Radiative energy transport & conduction

Notes following Ch. 5 of Pols Lectures here & HKT, Ch. 4; Ch. 3: LeBlanc 2011

Day 6 - September, 16, 2025

Agenda:

• Updates/Reminders - HW1 Due Thur. Sept. 18 before class (2m)
• Lecture (25m)
• In-Class Activity 6 - 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.

Global energy budget of a star

What do we know so far?

• We have seen that the "surface" temperature, and the average temperature estimated using the virial theorem are not the same; the star is not in global thermal equilibrium.

• We also know, from similar arguments, that the average temperature is higher than the surface temperature, so there should be an energy flow from the interior outwards.

• Goal: Take into account the conservation of energy on a local scale in the stellar interior.

Local energy conservation

We start by beginning with the first law of thermodynamics Pols (5.1) which states the internal () of a system can be changed by two forms of energy transfer: heat () and work (:

• the first term is the heat added or extracted
• the work done on (or performed by) the element.

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 .

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 .
     in spherical symmetry:

with at the surface and at the center and is the radial energy flux F in (.

Local energy conservation

We can now write our total change in heat content as:

Simplyfying, we have

next, we want to take the limit at and plug in our Eqn for (Pols 5.1)

Local energy conservation

This gives us our third stellar structure equation:

where is the specific entropy of the gas and we have used the Second law of Thermodynamics.

Local energy conservation

This allows us the write the third stellar structure equation as:

• 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.

Energy transport in stellar interiors

First, lets go over some sources of energy flow or transport:

1. Diffusion: thermal energy can be moved by the random motion of particles. Often the dominant energy transport mechanism:
   • Radiative, if the particles carrying the energy through their random motion are photons
   • Conductive: thermal motion of gas particles (typically electrons, unless you are in a neutron star) that carries the energy

Energy transport in stellar interiors

2. Convection: energy is transported by bulk motion of matter. Occurs as an instability if other means of energy transport are insufficient to carry the energy flux required.

   • Convection plays a significant role in the explosion of massive stars via the delayed neutrino driven mechanism.

Convection in a massive star

Diffusion processes

In general, the "diffusion approximation" is useful to describe the net flux of "something" when the average path of the carrier of said "something" is small compared to the lengthscale over which the "something" is transported.

• Such as when mean free path is much smaller than the size of the region .

In this approximation, the net flux of this "something" is related to the density of "something" by Fick's law:

Diffusion processes

which states when there is a gradient in the density of particles, the diffusive flux – i.e. the net flux of such particles per unit area per second is given by the above equation with .

• [] = [something]/(L²t) with L length dimension and t time;
• [] ≡ [] = 1/L;
• [] = [something]/L³

Our Diffusion Coefficient has units of [] = .

Diffusion processes

Suppose now that in addition to a , that there is also a gradient in the energy density carried by these particles (photons and gas particles in this case).

• This change in energy density can give rise to a net energy flux:

Using the relation that a change in the energy density is related to the temperature (Pols 3.51),

We can finally use this to write an equation for heat conduction

Diffusion processes

where and is the conductivity of the gas. This equation for all particles in LTE, photons as well as gas particles.

• Now, let's shift to a description of Radiative diffusion of energy

Radiative diffusion of energy

If the something in our diffusion equation is energy then:

1. then is a energy flux of radiative energy
2. is the energy density
3. diffusion coefficient the mean velocity of photons is ,
4. mean free path of photons Pols (Eqn 5.13) - where is mass absorption coefficient or opacity coefficient (Stay tuned).
5. Since we know (Eqn. 3.43), .

Now, lets compute our conductivity and plug it all in.

Radiative diffusion of energy

Recall

Plugging into our eqaution for the Flux:

where we use the spherical symmetry of the problem to explicit the
gradient and turn it into a total derivative

Radiative diffusion of energy

Relating the total flux , we arrive at our 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.

Conductive transport of energy

Collisions between the gas particles (ions and electrons) can also transport heat.

• Under normal (ideal gas) conditions, however, the collisional conductivity is much smaller than the radiative conductivity.
• the cross sections give mean free path for collisions that is several orders of magnitude smaller than
• the average particle velocity

So normally we can neglect heat conduction compared to radiative diffusion of energy.

Conductive transport of energy

Q: When could energy transport via conduction become relevant?

• A: When electrons become degenerate!

When electrons become degenerate:

1. Their velocities increase as their momenta reach the Fermi momentum (Sec. 3.3.5) .
2. Their mean free path increases significantly, (most of the quantum cells of phase spaceare occupied, so an electron has to travel further to find an empty cell and transfer its momentum).

Conductive 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:

in the absence of convection and neutrinos.

Conductive transport of energy

where can define similarly to that of radiative diffusion with

which allows us to write our Total Flux equation as

Summary: Energy transport in stellar interiors

1. The transport mechanism with the largest flux will dominate, that is the mechanism for which the stellar matter has the highest transparency, smallest , least opaque.

2. Radiative transport of energy via photons accurate in ideal and non-degenerate gases.

3. Description provides the fourth stellar structure equation for temperature gradient.

In-Class Assignment 6

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.