Neutrino Losses - Advanced Stages of Stellar Evolution and Nucleosynthesis - Lecture Slides, Slides of Advanced Physics

Download Neutrino Losses - Advanced Stages of Stellar Evolution and Nucleosynthesis - Lecture Slides and more Slides Advanced Physics in PDF only on Docsity! Lecture 11 Neutrino Losses and Advanced Stages of Stellar Evolution - I docsity.com The late stages (> helium burning) of evolution in massive stars are characterized by huge luminosities, carried away predominantly by neutrinos, and consequently by short time scales. The nuclear physics can become quite complicated. docsity.com Want energy loss per cm3 per second. Integrate over thermal distribution of e+ and e- velocities. These have, in general, a Fermi-Dirac distribution. - 3 2 2 2 9 2 - 1 ( 1) exp( ) 1 5.93/ c/m energy = Chemical potential/kT (determined by the condition that n (matter) = e e e e e e P n n vE m c W W dW n W m c E T W kT m c n n σ π θ φ θ φ + −± +∞ ± −∞ + = < > −⎛ ⎞ = ⎜ ⎟ ± −⎝ ⎠ = = = − = ∫  A N )eYρ Fermi Integral E = total energy including rest mass docsity.com Clayton (Chap 4) and Lang in Astrophysical Formulae give some approximations (not corrected for neutral currents) (NDNR) P± ≈ 4.9×10 18 T9 3 exp(−11.86 / T9 ) erg cm -3 s-1 2mec 2 / kT (NDR) P± ≈ 4.6×10 15 T9 9 erg cm-3 s-1 (better is 3.2×1015) Note origin of T9 : If n± is relativistic, n± ∝ T 3 (like radiation) σ ∝ E 2 v ∝ (kT ) 2 v energy carried per reaction ~ kT ( )( )( )6 2 9P T T T T n n v Eσ ± + − ≈ = T 9 > 3, but not too bad at T 9 > 2 T9 < 2 v cancels 1/v in σ These formulae are very crude; for more accurate results use subroutine neut01.f on the class website. (2 pages back) docsity.com More frequently we use the energy loss rate per gram per second -1 -1 erg gm s Pν νε ρ = In the non-degenerate limit εν from pair annihilation declines as ρ -1. In degenerate situations, the filling of phase space suppresses the creation of electron-positron pairs and the loss rate plummets. Usually pair annihilation neutrino emission dominates other processes when the matter is non-degenerate. This includes most of the advanced stages of stellar evolution (especially when electron capture on nuclei is negligible). docsity.com relativistic Gewmdet, Pekrosian, + Salpeter (1967) 10 p/p ham/em? ) ~non-relativistic docsity.com 3) Plasma Neutrino Process: (Clayton 275ff) plasma This process is important at high densities where the plasma frequency is high and can be comparable to or greater than kT. This limits its applicability to essentially white dwarfs, and to a le ω sser extent, the evolved cores of massive stars. It is favored in degenerate environments. A”plasmon” is a quantized collective charge oscillation in an ionized gas. For our purposes it behaves like a photon with rest mass. docsity.com ( ) 2 4 1/ 2 p 2 2 2 p 2 1/2 1/2 2/3 1/2 4 ND 5.6 10 4 D 1 3 /2 e e e e e e e F e n e n m n e n m m c m c π ω π ω π ε − ⎛ ⎞ = = ×⎜ ⎟ ⎝ ⎠ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎢ ⎥= +⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦  suppression for degeneracy increases with density Consider a neutral plasma, consisting of a gas of positively charged ions and negatively charged electrons. If one displaces by a tiny amount all of the electrons with respect to the ions, the Coulomb force pulls back, acting as a restoring force. If the electrons are cold it is possible to show that the plasma oscillates at the plasma frequency.! docsity.com 6 3 2 21 -3 -1 2 7.5 3/ 2 2 21 -3 -1 2 ) 7.4 10 erg cm s ) 3.3 10 exp ( / ) erg cm s p p e plasma e p p e plasma p e a kT m c P m c kT b kT m c P kT m c kT ω ω ω ω ω − − << ⎛ ⎞ ⎛ ⎞ ≈ × ⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠ >> ⎛ ⎞ ⎛ ⎞ ≈ × −⎜ ⎟ ⎜ ⎟ ⎝ ⎠⎝ ⎠      For moderate values of temperature and density, raising the density implies more energy in the oscillations and raising the temperature excites more oscillations. Hence the loss rate increases with temperature and density. p However, once the density becomes so high that, for a given temperature , raising the density still further freezes out the oscillations. The thermal plasma no longer has enough energy to exite t kTω > hem. The loss rate plummets exponentially. docsity.com This is a relevant temperature for Type Ia supernovae and the red line a relevant density docsity.com 4) Ordinary weak interactions – neutrinos from the decay of unstable nuclei Fuller, Fowler, & Newman, ApJS, 48, 27 (1982a) ApJ, 252, 715, (1982b) ApJ, 293, 1, (1985) Oda et al, Atom. Data and Nuc. Data Tables, 56, 231, (1996) Langanke & Martinez-Pinedo, Nuc Phys A, 673, 481 (2000) •  Beta-decay •  Electron capture •  Positron emission Electron capture – and to a lesser extent beta-decay can be very important in the final stages of stellar evolution – especially during silicon burning and core collapse. Typically these are included by studying each nucleus individually, its excited state distribution, distribution of weak strength, etc. The results are then published as fitting functions at f(T,r). docsity.com Approximate initial conditions: As we shall see, the temperature at which carbon burns in a massive star is determined by a state of balanced power between neutrino losses by the pair process and nuclear energy generation. This gives 8 x 108 K for carbon core burning. Burning in a shell is usually a little hotter at each step, about 1.0 x 109K for carbon burning. Assuming that T3/ scaling persists at the center, and that helium burned at 2 x 108 K and 1000 gm cm-3, this implies a carbon burning density around a few x 105 gm cm-3. The initial composition is the ashes of helium burning, chiefly C and O in an approximate 1 : 4 ratio (less carbon in more massive stars). There are also many other elements present in trace amounts: •  22Ne, 25,26Mg from the processing of CNO elements in He-burning •  The light s-process •  Traces of other heavy elements present in the star since birth •  Up to ~1% 20Ne from 16O(,)20Ne during He-burning docsity.com Principal nuclear reaction Bo + PC + Met + Me + n —2.62 MeV —+ Ne + a + 4.62 MeV + Na + p + 2.24 MeV ® docsity.com many resonances in Gamow window. Measured to about 2.5 MeV and S-factor is overall smooth but shows poorly understood broad “structures” at the factor of 2 level. See Rolfs and Rodney, p 419 ff - alpha cluster? Not seen in 16O + 16O docsity.com There are also some important weak interactions that can change the neutron excess . •  The neutron branch of 12C + 12C itself makes 23Mg. At lower temperature this decays by 23Mg(e+)23Na. At higher temperature it is destroyed by 23Mg(n,p)23Na. The former changes ; the latter does not, so there is some temperature, hence mass dependence of the result. •  20Ne(p,)21Na(e+)21Ne •  21Ne(p,)22Na(e+)22Ne Together these reactions can add - a little - to the neutron excess that was created in helium burning by 14N(,)18F(e+)18O or, in stars of low metallicity they can create a neutron excess where none existed before. docsity.com i % Viftenn ch \e  SS —————— — 25 Mo Por IT &@ “O-shell burning 1.0 20 m/Mo 6 + T T ——. = [ Z+0.01Z6] to 4 \2 docsity.com D. Energy Generation Suppose we make 20Ne and 24Mg in a 3:1 ratio (approximately solar) 7 12C( ) → 3 20 Ne( ) + 24Mg ε nuc = 9.65 × 1017 dY i dt ⎛ ⎝⎜ ⎞ ⎠⎟ ∑ BEi erg g-1 s-1 dY(20Ne) dt = - 3 7 dY(12C) dt dY(24Mg) dt = - 1 7 dY(12C) dt dY(12C) dt = - 2ρY2(12C)λ 12,12 /2 ε nuc = 9.65× 1017 - 3 7 (160.646) - 1 7 (198.258)+1(92.160) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ dY(12C) dt = − (9.65×1017 )(5.01) dY(12C) dt ε nuc ≈ 4.84×1018 ρY 2 (12C)λ 12,12 erg g-1 s-1 where λ 12,12 was given a few pages back. docsity.com 17 12 12 17 20 12 12 24 12 17 -1 12 9.65 10 ( ) 1 12 3 9.65 10 (5.01) 7 12 1 7 4.03 10 erg g nuc i i nuc nuc q Y BE Y X Y Y q X Y Y q X = × Δ Δ = Δ × Δ =− Δ = Δ Δ =− Δ = × Δ ∑ The total energy released during carbon burning is Since X12 << 1, this is significantly less than helium burning docsity.com ε nuc ≈ εν since Lν = εν dM >> Lγ∫ Neutrino losses in carbon burning are due to pair annihilation. Near T9 =1the non-relativistic, non-degenerate formula applies and εν is approximately proportional to T 16 (at ρ~105 gm cm-3) E. Balanced Power Averaged over the burning region, which is highly centrally concentrated Fowler and Hoyle (1964) showed that averaged over an n = 3 polytrope a density and temperature sensitive function has an average: ε = ε dM∫ dM∫ = ε o 3.2 3u + s( ) 3/2 where ε o is the central value of ε, and ε ∝ ρu-1T s docsity.com C C C O O O Si Si Convection docsity.com Carbon core burning not centrally convective in more massive stars. O O C Si CNeO docsity.com F. Lhekimne Na (%e) = (e%a 2.) us (% ae" © T+0.80 Dia,w = F9 xio7"* evaluate D Yu * ON/ 2 Xa (ec) = [@xvw? XS \(7.9m07')} 7 =240 years Rckual Ifetime is lengthened by convection whida binngs Sresh fuck te ‘he center, The ackual life tae Fange> from 3 few hundred to a Sem wlO* yr. More massive stars have the shorter lifetimes. ® docsity.com 15 solar mass star Stage T9 Radius L L) H-burn 0.03 4.36(11) 1.06(38) 7.0(36) He-burn 0.18 3.21(13) 1.73(38) 7.4(36) C-ign 0.50 4.76(13) 2.78(38) 7.1(37) C-dep 1.2 5.64(13) 3.50(38) 3.5(41) O-dep 2.2 5.65(13) 3.53(38) 3.8(43) Si-dep 3.7 5.65(13) 3.53(38) 2.3(45) PreSN 7.6 5.65(13) 3.53(38) 1.9(49) docsity.com Burning Stages in the Life of a Massive Star 0 docsity.com Entropy (k/baryon) roriil 1 boil Let » 5 - Log10(| Time - Timeref (sec) |) docsity.com  α +16 O 20 Ne+γ 20 Ne+α → 24 Mg +γ ( )20 16 24 20 16 20 The net result is that 2 Ne O + Mg at a rate that is determined by how fast Ne captures alpha particles from the equilibrium concentration set up by O and Ne. → Other secondary reactions: 24Mg(,)28Si 27Al(,p)30Si 25Mg(.n)29Si 30Si(p,)31P 26Mg(,n)30Si etc. Products: some more 16O and 24Mg, 29,30Si, 31P, 26Al and a small amount of s-process. fast slow docsity.com B. Photodisintegration Reaction Rates At high temperatures, the inverse reaction to radiative capture, [(n,),(p,),(,)] becomes important as there exists an appreciable abundance of -rays out on the tail of the Bose-Einstein distribution that have energy in excess of several MeV. The reactions these energetic photons induce are called photodisintegration reactions – the major examples being (,n),(,p), and (,) Consider + and I j L L I j γ γ + → + → + docsity.com In equilibrium, the abundances must obey the Saha equation For the reaction I + j L+ γ nI nj nL = gI g j gL ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ AI Aj AL ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 3/2 2π kT h2 N A ⎛ ⎝⎜ ⎞ ⎠⎟ 3/2 exp(−Qjγ / kT ) (deriveable from considerations of entropy and the chemical potential and the fact that the chemical potential of the photon is zero). Thus, in equilibrium (a more stringent condition than "steady state") nI nj nL ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 5.942×10 33 T9 3/2 gI g j gL ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ AI Aj AL ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 3/2 exp(−11.60485Qjγ / T9 ) for Qij measured in MeV docsity.com And since aoe = -2YC*Ne) Yu @ Mar (Ne), = - 26 a s a) 44 >) af dt ~% \ dh 1.405)4.794, and Ayu(Ne) = 9.87107 TM at) . rugtto)e* Ne $.65x\0'° 7 XC" Ne) o MPA, (previous page) Th la t very T sensitive = 9.gsnio' [4 (199.268) + 4 (127.62) - Wo.c4e] - 2F Nicyos? are — monn “9 Ne ® docsity.com Enuc Erwe® (9.65 10%) (2.2926) © 4.41xi0$ erg g's! Ne a9 Ma XY" (* Ne) -$4.69/T, E ™ 2.49 x\0 Ty Yo) rAur(*Ne) e ery gts? —— — — Ne Grae * 9.65x10'% 2:29 ay (ne) Tene = \.\ x\o* Ox (**Ne) ery g a relatively small Note that the eneryy generation rate is independent of =the density. ® docsity.com  Balanced Power Condihion : w(*Ne)= 0.2 x(*O)z0F Q~ 10-42? near Ty 2hS Aux (Ne) © 3.434107 TE s S ~54.$9 “, Care % ‘ "=" e > (t; 43 “« T, ® Weis very T- senntive Above eq” 4 conditions give Ne Enue = 2-9 * ory 7's! \.5 docsity.com Nucleosynthesis from neon burning The principal nuclei with major abundances at the end of neon burning are 16O and 24Mg. Most of the neutron excess resides in 25,26Mg. Most of the 16O has in fact survived even since helium burning. In terms of major production of solar material, important contributions are made to [16O], 24,25,26Mg, (26),27Al, 29,30Si, and 31P docsity.com Oxygen Burning: After neon burning the lightest nucleus remaining with appreciable abundance is 16O. This not only has the lowest Coulomb barrier but because of its double magic nature, has a high -particle separation energy. It is the next to burn. Because of its large abundance and the fact that it is a true fusion reaction, not just a rearrangement of light nuclei, oxygen burning releases a lot of energy and is a very important part of the late stages of stellar evolution in several contexts (e.g., pair-instability supernovae). It is also very productive nucleosynthetically. It’s chief products being most of the isotopes from 28Si to 40Ca as well as (part of) the p-process. docsity.com 16 16 32 * 31 30 31 28 ( ) 1.45MeV 5% 2.41MeV 5% 7.68MeV 56% 9.59MeV 34% O O S S n P d P p Si α + → → + + → + − ≤ → + + → + + Initial composition: 16O, 24Mg, 28Si Nuclear reactions: The deuteron, d, is quickly photodisintegrated into a free neutron and proton. proceeds through the 32S compound nucleus with a high density of resonances. Very like carbon burning. docsity.com 3) Onset of "quasi-equilibrium" clusters e.g. 28Si + n 29Si + γ 29Si + p 30 P+γ etc. These clusters apear and grow as oxygen burning proceeds (Woosley, Arnett, & Clayton, ApJS, 26, 271 (1973)) 4) Weak interactions increase  markedly during oxygen core burning (much less so during oxygen shell burning where the density is less and the time scale shorter). 33 33 37 37 35 35 ( , ) ( , ) ( , ) e e e S e P Ar e Cl Cl e S ν ν ν − − − docsity.com Nuclear Energy Generation Agproxmation %O+"“o-— “Sr usd My (actually 2851, 5, “Ar, *¢y i rough proportions worSsist ) % Sts Av) Man /a avi@s) 2 Ly, dy(*o) at at 9.271 Mev Ene * 2OSKI0'" Y*C“o)e Any [E(241.799)- 129.017] Rs Wo fo. = B.oxlo™® ¥*("o)@ Aun erg pe docsity.com \o$ -2 Au a“ — ne ay c = «35 gig? Heit « 4.24% Se, Ys x= 4.24% ( 3 ) Mace i = 38 2.60E-1 1.250 We = F.%xl0 . (3) apes 13 *BIE- ‘ For (0) ~ 0.5 ; set 4500 ae 1 1s :500 e ’ 35 Pay | a Ly 3 B09 Enc * © KIO s@) wy gs i eo ae 3 :000 Gus * 9. 6Sxi0"7 (2) ax, . Gras * S.0x10'? AXKy erg gu! substenti ab OXKu is larse docsity.com Whole star production factors near oxygen depletion (5%) in a 25 solar mass star. s25a28 docsity.com Srwmelreus Coe cerbury AD Commenthng on work Leucigpus (oth adidas ec) They [the atoms] move in the void and catching each other up and jostle together, and some recoil in any direction that may chance, and others become entangled with one another in various degrees according to the symmetry of their shapes and sizes and positions and order, and they re- main together and thus the coming into being of composite things is effected. ® docsity.com Sir Arthur Eddington The Internal Constitution of Stars (The Observatory, Vol. 43, p. 341-358 (1920)) p 354. docsity.com