Joel2: /* Tabulated Global Properties */

2024-07-18T17:12:48Z

Tabulated Global Properties

← Older revision		Revision as of 13:12, 18 July 2024
Line 242:		Line 242:
	====Tabulated Global Properties====		====Tabulated Global Properties====

	Here, drawing from tables that have been previously published by other authors, we record numerically determined properties of polytropic models having a fairly wide range of polytropic indexes. First, we draw from Table 4 (p.96) of [[Appendix/References#C67\|[<b><font color="red">C67</font></b>] ]]. To convert from his tabulated variables to our desired [[SSCpt1/Virial/FormFactors#PTtable\|3 structural form-factors]], our normalized equilibrium radius (see [[#~~Lane-Emden_Equation~~\|~~above~~ ASIDE]]), and the "virial" (drawn from a [[SSC/FreeEnergy/PowerPoint#Pressure-Truncated_Polytropes\|more general overview]]), note that for ''isolated'' polytropes,		Here, drawing from tables that have been previously published by other authors, we record numerically determined properties of polytropic models having a fairly wide range of polytropic indexes. First, we draw from Table 4 (p.96) of [[Appendix/References#C67\|[<b><font color="red">C67</font></b>] ]]. To convert from his tabulated variables to our desired [[SSCpt1/Virial/FormFactors#PTtable\|3 structural form-factors]], our normalized equilibrium radius (see [[SSC/Structure/Polytropes#MassRadiusRelation\|earlier ASIDE]]), and the "virial" (drawn from a [[SSC/FreeEnergy/PowerPoint#Pressure-Truncated_Polytropes\|more general overview]]), note that for ''isolated'' polytropes,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Joel2: Created page with "==Numerical Solutions== Here we explain how an Excel workbook can be used to numerically solve the Lane-Emden equation, evaluating the Lane-Emden function across a one-dimensional, discrete grid. ===Techniques=== ====HSCF Technique==== {| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:white;" | H_BookTiledMenu#Equilibrium_St..."

2023-12-18T17:33:32Z

Created page with "==Numerical Solutions== Here we explain how an Excel workbook can be used to numerically solve the Lane-Emden equation, evaluating the Lane-Emden function across a one-dimensional, discrete grid. ===Techniques=== ====HSCF Technique==== {| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:white;" | <font size="-1">H_BookTiledMenu#Equilibrium_St..."

New page

==Numerical Solutions==
Here we explain how an Excel workbook can be used to numerically solve the Lane-Emden equation, evaluating the Lane-Emden function across a one-dimensional, discrete grid.

===Techniques===
====HSCF Technique====
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>via<br />Self-Consistent<br />Field (SCF)<br />Technique</b>]]</font>
|}
On the first spreadsheet within the workbook, we establish the following columns of number:
<ul>
<li>'''Column A:'''   Labeled <math>~r_i/R</math> (for ''i'' between 1 and N), that represents a discrete radial grid of spacing, <math>~~\Delta = (N-1)^{-1}</math>; each row gives the radial coordinate location of the ''i''<sup>th</sup> zone, starting from <math>~r_1/R = 0</math> and ending at <math>~r_N/R = 1</math>.</li>
<li>'''Column B:'''   Labeled <math>~\mathrm{rhf}_i</math> (for ''i'' between 1 and N-1); each row gives the radial coordinate of the mid-point of a grid zone.</li>
<div align="center">
<math>~\mathrm{rhf}_i \equiv \frac{1}{2}\biggr[\frac{r_i}{R} + \frac{r_{i+1}}{R} \biggr] \, .</math>
</div>
<li>'''Column C:'''   Labeled <math>~\rho_i</math> (for ''i'' between 1 and N-1); each row provides an initial ''guess'' for the mass-density of the grid zone. Usually it is sufficient to guess, <math>~\rho_i = 1</math> throughout. For an <math>~n=0</math> polytrope, this proves also to be the correct ''final'' density profile.</li>
<li>'''Column D:'''   Labeled <math>~M_i</math> (for ''i'' between 1 and N); the ''i''<sup>th</sup> row gives the integrated mass enclosed interior to the radial grid coordinate, <math>~r_i/R</math>. Specifically, <math>~M_1 = 0</math>, and thereafter, beginning with zone, <math>~i = 2</math>,</li>
<div align="center">
<math>~M_i = M_{i-1} + \frac{4\pi \rho_{i-1}}{3}\biggl[ \biggl(\frac{r_{i}}{R} \biggr)^3 - \biggl(\frac{r_{i-1}}{R} \biggr)^3 \biggr] \, .</math>
</div>
<li>Note that, <math>~M_\mathrm{tot} = M_N \, .</math></li>
<li>'''Column E:'''   Labeled <math>~g_i</math> (for ''i'' between 2 and N); each row tabulates the inwardly directed gravitational acceleration that is felt at the outer edge of each grid zone. Specifically,</li>
<div align="center">
<math>g_i = \frac{GM_i}{ (r_i/R)^2} \, .</math>
</div>
<li>'''Column F:'''   Labeled <math>~\Phi_i</math> (for ''i'' between 1 and N); each row gives the value of the gravitational potential at the mid-point of a grid zone. Here, we start by specifying the ''value'' of the potential just (specifically, half a radial grid-zone) outside the surface of the configuration, where it should be, <math>~\Phi_N = -GM_\mathrm{tot}/(1+\Delta/2)</math>. Then, working from the surface, inward and, given that, <math>~g = d\Phi/dr</math>, we use the corresponding finite-difference representation of the radial derivative and set,</li>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{\Phi_i - \Phi_{i-1}}{\Delta} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~g_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \Phi_{i-1} </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \Phi_i - g_i \Delta \, .</math>
</td>
</tr>
</table>
</div>
<li>Note that the value of the gravitational potential ''at'' the surface is not <math>~\Phi_N</math> but, rather, must be <math>~\Phi_\mathrm{surf} = -GM_\mathrm{tot}/R</math>.</li>
<li>Furthermore, note that a lop-sided Taylor-series expansion about the center of the configuration provides the following good approximation to the gravitational potential ''at'' the center:   <math>~\Phi_c \approx (9\Phi_1 - \Phi_2)/8</math>.</li>
<li>Note as well that all of these numerically determined values of the gravitational potential can be checked against the [[SSC/Structure/UniformDensity#UniformSpherePotential|known analytic expression]] for the radial profile of the potential in a uniform-density sphere.</li>
<li>'''Column G:'''   Labeled <math>~H_i</math> (for ''i'' between 1 and N-1); each row provides the value of the fluid enthalpy at the center of a grid cell. Adopting the convention that the enthalpy is zero at the surface of the configuration, and given that [[SSCpt2/SolutionStrategies#Technique_3|the enthalpy and the gravitational potential must sum to zero]] throughout the configuration, we have,</li>
<div align="center">
<math>H_i = \Phi_\mathrm{surf} - \Phi_i \, .</math>
</div>
<li>At the center of the configuration, we have, <math>~H_c = \Phi_\mathrm{surf} - \Phi_c</math>. </li>
<li>'''Column H:'''   Labeled <math>~H_\mathrm{norm}</math> (for ''i'' between 1 and N-1); each row provides the value of the fluid enthalpy, renormalized to the central value, specifically,</li>
<div align="center">
<math>~[H_\mathrm{norm}]_i = \frac{H_i}{H_c} \, .</math>
</div>
</ul>

The second spreadsheet within the workbook should be initially created by generating a copy of the first spreadsheet. Then:
<ul>
<li>'''Column C:'''   Labeled <math>~\rho_i</math> (for ''i'' between 1 and N-1); generate a new, improved ''guess'' for the normalized mass-density at each grid zone based on the corresponding value of the normalized enthalpy from the previous spreadsheet/iteration. Specifically, given that the [[SR#Barotropic_Structure|relationship between the density and enthalpy in a polytrope]] of index, <math>~n</math>, is, <math>~\rho \propto H^n</math>, we should set,</li>
<div align="center">
<math>~\biggl\{ \frac{\rho_i}{\rho_c} \biggr\}_\mathrm{sheet2}= \biggr\{[H_\mathrm{norm}]_i^n \biggr\}_\mathrm{sheet1} \, .</math>
</div>
</ul>

====Straight Numerical Integration====
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>via<br />Direct<br />Numerical<br />Integration</b>]]</font>
|}
The [[#LaneEmdenEquation|above governing relation]] may be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~
\xi \theta^{''} + 2 \theta^'
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \xi \theta^n \, .</math>
</td>
</tr>
</table>
</div>

We'll adopt the following finite-difference approximations for the first and second derivatives on a grid of radial spacing, <math>~\Delta_\xi</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta_i'</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\frac{\theta_+ - \theta_-}{2\Delta_\xi}</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta_i''</math>
</td>
<td align="center">
<math>~\approx</math>
</td>
<td align="left">
<math>~\frac{\theta_+ - 2\theta_i +\theta_-}{\Delta_\xi^2} \, .</math>
</td>
</tr>
</table>
</div>
Our finite-difference approximation of the governing equation is, then,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\xi_i \biggl[ \frac{\theta_+ - 2\theta_i +\theta_-}{\Delta_\xi^2} \biggr] + 2\biggl[ \frac{\theta_+ - \theta_-}{2\Delta_\xi}\biggr]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \xi_i \theta_i^n </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~ \xi_i [ \theta_+ - 2\theta_i +\theta_-] + \Delta_\xi [ \theta_+ - \theta_- ]
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~- \Delta_\xi^2\xi_i \theta_i^n </math>
</td>
</tr>

<tr>
<td align="right">
<math>~\Rightarrow ~~~
\theta_+
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{2\xi_i \theta_i + \theta_-(\Delta_\xi - \xi_i) - \Delta_\xi^2\xi_i \theta_i^n }{\Delta_\xi + \xi_i} \, .</math>
</td>
</tr>
</table>
</div>

Now, for the first two steps away from the center — where, <math>~\theta_i = \theta_0 = 1</math> and <math>~\xi_i = \xi_0 = 0</math> — we will use the following [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicLaneEmden|power-series expansion]] (see, for example, eq. 62 from §5 in Chapter IV of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>]) to determine the value of <math>~\theta_i</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta_1
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>1 - \frac{\Delta_\xi^2}{6} + \frac{n \Delta_\xi^4}{120} - \frac{n}{378} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) \Delta_\xi^6 \, ,</math>
</td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\theta_2
</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>1 - \frac{(2\Delta_\xi)^2}{6} + \frac{n (2\Delta_\xi)^4}{120} - \frac{n}{378} \biggl( \frac{n}{5} - \frac{1}{8} \biggr) (2\Delta_\xi)^6 \, .</math>
</td>
</tr>
</table>
</div>

===Results===
====Tabulated Global Properties====

Here, drawing from tables that have been previously published by other authors, we record numerically determined properties of polytropic models having a fairly wide range of polytropic indexes. First, we draw from Table 4 (p.96) of [[Appendix/References#C67|[<b><font color="red">C67</font></b>] ]]. To convert from his tabulated variables to our desired [[SSCpt1/Virial/FormFactors#PTtable|3 structural form-factors]], our normalized equilibrium radius (see [[#Lane-Emden_Equation|above ASIDE]]), and the "virial" (drawn from a [[SSC/FreeEnergy/PowerPoint#Pressure-Truncated_Polytropes|more general overview]]), note that for ''isolated'' polytropes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathfrak{f}_\mathrm{M}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl( \frac{\rho_c}{\bar\rho}\biggr)^{-1} = \biggl[ - \frac{3\theta^'}{\xi}\biggr]_{\xi_1} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~(5-n)\mathfrak{f}_\mathrm{W}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~5 \mathfrak{f}_\mathrm{M}^2 \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~(5-n)\mathfrak{f}_\mathrm{A}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[\biggl( \frac{4\pi}{3}\biggr) W_n\biggr]^{-1} = 3(n+1)(-\theta^')^2_{\xi_1} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>~x_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \xi_1 (- \xi^2 \theta^')^{(1-n)/(n-3)}_{\xi_1} \, ,</math>
</td>
</tr>

<tr>
<td align="right">
Virial
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~(5-n)\biggl[ \frac{b}{n} \cdot x_\mathrm{eq}^{(n-3)/n} - \frac{a}{3} \biggr]</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ \biggl(\frac{3}{4\pi} \biggr)^{1/n} \frac{(5-n)\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}} \cdot x_\mathrm{eq}^{(n-3)/n} - 1 \, .</math>
</td>
</tr>
</table>
</div>

<div align="center" id="Chandrasekhar">
<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="1">From Table 4 of [[Appendix/References#C67|[<font color="red">C67</font>] ]]</th></tr>
<tr>
<th align="center">Copied Directly from Table (1<sup>st</sup> 5 columns)     …     Implied Values of 3 Structural Form Factors, <math>x_\mathrm{eq}</math>, and Virial (last 5 columns)</th>
</tr>
<tr>
<td align="left" colspan="1">
<pre>
n xi_1 "mass" rho_c/rho_avg W_n f_M (5-n)f_W (5-n)f_A x_eq Virial

0 2.4494 4.8988 1 0.119366 1 5 2.00000 0.620335 ---
0.5 2.7528 3.7871 1.8361 0.26227 0.544632645 1.483123592 0.910254 0.831089 -0.19009
1 3.14159 3.14159 3.28987 0.392699 0.303963378 0.461968677 0.607927 1.253313 3.0E-06
1.5 3.65375 2.71406 5.99071 0.77014 0.166925122 0.139319982 0.309857 2.357285 2.4E-06
2 4.35287 2.41105 11.40254 1.63818 0.087699758 0.038456238 0.145730 7.517481 1.4E-06
2.5 5.35528 2.18720 23.40646 3.90906 0.042723248 0.00912638 0.061072 186.3666 1.6E-08
3 6.89685 2.01824 54.1825 11.05066 0.018456144 0.001703146 0.0216035 --- ---
3.25 8.01894 1.94980 88.153 20.365 0.011343913 0.000643422 0.0117227 3.3265E-06 2.9E-06
3.5 9.53581 1.89056 152.884 40.9098 0.006540907 0.000213917 0.00583558 0.00166854 2.2E-06
4 14.97155 1.79723 622.408 247.558 0.001606663 1.29068E-05 0.00096435 0.051854 6.1E-06
4.5 31.83646 1.73780 6189.47 4922.125 0.000161565 1.30516E-07 4.8502E-05 0.284868 -5.9E-05
4.9 169.47 1.73205 9.348E+05 3.693E+06 1.06975E-06 5.7218E-12 6.4645D-08 2.129056 1.5E-04
</pre>
</td>
</tr>
<tr>
<td align="left">
The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')_{\xi_1}</math>.
</td>
</tr>
</table>
</div>

<div align="center" id="Horedt2004">
<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="1">From Table 2.5.2 (p. 77) of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)]
  —   "Polytropic Spheres (N = 3)"</th></tr>
<tr>
<th align="center">Copied Directly from Table (1<sup>st</sup> 3 columns)     …     Implied Values (last 7 columns)</th>
</tr>
<tr>
<td align="left" colspan="1">
<pre>
n xi_1 theta' "mass" rho_c/rho_avg W_n f_M (5-n)f_A x_eq Virial
0 2.44948974 -8.164966E-01 4.898980 1.000000 1.193662E-01 1.000000E+00 2.000000E+00 6.2035049E-01
0.5 2.75269805 -4.999971E-01 3.788651 1.835143 2.122091E-01 5.449168E-01 1.124987E+00 8.3099030E-01 0.0E+00
1 3.14159265 -3.183099E-01 3.141593 3.289868 3.926990E-01 3.039636E-01 6.079272E-01 1.2533141E+00 0.0E+00
1.5 3.65375374 -2.033013E-01 2.714055 5.990704 7.701402E-01 1.669253E-01 3.099856E-01 2.3572860E+00 0.0E+00
2 4.35287460 -1.272487E-01 2.411047 1.140254E+01 1.638182E+00 8.769977E-02 1.457301E-01 7.5164793E+00 0.0E+00
2.5 5.35527546 -7.626491E-02 2.187199 2.340646E+01 3.909062E+00 4.272324E-02 6.107153E-02 1.8636634E+02 0.0E+00
3 6.89684862 -4.242976E-02 2.018236 5.418248E+01 1.105068E+01 1.845615E-02 2.160341E-02
3.5 9.53580534 -2.079098E-02 1.890557 1.528837E+02 4.090983E+01 6.540920E-03 5.835575E-03 1.6685566E-03 0.0E+00
4 1.49715463E+01 -8.018079E-03 1.797230 6.224079E+02 2.475594E+02 1.606664E-03 9.643439E-04 5.1854394E-02 0.0E+00
4.5 3.18364632E+01 -1.714549E-03 1.737799 6.189473E+03 4.921842E+03 1.615646E-04 4.850469E-05 2.8486849E-01 0.0E+00
4.99 1.75818915E+03 -5.598955E-07 1.730765 1.046736E+09 4.237887E+10 9.553503E-10 5.633289E-12 2.3460204E+01 2.2E-15</pre>
</td>
</tr>
<tr>
<td align="left">
The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')_{\xi_1}</math>.
</td>
</tr>
</table>
</div>

====Plotted Structural Profiles====
Using the just-described numerical techniques, we have solved the polytropic Lane-Emden equation on a 200-zone, uniform grid for a variety of values of the polytropic index. In each case we have recorded how the dimensionless enthalpy, <math>~\theta_n(\xi)</math>, and its first radial derivative, <math>~\theta_n^'(\xi) \equiv d\theta_n/d\xi</math>, vary with <math>~\xi</math>, from the center of the polytropic configuration to its surface. For the record, these tabulated results reside in the following DropBox files:

* <b>n = 2.5:</b> (10 SCF iterations)   WorkFolder/Wiki edits/HSCF/n25.xlsx
* <b>n = 3:</b> (19 SCF iterations)   WorkFolder/Wiki edits/HSCF/n300.xlsx
* <b>n = 3.005:</b> (15 SCF iterations)   WorkFolder/Wiki edits/HSCF/n3005.xlsx
* <b>n = 3.05:</b> (15 SCF iterations)   WorkFolder/Wiki edits/HSCF/n305.xlsx
* <b>n = 3.5:</b> (18 SCF iterations)   WorkFolder/Wiki edits/HSCF/n25.xlsx
* <b>n = 6:</b> (direct integration)   WorkFolder/Wiki edits/EmbeddedPolytropes/N6.xlsx

For each of these models, as indicated (n = 2.5, 3, 3.005, 3.05, 3.5, 6), [[#Fig4|Figure 4]] illustrates how the normalized mass, <math>~M/M_\mathrm{SWS}</math>, varies with the normalized radius, <math>~R/R_\mathrm{SWS}</math>, where the definition of these two functions,
<div align="center">
<table border="0" cellpadding="3">

<tr>
<td align="right">
<math>
~\frac{M}{M_\mathrm{SWS}}
</math>
</td>
<td align="center">
<math>~\equiv~</math>
</td>
<td align="left">
<math>
\biggl( \frac{n^3}{4\pi} \biggr)^{1/2} \theta_n^{(n-3)/2} \xi^2
\biggl| \frac{d\theta_n}{d\xi} \biggr| \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>
~\frac{R}{R_\mathrm{SWS}}
</math>
</td>
<td align="center">
<math>~\equiv~</math>
</td>
<td align="left">
<math>
\biggl( \frac{n}{4\pi} \biggr)^{1/2} \xi \theta_n^{(n-1)/2} \, ,
</math>
</td>
</tr>
</table>
</div>

has been drawn from an [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|accompanying discussion of pressure-truncated polytropic configurations]]. In four of the Figure 4 panels, we have compared the profile of our numerically determined polytropic function (curve defined by <math>~\sim 200</math> small, black circular markers) to results (7 - 9 larger, blue circular markers) taken from Table 2.5.1 of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)] — see, specifically the segment of his table on pp. 74 - 75 that applies to polytropic spheres — in an effort to demonstrate that our numerically determined solutions are accurate.

<div align="center" id="Fig4">
<table border="1" align="center" cellpadding="8">
<tr>
<th align="center" colspan="3">
Figure 4:   Numerically Determined Solutions to the Polytropic Lane-Emden Equation
</th>
</tr>
<tr>
<td align="center">[[File:n25SequenceA.png|250px|n = 2.5 equilibrium sequence]]</td>
<td align="center">[[File:n30SequenceA.png|250px|n = 3 equilibrium sequence]]</td>
<td align="center">[[File:n3005SequenceA.png|250px|n = 3.005 equilibrium sequence]]</td>
</tr>
<tr>
<td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n25.xlsx --- worksheet = Horedt_n25]] <font size="+2">↲</font></td>
<td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/LinearPerturbation/n300.xlsx --- worksheet = Horedt_n300]] <font size="+2">↲</font></td>
<td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n3005.xlsx --- worksheet = Horedt_n3005]] <font size="+2">↲</font></td>
</tr>
<tr>
<td align="center">[[File:n305SequenceA.png|250px|n = 3.05 equilibrium sequence]]</td>
<td align="center">[[File:n35SequenceA.png|250px|n = 3.5 equilibrium sequence]]</td>
<td align="center">[[File:n600SequenceA.png|250px|n = 6 equilibrium sequence]]</td>
</tr>
<tr>
<td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n305.xlsx --- worksheet = Horedt_n305]] <font size="+2">↲</font></td>
<td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/HSCF/n35.xlsx --- worksheet = Horedt_n35]] <font size="+2">↲</font></td>
<td align="center" colspan="1" align="center">[[File:DataFileButton02.png|75px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/N6.xlsx --- worksheet = PolytropeN6 (2)]] <font size="+2">↲</font></td>
</tr>
<tr>
<td align="left" colspan="3">
Data examples from Table 2.5.1 (pp. 74 - 75) of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)]:
<table border="0" align="center" cellpadding="5">
<tr>
<td align="center"><math>~n</math></td>
<td align="center"><math>~\xi</math></td>
<td align="center"><math>~\theta_n</math></td>
<td align="center"><math>~\frac{d\theta_n}{d\xi}</math></td>
<td align="center"><math>~\frac{R}{R_\mathrm{SWS}}</math></td>
<td align="center"><math>~\frac{M}{M_\mathrm{SWS}}</math></td>
</tr>
<tr>
<td align="center">2.5</td>
<td align="right">4.00000</td>
<td align="right">0.1376807</td>
<td align="left">- 0.1340534</td>
<td align="right">0.4032551</td>
<td align="right">3.926310</td>
</tr>
<tr>
<td align="center">3.0</td>
<td align="right">5.00000</td>
<td align="right">0.1108198</td>
<td align="left">- 0.08012604</td>
<td align="right">0.2707342</td>
<td align="right">2.936234</td>
</tr>
<tr>
<td align="center">3.5</td>
<td align="right">5.00000</td>
<td align="right">0.1786843</td>
<td align="left">- 0.07362030</td>
<td align="right">0.3065541</td>
<td align="right">2.210326</td>
</tr>
<tr>
<td align="center">6.0</td>
<td align="right">5.00000</td>
<td align="right">0.3973243</td>
<td align="left">- 0.05113662</td>
<td align="right">0.3437981</td>
<td align="right">1.327430</td>
</tr>
</table>
</td>
</tr>
</table>
</div>

====Emden's (1907) Tabulated Data====

<div align="center" id="Horedt2004">
<table border="1" cellpadding="5" align="center">
<tr><th align="center" colspan="1">From Table 13 (p. 84) of [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)]
  —   "Global Properties"</th></tr>
<tr>
<td align="left" colspan="1">
<pre>
n xi_1 - theta' 2nd deriv. "mass" rho_c/rho_avg
0 2.4494 0.81647 -0.33333 4.8988 1
0.5 2.7528 0.49975 0.36309 3.7871 1.8361
1 3.14159 0.31831 0.20264 3.14159 3.2899
1.5 3.6571 0.20316 0.11355 2.7176 6.0003
2 4.3518 0.12729 0.06262 2.4107 11.396
2.5 5.4172 0.075 0.02795 2.201 24.076
3 6.9011 0.04231 0.01282 2.015 54.36
4 14.999 0.00803 0.00107 1.8064 623.4
4.5 32.14 0.00168 0.000104 1.7354 6377.7
4.9 169.47 6.04E-05 4.208E-07 1.73554 9.485E+05
5 infinity 0 0 sqrt(3) infinity
</pre>
</td>
</tr>
<tr>
<td align="left">
The column labeled "mass" contains the tabulated quantity, <math>(-\xi^2 \theta^')_{\xi_1}</math>.
</td>
</tr>
</table>
</div>

====Horedt's (1986) Tabulated Data====

[https://ui.adsabs.harvard.edu/abs/1986Ap%26SS.126..357H/abstract G. P. Horedt (1986)], Astrophysics and Space Science, Vol. 126, Issue 2, pp. 357 - 408: ''Seven-digit tables of Lane-Emden functions''

<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%"> </td><td align="left">
<font color="green">
In Table I we present seven digit numerical solutions of the Lane-Emden equation for the plane-parallel (N = 1), cylindrical (N = 2), and spherical (N = 3) case for polytropic indices of <math>~n = -10, -5, -4, -3, -2, -1.5, -1.01, -0.9, -0.5, 0, 0.5, 1, 1.5, 2, 3, 4, 5, 6, 10, 20, \pm \infty</math>, supplemented by <math>~n = 2.5, 3.5, 4.5,</math> and <math>~4.99</math> for the spherical case.

In Table II some finite boundary values of polytropic slabs, cylinders, and spheres are summarized. For polytropic spheres (N = 3) we have also quoted boundary values near the minimum of the dimensionless mass <math>~-\xi_1^2\theta_1</math>, occurring at n ≈ 4.823 [https://ui.adsabs.harvard.edu/abs/1978SvA....22..711S/abstract (Z. F. Seidov and R. Kh. Kuzakhmedov, 1978)].
</font>
</td></tr></table>

Focusing specifically on the spherically symmetric (N = 3) configurations, we list here the page number(s) on which the table associated with each individual polytropic index can be found in [https://ui.adsabs.harvard.edu/abs/1986Ap%26SS.126..357H/abstract Horedt (1986)].
<table border="1" align="center" cellpadding="8">
<tr>
<th align="center" colspan="3">Spherical (N = 3)<br />Configurations</th>
</tr>
<tr>
<td align="center">n</td>
<td align="center">page(s)</td>
<td align="center" colspan="1"><math>~\xi_1</math></td>
</tr>
<tr>
<td align="center">- 10</td>
<td align="center">386 → 387</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
<tr>
<td align="center">- 5</td>
<td align="center">387 → 388</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
<tr>
<td align="center">- 4</td>
<td align="center">388 → 389</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
<tr>
<td align="center">- 3</td>
<td align="center">389</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
<tr>
<td align="center">- 2</td>
<td align="center">390</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
<tr>
<td align="center">- 1.5</td>
<td align="center">390 → 391</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
<tr>
<td align="center">- 1.01</td>
<td align="center">391 → 392</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
<tr>
<td align="center">- 0.9</td>
<td align="center">392 → 393</td>
<td align="center">2.05040073E+00</td>
</tr>
<tr>
<td align="center">- 0.5</td>
<td align="center">393</td>
<td align="center">2.20858842E+00</td>
</tr>
<tr>
<td align="center">0</td>
<td align="center">393 → 394</td>
<td align="center">√6 = 2.44948974E+00</td>
</tr>
<tr>
<td align="center">0.5</td>
<td align="center">394</td>
<td align="center">2.75269805E+00</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">394 → 395</td>
<td align="center">π = 3.14159265E+00</td>
</tr>
<tr>
<td align="center">1.5</td>
<td align="center">395</td>
<td align="center">3.65375374E+00</td>
</tr>
<tr>
<td align="center">2</td>
<td align="center">395 → 396</td>
<td align="center">4.35287460E+00</td>
</tr>
<tr>
<td align="center">2.5</td>
<td align="center">396 → 397</td>
<td align="center">5.35527546E+00</td>
</tr>
<tr>
<td align="center">3</td>
<td align="center">397 → 398</td>
<td align="center">6.89684862E+00</td>
</tr>
<tr>
<td align="center">3.5</td>
<td align="center">398 → 399</td>
<td align="center">9.53580534E+00</td>
</tr>
<tr>
<td align="center">4</td>
<td align="center">399</td>
<td align="center">1.49715463E+01</td>
</tr>
<tr>
<td align="center">4.5</td>
<td align="center">399 → 400</td>
<td align="center">3.18364632E+01</td>
</tr>
<tr>
<td align="center">4.99</td>
<td align="center">400 → 401</td>
<td align="center">1.75818915+03</td>
</tr>
<tr>
<td align="center">5</td>
<td align="center">401 → 402</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
<tr>
<td align="center">6</td>
<td align="center">402 → 403</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
<tr>
<td align="center">10</td>
<td align="center">403 → 404</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
<tr>
<td align="center">20</td>
<td align="center">404 → 405</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
<tr>
<td align="center"><math>~\pm \infty</math></td>
<td align="center">405 → 406</td>
<td align="center"><font size="+2">∞</font></td>
</tr>
</table>

← Older revision		Revision as of 14:25, 18 December 2023
Line 2:		Line 2:
	<table border="1" align="center" width="100%" colspan="8">		<table border="1" align="center" width="100%" colspan="8">
	<tr>		<tr>
	<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/Lane1870\|Lane (~~1980~~)]]		<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/Lane1870\|Lane (1870)]]

	</td>		</td>

← Older revision		Revision as of 14:12, 18 December 2023
Line 2:		Line 2:
	<table border="1" align="center" width="100%" colspan="8">		<table border="1" align="center" width="100%" colspan="8">
	<tr>		<tr>
	<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/~~Polytropes~~\|~~Part I:  Isolated Polytropes~~]]		<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/Lane1870\|Lane (1980)]]

	</td>		</td>
	<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/Polytropes~~/Analytic~~\|Part II:  ~~Known Analytic Solutions~~]]		<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/Polytropes\|Part I:  Isolated Polytropes]]

	</td>		</td>
	<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/Polytropes/Numerical\|Part II:  Numerical Solutions]]		<td align="center" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/Polytropes/Analytic\|Part II:  Known Analytic Solutions]]

			</td>
			<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/Polytropes/Numerical\|Part III:  Numerical Solutions]]

	</td>		</td>

SSC/Structure/Polytropes/Numerical - Revision history

Joel2: /* Tabulated Global Properties */

Joel2: /* Polytropic Spheres */

Joel2 at 18:12, 18 December 2023

Joel2 at 18:04, 18 December 2023

← Older revision		Revision as of 14:04, 18 December 2023
Line 1:		Line 1:
			=Polytropic Spheres=
			<table border="1" align="center" width="100%" colspan="8">
			<tr>
			<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/Polytropes\|Part I:  Isolated Polytropes]]

			</td>
			<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/Polytropes/Analytic\|Part II:  Known Analytic Solutions]]

			</td>
			<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/Polytropes/Numerical\|Part II:  Numerical Solutions]]

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

	==Numerical Solutions==		==Numerical Solutions==
	Here we explain how an Excel workbook can be used to numerically solve the Lane-Emden equation, evaluating the Lane-Emden function across a one-dimensional, discrete grid.		Here we explain how an Excel workbook can be used to numerically solve the Lane-Emden equation, evaluating the Lane-Emden function across a one-dimensional, discrete grid.