SSC/Structure/WhiteDwarfs

White Dwarfs

Zero-Temperature White Dwarf

(More introductory material is needed here.)  
 
 
 
 
 

Mass-Radius Relationships

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

Chandrasekhar mass

Chandrasekhar Limiting Mass (1935)

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,

	$P_{\mathrm{d}\mathrm{e}\mathrm{g}}=A_{\mathrm{F}}F(\chi )$
where:  $F(\chi )\equiv \chi (2{\chi }^2-3)({\chi }^2+1{)}^{1/2}+3{\sinh }^{-1}\chi$
and:	$\chi \equiv (\rho /B_{\mathrm{F}}{)}^{1/3}$

In so doing, he demonstrated that the maximum mass of an isolated, nonrotating white dwarf is $M_{\mathrm{C}\mathrm{h}}=1.44({\mu }_e/2)M_{\odot }$, where ${\mu }_e$ is the number of nucleons per electron and, hence, depends on the chemical composition of the white dwarf. A concise derivation of $M_{\mathrm{C}\mathrm{h}}$ (although, at the time, it was referred to as $M_3$) is presented in Chapter XI of Chandrasekhar (1967), where we also find the expressions for the characteristic Fermi pressure, $A_{\mathrm{F}}$, and the characteristic Fermi density, $B_{\mathrm{F}}$. The derived analytic expression for the limiting mass is,

${\mu }_e^2{M}_{\mathrm{C}\mathrm{h}}=4\pi m_3(\frac{2A_{\mathrm{F}}}{\pi G}{)}^{3/2}\frac{{\mu }_e^2}{B_{\mathrm{F}}^2}=1.14205\times 10^{34}\mathrm{g}$,

where the coefficient,

$m_3\equiv (-{\xi }^2\frac{d{\theta }_3}{d\xi }{)}_{\mathrm{\xi }\mathrm{=}{\mathrm{\xi }}_{\mathrm{1}}\mathrm{(}{\mathrm{\theta }}_{\mathrm{3}}\mathrm{)}}=2.01824$,

represents a structural property of $n=3$ polytropes ($\gamma =4/3$ 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, ${\ell }_1$, for white dwarfs given by the expression,

${\ell }_1{\mu }_e\equiv (\frac{2A_{\mathrm{F}}}{\pi G}{)}^{1/2}\frac{{\mu }_e}{B_{\mathrm{F}}}=7.71395\times 10^8\mathrm{c}\mathrm{m}.$

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,

$R=R_0\left[\frac{(1-n^{4/3}{)}^{1/2}}{n^{1/3}}\right],$

where,

$n$	$\equiv$	$\frac{M}{(\overline{\mu }{m}_u)N_0},$
$N_0$	$\equiv$	$\frac{(3{\pi }^2\zeta {)}^{1/2}}{{\nu }^{3/2}}[\frac{hc}{2\pi G(\overline{\mu }{m}_u{)}^2}{]}^{3/2}=\frac{{\mu }_e^2{m}_p^2}{(\overline{\mu }{m}_u{)}^3}[\frac{4\pi \zeta }{m_3^2{\nu }^3}{]}^{1/2}{M}_{\mathrm{C}\mathrm{h}},$
$R_0$	$\equiv$	$(3{\pi }^2\zeta {)}^{1/3}[\frac{h}{2\pi m_{e}c}]N_0^{1/3}=\frac{{\mu }_e{m}_p}{\overline{\mu }{m}_u}[\frac{4\pi \zeta }{\nu }{]}^{1/2}{\ell }_1,$

$m_u$ is the atomic mass unit, $\overline{\mu }$ is the mean molecular weight of the gas, and $\zeta$ and $\nu$ 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 $({\mu }_e{m}_p)$ is identical to the average particle mass specified by Nauenberg (1972) as $(\overline{\mu }{m}_u)$ and, following Nauenberg's lead, by setting $\nu =1$ and,

$\zeta =\frac{m_3^2}{4\pi }=0.324142$,

in the above expression for $N_0$, we see that,

$(\overline{\mu }{m}_u)N_0=M_{\mathrm{C}\mathrm{h}}.$

Hence, the denominator in the above expression for $n$ becomes the Chandrasekhar mass. Furthermore, the above expressions for $R_0$ and $R$ become, respectively,

${\mu }_e{R}_0=m_3({\ell }_1{\mu }_e)=1.55686\times 10^9\mathrm{c}\mathrm{m},$

and,

$R=R_0\left\{\frac{[1-(M/M_{\mathrm{C}\mathrm{h}}{)}^{4/3}{]}^{1/2}}{(M/M_{\mathrm{C}\mathrm{h}}{)}^{1/3}}\right\}.$

Finally, by adopting appropriate values of $M_{\odot }$ and $R_{\odot }$, we obtain essentially the identical approximate, analytic mass-radius relationship for zero-temperature white dwarfs presented in Eqs. (27) and (28) of Nauenberg (1972):

$\frac{R}{R_{\odot }}=\frac{0.0224}{{\mu }_e}\left\{\frac{[1-(M/M_{\mathrm{C}\mathrm{h}}{)}^{4/3}{]}^{1/2}}{(M/M_{\mathrm{C}\mathrm{h}}{)}^{1/3}}\right\},$

where,

$\frac{M_{\mathrm{C}\mathrm{h}}}{M_{\odot }}=\frac{5.742}{{\mu }_e^2}.$

Eggleton Mass-Radius Relationship

Verbunt & Rappaport (1988) introduced the following approximate, analytic expression for the mass-radius relationship of a "completely degenerate $\dots$ star composed of pure helium" (i.e., ${\mu }_e=2$), attributing the expression's origin to Eggleton (private communication):

$\frac{R}{R_{\odot }}=0.0114\left[\right(\frac{M}{M_{\mathrm{C}\mathrm{h}}}{)}^{-2/3}-\left(\frac{M}{M_{\mathrm{C}\mathrm{h}}}{)}^{2/3}{]}^{1/2}\right[1+3.5(\frac{M}{M_p}{)}^{-2/3}+(\frac{M}{M_p}{)}^{-1}{]}^{-2/3},$

where $M_p$ is a constant whose numerical value is $0.00057M_{\odot }$. 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 $M/M_p\gg 1$, it seems clear that Eggleton's contribution was the insertion of the term in square brackets involving the ratio $M/M_p$ 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,

$R_{\mathrm{e}\mathrm{q}}$

$=$

$\left[\right(\frac{G}{K_n}{)}^n{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{n-1}{]}^{1/(n-3)}\left[\frac{4\pi }{(n+1{)}^n}{]}^{1/(n-3)}\right[(-{\theta }^{\text{'}}){\xi }^2{]}_{{\xi }_1}^{(1-n)/(n-3)}{\xi }_1.$

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

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{1-n}$	$=$	$R_{\mathrm{e}\mathrm{q}}^{3-n}\left(\frac{G}{K_n}{)}^n\right[\frac{4\pi }{(n+1{)}^n}\left]\right[(-{\theta }^{\text{'}}){\xi }^2{]}_{{\xi }_1}^{1-n}{\xi }_1^{n-3}$
$\Rightarrow M_{\mathrm{t}\mathrm{o}\mathrm{t}}$	$=$	$R_{\mathrm{e}\mathrm{q}}^{(3-n)/(1-n)}[\frac{G}{(n+1)K_n}{]}^{n/(1-n)}(4\pi {)}^{1/(1-n)}[(-{\theta }^{\text{'}}){\xi }^2{]}_{{\xi }_1}{\xi }_1^{(n-3)/(1-n)}$
	$=$	$4\pi R_{\mathrm{e}\mathrm{q}}^{(3-n)/(1-n)}\left[\frac{(n+1)K_n}{4\pi G}{]}^{n/(n-1)}\right[(-{\theta }^{\text{'}}){\xi }^2{]}_{{\xi }_1}{\xi }_1^{(n-3)/(1-n)}$

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 ${\xi }_1$, 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)

$M$ $=$ $4\pi R^{(3-n)/(1-n)}[\frac{(n+1)K}{4\pi G}{]}^{n/(n-1)}{\xi }_1^{(3-n)/(1-n)}{\xi }_1^2|{\theta }^{\text{'}}({\xi }_1)|.$       (Eq. 3.3.11)

$\mathrm{\Gamma }=\frac{4}{3},$       $n=3,$       ${\xi }_1=6.89685,$       ${\xi }_1^2|{\theta }^{\text{'}}({\xi }_1)|=2.01824.$               (Eq. 3.3.12)

$\mathrm{\Gamma }=\frac{4}{3},$       $K=\frac{3^{1/3}{\pi }^{2/3}}{4}\frac{\hslash c}{m_u^{4/3}{\mu }_e^{4/3}}=\frac{1.2435\times 10^{15}}{{\mu }_e^{4/3}}\mathrm{c}\mathrm{g}\mathrm{s}.$                   (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, $\mathrm{\Gamma }={\gamma }_g=4/3$ — that is, $n=3$ — and acknowledging as we have above that, for isolated $n=3$ polytropes,

$m_3\equiv (-{\xi }^2\frac{d{\theta }_3}{d\xi }{)}_{\mathrm{\xi }\mathrm{=}{\mathrm{\xi }}_{\mathrm{1}}\mathrm{(}{\mathrm{\theta }}_{\mathrm{3}}\mathrm{)}}=2.01824$,

this polytropic expression for the mass becomes,

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$

$=$

$4\pi m_3[\frac{K_3}{\pi G}{]}^{3/2}.$

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),

$K_3$

$=$

$\left(\frac{3{\pi }^2}{2^6}{)}^{1/3}\right[\frac{{\hslash }^3{c}^3}{m_u^4{\mu }_e^4}{]}^{1/3}=[\frac{3h^3{c}^3}{2^9\pi m_u^4{\mu }_e^4}{]}^{1/3}.$

Together, then, the [ST83] analysis gives,

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$

$=$

$4\pi m_3\left[\frac{1}{\pi G}{]}^{3/2}\right[\frac{3h^3{c}^3}{2^9\pi m_u^4{\mu }_e^4}{]}^{1/2}.$

Given that the definitions of the characteristic Fermi pressure, $A_{\mathrm{F}}$, and the characteristic Fermi density, $B_{\mathrm{F}}$, are,

$A_{\mathrm{F}}\equiv \frac{\pi m_e^4{c}^5}{3h^3}$

$\frac{B_{\mathrm{F}}}{{\mu }_e}\equiv \frac{8\pi m_p}{3}(\frac{m_{e}c}{h}{)}^3,$

we have,

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$	$=$	$4\pi m_3\left[\frac{2A_F}{\pi G}{]}^{3/2}\frac{{\mu }_e^2}{B_F^2}\right[\frac{3h^3{c}^3}{2^9\pi m_u^4{\mu }_e^4}{]}^{1/2}\left[\frac{8\pi m_p}{3}\right(\frac{m_{e}c}{h}{)}^3{]}^2[\frac{3h^3}{2\pi m_e^4{c}^5}{]}^{3/2}$
	$=$	$4\pi m_3\left[\frac{2A_F}{\pi G}{]}^{3/2}\frac{{\mu }_e^2}{B_F^2}\right\{\frac{3h^3{c}^3}{2^9\pi m_u^4{\mu }_e^4}\cdot \frac{2^{12}{\pi }^4{m}_p^4}{3^4}(\frac{m_{e}c}{h}{)}^{12}\cdot \frac{3^3{h}^9}{2^3{\pi }^3{m}_e^{12}{c}^{15}}{\}}^{1/2}$
	$=$	$4\pi m_3\left[\frac{2A_F}{\pi G}{]}^{3/2}\frac{{\mu }_e^2}{B_F^2}\right[\frac{m_p}{m_u{\mu }_e}{]}^2$
$\Rightarrow {\mu }_e^2{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}$	$=$	$4\pi m_3\left[\frac{2A_F}{\pi G}{]}^{3/2}\frac{{\mu }_e^2}{B_F^2}\right[\frac{m_p}{m_u}{]}^2,$

which matches the expression presented above for the Chandrasekhar mass if we set $m_u=m_p$.

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

• S. Chandrasekhar (1931), ApJ, 74, p. 81: The Maximum Mass of Ideal White Dwarfs

• J. P. Ostriker, P. Bodenheimer & D. Lynden-Bell (1966), Phys. Rev. Letters, 17, 816: Equilibrium Models of Differentially Rotating Zero-Temperature 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 $M_3$ 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 $M_3$ 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.