Lecture 12. The solution of reaction networks. Advanced Stages of Stellar Evolution II Silicon Burning and NSE ( ) = Y 2.

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1 Lctur 1 Advancd Stags of Stllar Evolution II Silicon Burning and NSE Th solution of raction ntworks. Considr th raction pair: On th othr hand, forward or "implicit" diffrncing would giv dy 1 dt = Y 1λ 1 + Y λ dy dt = Y 1λ 1 Y λ δy 1 Δt δy Δt = ( Y 1 + δy 1 )λ 1 + ( Y + δy )λ = ( Y + δy )λ + ( Y 1 + δy 1 )λ 1 An xplicit solution of th linarizd quations would b Y nw =Y old +δy whr δy 1 =( Y 1 λ 1 + Y λ ) Δt δy =(Y 1 λ 1 Y λ ) Δt For larg t, th answr could oscillat. In th usual cas that th two spcis wr not in qulibrium Y 1 λ 1 Y λ, a larg tim stp, Δt, would lad to a divrgnt valu for th chang in Y, including ngativ valus. 1 δy 1 Δt + λ 1 +δy λ 1 ( ) +δy δy 1 λ 1 ( ) = Y λ + Y 1 λ 1 Δt + λ = Y λ Y 1 λ 1 Add quations δy 1 = δy ; substituting, on also has : Y δy 1 = 1 λ 1 Y λ 1/ Δt + λ 1 + λ (if 1/Δt >> λ, sam as th xplicit solution) Evn if Δt th chang in Y is finit and convrgs on th quilibrium valu Y 1 λ 1 =Y λ In gnral an n x n matrix n = hr

2 Silicon Burning Silicon burning procds in a way diffrnt from any nuclar procss discussd so far. It is analogous, in ways, to non burning in that it procds by photodisintgration and rarrangmnt, but it involvs many mor nucli and is quit complx. Th raction 8 Si + 8 Si! ( 56 Ni) * dos not occur owing to th larg Coulomb inhibition. Rathr a portion of th silicon (and sulfur, argon, tc.) mlt by photodisintgration ractions into a sa of nutrons, protons, and alpha-particls. Ths lightr constitunts add onto th rmaining silicon and havir lmnts, gradually incrasing th man atomic wight until spcis in th iron group ar most abundant. Initial Composition for Silicon Burning Th initial composition dpnds on whthr on is discussing th innr cor or locations farthr out in th star. It is quit diffrnt,.g., for silicon cor burning in a prsuprnova star and th xplosiv varity of silicon burning w will discuss latr that gos on in th shock wav and gts jctd. In th cntr of th star, on typically has, aftr oxygn burning, and a phas of lctron captur that gos on btwn oxygn dpltion and silicon ignition: 30 Si, 34 S, 38 Ar and a lot of othr lss abundant nucli Farthr out in th silicon shll - if thr is on - on has: 8 Si, 3 S, 36 Ar, 40 Ca, tc. Historically, Si burning has only bn discussd for a 8 Si rich composition 15 5 Nutron xcss aftr oxygn cor dpltion in 15 and 5 solar mass stars. Th innr cor is bcoming incrasingly nutronizd, spcially th lowr mass stars. Quasi-quilibrium This trm is usd to dscrib a situation whr groups of adjacnt isotops, but not all isotops globally hav com into quilibrium with rspct to th xchang of n, p,, and. It bgan in non burning with 0 N + γ 16 O + α and continud to charactriz an incrasing numbr of nucli during oxygn burning. In silicon burning it bcoms th rul rathr than th xcption. A typical quasiquilibrium clustr might includ th quilibratd ractions: 8 Si 9 Si 30 Si 31 P 3 S 8 Si n n p p α By which on mans Y( 8 Si) Y n ρ λ nγ ( 8 Si) Y( 9 Si) λ γ n ( 9 Si) Y( 30 Si)Y n ρ λ nγ ( 9 Si) Y( 30 Si)λ γ n ( 30 Si) tc.

3 Lat during oxygn burning, many isolatd clustrs grow and mrg until, at silicon ignition, thr xist only two larg QE groups In th group that contains 8 Si, on can writ any abundanc 4 A A 60 (at last) Ractions blow 4 Mg,.g., 0 N(, ) 4 Mg and 1 C(, ) 16 O ar, in gnral, not in quilibrium with thir invrss (xcption, 16 O(, ) 0 N which has bn in quilibrium sinc non burning). Within th groups havir than A = 4, xcpt at th boundaris of th clustrs, th abundanc of any spcis is rlatd to that of anothr by succssiv application of th Saha quation Y ( Ca) Y ( S) Y ( Ar Y ( Ca). g., = Y ( Si) Y ( Si) Y ( S) Y ( Ar) ρyα λ ( Si) αγ ρyα λ ( S) αγ ρyα λ ( Ar) αγ = λγα ( S) λγα ( Ar) λγα ( Ca) = f T Q Y tc 3 3 (, αγ ) ρ α. whr Z-14 δα = largst intgr δ = N 14 δ δ n p δ p p δα Y Z C Z T Y Si Y Y Y A A 8 ( ) = (, ρ, 9) ( ) α = α 35 Z 14 δα. g., Cl Nd 6 paramtrs: Y, Y p, Y n and Y( 8 Si) plus T and. δ = 1 δ = 1 δ = α 40 K δn n p δ = δ = 1 δ = 3 α p n n C( A Z) = ρn A ( ) δ α +δ p +δ n C '( A Z) ( ) C '( A Z) = / ( T 9 ) δ α +δ p +δ n whr G( A Z) ( G( 8 Si) δ p +δn ) A 8 3/ 1 4 3δ α / Q = BE( A Z) BE( 8 Si) δ α BE(α) xp(q / kt ) Morovr thr xist loops lik: 3 S p % 31 P 8 Si 9 Si 30 Si n n p For constant ρ and T Y ( S) = const iy ( Si) Y 3 30 p Y ( Si) = const ' iy ( Si) Y 30 8 n 3 8 Y ( S) = const '' iy ( Si) Yα Y ( S) Y ( S) Y ( Si) Yα Y Y Y ( Si) Y ( Si) Y ( Si) p n i.., th nrgy rquird to dissociat th nuclus A Z into 8 Si and alpha particls. Th binding nrgy of a nutron or proton is zro. Y α = C α (T 9,ρ) Y n Y p C α = 1 (5.94 ) 3 ρ 3 9/ T xp(38.36 / T 9 ) whr = BE(α )/kt This rducs th numbr of indpndnt variabls to 5, but wait

4 Th situation at th nd of oxygn burning is that thr ar two larg QE groups coupld by non-quilibratd links nar A = 45. Th clustr volvs at a rat givn by 4 Mg(, ) 0 N A > 45 QE 4 A 60 4 A 45 Th non-quilibratd link has to do with th doubl shll closur at Z = N = 0 7! 0 N+ % 8 Si 4 Mg+ % ρ η 8, T9,, Y ( Si) QE Early during silicon burning ths two groups mrg and th only rmaining non-quilibratd ractions ar for A < 4. 3 % 1 C+ % 16 O+ % Th photodisintgration of 4 Mg provids a s (and n s and p s sinc Y a =C a Y n Y p ) which add onto th QE group gradually incrasing its man atomic wight. As a rsult th intrmdiat mass group, Si-Ca gradually mlts into th iron group. Th larg QE clustr that includs nucli from A = 4 through at last A = 60 contains most of th mattr ( 0 N, 16 O, 1 C, and! ar all small), so w hav th additional two constraints Natur of th burning: Lightr spcis mlt away whil th iron group grows 60 A 4 60 A 4 AY i i 1 ( N Z ) Y η i i i mass consrvation charg consrvation Th first quation can b usd to liminat on mor unknown, say Y p, and th scond can b usd to rplac Y n with an asir to us variabl,. Thus 4 variabls now spcify th abundancs of all nucli havir than magnsium. Ths ar, T 9,, and Y( 8 Si)

5 This is all a bit mislading bcaus, xcpt xplosivly (latr), silicon burning dos not produc 56 Ni. Thr has bn a lot of lctron captur during oxygn burning and mor happns in silicon burning. E.g., Si ignition in a 15 M star η c 0.07 Y 0.46 Si dpltion η c 0.13 Y 0.44 Undr ths conditions silicon burning producs 54 F, 56 F, 58 F and othr nutron rich spcis in th iron group. But this assums 8 Si burns to 56 Ni ( = 0.00 approximation) 56 Suppos Si 56 F q nuc = ( ) / 56 ( 55.6) / 30 = rg g -1 which is closr to corrct for Si cor burning than rg g 1 i.., q nuc = X i BE(A i ) A i Raction rats govrning th rat at which silicon burns: Gnrally spaking, th most critical ractions will b thos conncting quilibratd nucli with A > 4 (magnsium) with alpha-particls. Th answr dpnds on tmpratur and nutron xcss: Most frquntly, for small, th critical slow link is 4 Mg(, ) 0 N Th raction 0 N(, ) 16 O has bn in quilibrium with 16 O(, ) 0 N vr sinc non burning. At high tmpraturs and low Si-mass fractions, 0 N(, ) 4 Mg quilibrats with 4 Mg(, ) 0 N and 16 O(, ) 1 C bcoms th critical link. This is vry lik non burning xcpt that 7 alpha-particls ar involvd instad of on. Howvr for th valus of actually appropriat to silicon burning in a massiv stllar cor, th critical rat is 6 Mg(p, ) 3 Na(p, ) 0 N

6 To gt 4 Mg To gt %

7 Nuclosynthsis Basically, silicon burning in th star s cor turns th products of oxygn burning (Si, S, Ar, Ca, tc.) into th most tightly bound nucli (in th iron group) for a givn nutron xcss,.!! Th silicon-burning nuclosynthsis that is jctd by a suprnova is producd xplosivly, and has a diffrnt composition dominatd by 56 Ni. Th products of silicon-cor and shll burning in th cor ar both so nutronrich ( so larg) that thy nd to b lft bhind in a nutron star or black hol. Howvr, vn in that cas, th composition and its volution is critical to stting th stag for cor collaps and th suprnova xplosion that follows. Silicon burning nuclosynthsis Following Si-burning at th middl of a 5 solar mass star: 54 F Ni F F Co Nutron-rich nucli in th iron pak. Y = Following xplosiv Si-burning in a 5 solar mass suprnova, intrsting spcis producd at Y = to Solar Mass Pop I PrSuprnova (Woosly and Hgr 007) product parnt 44 Ca 44 Ti 47,48,49 Ti 48,49 Cr 51 V 51 Cr 55 Mn 55 Co 50,5,53 Cr 5,53 F 54,56,57 F 56,57 Ni 59 Co 59 Cu 58,60,61,6 Ni 60,61,6 Zn 44 Ti and Ni ar important targts of -ray astronomy This part will collaps and mak a shock Explosiv burning will happn hr 5 M η = nutron xcss

8 Nuclar Statistical Equilibrium As th silicon abundanc tnds towards zro (though it nvr bcoms microscopically small), th unquilibratd ractions blow A = 4 finally com into quilibrium N( α, γ ) Mg Mg( γ, α) N O( α, γ ) N N( γ, α) O (for a long tim alrady) C( α, γ ) O O( γ, α) C C C(, ) α γ α α Th 3 raction is th last to quilibrat. Onc this occurs, vry isotop is in quilibrium with vry othr isotop by strong and lctromagntic ractions (but not by wak intractions) In particular, Y( 8 Si) = f (T,ρ)Y 7 α with th rsult that now only 3 variabls, ρ, T 9,and η now spcify th abundancs of vrything Y( A Z) = C( A Z,ρ,T 9 ) Y n N Y p Z C( A Z,ρ,T 9 ) = ( ρn A ) A 1 C ' ( A Z,T 9 ) C ' ( A Z,T 9 )= G( A Z,T 9 ) θ 1 A xp BE( A Z) / kt A θ = / T 9 G( A Z,T 9 )is th tmpratur-dpndnt At low T partition function. ( ) G( A Z,T 9 ) = J i +1 E i /kt At high T, though (s arlir discussion of nuclar lvl dnsity) G( A Z,T 9 ) π 6akT a(kt ) a A 9 MV-1 Until th tmpratur bcoms vry high (kt >> 1 MV) Th most abundant nucli ar thos with larg binding nrgy pr nuclon and "natural" valus of η. For xampl, η = tc. 56 Ni 54 F 56 F In gnral, th abundanc of an isotop paks at its natural valu for η. E. g., η( 56 F) = N Z A = = Th rsultant nuclosynthsis is most snsitiv to.!

9 Tru Equilibrium If th wak intractions wr also to b balancd, (.g., nutrino captur occurring as frquntly on th daughtr nuclus as lctron captur on th parnt), on would hav a stat of tru quilibrium. Only two paramtrs, and T, would spcify th abundancs of vrything. Th last tim this occurrd in th univrs was for tmpraturs abov 10 billion K in th Big Bang. Howvr, on can also hav a dynamic wak quilibrium whr nutrino mission balancs anti-nutrino mission, i.., whn dy dt = 0 This could occur, and for som stars dos, whn lctron-captur balancs bta-dcay globally, but not on individual nucli. Th abundancs would b st by and T, but would also dpnd on th wak intraction rat st mployd. Wak Intractions Elctron captur, and at lat tims bta-dcay, occur for a varity of isotops whos idntity dpnds on th star, th wak raction rats mployd, and th stag of volution xamind. During th lat stags it is most snsitiv to ta, th nutron xcss. Asid from thir nuclosynthtic implications, th wak intractions dtrmin Y, which in turn affcts th structur of th star. Th most important isotops ar not gnrally th most abundant, but thos that hav som combination of significant abundanc and favorabl nuclar structur (spcially Q-valu) for wak dcay. From silicon burning onwards ths wak dcays provid nutrino mission that compts with and ultimatly dominats that from thrmal procsss (i.., pair annihilation). log % H dpltion O dpltion PrSN 5 M solar mtallicity Th distribution of nutron xcss,, within two stars of 5 solar masss (8 solar mass hlium cors) is rmarkably diffrnt. In th Pop I star, is approximatly 1.5 x 10-3 vrywhr xcpt in th innr cor (dstind to bcom a collapsd rmnant) O-dpltion O-shll Si-ignition Si-shll T(10 9 K) (g cm -3 ) 1. x x x x 10 7 Y captur 35 Cl, 37 Ar 35 Cl, 33 S 33 S, 35 Cl 54,55 F -dcay 3 P, 36 Cl 3 P, 36 Cl 3 P, 8 Al 54,55 Mn Intrior mass In th Pop III (Z = 0) star th nutron xcss is ssntially zro at th nd of hlium burning (som primordial nitrogn was cratd) Outsid of th cor is a fw x 10-4, chifly from wak intractions during carbon burning. Not som primary nitrogn production at th outr dg whr convction has mixd 1 C and protons. log % 5 M zro initial mtalicity H dpltion O dpltion PrSN Si-dpltion Si-shll burn Cor contraction PrSN T(10 9 K) (g cm -3 ) 5.9 x x x x 10 9 Y captur 54,55 F 57 F, 61 Ni 57 F, 55 Mn 65 Ni, 59 F -dcay 54 Mn, 53 Cr 56 Mn, 5 V 6 Co, 58 Mn 64 Co, 58 Mn Intrior mass 15 solar mass star (Hgr t al 001)

10 5 M Prsuprnova Star (typical for M ) 1400 R 0.5 R H, H H 40,000 L C,O O,Mg,N Si, S, Ar, Ca F H, C 0.1 R R (not scal braks)

11 As silicon shlls, typically on or at most two, burn out, th iron cor grows in discontinuous spurts. It approachs instability. Prssur is prdominantly du to rlativistic lctrons. As thy bcom incrasingly rlativistic, th structural adiabatic indx of th iron cor hovrs prcariously nar 4/3. Th prsnc of nondgnrat ions has a stabilizing influnc, but th cor is rapidly losing ntropy to nutrinos making th concpt of a Chandraskhar Mass rlvant. In addition to nutrino losss thr ar also two othr important instabilitis: Elctron captur sinc prssur is dominantly from lctrons, rmoving thm rducs th prssur. Photodisintgration which taks nrgy that might hav providd prssur and uss it instad to pay a dbt of ngativ nuclar nrgy gnration. Illiadis S also Clayton Fig 7-9 and discussion Dominant constitunt in NSE ( approximatly 0) Typical prsuprnova cntral conditions Th transition from th iron group to H and spcially from H to n + p absorbs an normous amount of nrgy. Figur nds corrcting for high tmpratur nuclar partition functions. Photodisintgration: from pag 1 of this lctur Y α = C α (T 9,ρ) Y n Y p C α = 1 (5.94 ) 3 ρ 3 9/ T xp(38.36 / T 9 ) whr = BE(α )/kt Stting Y = 1/8 (i.., X = ½ dividd by 4) and Y p =Y n =1/4 givs th lin for th hlium-nuclon transition on th prvious pag. Th Ni- transition is givn by Clayton problm 7-11 and qn 7-. It coms from solving th Saha quation for NSE (pag 30 of ths nots) for th cas X( 56 Ni)= X α = 0.5 Y ( 56 Ni)=1/11; Y α =1/ 8 Y ( 56 Ni)=C 56 (T 9 )ρ 1 Y α 13 Th photodisintgration of on gram of 56 Ni to on gram of -particls absorbs: q nuc = N A (δy i )(BE i ) q ν rg/gm = ( )+ 1 4 ( 8.96 ) = rg g 1 Similarly, th photodisintgration of on gram of s to on gram of nuclons absorbs: q nuc = = rg g 1 ( ) + 0 ssntially undoing all th nrgy rlasd by nuclar ractions sinc th zro ag main squnc.

12 What about dgnracy? Can th cor b supportd by lctron dgnracy prssur and form a stabl (F) dwarf? Entropy (S/N A k) Entropy Bcaus of incrasing dgnracy th concpt of a Chandraskhar Mass for th iron cor is rlvant but it must b gnralizd. Th Chandraskhar Mass Traditionally, for a fully rlativistic, compltly dgnrat gas: η >0 implis dgnracy P K ρ Y G M ρ c 4/3 c = = 4 4 c ρc K 4/3 = dyn cm 15 - M Ch = Y = M at Y = 0.50 This is oftn rfrrd to loosly as 1.4 or vn 1.44 solar masss

13 BUT 1) Y hr is not 0.50 (Y is actually a function of radius or intrior mass) ) Th lctrons ar not fully rlativistic in th outr layrs ( is not 4/3 vrywhr) 3) Gnral rlativity implis that gravity is strongr than classical and an infinit cntral dnsity is not allowd (thr xists a critical for stability) 4) Th gas is not idal. Coulomb intractions rduc th prssur at high dnsity 5) Finit tmpratur (ntropy) corrctions 6) Surfac boundry prssur (if WD is insid a massiv star) 7) Rotation Effct on M Ch Rlativistic corrctions, both spcial and gnral, ar tratd by Shapiro and Tukolsky in Black Hols, Whit Dwarfs, and Nutron Stars pags 156ff. Thy find a critical dnsity (ntropy = 0) ρc = gm cm 10-3 Y Abov this dnsity th whit dwarf is (gnral rlativistically) unstabl to collaps. For Y = 0.50 this corrsponds to a mass M Ch 1 /3 ΔM 0.50 = M Y =.87% Y = 0.50 = M in gnral, th rlativistic corrction to th Nwtonian valu is =.67% Y = 0.45 Coulomb Corrctions Thr ffcts must b summd lctron-lctron rpulsion, ion-ion rpulsion and lctron ion attraction. Clayton p givs a simplifid tratmnt and finds, ovr all, a dcrmnt to th prssur (q. -75) ΔP Coul = π 3 Z /3 4/3 n Fortunatly, th dpndnc of this corrction on n is th sam as rlativistic dgnracy prssur. On can thn just procd to us a corrctd K = K 1 Z 0 /3 4/3 4/3 = K /3 3 c 5 π Z 0 3 /3 4/3 1/3 c whr K = 3 4 and hnc ( π ) 1/3 0 4/3 4/3 A M M K = Ch 0 0 Ch K 3/ N Z 0 MCh = M Ch Putting th rlativistic and Coulomb corrctions togthr with th dpndnc on Y on has /3 1 MCh = 1.38 M for C (Y = 0.50) 56 6 = 1.15 M for F (Y = = 0.464) 56 = 1.08 M for F-cor with <Y > 0.45 So why ar iron cors so big at collaps ( M ) and why do nutron stars hav masss 1.4 M?

14 Finit Entropy Corrctions Chandraskhar (1938) Fowlr & Hoyl (1960) p 573, q. (17) Baron & Cooprstin, ApJ, 353, 597, (1990) For n = 3, γ = 4 / 3, rlativistic dgnracy 0 1 π k T Pc = K4/ Y 4 + x m c 1/3 ( ρ ) 4/3 h 3 pf x = n = = Y mc 8π mc (Clayton -48) ( ρ ) 1/3 In particular, Baron & Cooprstin (1990) show that π kt P = P o ε F F and sinc M M 3 3h n ε F = pfc = 8π ε = Ch 1/3 1/3 1.11( ρ7 Y ) MV K 3/ 4/3 0 Ch M Ch 1+ ε F a first ordr xpansion givs π kt And sinc, in th appndix to ths nots w show on also has s = π kty ε F 0 M Ch M Ch (rlativistic dgnracy) s 1+ πy +... Th ntropy of th radiation and ions also affcts M Ch, but much lss. This finit ntropy corrction is not important for isolatd whit dwarfs. Thy r too cold. But it is vry important for undrstanding th final volution of massiv stars. Bcaus of its finit ntropy (i.., bcaus it is hot) th iron cor dvlops a mass that, if it wr cold, could not b supportd by dgnracy prssur. Bcaus th cor has no choic but to dcras its lctronic ntropy (by nutrino radiation, lctron captur, and photodisintgration), and bcaus its (hot) mass xcds th Chandraskhar mass, it must vntually collaps.

15 E.g., on th following pags ar xcrpts from th final day in th lif of a 15 M star. During silicon shll burning, th lctronic ntropy rangs from 0.5 to 0.9 in th F cor and is about 1.3 in th convctiv shll. 0.7 M Ch 1.08 M 1+ π(0.45) = 1.34 M > 0.95 M Th F cor plus Si shll is also stabl bcaus 1.0 M Ch 1.08 M 1+ π(0.47) = 1.57 M > 1.3M But whn Si burning in this shll is complt: 3) Th F cor is now ~1.3 M. s cntral = 0.4 s at dg of F cor = 1.1 avrag 0.7 M Ch now about 1.34 M (uncrtain to at last a fw tims 0.01 M Nutrino losss farthr rduc s. So too do photodisintgration and lctron captur as w shall s. And th boundry prssur of th ovrlying silicon shll is not ntirly ngligibl. Elctron Th cor collapss

16 Th collaps bgins on a Klvin-Hlmholz (nutrino) tim scal and acclrats to a dynamic implosion as othr instabilitis ar ncountrd. Photodisintgration: As th tmprat and dnsity ris, th star sks a nw ful to burn, but instad ncountrs a phas transition in which th NSE distribution favors particls ovr bound nucli. In fact, this transition nvr gos to compltion owing to th larg statistical wight affordd th xcitd stats of th nucli. But considrabl nrgy is lost in a partial transformation. Rcall for s. not rally fr nutrons. Thy stay lockd insid bound nucli that ar 56 F 13 α + "4 n" progrssivly mor nutron rich qnuc ΔX q photo = rg g 18-1 i.., BE/A for a typical iron group nuclus What happns? As th dnsity riss, so dos th prssur (it nvr dcrass at th middl), but so dos gravity. Th ris in prssur is not nough to maintain hydrostatic quilibrium, i.., rmains slightly lss than 4/3. Th collaps acclrats. Photodisintgration dcrass s bcaus at constant total ntropy (th collaps is almost adiabatic), s i incrass sinc 14 particls hav mor statistical wight than on nuclus. Th incras in s i coms at th xpns of s. Elctron captur Th prssur and ntropy com mainly from lctrons, but as th dnsity incrass, so dos th Frmi nrgy, F. Th ris in F mans mor lctrons hav nough nrgy to captur on nucli turning protons to nutrons insid thm. This rducs Y which in turn maks th prssur and ntropy at a givn dnsity smallr. F ( ρ Y ) 1/3 ε = 1.11 MV 7 By x g cm -3, F = 10 MV which is abov th captur thrshold for all but th most nutron-rich nucli. Thr is also brifly a small abundanc of fr protons (up to 10-3 by mass) which capturs lctrons. But th star dos not a) photodisintgrat to nutrons and protons; thn b) captur lctrons on fr protons; and c) collaps to nuclar dnsity as a fr nutron gas as som txts naivly dscrib. Bound nucli prsist, thn finally touch and mlt into on gigantic nuclus with nuclons th nutron star. Y dclins to about 0.37 bfor th cor bcoms opaqu to nutrinos. (Y for an old cold nutron star is about 0.05; Y for th nutron star that bouncs whn a suprnova occurs is about 0.9). Th ffcts of a) xcding th Chandraskhar mass, b) photodisintgration and c) lctron captur oprat togthr, not indpndntly.

17 Zntralfridhof Vinna, Austria S = k log W Appndix on Entropy T As discussd prviously P = at + ρn A k T µ for idal gas plus radiation ( T / ρ) /( T / ρ ) = T 3/ 3 3/ dividing by k maks s dimnsionlss Eg., just th radiation part V 1/ ρ T ds = dε + P dv 3 4aT T ds = dt + at d ρ at dv + ρ ρ 3 3 4aT 4 1 dt at dv at 4 = + + dv ρ 3 3 ds = 4aT V dt + at dv = d at V 4 3 So S rad 3 4 at = 3 ρ

18 Rif Fundamntals of Statistical and Thrmal Physics McGraw Hill η = µ kt whr µ, th chmical potntial is dfind by T S = E + PV µ N (CG10.0) hnc S /V =(ε ρ + P µ n ) / T Cox and Guili Principls of Stllar Structur Scond dition A. Wiss t al Cambridg Scintific Publishrs xprssion Cox and Guili (4.76b) S = ε + P ρ µn ρ / T s = S / N A k = ε ρ + P ρn A kt ηy = 5 η Y η 0 n Y = ρ N A For an idal gas ε ρ + P = 3 +1 ρn kt A For a non-rlativistic, non-dgnrat lctron gas, Clayton -63 and -57 imply (for η<< 0) which implis n = ( πm kt )3/ h 3 ( )3/ η η / -η ln πm kt n h 3 which givs th idal gas limit for lctron ntropy (similar to ions but has Y and m ) For η>>1 (grat dgnracy) n 8π kt c 3 h 3 P 8π kt 3c 3 h 3 ( ) η3 1+ π η ( ) η4 1+ π P n kt 1 1+ π 4 η η 1+ π >>1 η η π and kping only trms in η P n kt η 1+ π 4 η 1 π η η π 1+ 4 η π η = η 4 + π 4η ( ) + 7π 4 4/3 η 15η 4 P n

19 u =3P η = µ kt µ ε F S ( ) V =(u + P µ n ) / T 4P µn T n kt η + π η ηn kt = π n k T η = π ρn A Y k η s = S ρn A k π Y η s π kty ε F and ngligibl radiation prssur (ntropy)

20