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<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>, | ||
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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). | 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, | ||
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\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} . | |||
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Revision as of 03:15, 1 August 2010
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White Dwarfs
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>
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