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The Chandrasekhar limit

One curious feature of white-dwarf stars is that their radius decreases as their mass increases [see Eq. (696)]. It follows, from Eq. (689), that the mean energy of the degenerate electrons inside the star increases strongly as the stellar mass increases: in fact, $K\propto M^{4/3}$. Hence, if $M$ becomes sufficiently large the electrons become relativistic, and the above analysis needs to be modified. Strictly speaking, the non-relativistic analysis described in the previous section is only valid in the low mass limit $M\ll M_\odot$. Let us, for the sake of simplicity, consider the ultra-relativistic limit in which $p\gg m\,c$.

The total electron energy (including the rest mass energy) can be written

K = \frac{3\,V}{{\mit\Lambda}^3}\int_0^{p_F} (p^2\,c^2
\end{displaymath} (697)

by analogy with Eq. (688). Thus,
K\simeq \frac{3\,V\,c}{{\mit\Lambda}^3}\int_0^{p_F} \left(p^3
+ \frac{m^2\,c^2}{2}\,p + \cdots\right)dp,
\end{displaymath} (698)

K \simeq \frac{3}{4}\,\frac{V\,c}{{\mit\Lambda}^3}
\left[p_F^{\,4} + m^2\,c^2\,p_F^{\,2}+\cdots \right].
\end{displaymath} (699)

It follows, from the above, that the total energy of an ultra-relativistic white-dwarf star can be written in the form

E \simeq \frac{A-B}{R} + C\,R,
\end{displaymath} (700)

$\displaystyle A$ $\textstyle =$ $\displaystyle \frac{3}{8}\left(\frac{9\pi}{8}\right)^{1/3} \!\hbar\,c\left(\frac{M}{m_p}
\right)^{4/3},$ (701)
$\displaystyle B$ $\textstyle =$ $\displaystyle \frac{3}{5}\,G\,M^2,$ (702)
$\displaystyle C$ $\textstyle =$ $\displaystyle \frac{3}{4}\,\frac{1}{(9\pi)^{1/3}}\,
\frac{m^2\,c^3}{\hbar} \left(\frac{M}{m_p}\right)^{2/3}.$ (703)

As before, the equilibrium radius $R_\ast$ is that which minimizes the total energy $E$. However, in the ultra-relativistic case, a non-zero value of $R_\ast$ only exists for $A-B>0$. When $A-B<0$ the energy decreases monotonically with decreasing stellar radius: in other words, the degeneracy pressure of the electrons is incapable of halting the collapse of the star under gravity. The criterion which must be satisfied for a relativistic white-dwarf star to be maintained against gravity is that
\frac{A}{B} > 1.
\end{displaymath} (704)

This criterion can be re-written
M< M_C,
\end{displaymath} (705)

M_C = \frac{15}{64}\,(5\pi)^{1/2} \,\frac{(\hbar\,c/G)^{1/2}}{m_p^{\,2}}=
\end{displaymath} (706)

is known as the Chandrasekhar limit, after A. Chandrasekhar who first derived it in 1931. A more realistic calculation, which does not assume constant density, yields
M_C = 1.4\,M_\odot.
\end{displaymath} (707)

Thus, if the stellar mass exceeds the Chandrasekhar limit then the star in question cannot become a white-dwarf when its nuclear fuel is exhausted, but, instead, must continue to collapse. What is the ultimate fate of such a star?

next up previous
Next: Neutron stars Up: Quantum statistics Previous: White-dwarf stars
Richard Fitzpatrick 2006-02-02