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One curious feature of whitedwarf 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 nonrelativistic
analysis described in the previous section
is only valid in the low mass limit .
Let us, for the sake of simplicity, consider the ultrarelativistic
limit in which .
The total electron energy (including the rest mass energy) can be
written

(697) 
by analogy with Eq. (688). Thus,

(698) 
giving

(699) 
It follows, from the above, that the total energy of an ultrarelativistic
whitedwarf star can be written
in the
form

(700) 
where
As before, the equilibrium radius is that which minimizes the
total energy .
However, in the ultrarelativistic case, a nonzero 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 whitedwarf
star to be maintained against gravity is that

(704) 
This criterion can be rewritten

(705) 
where

(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

(707) 
Thus, if the stellar mass exceeds the Chandrasekhar limit then the star in question
cannot become a whitedwarf 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: Whitedwarf stars
Richard Fitzpatrick
20060202