Stellar Energy Sources

Ch. 6 of Pols here & HKT, Ch. 6; Ch. 6: LeBlanc 2011.

Day 10 - September, 30, 2025

Agenda:

• Updates/Reminders - HW2 - Due: Before Class, Oct. 9 (2m)
• Lecture (25m)
• ICA 10 - Not for Credit (30m)
• Report out in ica on Slack Channel (10m)

Recap - Local energy conservation

Changes in the heat content in a Lagrangian shell can occur for a various number of sources or sinks:

1. Heat is added by the release of nuclear energy, if available.
   • The rate at which nuclear energy is produced per unit mass and per second is written as . Stay tuned.
2. Heat can be removed by the release of energetic neutrinos, which escape from the stellar interior without interaction.
   • Nuclear burning or weak interaction processes: the rate at which these neutrinos take away energy per unit mass is written as .

Recap - Local energy conservation

3. Heat is absorbed or emitted according to the balance of heat fluxes flowing into and out of the shell.
   • Regulated via the local luminosity : the rate at which energy in the form of heat flows outward through a sphere of radius .

Goal: Describe the various sources of stellar energy and their relation to local energy conservation in stellar interiors.

Gravitational Energy Sources

We already introduced our gravitational energy generation rate:

• If > 0 (positive), then energy is released by the shell, typically in the case of contraction.
• If < 0 (negative), then energy is absorbed by the shell, typically in the case of expansion.
• In thermal equilibrium, = 0.

Thermonuclear Energy Sources

Charged Particle Thermonuclear Reactions

Consider a thermonuclear reaction of the form

or often written as

where the left handside is called the "entrance channel", 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

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

An example reaction

can decay to a variety of exit channels:

where () is another excited (or ground) state of C, is photon, is proton, is neutron, is electron, is an anti-electron type neutrino, and is a He nucleus.

Nuclear Energetics

Total Binding Energy: the energy required to break up and disperse to infinity all the constituent nucleons () in that nucleus.

• we can define the average binding energy per nucleon, , where 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 nucleus in its ground state.

Nuclear Energetics

Average Binding Energy Per Nucleon

Wapstra et al. (1988) - Binding energy per nucleon, , as a function of atomic mass number for the most stable isobar of .

• For nuclei with mass of 60, we can fuse two light nuclei and energy is released.

Nuclear Energetics

Average Binding Energy Per Nucleon

In particular, the energy requirement for fusion is

Example: The triple-alpha reaction ()

Consider the fusion of 3 particles to produce a C nucleus,
written to account for total binding energy (),

• Mass excess () tables are used to find -values.

Example: The triple-alpha reaction ()

Fission occurs, reaction in which the nucleus of an atom splits into two or more smaller nuclei, on the branch with 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.

• Thus any star that ends up with nuclei in the iron peak has lost potential fuel and this is a matter of grave consequence for the star.

Astrophysical Thermonuclear Cross Sections and Reaction Rates

We want to measure the cross section for the reaction :

define the incident projectile flux as such that the reaction rate per target nucleus is , leading to the total reaction rate:

Astrophysical Thermonuclear Cross Sections and Reaction Rates

Here, 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:

Astrophysical Thermonuclear Cross Sections and Reaction Rates

We have introduced the reduced mass as .

• The cross section can be expressed in the form (usually) as a function of center of mass energy as

Astrophysical Thermonuclear Cross Sections and Reaction Rates

Where we have is the reduced DeBroglie wavelength, is a shape factor, is the joint probability of forming and then through the compound state .

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

Astrophysical Thermonuclear Cross Sections and Reaction Rates

• Resonant - Varies rapidly with energy over some interesting energy range and is strongly peaked at a resonant energy .

• 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 or when the reaction is intrinsically nonresonant.

We will explore these two types of Thermonuclear Reactions.

Nonresonant Reactions

Nuclear reactions of major astrophysical interest are exothermic: they produce energy and the -value is positive in 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

Nonresonant Reactions

Here is the dimensionless Sommerfield factor:

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

Nonresonant Reactions

A common procedure is to extract experimentally at accessible laboratory energies (usually, and unfortunately, at energies not much below 100 keV) and plot the astrophysical S factor:

and then extrapolate to lower energies expected in stellar environments.

• This is a major source of uncertainty in stellar nuclear reaction rates.

Nonresonant Reactions

Assuming a constant astrophysical factor, we can compute a numerical form for the averaged cross section:

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

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

Nonresonant Reactions

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

• the MB 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.

Nonresonant Reactions

Example Gamow Integrand

The integrand plotted against center-of-mass energy (in keV) for the temperatures and . Here, .

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

Nonresonant Reactions

Once the cross section is determined,

the total reaction rate:

nuclear energy generation rate:

Nonresonant Reactions

Example Astrophysical factor measurement and extrapolation

Fowler et al., 1967 The nonresonant for the reaction with an extrapolation to low energies.

Resonant Reactions

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

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

We can simplify this further by evaluating the delta function.

Resonant Reactions

The term as often tabulated and values for specific reactions can be plugged in to further reduce the cross section equation.

• 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 and dependence on different reactions, it is important to also consider the half-life of the nuclei in question.

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

For example, the reaction produces the nucleus in the ground state.

Weak Interactions

However, the half-life for beta decay via emission of a positron () is much shorter for temperatures like those on the main-sequence:

and the decay may be the more likley channel, as opposed to the channel.

• At higher temperatures and densities where the resonant component can contribute, and the capture reaction is comparable to the beta decay timescale.

Weak Interactions

Another example is electron capture, consider the example of

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

• These are particularly important at high density/degenerate 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:

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

In-Class Assignment 10

In class: Work on ICA here with partner, I will ask one or two people to share and describe plots at the end of class.

After Class: Not for Credit

ICAs are not always designged to be completed but rather worked on in class, submit what you have when you leave the class even if you did not make much progress.