Editing SSC/Structure/WhiteDwarfs (section)

==Highlights from Discussion by Shapiro &amp; Teukolsky (1983)==

Here we interleave our own derivations and discussions with the presentation found in [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>].

In our [[SSC/Structure/Polytropes#MassRadiusRelation|accompanying discussion]], we have shown that the equilibrium radius of an isolated polytrope is given, quite generally, by the expression,
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<math>~R_\mathrm{eq} </math>
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<math>~=</math>
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<math>~\biggl[\biggl(\frac{G}{K_n}\biggr)^n M_\mathrm{tot}^{n-1}  \biggr]^{1/(n-3)} 
\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggl[(-\theta^') \xi^2\biggr]_{\xi_1}^{(1-n)/(n-3)} \xi_1 \, .
</math>
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</div>
Inverting this provides the following expression for the total mass in terms of the equilibrium radius:
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<math>~ M_\mathrm{tot}^{1-n} </math>
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<math>~=</math>
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<math>~R_\mathrm{eq}^{3-n}\biggl(\frac{G}{K_n}\biggr)^n 
\biggl[ \frac{4\pi}{(n+1)^n} \biggr] \biggl[(-\theta^') \xi^2\biggr]_{\xi_1}^{1-n} \xi_1^{n-3} 
</math>
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<math>~\Rightarrow~~~ M_\mathrm{tot}</math>
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<math>~=</math>
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<math>~R_\mathrm{eq}^{(3-n)/(1-n)}\biggl[\frac{G}{(n+1)K_n}\biggr]^{n/(1-n)} 
( 4\pi )^{1/(1-n)} \biggl[(-\theta^') \xi^2\biggr]_{\xi_1} \xi_1^{(n-3)/(1-n)} 
</math>
  </td>
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&nbsp;
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<math>~=</math>
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<math>~4\pi R_\mathrm{eq}^{(3-n)/(1-n)}\biggl[\frac{(n+1)K_n}{4\pi G}\biggr]^{n/(n-1)}  
\biggl[(-\theta^') \xi^2\biggr]_{\xi_1} \xi_1^{(n-3)/(1-n)} 
</math>
  </td>
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</table>
</div>

As is shown by the following boxed-in equation table, this expression matches equation (3.3.11) from [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], except for the sign of the exponent on <math>~\xi_1</math>, which is demonstratively correct in our expression.

<!-- <table border="1" cellpadding="5" align="center">
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Equation extracted from &sect;3.3 (p. 63) of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]
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[[File:ST83Eq3_3_11.png|600px|Equation 3.3.11 from ST83]]
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-->

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Equations extracted<sup>&dagger;</sup> from &sect;3.3 (p. 63) and &sect;2.3 (p. 27) of [http://adsabs.harvard.edu/abs/1983bhwd.book.....S Shapiro &amp; Teukolsky (1983)]<p></p>
"''Black Holes, White Dwarfs, and Neutron Stars: &nbsp; The Physics of Compact Objects''"<p></p>
(New York:  John Wiley &amp; Sons)
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<!-- [[File:ST83Eq3_3_11.png|600px|Equation 3.3.11 from ST83]] -->
<!-- [[Image:AAAwaiting01.png|400px|center|Norman &amp; Wilson (1978)]] -->
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<math>~M</math>
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<math>~=</math>
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<math>~
4\pi R^{(3-n)/(1-n)} \biggl[ \frac{(n+1)K}{4\pi G} \biggr]^{n/(n-1)} \xi_1^{(3-n)/(1-n)} \xi_1^2 |\theta^'(\xi_1)| \, .
</math>
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&nbsp; &nbsp; &nbsp; (Eq. 3.3.11)
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</table>

  </td>
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<!-- [[File:ST83Eq3_3_12.png|600px|Equation 3.3.12 from ST83]] -->
<!-- [[Image:AAAwaiting01.png|400px|center|Norman &amp; Wilson (1978)]] -->
<math>~\Gamma = \tfrac{4}{3} \, ,</math> &nbsp; &nbsp; &nbsp;
<math>~n = 3 \, ,</math> &nbsp; &nbsp; &nbsp;
<math>~\xi_1 = 6.89685 \, ,</math> &nbsp; &nbsp; &nbsp;
<math>~\xi_1^2|\theta^'(\xi_1)| = 2.01824 \, .</math> &nbsp; 
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (Eq. 3.3.12)
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<!-- [[File:ST83Eq2_3_23.png|600px|Equation 2.3.23 from ST83]]<br />-->
<!-- [[Image:AAAwaiting01.png|400px|center|Norman &amp; Wilson (1978)]] -->
<math>~\Gamma = \frac{4}{3} \, ,</math> &nbsp; &nbsp; &nbsp;
<math>~K = \frac{3^{1 / 3} \pi^{2 /3}}{4} \frac{\hbar c}{m_u^{4 / 3}\mu_e^{4 / 3}} = \frac{1.2435 \times 10^{15}}{\mu_e^{4 /3}} ~\mathrm{cgs} \, .</math> &nbsp; &nbsp; &nbsp;
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (Eq. 2.3.23)
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<tr><td align="left"><sup>&dagger;</sup>Each equation has been retyped here exactly as it appears in the original publication.</td></tr>
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Given that (see equation 3.3.12 of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]; see the boxed-in equation table) in the relativistic limit, <math>~\Gamma = \gamma_g = 4/3</math> &#8212; that is, <math>~n=3</math> &#8212; and acknowledging as [[#m3|we have above]] that, for isolated <math>~n = 3</math> polytropes, 
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<math>m_3 \equiv \biggl(-\xi^2 \frac{d\theta_3}{d\xi} \biggr)_\mathrm{\xi=\xi_1(\theta_3)} = 2.01824</math>,
</div>

<!-- <table border="1" cellpadding="5" align="center">
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Equation extracted from &sect;3.3 (p. 63) of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]
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[[File:ST83Eq3_3_12.png|600px|Equation 3.3.12 from ST83]]
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-->

this polytropic expression for the mass becomes,
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<math>~ M_\mathrm{tot}</math>
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<math>~=</math>
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<math>~4\pi m_3 \biggl[\frac{ K_3 }{\pi G}\biggr]^{3/2}  \, .
</math>
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</table>
</div>

Separately, [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>] show that the effective polytropic constant for a relativistic electron gas is (see their equation 2.3.23, reprinted above in the boxed-in equation table),

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<math>~K_3</math>
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<math>~=</math>
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<math>~
\biggl( \frac{3\pi^2}{2^6}\biggr)^{1/ 3} \biggl[ \frac{ \hbar^3 c^3  }{ m_u^4 \mu_e^4  } \biggr]^{ 1 / 3}
= \biggl[ \frac{3 h^3 c^3  }{2^9\pi m_u^4 \mu_e^4  } \biggr]^{ 1 / 3} \, .
</math>
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<!-- <table border="1" cellpadding="5" align="center">
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  <td align="center">
Equation extracted from &sect;2.3 (p.27) of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]
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[[File:ST83Eq2_3_23.png|600px|Equation 2.3.23 from ST83]]
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-->

Together, then, the [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>] analysis gives,

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<math>~ M_\mathrm{tot}</math>
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<math>~=</math>
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<math>~4\pi m_3 \biggl[\frac{ 1 }{\pi G}\biggr]^{3/2}  \biggl[ \frac{3 h^3 c^3  }{2^9\pi m_u^4 \mu_e^4  }  \biggr]^{ 1 / 2} \, .
</math>
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Given that the definitions of the characteristic Fermi pressure, {{ Math/C_FermiPressure }}, and the characteristic Fermi density,  {{ Math/C_FermiDensity }}, are,

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<math>A_\mathrm{F} \equiv \frac{\pi m_e^4 c^5}{3h^3} </math><br /><p></p>

<math>\frac{B_\mathrm{F}}{\mu_e} \equiv \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 \, ,</math>
</div>

we have,
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<math>~ M_\mathrm{tot}</math>
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<math>~=</math>
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<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2}  \frac{\mu_e^2}{B_F^2} \biggl[ \frac{3 h^3 c^3  }{2^9\pi m_u^4 \mu_e^4  }  \biggr]^{ 1 / 2} 
\biggl[ \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 \biggr]^2 \biggl[\frac{3h^3}{2\pi m_e^4 c^5} \biggr]^{3 / 2}
</math>
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&nbsp;
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<math>~=</math>
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<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2}  \frac{\mu_e^2}{B_F^2} 
\biggl\{  \frac{3 h^3 c^3  }{2^9\pi m_u^4 \mu_e^4  }  \cdot
\frac{2^{12} \pi^4 m_p^4}{3^4} \biggl( \frac{m_e c}{h} \biggr)^{12} \cdot \frac{3^3h^9}{2^3\pi^3 m_e^{12} c^{15}} 
\biggr\}^{1 / 2}
</math>
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&nbsp;
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<math>~=</math>
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<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2}  \frac{\mu_e^2}{B_F^2} 
\biggl[ \frac{ m_p  }{m_u \mu_e  } \biggr]^{2} 
</math>
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<math>~\Rightarrow ~~~ \mu_e^2 M_\mathrm{tot}</math>
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<math>~=</math>
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<math>~4\pi m_3 \biggl[\frac{ 2A_F }{\pi G}\biggr]^{3/2}  \frac{\mu_e^2}{B_F^2} 
\biggl[ \frac{ m_p  }{m_u  } \biggr]^{2} \, ,
</math>
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</div>

which matches the [[#Chandrasekhar_mass|expression presented above]] for the Chandrasekhar mass if we set <math>~m_u = m_p</math>.