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==Models Examined by Ou (2006)== In §2 of {{ Ou2006 }}, immediately after equation (6), we find the following declaration: In ''direct'' configurations, ω > λ so the fluid motion is dominated by figure rotation; conversely, in an ''adjoint'' configuration, ω < λ so the fluid motion is dominated by internal motions. ===His Tabulated Model Parameters=== Table 1 (see below) lists a subset of the Riemann S-type ellipsoids that were studied by {{ Ou2006 }}; properties of various so-called ''Direct'' configurations can be found in Ou's Table 1, while properties of various ''Adjoint'' configurations can be found in his Table 5. Each row of ''our'' Table 1 was constructed as follows: <ul> <li>The pair of axis ratios <math>~(\tfrac{b}{a}, \tfrac{c}{a} )</math> associated with one of Ou's (2006) uniform-density, incompressible <math>~(n=0)</math> ellipsoid models (columns 1 and 2 from Ou's Table 1) has been copied into columns 1 and 2 of ''our'' table.</li> <li>Properties of ''Direct Configurations'' …</li> <ul> <li>The pair of parameter values <math>~(\omega_\mathrm{analytic}, \lambda_\mathrm{analytic})</math> that is required in order for this to be an <b>equilibrium</b> configuration — as specified by the above set of analytical expressions from EFE — is copied from, respectively, columns 11 and 13 of Ou's Table 1 into columns 3 and 4 of ''our'' table; in our table, the "analytic" subscript has been dropped from the column headings.</li> <li>The value of the equilibrium configuration's vorticity, <math>~\zeta</math> — see column 5 of our table — has been determined from the expression,<br /><table border="0" align="center"><tr><td align="center"><math>~\zeta = - \biggl[ \frac{1 + (b/a)^2}{b/a} \biggr] \lambda \, .</math></td></tr></table></li> <li>Column 6 of our table lists the value of the frequency ratio, <math>~f \equiv \zeta/\omega</math>. </ul> <li>Properties of ''Adjoint Configurations'' [in order to distinguish from ''Direct'' configuration properties, a superscript † has been attached to each parameter name] …</li> <ul> <li>As listed in column 7 of our Table, the "spin" angular velocity of the ''adjoint'' equilibrium configuration has been determined from the vorticity of the ''direct'' configuration via the relation, <br /><table border="0" align="center"><tr><td align="center"><math>~\omega^\dagger = \zeta \biggl[\frac{b/a}{1 + (b/a)^2}\biggr] \, .</math></td></tr></table></li> <li>As listed in column 10 of our Table, the ratio <math>~(f^\dagger)</math> of the vorticity to the angular velocity in the ''adjoint'' equilibrium configuration has been determined from the same ratio <math>~(f)</math> in the ''direct'' configuration via the relation, <br /><table border="0" align="center"><tr><td align="center"><math>~f^\dagger = \frac{1}{f} \biggl\{ \frac{[1 + (b/a)^2]^2}{(b/a)^2} \biggr\} \, .</math></td></tr></table></li> <li>As indicated, the value of the vorticity in the ''adjoint'' equilibrium configuration (column 9 of our table) has been determined from a product of <math>~\omega^\dagger</math> and <math>~f^\dagger</math>.</li> <li>As listed in column 8 of our table, the value of the parameter, <math>~\lambda^\dagger</math>, has been determined from the vorticity in the ''adjoint'' equilibrium configuration via the relation, <br /><table border="0" align="center"><tr><td align="center"><math>~\lambda^\dagger = -~ \zeta^\dagger \biggl[ \frac{b}{a} + \frac{a}{b}\biggr]^{-1} \, .</math></td></tr></table></li> </ul> </ul> <table border="1" align="center" cellpadding="8" width="90%"> <tr> <td align="center" colspan="10"> <b>Table 1: Example Riemann S-type Ellipsoids</b><br /> [Cells with a pink background contain numbers copied directly from Table 1 of {{ Ou2006 }}]<br /> [Cells with a yellow background contain numbers drawn from Table IV (p. 103) of EFE] </td> </tr> <tr> <td align="center" rowspan="2"><math>~\frac{b}{a}</math></td> <td align="center" rowspan="2"><math>~\frac{c}{a}</math></td> <td align="center" rowspan="1" colspan="4"> Properties of<br /><b>''Direct'' Configurations</b> </td> <td align="center" rowspan="1" colspan="4"> Properties of<br /><b>''Adjoint'' Configurations</b> </td> </tr> <tr> <td align="center" rowspan="1"><math>~\omega = \frac{\Omega}{\sqrt{G \rho}}</math></td> <td align="center" rowspan="1"><math>~\lambda</math></td> <td align="center" rowspan="1"><math>~\zeta </math></td> <td align="center" rowspan="1"><math>~f \equiv \frac{\zeta}{\omega}</math></td> <td align="center" rowspan="1"><math>~\omega^\dagger </math></td> <td align="center" rowspan="1"><math>~\lambda^\dagger </math></td> <td align="center" rowspan="1"><math>~\zeta^\dagger = \omega^\dagger f^\dagger</math></td> <td align="center" rowspan="1"><math>~f^\dagger </math></td> </tr> <tr> <td align="center">(1)</td> <td align="center">(2)</td> <td align="center">(3)</td> <td align="center">(4)</td> <td align="center">(5)</td> <td align="center">(6)</td> <td align="center">(7)</td> <td align="center">(8)</td> <td align="center">(9)</td> <td align="center">(10)</td> </tr> <tr> <td align="center" rowspan="7" bgcolor="white">0.90</td> <td align="center" bgcolor="pink">0.795</td> <td align="center" bgcolor="pink">1.14704</td> <td align="center" bgcolor="pink">0.43181</td> <td align="center">-0.86842</td> <td align="center">-0.75709</td> <td align="center">-0.43181</td> <td align="center">-1.14704</td> <td align="center">+2.30682</td> <td align="center">-5.3422</td> </tr> <tr> <td align="center" bgcolor="pink">0.641</td> <td align="center" bgcolor="pink">1.13137</td> <td align="center" bgcolor="pink">0.15077</td> <td align="center"> - 0.30322</td> <td align="center">- 0.26801</td> <td align="center">- 0.15077</td> <td align="center">-1.13137</td> <td align="center">2.27531</td> <td align="center">- 15.0913</td> </tr> <tr> <td align="center" bgcolor="pink">0.590</td> <td align="center" bgcolor="pink">1.10661</td> <td align="center" bgcolor="pink">0.06406</td> <td align="center">-0.12883</td> <td align="center">-0.11642</td> <td align="center">-0.06406</td> <td align="center">-1.10661</td> <td align="center">+2.22552</td> <td align="center">-34.7411</td> </tr> <tr> <td align="center" bgcolor="pink">0.564</td> <td align="center" bgcolor="pink">1.09034</td> <td align="center" bgcolor="pink">0.02033</td> <td align="center">-0.04089</td> <td align="center">-0.03750</td> <td align="center">-0.02033</td> <td align="center">-1.09034</td> <td align="center">+2.19279</td> <td align="center">-107.86</td> </tr> <tr> <td align="center" bgcolor="pink">0.538</td> <td align="center" bgcolor="pink">1.07148</td> <td align="center" bgcolor="pink">- 0.02324</td> <td align="center">+0.04674</td> <td align="center">+0.04362</td> <td align="center">+0.02324</td> <td align="center">- 1.07148</td> <td align="center">+2.15487</td> <td align="center">+92.722</td> </tr> <tr> <td align="center" bgcolor="pink">0.487</td> <td align="center" bgcolor="pink">1.02639</td> <td align="center" bgcolor="pink">- 0.10880</td> <td align="center">+0.21881</td> <td align="center">+0.21318</td> <td align="center">+0.10880</td> <td align="center">-1.02639</td> <td align="center">+2.06418</td> <td align="center">+18.972</td> </tr> <tr> <td align="center" bgcolor="pink">0.333</td> <td align="center" bgcolor="pink">0.79257</td> <td align="center" bgcolor="pink">- 0.39224</td> <td align="center">+0.78884</td> <td align="center">+0.99529</td> <td align="center">+0.39224</td> <td align="center">-0.79257</td> <td align="center">+1.59395</td> <td align="center">+4.06370</td> </tr> <tr> <td align="center" rowspan="4" bgcolor="white">0.28</td> <td align="center" bgcolor="pink">0.256</td> <td align="center" bgcolor="pink">0.80944</td> <td align="center" bgcolor="pink">0.03668</td> <td align="center">-0.14127</td> <td align="center">-0.17453</td> <td align="center">-0.03668</td> <td align="center">-0.80944</td> <td align="center">+3.11750</td> <td align="center">-84.992</td> </tr> <tr> <td align="center" bgcolor="yellow">0.245083</td> <td align="center" bgcolor="yellow">0.796512<sup>a</sup></td> <td align="center" bgcolor="yellow">0.0</td> <td align="center">0.0</td> <td align="center">0.0</td> <td align="center">0.0</td> <td align="center">…</td> <td align="center">…</td> <td align="center"><math>~\infty</math></td> </tr> <tr> <td align="center" bgcolor="pink">0.231</td> <td align="center" bgcolor="pink">0.77651</td> <td align="center" bgcolor="pink">- 0.04714</td> <td align="center">+0.18156</td> <td align="center">+0.23381</td> <td align="center">+0.04714</td> <td align="center">-0.77651</td> <td align="center">+2.99067</td> <td align="center">+63.442</td> </tr> <tr> <td align="center" bgcolor="pink">0.205</td> <td align="center" bgcolor="pink">0.72853</td> <td align="center" bgcolor="pink">- 0.13511</td> <td align="center">+0.52037</td> <td align="center">+0.71427</td> <td align="center">+0.13511</td> <td align="center">-0.72853</td> <td align="center">+2.80588</td> <td align="center">+20.7674</td> </tr> <tr> <td align="left" colspan="10"> <sup>a</sup>According to Table IV (p. 103) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], the square of the angular velocity of this Jacobi ellipsoid is, <math>~\Omega^2/(\pi G\rho) = 0.201946</math>; from this value, we find that, <math>~\omega = \sqrt{\pi} \cdot \sqrt{0.201946} = 0.796512</math>. </td> </tr> </table> ===Our Parameter Determinations=== The parameter values that have been posted above in our Table 1 are typically given with five digits of precision. This is because, as explained, the values were determined from the ''analytically determined'' values, <math>~\omega_\mathrm{analytic}</math> and <math>~\lambda_\mathrm{analytic}</math>, that were provided by {{ Ou2006 }} with only five digit accuracy. Our Table 2 (shown immediately below) provides values of this same set of model parameters to better than eleven digits accuracy. We calculated these parameter values by following the steps detailed in earlier subsections of this chapter and, as a foundation, using double-precision versions of ''Numerical Recipes'' algorithms to evaluate the special functions, <math>~F(\phi,k)</math> and <math>~E(\phi,k)</math>. As an example, the above pair of brief tables titled, ''TEST (part 1)'' and ''TEST (part 2)'' detail all of the intermediate steps that were used in order to determine the high-precision parameter values specifically for the model having the axis-ratio pair <math>~(0.9,0.641)</math>. This table of higher precision parameter values was primarily generated in order to convince ourselves that we understood from first principles how to accurately determine the properties of Riemann S-type ellipsoids; the lower-precision parameter values that we derived from Ou's work provided a handy means of cross-checking these "first principles" determinations. <span id="Table2">In generating our Table 2, we wondered what the approriate ''signs'' were of the various model parameters — especially when part of our objective is to distinguish between ''direct'' and ''adjunct'' configurations. We took the following approach: First we decided that the spin frequency of every ''direct'' configuration should be positive. (Evidently, Ou made this same choice.)</span> <table border="1" align="center" cellpadding="8" width="90%"> <tr> <td align="center" colspan="10"> <b>Table 2: Example Riemann S-type Ellipsoids</b> (double-precision evaluation) </td> </tr> <tr> <td align="center" rowspan="2"><math>~\frac{b}{a}</math></td> <td align="center" rowspan="2"><math>~\frac{c}{a}</math></td> <td align="center" rowspan="1" colspan="4"> Properties of<br /><b>''Direct'' Configurations</b> </td> <td align="center" rowspan="1" colspan="4"> Properties of<br /><b>''Adjoint'' Configurations</b> </td> </tr> <tr> <td align="center" rowspan="1"><math>~\omega = \frac{\Omega}{\sqrt{G \rho}}</math></td> <td align="center" rowspan="1"><math>~\lambda</math></td> <td align="center" rowspan="1"><math>~\zeta </math></td> <td align="center" rowspan="1"><math>~f \equiv \frac{\zeta}{\Omega}</math></td> <td align="center" rowspan="1"><math>~\omega^\dagger </math></td> <td align="center" rowspan="1"><math>~\lambda^\dagger </math></td> <td align="center" rowspan="1"><math>~\zeta^\dagger = \omega^\dagger f^\dagger</math></td> <td align="center" rowspan="1"><math>~f^\dagger </math></td> </tr> <tr> <td align="center">(1)</td> <td align="center">(2)</td> <td align="center">(3)</td> <td align="center">(4)</td> <td align="center">(5)</td> <td align="center">(6)</td> <td align="center">(7)</td> <td align="center">(8)</td> <td align="center">(9)</td> <td align="center">(10)</td> </tr> <tr> <td align="center" rowspan="7" bgcolor="white">0.90</td> <td align="center" bgcolor="pink">0.795</td> <td align="center" bgcolor="white">+1.147036091720</td> <td align="center" bgcolor="white">+0.431809451699</td> <td align="center">-0.868416786194</td> <td align="center">-0.757096320116</td> <td align="center">-0.431809460593</td> <td align="center">-1.147036104571</td> <td align="center">+2.306817054749</td> <td align="center">-5.342210487323</td> </tr> <tr> <td align="center" bgcolor="pink">0.641</td> <td align="center" bgcolor="white">+1.131374738327</td> <td align="center" bgcolor="white">+0.150771621841</td> <td align="center"> -0.303218483925</td> <td align="center">-0.268008886644</td> <td align="center">-0.150771621877</td> <td align="center">-1.131374730590</td> <td align="center">+2.275320291519</td> <td align="center">-15.091170863305</td> </tr> <tr> <td align="center" bgcolor="pink">0.590</td> <td align="center" bgcolor="white">+1.106612583610</td> <td align="center" bgcolor="white">+0.064060198174</td> <td align="center">-0.128832176328</td> <td align="center">-0.116420305902</td> <td align="center">-0.064060197762</td> <td align="center">-1.106612576964</td> <td align="center">+2.225520849228</td> <td align="center">-34.741086358509</td> </tr> <tr> <td align="center" bgcolor="pink">0.564</td> <td align="center" bgcolor="white">+1.090339840378</td> <td align="center" bgcolor="white">+0.020334563779</td> <td align="center">-0.040895067155</td> <td align="center">-0.037506716440</td> <td align="center">-0.020334563809</td> <td align="center">-1.090339837153</td> <td align="center">+2.192794561386</td> <td align="center">-107.8358300897</td> </tr> <tr> <td align="center" bgcolor="pink">0.538</td> <td align="center" bgcolor="white">+1.071485625744</td> <td align="center" bgcolor="white">-0.023236834336</td> <td align="center">+0.046731855720</td> <td align="center">+0.043614077664</td> <td align="center">+0.023236835120</td> <td align="center">-1.071485656401</td> <td align="center">+2.154876708984</td> <td align="center">+92.735376233270</td> </tr> <tr> <td align="center" bgcolor="pink">0.487</td> <td align="center" bgcolor="white">+1.026387311947</td> <td align="center" bgcolor="white">-0.108799837242</td> <td align="center">+0.218808561563</td> <td align="center">+0.213183225210</td> <td align="center">+0.108799835209</td> <td align="center">-1.026387320039</td> <td align="center">+2.064178943634</td> <td align="center">+18.972261524065</td> </tr> <tr> <td align="center" bgcolor="pink">0.333</td> <td align="center" bgcolor="lightgreen">+0.792566980901</td> <td align="center" bgcolor="lightgreen">-0.392440787995</td> <td align="center">+0.789242029190</td> <td align="center">+0.995804846843</td> <td align="center">+0.392440793882</td> <td align="center">-0.792566979129</td> <td align="center">+1.593940258026</td> <td align="center">+4.061606964516</td> </tr> <tr> <td align="center" rowspan="1" bgcolor="white">0.74</td> <td align="center" bgcolor="pink">0.692</td> <td align="center" bgcolor="lightgreen">+1.132148956838</td> <td align="center" bgcolor="lightgreen">+0.385991660900</td> <td align="center">-0.807244181633</td> <td align="center">-0.713019398562</td> <td align="center">-0.385991654519</td> <td align="center">-1.132148989537</td> <td align="center">2.367721319199</td> <td align="center">-6.134125500116</td> </tr> <tr> <td align="center" rowspan="2" bgcolor="white">0.41</td> <td align="center" bgcolor="pink">0.385</td> <td align="center" bgcolor="lightgreen">+0.971082162758</td> <td align="center" bgcolor="lightgreen">+0.141593941719</td> <td align="center">-0.403404593468</td> <td align="center">-0.415417564427</td> <td align="center">-0.141593939418</td> <td align="center">-0.971082191477</td> <td align="center">2.766636848450</td> <td align="center">-19.539231537777</td> </tr> <tr> <td align="center" bgcolor="pink">0.333</td> <td align="center" bgcolor="white">+0.929630138695</td> <td align="center" bgcolor="white">+0.003311666790</td> <td align="center">-0.009435019456</td> <td align="center">-0.010149218281</td> <td align="center">-0.003311666699</td> <td align="center">-0.929630099681</td> <td align="center">+2.648538827896</td> <td align="center">-799.7601146950</td> </tr> <tr> <td align="center" rowspan="4" bgcolor="white">0.28</td> <td align="center" bgcolor="pink">0.256</td> <td align="center" bgcolor="lightgreen">+0.809436834686</td> <td align="center" bgcolor="lightgreen">+0.036676037913</td> <td align="center">-0.141255140305</td> <td align="center">-0.174510396110</td> <td align="center">-0.036676038521</td> <td align="center">-0.809436833116</td> <td align="center">+3.117488145828</td> <td align="center">-85.000678306244</td> </tr> <tr> <td align="center" bgcolor="yellow">0.245083</td> <td align="center" bgcolor="white">0.796512<sup>a</sup></td> <td align="center" bgcolor="white">0.0</td> <td align="center">0.0</td> <td align="center">0.0</td> <td align="center">0.0</td> <td align="center">…</td> <td align="center">…</td> <td align="center"><math>~\infty</math></td> </tr> <tr> <td align="center" bgcolor="pink">0.231</td> <td align="center" bgcolor="white">+0.776514825339</td> <td align="center" bgcolor="white">-0.047142035397</td> <td align="center">+0.181564182043</td> <td align="center">+0.233819345828</td> <td align="center">+0.047142037070</td> <td align="center">-0.776514835457</td> <td align="center">+2.990691423416</td> <td align="center">+63.440011724689</td> </tr> <tr> <td align="center" bgcolor="pink">0.205</td> <td align="center" bgcolor="white">+0.728526018042</td> <td align="center" bgcolor="white">-0.135108121071</td> <td align="center">+0.520359277725</td> <td align="center">+0.714263156392</td> <td align="center">+0.135108125079</td> <td align="center">-0.728526039364</td> <td align="center">+2.805866003036</td> <td align="center">+20.767558718483</td> </tr> <tr> <td align="left" colspan="10"> <sup>a</sup>According to Table IV (p. 103) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], the square of the angular velocity of this Jacobi ellipsoid is, <math>~\Omega^2/(\pi G\rho) = 0.201946</math>; from this value, we find that, <math>~\omega = \sqrt{\pi} \cdot \sqrt{0.201946} = 0.796512</math>. </td> </tr> </table> <span id="Fig2"> </span> <table border="1" cellpadding="5" width="90%" align="center"> <tr><td align="center" colspan="1">'''Figure 2: EFE Diagram'''</td> <td align="left" rowspan="2"> In the context of our broad discussion of ellipsoidal figures of equilibrium, the label "EFE Diagram" refers to a two-dimensional parameter space defined by the pair of axis ratios (b/a, c/a), ''usually'' covering the ranges, 0 ≤ b/a ≤ 1 and 0 ≤ c/a ≤ 1. The classic/original version of this diagram appears as Figure 2 on p. 902 of {{ Chandrasekhar65_XXV }}; a somewhat less cluttered version appears on p. 147 of Chandrasekhar's [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]. The version of the EFE Diagram shown here, on the left, highlights four model ''sequences'', all of which also can be found in the original version: <ul> <li>''Jacobi'' sequence — the smooth curve that runs through the set of small, dark-blue, diamond-shaped markers; the data identifying the location of these markers have been drawn from §39, Table IV of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]. The small red circular markers lie along this same sequence; their locations are taken from our own determinations, as detailed in [[ThreeDimensionalConfigurations/JacobiEllipsoids#Table2|Table 2]] of our accompanying discussion of Jacobi ellipsoids. All of the models along this sequence have <math>~f \equiv \zeta/\Omega_f = 0</math> and are therefore solid-body rotators, that is, there is no internal motion when the configuration is viewed from a frame that is rotating with frequency, <math>~\Omega_f</math>.</li> <li>''Dedekind'' sequence — a smooth curve that lies precisely on top of the ''Jacobi'' sequence. Each configuration along this sequence is ''adjoint'' to a model on the ''Jacobi'' sequence that shares its (b/a, c/a) axis-ratio pair. All ellipsoidal figures along this sequence have <math>~1/f = \Omega_f/\zeta = 0</math> and are therefore stationary as viewed from the ''inertial'' frame; the angular momentum of each configuration is stored in its internal motion (vorticity).</li> <li>The X = -1 ''self-adjoint'' sequence — At every point along this sequence, the value of the key frequency ratio, <math>~\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>~(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration <math>~(f_-)</math>; specifically, <math>~f_+ = f_- = -(a^2+b^2)/(ab)</math>. The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].</li> <li>The X = +1 ''self-adjoint'' sequence — At every point along this sequence, the value of the key frequency ratio, <math>~\zeta/\Omega_f</math>, in the ''adjoint'' configuration <math>~(f_+)</math> is identical to the value of the frequency ratio in the ''direct'' configuration <math>~(f_-)</math>; specifically, <math>~f_+ = f_- = +(a^2+b^2)/(ab)</math>. The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>].</li> </ul> Riemann S-type ellipsoids all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram. The yellow circular markers in the diagram shown here, on the left, identify four Riemann S-type ellipsoids that were examined by [https://ui.adsabs.harvard.edu/abs/2006ApJ...639..549O/abstract Ou (2006)] and that we have also chosen to use as examples. </td> </tr> <tr> <td align="left"> [[File:EFEdiagram4.png|left|500px|EFE Diagram identifying example models from Ou (2006)]] </td> </tr> </table> <div id="SAdata"> <table border="1" align="center" cellpadding="10"> <tr> <td align="center">Self-Adjoint Sequences<br /><b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, Β§48, p. 142, Table VI<br />Also, Howard Cohl's [[Appendix/Ramblings/ForCohlHoward#HowardHighResolution|High-Resolution Analysis]]</td> </tr> <tr><td align="center"> <pre> c/a for Upper (x = - 1) Sequence c/a for Lower (x = + 1) Sequence b/a (EFE) (20-digit accuracy) (EFE) (20-digit accuracy) 0.00 0.00000 0.00000000000000000000 0.00000 0.000000000000000000000 0.04 --- 0.04812224688754297818 --- 0.032492366749482199925 0.08 0.10361 0.10361306225462674898 0.057817 0.057816723024001224133 0.12 0.16154 0.16153789497880627606 0.079299 0.079298698625164323530 0.16 0.21970 0.21969876513711574739 0.098178 0.098178582821665570336 0.20 0.27691 0.27691183106981782343 0.11513 0.11512631936916250684 0.24 0.33250 0.33250375904983621093 0.13056 0.13056017471495792387 0.28 0.38609 0.38608942382754657558 0.14476 0.14476444864794971656 0.32 0.43746 0.43745820820586512706 0.15794 0.15794374720008968932 0.36 0.48651 0.48651009591845463512 0.17025 0.17025155740243264509 0.40 0.53322 0.53321717007090044978 0.18181 0.18180672031800805031 0.44 0.57760 0.57759916496531807675 0.19270 0.19270358265482766408 0.48 0.61971 0.61970731624526462073 0.20302 0.20301858308989486955 0.52 0.65961 0.65961338021128620371 0.21281 0.21281470455368433422 0.56 0.69740 0.69740201597965416041 0.22214 0.22214458680677346510 0.60 0.73316 0.73316543250962834935 0.23105 0.23105276438034978438 0.64 0.76700 0.76699960292399202953 0.23958 0.23957731445406474795 0.68 0.79900 0.79900158609578752317 0.24775 0.24775109537636838357 0.72 0.82927 0.82926764261302051281 0.25560 0.25560269426540122707 0.76 0.85789 0.85789192689326638880 0.26316 0.26315716347079152387 0.80 0.88497 0.88496560014936540261 0.27044 0.27043660093734799732 0.84 0.91058 0.91057625192549140531 0.27746 0.27746061325324716171 0.88 0.93481 0.93480754802412975193 0.28425 0.28424668922671777898 0.92 0.95774 0.95773904411679071746 0.29081 0.29081050432029451542 0.96 0.97945 0.97944611989047804643 0.29714 0.29716617101118183175 1.00 1.00000 1.00000000000000000000 0.30333 0.30332644640104007578 </pre> </td></tr></table> </div>
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