Convection

Ch. 5.5 of Pols here & HKT, Ch. 5; Ch. 5.3.3: LeBlanc 2011, Jermyn et al. 2022, Matteo Cantiello's talk at KITP in 2017, review by Joyce & Tayar 2023.

Day 8 - September, 23, 2025

Agenda:

• Updates/Reminders - Presentation Tips - Schedule on D2L (2m)
• Lecture (25m)
• In-Class Activity 8 - Due: End of Day, Sept. 23, 2025 (30m)
• Report out on ICA in ica Slack Channel (10m)

Recap

Rosseland mean opacity:

• Weighted harmonic mean of with the weighting .
• favors the frequency range where the flux is large.
• represents the average transparency of the stellar gas.
• weighting function peaks at

Recap

Conductive and Radiative Transport of Energy:

At high densities, electron conduction becomes a much more efficient way of transporting energy than radiative diffusion.

We can write the heat flux due to this heat conduction as:

such that the such as radiative fluxes can be written as:

Recap

Fourth stellar structure equation:

• For radiative diffusion to transport energy outwards, a certain temperature gradient is needed

• The larger the luminosity that has to be carried, the larger the temperature gradient required

• an upper limit to the temperature gradient inside a star – if this limit is exceeded an instability in the gas sets in.

Recap

Fourth stellar structure equation:

• This instability leads to cyclic macroscopic motions of the gas, known as convection.

Goal: derive a more stringent criterion for convection to occur, based on considerations of dynamical stability

Detour: Adiabatic Processes

We introduced the Second Law of Thermodynamics, for a reversible process, the change in entropy equals the change in the heat content divided by the temperature (Pols 3.47):

where .

• Now, let's explore processes for which there is no change in the heat content (), such processes are adiabatic.

Detour: Adiabatic Processes

The thermodynamic response of a system to adiabatic changes is measured by the so-called adiabatic derivatives:

Adiabatic exponent:

which defines the response of the pressure to adiabatic compression or expansion, i.e. to a change in the density. See Pols 3.56, HKT 3.93 If is constant then, for adiabatic changes.

Detour: Adiabatic Processes

For an adiabatic process, we have ,

we can use our general relation for non-ideal EoS (Pols 2.30),

differentiating this and substituting in we find

Detour: Adiabatic Processes

For an adiabatic process, we have ,

we can therefore define our adiabatic exponent as

Detour: Adiabatic Processes

Adiabatic Exponent -

• non-relativistic particles (e.g. a classical ideal gas, NR degenerate electrons) and

• extremely relativistic particles (e.g. photons, ER degenerate electrons), and

• for a mixture of gas and radiation and/or moderately relativistic degenerate electrons,

Detour: Adiabatic Processes

Adiabatic Temperature Gradient:

the behaviour of the temperature under adiabatic compression or expansion. Where if is constant.

In a similar way, we can rewrite this in terms of our adiabatic exponent and other variables.

Detour: Adiabatic Processes

Adiabatic Temperature Gradient:

where we have introduced two new variables,

measure the response of the pressure to changes in density or temperature, not assuming adiabiatic.

Detour: Adiabatic Processes

Adiabatic Temperature Gradient:

• for an ideal gas without radiation, and , we find

• for an ideal radiation-dominated gas, and and , we find

Criteria for stability against convection

• A gas element is perturbed and displaced upwards from position 1 to position 2,
• the blob expands adiabatically (no heat exchange ) to maintain pressure equilibrium with its surroundings. .

Schematic Diagram of Convection

Credit: Onno Pols

Criteria for stability against convection

• If - the blob will sink back down to position 1.

• If - buoyancy forces will accelerate it upwards, convection occurs.

• Blob expainsion over takes place on the local dynamical timescale (i.e. with the speed of sound).

Schematic Diagram of Convection

Criteria for stability against convection

We can express our stability against convection ()
as

plugging in for we have

simplifying we have the limit for the density gradient for which a layer inside
the star is stable against convection

Criteria for stability against convection

• This means that the density gradient must be steeper than a critical value, determined by for convection to occur.

Schwarzschild Criterion

The simplest form of criteron for stability against convection is known as the Schwarzschild Criterion:

• a region is stable against convection (non-convective) if the actual temperature gradient (recall ) is less than the adiabatic temperature gradient.

Schwarzschild Criterion

If all of the energy is presently transported by radiation then we can replace with to arrive at:

Stable to convection:

Unstable to convection:

Ledoux Criterion

The Schwarzschild Criterion assumes a chemically homogeneous star. We could, instead, have changes in the mean molecular weight .

• Consider a change in the mean molecular weight , with a non-zero (). This leads us to what is know as the Ledoux criterion for convection. A region is stable against convection if the following is true

where .

Ledoux Criterion

Because normally increases inward, , the result is that gradients can lead to a stabilizing effect.

For an ideal gas, with all energy transported by radiation we have () and :

For , this reduces to the Schwarzschild Criterion.

Criteria for stability against convection

For convection to occur, you typically will need one of the following:

• a large , convection occurs in opaque regions of a star.
• region with large energy flux, stars with values near the center are expected to have convective cores - massive stars.
• a small value of such as at partial ionization zones at relatively low temperatures. stars of all masses have shallow surface convection zones at temperatures where hydrogen and helium are partially ionized.

In-Class Assignment 8

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: End of day today, September, 23, 2025

• Submit as a PDF, preferrably using nbconvert to D2L, the progress you have made.

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.