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=White Dwarfs=
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<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures|<b>Zero-Temperature<br />White Dwarf</b>]]</font>
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==Mass-Radius Relationships==

The following summaries are drawn from Appendix A of [http://adsabs.harvard.edu/abs/2009ApJS..184..248E Even &amp; Tohline (2009)].
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />

===Chandrasekhar mass===

[http://adsabs.harvard.edu/abs/1935MNRAS..95..207C Chandrasekhar (1935)] was the first to construct models of spherically symmetric stars using the [[SR#Time-Independent_Problems|barotropic equation of state appropriate for a degenerate electron gas]], namely,
<div align="center">
{{Math/EQ_ZTFG01}}
</div>
In so doing, he demonstrated that the maximum mass of an isolated, nonrotating white dwarf is <math>M_\mathrm{Ch} = 1.44 (\mu_e/2)M_\odot</math>, where {{Math/MP_ElectronMolecularWeight}} is the number of nucleons per electron and, hence, depends on the chemical composition of the white dwarf.  A concise derivation of <math>M_\mathrm{Ch}</math> (although, at the time, it was referred to as <math>M_3</math>) is presented in Chapter ''XI'' of [[Appendix/References#C67|Chandrasekhar (1967)]], where we also find the expressions for the characteristic Fermi pressure, {{Math/C_FermiPressure}}, and the characteristic Fermi density, {{Math/C_FermiDensity}}.  The derived analytic expression for the limiting mass is,
<div align="center">
<math>\mu_e^2 M_\mathrm{Ch} = 4\pi m_3 \biggl( \frac{2A_\mathrm{F}}{\pi G} \biggr)^{3/2} \frac{\mu_e^2}{B_\mathrm{F}^2} = 1.14205\times 10^{34} ~\mathrm{g}</math>,
</div>
<span id="m3">where the coefficient,</span>
<div align="center">
<math>m_3 \equiv \biggl(-\xi^2 \frac{d\theta_3}{d\xi} \biggr)_\mathrm{\xi=\xi_1(\theta_3)} = 2.01824</math>,
</div>
represents a structural property of <math>n = 3</math> polytropes (<math>\gamma = 4/3</math> gasses) whose numerical value can be found in Chapter ''IV'', Table 4 of [[Appendix/References#C67|Chandrasekhar (1967)]].  We note as well that [[Appendix/References#C67|Chandrasekhar (1967)]] identified a characteristic radius, <math>\ell_1</math>, for white dwarfs given by the expression,
<div align="center">
<math>
\ell_1 \mu_e \equiv \biggl( \frac{2A_\mathrm{F}}{\pi G} \biggr)^{1/2} \frac{\mu_e}{B_\mathrm{F}} = 7.71395\times 10^8~\mathrm{cm} .
</math>
</div>

===The Nauenberg Mass-Radius Relationship===

[http://adsabs.harvard.edu/abs/1972ApJ...175..417N Nauenberg (1972)] derived an analytic approximation for the mass-radius relationship exhibited by isolated, spherical white dwarfs that obey the zero-temperature white-dwarf equation of state.  Specifically, he offered an expression of the form,
<div align="center">
<math>
R = R_0 \biggl[ \frac{(1 - n^{4/3})^{1/2}}{n^{1/3}} \biggr] ,
</math>
</div>
where,
<table align="center" border="0" cellpadding="8">
<tr>
  <td align="right">
<math>
n 
</math>
  </td>
  <td align="center">
<math>
\equiv 
</math>
  </td>
  <td align="left">
<math>
\frac{M}{(\bar{\mu} m_u) N_0} ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
N_0 
</math>
  </td>
  <td align="center">
<math>
\equiv 
</math>
  </td>
  <td align="left">
<math>
\frac{(3\pi^2\zeta)^{1/2}}{\nu^{3/2}} \biggl[ \frac{hc}{2\pi G(\bar\mu m_u)^2} \biggr]^{3/2} = \frac{\mu_e^2 m_p^2}{(\bar\mu m_u)^3} \biggl[ \frac{4\pi \zeta}{m_3^2 \nu^3} \biggr]^{1/2} M_\mathrm{Ch} ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
R_0
</math>
  </td>
  <td align="center">
<math>
\equiv 
</math>
  </td>
  <td align="left">
<math>
(3\pi^2 \zeta)^{1/3} \biggl[ \frac{h}{2\pi m_e c} \biggr] N_0^{1/3} = \frac{\mu_e m_p}{\bar\mu m_u} \biggl[ \frac{4\pi \zeta}{\nu} \biggr]^{1/2} \ell_1 ,
</math>
  </td>
</tr>
</table>
{{Math/C_AtomicMassUnit}} is the atomic mass unit, {{Math/MP_MeanMolecularWeight}} is the mean molecular weight of the gas, and <math>\zeta</math> and <math>\nu</math> are two adjustable parameters in Nauenberg's analytic approximation, both of which are expected to be of order unity.  By assuming that the average particle mass denoted by [[Appendix/References#C67|Chandrasekhar (1967)]] as <math>(\mu_e m_p)</math> is identical to the average particle mass specified by [http://adsabs.harvard.edu/abs/1972ApJ...175..417N Nauenberg (1972)] as <math>(\bar\mu m_u)</math> and, following Nauenberg's lead, by setting <math>\nu = 1</math> and,
<div align="center">
<math>\zeta = \frac{m_3^2}{4\pi} = 0.324142</math>,
</div>
in the above expression for <math>N_0</math>, we see that,
<div align="center">
<math>
(\bar\mu m_u)N_0 = M_\mathrm{Ch} .
</math>
</div>
Hence, the denominator in the above expression for <math>n</math> becomes the Chandrasekhar mass.  Furthermore, the above expressions for <math>R_0</math> and <math>R</math> become, respectively,
<div align="center">
<math>
\mu_e R_0 = m_3(\ell_1 \mu_e) = 1.55686\times 10^9~\mathrm{cm} ,
</math>
</div>
and,
<div align="center">
<math>
R = R_0 \biggl\{ \frac{[1 - (M/M_\mathrm{Ch})^{4/3} ]^{1/2}}{(M/M_\mathrm{Ch})^{1/3}} \biggr\} .
</math>
</div>
Finally, by adopting appropriate values of <math>M_\odot</math> and <math>R_\odot</math>, we obtain essentially the identical approximate, analytic mass-radius relationship for zero-temperature white dwarfs presented in Eqs. (27) and (28) of [http://adsabs.harvard.edu/abs/1972ApJ...175..417N Nauenberg (1972)]:
<div align="center">
<math>
\frac{R}{R_\odot} = \frac{0.0224}{\mu_e} \biggl\{ \frac{[1 - (M/M_\mathrm{Ch})^{4/3} ]^{1/2}}{(M/M_\mathrm{Ch})^{1/3}} \biggr\} ,
</math>
</div>
<span id="ChandrasekharMass">where,</span>
<div align="center">
<math>
\frac{M_\mathrm{Ch}}{M_\odot} = \frac{5.742}{\mu_e^2} .
</math>
</div>

===Eggleton Mass-Radius Relationship===
[http://adsabs.harvard.edu/abs/1988ApJ...332..193V Verbunt &amp; Rappaport (1988)] introduced the following approximate, analytic expression for the mass-radius relationship of a "completely degenerate <math>\ldots</math> star composed of pure helium" (''i.e.,'' <math>\mu_e = 2</math>), attributing the expression's origin to Eggleton (private communication):
<div align="center">
<math>
\frac{R}{R_\odot} = 0.0114 \biggl[ \biggl(\frac{M}{M_\mathrm{Ch}}\biggr)^{-2/3} - \biggl(\frac{M}{M_\mathrm{Ch}}\biggr)^{2/3} \biggr]^{1/2} \biggl[ 1 + 3.5 \biggl(\frac{M}{M_p}\biggr)^{-2/3} + \biggl(\frac{M}{M_p}\biggr)^{-1} \biggr]^{-2/3} ,
</math>
</div>
where <math>M_p</math> is a constant whose numerical value is <math>0.00057 M_\odot</math>.  This "Eggleton" mass-radius relationship has been used widely by researchers when modeling the evolution of semi-detached binary star systems in which the donor is a zero-temperature white dwarf.  Since the [http://adsabs.harvard.edu/abs/1972ApJ...175..417N Nauenberg (1972)] mass-radius relationship discussed above is retrieved from this last expression in the limit <math>M/M_p \gg 1</math>, it seems clear that Eggleton's contribution was the insertion of the term in square brackets involving the ratio <math>M/M_p</math> which, as [http://adsabs.harvard.edu/abs/2004MNRAS.350..113M Marsh, Nelemans &amp; Steeghs (2004)] phrase it, "allows for the change to be a constant density configuration at low masses ([http://adsabs.harvard.edu/abs/1969ApJ...158..809Z Zapolsky &amp; Salpeter 1969])."

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

<tr>
  <td align="right">
<math>~R_\mathrm{eq} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>
Inverting this provides the following expression for the total mass in terms of the equilibrium radius:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ M_\mathrm{tot}^{1-n} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
</tr>
</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">
<tr>
  <td align="center">
Equation extracted from &sect;3.3 (p. 63) of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]
  </td>
</tr>
<tr>
  <td align="center">
[[File:ST83Eq3_3_11.png|600px|Equation 3.3.11 from ST83]]
  </td>
</tr>
</table>
-->

<div align="center">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center">
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)
</td></tr>
<tr>
  <td align="center">
<!-- [[File:ST83Eq3_3_11.png|600px|Equation 3.3.11 from ST83]] -->
<!-- [[Image:AAAwaiting01.png|400px|center|Norman &amp; Wilson (1978)]] -->
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
  <td align="right">
&nbsp; &nbsp; &nbsp; (Eq. 3.3.11)
  </td>
</tr>
</table>

  </td>
</tr>
<tr>
  <td align="center">
<!-- [[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)
  </td>
</tr>
<tr>
  <td align="center">
<!-- [[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)
  </td>
</tr>
<tr><td align="left"><sup>&dagger;</sup>Each equation has been retyped here exactly as it appears in the original publication.</td></tr>
</table>
</div>


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

this polytropic expression for the mass becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi m_3 \biggl[\frac{ K_3 }{\pi G}\biggr]^{3/2}  \, .
</math>
  </td>
</tr>
</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),

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

<tr>
  <td align="right">
<math>~K_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>

<!-- <table border="1" cellpadding="5" align="center">
<tr>
  <td align="center">
Equation extracted from &sect;2.3 (p.27) of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]
  </td>
</tr>
<tr>
  <td align="center">
[[File:ST83Eq2_3_23.png|600px|Equation 2.3.23 from ST83]]
  </td>
</tr>
</table>
-->

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

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

<tr>
  <td align="right">
<math>~ M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>
Given that the definitions of the characteristic Fermi pressure, {{ Math/C_FermiPressure }}, and the characteristic Fermi density,  {{ Math/C_FermiDensity }}, are,

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

<tr>
  <td align="right">
<math>~ M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \mu_e^2 M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>

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

=See Also=

* [https://ui.adsabs.harvard.edu/abs/1930LEDPM...9..944S/abstract Edmund C. Stoner (1930)], The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series 7, Volume 9, Issue 60, p. 944-963:  ''The Equilibrium of Dense Stars''

* [https://ui.adsabs.harvard.edu/abs/1931ApJ....74...81C/abstract S. Chandrasekhar (1931)], ApJ, 74, p. 81:  ''The Maximum Mass of Ideal White Dwarfs''

* [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract J. P. Ostriker, P. Bodenheimer &amp; D. Lynden-Bell (1966)], Phys. Rev. Letters, 17, 816:  ''Equilibrium Models of Differentially Rotating Zero-Temperature Stars''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%">&nbsp;</td><td align="left">
<font color="green">&hellip; work by Roxburgh (1965, Z. Astrophys., 62, 134), Anand (1965, Proc. Natl. Acad. Sci. U.S., 54, 23), and James (1964, ApJ, 140, 552) shows that the</font> [Chandrasekhar (1931, ApJ, 74, 81)] <font color="green">mass limit <math>~M_3</math> is increased by only a few percent when uniform rotation is included in the models, &hellip;</font>

<font color="green">In this Letter we demonstrate that white-dwarf models with masses considerably greater than  <math>~M_3</math> are possible if differential rotation is allowed &hellip; models are based on the physical assumption of an axially symmetric, completely degenerate, self-gravitating fluid, in which the effects of viscosity, magnetic fields, meridional circulation, and relativistic terms in the hydrodynamical equations have been neglected.</font>
</td></tr></table>

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