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=White Dwarfs=
{{LSU_HBook_header}}
{{LSU_HBook_header}}
==Mass-Radius Relationships==


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


The following summaries are drawn from Appendix A of Even &amp; Tohline (2009, ApJS, 184, 248,263).
===Chandrasekhar mass===


==Mass-Radius Relation Relationship==
[http://adsabs.harvard.edu/abs/1935MNRAS..95..207C Chandrasekhar (1935)] was the first to construct models of spherically symmetric stars using the [[User:Tohline/SR#Time-Independent_Problems|barotropic equation of state appropriate for a degenerate electron gas]], namely,
 
<div align="center">
Chandrasekhar (1935) was the first to construct models of spherically symmetric stars using the equation of state defined by XXX and, in so doing, 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 {{User:Tohline/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 Chandrasekhar (1967), where we also find the expressions for the characteristic Fermi pressure, {{User:Tohline/Math/C_FermiPressure}}, and the characteristic Fermi density, {{User:Tohline/Math/C_FermiDensity}}.  The derived analytic expression for the limiting mass is,
{{User:Tohline/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 {{User:Tohline/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 [[User:Tohline/Appendix/References#C67|Chandrasekhar (1967)]], where we also find the expressions for the characteristic Fermi pressure, {{User:Tohline/Math/C_FermiPressure}}, and the characteristic Fermi density, {{User:Tohline/Math/C_FermiDensity}}.  The derived analytic expression for the limiting mass is,
<div align="center">
<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>,
<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>
</div>
where the coefficient,
<span id="m3">where the coefficient,</span>
<div align="center">
<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>,
<math>m_3 \equiv \biggl(-\xi^2 \frac{d\theta_3}{d\xi} \biggr)_\mathrm{\xi=\xi_1(\theta_3)} = 2.01824</math>,
</div>
</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 Chandrasekhar (1967).  We note as well that Chandrasekhar (1967) identified a characteristic radius, <math>\ell_1</math>, for white dwarfs given by the expression,
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 [[User:Tohline/Appendix/References#C67|Chandrasekhar (1967)]].  We note as well that [[User:Tohline/Appendix/References#C67|Chandrasekhar (1967)]] identified a characteristic radius, <math>\ell_1</math>, for white dwarfs given by the expression,
<div align="center">
<div align="center">
<math>
<math>
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</div>
</div>


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


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,
[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">
<div align="center">
<math>
<math>
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</tr>
</tr>
</table>
</table>
{{User:Tohline/Math/C_AtomicMassUnit}} is the atomic mass unit, {{User:Tohline/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 Chandrasekhar (1967) as <math>(\mu_e m_p)</math> is identical to the average particle mass specified by Nauenberg (1972) as <math>(\bar\mu m_u)</math> and, following Nauenberg's lead, by setting <math>\nu = 1</math> and,
{{User:Tohline/Math/C_AtomicMassUnit}} is the atomic mass unit, {{User:Tohline/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 [[User:Tohline/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">
<div align="center">
<math>\zeta = \frac{m_3^2}{4\pi} = 0.324142</math>,
<math>\zeta = \frac{m_3^2}{4\pi} = 0.324142</math>,
Line 110: Line 114:
</math>
</math>
</div>
</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 Nauenberg (1972):
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">
<div align="center">
<math>
<math>
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</math>
</math>
</div>
</div>
where,
<span id="ChandrasekharMass">where,</span>
<div align="center">
<div align="center">
<math>
<math>
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</div>
</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>[[User:Tohline/Appendix/References#ST83|<font color="red">ST83</font>]]</b>].
In our [[User:Tohline/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>[[User:Tohline/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>[[User:Tohline/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>[[User:Tohline/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>[[User:Tohline/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>[[User:Tohline/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>[[User:Tohline/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>[[User:Tohline/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, {{ User:Tohline/Math/C_FermiPressure }}, and the characteristic Fermi density,  {{ User:Tohline/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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Latest revision as of 15:34, 9 August 2019

White Dwarfs

Whitworth's (1981) Isothermal Free-Energy Surface
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Mass-Radius Relationships

The following summaries are drawn from Appendix A of Even & Tohline (2009).

Chandrasekhar mass

Chandrasekhar (1935) was the first to construct models of spherically symmetric stars using the barotropic equation of state appropriate for a degenerate electron gas, namely,

LSU Key.png

<math>~P_\mathrm{deg} = A_\mathrm{F} F(\chi) </math>

where:  <math>F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi</math>

and:   

<math>\chi \equiv (\rho/B_\mathrm{F})^{1/3}</math>

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>~\mu_e</math> 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 Chandrasekhar (1967), where we also find the expressions for the characteristic Fermi pressure, <math>~A_\mathrm{F}</math>, and the characteristic Fermi density, <math>~B_\mathrm{F}</math>. The derived analytic expression for the limiting mass is,

<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>,

where the coefficient,

<math>m_3 \equiv \biggl(-\xi^2 \frac{d\theta_3}{d\xi} \biggr)_\mathrm{\xi=\xi_1(\theta_3)} = 2.01824</math>,

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 Chandrasekhar (1967). We note as well that Chandrasekhar (1967) identified a characteristic radius, <math>\ell_1</math>, for white dwarfs given by the expression,

<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>

The Nauenberg Mass-Radius Relationship

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,

<math> R = R_0 \biggl[ \frac{(1 - n^{4/3})^{1/2}}{n^{1/3}} \biggr] , </math>

where,

<math> n </math>

<math> \equiv </math>

<math> \frac{M}{(\bar{\mu} m_u) N_0} , </math>

<math> N_0 </math>

<math> \equiv </math>

<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>

<math> R_0 </math>

<math> \equiv </math>

<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>

<math>~m_u</math> is the atomic mass unit, <math>~\bar{\mu}</math> 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 Chandrasekhar (1967) as <math>(\mu_e m_p)</math> is identical to the average particle mass specified by Nauenberg (1972) as <math>(\bar\mu m_u)</math> and, following Nauenberg's lead, by setting <math>\nu = 1</math> and,

<math>\zeta = \frac{m_3^2}{4\pi} = 0.324142</math>,

in the above expression for <math>N_0</math>, we see that,

<math> (\bar\mu m_u)N_0 = M_\mathrm{Ch} . </math>

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,

<math> \mu_e R_0 = m_3(\ell_1 \mu_e) = 1.55686\times 10^9~\mathrm{cm} , </math>

and,

<math> R = R_0 \biggl\{ \frac{[1 - (M/M_\mathrm{Ch})^{4/3} ]^{1/2}}{(M/M_\mathrm{Ch})^{1/3}} \biggr\} . </math>

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 Nauenberg (1972):

<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>

where,

<math> \frac{M_\mathrm{Ch}}{M_\odot} = \frac{5.742}{\mu_e^2} . </math>

Eggleton Mass-Radius Relationship

Verbunt & 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):

<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>

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 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 Marsh, Nelemans & Steeghs (2004) phrase it, "allows for the change to be a constant density configuration at low masses (Zapolsky & Salpeter 1969)."

Highlights from Discussion by Shapiro & Teukolsky (1983)

Here we interleave our own derivations and discussions with the presentation found in [ST83].

In our accompanying discussion, we have shown that the equilibrium radius of an isolated polytrope is given, quite generally, by the expression,

<math>~R_\mathrm{eq} </math>

<math>~=</math>

<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>

Inverting this provides the following expression for the total mass in terms of the equilibrium radius:

<math>~ M_\mathrm{tot}^{1-n} </math>

<math>~=</math>

<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>

<math>~\Rightarrow~~~ M_\mathrm{tot}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<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>

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


Equations extracted from §3.3 (p. 63) and §2.3 (p. 27) of Shapiro & Teukolsky (1983)

"Black Holes, White Dwarfs, and Neutron Stars:   The Physics of Compact Objects"

(New York: John Wiley & Sons)

<math>~M</math>

<math>~=</math>

<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>

      (Eq. 3.3.11)

<math>~\Gamma = \tfrac{4}{3} \, ,</math>       <math>~n = 3 \, ,</math>       <math>~\xi_1 = 6.89685 \, ,</math>       <math>~\xi_1^2|\theta^'(\xi_1)| = 2.01824 \, .</math>               (Eq. 3.3.12)

<math>~\Gamma = \frac{4}{3} \, ,</math>       <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>                   (Eq. 2.3.23)

Each equation has been retyped here exactly as it appears in the original publication.


Given that (see equation 3.3.12 of [ST83]; see the boxed-in equation table) in the relativistic limit, <math>~\Gamma = \gamma_g = 4/3</math> — that is, <math>~n=3</math> — and acknowledging as we have above that, for isolated <math>~n = 3</math> polytropes,

<math>m_3 \equiv \biggl(-\xi^2 \frac{d\theta_3}{d\xi} \biggr)_\mathrm{\xi=\xi_1(\theta_3)} = 2.01824</math>,


this polytropic expression for the mass becomes,

<math>~ M_\mathrm{tot}</math>

<math>~=</math>

<math>~4\pi m_3 \biggl[\frac{ K_3 }{\pi G}\biggr]^{3/2} \, . </math>

Separately, [ST83] 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),

<math>~K_3</math>

<math>~=</math>

<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>


Together, then, the [ST83] analysis gives,

<math>~ M_\mathrm{tot}</math>

<math>~=</math>

<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>

Given that the definitions of the characteristic Fermi pressure, <math>~A_\mathrm{F}</math>, and the characteristic Fermi density, <math>~B_\mathrm{F}</math>, are,

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

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

we have,

<math>~ M_\mathrm{tot}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<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>

<math>~\Rightarrow ~~~ \mu_e^2 M_\mathrm{tot}</math>

<math>~=</math>

<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>

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

See Also

  • 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
 

… 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 [Chandrasekhar (1931, ApJ, 74, 81)] mass limit <math>~M_3</math> is increased by only a few percent when uniform rotation is included in the models, …

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 … 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.


Whitworth's (1981) Isothermal Free-Energy Surface

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