Topics in Nuclear strophysics II Stellar Reaction Rates definition of a reaction rate Gamow window lifetimes of isotopes at stellar conditions nuclear energy production rate introduction to network simulations input parameters for calculating a reaction rate uncertainties in stellar reaction rates
Reaction Rate Definition For a given relative velocity v with projectile number density n p λ σ n p v [ ] s 1 reaction/target particle energy/temperature dependent decay constant λ R σ n p v n T V [ ] s 1 reaction rate in volume V
Reaction Rate in Stellar Environment reaction rate per second and cm 3 : r n n σ p T v Reaction rate for particles with velocity distribution Φ(v) r 1 n n σ v Φ( v) dv p T 1+ δ pt ccounting for reactions Between identical particles
Maxwell Boltzmann Distribution In stellar material of temperature T particles follow ideal gas law Φ( v) 4π m π kt 3/ m v kt v e with Φ( v) dv 1 4 arbitrary units 3 1 max at EkT example: in terms of energy E1/ m v 0 0 0 40 60 80 energy (kev)
Temperature in Stars
Gamow Window & Reaction Rate The Gamow window is the energy range for which the cross section needs to be experimentally or theoretically known!
The Gamow Range of Stellar Burning The Gamow window or the range of relevant cross section is calculated: 3/ bkt E0 0.1 1 T ( ) 1/3 /3 Z Z MeV 9 E 4 6 E0kT 0.368 1 T9 3 ( ) 1/ 6 5/ Z Z MeV with reduced mass number and T 9 the temperature in GK
The Gamow peak for 1 C(p,γ) 13 13 N Note: kt.5 kev!
Examples of Gamow window energies EG amow [M ev] 10.00 1.00 0.10 0.01 0.0 0.1 1.0 10.0 temperature [GK] p+p 1C+p 1C+a 1C+1C strong dependence on Z & temperature
Gamow-Range & Reaction Rate σ: cross section ωγ: res. strength E R : res. energy
Typical temperature dependence of 19 F( Ne reaction rate of 19 F(α,p) Ne 1.00E+10 reaction rate [cm3/s/mole] 1.00E+05 HF-rate exp Rate 1.00E+00 1.00E-05 1.00E-10 1.00E-15 1.00E-0 19 F(α,p) Ne 0.1 1.0 10.0 temperature [GK] exponential increase with temperature
Stellar reaction rates r 1 YT Ypρ N < σv 1+ δ pt > reactions per s and cm 3 1 λ Y p ρ N < σv 1+ δ pt > reactions per s & Target nucleus this is usually referred to as the stellar reaction rate units of stellar reaction rate N <σv>: usually cm 3 /s/mole n T X T ρ N ρ T N Y T X T ; mass fraction Y T : abundance
Change in bundance + a B reaction is a random process with a reaction probability (reaction rate) and follows the laws of radioactive decay: Depletion of Formation of B dn dt dn dt B n λ n λ ρ N a + n Y < σ v > consequently: n ( t) n 0 e λ t n B ( t) n 0 (1 e λ t )
Stellar lifetime of nuclei 0.007 0.006 abundance 0.005 0.004 0.003 0.00 Y 0 Y 0 /e B same abundance level Y 0 0.001 Y Y B 0 10-10 -1 10 0 10 1 10 10 3 10 4 10 5 ( t) Y ( t) Y 0 0 e λ t (1 e λ t ) τ time 1 τ λ 1 Y a ρ < σ v N >
Energy production Reaction Q-value: Q Energy generated (if >0) by a single reaction Q c m i m j initial nuclei i final nuclei j Difference between masses in entrance and exit channel Energy generation: Energy generated per g and sec by a reaction: ε r Q 1 Q Y Y ρ N < σ v a ρ 1+ δ a >
Reaction Flow Reaction flow is defined as the net # of nuclei converted in time T from species to B via a specific reaction F T dy dt dt 0 via specific reaction T 0 λ( t) Y ( t) dt Reaction path is usually defined as the sequence of reactions with maximum reaction flow in a certain stellar environment!
Reaction Network Simulations Reaction Network Simulations Change of istopic abundances: d Y d t i + i j,k,l of Stellar Nucleosynthesis Σ N λ Y + N ρ N < j,k > j j k l Σ N ρ N < j,k,l > Y Y Y j,k,l i j j j Σ j,k i j,k j k Y Y + (α,γ) (p,γ) + (γ,p) (β ν) Coupled system of nonlinear differential equations Reaction flux net number of reactions:
Cross Cross-Section and S Section and S-Factor Factor ), ( ), ( 1 ), ( ) ( ) ( 0 0 0 0 R E G R E F R E P R r r P l l l l l l + ϕ ϕ πη 0 ) ( 0, << e E P E E C l S-factor to correct for Coulomb barrier: S(E) σ(e) E e πη
Example: 1 C(p,γ) cross section need cross section here! [ ] MeV E Z Z E e Z Z cm cm µ µ π η π σ η π 1 1 31.38 e S(E) h
S-factor Conversion From the NCRE compilation of charged particle induced reaction rates on stable nuclei from H to Si (ngulo et al. Nucl. Phys. 656 (1999) 3
Reaction rate from S-factorS If S-factor ~ constant over the Gamow range the rate is calculated in terms of the S-factor S(E)S(E 0 ) N < σv > 7.8310 9 Z1Z µ T 9 1/3 S( E 0 )[MeV barn] e 4.487 1 Z Z T 9 µ 1/ 3
Example: 1 C(p,γ) 13 13 N Calculate the reaction rate as function of temperature! N<σv> cm 3 /s/mole 1.E+10 1.E+05 1.E+00 1.E-05 1.E-10 1.E-15 1.E-0 0.01 0.1 1 10 temperature [GK]
Calculate life time of 1 C life time [s] 1.E+0 1.E+15 1.E+10 1.E+05 1.E+00 1.E-05 1 1 τ λ Y ρ < σ v > with ρ100 g/cm 3 and X H 0.5 nonres 0.01 0.1 1 10 a N temperature [GK] Determine 1 C life time in the sun T0.015 GK! τ 6 10 13 s 10 6 y
Resonant Reaction Rate N σv 3/ 11.605E 11 1 T9 1.5410 ωγ [ MeV ] e µ T9 R [ MeV ] ωγ ( J + 1) Γ Γ in out Γ ( ) ( ) tot Γ i i j + 1 j + 1 Γ p T tot Γ in Θ p, α at low energy astrophysical conditions: Γ in << Γ out Γ tot Γ p, α << 1 TV, Θ c V l p, α spectroscopic factor, depends on resonant state configuration
Calculate 1 C(p,γ) 13 N resonance ωγ 0.65 ev 6.5 10-7 MeV experimental value: E cm r 0.45 MeV assumed value: E cm r 0.05 MeV 1.00E+10 N<sv> cm3/s/mole 1.00E+05 1.00E+00 1.00E-05 1.00E-10 1.00E-15 1.00E-0 0.01 0.1 1 10 temperature [GK]
Comparison between resonant and non-resonant contributions N<sv> cm3/s/mole 1.00E+10 1.00E+05 1.00E+00 1.00E-05 1.00E-10 1.00E-15 fictional resonance at 0.05 MeV non-resonant term resonance at 0.45 MeV 1.00E-0 0.01 0.1 1 10 temperature [GK] strong dependence in resonance energy!!!
with fictional resonance at 0.05 MeV life time of solar 1 C would drop to 1 s only! Impact on life time of 1 C life time [s] 1.00E+0 1.00E+15 1.00E+10 1.00E+05 1.00E+00 τ 1 1 λ Y ρ > a N ( < σ v > + < σ v ) nonres resonance at 0.45 MeV res 1.00E-05 1.00E-10 resonance at 0.05 MeV 0.01 0.1 1 10 temperature [GK] ]
Reaction Rates for inverse Processes σ σ a Bb ( ( j j B + 1)( + 1)( j j b a + 1) + 1) m m B m m b a E E Bb b detailed balance The ratio of reaction rate to inverse reaction rate depends on Q and T! συ συ E Bb συ a Bb Bb Q + 8π µ E a a 8π µ ( ( Bb j j B ( kt ) ( kt ) + 1)( + 1)( 3 3 j j a b 0 0 σ σ a Bb E E + 1) µ + 1) µ a a a Bb e e 3 E kt a EBb kt e de de Q kt a Bb συ a
Photodisintegration Processes γ + a Q + a C + γ γ + C + a equilibrium conditions between forward and reverse reaction! N N a συ a N C λ γ C Saha Equation photo disintegration rate in units [1/s] N N NC λ γ a 1 τ γ eq ( π µ kt ) ( π µ kt ) a 3 h a 3 h 3 3 ( j + 1)( ja ( jc + 1) ( j + 1)( ja ( jc + 1) + 1) e + 1) e Q kt Q kt συ a
Conclusion stellar reaction rates depend on low energy cross sections reaction rates determine stellar nucleosynthesis timescale of nuclear burning processes stellar energy production reaction rates depend on cross section and resonance parameters in Gamow range