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Nuclear Reactions

Binding Energies

- The mass law below represents the masses of thousands of nuclei with a few parameters
- B=(Z(mp+me)+(A-Z)mn - M(A,Z))c2
- Mass Excess M= 9.31.478MeV (M(A,Z)-A) ; M in AMU
- Q value - energy released in exit channel of rxn assuming incoming kinetic energy small Min - Mout
- B/A binding energy per nucleon

Nuclear Reactions

Mass terms

- M(A,Z) = Zmp + (A-Z)mn
- m1 = -a1A volume term
- m2 = a2A2/3 surface tension
- m3 = a3(A/Z - Z)2/A symmetry term from Fermi energy of p&n Fermi-Dirac gases
- m4 = a4 Z2/A1/3 Coulomb repulsion of protons
- m5 = (A) pairing energy - paired p or n more tightly bound
- set to find minimum in mass for a given A - valley of stability

Nuclear Reactions

- valley of stability - At high Z, nuclei are stable only if neutron # > proton # - coulomb term otherwise too large
- High Z elements neutron rich - initla stellar composition n poor - need rxns which are n sources

Nuclear Reactions

The Coulomb barrier

- Classical limit
- Rnucleus ~ r0A1/3 ; r0=1.2x10-13cm
- r >> = h/mc x c/v

- QM limit
- compton = h/mc = 1.13x10-13cm
- for v/c ~ 0.25, ~ 4.5x10-13cm

- Rxn rate for flux of particles Npv into a target of area a, thickness x, and density Nt

Nuclear Reactions

In center of mass frame

Nuclear Reactions

Assuming Boltzmann dist.

Integrand max when =E/kT+b/√E is a minimum

gives shape of nuclear potential

Coulomb part of potential

v2/2

nuc. pot.

Nuclear Reactions

Resonances

After capture the new particle may be in an excited state of the compound nucleus. This increases the cross section for capture in a narrow energy range around the excited state with width E= /state

Network equations

A term exists for every possible rxn channel which creates or destroys j

finite difference approx

Nuclear Reactions

Terms such as Yj(t+t)Yk(t+t) go to Yj(t)k+ Yk(t)j+Yj(t)Yk(t)

linearize - discard higher order terms in

An eqn linear in unknowns can be written for each species

The eqn for each species j contains a term k for each species k connected to j by a rxn

Write as a matrix A=B where is a column of ’s

A is a JxK matrix for J species with K terms - generally J=K with most entries = 0

B is a column of RHS rxn coefficients YaYbNA<v>

Want ’s = Y to determine change in Y

Solve =BA-1

This formulation automatically includes reverse rates for rxns since for every matrix element j,k there is an element k,j which describes reverse rxn

Nuclear Reactions

Nuclear rxns in stars can progress down three paths

- Complete burning - most familiar HHe, HeCO ash is a minimum energy state
- Steady state - dYi/dt=0 from contributions of several channels - CNO in H burning reach steady state abundances for a given T,
- Equilibrium - forward/reverse rates balance. Get broad distribution of abundances determined by chemical potentials - minimize thermodynamic free energy of system
Limiting rates determine speed of reaction - often weak interactions e.g. in PP chain 1H(p,+)2D ~109yr

The Asymptotic Giant Branch

When He core exhausted He shell burning begins

Like H shell burning He shell drives the star redward - moves star along the Asymptotic Giant Branch roughly parallel to but higher in luminosity than the RGB

The Asymptotic Giant Branch

When He core exhausted He shell burning begins

Like H shell burning He shell drives the star redward - moves star along the Asymptotic Giant Branch roughly parallel to but higher in luminosity than the RGB

Second dredge-up brings H burning products to surface

H shell quenched until He shell moves out far enough to heat shell to burning T

# of stars on AGB/# of stars on HB gives constraint on amount of time star spends in core He burning

The Asymptotic Giant Branch

Extreme density gradients outside degenerate corre and burning shells

The Asymptotic Giant Branch

Center of star is degenerate and cooling from weak emission - peak T not in core

The Asymptotic Giant Branch

Star has extremely compact core - most of radius is extended envelope

The Asymptotic Giant Branch

Star has extremely compact core - most of radius is extended envelope

The Asymptotic Giant Branch

Double shell burning or Thermal Pulse AGB

- q(He) ~ 0.1q(H) so He shell catches up to H shell
- As He shell approaches H shell material expands, H shell quenched
- He burns outward, runs out of fuel, quenched H shell restarts, eats outward, ash builds up
- He shell ignites, repeat

The Asymptotic Giant Branch

Double shell burning or Thermal Pulse AGB

- During He shell phases envelope convection penetrates deeply into star
- He shell produces small convective shell
- Non-convective mixing allows transport between shells
- mixing 12C into H flame zone or p into He flame gives 12C(p,)13N(+decay)13C
- 13C(,n)16O is a neutron source - only works when p and He burning can mix

The Asymptotic Giant Branch

Double shell burning or Thermal Pulse AGB

- s-process - slow n capture onto Fe peak seed nuclei - each n captured has time to decay to a proton, increasing Z
- s-process takes place in intershell region where n produced primarily in intermediate mass stars just above maximum mass for He flash
- Produces species with A>90
- 3rd dredge-up (actually numerous dredge-ups for each thermal pulse cycle) brings partial He burning products to surface with s-process enhancements - most efficient at low metallicity - C stars, Sr stars

The Asymptotic Giant Branch

s-process peaks where p & n form closed shells - p and n magic numbers

i.e. 208Pb with Z=82, N=126, both magic numbers

even Z and even A nuclei more abundant

Double shell burning or Thermal Pulse AGB

- Produces species with A>90

AGB Mass Loss

- Often highly asymmetric (bipolar)
- AGB stars generally very cool - spectra dominated by molecular species
- H2O, TiO, VO, Sr, Ba compounds, Si2O3; SiC, C2, Buckyballs in carbon stars

- Complex molecular spectra and low T allow line blanketing - much of high L goes into accelerating wind
- Atmospheres of cool stars dust rich
- winds from direct radiation pressure
- dust formation region can act like mechanism - drive pulsations which become non-linear and create shocks in low stellar atmosphere

AGB Mass Loss

- Thermal pulses during double shell burning can drive mass loss episodes
- Shell flashes - if H or He shell is degenerate when it ignites small explosion drives mass loss, may revivify proto-WD as red giant (Sakurai’s object)
- Small envelope above a burning shell can be removed in a short event - planetary nebula
- Fast wind from proto-WD evacuates bubble, causes Rayleigh-Taylor instabilities in swept-up shell
- add ionizing radiation from central star and get planetary nebula
- Low mass stars have only compact ionized bubble, high mass disperse envelope very quickly - only intermediate masses have visible PN with lifetime ~ 10,000yr

Morphology of Planetary Nebulae

- Many PN/proto-PN strongly bipolar
- IR and polarization show thick dusty torus
- Some axisymmetry due to rotation
- Mechanism for tight collimation unknown - B fields or companions possible
- Fliers - Fast, low ionization emission regions - clumps moving at hundreds of km s-1 near symmetry axis - mechanism unknown

Morphology of Planetary Nebulae

- Clumping - Shell of swept-up material breaks into dense clumps (n~104-6cm-3)
- Two possible mechanisms - Rayleigh-Taylor instability from fast, low density wind impacting shell
- Or thermal instabilities - rapid efficient cooling ahead of shock causes fragmentation on scale where sound crossing time = cooling time

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