Stellar Atmospheres

Materials: Mihalas Stellar atmospheres book, Chapter 3. de Koter's lecture notes on radiative transfer and stellar atmospheres, Pols Chapter 7.2.

Day 12 - October, 7, 2025

Agenda:

• Updates/Reminders - HW2 - Due: Before Class, Oct. 9 (2m)
• Lecture (30m)
• ICA 11 - 4 Groups - Not for Credit (20m)
• ICA 11 Group Discussion (15m)

Stellar Atmospheres

• We have mostly dealt with the interior of stars and derived equation to determine the stellar structure and what drives their evolution (which is typically "slow", ).

• The nuclear burning can be seen just as an attempt by the star to delay its gravitational collapse: as nuclear fuel is depleted the star is forced to evolve (until either gravity "wins" or the EOS deviates from the classical ideal gas, and quantum mechanical effects provide sufficient pressure that win over gravity).

• Q: Which assumptions made in the stellar interior fail when applied to the stellar atmosphere?

Some Definitions

• stellar atmosphere is the layer where the radiation field is not isotropic, and the photon flux has a net radial component. this occurs where , the diffusion approximation/LTE fails.

• To make models of stellar spectra and determine the outer boundary conditions we need to consider how radiation from the bottom flows through the atmospheric layer and consequently how this layer stratifies, determining the outer boundary pressure and temperature.

This requires us to look at the Radiative Transfer of anisotripc radiation fields.

Some Definitions

Photosphere - surface where the bulk of the radiation escapes and which corresponds with the visible surface of the star. Approximated as a single surface at optical depth of . T_eff is measured at the photosphere.

Here we have defined the optical depth as

where we have the average opacity over the atmosphere (r > ).

Radiative Transfer

• We need to consider in the stellar atmosphere is how radiation coming from the photosphere flows through the overlaying gas, and how this impacts the observable radiation itself.

• The radiation coming out from the photosphere is a black body, it is isotropic.

• so if you think of the atmosphere as a slab of gas with a thickness , the radiation illuminating it from below may come at an angle w.r.t. the slab.

Radiative Transfer

• We can define the specific intensity (per unit frequency or wavelength) as the amount of energy flowing through a surface element (our slab) in a time interval and coming within a solid angle around the direction with frequency in the range between and :

which has the dimensions of []/([][][][solid angle]).

Radiative Transfer

This would be constant as photons propagate along a path of length ds along however there can be processes that add photons:

• scattering from another direction onto the direction of interest;
• emission processes;

and process that can remove photons:

• scattering from the direction of propagation onto another direction;
• absorption processes.

Radiative Transfer

Moreover, itself may in general be time dependent (although
this is not the case for stellar atmosphere), so we can write down the
equation of radiative transfe (compare with HKT 4.4) as

• The l.h.s. expresses the total change in specific intensity along the direction due to the intrinsic time-dependence () of and the spatial dependence along the direction we are considering ().

Radiative Transfer

• The r.h.s. expresses the loss of radiation intensity due to scattering and absorption processes, which depends on and is proportional to itself (you can't lose photons you don't have!),

• and the addition of radiation intensity from emission processes and scattering along the line of sight which depends on the emission coefficient .

Radiative Transfer

• Dimensional analysis reveals that each side has the units of []/[L], this equation describes how the intensity changes along its path.

• The fact that photons propagate at the speed of light make the leftmost factor of appear: .

• The density on the r.h.s. expresses that the more matter there is (per unit volume), the more likely there will be absorption and emission.

Radiative Transfer

The specific intensity at the bottom of the atmosphere is related to
the photospheric emission by:

that is the black body flux is obtained by integrating the specific
intensity over the solid angles.

• Note the factor that arises because is a vector and we only want the component normal to the surface element .

Radiative Transfer

This last expression is going to be useful to connect the physics in
the atmosphere with the interior, since we define the photosphere to
have a flux .

• The photosphere flux above acts as inner boundary condition for the problem of the radiative transfer through the atmosphere. We have already seen that it acts as outer boundary condition for the stellar interior.

Solutions of radiative transfer equation

1. Steady state without emission - Assume: absence of an explicit time dependence () and emission processes ().

• this equation is solved calling the length element along the direction so that , and the solution becomes:

where we introduce the definition of specific optical depth . This variable is useful because it gives the scale-length of the problem as depending on . Effectively, this allows us to use as the independent coordinate for the propagation.

Solutions of radiative transfer equation

Steady state with emission and absorption cancelling each other

With the definition of , we can re-write (still assuming ):

where in the last step we define the source function . In LTE and at high optical depth, such as the stellar interior, and is the black body function for the intensity, and this equation states .

Solutions of radiative transfer equation

Steady state with emission and absorption cancelling each other

• This effectively is a statement that at thermal equilibrium, the emission processes, the absorption processes, and scattering in and out of the direction of interest all cancel each other out.

Eddington Atmosphere

Eddington gray atmosphere - provides an analytic relation in the atmosphere that can be smoothly attached to the stellar interior where .

• Also can calculate the pressure needed at such boundary to have hydrostatic equilibrium.

• plane parallel atmosphere, that is we neglect the curvature of the stellar atmosphere, which is acceptable if its radius is much larger than the length scale of interest at any point in it.

Eddington Atmosphere

• This assumption reduces the problem to a one-dimensional problem along the vertical direction, and for the element of length along a generic photon path , and rewrite the steady state () radiative transfer equation as:

• The second approximation of the Eddington atmosphere is that we assume a "gray" radiative transfer, meaning the opacity (and ) is independent of frequency , thus .

Eddington Atmosphere

We can now integrate this in from to :

which can be solved analytically (multiply by , rewrite the l.h.s. as a total derivative and integrate in ) getting

where the r.h.s. is integrated from a certain optical depth outwards.

Eddington Atmosphere

We can recover the dependence of as an optical depth dependence in this integral.

We can also define the radiation energy density , the total flux
, and the radiation pressure as moments of the intensity
w.r.t. (since always appears in a cosine, it is usual to
change variable to in radiative transfer calculations):

Eddington Atmosphere

We can also define the average specific intensity as

so that . and we have

Eddington Atmosphere

Now the total radiative gray flux in the atmosphere has to be
constant, : there is radiative equilibrium and what goes in
must come out! So this equations tells us .

We can also take and multiply it by and integrate between -1 and 1 in to obtain:

Eddington Atmosphere

The r.h.s. is constant, so this can be integrated to give .

• One more hypothesis of the Eddington approximation is to assume that the gas is radiation pressure dominated .
• we also know that (using the definition of and its relation with the radiation energy density ).

Eddington Atmosphere

We have an expression for the source function:

Substituting for S in the solution for I we get:

Eddington Atmosphere

To determine the constant of integration, we can our definition of F using the solution for in the integral:

• We obtained this constant imposing the flux to come from , so from the layer after which there is nothing impeding the photons anymore.

• With this constant, we specified the source function and we can obtain and use it to calculate the pressure!

Outer boundary conditions: and

From our source function expression we have:

but also, assuming that the atmosphere is also in LTE (including radiation!), , so using that we obtain:

Outer boundary conditions: and

• this is the Eddington relation which connects the effective temperature of the black body to the outer temperature under the approximations for the atmosphere:

  • plane parallel
  • gray (i.e., independent on frequency )
  • radiation dominated
  • Local thermal equilibrium.

• In the stellar atmosphere, is a steep function of .

Outer boundary conditions: and

• in this approach (the Eddington approximation) the photosphere correspond to , this factor comes from imposing in the radiation dominated, gray, plane parallel atmosphere.

• it is important to remember that the photosphere is an idealization, and nothing that special occurs at , it's just a convenient location where we can stitch the Eddington gray atmospheric model to the interior model.

Outer boundary conditions: and

To find the outer boundary pressure, we need to integrate downward from to the HSE equation. We assume:

• the gravity is constant, or in other words, we neglect the atmosphere's "self-gravity" since the bulk of the mass is inside its inner boundary.

• also assume that is constant throughout the atmosphere:

Outer boundary conditions: and

where is the total mass of the star, is the radius such that , is the opacity assumed constant in the atmosphere, and we define the bottom of the atmosphere at because of the Eddington relation.

• Alternatively, one could use tabulated values of and a to perform the integral.

Outer boundary conditions: and

• Together with , we now have specified the outer boundary conditions fixing and at and completely determined the mathematical problem of the structure and evolution of a single, non-rotating, non-magnetic star of known total (initial) mass and composition.

• A "classic" generalization of this atmospheric model is the generic class of gray atmospheres where the constant of integration is not a constant, but a function of itself.

In-Class Assignment 11

In class: Work on ICA here with groups per usual. 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: Not for Credit

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.