Editing AxisymmetricConfigurations/SolutionStrategies (section)

===Prescribing Mass-Dependent Rotation Profile Based on an Initial Spherical Configuration===

Each of the ''simple rotation profiles'' listed in Table 1 has been defined by specifying the radial distribution of the specific angular momentum, <math>j(\varpi)</math>, ''in the rotationally flattened equilibrium configuration.''  Here we follow the lead of {{ Stoeckly65full }}, of {{ BO73full }} and of {{ MPT77full }} and, instead, present rotation profiles that are defined by specifying the function, <math>j(m_\varpi)</math>, where <math>m_\varpi</math> is a function describing how the fractional mass enclosed inside <math>\varpi</math> varies with <math>\varpi</math>.

To better clarify what is meant by the function, <math>m_\varpi</math>, consider a configuration (not necessarily in equilibrium) that is spherically symmetric and that exhibits an &#8212; as yet unspecified &#8212; mass-density profile, <math>\rho(r)</math>.  The [[SSCpt2/SolutionStrategies#Solution_Strategies|mass enclosed within each spherical radius]] is,
<div align="center">
<math>M_r = \int_0^r 4\pi r^2 \rho( r ) dr \, ,</math>
</div>
and, if the radius of the configuration is <math>R</math>, then the configuration's total mass is,
<div align="center">
<math>M = \int_0^R 4\pi r^2 \rho( r ) dr \, .</math>
</div>

In contrast, the mass enclosed within each ''cylindrical'' radius, <math>\varpi</math>, is
<div align="center">
<math>M_\varpi = 2\pi \int_0^\varpi \varpi d\varpi \int_0^{\sqrt{R^2 - \varpi^2}} \rho( r ) 2dz \, ,</math>
</div>
where it is understood that the argument of the density function is, <math>r = \sqrt{\varpi^2 + z^2} </math>.  

<span id="Example1">'''Example #1''':</span>  If the configuration has a uniform density, <math>\rho_0</math>, then its total mass is, <math>M = 4\pi \rho_0 R^3/3</math>, and

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \rho_0 \int_0^\varpi \varpi [R^2 - \varpi^2]^{1 / 2}d\varpi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} \rho_0 \biggl[R^3 - (R^2 - \varpi^2)^{3 / 2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>M -
\frac{4\pi}{3} \rho_0 \biggl[(R^2 - \varpi^2)^{3 / 2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~m_\varpi \equiv \frac{M_\varpi}{M}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 -
\biggl[1 - \frac{\varpi^2}{R^2}\biggr]^{3 / 2} \, .
</math>
  </td>
</tr>
</table>
</div>

'''Example #2''':  If the spherically symmetric configuration has a density profile given by the function,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho(r)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \biggl[\frac{\sin (\pi r/R)}{\pi r/R}  \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>

then [[SSC/Structure/Polytropes#n_.3D_1_Polytrope|its total mass]] is, <math>M = 4 \rho_0 R^3/\pi</math>, and

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \rho_0\int_0^\varpi \varpi d\varpi \int_0^{\sqrt{R^2 - \varpi^2}} 
\biggl\{ \frac{\sin (\pi \sqrt{\varpi^2 + z^2} /R)}{\pi \sqrt{\varpi^2 + z^2} /R}  \biggr\} dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4 \rho_0 R^3\int_0^\chi \chi d\chi \int_0^{\sqrt{1 - \chi^2}} 
\biggl\{ \frac{\sin (\pi \sqrt{\chi^2 + \zeta^2} )}{\sqrt{\chi^2 + \zeta^2}}  \biggr\} d\zeta
</math>
  </td>
</tr>
</table>
</div>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \rho_0 \biggl\{ \int_{\sqrt{R^2 - \varpi^2}}^R dz 
\int_0^\sqrt{R^2-z^2} \biggl[ \frac{\sin (\pi \sqrt{\varpi^2 + z^2} /R)}{\pi \sqrt{\varpi^2 + z^2} /R}  \biggr] \varpi d\varpi
+ \int_0^{\sqrt{R^2 - \varpi^2}} dz 
\int_0^\varpi \biggl[ \frac{\sin (\pi \sqrt{\varpi^2 + z^2} /R)}{\pi \sqrt{\varpi^2 + z^2} /R}  \biggr] \varpi d\varpi
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4 \rho_0 R^3 \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 d\zeta 
\int_0^\sqrt{1-\zeta^2} \biggl[ \frac{\sin (\pi \sqrt{\chi^2 + \zeta^2})}{ \sqrt{\chi^2 + \zeta^2}}  \biggr] \chi d\chi
+ \int_0^{\sqrt{1 - \chi^2}} d\zeta 
\int_0^\chi \biggl[ \frac{\sin (\pi \sqrt{\chi^2 + \zeta^2})}{ \sqrt{\chi^2 + \zeta^2}}  \biggr] \chi d\chi
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4 \rho_0 R^3 \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 
\biggl[  - \frac{ \cos(\pi\sqrt{\zeta^2 + \chi^2})}{\pi}    \biggr]_0^\sqrt{1-\zeta^2} d\zeta
+ \int_0^{\sqrt{1 - \chi^2}} 
\biggl[ - \frac{ \cos(\pi\sqrt{\zeta^2 + \chi^2})}{\pi}  \biggr]_0^\chi  d\zeta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4 \rho_0 R^3}{\pi} \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 
\biggl[  - \cos(\pi) + \cos(\pi\zeta)  \biggr] d\zeta
+ \int_0^{\sqrt{1 - \chi^2}} 
\biggl[ \cos(\pi\zeta ) - \cos(\pi\sqrt{\zeta^2 + \chi^2}) \biggr]  d\zeta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4 \rho_0 R^3}{\pi} \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 d\zeta
+  \int_0^1 \cos(\pi\zeta)   d\zeta
- \int_0^{\sqrt{1 - \chi^2}} 
\cos(\pi\sqrt{\zeta^2 + \chi^2})  d\zeta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4 \rho_0 R^3}{\pi} \biggl\{ \biggl[ z \biggr]_{\sqrt{1 - \chi^2}}^1
+  \frac{1}{\pi} \int_0^\pi \cos(u)   du
- \int_0^{\sqrt{1 - \chi^2}} 
\cos(\pi\sqrt{\zeta^2 + \chi^2})  d\zeta
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

====Uniform-Density Initially (n' = 0)====

Drawing directly from &sect;IIc of {{ Stoeckly65 }}, <font color="orange">&hellip; consider a large, gaseous mass, initially a homogeneous sphere of mass <math>M</math> and angular momentum <math>J</math> rotating as a solid body, and suppose it contracts in such a way that cylindrical surfaces remain cylindrical and each such surface retains its angular momentum.  Let <math>\rho_0</math>, <math>R_0</math>, and <math>\dot\varphi_0</math> denote the initial density, radius, and angular velocity of the</font> [initially unstable configuration]<font color="orange">, <math>\varpi_0(\varpi)</math> the initial radius of the surface now at radius <math>\varpi</math>, and <math>M_\varpi(\varpi)</math> the mass inside this surface.  The conditions on the contraction are then</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M - M_\varpi(\varpi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \rho_0 \int_{\varpi_0(\varpi)}^{R_0} \biggl[ \biggl(R_0^2 - (\varpi_0^')^2\biggr) \biggr]^{1 / 2} \varpi_0^' d\varpi_0^' \, ,
</math>
  </td>
</tr>
</table>
</div>
<font color="orange">and</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\dot\varphi(\varpi) \varpi^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\dot\varphi_0 [\varpi_0(\varpi)]^2 \, .</math>
  </td>
</tr>
</table>
</div>
<font color="orange">By integrating, eliminating <math>\varpi_0(\varpi)</math> between these equations, and eliminating <math>\rho_0</math>, <math>R_0</math>, and <math>\dot\varphi_0</math> in favor of <math>M</math> and <math>J</math>, one finds the relation of <math>\dot\varphi(\varpi)</math> to the mass distribution to be</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\dot\varphi(\varpi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5J}{2M\varpi^2}\biggl\{ 1 - [1 - m(\varpi) ]^{2 / 3} \biggr\} \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Stoeckly65 }}, &sect;II.c, eq. (12)
  </td>
</tr>
</table>
</div>
where, the mass fraction,
<div align="center">
<math>m(\varpi) \equiv \frac{M_\varpi(\varpi)}{M} \, .</math>
</div>

As noted, this is equation (12) of {{ Stoeckly65 }}; it also appears, for example, as equation (45) in {{ OM68 }}, as equation (12) in {{ BO70full }}, and in the sentence that follows equation (3) in {{ BO73 }}.  As Stoeckly points out, the angular momentum distribution implied by this functional form of <math>~\dot\varphi</math> satisfies the [[2DStructure/AxisymmetricInstabilities#Solberg.2FRayleigh_Criterion|Solberg/Rayleigh stability criterion]] &#8212; that is,
<div align="center">
<math>\frac{dj^2}{d\varpi} > 0 </math>
</div>
&#8212; initially, and also in the final equilibrium configuration because every cylindrical surface conserves specific angular momentum and the surfaces do not reorder themselves.

<table border="1" width="80%" align="center" cellpadding="5"><tr><td align="left">
We should be able to obtain the identical result by extending [[#Example1|Example 1]] above.  Attaching the subscript "0" to <math>\varpi</math> in order to acknowledge that, here, the initial configuration is a uniform-density sphere (n' = 0), our derivation gives,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>m_\varpi \equiv \frac{M_\varpi}{M}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 -
\biggl[1 - \frac{\varpi_0^2}{R^2}\biggr]^{3 / 2} \, ,
</math>
  </td>
</tr>
</table>
from which we see that,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>\frac{\varpi_0^2}{R^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 - \biggl[1 - m_\varpi \biggr]^{2 / 3} \, .
</math>
  </td>
</tr>
</table>
Now, the total angular momentum, <math>J</math>, of this initial configuration &#8212; a uniformly rotating <math>(\dot\varphi_0)</math>, uniform-density sphere &#8212; is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>J = I{\dot\varphi}_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{5}MR^2{\dot\varphi}_0
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ {\dot\varphi}_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5J}{2MR^2} \, ,
</math>
  </td>
</tr>
</table>
in which case, the specific angular momentum of each fluid element &#8212; which is conserved as the configuration contracts or expands &#8212; is given by the expression,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\dot\varphi \varpi^2 = {\dot\varphi}_0 \varpi_0^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5J}{2MR^2} \cdot \varpi_0^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5J}{2M} \biggl\{ 1 - \biggl[1 - m_\varpi \biggr]^{2 / 3}\biggr\} \, .
</math>
  </td>
</tr>
</table>
Q.E.D.
</td></tr></table>

Now, just as the fraction of the configuration's mass that lies ''interior to'' radial position, <math>\varpi</math>, is detailed by the function, <math>m_\varpi</math>, let's use <math>\ell_\varpi</math> to detail what fraction of the configuration's angular momentum lies ''interior to'' <math>m_\varpi</math>.  We have,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>J \ell_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\int_0^{m_\varpi} (\dot\varphi \varpi^2) M \cdot dm_\varpi
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \ell_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5}{2} \int_0^{m_\varpi} \biggl\{ 1 - \biggl[1 - m_\varpi \biggr]^{2 / 3}\biggr\} dm_\varpi
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{2}{5} \cdot \ell_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\int_0^{m_\varpi} dm_\varpi
-
\int_0^{m_\varpi} \biggl[1 - m_\varpi \biggr]^{2 / 3}dm_\varpi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m_\varpi
+
\biggl[
\frac{3}{5}\biggl(1 - m_\varpi\biggr)^{5/3}
\biggr]_0^{m_\varpi}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m_\varpi
+
\frac{3}{5}\biggl(1 - m_\varpi\biggr)^{5/3}
-\frac{3}{5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\biggl(1 - m_\varpi\biggr)
+
\frac{3}{5}\biggl(1 - m_\varpi\biggr)^{5/3}
+ \biggl(1-\frac{3}{5}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow \ell_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 - \frac{5}{2}\biggl(1 - m_\varpi\biggr)
+
\frac{3}{2}\biggl(1 - m_\varpi\biggr)^{5/3} \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ MPT77 }}, &sect;IV.a, eq. (4.3)
  </td>
</tr>
</table>

====Centrally Condensed Initially (n' > 0)====

<!--
Here, following [http://adsabs.harvard.edu/abs/1973ApJ...180..159B Bodenheimer &amp; Ostriker (1973)], we introduce an approach to specifying a wider range of physically reasonable angular momentum distributions; text that appears in an dark green font has been taken ''verbatim'' from this foundational paper.
-->
In &sect;III.d (pp. 1084 - 1086) of {{ OM68 }}, we find the following relations:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>h(m) \equiv \biggl[\frac{M}{J}\biggr] j(m)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a_1 + a_2(1-m)^{\alpha_2} + a_3(1-m)^{\alpha_3} \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ OM68 }}, &sect;III.d, Eq. (50)<br />
{{ OB68 }}, p. 1090, Eq. (6)<br />
{{ BO73 }}, &sect;II, Eq. (4)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;10.4 (p. 254), Eq. (44)<br />
{{ PDD96 }}, &sect;2.1, Figure 1
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{1}{\alpha_2} = q_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{2\beta - \alpha \beta(2n+5)}{\alpha \beta(2n+5) - (2n + 3)} \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>\frac{1}{\alpha_3} = q_2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{2n+3}{2} \, ,
</math>
  </td>
  <td align="center" colspan="4">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>b_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{\alpha (q_2 + 1) - 1}{\alpha (q_2 - q_1)} \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>b_2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{ 1 - \alpha (q_1+1)}{\alpha (q_2 - q_1)} \, ,
</math>
  </td>
  <td align="center" colspan="4">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>a_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
b_1(q_1+1) + b_2(q_2+1) \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>a_2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
-b_1(q_1+1) \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>a_3</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
- b_2(q_2+1) \, .
</math>
  </td>
</tr>

</table>

Ostriker &amp; Mark claim that the analytical expression for <math>\dot\varphi (\varpi) = j[m(\varpi)]/\varpi^2</math> that was derived by {{ Stoeckly65 }} for a uniform-density sphere, is retrieved by setting, <math>(n, \alpha, \beta) = (0, \tfrac{2}{5}, \tfrac{3}{2}) \, .</math>  Let's see &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lim_{n\rightarrow 0} \biggl[ q_1 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lim_{n\rightarrow 0} \biggl[ \frac{-\tfrac{6n}{5} }{-\tfrac{4n}{5}}\biggr] = + \frac{3}{2} \, ;</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~q_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+\frac{3}{2} \, ;
</math>
  </td>
<td align="center" colspan="3">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>~b_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\tfrac{2}{5} (\tfrac{3}{2} + 1) - 1}{\tfrac{2}{5} (\tfrac{3}{2} - \tfrac{2}{3})} 
=
\frac{0}{\tfrac{1}{3}} =0 \, ;
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~b_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ 1 - \tfrac{2}{5} (\tfrac{2}{3}+1)}{\tfrac{2}{5} (\tfrac{3}{2} - \tfrac{2}{3})} 
=
\frac{ \tfrac{1}{3}}{\tfrac{1}{3} } = 1 \, ;
</math>
  </td>
<td align="center" colspan="3">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>~a_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tfrac{5}{2}  \, ;
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~a_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, ;
</math>
  </td>
  <td align="right">
<math>~a_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \tfrac{5}{2} \, .
</math>
  </td>
</tr>
</table>

This implies,
<table border="1" cellpadding="10" align="center"><tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h(m)\biggr|_{n' = ~0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{5}{2}\biggl[1 - (1-m)^{2/3} \biggr] \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

Q. E. D.

In addition, from p. 163 (Table 1) of {{ BO73 }} we find the following table of coefficient values.

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="6">
<b>Coefficients for <math>~h(m)</math> Expression</b><br />
[from K. Braly's (1969) unpublished undergraduate thesis, Princeton University]
</td>
<td align="center">
Figure &amp; caption extracted from p. 715 of<br />{{ PDD96figure }}<br />&copy; American Astronomical Society
</td>
</tr>
<tr>
  <td align="center"><math>~n^'</math></td>
  <td align="center"><math>~a_1</math></td>
  <td align="center"><math>~a_2</math></td>
  <td align="center"><math>~a_3</math></td>
  <td align="center"><math>~\alpha_2 = \frac{1}{q_1}</math></td>
  <td align="center"><math>~\alpha_3 = \frac{1}{q_2}</math></td>
  <td align="center" rowspan="10">[[File:PickettDurisenDavis96Fig1.png|400px]]</td>
</tr>
<tr>
  <td align="center">0</td>
  <td align="center">+2.5</td>
  <td align="center"><math>~\cdots</math></td>
  <td align="center">-2.5</td>
  <td align="center"><math>~\cdots</math></td>
  <td align="center"><math>~\tfrac{2}{3}</math></td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="center">+3.068133</td>
  <td align="center">+0.203667</td>
  <td align="center">-3.271800</td>
  <td align="center">+0.801297</td>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">+3.825819</td>
  <td align="center">+0.857311</td>
  <td align="center">-4.68313</td>
  <td align="center">+0.650981</td>
  <td align="center"><math>~\tfrac{2}{5}</math></td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{3}{2}</math></td>
  <td align="center">+4.887588</td>
  <td align="center">+2.345310</td>
  <td align="center">-7.232898</td>
  <td align="center">+0.525816</td>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">+6.457897</td>
  <td align="center">+6.018111</td>
  <td align="center">-12.476007</td>
  <td align="center">+0.417472</td>
  <td align="center"><math>~\tfrac{2}{7}</math></td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{5}{2}</math></td>
  <td align="center">+8.944150</td>
  <td align="center">+18.234305</td>
  <td align="center">-27.178455</td>
  <td align="center">+0.321459</td>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">+13.270061</td>
  <td align="center">+163.26149</td>
  <td align="center">-176.53154</td>
  <td align="center">+0.235287</td>
  <td align="center"><math>~\tfrac{2}{9}</math></td>
</tr>
<tr>
  <td align="center" colspan="6">
<b>Coefficients for <math>~h(m)</math> Expression</b><br />
used by {{ OB68 }}, p. 1090, Eq. (6)
</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{3}{2}</math></td>
  <td align="center">+4.8239</td>
  <td align="center">+1.8744</td>
  <td align="center">-6.6983</td>
  <td align="center">+0.5622</td>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
</tr>
</table>