Reaction Rates Marialuisa Aliotta School of Physics University of Edinburgh principles of stellar structure and evolution general features of thermonuclear reactions experimental approach Second European Summer School on Experimental Nuclear Astrophysics St. Tecla, Sept. 28th – Oct. 5th 2003

The Macro-cosmos: some observables Luminosity vs. surface temperature Hertzsprung-Russel (HR) Diagram Surprise! SUPERGIANTS No chaos, but order! GIANTS Stefan´s law: L = 4R2T4 Luminosity sun WHITE DWARFS MAIN SEQUENCE  Temperature [K] ~ 95% of all stars in MAIN SEQUENCE highest probability of observing them in this stage (cf. adulthood for human beings) Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

The Macro-cosmos: some observables Mass-Luminosity relationship (for main sequence stars only) L ~ M4  more massive stars evolve more rapidly mass (Msun) lifetime (years) 1 ~1010 5 ~108 10 ~107 L, T, M cannot take up ANY values ORDER! Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

The Macro-cosmos: some observables Abundance curve of the elements Data sources: Earth, Moon, meteorites, stellar (Sun) spectra, cosmic rays... Features: distribution everywhere similar 12 orders-of-magnitude span H ~ 75% He ~ 23% C  U ~ 2% (“metals”) D, Li, Be, B under-abundant exponential decrease up to Fe peak near Fe almost flat distribution beyond Fe Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Experimental nuclear astrophysics study energy generation processes in stars study nucleosynthesis of the elements What is the origin of the elements? How do stars and galaxies form and evolve? What powers the stars? How old is the universe? … MACRO-COSMOS intimately related to MICRO-COSMOS NUCLEAR PHYSICS KEY for understanding Courtesy: M. Arnould Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Quiescent stages of stellar evolution Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Principles of stellar structure and evolution: quiescent evolution Stellar structure and evolution controlled by: Gravity  collapse Internal pressure  expansion Star composed of many particles (~1057 in the Sun) Total energy: a) mutual gravitational energy of particles () b) internal (kinetic) energy of particles (including photons) (U) For an ideal gas in hydrostatic equilibrium: 2U +  = 0 virial theorem Assume pressure imbalance gravitational contraction sets in amount of energy released - internal energy change to restore equilibrium U = - ½  gas temperature increases energy excess - ½  lost from star in form of radiation Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Principles of stellar structure and evolution: quiescent evolution gravitational contraction of gas (mainly H) increase of central temperature if T high enough “nuclear burning” takes place HYDROGEN BURNING (1st equilibrium) 4H  4He + 2b+ + 2n + 26 MeV ash of nuclear burning energy source gravitational collapse is halted  star undergoes phase of hydrostatic equilibrium MAIN SEQUENCE STARS Here: T ~ 10 – 15 X106 K and r ~ 102 gcm-3 are required M > 0.1 M (Jupiter = failed star) Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

(also depends on CNO abundance) Principles of stellar structure and evolution: quiescent evolution Hydrogen burning Two main mechanisms: proton-proton chain and CNO cycle Energy production rate M < 1.5 M  T6 < 30  p-p chain M  1.5 M  T6 > 30  CNO cycle (also depends on CNO abundance) (Fiorentini’s lecture) Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Principles of stellar structure and evolution: quiescent evolution H exhausted in core isothermal He core contraction sets in temperature increases H-burning shell R ~ 10-100 Ri  Ts ~ 3-4x103 K Wien’s law: maxT = const. RED GIANT STARS contracting core expanding envelope when T ~ 108 K and r ~ 103 gcm-3 (minimum mass ~ 0.5 M) HELIUM BURNING (2nd equilibrium) 3a  12C 12C(a,g)16O + 8 MeV nuclear burning ashes energy source Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Principles of stellar structure and evolution: quiescent evolution 12C/16O BURNING … 12C ashes = Ne, Na, Mg … 16O ashes = Al, … Si major ash = 28Si SUPER RED-GIANT STARS 28Si MELTING … A = 40-65 major ash = 56Fe PRE-SUPERNOVA STARS further reactions become endothermic final gravitational collapse SUPERNOVA EXPLOSION (type II) T, r M  8 M remnant: neutron star or black hole Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Principles of stellar structure and evolution: summary Stellar mass (M) Stage reached < 0.08 no thermonuclear fusion 0.1 -0.5 H burning 0.5 - 8 He burning 8 - 11 C burning > 11 all stages Evolution stages of a 25 M star Stage reached Timescale Tcore (109 K) Density (g cm-3) H burning 7x106 y 0.06 5 He burning 5x105 y 0.23 7x102 C/O burning 600 y / 6 months 0.93 – 2.3 2x105 – 1x107 Si melting 1 d 4.1 3x107 Explosive burning 0.1 – 1 s 1.2 - 7 varies Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Principles of stellar structure and evolution: quiescent evolution Main parameters: 1) initial mass (  central temperature) 2) initial chemical composition (  nuclear processes) Energy generation rate  ~ Tn n ~ 4 (H-burning) n ~ 30 (C-burning) Quiescent burning Main Sequence ~ 1010 y Red Giant ~ 3x108 y Super Giant ~ 3x104 y Pre-Supernova Gravitational Contraction H He C O, Ne … innermost regions only contribute to nuclear burning e.g. 1/10 M for H-burning less for subsequent stages H-burning  MAIN SEQUENCE longest stage of star’s lifetime Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Explosive stages of stellar evolution Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Principles of stellar structure and evolution: explosive evolution NOVAE = sudden increase in star’s luminosity (L ~ 104 – 106 Li and t ~ 1 h – 1 d) semi-detached binary system: White Dwarf + less evolved star (e.g. Red Giant) H-rich mass transfer from RG to WD degenerate matter  P and T uncoupled thermonuclear runaway  cataclysmic explosion temperature and density increase on WD’s surface (p,) and (,p) reactions on proton-rich nuclei T > 108 K  > 103 g cm-3 determine nature of nova phenomenon nucleosynthesis up to A ~ 60 mass region Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

“seed” nuclei in Fe region Principles of stellar structure and evolution: explosive evolution X-RAY BURSTERS & X-RAY PULSARS semi-detached binary system: Neutron star + less evolved star T ~ 109 K  ~ 106 g cm-3 (,p) and (p,) reactions on proton-rich nuclei nucleosynthesis up to A ~ 80-100 mass region intense X-ray fluxes CORE-COLLAPSE SUPERNOVAE end stage of M ~ 8-30 M stars core collapse & rebound shock wave outer layers blown off Neutron Star or Black Hole remnants T > 109 K n > 1020 g cm-3 “seed” nuclei in Fe region (n,) reactions on neutron-rich nuclei followed by  decays nucleosynthesis of n-rich elements through r-process Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

gravitational contraction Stellar life cycle BIRTH gravitational contraction Interstellar gas Stars mixing of interstellar gas thermonuclear reactions energy production stability against collapse synthesis of “metals” abundance distribution explosion DEATH Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Thermonuclear reactions in stars Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Thermonuclear reactions in stars: properties of nuclei Aston: measurements of atomic masses E = Mnc2 Mnucl < mp + mn enormous energy stored in nuclei! Rutherford (1919): discovery of nuclear reactions liberate nuclear energy source complex nuclides formed through reactions amount of energy liberated in nuclear reaction: Q =[(m1+m2)-(m3+m4)]c2 > 0 Binding energy curve spontaneous nuclear processes: Q > 0 Q < 0 Q > 0 fusion up to Fe region fission of heavy nuclei H most abundant element in the Universe FUSION reactions most effective in stars Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Thermonuclear reactions in stars: general features & definitions Consider reaction: 1 + 2  3 + 4 Q12 > 0 (  known from atomic mass tables) Reaction cross section   probability for a reaction to occur Dimension: area Unit: barn (b) = 10-24 cm2 In general: not possible to determine reaction cross section from first principles However: cross sections depend on nature of force involved cross sections are energy (i.e. velocity) dependent Reaction Force  (barn) Eproj (MeV) 15N(p,)12C strong 0.5 2.0 3He(,)7Be electromagnetic 10-6 p(p,e+)d weak 10-20 Reaction rate: r = N1N2 v(v) Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Maxwell-Boltzmann distribution Thermonuclear reactions in stars: general features & definitions In stellar plasma: velocity of particles varies over wide range (v) velocity distribution <v>12 = Reaction rate per particle pair: v(v)(v)dv Quiescent stellar burning: non-relativistic, non-degenerate gas in thermodynamic equilibrium at temperature T Maxwell-Boltzmann distribution (v)  exp = exp Probability (E) (E)  exp(-E/kT) = reduced mass v = relative velocity (E)  E kT Energy <v>12 = (E) exp E dE Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Thermonuclear reactions in stars: general features & definitions Ni = number density Total reaction rate: R12 = (1+12)-1 N1N2 <v>12 reactions cm-3 s-1 Energy production rate: 12 = R12 Q12 Mean lifetime of nuclei X against destruction by nuclei a = energy production as star evolves change in abundance of nuclei X <v> = KEY quantity to be determined from experiments and/or theoretical considerations as star evolves, T changes  evaluate <sv> for each temperature NEED ANAYLITICAL EXPRESSION FOR ! Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Non-resonant process Resonant process Thermonuclear reactions in stars: reaction mechanisms Consider reaction: a + X  b + Y (b = particle or photon) Non-resonant process One-step process leading to final nucleus Y   |<b+Y lHl a+X>|2 single matrix element occurs at all interaction energies cross section has WEAK energy dependence Resonant process Two-step process: 1) compound nucleus formation a + X  C* 2) decay of compound nucleus C*  b + Y V r r0 E incident nucleus Er Er+1 E1 E2    |<b+Y lH’l C*>|2 |<C* lHl a+X>|2 two matrix elements occurs at specific energies cross section has STRONG energy dependence Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Reactions between charged particles nucleosynthesis up to Fe typically quiescent stages Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

determines exponential drop in abundance curve! Thermonuclear reactions in stars: charged particles charged particles Coulomb barrier V Coulomb potential energy available: from thermal motion Ekin ~ kT (keV) Ecoul ~ Z1Z2 (MeV) kT ~ 8.6 x 10-8 T[K] keV tunnel effect r0 r nuclear well T ~ 15x106 K (e.g. our Sun)  kT ~ 1 keV T ~ 1010 K (Big Bang)  kT ~ 2 MeV reactions occur through TUNNEL EFFECT tunneling probability P  exp(-2) during quiescent burnings: kT << Ec in numerical units: 2ph = 31.29 Z1Z2(m/E)½ m in amu and Ecm in keV 2ph = GAMOW factor determines exponential drop in abundance curve! Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Thermonuclear reactions in stars: non-resonant reactions geometrical factor (particle’s de Broglie wavelength) interaction matrix element penetrability probability depends on projectile’s angular momentum  and energy E  (E) = exp(-2) S(E) (for s-waves only!) non-nuclear origin STRONG energy dependence nuclear origin WEAK energy dependence Above relation defines ASTROPHYSICAL S(E)-FACTOR N.B. If angular momentum is non zero  centrifugal barrier must also be taken into account Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Thermonuclear reactions in stars: Gamow peak With above definition of cross section: <v>12 = S(E) exp dE f(E) varies smoothly with energy governs energy dependence MAXIMUM reaction rate: Gamow peak tunnelling through Coulomb barrier  exp(- ) Maxwell-Boltzmann distribution  exp(-E/kT) relative probability energy kT E0 E0 E0 < E0 only small energy range contributes to reaction rate  OK to set S(E) ~ S(E0) = const. Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Thermonuclear reactions in stars: Gamow peak most effective energy region for thermonuclear reactions E0 ± E0/2 energy window of astrophysical interest E0 = f(Z1, Z2, T) varies depending on reaction and/or temperature Examples: T ~ 15x106 K (T6 = 15) reaction Coulomb barrier (MeV) E0 (keV) exp(-3E0/kT) E0 p + p 0.55 5.9 7.0x10-6  + 12C 3.43 56 5.9x10-56 16O + 16O 14.07 237 2.5x10-237 area of Gamow peak (height x width) ~ <v> STRONG sensitivity to Coulomb barrier separate stages: H-burning He-burning C/O-burning … Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

insert in expression for reaction rate, Thermonuclear reactions in stars: resonant reactions Resonant reactions 1. Narrow resonances  << ER Breit-Wigner formula (E)BW = π2(1+12) a b (E-Er)2 + (/2)2 insert in expression for reaction rate, integrate and get: <v>12 = exp (for single resonance) resonance strength (integrated cross section over resonant region) low-energy resonances (ER  kT) dominate reaction rate Experiment: determine and ER Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

energy dependence of partial a(E), b(E) and total (E) widths Thermonuclear reactions in stars: resonant reactions 2. Broad resonances  ~ ER Breit-Wigner formula + energy dependence of partial a(E), b(E) and total (E) widths N.B. Overlapping broad resonances of same Jπ  interference effects Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Thermonuclear reactions in stars: resonant reactions 3. Sub-threshold resonances any exited state has a finite width  ~ h/ high energy wing can extend above particle threshold cross section can be entirely dominated by contribution of sub-threshold state(s) Example: 12C(,)16O (Gialanella’s lecture & Schürmann’s talk) TOTAL REACTION RATE vtot = vr + vnr Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Reactions with neutrons nucleosynthesis beyond Fe typically explosive stages Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Thermonuclear reactions in stars: neutron captures NO Coulomb barrier neutrons produced in stars are quickly thermalised E0 ~ kT = relevant energy (e.g. T ~ 1-6x108 K  E0 ~ 30 keV) accounts for almost flat abundance distribution beyond iron peak Typically: <sv> ~ const = <sTvT> neutron-capture cross sections can be measured DIRECTLY at relevant energies Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Experimental approach & laboratory requirements Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Quiescent burning stages of stellar evolution Experimental approach: general features Quiescent burning stages of stellar evolution FEATURES T ~ 106 - 108 K  E0 ~ 100 keV << Ecoul  tunnel effect  10-18 barn <  < 10-9 barn  average interaction time  ~ <v>-1 ~ 109 y unstable species DO NOT play significant role PROBLEMS 10-18 b <  < 10-9 b  poor signal-to-noise ratio  major experimental challenge  extrapolation procedure required REQUIREMENTS poor signal-to-noise ratio  long measurements  ultra pure targets  high beam intensities  high detection efficiency Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Experimental procedure Experimental approach: extrapolation Experimental procedure measure (E) over as wide a range as possible, then EXTRAPOLATE down to Gamow energy region around E0! CROSS SECTION LOG SCALE  direct measurements E0 Ecoul Coulomb barrier (E) non-resonant resonance extrapolation needed ! many orders of magnitude Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Thermonuclear reactions in stars: non resonant reactions Example: cross section S-factor Data EXTRAPOLATION down to astrophysical energies REQUIRED! Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Experimental approach: extrapolation S(E)-FACTOR S(E) extrapolation LINEAR SCALE direct measurement low-energy tail of broad resonance Er non resonant process sub-threshold resonance -Er interaction energy E DANGER OF EXTRAPOLATION ! Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Experimental approach: extrapolation ALTERNATIVE SOLUTIONS Go UNDERGROUND  reduce (cosmic) background example: LUNA facility  (Junker’s lecture) Use INDIRECT methods  (Figuera’s lecture) INTRINSIC LIMITATION At lower and lower energies ELECTRON SCREENING EFFECT sets in Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

The electron screening Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

penetration through Coulomb barrier between BARE nuclei Experimental approach: electron screening (E) = S(E) exp(-2) penetration through Coulomb barrier between BARE nuclei in stellar plasmas: ions in sea of free electrons Ec bare Coulomb potential Debye-Hückel radius RD ~ (kT/)½ E + Ue screened E Rn RD Rt Ue = electron screening potential Similarly: in terrestrial laboratories: interaction between ions (projectiles) and atoms or molecules (target) Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Experimental approach: electron screening cross-section enhancement factor: fplasma(E) =  exp(Ue/E)  1 plasma(E) bare(E)  (E) screened bare E E bare S(E) S(E) high-energy data extrapolation screened S(E) fit to measured low-energy data  Ue BUT: electron screening in lab DIFFERENT from electron screening in plasma need to understand flab(E) improve calculation of fplasma(E) PROBLEM: experimental Ue >> theoretical Ue Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Explosive burning stages of stellar evolution Experimental approach: general features Explosive burning stages of stellar evolution FEATURES T > 108 K  E0 ~ 1 MeV ~ Ecoul  10-6 barn <  < 10-3 barn  NO extrapolation needed  average interaction time  ~ <v>-1 ~ seconds unstable species DO GOVERN nuclear processes PROBLEMS  ~ <v>-1 ~ seconds  unknown nuclear properties  low beam intensities (5-10 o.d.m. lower than for stable beams)  beam-induced background REQUIREMENTS unstable species  RIBs production and acceleration  large area detectors  high detection efficiency Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Experimental approach: data needs NUCLEAR DATA NEEDS reactions involving: A < 30 A > 30 cross-section dependence: individual resonances nuclear properties statistical properties Hauser-Feshbach calculations knowledge required: excitation energies spin-parity & widths decay modes masses level densities part. separation energy experimental constraints wherever possible Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Experimental approach: explosive nucleosynthesis EXAMPLES rp-process r-process rapid proton captures X(p,)Y rapid neutron captures X(n,)Y synthesis of neutron-rich nuclei A > 60 synthesis of proton-rich nuclei A ~ 100 proton capture neutron capture - decay + decay Z stable N unstable (Shotter’s lecture) Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Overview of main astrophysical processes M.S. Smith and K.E. Rehm, Ann. Rev. Nucl. Part. Sci, 51 (2001) 91-130 Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

stellar reactions take place through TUNNEL effect Thermonuclear reactions in stars: overview Hydrostatic equilibrium T ~ 106 - 108 K  average interaction time  ~ <v>-1 ~ 109 y unstable species do not play significant role stellar reactions take place through TUNNEL effect Gamow peak tunnelling through Coulomb barrier  exp(- ) Maxwell-Boltzmann distribution  exp(-E/kT) relative probability energy kT E0 E0 kT << E0 << Ecoul 10-18 barn <  < 10-9 barn Extrapolation NEEDED Solutions: underground measurements indirect approaches BUT! Electron screening problem Explosive phenomena T > 108 K  average interaction time  ~ <v>-1 ~ seconds unstable nuclei govern nuclear reaction processes Sophisticated techniques for RIBs production and acceleration Ad-hoc detection systems required E0 ~ Ecoul 10-6 barn <  < 10-3 barn Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003

Further reading W.D. Arnett and J.W. Truran Nucleosynthesis The University of Chicago Press, 1968 J. Audouze and S. Vauclair An introduction to Nuclear Astrophysics D. Reidel Publishing Company, Dordrecth, 1980 E. Böhm-Vitense Introduction to Stellar Astrophysics, vol. 3 Cambridge University Press, 1992 D.D. Clayton Principles of stellar evolution and nucleosynthesis The University of Chicago Press, 1983 H. Reeves Stellar evolution and Nucleosynthesis Gordon and Breach Sci. Publ., New York, 1968 C.E. Rolfs and W.S. Rodney Cauldrons in the Cosmos The University of Chicago Press, 1988 (…the “Bible”) Copies of this lecture at: www.ph.ed.ac.uk/~maliotta/teaching Second European Summer School on Experimental Nuclear Astrophysics - St. Tecla, Sept. 28th – Oct. 5th 2003