Opacity

Notes following Ch. 5 of Pols Lectures here & HKT, Ch. 4.4; Ch. 3.2: LeBlanc 2011

Day 7 - September, 18, 2025

Agenda:

• Updates/Reminders - Oral Presentation Tips - Schedule soon (2m)
• Lecture (25m)
• In-Class Activity 7 - Not for Credit (30m)
• Report out on ICA in ica Slack Channel (10m)

Recap

Mass continuity:

Hydrostatic equilibrium:

Recap

Pressure of a mixture of gas and radiation

If the electrons are not degenerate (they can be described classically as in Pols 3.19),

In practice, stellar evolution code often rely on tabulated EOS, which account for many non-ideal effects.

Today, we will talk about additional terms to the total pressure and when quantum mechanic effects might play a role.

Recap

Local energy conservation:

• 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.

Recap

Fourth stellar structure equation:

This is a local quantity and it is valid in a region of the star where
the dominant energy transport is radiative diffusion only.

Recap

We also introduced , mass absorption coefficient or opacity coefficient.

determines the flux that can be transported by radiation for a certain temperature gradient.

• Goal: Describe the different physical processes that contribute to the opacity in stellar interiors and some approximations.

The Rosseland mean opacity

In our previous discussion, our flux was independent of frequency because it represented an integration over all frequencies with

• However, in general, the can depend on the frequency of the radiation, so we need a way for to represent a proper average of over all frequencies with .

In the next few slides we will work to compute this average opacity.

The Rosseland mean opacity

For the specific flux, we also should put in the equation the specific opacity and consider only the energy density of radiation between and :

where we have

The Rosseland mean opacity

Recall from Pols Eqn 3.41 that for photons and in frequency space and in LTE, their distrubution follows the Planck distribution for blackbody radiation and that :

where we have defined frequency-dependent Planck function for the intensity of black-body radiation

The Rosseland mean opacity

We can use the chain rule to write as

to rewrite the total flux as

Following these steps we have computed our new estimate for the radiative conductivity - recall - (Pols 5.15)

The Rosseland mean opacity

We find radiative conductivity :

• Compare to the we found in the last lecture and equate them

The Rosseland mean opacity

We can simplify this equality to arrive at a proper averaging of the absorption coefficient called the Rosseland mean opacity:

We know that

where , allowing us to rewrite this opacity as:

The Rosseland mean opacity

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

Sources of opacity

Now, let's consider the interactions that can be source of opacity (determine the that we put in the Rosseland mean to obtain ).

• Electron Scattering
• Free-Free Absorption
• Bound-Bound Absorption
• Bound-Free Absorption
• The negative hydrogen ion - (H-minus)
• Molecules and dust
• Conductive opacities

Electron scattering

Scattering does not lead to the "disappearance" of a photon, but can still change its energy (and direction of propagation), thus affecting its ability to carry flux.

Photons (orange wiggly line), electrons (green little guys)

• At high temperatures, scattering off free electrons is the main source of opacity.

Electron scattering

• The scattering of a classical electromagnetic wave off-an electron can be described by the Thomson scattering cross section, which divided by the gives the opacity. For K,

• For higher energy, one needs to account also for the momentum exchange between radiation and the electron (Compton scattering), which decreases the opacity. At even higher energies of the photons, one may need to use the Klein-Nishina formula.

Free-Free Absorption

Absorption of a photon by a free electron in the presence of an ion.

• The inverse of bremstrahlung radiation, where the acceleration of an unbound electron results in the production of photons

Desribed as a Kramers Opacity

Bound-Bound Absorption

Photon-induced transitions between bound states in atoms or ions.

• The photon is absorbed by the ion and its energy goes into the energy level of the electron, which was bound to the nucleus before and after the interaction with the photon (hence the name).

Important for . Need bound electrons, fewer at high due to ionization.

Bound-Free Absorption

An incoming photon may have sufficient energy to photoionize an atom/ion and make an electron jump from a bound energy level to an unbound energy level.

• Its contribution to the opacity decreases at very high temperatures, when bound electrons are absent.

The negative hydrogen ion ()

An important source of opacity in cool stars is the bound-free absorption of the negative hydrogen ion (i.e., a proton with 2 bound electrons).

H^- is the dominant source of opacity in cool stars, such as the Sun, red giants and supergiants, but for this ion to form metals able to lose an electron are required.

Molecules and dust

• At K, atoms may bound together and form molecules
• At even lower ( K) dust grains may form.
• When dust grains form, they are very effective absorbers in the outer atmospheres of very cool stars.

Molecular opacity is a field of research in laboratory astrophysics, when the relevant molecules can be synthesized and kept at the relevant and one can experimentally measure their which is extremely complicated to calculate from first principles.

Conductive opacities

For an ideal gas, making conduction irrelevant.

• Only for degenerate gas (at least partially), diffusion of energy through the thermal motion of particles (electrons, because of their lower mass) is important.

At very high densities, the electron mean-free path are very long (since collisions are forbidden by not having any level available below the Fermi energy), making conduction very efficient and allowing high density degenerate cores to become effectively isothermal ( = constant, ).

Tabulated Opacities in Stellar Models

In stellar evolution we use tabulated opacities that (try to) account for all these effects.

combining all the sources of opacities we discussed (and more) from Farag et al. 2024.

• At fixed , there is more structure as a function of (because determines the ionization levels).

Tabulated Opacities in Stellar Models

for various fixed densities (as indicated by the colorbar).

• This plot effectively shows where powerlaw approximations can be used in certain regimes. From Farag et al. 2024."

In-Class Assignment 7

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.

Not for Credit

If you've missed a previous ICA, but make progress on this one today, you can upload this one in place of the missed one and recieve half credit.