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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,
. Hence, if
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
.
Let us, for the sake of simplicity, consider the ultra-relativistic
limit in which
.
The total electron energy (including the rest mass energy) can be
written
![\begin{displaymath}
K = \frac{3\,V}{{\mit\Lambda}^3}\int_0^{p_F} (p^2\,c^2
+m^2\,c^4)^{1/2}\,p^2\,dp,
\end{displaymath}](img1622.png) |
(697) |
by analogy with Eq. (688). Thus,
![\begin{displaymath}
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}](img1623.png) |
(698) |
giving
![\begin{displaymath}
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}](img1624.png) |
(699) |
It follows, from the above, that the total energy of an ultra-relativistic
white-dwarf star can be written
in the
form
![\begin{displaymath}
E \simeq \frac{A-B}{R} + C\,R,
\end{displaymath}](img1625.png) |
(700) |
where
As before, the equilibrium radius
is that which minimizes the
total energy
.
However, in the ultra-relativistic case, a non-zero value of
only exists
for
. When
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
![\begin{displaymath}
\frac{A}{B} > 1.
\end{displaymath}](img1632.png) |
(704) |
This criterion can be re-written
![\begin{displaymath}
M< M_C,
\end{displaymath}](img1633.png) |
(705) |
where
![\begin{displaymath}
M_C = \frac{15}{64}\,(5\pi)^{1/2} \,\frac{(\hbar\,c/G)^{1/2}}{m_p^{\,2}}=
1.72\,M_\odot
\end{displaymath}](img1634.png) |
(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
![\begin{displaymath}
M_C = 1.4\,M_\odot.
\end{displaymath}](img1635.png) |
(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: Neutron stars
Up: Quantum statistics
Previous: White-dwarf stars
Richard Fitzpatrick
2006-02-02