262 CHAPTER16. LIVEWIRESANDDEADSTARS
Thentheinwardgravitationalpressureis
pG = −
dEG
dV
= −
1
5
( 4 π
3
) 1 / 3
GM^2 V−^4 /^3 (16.47)
Finally,requiringthattheinward(pG)andoutward(pD)pressuresbalance,
1
5
( 4 π
3
) 1 / 3
GM^2 V−^4 /^3 =
̄h^2 π^3
15 mn
( 3 N
π
) 5 / 3
V−^5 /^3 (16.48)
solvingforV andthenforR=(3V/ 4 π)^1 /^3 ,weget
R=
(
81 π^20
16
) 1 / 3
h ̄^2
Gm^3 n
N−^1 /^3 (16.49)
Foratwosolarmassstar,N=M/mn,Rturnsouttobeabout 10 km.
Itseemsfromeq. (16.49)thatno matterhowmassivethestar(i.e. nomatter
howlargeN)thedegeneracypressurewillalwaysendthecollapseatafinitevalueof
R. Actuallythisisnotquiteright,becausewehaveignoredrelativisticeffects. The
freeSchrodingerequationisbasedonE=p^2 / 2 m,i.e. Egrowsquadraticallywith
p.Infact,thecorrectexpressioninrelativityisE=
√
p^2 c^2 +m^2 c^4 ,andatvelocities
veryclosetothespeedoflight,theenergyisapproximatelyE≈pc;i.e. theenergy
growsonlylinearlywithp. Thismeansthatthedegeneracypressureforrelativistic
neutronsislessthatonewouldexpectfromthenon-relativisticcalculation. Infact,
forstarsataboutfoursolarmasses,theneutronsinthestardobecomerelativistic,
andthedegeneracypressureisinsufficienttocounterthegravitationalpressure. For
starsofthatmassandgreater,no knownphysicalprocess canstopthecontinuing
collapse,andthestarbecomesablackhole.
Forastarwithamasssomewhatlessthanthatofthesun,thecoldstarnever
reachestheneutronstarstage,sincetheelecrondegeneracypressureissufficientto
stabilizethestaratradiiontheorderof 10 , 000 km. Formassessomewhatgreater
thanthatofthesun,theelectronsbecomesrelativistic,andthecollapsecontinues
until,formasseslessthanfoursolarmasses,thestarbecomesaneutronstar.
Beforeleavingthetopic,itishardtoresistmentioningonelastaspectofelectron
spin. Inthe1920’sDiracwrotedownarelativisticequationforelectrons,whichnot
onlypredictedthatelectronsshouldhavespin^12 ,butalsopredictedthemagnitude
of theelectron magneticmoment,with g-factorg = 2. Therewas just onelittle
problem:inadditiontofreeelectronswithpositiveenergy
E=
√
p^2 c^2 +m^2 c^4 (16.50)