Introduction to stellar reaction rates

1. Introduction to stellar reaction rates • Nuclear reactions • generate energy • create new isotopes and elements Notation for stellar rates: 12C 13N p g 12C(p,g)13N The heavier“target” nucleus(Lab: target) the lighter“incoming projectile”(Lab: beam) the lighter “outgoingparticle”(Lab: residualof beam) the heavier “residual”nucleus(Lab: residual of target) (adapted from traditional laboratory experiments with a target and a beam)

2. Typical reactions in nuclear astrophysics: (p,g) (p,a) (p,n) and their inverses (n,g) (n,a): A(B,g) A(B,p) A(B,a) A(B,n)

3. cross section s bombard target nuclei with projectiles: relative velocity v Definition of cross section: # of reactions = . # of incoming projectiles per second and target nucleus per second and cm2 with j as particle number current density.Of course j = n v withparticle number density n) l = s j or in symbols: Units for cross section: 1 barn = 10-24 cm2 ( = 100 fm2 or about half the size (cross sectional area) of a uranium nucleus)

4. Reaction rate in stellar environment Mix of (fully ionized) projectiles and target nuclei at a temperature T Reaction rate for relative velocity v in volume V with projectile number density np Reactions per second so for reaction rate per second and cm3: This is proportional to the number of p-T pairs in the volume. If projectile and target are identical, one has to divide by 2 to avoid double counting

5. Relative velocities in stars: Maxwell Boltzmann distribution for most practical applications (for example in stars) projectile and target nucleiare always in thermal equilibrium and follow a Maxwell-Bolzmann velocity distribution: then the probability F(v) to find a particle with a velocity between v and v+dv is with example: in termsof energyE=1/2 m v2 max atE=kT

6. one can show (Clayton Pg 294-295) that the relative velocities between two particlesare distributed the same way: with the mass m replaced by the reduced massm of the 2 particle system the stellar reaction rate has to be averaged over the distribution F(v) typical strong velocity dependence! or short hand:

7. expressed in terms abundances reactions per s and cm3 reactions per s and targetnucleus this is usually referred to as the stellar reaction rateof a specific reaction units of stellar reaction rate NA<sv>: usually cm3/s/mole, though in fact cm3/s/g would be better (and is needed to verify dimensions of equations) (Y does not have a unit)

8. Abundance changes, lifetimes Lets assume the only reaction that involves nuclei A and B is destruction(production) of A (B) by A capturing the projectile a: A + a -> B Again the reaction is a random process with const probability (as long as the conditions are unchanged) and therefore governed by the same laws as radioactive decay: consequently:

9. and of course after some time, nucleus Ais entirely converted to nucleus B Example: A B Y0A sameabundancelevel Y0A abundance Y0A/e t time Lifetime of A (against destruction via the reaction A+a) : (of course half-life of A T1/2=ln2/l)

10. Energy generation through a specific reaction: Again, consider the reaction A+a->B Reaction Q-value:Energy generated (if >0) by a single reaction in general, for any reaction (sequence) with nuclear masses m: Energy generation: Energy generated per g and second by a reaction:

11. Reaction flow abundance of nuclei of species A converted in time in time interval [t1,t2] intospecies B via a specific reaction AB is called reaction flow For Net reaction flow subtract the flow via the inverse of that specific reaction (Sometimes the reaction flow is also called reaction flux)

12. Multiple reactions destroying a nuclide 14O example: in the CNO cycle, 13N can either capture a proton or b decay. (p,g) 13N each destructive reaction i has a rate li (b+) Total lifetime 13C the total destruction rate for the nucleus is then its total lifetime Branching the reaction flow branching into reaction i, bi is the fraction of destructive flowthrough reaction i. (or the fraction of nuclei destroyed via reaction i)