Outline

7. Energy generation

1. Gravitational energy
The expression for the gravitational energy generation per gram of material is deceptively simple:

ε_grav = − dQ /dt = − [dE /dt + Pd(1/ρ) /dt]
Q: Heat added per gram
E: Internal energy per gram
and the derivatives are understood to be Lagrangian, e.g. following the matter (automatic on a mass scale).

Mushing things about with thermodynamic identities and adiabatic coeficients, we may write in terms of our model state variables:

ε_grav = − [P/ρ(Γ₃ − 1)] [d lnP/dt − Γ₁ d lnρ/dt] .

At first, this expression seems not useful, because we are used to thinking 'what is the temperature profile if ε is such-and-such?' How are we to determine the time derivatives as functions of P and T? This thinking is the wrong approach. Instead, we should think of this equation as 'how the state variables will change with time if the local divergence of the flux is so-and-so'. Somewhere there must be a divergence of flux. The surface radiates l(m) = L, and l(m) = 0 at the center. Time evolution evens out the divergence over m until a relatively steady structure is achieved. Ultimately, the logarithmic derivatives will try to equalize throughout the star; all the star must age at roughly the same rate. Such equality is called homologous change; all of the layers change in proportion. Gravitational collapse is time-dependent at its core, despite these relatively stable configurations.

In radiative regions, the divergence of flux is controlled by the opacity. Very opaque regions will insulate the interior, preventing transport so luminosities will be low and time rates of change will be long.

In convective regions, things are different. Remember that the adiabatic gradient says nothing about flux or flux divergence. Technically, a purely adiabatic region cannot transport flux at all, but a slight super-adiabaticity can carry nearly infinite flux if necessary. Thus, in regions of efficient convection the structure remains very nearly adiabatic. ε_grav is determined by the global contraction/expansion controlled by the radiative surface and/or other radiative regions.

2. Nuclear Reactions
1. Introduction

General process: α + X → Z* → Y + β, xxxx or X(α,β)Y .
α, X, Y, β : nuclear species,
Z* : intermediate, short-lived, usually excited, compound nucleus.
α, β are often smaller particles such as α n p e ν γ, etc.

The intermediate Z* often has many decay channels Y + β.

Nuclear reactions are usually treated with the same formalisms used for atomic excitation/ionization/recombination/molecular reactions; e.g. cross sections, lifetimes.

For an excited state, the natural width Γ_i = ΔE_i = (h/2π) / τ_i
[(1/τ_i) ≡ A_ji in atomic transitions.]

2. Energetics

Total binding energy: energy required to disperse all nucleons to infinity.
The usual graph of binding energies per nucleon as a function of the mass number A:

For example:
Total BE of ⁴He is 4 × 7.074 = 28.3 MeV,
total BE of ¹²C is 12 × 7.680 = 92.2 MeV, so
3 (⁴He) − ¹²C = 7.26 MeV, a typical Q-value.

Radii: R ≈ 1.4 A^1/3 femtometers.
Minimum interparticle separation: D ≈ 1.4 (A_α^1/3 + A_X^1/3) fm.
Coulomb barrier: Z_α Z_X e ² / D ≈ 1 MeV.

3. Cross sections and rates

The fuzzy quantum mechanical process is treated in bulk with cross sections:
Total rate in interval dv : R_αβ(v) dv = n_α(v)dv σ_αβ(v) v n_X cm⁻³s⁻¹,
where v is the relative speed in the rest frame of the α-X center of mass. We may write the rate integrated over all speeds as

R_αβ = ∫ n_α(v) σ_αβ(v) v n_X dv = n_α n_X <σv>_αβ cm⁻³s⁻¹,

partly for convenience and partly because usually the range of thermal kintetic energies is small relative to the energy range of nuclear excitations.

The Maxwell distributrion function of kinetic energies is:

n_α(E)dE = n_αΨ(E)dE = n_α (2/√π) (kT)^−3/2 e^−E/kT E ^1/2 dE ,

normalized so ∫Ψ(E)dE = 1. Non relativistic, E = mv ²/2.
So,

<σv>_αβ = (8/πm)^1/2(kT)^−3/2 ∫σ_αβ (E) e^−E/kT E dE cm⁻³s⁻¹,
m = m_αm_X/(m_α + m_X).

The usual expression for σ_αβ (E) is

σ_αβ (E) = π (λ/2π)² g (Γ_αΓ_β /Γ²) f (E),
π (λ/2π)² = π(h/2π)²/2Em = 0.657/Eμ barns (the 'deBroglie cross section'); E in MeV, μ reduced mass in AMU.
g : statistical weight factor.
(Γ_αΓ_β /Γ²) : joint probablility of α-channel in, β-channel out.
f (E) : profile function; resonant and/or non-resonant.

Resonant reactions

The compound nucleus Z* has a discrete energy state E* so the energy equation reads

m_Z*c ² + E* = m_αc ² + m_Xc ² + E_r ,

and the kinetic energy range significantly overlaps E_r. For well-separated nuclear states, f (E) becomes the Breit-Wigner (Lorenzian)

f (E) = Γ²/[(E − E_r)² + (Γ/2)²].

If we assume that the width of the resonance Γ is much narrower than any variations of other terms over energy, we may replace those other terms with their values at E_r, and carry out the integral of f (E) to get

<σv>_αβ = (h/2π)² (2π/mkT)^3/2g (Γ_αΓ_β /Γ ) e^−E_r /kT.

The expression g (Γ_αΓ_β /Γ ) is written (ωγ)_r by Fowler, et al. and tabulated for many reactions. Then

<σv>_αβ = 2.56×10⁻¹³ (ωγ)_r (μT₉)^−3/2 e^{−11.605E_r /T₉}; ωγ and E_r in MeV.

Expressing the above temperature dependance as a power law T ^ν, ν = (11.605E_r /T₉) − 3/2.

Homework: Show that the formula for the resonant <σv>_αβ above is correct (i.e. integrate the profile) and that the power law exponent ν is correct.

Non-resonant reactions

When there is no energetically favorable resonance, the cross section takes the form

σ_αβ (E) = S(E ) e^−2πη /E .

The 1/E factor comes from the deBroglie cross section,
The e^−2πη factor comes from the Coulomb barrier penetration,
and the S(E ) contains all the other miscelaneous constants and weakly energy dependent factors.

Here η = Z_αZ_Xe ²/(vh/2π) = 0.1574 Z_αZ_X (μ/E )^1/2.

Substituting into the formula for <σv>_αβ averaged over the Mawell distribution of speeds gives

<σv>_αβ = 1.6×10⁻¹⁵ μ^−1/2 (kT)^−3/2 S_o ∫ exp[ − (E/kT + b/E^1/2)] dE cm³s⁻¹,
with b = 0.99 Z_αZ_X μ^1/2.

Everything is in AMU and MeV. S_o is the value of S(E ) at E_o described below. The figure below shows the terms in the exponential, their sum, and a scaled to fit plot of the exponent function itself for the reaction p + ¹²C at T₆ = 20.

The exponent function forms a nearly Gaussian peak, called the Gamow peak. The maximum is at

E_o = 1.22 (Z_α² Z_X ² μT₆² )^1/3 keV

and has a full-width at 1/e of maximum of about Δ ≈ 2.3 (E_okT)^1/2.

Assuming a Gaussian shape, the average rate becomes:

<σv>_αβ = 0.72×10⁻¹⁸ S_o (μZ_αZ_X)⁻¹ e^−τ τ ² cm³s⁻¹,
τ = 3E_o/kT.

Expressed as a power law T ^ν, ν = (1/3) (a T₆^−1/3 − 2),
a = 42.49 (Z_α² Z_X ² μ)^1/3.

4. Summary

The rate for an αβ reaction is
R_αβ = ∫ n_α(v) σ_αβ(v) v n_X dv = n_α n_X <σv>_αβ = ρ ² N_Av² (X_α /A_α) (X_X /A_X) <σv>_αβ cm⁻³s⁻¹,
where the X 's are the mass fractions. The energy generation is
ε_αβ = R_αβ Q_αβ /ρ erg g⁻¹ s⁻¹.

The energy generation is usually espressed as a power law,
ε = ε_o (ρ/ρ_o)^λ (T/T_o)^ν,
λ = 1,
ν = (11.605E_r /T₉) − 3/2, resonant,
ν = (1/3) [42.49 (Z_α² Z_X ² μ)^1/3 T₆^−1/3 − 2] non-resonant.

Reaction rates also contribute to abundance equilibrium and time rates of change equations, of course.

5. Electron screening

The presence of electrons lowers the Coulomb barrier, particularly at distances greater than the inter-electron distance. The electrons tend to cluster in the neighborhood of nuclei. An approximation for the reduction of the potential comes from Debye-Hückel theory:

φ(r) = (Ze /r) e^−r/r_D,
r_D = [(4πe ²/kT) (Z ²n_I + n_e)] ^−1/2.

Various further approximations involving the classical turning radius r_t = (Ze)²/E_GamowPeak leads to a reduction of the barrier by U_o = (Ze)²/r_D , which in turn increases the velocity-averaged cross section by a factor of e^{U_o /kT}. If this factor is significant, some care must be taken to do the screening right.

6. Specific cases

The proton-proton chain and deuterium burning
The CNO cycles
Helium burning
C, O, Ne, Si burning and photo-disintegration processes
Nuclear statistical equilibrium among the iron group