.58</td>		<td align="right">0.5800</td>
	<td align="center"> </td>		<td align="center"> </td>
	<td align="left">1.90201</td>		<td align="left">1.90201</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="right">0.~~578~~</td>		<td align="right">0.5780</td>
	<td align="center"> </td>		<td align="center"> </td>
	<td align="left">2.01647</td>		<td align="left">2.01647</td>
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	</tr>		</tr>
	<tr>		<tr>
	<td align="right">0.~~500~~</td>		<td align="right">0.5000</td>
	<td align="center"> </td>		<td align="center"> </td>
	<td align="left">2.75557</td>		<td align="left">2.75557</td>

Tohline: /* Plotting Concise Mass-Radius Relation */ Add columns of data for n = 3.5 virial -equilibrium configurations

2017-04-05T00:09:34Z

Plotting Concise Mass-Radius Relation: Add columns of data for n = 3.5 virial -equilibrium configurations

← Older revision		Revision as of 00:09, 5 April 2017
Line 1,849:		Line 1,849:
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
	<tr>		<tr>
			<td align="center" colspan="1"><math>~n =3.5</math></td>
	<td align="center" colspan="1"><math>~n = 4</math></td>		<td align="center" colspan="1"><math>~n = 4</math></td>
	<td align="center" colspan="1"><math>~n = 5</math></td>		<td align="center" colspan="1"><math>~n = 5</math></td>
	</tr>		</tr>
	<tr>		<tr>
			<td align="center">
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
			<td align="center"> </td>
			<td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
			</tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr>
			<td align="right">0.3800</td>
			<td align="center"> </td>
			<td align="left">0.26639</td>
			</tr>
			<tr>
			<td align="right">0.4500</td>
			<td align="center"> </td>
			<td align="left">0.481072</td>
			</tr>
			<tr>
			<td align="right">0.5000</td>
			<td align="center"> </td>
			<td align="left">0.722406</td>
			</tr>
			<tr>
			<td align="right">0.5250</td>
			<td align="center"> </td>
			<td align="left">0.89152</td>
			</tr>
			<tr>
			<td align="right">0.5600</td>
			<td align="center"> </td>
			<td align="left">1.246123</td>
			</tr>
			<tr>
			<td align="right">0.5650</td>
			<td align="center"> </td>
			<td align="left">1.32113</td>
			</tr>
			<tr>
			<td align="right">0.5750</td>
			<td align="center"> </td>
			<td align="left">1.52651</td>
			</tr>
			<tr>
			<td align="right">0.58</td>
			<td align="center"> </td>
			<td align="left">1.745165</td>
			</tr>
			<tr>
			<td align="right"><font color="darkgreen">0.5803836</font></td>
			<td align="center"> </td>
			<td align="left"><font color="darkgreen">1.823995</font></td>
			</tr>
			<tr>
			<td align="right">0.58</td>
			<td align="center"> </td>
			<td align="left">1.90201</td>
			</tr>
			<tr>
			<td align="right">0.578</td>
			<td align="center"> </td>
			<td align="left">2.01647</td>
			</tr>
			<tr>
			<td align="right">0.5750</td>
			<td align="center"> </td>
			<td align="left">2.11019</td>
			</tr>
			<tr>
			<td align="right">0.5600</td>
			<td align="center"> </td>
			<td align="left">2.35906</td>
			</tr>
			<tr>
			<td align="right">0.5400</td>
			<td align="center"> </td>
			<td align="left">2.543602</td>
			</tr>
			<tr>
			<td align="right">0.500</td>
			<td align="center"> </td>
			<td align="left">2.75557</td>
			</tr>
			<tr>
			<td align="right">0.4500</td>
			<td align="center"> </td>
			<td align="left">2.543602</td>
			</tr>
			<tr>
			<td align="right">0.3800</td>
			<td align="center"> </td>
			<td align="left">2.94829</td>
			</tr>
			<tr>
			<td align="right" bgcolor="yellow"><font color="black">0.3749583</font></td>
			<td align="center"> </td>
			<td align="left" bgcolor="yellow"><font color="black">2.948526</font></td>
			</tr>
			<tr>
			<td align="right">0.3300</td>
			<td align="center"> </td>
			<td align="left">2.93161</td>
			</tr>
			<tr>
			<td align="right">0.2500</td>
			<td align="center"> </td>
			<td align="left">2.829401</td>
			</tr>
			<tr>
			<td align="right">0.1500</td>
			<td align="center"> </td>
			<td align="left">2.578605</td>
			</tr>
			<tr><td align="center" colspan="3"> </td></tr>
			<tr><td align="center" colspan="3"> </td></tr>
			</table>
			</td>
	<td align="center">		<td align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Plotting Concise Mass-Radius Relation */ Finished typing in coordinates for n=4 virial-equil. sequence

2017-04-04T22:28:30Z

Plotting Concise Mass-Radius Relation: Finished typing in coordinates for n=4 virial-equil. sequence

Show changes

Tohline: /* Plotting Concise Mass-Radius Relation */ Finish generating table of numbers for n=5 virial-equil. sequence

2017-04-04T22:12:49Z

Plotting Concise Mass-Radius Relation: Finish generating table of numbers for n=5 virial-equil. sequence

← Older revision		Revision as of 22:12, 4 April 2017
Line 1,849:		Line 1,849:
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
	<tr>		<tr>
	<td align="center" colspan="2"><math>~n = 3</math></td>		<td align="center" colspan="1"><math>~n = 5</math></td>
	</tr>		</tr>
			<tr><td align="center">
			<table border="0" cellpadding="5" align="center">
	<tr>		<tr>
	<td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>		<td align="center"><math>~ \frac{R}{R_\mathrm{SWS}}</math></td>
			<td align="center"> </td>
	<td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>		<td align="center"><math>~ \frac{M}{M_\mathrm{SWS}}</math></td>
	</tr>		</tr>
			<tr>
			<td align="right">0.1000</td>
			<td align="center"> </td>
			<td align="left">0.004224</td>
			</tr>
			<tr>
			<td align="right">0.2000</td>
			<td align="center"> </td>
			<td align="left">0.03471</td>
			</tr>
			<tr>
			<td align="right">0.3000</td>
			<td align="center"> </td>
			<td align="left">0.123115</td>
			</tr>
			<tr>
			<td align="right">0.4000</td>
			<td align="center"> </td>
			<td align="left">0.31766</td>
			</tr>
			<tr>
			<td align="right">0.4500</td>
			<td align="center"> </td>
			<td align="left">0.48278</td>
			</tr>
			<tr>
			<td align="right">0.5000</td>
			<td align="center"> </td>
			<td align="left">0.72866</td>
			</tr>
			<tr>
			<td align="right">0.5250</td>
			<td align="center"> </td>
			<td align="left">0.905006</td>
			</tr>
			<tr>
			<td align="right">0.5400</td>
			<td align="center"> </td>
			<td align="left">1.042907</td>
			</tr>
			<tr>
			<td align="right">0.5500</td>
			<td align="center"> </td>
			<td align="left">1.15886</td>
			</tr>
			<tr>
			<td align="right">0.5600</td>
			<td align="center"> </td>
			<td align="left">1.313712</td>
			</tr>
			<tr>
			<td align="right"><font color="darkgreen">0.5692185</font></td>
			<td align="center"> </td>
			<td align="left"><font color="darkgreen">1.657839</font></td>
			</tr>
			<tr>
			<td align="right">0.5675</td>
			<td align="center"> </td>
			<td align="left">1.511304</td>
			</tr>
			<tr>
			<td align="right">0.5675</td>
			<td align="center"> </td>
			<td align="left">1.798532</td>
			</tr>
			<tr>
			<td align="right">0.5600</td>
			<td align="center"> </td>
			<td align="left">1.97061</td>
			</tr>
			<tr>
			<td align="right">0.5400</td>
			<td align="center"> </td>
			<td align="left">2.17282</td>
			</tr>
			<tr>
			<td align="right">0.5250</td>
			<td align="center"> </td>
			<td align="left">2.25888</td>
			</tr>
			<tr>
			<td align="right">0.5000</td>
			<td align="center"> </td>
			<td align="left">2.34793</td>
			</tr>
			<tr>
			<td align="right">0.4600</td>
			<td align="center"> </td>
			<td align="left">2.410374</td>
			</tr>
			<tr>
			<td align="right" bgcolor="yellow"><font color="black">0.4391754</font></td>
			<td align="center"> </td>
			<td align="left" bgcolor="yellow"><font color="black">2.417330</font></td>
			</tr>
			<tr>
			<td align="right">0.4000</td>
			<td align="center"> </td>
			<td align="left">2.396465</td>
			</tr>
			<tr>
			<td align="right">0.3000</td>
			<td align="center"> </td>
			<td align="left">2.19848</td>
			</tr>
			<tr>
			<td align="right">0.2000</td>
			<td align="center"> </td>
			<td align="left">1.84195</td>
			</tr>
			<tr>
			<td align="right">0.1000</td>
			<td align="center"> </td>
			<td align="left">1.31421</td>
			</tr>
			<tr>
			<td align="right">0.0500</td>
			<td align="center"> </td>
			<td align="left">0.930314</td>
			</tr>
			<tr>
			<td align="right">0.0200</td>
			<td align="center"> </td>
			<td align="left">0.58847</td>
			</tr>
			</table>
			</td></tr>
	</table>		</table>
	</div>		</div>