QMGreensite_merged

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)