Stellar Energy Sources II

Ch. 6 of Pols here & HKT, Ch. 6.3-6.8; Ch. 6: LeBlanc 2011.

Day 11 - October, 2, 2025

Agenda:

• Updates/Reminders - HW2 - Due: Before Class, Oct. 9 (2m)
• Lecture (25m)
• ICA 10 - 4 Groups - Due: End of Day (15m)
• ICA 10 Group Discussion (15m)
• Groups Report out in ica on Slack Channel (10m)

Recap - 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.

Recap - Thermonuclear Reactions

Total reaction rate:

nuclear energy generation rate:

Recap - Thermonuclear Reactions

• 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.

• The non-resonant and resonant estimates for the cross section are added together to provide the total cross section as function of .

Goal: Describe the various burning cycles in stars (and nuetrinos).

The Proton–Proton Chains

The Proton–Proton () Chains - A series of reactions that lead to the production of .

• The subsequent chains become more important as the temperature increases.

Another way to visualize these reactions is via a mass number versus charge .

The Proton–Proton Chains

Proton–Proton Chain Reaction Flow

• Starting from , the reaction sequences in the three -chains all end up at .

• The slowest reaction in the chain is the -reaction itself, leading to it being a bottleneck and controlling the lifetime of the star on the main-sequence.

The Proton–Proton Chains

We can define an effective -value based on weights of the contribution for each subchain:

This allows use to also compute an effective energy generation rate:

From Table 1.1 in HKT, the temperature dependence for .

The Carbon–Nitrogen–Oxygen (CNO) Cycles

CNO cycles: series of proton captures on isotopes of CNO, positron decays and ending with a proton capture to produce

CNO Cycle Reactions

• The slowest step (lowest reaction rate) in the sequence is the . This reaction is often referred to as the bottleneck reaction rate for stars that burn H via the CNO cycles.

The Carbon–Nitrogen–Oxygen (CNO) Cycles

Similar to the -chains we can determine an effective energy generation rate:

The temperature exponent is significantly larger than that of the -chains with .

The Carbon–Nitrogen–Oxygen (CNO) Cycles

Cross over temperature for pp-chains to CNO cycles

• In a solar like star with with , and , , as we saw in ICA9.

• For more massive stars with higher central the larger dependence of dominates the H-burning on the main-sequence.

Helium-Burning Reactions

Helium burning in stars will begin in earnest at sufficient tempratures (> K) via the first step in the "triple-alpha" reaction:

However, recall that has a lifetime of only seconds! So, the next stage of the reaction can only proceed at sufficient and seed nuclei available. The next reaction in the sequence is

Note an intermediate step via the creation of an excited state of
which can decay back to . The main point being that not all forward reactions will lead to creation of .

Helium-Burning Reactions

Note: an intermediate step via the creation of an excited state of which can decay back to .

We can similarly determine an energy generation rate for triple-:

At a temperature of , the temperature exponent is !

Helium-Burning Reactions

Next, we have the reaction.

• “If users find that their results in a given study are sensitive to the rate of the reaction, then they should repeat their calculations with 0.5 times and 2 times the values recommended here.” - Fowler (1985)

• Operates at tempratures of around .

One of the most important and difficult cross-sections to measure experimentally e.g. deBoer et al. 2017.

Helium-Burning Reactions

Finally, we have the reaction.

• The race between how quickly is produced via triple- and how quickly it is consumed via (or ) is of significant consequence for many different subsequent stellar evolution consequence.

• For example, the final C/O ratio of white dwarf star can lead to different observational properties.

Carbon Burning

Carbon Burning The first of these advanced burning stages is carbon-burning via and their two exit channels.

• These creactions are followed by and reactions to produce primarly and at lesser amounts and .

The energy generation rate for these two reactions is given by

These reactions are susceptible to strong electron screening effects.

Neon Burning

Neon Burning - takes place via photodisintegration the use of high-energy photons to break up via the .

However, temperatures are also high enough to allow the reaction sequence, .

The net result of neon burning is , , and .

Oxygen Burning

Oxygen Burning proceeds in a similar fashion as carbon this time with three exit channels summarized in the Table below:

Table of Carbon- and Oxygen-Burning Reactions

The energy generation rate for these reactions is given by

Oxygen Burning

C- and O-Burning Reactions

• We note that are possible by the seed is used up quickly by and the reaction is intrinsically slow.

The final result of Oxygen burning is the production of , , and depending on the conditions in the core.

Silicon “Burning” or "Melting"

At temperatures , many reactions are possible.

Photodisintigration plays a role in Silicon burning via the path:

• In this process, the photodisintegration has essentially add two neutrons to produce . Many similar pathways comprise the collection of silicon burning.

Silicon “Burning” or "Melting"

Adapted by Clayton (1968) - Sample reaction network for silicon burning that also shows the reactions possible between nuclei in the network.

• As burning advances you can reach quasi-statistical equilibrium (QSE) where the photodisintigration rates roughly match the capture rates.

The result of silicon burning is production of nuclei in the iron peak. For quiescient burning, where much time is allowed to pass, the most abundant nuceli is . For short timescales such as in CCSNe, the electron/positron decay and electron capture rates are insufficient and the primary product is .

Silicon “Burning” or "Melting"

• At quasi-statistical equilibrium (QSE), abundances of most nuclei may approximated by a nuclear version of the Saha equation.

• The result of silicon burning is production of nuclei in the iron peak.

  • For quiescient burning, where much time is allowed to pass, the most abundant nuceli is .
  • For short timescale (explosive burning) such as in CCSNe, the electron/positron decay and electron capture rates are insufficient and the primary product is .

Neutrino Emission Mechanisms

Neutrino absorption or scattering requires high density or neutrino energies.

• We can determine this value by looking at the mean free path for a neutrino as .
  • This can typically occur in the proto-neutron star, the collapsed iron core of a massive star where nuclear densities are reached, and the neutrinos can become "trapped".

In less extreme stellar environments, we can think of neutrinos as a power drain (or sink) removing energy from the system.

Pair Annihilation Neutrinos

Produced by the annihilation of an electron by a positron

however, this reaction requires positrons ().

At sufficient temperatures, () ambient photons can undergo pair creation (aka called pair production):

Photoneutrinos and Bremsstrahlung Neutrinos

Photoneutrinos
Similar to electron-photon scattering but not producing a gamma ray:

Bremsstrahlung (braking radiation) Neutrinos

Yields a photon when an electron is scattered off and accelerated (positive or negative) by an ion. This is an important energy loss mechanism for hot white dwarfs.

Plasma Neutrinos

In a very dense plasma, electromagnetic waves can be quantized in such a way that they behave like relativistic Bose particles with finite mass, plasmons, or heavy photons.

• These can decay into or pairs.

Total Neutrino Energy Loss Rates

Combined Neutrino Loss Rates

Combined losses from pair annihilation, photo-, and plasma neutrinos versus and temperature.

Adapted from the calculations of Itoh and collaborators.

In-Class Assignment 10

In class: Work on ICA here with groups per usual. Discuss conceptual questions together and prepare answers to share at the end of class.

After Class: Due to D2L, by End of Day

Note: ICAs will be shorter with the goal of: reducing focus on coding, increasing time for discussion and interpretation of results / plots in groups and as a class.