User:Tohline/SSC/Structure/WhiteDwarfs

White Dwarfs

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

Mass-Radius Relation Relationship

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 <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>

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation