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Nuclear Reactions. Nuclear Reactions. Binding Energies The mass law below represents the masses of thousands of nuclei with a few parameters B=(Z(m p +m e )+(A-Z)m n - M(A,Z))c 2 Mass Excess M = 9.31.478MeV (M(A,Z)-A) ; M in AMU

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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


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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


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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


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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


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Nuclear Reactions

In center of mass frame


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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.


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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


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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 YaYbNA<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


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Nuclear Reactions

Nuclear rxns in stars can progress down three paths

  • Complete burning - most familiar HHe, HeCO 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


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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


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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


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The Asymptotic Giant Branch

Extreme density gradients outside degenerate corre and burning shells


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The Asymptotic Giant Branch

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


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The Asymptotic Giant Branch

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


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The Asymptotic Giant Branch

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


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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


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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


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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


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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



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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


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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


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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


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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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