Post-MS Evolution through Helium Burning

Ch. 10 of Notes by Onno Pols. HKT Chapter 2.

Day 14 - October, 14, 2025

Agenda:

• Reminders - Final Project Report Draft - Due: Nov. 11 (2m)
• Lecture (20m)
• HW2 Review - 3/4 Groups - Due: Corrections, Tue. Oct 21. (30m)
• HW2 Review Report out (10m)

• the maximum mass of a non-fusing, isothermal core that can support an enclosing envelope.

The maximum core mass fraction can be defined from which we can describe the limit as

this value is appropriate for a helium core with and a H-rich envelope.

• Stars that leave the main-sequence with a helium core mass below the SC limit can therefore remain in HSE during H-shell burning. Typically this applies to stars with .

  • Low mass -: Can develop degenerate He cores, leading to a larger allowed core and the SC no longer applies.
    • Electron conduction helps keep the core isothermal and HSE.
  • Intermediate mass - to : When the SC limit is exceeded, the helium core begins to contract, and a gradient develops in the core.

Evolution tracks for stars of quasi-solar composition.

• The 1 model is characteristic of low-mass stars: the central core becomes degenerate soon after leaving the main sequence and helium is ignited in an unstable flash at the top of the red giant branch.

• We observe the mirror principle during thick H-shell burning.

• Over time, the He core mass grows and the shell begins to occupy less and less mass leading to thin H-shell burning phase.

• The rapid evolution on a thermal timescale across the H-R diagram from the end of the MS is due to this thick hydrogen shell burning.

Because the helium core in these stars have become begenerate a large density jump is developed and the envelope is quite extended. The structure of a low mass red-giant is essentially a function of its core mass:

Core-Mass Luminosity Relation:

• The luminosity of a low-mass red giant is independent of its total mass for core masses !

• First dredge-up and the luminosity bump the convective envelope has penetrated to the location of the processed material of the H-core.
  • The first dredge up can occur and lead to an increase in the C/N ratio and He surface abundances.

luminosity when the H-burning shell crosses the hydrogen discontinuity left by the first dredge-up.

• Once the thin H-shell has processed this material mixed by the FDU/1DU, it reaches again H-rich material and this leads to slower burning rate and reduction in luminosity.

Inset shows the temporary decrease of luminosity when the H-burning shell crosses the hydrogen discontinuity left by the first dredge-up.

Mass loss on the red giant branch- As the stellar luminosity and radius is increased along the giant branch, the envelope becomes loosely bound and is becomes possible for large photon flux to remove material from the surface.

One example of a mass loss rate for stellar models based on empirical data is given by Reimers:

where is a free parameter of order unity.

An example 1 stellar model loses about 0.3 of its envelope by the time it reaches the tip of the giant branch.

Two main characteristics:

• These stars ignite helium under degenerate conditions leading to a Helium Core Flash

• All these stars start helium burning at nearly the same core mass of - .

Helium Core flash - a star as an example:

• helium is ignited in a strongly degenerate core at and .

• the burning is unstable that is, it causes a temperature increase instead of decrease leading to thermonuclear runaway.

  • the electron degeneracy pressure is basically independent of so the ignition doesnt change the pressure and hence the corresponding work done as required in a stable burning event.

Helium Core flash - a star as an example:

• The nuclear energy goes into raising the internal energy of the ions leading to an increase of the temperature, but not an increase in the density of the gas - dictacted by the degenerate electrons.

  • material is nearly vertically upward in the diagram.

• the thermonuclear runaway leads to a main helium flash producing a local luminosity of - similar to a small galaxy, but lasting for only a few seconds.

• Stellar models suggest a series of flashes occur until the core has sufficiently expanded,
• degeneracy lifted and burning proceeds in stable convective core.

Evolution with time of the surface luminosity, the He-burning luminosity and the H-burning luminosity during core He-flash low-mass star.

The Horizontal Branch (HB)

• Described as the location in the HR diagram for stars that have just undergone helium flash and settled into stable burning and thermodynamic equilibrium.

  • For a fixed helium core ignition mass, the radius and effective temperature depends on the envelope mass.

• Stars with less envelope mass at the ZAHB can be substantially hotter than those with much more of their envelope remaining.

The Horizontal Branch (HB)

• zero-age horizontal branch (think gray line) for a metallicity Z = 0.001 typical of globular clusters.
• models have the same core mass (0.489 ) but varying total (i.e. envelope) mass, which determines their position in the H-R diagram.

The Horizontal Branch (HB)

• Globular cluster, M3, an example of the horizontal branch.

• models have the same core mass (0.489 ) but varying total (i.e. envelope) mass, which determines their position in the H-R diagram.

• Recall that helium burning begins when the temparature in the core has reached K via the triple- reaction.

• Core helium burning proceeds in a stable fashion, causing a large focus of energy production near the center that leads to formation of a convective core that grows with time.

• We will explore the stages of core He-burning for these intermediate mass stars.

Radial variation of various mass shells (solid lines) in a 5 ( = 0.02) during the early post-main sequence evolution.

• As point D is approached the envelope temperature decreases and the opacity in the envelope rises. The envelope grows increasingly unstable to convection.

Blue Loop: an evolved star changes from a cool star to a hotter one before cooling again.

• E - at the tip of the RGB, helium is ingnited in the core as stable.
  • luminosity decreases while the envelope is convective - the star moves along the Hayashi line.

• F - the envelope is mostly radiative and luminosity stops decreasing, the star leaves the RGB and starts the blue loop - its increases.

• G - at , the star reaches its minimum stellar radius and its maximum effective temperature, the envelope again begins to expand and the star cools.

• H - the star has exhausted its helium in the core and the star is realigned with the Hayashi line.

  • The Blue Loop increases in width (extend to high ) for stars with . Stars below tend to have loops that stay close to the RGB and are not very blue after all.

Homework 2 Review

In class: Review HW responses, compare and converge with groups per usual. 3 (4?) Groups: Choose one of the HW problems.

Discuss conceptual questions together and prepare answers to share at the end of class.

• Choose someone that will report out the groups responses ahead of time!

After Class: Due: Tue. Oct 21, before Class to D2L to original submission Corrections for up to 50% credit.