Lecture 12 Advanced Stages of Stellar Evolution – II Silicon Burning and NSE Silicon Burning Silicon burning proceeds in a way different from any nuclear process discussed so far. It is analogous, in ways, to neon burning in that it proceeds by photodisintegration and rearrangement*, but it involves many more nuclei and is quite complex. The reaction 28Si + 28Si à (56Ni)* does not occur owing to the large Coulomb inhibition. Rather a portion of the silicon (and sulfur, argon, etc.) melt by photodisintegration reactions into a sea of neutrons, protons, and alpha-particles. These lighter constituents add onto the remaining silicon and heavier elements, gradually increasing their mean atomic weight until species in the iron group are most abundant. Carbon burning Heavy ion fusion Neon Burning Photodisintegration rearrangement Oxygen burning Heavy ion fusion Silicon burning Photodisintegration rearrangement *Basically the temperature threshold for removing an alpha from 24Mg is reached before that of 28Si+28Si Initial Composition The initial composition is mostly Si and S, but which isotopes of Si and S dominate depends upon whether one is discussing the inner core or less dense locations farther out in the star. It is quite different for silicon core burning in a presupernova star and the explosive variety of silicon burning we shall discuss later. In the center of the star, one typically has, after oxygen burning, and a phase of electron capture that goes on between oxygen depletion and silicon ignition: 30Si, 34S, 38Ar and a lot of other less abundant nuclei. High η Farther outside of the core where silicon might burn explosively, one has species more characteristically with Z = N 28Si, 32S, 36Ar, 40Ca, etc. Historically, Si burning was discussed for a 28Si rich composition. Low η 15 Neutron excess after oxygen core depletion η in 15 and 25 solar mass stars. The inner core is becoming increasingly neutronized, especially for the lower mass stars. The process accelerates during silicon burning 25 The nucleosynthesis of the η inner core would be very strange were it to be ejected (it is not). η ~ 0.002 – 0.004 is good. 0.01 is not Quasi-equilibrium This term is used to describe a situation where groups of adjacent isotopes, but not all isotopes globally, have come into equilibrium with respect to the exchange of n, p, a, and g. It began in neon burning with 20Ne + γ ! 16O + α and continues to characterize an increasing number of nuclei during oxygen burning. In silicon burning, it becomes the rule rather than the exception. A typical "quasiequilibrium cluster" might include the equilibrated reactions : 28Si ! 29Si ! 30Si ! 31P ! 32S ! 28Si n n p p α 28 28 29 29 By which one means Y( Si) Yn ρ λnγ ( Si) ≈ Y( Si) λγ n ( Si) 30 29 30 30 Y( Si)Yn ρ λnγ ( Si) ≈ Y( Si)λγ n ( Si) etc. Late during oxygen burning, many isolated clusters grow and merge until, at silicon ignition, there exist only two large QE groups 24 ≤ A≤ 45 46 ≤ A≤60 (at least) Reactions below 24Mg, e.g., 20Ne(a,g)24Mg and 12C(a,g)16O are, in general, not in equilibrium with their inverses at oxygen depletion (exception, 16O(a,g)20Ne which has been in equilibrium since neon burning). Within the groups heavier than A = 24, except at the boundaries of the clusters, the abundance of any species is related to that of another by successive application of the Saha equation. Y (40 Ca) ⎛ Y (32 S) ⎞⎛ Y (36 Ar ⎞⎛ Y (40 Ca) ⎞ e.g., 28 = ⎜ 28 ⎟⎜ 32 ⎟⎜ 36 ⎟ Y ( Si) ⎝ Y ( Si) ⎠⎝ Y ( S) ⎠⎝ Y ( Ar) ⎠ 28 32 36 ⎛ ρYα λαγ ( Si) ⎞⎛ ρYα λαγ ( S) ⎞⎛ ρYα λαγ ( Ar) ⎞ = ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ (32 S) ⎟⎜ (36 Ar) ⎟⎜ (40 Ca) ⎟ ⎝ λγα ⎠⎝ λγα ⎠⎝ λγα ⎠ 3 3 = f (T,Qαγ ) ρ Yα etc. The situation at the end of oxygen burning is that there are two large QE groups coupled by non-equilibrated links near A = 45. A > 45 The non-equilibrated link 24 ≤ A ≤ 45 has to do with the double shell closure at Z = N = 20 Early during silicon burning these two groups merge and the only remaining non-equilibrated reactions are for A < 24 (Mg). Within that one group,(A > 23), which contains 28Si and the vast majority of the mass, one can evaluate any abundance relative to e.g., 28Si δ A A 28 δα p δ n Y( Z) = C( Z,ρ,T9 )Y( Si)Yα Yp Yn Z-14 where δ = largest integer ≤ Need 6 parameters: Ya, Yp, Yn and α 2 Y(28Si) plus T and r, but … δ n = N −14 − 2δα 35 δ p = Z −14− 2δα e.g., Cl 17 protons; 20 neutrons δα =1 δ p = 1 δ n = 2 40 K 19 protons 21 neutrons δ = 2 δ = 1 δ = 3 α p n A δα +δ p +δn A C( Z) = (ρN A ) C '( Z) A 33 3/2 −(δα +δ p +δn ) C '( Z) = (5.942×10 T9 ) A 3/2 3δα /2 G( Z) − δ p +δn ⎛ A ⎞ ⎛ 1 ⎞ 2 ( ) exp(Q / kT ) G(28 Si) ⎝⎜ 28⎠⎟ ⎝⎜ 4 ⎠⎟ where A 28 Q = BE( Z) − BE( Si) − δα BE(α) i.e., the energy required to dissociate the nucleus AZ into 28 Si and d a alpha particles. The binding energy of a neutron or proton is zero. Moreover there exist loops like: 32S For constant ρ and T 32 30 2 p Y ( S) = const iY ( Si)Yp a 31 30 28 2 P Y ( Si) =const 'iY ( Si)Yn 32 28 p Y ( S) =const ''iY ( Si)Yα 28 29 30 32 32 30 Si Si Si ⎛ Y ( S) ⎞ ⎛ Y ( S) ⎞⎛ Y ( Si) ⎞ 2 2 n n Yα ∝⎜ 28 ⎟ ∝⎜ 30 ⎟⎜ 28 ⎟∝Yp Yn ⎝ Y ( Si) ⎠ ⎝ Y ( Si) ⎠⎝ Y ( Si) ⎠ 2 2 Yα = Cα (T9 ,ρ) Yn Yp 1 C = (5.94 ×1033 )−3 ρ 3T 9/2 exp(328.36 / T ) α 2 9 9 where 328.36 = BE(α )/kT This reduces the number of independent variables to 5, but wait … The large QE cluster that includes nuclei from A = 24 through at least A = 60 contains most of the matter (20Ne, 16O, 12C, and a are all small), so we have the additional two constraints AY 1 mass conservation ∑ i i ≈ 60≥ A≥24 (N Z ) Y ∑ i − i i ≈ η charge conservation 60≥ A≥24 The first equation can be used to eliminate one more unknown, say Yp, and the second can be used to replace Yn with an easier to use variable, η. Thus 4 variables now specify the abundances of all nuclei heavier than magnesium . These are 28 r, T9, h, and Y( Si) For low η, the cluster evolves at a rate given by 24Mg(g,a)20Ne A A 28 δ δ δ Y( Z) = C( Z,ρ,T )Y( Si)Y α Y pY n 9 α p n QE 24 ≤ A ≤ 60 28Si 24 28 Mg+α ρ,T9 ,η,Y ( Si)⇒QE 7α 20Ne+α The photodisintegration of 24Mg 16O+α provides as (and ns and ps since 2 2 12C+α Ya =CaYn Yp ) which add onto the QE group gradually increasing its mean atomic weight. As a 3α result the intermediate mass group, Si-Ca gradually melts into the iron group. Nature of the burning: Lighter species melt away while the iron group grows Energetics: Suppose 28Si burns to 56Ni. To rough approximation 2(28Si) -> 56Ni (n.b. not fusion of 2 silicons) 17 qnuc = 9.65 x 10 [1/2 (483.982 -236.536)/28] = 1.9 x 1017 erg g-1 (not much) But this assumes 28Si burns to 56Ni ( small η approximation) This is misleading because, except explosively (later), silicon burning does not produce 56Ni. There has been a lot of electron capture during oxygen burning and more happens in silicon burning. The silicon that burns is not 28Si, but more typically 30Si. E.g., Si ignition in a 15 M star ηc≈0.07 Ye ≈ 0.46 Si depletion 0.13 Y 0.44 ηc ≈ e ≈ Under these conditions silicon burning produces 54Fe, 56Fe, 58Fe and other neutron rich species in the iron group. ⎛ 56⎞ 30 56 X Suppose ⎜ ⎟ Si → Fe i.e., q = 9.65×1017 i BE(A ) ⎝ ⎠ nuc ∑ i 30 Ai 17 qnuc = 9.65 × 10 ⎣⎡ (492.26) / 56−(255.62) / 30⎦⎤ = 2.6 × 1017 erg g-1 which is closer to correct for Si core burning than 1.9 × 1017 erg g−1 An approximation to energy generation is derived in Appendix 2. See also next page It depends on T47 This approximation implies that Si buring will achieve balanced power at 3.5 billion degrees with a generation rate approximately 1013 erg g-1 s-1 The approximate lifetime is thus q ΔX(28Si) 2.6 ×1017 (1) ∼ 13 ~ 7 hours εnuc 10 Shell burning and convection can lengthen this to days to weeks Nucleosynthesis Basically, silicon burning in the stars core turns the products of oxygen burning (Si, S, Ar, Ca, etc.) into the most tightly bound nuclei (in the iron group) for a given neutron excess, η. The silicon-burning nucleosynthesis that is ejected by a super- nova is produced explosively, and has a different composition dominated by 56Ni. The products of silicon-core and shell burning in the core are both so neutron- rich (η so large) that they need to be left behind in a neutron star or black hole. However, even in that case, the composition and its evolution is critical to setting the stage for core collapse and the supernova explosion that follows.