Editing SSC/Structure/WhiteDwarfs (section)

==Mass-Radius Relationships==

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

===Chandrasekhar mass===
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<font size="-1">[[H_BookTiledMenu#MoreModels|<b>Chandrasekhar<br />Limiting<br />Mass</b>]]<br />(1935)</font>
|}

[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])."