1.2 Neutron Stars

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The electron energies available in a White Dwarf are of the order of the Fermi

energy. Necessarily EF

<_

mec2 since the electrons are otherwise relativistic

and cannot support the star. A White Dwarf is therefore stable against

inverse _-decay

e􀀀 + p+ ! n + _e (1.20)

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since the reaction needs energy of at least (_mn)c2 where _mn is the

neutron-proton mass di_erence. Clearly _m > me (_-decay would otherwise

be impossible) and in fact _m _ 3me. So we need energies of order of

3mec2 for inverse _-decay. This is not available in White Dwarf stars but for

M > MC the star must continue to contract until EF _ (_mn)c2. At this

point inverse _-decay can occur. The reaction cannot come to equilibrium

with the reverse reaction

n + _e ! e􀀀 + p+ (1.21)

because the neutrinos escape from the star, and _-decay,

n ! e􀀀 + p+__e (1.22)

cannot occur because all electron energy levels below E < (_mn)c2 are

_lled when E > (_mn)c2. Since inverse _-decay removes the electron degeneracy

pressure the star will undergo a catastrophic collapse to nuclear

matter density, at which point we must take neutron-degeneracy pressure

into account.

Can neutron-degeneracy pressure support the star against collapse?

The ideal gas approximation would give same result as before but with

me ! mp. The critical mass MC is independent of me and so is una_ected,

but the critical radius is now

_

me

mp

_

RC _

1

m2p

_

~3

Gc

_1=2

_

GMC

c2 (1.23)

which is the Schwarzschild radius, so the neglect of GR e_ects was not

justi_ed. Also, at nuclear matter densities the ideal gas approximation is

not justi_ed. A perfect uid approximation is reasonable (since viscosity

can't help). Assume that P(_) (_ = density of uid) satis_es

i) P _ 0 (local stability). (1.24)

ii) P0 < c2 (causality). (1.25)

Then the known behaviour of P(_) at low nuclear densities gives

Mmax _ 3M_: (1.26)

More massive stars must continue to collapse either to an unknown new

ultra-high density state of matter or to a black hole. The latter is more

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likely. In any case, there must be some mass at which gravitational collapse

to a black hole is unavoidable because the density at the Schwarzschild

radius decreases as the total mass increases. In the limit of very large mass

the collapse is well-approximated by assuming the collapsing material to be

a pressure-free ball of uid. We shall consider this case shortly.

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