Editing SSC/Structure/WhiteDwarfs (section)

===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>