QMGreensite_merged

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)