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Nucleosynthesis Flowchart BB 1,2H 3,4He 7Li Intergalactic medium Galaxy formation inflow Gal. winds, stripping, mergers Interstellar medium Star formation Small stars D, Li ? Winds, PN, Novae He, 7Li, C, N D, Li, Be, B Middling stars Big stars WD NS BH Explosive r-process SN Spallation 6Li, Be, B Cosmic rays Lecture 9: Supernova Rates Star-Formation Efficiency, Yield How many supernovae per year for each galaxy type ? Use power-law IMF, Salpeter slope -7/3 = -2.33 N(M ) µ M logN(M) -7/3 20 M. Limits of validity, not well known 0.1M. slope = -7/3 = -2.33 log M 0.1 M 8 M 20 M Supernova limit “Universal” IMF (Kroupa 2002) N(M ) µ M a -a a ~ - 7/3 M > 1 M - 4/3 0.1 - 1 M - 1/3 M < 0.1 M M42 M35 Pleiades local MW MC GC local log( M / M) log( M / M ) Integrating a Power-Law IMF Number of stars : N= ò N(M) dM = A ò M B dM = A M B +1 ( if B ¹ -1) B +1 Fraction of stars with M > 8 M ( for B = -7/3 ) number of SNe fN º = number of stars ò ò 20 8 20 0.1 M B dM M B dM A Most stars at B+1 20 -4/3 20 M M low-mass end! 0.018 - 0.063 8 B +1 8 fN = = = 20 20 -4/3 A B+1 0.018 - 21.544 M M 0.1 0.1 B +1 500 stars --> 1 supernova! Þ f N = 0.2% SN Mass Fraction Supernovae are rare, but each is very massive. What fraction of the mass goes into SNe? fM ò = ò = Þ 20 8 20 0.1 M M M ´ M -7/3 dM M ´ M -7/3 dM -1/3 20 8 -1/3 20 0.37-0.50 = 0.37 - 2.15 0.1 f M = 7.2% Most of mass is in low-mass stars. “Typical” SN Mass Median mass: ò ò M SN 1 = 2 Þ M ´ M -7/3 dM -1/3 M SN - 0.50 8 = 20 0.37 - 0.50 M ´ M -7/3 dM 8 M SN = 12.2 M . Mean mass: 20 1 -1/3 -7/3 M M ´ M dM ò 8 8 -1/3 M = = 20 -7/3 1 -4/3 20 M dM ò8 M 8 -4/3 4 ´ (20-1/3 - 8-1/3 ) 4 ´ (0.37 - 0.50) = = = 12 M . -4/3 -4/3 20 - 8 0.018 - 0.062 20 SN Rates vs Galaxy Type Spiral Galaxy: SFR: ~ 8 M yr-1. Þ SN rate: 7.2% have M > 8 M . (8 M yr-1) x 0.072 ~ 0.6 M yr-1 go into SNe 0.6 M . yr -1 1 ~ yr -1 (fewer seen due to dust) 12.2 M . 20 Irregular Galaxy: ~10x this rate during bursts (1 SN per 2 yr)! No SNe between bursts. SN Rates: Ellipticals t* = 1 Gyr e-folding time t = 10 Gyr age a = 0.95 efficiency M0 = 1011 M total mass = initial gas mass Gas consumption: MG (t) = M0 e-t / t* = M0 - a MS (t) Star formation: M S (t) = M0 a ( 1- e ) m(t) = e-t / t * 1 gas 1 S(t) = (1- e-t/t* ) / a a -t / t* stars -t / t* 10 M . ) e ( dM S M e -3 -1 0 ˙ º MS = = = 5 ´10 M yr . dt a t* (0.95) (10 9 yr) -10 11 SN rate: (0.072) (5 ´10 -3 M. yr -1) 12.2 M . t* » 3 ´10 -5 yr -1 3 SN per 105 yr. t t* Negligible! t What Star Formation Efficiency a and Yields of H, He and Metals ? X = 0.75 Y = 0.25 Z = 0.00 MG = M0 MS = 0 MG = 0 MS = M0 KABOOM! KABOOM! MG = (1-a) M0 MS = a M0 a=? X=? Y=? Z=? Estimates for efficiency a , yield in X, Y, Z Assume: 1. Type-II SNe enrich the ISM. (Neglect: Type-I SNe, stellar winds, PNe, ....) 2. Closed Box Model: (Neglect: Infall from the IGM, outflow to the IGM) 3. SN 1987A is typical Type-II SN. Better models include these effects. What do we know about SN 1987A? SN 1987A 23 Feb 1987 in LMC Brightest SN since 1604! First SN detected in neutrinos. Visible (14 --> 4.2 mag) to naked eye, in southern sky. Progenitor star visible: ~20 Msun blue supergiant. 3- ring structure (pre-SN wind) UV flash reached inner ring in 80 d. Fastest ejecta reached inner ring in ~6 yr. Fast ejection velocity v~c/30~11,000 km/s. Slower (metal-enriched) ejecta asymmetric. SN 1987A 23 Feb 1987 in LMC Brightest SN since 1604! 2010 First SN detected in neutrinos. Visible (14 --> 4.2 mag) to naked eye, in southern sky. Progenitor star visible: ~20 Msun blue supergiant. 3- ring structure (pre-SN wind) Shockwave reaches inner ring 2003. 2003 Star Formation Efficiency Use SN 1987A to calculate a and yield. SN 1987A: progenitor star mass = 20 M remnant neutron star mass = 1.6 M mass returned to the ISM = 18.4 M From IMF, 7.2% of MS is in stars with M > 8 M = Fraction of MS returned to ISM: b= mass returned to gas 18.4 = 0.072 ´ » 6.6% mass turned into stars 20 Star Formation Efficiency a = fraction of MS retained in stars: a =1- b = 93% SN 1987A Lightcurve Powered by radioactive decay of r-process nuclei. Use to measure metal abundances in ejected gas. 56Ni => 56 Co 56Co => 56 Fe 6d half-life 78d half-life X, Y, Z of ejecta from SN1987A Initial mass ~ 20 M NS mass ~ 1.6 M Mass ejected ~ 18.4 M Þ in H 9.0 M He 7.0 M Z 2.4 M 9 X= » 0.49 18.4 2.4 Z= » 0.13 18.4 } = 18.4 M 7 Y= » 0.38 18.4 Q1: What changes to the particle content of the expanding Universe occur at the epochs of: • Annihilation: – pair soup -> quark soup (109 photons/quark) • Baryogenesis: – quarks bound (by strong force) into baryons. – UUD = proton DDU = neutron • Nucleosynthesis: – Atomic nuclei: 75% H, 25% He, traces of Li, Be • Recombination: – Neutral atoms form as free electrons recombine – photons fly free Q2: Given present-day density parameters WM = 0.3 for matter and WR = 5x10-5 for radiation, at what redshift z were the energy densities equal ? volume R3 N particles of mass m photon wavelengths stretch: 1 l µRµ 1+ z e M = r M c 2 = WM ( rcrit c 2 ) (1+ z ) 3 rM = Nb m 3 µ (1+ z) R3 Ng h n 4 -4 e R = WR ( rcrit c ) (1+ z ) eR = µ R µ 1+ z ( ) 3 R e M WM 1 WM 0.3 1= = Þ 1+ z = = = 6000 -5 e R WR 1+ z WR 5´10 2 4 Q3 a) Evaluate the neutron/proton ratio in thermodynamic equilibrium at high and low T. æ Dm c 2 ö N n æ mn ö = çç ÷÷ exp ç ÷ N p è mp ø è kT ø 3/2 mn = m p + Dm = 1.0014 m p T ®¥ æm ö Nn 3/2 n ® çç ÷÷ exp ( 0 ) = (1.0014) » 1 N p è mp ø T ®0 N n æ mn ö ® çç ÷÷ exp ( -¥) = 0 N p è mp ø 3/2 3/2 b) Evaluate the n/p ratio and Yp if mn = mp. mn =1 mp mn =m p Þ Dm = 0 Nn = N p Þ 100% He Yp =1 Nn 3/2 ® (1) exp ( 0) =1 Np Q4 Alien’s CMB-meter reads 5.1K and 4.9K in the fore and aft directions. Evaluate the velocity. V DT 0.1K = = c T 5K ÞV = c = 6000 km/s 50 Are humans present on Earth at this time? T = 5K T0 = 2.7K lµR Þ T µ 1 R mater dominated expansion: R µ t 2/3 3/2 3/2 æ ö æ ö æ ö t R T 2.7K time : =ç ÷ = ç 0 ÷ = ç ÷ = 0.40 èT ø è 5K ø t0 è R0 ø 1 now: t0 ~ ~ 13´10 9 yr Age of Sun: ~5´10 9 yr H0 3/2 look-back time: t0 - t = 0.6t0 ~ 8 ´10 9 yr ( Before Sun was born! ) Cosmological Models Assume a Universe filled with uniform density fluid. [ OK on large scales > 100 Mpc ] Density: Energy density: e = r c2 11 3 H 1.4 ´10 Msun -26 -3 0 Critical density: r º » 10 kg m » c 8p G (Mpc) 3 2 3 components: 1. Radiation 2. Matter 3. “Dark Energy” Total { “Dark Matter” baryons WB ~ 0.04 Only ~4% is matter as we know it! Energy Density of expanding box volume R 3 N particles particle mass m momentum p 2 p energy E = hn = m 2c 4 + p 2c 2 = m c 2 + + ... 2m Cold Matter: ( m > 0, p << mc ) E » m c 2 = const N m c2 -3 eM » µ R R3 Radiation: ( m = 0 ) Hot Matter: ( m > 0, p >> mc ) l µ R (wavelengths stretch) : E =hn = eR = hc l N hn -4 µ R R3 µ R-1 3 Eras: radiation…matter…vacuum radiation : rR µ R matter : r M µ R-3 r L = const vacuum : R 1 aº = R0 1+ z r= r R,0 a 4 + r M ,0 a 3 log r -4 rR rM z = redshift + rL rR = rM at a ~ 10 log R e log R t -4 t ~ 10 yr 4 rM = r L at a ~ 0.7 t ~ 1010 yr rL t 2/3 1/ 2 log t t • Q1: Given the density parameters W=0.3 for matter and W=0.7 for Dark Energy, evaluate the redshift z at which the energy densities of matter and Dark Energy are equal. W= crit ~ R-3 W= crit ~ R0 1 + z = R0 / R when W z W • 1+z = ( W / W )1/3 = ( / )1/3 = 1.326 • z = 0.326 • Q2: What changes to the particle content of the expanding Universe occur at the following epochs: • Annihilation: particles and anti-particles annihilate, producing photons. Small excess of particles (~1 per 109 photons) • Baryogenesis: free quarks confined by strong force in (colourless) groups of 3 producing neutrons (ddu) and protons (uud). • Nucleosynthesis: protons and neutrons bind to form 2D, then 4He. Yp set by p/n ratio, as virtually all n go into 4He leaving residual p as H. • Recombination: H and He nuclei capture free electrons. Universe now transparent to photons. • Q3: If the neutron decay time were 1 s, rather than 900s, what primordial helium abundance Yp would emerge from Big Bang Nucleosynthesis? • n(t) = n(0) exp(- t / ) • p(t) = p(0)+(n(0)-n(t)) • t~300s = 900s => 1s • Yp = 2n/(p+n) => 0 since virtually all neutrons decay. • Q4: Name and describe three effects that give rise to anisotropy in the Cosmic Microwave Background, indicating which are most important on angular scales of 10, 1 and 0.1 degrees. • 10o Sachs-Wolf effect - photons last scattered from higher-density regions lose energy climbing out of the potential well. • 1o Doppler effect - velocity of gas on last-scattering surface shifts photon wavelengths. • 0.1o Sunyaev-Zeldovich effect - re-ionised gas (e.g. X-ray gas in galaxy clusters) scatters CMB photons passing thru, changing photon direction and energy. In dimensionless form MG (t) = fraction of M0 in gas m(t) º M0 M S (t) = fraction of M0 that has been S(t) º turned into stars M0 m(t) MG = M 0 - a M S 1 m = 1- a S slope = - a OK, since some gas is recycled. 1 S¥ S(t)