16.2. THEFREEELECTRONGAS 261
Thepressurewithwhichanyobjectresistscompressionisgivenby
p=−
dET
dV
(16.42)
whereET istheenergyoftheobject. Foracoldelectrongas,thisisknownasthe
degeneracypressure. ThebulkmodulusBdescribesthechangeinpressureper
fractionaldecreaseinvolume,i.e.
B = −V
dp
dV
= −V
d^2 ET
dV^2
=
π^3 ̄h^2
9 m
( 3 N
π
) 5 / 3
V−^5 /^3
=
π^3 ̄h^2
9 m
(
3 ne
π
) 5 / 3
(16.43)
Rathersurprisingly, givenallthe unrealisticassumptions that havegone into this
expression,itgivesanswersthatcomparereasonablywell(within30%insomecases)
totheexperimentalresults.
Finally,weturntotheneutronstar. Theradiusofthedeadstarisdetermined
byabalancebetweentheinwardgravitationalpressure,andtheoutwarddegeneracy
pressurewhichisdue tothefact thatneutrons, like electrons,arespin^12 objects.
Thedegeneracypressureisthesameexpressionforneutronstarsasitisforordinary
solids,theonlydifferencebeingthattheelectronmassmshouldbereplacedbythe
neutronmassmn,theelectronnumberdensitynebytheneutronnumberdensitynn,
andNisthenumberofneutrons. Thedegeneracypressureis
pD = −
dET
dV
=
̄h^2 π^3
15 mn
( 3 N
π
) 5 / 3
V−^5 /^3
=
̄h^2 π^3
15 mn
(
3 nn
π
) 5 / 3
(16.44)
whereET isthetotalkineticenergyoftheneutronsuptotheFermilevel. Onthe
otherhand,thegravitationalpotentialenergyofasphereofconstantdensityρand
radiusRisgivenby
EG=−
(4π)^2
15
Gρ^2 R^5 (16.45)
Usingρ=M/V,whereMisthemassofthestar,andV =^43 πR^3 ,wegetanexpression
forEGintermsofV
EG=−
3
5
(
4 π
3
) 1 / 3
GM^2 V−^1 /^3 (16.46)