Editing SSC/Structure/Lane1870 (section)

==SEGMENT III==

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{{ Lane1870figure }}

'''SEGMENT III''' (pp. 60 - 64)
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<font color="darkgreen">
In equations (6) and (7) it is plain that upon the value of <math>~k</math> alone depends:  first the form of the curve that expresses the value of <math>~\rho/\rho_0</math> for each value of <math>~x</math>; secondly, the value of the upper limit of <math>~x</math> corresponding to <math>~\rho/\rho_0 = 0</math></font> [henceforth this value of the dimensionless radius at the ''surface'' will be labeled <math>~x'</math>]<font color="darkgreen">; and thirdly, the corresponding value of <math>~\mu</math></font> [henceforth this value of the dimensionless total mass will be labeled <math>~\mu'</math>]<font color="darkgreen">.  These limiting, or terminal, value of <math>~x</math> and <math>~\mu</math>, cannot be found except by calculating the curve, for equations (6) and (7) seem incapable of being reduced to a complete general integral.  But when these values have been found for any proposed value of <math>~k</math>, they may be introduced once for all into equations (4) and (5), from which the values of <math>~\rho_0</math>, and <math>~\sigma t_1</math>, are at once deduced.

I have made these calculations for two different assumed values of <math>~k</math>, viz., <math>~k = 1.4</math> &hellip; and <math>~k = 1\tfrac{2}{3}</math> &hellip;
</font>

<table border="0" align="center" width="60%" cellpadding="8"><tr><td align="left">[In the next few paragraphs of his paper, Lane describes in detail how he numerically integrated the governing equations in order to determine how the density varies with radius as one moves from the center to the surface of the model.]</td></tr></table>

Upon <font color="darkgreen">''[d]etermination of the curve of density for'' <math>~k = 1.4</math> &hellip; we find</font>
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<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~5.355 \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\mu'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2.188 \, .</math>
  </td>
</tr>
</table>

And a determination of the<font color="darkgreen"> ''[c]urve of density for'' <math>~k = 1\tfrac{2}{3}</math> &hellip; gives us</font>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3.656 \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\mu'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2.741 \, .</math>
  </td>
</tr>
</table>
</td></tr>
</table>


In an effort to acknowledge the strikingly high precision with which Lane was able to determine the "curves of density" for this pair of models, the following table shows the values of <math>~x'</math> and <math>~\mu'</math> taken directly from Lane's publication (see the table cells with light green backgrounds) along with the corresponding values of the surface radius and total mass as has been calculated and published independently by [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)] and by [https://link.springer.com/book/10.1007/1-4020-2351-0 Horedt (2004)].  (Additional details from the published tabulations of these two authors can be found in the subsections that follow.)


<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" rowspan="2">&nbsp;</td>
  <td align="center" colspan="3"><math>~k = 1\tfrac{2}{3} ~~\Rightarrow ~~n = \tfrac{3}{2}</math></td>
  <td align="center" colspan="3"><math>~k = 1.4 ~~\Rightarrow ~~n = \tfrac{5}{2}</math></td>
</tr>
<tr>
  <td align="center"><math>~x' = \xi_1</math><br />(radius)</td>
  <td align="center"><math>~\mu'</math><br />(total mass)</td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho} = \frac{(x')^3}{3\mu'}</math></td>
  <td align="center"><math>~x' = \xi_1</math><br />(radius)</td>
  <td align="center"><math>~\mu'</math><br />(total mass)</td>
  <td align="center"><math>~\frac{\rho_c}{\bar\rho} = \frac{(x')^3}{3\mu'}</math></td>
</tr>
<tr>
  <td align="left">Lane (1870)</td>
  <td align="left" bgcolor="lightgreen">3.656</td>
  <td align="left" bgcolor="lightgreen">2.741</td>
  <td align="left">5.943</td>
  <td align="left" bgcolor="lightgreen">5.355</td>
  <td align="left" bgcolor="lightgreen">2.188</td>
  <td align="left">23.39</td>
</tr>
<tr>
  <td align="left">Emden (1907)</td>
  <td align="left">3.6571</td>
  <td align="left">2.7171</td>
  <td align="left">6.0004</td>
  <td align="left">5.4172</td>
  <td align="left">2.2010</td>
  <td align="left">24.08</td>
</tr>
<tr>
  <td align="left">Horedt (2004)</td>
  <td align="left">3.65375374</td>
  <td align="left">2.714055</td>
  <td align="left">5.990705</td>
  <td align="left">5.35527546</td>
  <td align="left">2.187200</td>
  <td align="left">23.40645</td>
</tr>
</table>

According to our [[SSCpt1/Virial/FormFactors#PTtable|accompanying summary of structural form factors]], the central-to-mean density of an isolated polytropic configuration is,
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  <td align="right">
<math>~\frac{\rho_c}{\bar\rho} = \frac{1}{\mathfrak{f}_M}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[- \frac{3}{\xi} \frac{d\Theta_H}{d\xi}\biggr]^{-1}_{\xi_1}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(x')^3}{3\mu^'} \, .</math>
  </td>
</tr>
</table>
The value of this density ratio, <math>~\rho_c/\bar\rho</math>, as determined for both models from the dimensionless masses and radii calculated by all three researchers has also been included for comparison in this table.

<table border="1" width="80%" align="center" cellpadding="10">
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  <td align="left">
<font color="red">'''NOTE:'''</font>&nbsp;  In his equation (14), [https://archive.org/details/mobot31753002152772/page/56 Lane (1870)] states that the central density is given by the expression,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="2">&nbsp; &nbsp; &nbsp; </td>
  <td align="center" colspan="1">'''Lane's Notation'''<br />
----
  </td>
  <td align="center" colspan="2">&nbsp; &nbsp; &nbsp; </td>
  <td align="center" colspan="1">'''Modern Notation'''<br />
----
  </td>
</tr>
<tr>
  <td align="right">
<math>~\rho_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="center">
<math>~\frac{m' (x')^3}{4\pi \mu^' (r')^3}</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="center">
<math>~\frac{3M_\odot}{4\pi R_\odot^3} \cdot \frac{(x')^3}{3\mu^'} = \bar\rho_\odot \cdot  \frac{(x')^3}{3\mu^'} \, .</math>
  </td>
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</table>

Lane concludes (see near the bottom of his p. 63) that, <math>~\rho_0 = 28.16</math> for the model with <math>~k = 1.4</math>, and (see near the top of his p. 64) that, <math>~\rho_0 = 7.11</math> for the model with <math>~k = 1\tfrac{2}{3}</math>.  This implies that he adopted a value of the Sun's mean density of, <math>~\bar\rho_\odot = 1.20</math> &#8212; apparently measured, in some manner we have as yet been unable to decipher, in units of the specific gravity of the Earth.  These calculated values &#8212; 28.16 and 7.11, respectively &#8212; of the central specific gravity are the only ''quantitative'' labels that appear in Lane's diagram that shows the variation with radius of the "Absolute density" in his pair of models (see further elaboration, below).
  </td>
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</table>


[https://archive.org/details/mobot31753002152772/page/56 Lane (1870)] did not publish a table of numbers to detail how the density varied with radius throughout the interiors of his pair of polytropic models.  But both [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)] and [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)] did.  From the tabulated data provided by these two researchers &#8212; see below &#8212; we have constructed curves showing how the dimensionless density (abscissa),  
<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~\frac{\rho}{\bar\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho_c}{\bar\rho}\biggr) \Theta_H^n\, ,</math>
  </td>
</tr>
</table>

varies with the fractional radius (ordinate), <math>~\xi/\xi_1</math>, in both of [https://archive.org/details/mobot31753002152772/page/56 Lane's (1870)] models.  These curves are displayed in the right-hand panel of the accompanying figure:  The solid blue curve (and the accompanying small, filled, yellow circular markers) shows the density variation in the model for which <math>k = 1.4 = \tfrac{7}{5} ~\Rightarrow ~n = \tfrac{5}{2}</math> as documented by Emden (and by Horedt); the solid green curve (and the accompanying small, filled, purple circular markers) shows the density variation in the model for which <math>k = 1\tfrac{2}{3} = \tfrac{5}{3} ~\Rightarrow ~n = \tfrac{3}{2}</math> as documented by Emden (and by Horedt). 

<table border="1" align="center" cellpadding="0">
<tr>
  <td align="center">[[File:Lane1870Diagram0.png|400px|Lane Diagram]]</td>
  <td align="center">[[File:Our2019Diagram2.png|400px|Our version of Lane's diagram]]</td>
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</table>

The only diagram that appears in [https://archive.org/details/mobot31753002152772/page/56 Lane's (1870)] publication has been reproduced here, in the left-hand panel of our figure.  Our pair of plotted curves (right-hand panel of our figure) appear to match the pair of curves that have been drawn in the bottom, left-hand portion of Lane's diagram; the caption accompanying Lane's diagram identifies these curves as plots of the "Absolute density" for <math>~k = 1\tfrac{2}{3}</math> and <math>~k = 1.4</math>.  We conclude that, in essentially all respects, Lane's published models provide quantitatively accurate representations of spherical polytropes having indexes, <math>~n = \tfrac{3}{2}</math> and <math>~n = \tfrac{5}{2}</math>.


===Published n = 3/2 Tabulations===

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<!-- =========== Emden Table n 1.5 ========================= -->
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  <th align="center">
[https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden's (1907)] "n = 1,5" Tabelle 4 (p. 79)<br />
Copied Directly from Table (1<sup>st</sup> 3 columns) &hellip; Implied Values (last 3 columns)
  </th>
</tr>
<tr><td align="left">
<pre>
Emden (1907) Tabelle 4 [n = 3/2]					
1.5	<-- n				
6.000360963	<-- rho_c/rho_avg				
										
xi	theta	-theta'		theta^(n+1)	"mass"	rho/rho_avg
0.00	1.00000	0.00000		1.0000		0.00000	6.0004
0.25	0.98966	0.08268		0.9744		0.00517	5.9075
0.50	0.95911	0.16057		0.9009		0.04014	5.6361
0.75	0.91008	0.22988		0.7901		0.12931	5.2095
1.00	0.84516	0.28727		0.6567		0.28727	4.6621
1.25	0.76761	0.33061		0.5162		0.51658	4.0354
1.50	0.68132	0.35752		0.3832		0.80442	3.3745
1.75	0.58994	0.37168		0.2673		1.1383	2.7189
2.00	0.49670	0.37209		0.1739		1.4884	2.1005
2.25	0.40477	0.36119		0.1042		1.8285	1.5452
2.50	0.31678	0.34120		0.0565		2.1325	1.0698
2.75	0.23468	0.34750		0.0267		2.6280	6.8217E-01
2.8085	0.21617	0.30788		0.0217		2.4285	6.0307E-01
3.00	0.15972	0.28442		0.0102		2.5598	3.8302E-01
3.25	0.09258	0.25261		0.0026		2.6682	1.6903E-01
3.50	0.03335	0.22147		0.0002		2.7130	3.6544E-02
3.625	0.00659	0.20680		0.0000		2.7175	3.2100E-03
3.64	0.00350	0.20511		0.0000		2.7176	1.2425E-03
3.6571	0.00000	0.20316		0.0000		2.7171	0.0000E+00
</pre>
</td></tr>
<tr>
  <td align="left">
The quantity labeled "rho_c/rho_avg" is <math>[\xi/(-3\theta^')]</math>, evaluated at the surface; the column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "rho/rho_avg" is &theta;<sup>n</sup> &times;  rho_c/rho_avg.
  </td>
</tr>
</table>
<!-- =========== END ========================= -->

</td>
<td align="right">
<!-- =========== Horedt Table n 1.5 ========================= -->

<table border="1" align="right" cellpadding="10">
<tr>
  <th align="center">
[http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt's (2004)] "n = 1.5" Table (pp. 73-74)<br />
Copied Directly from Table (1<sup>st</sup> 5 columns) &hellip; Implied Values (last column)
  </th>
</tr>
<tr><td align="left">
<pre>
Horedt  2004 pp. 73-74  [n = 3/2]					
1.5	<-- n				
5.990704011	<-- rho_c/rho_avg				
						
xi		theta		-theta'		theta^(n+1)	"mass"		rho/rho_avg
0.00		1.00000		0.00000		1.0000		0.00000		5.990704
0.10		0.99833		0.03328		0.9958		0.00033		5.9757
0.50		0.95910		0.16054		0.9009		0.04014		5.6270
1.00		0.84517		0.28726		0.6567		0.2873		4.6547
3.00		0.15886		0.28425		1.006E-02	2.5583		3.793E-01
3.60		1.1091E-02	0.20939		1.295E-05	2.7137		6.997E-03
3.650		7.6392E-04	0.20372		1.613E-08	2.7141		1.265E-04
3.65375374	0.00000		2.033013E-01	0.0000		2.714055	0
</pre>
</td></tr>
<tr>
  <td align="left">
The quantity labeled "rho_c/rho_avg" is <math>[\xi/(-3\theta^')]</math>, evaluated at the surface; the column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "rho/rho_avg" is &theta;<sup>n</sup> &times;  rho_c/rho_avg.
  </td>
</tr>
</table>
<!-- =========== END ========================= -->

</td></tr></table>

===Published n = 5/2 Tabulations===

<table border="0" align="center" cellpadding="10">
<tr><td align="left">
<!-- =========== Emden Table n 2.5 ========================= -->

<table border="1" align="left" cellpadding="10">
<tr>
  <th align="center">
[https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden's (1907)] "n = 2,5" Tabelle 6 (pp. 79-80)<br />
Copied Directly from Table (1<sup>st</sup> 3 columns) &hellip; Implied Values (last 3 columns)
  </th>
</tr>
<tr><td align="left">
<pre>
Emden (1907) Tabelle 6 [n = 5/2]							
2.5	<-- n						
24.08	<-- rho_c/rho_avg						
								
xi	theta	-theta'		theta^(n+1)	"mass"	rho/rho_avg	
0.00	1.00000	0.00000		1.0000		0.00000	24.076
0.25	0.98971	0.08226		0.9644		0.00514	23.462
0.50	0.95961	0.15676		0.8656		0.03919	21.718
0.75	0.91242	0.21798		0.7256		0.12261	19.146
1.00	0.85196	0.26282		0.5708		0.26282	16.130
1.25	0.78246	0.29036		0.4238		0.45369	13.039
1.50	0.70809	0.30213		0.2988		0.67979	10.158
1.75	0.63246	0.29532		0.2012		0.9044	 7.659	
2.00	0.55961	0.28614		0.1311		1.1446	 5.640
2.35	0.46331	0.26290		6.769E-02	1.4516	 3.518
2.50	0.42473	0.25080		4.993E-02	1.5675	 2.831
3.00	0.31000	0.20793		1.659E-02	1.8714	 1.288
3.5000	0.21752	0.16783		4.800E-03	2.0559	 5.313E-01	
4.00	0.14300	0.13445		1.106E-03	2.1512	 1.862E-01	
4.50	0.08263	0.10813		1.622E-04	2.1896	 4.725E-02
5.00	0.03384	0.08796		7.129E-06	2.1990	 5.072E-03	
5.400	0.00128	0.07545		7.503E-11	2.2001	 1.411E-06	
5.4172	0.00000	0.07500		0.0000		2.2010	 0.000E+00	
</pre>
</td></tr>
<tr>
  <td align="left">
The quantity labeled "rho_c/rho_avg" is <math>[\xi/(-3\theta^')]</math>, evaluated at the surface; the column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "rho/rho_avg" is &theta;<sup>n</sup> &times;  rho_c/rho_avg.
  </td>
</tr>
</table>
<!-- =========== END ========================= -->

</td>
<td align="right">
<!-- =========== Horedt Table n 1.5 ========================= -->

<table border="1" align="right" cellpadding="10">
<tr>
  <th align="center">
[http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt's (2004)] "n = 2.5" Table (p. 74)<br />
Copied Directly from Table (1<sup>st</sup> 3 columns) &hellip; Implied Values (last 3 columns)
  </th>
</tr>
<tr><td align="left">
<pre>
Horedt  2004 pp. 73-74  [n = 5/2]					
2.5	<-- n				
23.40646	<-- rho_c/rho_avg				
					
xi		theta		-theta'		theta^(n+1)	"mass"		rho/rho_avg
0.00		1.00000		0.00000		1.0000		0.00000		23.4065
0.10		0.99834		0.03325		0.9942		0.00033		23.3092
0.50		0.95960		0.15670		0.8656		0.03917		21.1134
1.00		0.85194		0.26287		0.5707		0.2629		15.6806
4.00		0.13768		0.13405		9.684E-04	2.1449		 1.646E-01
5.00		2.9019E-02	0.08747		4.163E-06	2.1868		 3.358E-03
5.300		4.2594E-03	0.07786		5.044E-09	2.1872		 2.772E-05
5.3550		2.1009E-05	0.07627		4.250E-17	2.1872		 4.735E-11
5.35527546	0.00000		7.626491E-02	0.000E+00	2.187200	 0.000E+00
</pre>
</td></tr>
<tr>
  <td align="left">
The quantity labeled "rho_c/rho_avg" is <math>[\xi/(-3\theta^')]</math>, evaluated at the surface; the column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')</math>; and the column labeled "rho/rho_avg" is &theta;<sup>n</sup> &times;  rho_c/rho_avg.
  </td>
</tr>
</table>
<!-- =========== END ========================= -->

</td></tr></table>

<!-- ORIGINAL PRESENTATION OF TABULATED DATA ...
===Published n = 3/2 and n = 5/2 Tabulations===

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center" colspan="7">Published Tabulations of Numerically Constructed n = 3/2 Polytrope</th>
</tr>
<tr>
  <td align="center" colspan="3">
Copied from p. 79 of [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)]
  </td>
  <td align="center" width="10%">&nbsp;</td>
  <td align="center" colspan="3">
Copied from p. 73 of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)]
  </td>
</tr>
<tr>
  <td align="center"><math>~\mathfrak{r}_1 \leftrightarrow \xi</math></td>
  <td align="center"><math>~u_1 \leftrightarrow \Theta_\mathrm{H}</math></td>
  <td align="center"><math>~- \frac{du_1}{d\mathfrak{r}_1} \leftrightarrow - \frac{d\Theta_\mathrm{H}}{d\xi} </math></td>
  <td align="center">&nbsp;</td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~\theta \leftrightarrow \Theta_H</math></td>
  <td align="center"><math>~- \theta^' \leftrightarrow - \frac{d\Theta_H}{d\xi}</math></td>
</tr>
<tr>
  <td align="right"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~0</math></td>
  <td align="center"><math>~1</math></td>
  <td align="center"><math>~0</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~0.10</math></td>
  <td align="center"><math>~9.983346\mathrm{E-01}</math></td>
  <td align="center"><math>~3.328337\mathrm{E-02}</math></td>
</tr>
<tr>
  <td align="right"><math>~0.25</math></td>
  <td align="right"><math>~0.98966</math></td>
  <td align="right"><math>~0.08268</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~0.50</math></td>
  <td align="right"><math>~0.95911</math></td>
  <td align="right"><math>~0.16057</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~0.50</math></td>
  <td align="center"><math>~9.591039\mathrm{E-01}</math></td>
  <td align="center"><math>~1.605449\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~0.75</math></td>
  <td align="right"><math>~0.91008</math></td>
  <td align="right"><math>~0.22988</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~1</math></td>
  <td align="right"><math>~0.84516</math></td>
  <td align="right"><math>~0.28727</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~1.00</math></td>
  <td align="center"><math>~8.451698\mathrm{E-01}</math></td>
  <td align="center"><math>~2.872555\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~1.25</math></td>
  <td align="right"><math>~0.76761</math></td>
  <td align="right"><math>~0.33061</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~1.50</math></td>
  <td align="right"><math>~0.68132</math></td>
  <td align="right"><math>~0.35752</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2</math></td>
  <td align="right"><math>~0.49670</math></td>
  <td align="right"><math>~0.37209</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2.25</math></td>
  <td align="right"><math>~0.40477</math></td>
  <td align="right"><math>~0.36119</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2.50</math></td>
  <td align="right"><math>~0.31678</math></td>
  <td align="right"><math>~0.34120</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2.75</math></td>
  <td align="right"><math>~0.23468</math></td>
  <td align="right"><math>~0.31475</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~2.8085</math></td>
  <td align="right"><math>~0.21617</math></td>
  <td align="right"><math>~0.30788</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~3</math></td>
  <td align="right"><math>~0.15972</math></td>
  <td align="right"><math>~0.28442</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~3.00</math></td>
  <td align="center"><math>~1.588576\mathrm{E-01}</math></td>
  <td align="center"><math>~2.842527\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~3.25</math></td>
  <td align="right"><math>~0.09258</math></td>
  <td align="right"><math>~0.25261</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~3.50</math></td>
  <td align="right"><math>~0.03335</math></td>
  <td align="right"><math>~0.22147</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~3.60</math></td>
  <td align="center"><math>~1.109099\mathrm{E-02}</math></td>
  <td align="center"><math>~2.093927\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~3.625</math></td>
  <td align="right"><math>~0.00659</math></td>
  <td align="right"><math>~0.20680</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~3.64</math></td>
  <td align="right"><math>~0.00350</math></td>
  <td align="right"><math>~0.20511</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~</math></td>
  <td align="center"><math>~</math></td>
  <td align="center"><math>~</math></td>
</tr>
<tr>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="right"><math>~</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~3.65</math></td>
  <td align="center"><math>~7.639242\mathrm{E-04}</math></td>
  <td align="center"><math>~2.037196\mathrm{E-01}</math></td>
</tr>
<tr>
  <td align="right"><math>~3.6571</math></td>
  <td align="right"><math>~0</math></td>
  <td align="right"><math>~0.20316</math></td>
  <td align="center">&nbsp;</td>
  <td align="right"><math>~3.65375374</math></td>
  <td align="center"><math>~0</math></td>
  <td align="center"><math>~2.033013\mathrm{E-01}</math></td>
</tr>
</table>

... END OF ORIGINAL PRESENTATION OF TABULATED DATA -->

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