Stellar Energy Sources

Notes based on HKT Chapter 6.1-6.2.

Monday, Feb. 17, 2025

astrophysics of stars and planets - spring 2025 - university of arizona, steward observatory

Today’s Agenda

1. Announcements (2m)

2. Reading Overview/Key Points (10m)

3. In-Class Activity 12 (20m)

4. Debrief/Reminders (2m)

Gravitational Energy Sources

Recall our earlier definition for thermal balance

\[ \frac{\partial L}{\partial m} = \epsilon \]

where \(\epsilon\) is thermonuclear energy generation rate.

The difference \(\epsilon - \frac{\partial L}{\partial m}\) is then equivalent to the rate at which heat is added to, or removed from, each gram.

Equating this to \(dQ/dt\) and combining with the first law of thermodynamics, and rearranging a few terms, we arrive at our definition for the gravitational energy generation rate

Definition 87

\[ \epsilon_{\rm{grav}} = - \left [ \frac{\partial E}{\partial t} + P \frac{\partial}{\partial t}(\frac{1}{\rho}) \right ] \]

and leading to

\[ \frac{\partial L}{\partial m} = \epsilon + \epsilon_{\rm{grav}} \]

Observation 7

Note that \(\epsilon_{\rm{grav}}\) may be positive, negative, or zero.

Energy release in gravitational sources thus arises from departures from adiabaticity during contraction or expansion.

Thermonuclear Energy Sources

Charged Particle Thermonuclear Reactions

Consider a thermonuclear reaction of the form

\[ \large{\alpha} \ (\textrm{"projectile"}) + X \ (\textrm{"target nucleus"}) \rightarrow Z^{*} \ (\textrm{compound nucleus}) \rightarrow Y + \beta \ (\textrm{"products"}) \]

or often written as

\[ \Large X(\alpha,\beta)Y \]

where the left handside is called the “entrance channel”, \(Z^{*}\) is an intermediate (almost always) excited state as a result of the reaction, that leads to the “exit channel” and the corresponding products.

An example reaction

Consider the following thermonuclear reaction:

\[ \Large p + ^{11}\textrm{B} \rightarrow ^{12}\textrm{C}^{*} \]

this compound nucleus can then break up into a various number of products, or exit channels that include,

\[\begin{split} \Large \begin{align} ^{12}\textrm{C}^{*} & \rightarrow ^{12}\textrm{C} + \gamma \\ & \rightarrow ^{11}\textrm{B} + p \\ & \rightarrow ^{11}\textrm{C} + n \\ & \rightarrow ^{12}\textrm{N} + e^{-} + \bar{\nu_{\rm{e}}} \\ & \rightarrow ^{8}\textrm{Be} + \alpha \end{align} \end{split}\]

where (**) is another excited (or ground) state of \(^{12}\)C, \(\gamma\) is photon, \(p\) is proton, \(n\) is neutron, \(e^{-}\) is electron, \(\bar{\nu_{\rm{e}}}\) is an anti-electron type neutrino, and \(\alpha\) is a \(^{4}\)He nucleus.

Nuclear Energetics

We can begin by defining the total binding energy as the energy required to break up and disperse to infinity all the constituent nucleons (protons and neutrons) in that nucleus.

\[ B_{\epsilon} = \textrm{(mass of constituent nucleons − mass of bound nucleus)} \ c^{2} (\rm{MeV}) \]

similarly, we can define the average binding energy per nucleon, \(B_{\epsilon}/A\), where \(A\) is the total nucleon mass number, A (in integer amu).

This is often used as a measure of the energy required to remove the most energetic nucleon from a given nucleus in its ground state.

Average Binding Energy Per Nucleon

Credit: Data from Wapstra et al. (1988) - Plotted is the binding energy per nucleon, \(B_{\epsilon}/A\), as a function of atomic mass number \(A\) for the most stable isobar of \(A\).

Observation 8

For nuclei with mass of \(A\lesssim\) 60, we can fuse two light nuclei and energy is released.

In particular, the energy requirement for fusion is

\[ B_{\epsilon}(1) + B_{\epsilon}(2) \lt B_{\epsilon}(3) \]

Example: The triple-alpha reaction (\(3\alpha\rightarrow^{12}\textrm{C}\))

Consider the fusion of 3\(\alpha\) particles to produce a \(^{12}\)C nucleus, written to account for total binding energy,

\[\begin{split} \Large \begin{align} 3 \times {^{4}\textrm{He}} \rightarrow {^{12}\textrm{C}} \\ 3 \times {^{4}\textrm{He}} \ (\textrm{in MeV}) - {^{12}\textrm{C}} \ (\textrm{in MeV}) & = ? \\ 3 \times [4\times (-7.1 \ \textrm{MeV})] - 12\times (-7.7 \ \textrm{MeV}) & = ? \\ -85.2 \ \textrm{MeV} + 92.4 \ \textrm{MeV} & = 7.2 \ \textrm{MeV} = \bbox{Q-\textrm{value}} \\ \end{align} \end{split}\]

In practice, mass excess (\(\Delta = [M-A]c^2\)) tables are used to find \(Q\)-values.

Fission on the branch with \(A\) greater than about 60 achieves the same end as the above, slowly approaching a saturation value of 8 MeV.

nuclei around the iron peak, which are the most tightly bound of all nuclei (per constituent nucleon), are not of much use as an energy source. the end is near.

Astrophysical Thermonuclear Cross Sections and Reaction Rates

Lets start by determining the cross section for the reaction \(X(\alpha,\beta)Y\) denoted as

Definition 88

\[ \sigma_{\alpha \beta} = \frac{\textrm{number of reactions per unit time per target }X}{\textrm{incident flux of projectiles }\alpha} (\textrm{cm}^{2}) \]

we can define the incident projectile flux as \(n_{\alpha}v\) such that the reaction rate per target nucleus is \(n_{\alpha}v\sigma_{\alpha \beta}\), leading to the total reaction rate

Definition 89

\[ r_{\alpha \beta} = n_{\alpha} n_{X} \sigma_{\alpha \beta}(v)(v) \ (\textrm{cm}^{-3} \ \textrm{s}^{-1}) \]

Here, \(n_{X}\) is the target number density.

We can simplify this by switching to a center-of-mass system expression for integrating over all particles in their respective distributions provides a averaged product of cross section and velocity:

Definition 90

\[ r_{\alpha \beta} = n_{\alpha} n_{X} \left < \sigma_{\alpha \beta} \right > (\textrm{cm}^{-3} \ \textrm{s}^{-1}) \]

where here we have introduced the reduced mass as \(m=m_{\alpha}m_{X}/(m_{\alpha}+m_{X})\).

The cross section can be expressed in the form (usually) as a function of center of mass energy \(\mathcal{E}\) as

Definition 91

\[ \sigma_{\alpha \beta} = \pi \lambda^{2}_{dB} g (2\ell+1) \frac{\Gamma_{\alpha}\Gamma_{\beta} }{\Gamma^{2}} f(\mathcal{E}) \]

where \(\lambda^{2}_{dB}\) is the reduced DeBroglie wavelength, \(f(\mathcal{E})\) is a shape factor, \(\frac{\Gamma_{\alpha}\Gamma_{\beta} }{\Gamma^{2}}\) is the joint probability of forming \(\alpha+X\) and then \(\beta+Y\) through the compound state \(Z^{*}\).

The shape factor can take two forms: resonant or non-resonant.

Resonant - Varies rapidly with energy over some interesting energy range and is strongly peaked at a resonant energy \(\mathcal{E}_{r}\).

Nonresonant - Shape factory is constant or is slowly varying compared to other factors in the cross section. Occurs when the energy range of interest is far from \(\mathcal{E}_{r}\) or when the reaction is intrinsically nonresonant.

Nonresonant Reactions

Nuclear reactions of major astrophysical interest are exothermic: they produce energy and the \(Q\)-value is positive in the nuclear energy equation.

In the classical picture, the target and projectile could never combine because the Coulomb barrier cannot be penetrated. However, quantum mechanics can allow this to occur via tunneling with some barrier penetrability factor

\[ P_{\ell}(\mathcal{E}) \propto e^{-2\pi \eta} \]

where is the dimensionless Sommerfield factor:

\[ \eta = \frac{ Z_{\alpha} Z_{X} e^{2}}{\bar{h}v} = 0.1574 Z_{\alpha} Z_{X} \left (\frac{\mu}{\mathcal{E}} \right)^{1/2} \]

this factor depends strongly on the entrance channel kinetic energy.

This allows us to bring this together and write the non-resonant form of the cross section

Definition 92

\[ \sigma_{\alpha \beta} = \frac{S(\mathcal{E})}{\mathcal{E}}e^{-2 \pi \eta} \]

A common procedure is to extract \(\sigma_{\alpha \beta}\) experimentally, to energies non less than about 100 keV and plot the astrophysical S factor,

\[ S(\mathcal{E}) = \sigma_{\alpha \beta} \mathcal{E} e^{2 \pi \eta} \]

and then extract to lower energies expected in stellar environments. This is a major source of uncertainty in stellar nuclear reaction rates.

Assuming a constanct astrophysical \(S\) factor, we can compute a numerical form for the averaged cross section:

Definition 93

\[ \left<\sigma_{\alpha \beta}\right> = \frac{1.6\times10^{-15}}{\mu^{1/2}(kT)^{3/2}}S \int^{\inf}_{0} \textrm{exp} \left [ - \left ( \frac{\mathcal{E}}{kT} + \frac{b}{\mathcal{E}^{1/2}} \right ) \right ] \]

In the above equation the integrand is called the Gamow peak.

The structure of the integrand in reflects the combination of two strongly competing factors.

• The barrier penetration factor contributes the second term, which increases rapidly with increasing energy

• the Maxwell–Boltzmann exponential decreases rapidly as energy increases

The integrand thus increases as energy increases because the Coulomb barrier becomes more penetrable but, to offset that, the number of pairs of particles available for the reaction decreases in the exponential tail of the distribution.

Example Gamow Integrand

The integrand plotted against center-of-mass energy (in keV) for the temperatures \(T_6=20\) and \(T_6 = 22\). Here, \(T_{6}=T (\textrm{K}) / T \times 10^{6} (\textrm{K})\).

We note the significant dependence of temperature for the integrand and thus the nuclear reaction rate.

Once the cross section is determined the total reaction rate can be written as

Definition 94

\[ r_{\alpha \beta} = \rho^{2} N^{2}_{\rm{A}} \frac{X_{\alpha} X_{X}}{A_{\alpha} A_{X}} \left<\sigma_{\alpha \beta}\right> \]

and the resulting nuclear energy generation rate

Definition 95

\[ \epsilon_{\alpha \beta} = Q \frac{r_{\alpha \beta}}{\rho} \]

Example Astrophysical S factor measurement and extrapolation

Credit: Fowler et al., 1967 The nonresonant factor S(E) for the reaction 12C (p, γ) 13N with an ex- trapolation to low energies.

Resonant Reactions

To capture the resonant portion of the reaction, the form is often treated as a delta function.

This leads to a form of the resonant cross-section:

Definition 96

\[ \left<\sigma_{\alpha \beta}\right> = \bar{h}^{2} \left ( \frac{2 \pi}{mkT} \right )^{3/2} g(2\ell+1) \frac{\Gamma_{\alpha} \Gamma_{\beta} }{\Gamma} e^{-\mathcal{E}_{r}/kT} \]

Further simplification assuming the integrated cross section as a delta function leads to

Definition 97

\[ \left<\sigma_{\alpha \beta}\right> = \left ( \frac{2 \pi \bar{h}^{2}}{mkT} \right )^{3/2} \frac{(\omega \gamma)_{r}}{\bar{h}} e^{-\mathcal{E}_{r}/kT} \]

The term \((\omega \gamma)_{r}\) as often tabulated and values for specific reactions can be plugged in to further reduce the cross section equation.

Lastly, the non-resonant and resonant estimates for the cross section are added together to provide the total cross section as a function of temperature.

Other Forms of Reaction Rates

Neutron Capture and the S-Process

s-process - a slow process by which excess neutrons are captured onto “seed” nuclei in the iron range of elements. can occur for example in helium shell burning in low mass stars.

r-process - a rapid process by which a rapid succession of neutron captures lead to the formation of heavier and heavier nuclei and requiring a significant amount of neutron captures. can occur in core-collapse supernovae and neutron star mergers.

Weak Interactions

Because of the strong \(T\) and \(\rho\) dependence on different reactions, it is important to also consider the half-life of the nuclei in question.

It could be the case that beta-decay, when atomic nucleus emits a high-energy electron or positron, is the more likely next interaction for the created nucleus.

For example:

In the \({^{13}\textrm{N}}(p,\gamma){^{12}\textrm{C}}\) reaction we have been following,

\[ {^{13}\textrm{N}}\rightarrow{^{13}\textrm{C}} + e^{+} + \nu_{e}, \tau_{1/2} = 10 (\textrm{min}) \]

and the decay may be the more likley channel. At higher temperatures and densities where the resonant component can contribute, and the capture reaction is comparable to the beta decay timescale.

Another example is electron capture, consider the example of

\[ e^{-} + {^{7}\textrm{Be}}\rightarrow{^{7}\textrm{Li}}+\nu_{e} \]

where the reaction is the capture of a free electron or one in an atomic orbital.

These are particularly important at high density environments where the reduction of free electrons and reduce electron degenracy pressure support such as in the iron core of a massive star.

Electron Screening

The final consideration for modifications to the reaction rate cross section is an overall reduction to the Coloumb potential due to intervening electrons. The net result is an increase to the penetrability factor and thus the reaction rate:

Definition 98

\[ \left<\sigma_{\alpha \beta}\right> (\textrm{with screening}) = \left<\sigma_{\alpha \beta}\right> (\textrm{unscreened}) e^{U_{0}/kT} \]

Screening is strongly effected by the density of the stellar environment.

In-Class Assignment

In-Class Assignment 12 is here.