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Neutron Degeneracy and Neutron Stars

Pablo CalvaPhysics Department, Drexel University, Philadelphia, PA 19104

May 16, 010

Abstract

!ravity never stops attractin" matter to other massive ob#ects$ its p%ll can brin" "as to"ether &romacross space to &orm stars and is also responsible &or the stars demise' (e%trons in a collapsin" star )illprovide a *%ant%m mechanically derived press%re to maintain the star in e*%ilibri%m' De"eneracy o&ener"y states o& the ne%trons, explained by *%ant%m mechanics as a byprod%ct o& a predetermined )ayin )hich the particles can occ%py the states, )ill prevent the collapse o& stars )ith the re*%isite massinto blac+ holes'


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1' ntrod%ction

-iolent deaths on .arth, more o&ten than not lost in the &lo) o& history, have also been catalysts&or the &ormation o& ne) li&e as )ell as t%rnin" points in h%man history' /i"ni&icance in s%ch deathsd%rin" .arths history, &rom the incineration o& the dinosa%rs to the cr%ci&ixion o& es%s, pales in

comparison to that associated )ith the death o& the &irst stars ever &ormed, )hose demise pop%lated the%niverse )ith the materials necessary &or the emer"ence o& or"anic li&e' /tars all be"in to die as the sel&cannibali2ation o& their bodies, necessary as a &%el so%rce &or the n%clear &%sion processes at their core,becomes a &r%itless search &or &%sible elements' 3nce a star can no lon"er &%se eno%"h &%el toco%nteract the same "ravitational &orce that po)ered its i"nition, it )ill "o thro%"h several sta"es o&death dictated solely by ho) m%ch mass the star contains' otal death &or a star o& s%&&icient massinvolves a trans&ormation into a blac+ hole, a metamorphosis preceded by a sta"e d%rin" )hich theremnant core is composed entirely o& ne%trons' .xtreme press%re ca%sed by "ravitys ta+e over o& thestars li&e is a"ain the ca%se o& a co%nter5ins%r"ent press%re bro%"ht &orth by the *%ant%m mechanicale&&ects that dominate the behavior o& the ne%trons'

(e%tron stars represent a class o& stars in a speci&ic mass ran"e that, thro%"h the de"eneracypress%re exerted by the ne%trons, )ill o&&er eno%"h resistance to balance "ravitys death "rip' (e%troncreation in stars that have collapsed beyond the )hite d)ar& sta"e is d%e to the interaction o& electrons

and protons, modeled by the inverse beta decay reaction pe1'6Me-nve ' Altho%"hne%trons &ormed in this manner are %s%ally %nstable and decay in a matter o& min%tes, the )hite d)ar&sta"e sa) a de"eneracy in the electron "as that res%lted in all available electron states bein" &illed' Asno electrons can &orm )ith ener"y belo) or e*%al to 1'6Mev, the ne%trons )ill not decay via a betareaction to &orm an electron and a proton' Alon" )ith ne%trons, the inverse beta decay reaction also&orms ne%trinos' .no%"h ne%trons are &ormed so that the ne%trinos &l%x, co%pled )ith the ti"htlypac+ed matter in the s%rro%ndin" star, is eno%"h to be"in co%nteractin" the "ravitational collapse o& thestar' As the ne%trons become de"enerate and occ%py hi"her and hi"her ener"y states, they exert thede"eneracy press%re that i& prod%ced in a small eno%"h star )ill balance the e&&ects o& "ravity' n stars)hose mass prevents the e*%ilibri%m bet)een de"eneracy press%re and "ravity, the contin%o%scr%shin" o& "ravity leads inexorably to the next sta"e in star death, a blac+ hole'

n order to *%anti&y ne%tron de"eneracy press%re, a *%ant%m mechanical approach to a systemo& ne%trons inside a three dimensional box in )hich the potential is 2ero, described in section , )ill be%sed to prod%ce a model o& the ne%trons in a star' /ection )ill detail the "ravitational collapse o& astar and )ill contain the derivation o& an expression o& ener"y, )hich )ill be %sed to calc%late thepress%re d%e to "ravity in the star' /ection 4 )ill %se the expression obtained &rom section &or thetotal ener"y contained in the de"enerate system, )hose relevance is predicated by the &act that the

press%re exerted by a system on its s%rro%ndin"s can be calc%lated by dE

dV' 3& interest is the

radi%s o& a ne%tron star' Calc%lated in section 7, this radi%s )ill be the limit at )hich ne%tronde"eneracy s%ccess&%lly balances "ravity to prod%ce a stable star'


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' 8ree (e%tron !as De"eneracy

(e%trons constrained to a box o& len"th )ith vol%me V=L constit%te a system that can be st%died

by ma+in" the simpli&yin" ass%mptions that ne%trons cannot exist o%tside the box, experience none%tron5ne%tron interactions and are &ree o& any other potential' /%ch ass%mptions lead to the &ollo)in"

&orm o& the /chr:din"er e*%ation;

mx , y , z =Ex , y , z


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N=1

>

4

R

=

1

R


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' !ravitational Collapse

n order to stay a ne%tron star, the dyin" core needs to exert eno%"h press%re thro%"h the ne%tronde"eneracy to balance the "ravitational &orce drivin" the collapse' !ravity acts to "ather all material ina system to the center o& mass' 8or a star o& radi%s r , ass%med %ni&orm density , and mass

mstar=4 r

, the )or+ dW "ravity does to add mass dm=4 dr to the star can )ritten

in terms o& its radi%s, the n%mber o& ne%trons and the mass in the &ollo)in" )ay;

dW=Gmstar

r dm


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4' De"eneracy Press%re

he ne%tron de"eneracy press%re is "iven by applyin" the derivative on e*%ation 1 in the same)ay as the "ravitational press%re;

Pdegeneracy=d E degeneracydV =

17 mn (

7

V

7

=

17 mn nn

7

and 19 can be e*%ali2ed and solved &or the vol%me -;

Pgrav=Pdegeneracy=17 4

1 G M

V

4

=

17mn( 7 V

7


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6' Concl%sion

3%tlined in this paper is an example o& the connection bet)een *%ant%m mechanics and the&orce o& "ravity' n the trans&ormation o& a star &rom a &%sion reactor to its stint as a ne%tron star,"ravity )as opposed in its habit%al attractive tendencies by a press%re )hose ori"in rests solely in

*%ant%m mechanics' Ultimately dependent on the amo%nt o& matter present in the star, its path can leadto the point o& e*%ilibri%m bet)een ne%tron de"eneracy press%re and "ravitational press%re' As "ravitycr%shes the stellar material )ith no resistance &rom &%sion reactions, the electrons and protons arebro%"ht close eno%"h to &%se into ne%trons, )hose collective ener"y &ills every available state %p to the

8ermi ener"y E f=

m

nn ' 8ermi ener"y is o& the same so%rce as that &rom )hich ne%tron

de"eneracy press%re arises, namely the &ermion nat%re o& the ne%tron that allo)s t)o ne%trons o& thesame ener"y b%t )ith di&&erent spin to occ%py the same state' !ravity contin%es in its attempt tocompress the star, )hich serves to impart the ener"y on the ne%trons, ca%sin" the pop%lation o& hi"herener"y states that p%sh o%t a"ainst the collapse' A star that is in e*%ilibri%m )ith "ravity )ill have the

theoretical radi%s R=>10

16

1

GmnN

1/, )hich is determined by the amo%nt o& ne%trons present

and their mass'


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?' @e&erences

!ri&&iths, David ' ntrod%ction to %ant%m Mechanics' Upper /addle @iver, ($ Prentice Ball,007'

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