Radiative and Conductive Heat Transfer

Wednesday, Feb. 5, 2025

astrophysics of stars and planets - spring 2025 - university of arizona, steward observatory

Today’s Agenda

1. Announcements (2m)

2. Reading Overview/Key Points (10m)

3. In-Class Activity 7 (20m)

4. Debrief/Reminders (2m)

Radiative Transfer

In discussing blackbody radiation and equations of state we assumed LTE as a very good approximation. We do know, however, that LTE implies complete isotropy of the radiation field and this, in turn, means that radiant energy cannot be transported through the material of the star.

Anisotropy in the field is required for that to happen.

Source: HKT Fig. 4.1 - The geometry associated with the specific intensity \(I(\theta)\). The position coordinate may either be radius r in a spherically symmetric star or vertical distance z in a plane parallel “star.” In the latter case, symmetry in the transverse x- and y- coordinates is assumed. The properties of the stellar medium are then independent of azimuthal angle \(\phi\) for either choice of geometry.

This allows us to write the total flux in the \(z-\)direction as

Definition 49

\[ \mathcal{F} = \int_{4 \pi} I(\theta) \rm{cos}(\theta) \rm{d}\Omega = 2 \pi \int^{+1}_{-1} I(\mu)\mu \rm{d} \mu \ (\rm{ erg \ s^{-1} \ cm}^{-2}). \]

Observation 6

Note that if \(I\) is a constant, then the total flux is zero because the same amount of radiation comes in as goes out. \(I\) must vary with \(\mu\) (or \(\theta\)) for radiant energy to be transported; that is, \(I(\theta)\) must be anisotropic.

We can think about the ways in which a beam may be altered via mass emission coefficient \(j(\theta)\) with

Definition 50

\[ dI(\rm{put \ into \ \theta}) = j(\theta)\rho ds \]

where \(ds\) is a distance along \(I(\theta)\) for scattering and emission.

While we have

Definition 51

\[ dI(\rm{taken \ out \ \theta}) = -\kappa \rho I(\theta) ds \]

for photons removed from the beam via absorption and scattering.

This brings us to the net change in \(I(\theta)\) per unit path length,

Definition 52

\[ \frac{1}{\rho} \frac{dI(\theta)}{ds} = j - \kappa I(\theta) \]

also known as the equation of transfer.

Some definitions:

Source Function

Definition 53

\[ S_{\nu} = j_{\nu}/\kappa_{\nu} \]

Optical Depth

Definition 54

\[ \tau_{\nu}(z) = \tau_{\nu,0} - \int^{z}_{z_{0}} \kappa_{\nu} \rho dz \]

where \(z_{0}\) is a spatial reference value and \(\tau_{\nu,0}\) is optical depth at that location. For example, if \(z_{0}\) is the “true surface” of the star where \(\rho,P\rightarrow0\) then \(\tau_{\nu,0}=0\).

The Diffusion Equation

To derive the diffusion approximation we introduce the Rosseland mean opacity, \(\kappa\) allowing us to remove the frequency dependence and write the total flux as

Definition 55

\[ \mathcal{F}(r) = -\frac{4 a c}{3} \frac{1}{\kappa \rho} T^3 \frac{dT}{dr} \equiv -\frac{c}{3 \kappa \rho} \frac{d(aT^4)}{dr}. \]

The resulting total luminosity in the diffusion approximation to radiative transfer, \(\mathcal{L}=4 \pi r^2 \mathcal{F}\),

Definition 56

\[ \mathcal{L}(r) = -\frac{16 \pi a c r^2}{3 \kappa \rho} T^3 \frac{dT}{dr} \equiv -\frac{4 \pi a c r^2}{3 \kappa \rho} \frac{dT^4}{dr}. \]

The Lagrangian form of the equation can be expressed as

Definition 57

\[ \mathcal{L}(r) = -\frac{16 \pi a c G}{3} \frac{T^4}{P\kappa} M(r)\nabla. \]

Where we have introduced a new quantity, del,

Definition 58

\[ \nabla \equiv \frac{d \rm{ln} T}{d \rm{ln} P} = - \frac{r^2 P}{G M(r) \rho } \frac{1}{T} \frac{dT}{dr} \equiv \nabla_{\rm{actual}} \]

Lastly, we will define the last del, delrad, where

Definition 59

\[ \nabla_{\rm{rad}} \equiv \left ( \frac{d \rm{ln} \ T}{d \rm{ln} P} \right )_{\rm{rad}} = \frac{3}{16 \pi a c G} \frac{P \kappa}{T^4} \frac{L_{\rm{tot}}(r)}{M(r)} = \frac{3r^2}{4\pi a c G }\frac{P \kappa}{T^4} \frac{\mathcal{F}_{\rm{tot}}}{M(r)}. \]

A Simple Atmosphere

We can define the photosphere to be at the optical depth of \(\tau=2/3\). This allows us to write the pressure at this location as

Definition 60

\[ P(\tau_{\rm{P}}) = \frac{2}{3}\frac{g_{\rm{s}}}{\kappa_{\rm{P}}}(1 + \frac{\kappa_{\rm{P} L }}{4 \pi c G M}) \]

The second term is usually small by comparison and only relevant for the most massive and luminous stars, it can be approximated as

Definition 61

\[ \frac{\kappa_{\rm{P}} L}{4 \pi c G M} = 7.8\times10^{-5} \kappa_{\rm{P}} \left ( \frac{L}{L_{\odot}} \right ) \left ( \frac{M}{M_{\odot}} \right )^{-1} \]

We can also estimate how large the luminosity must be so that radiative forces exceed gravitational forces, known as the Eddington critical luminosity or Eddington limit

Definition 62

\[ L_{\rm{Edd}} = \frac{4 \pi c G M}{\kappa_{\rm{P}}} \]

Radiative Opacity Sources

Electron Scattering

If a beam of photons of a given flux—now defined as the number of photons per cm\(^2\) per second-is incident upon a collection of stationary electron targets, then the rate at which a given event (a photon scattered out of the beam) takes place per target is related to the cross section, \(\sigma\),

Definition 63

\[ \sigma = \frac{\rm{number \ of \ events \ per \ unit \ time \ per \ target}}{\rm{incident \ flux \ of \ photons}} \ (\rm{cm}^2) \]

This allows us to relate to the opacity and the number density of electrons as

Definition 64

\[ \kappa = \frac{\sigma n_{\rm{e}}}{\rho} \]

for electron or photon thermal energies below \(kT \ll m_{\rm{e}}c^2\) or \(T \ll 5.93\times10^{9} \ (\rm{K})\), then the frequency-dependent Thomson scattering process describes the cross section well,

Definition 65

\[ \sigma_{\rm{e}} = \frac{8 \pi}{3} \left ( \frac{e^2}{m_{\rm{e}}c^2} \right )^{2} = 0.6652\times10^{-24} \ (\rm{cm}^2) \]

Similarly, we assumed the material is completely ionized allowing us to write the electron scattering opacity as

Definition 66

\[ \kappa_{\rm{e}} = 0.2(1+X) \ (\rm{cm}^2 \ \rm{g}^{-1}) \]

Free-Free Absorption (inverse Bremsstrahlung)

We can assumed the radiation field is LTE and we have \(j/k=S=B(T)\), and \(\kappa\) is only due to free-free absorption such that, \(\kappa=j/B(T)=\pi j / \sigma T^4\). Putting in some numbers gives

Definition 67

\[ \kappa_{\rm{ff}} \approx 4\times10^{-24} \frac{ Z^2_{c} n_{\rm{e}} n_{\rm{I}} T^{-3.5}}{\rho} \propto \rho T^{-3.5} \]

where the above is basically correct. But to use this as a Rosseland mean opacity, we need to actually computed 4.22.

A simplified form is given by

Definition 68

\[ \kappa_{\rm{ff}} \approx 10^{23} \frac{\rho}{\mu_{\rm{e}}} \frac{Z^2_{c}}{\mu_{\rm{I}}} T^{-3.5} \]

and requiring the presence of free electrons.

For a mixture composed of hydrogen and some helium (and traces of metals), we expect the free–free opacity to be negligible below temperatures of around 10\(^4\) K (or perhaps a little higher if densities are relatively high: see the half-ionization curve for hydrogen of Fig. 3.10).

The strongest dependence in \(\kappa_{ff}\) is that of temperature.

Bound–Free and Bound–Bound Absorption

Bound–free absorption is absorption of a photon by a bound electron where the photon energy is sufficient to remove the electron from the atom or ion altogether.

The opacity for this absorption is given by

Definition 69

\[ \kappa_{\rm{bf}} \approx 4\times10^{25}Z(1+X)\rho T^{-3.5} \]

This expression should not be applied if temperatures are much below \(T \approx 10^4\) K because, as only part of the story, most photons are not energetic enough to ionize the electrons.

Bound–bound opacity is associated with photon-induced transitions between bound levels in atoms or ions. Usually of magnitude less than \(\kappa_{\rm{ff}}\) or \(\kappa_{\rm{bf}}\).

H\(^{-}\) Opacity and Others

Among the more important sources of opacity in cooler stars is that resulting from free–free and bound–free transitions in the negative hydrogen ion, H\(^{−}\) (“H-minus”) and is sensitive not only to temperature but also to metal abundance.

Because of the large polarizability of the neutral hydrogen atom, it is possible to attach an extra electron to it with an ionization potential of 0.75 eV.

We can make an estimate for this opacity,

Definition 70

\[ \kappa_{\rm{H}^{-}} \approx 2.5\times10^{-31}\left (\frac{Z}{0.02}\right ) \rho^{1/2} T^{9}. \]

Unlike Kramers’, it increases strongly with temperature until about 10\(^4\) K, above which Kramers’ and electron scattering take over (and, any case, mosto r all of the H\(^{−}\) is gone by this temperature).

For very cool stars with effective temperatures of less than about 3000 K, opacity sources due to the presence of molecules or small grains become important.

Heat Transfer by Conduction

Beginning from Fick’s law of diffusion,

\[ \mathcal{F} = -\mathcal{D}_{\rm{e}} \frac{dT}{dr} \]

we can recast this equation in a similar form as diffusive radiative transfer (Eq. 4.23 or 4.24) to define a conductive opacity,

Definition 71

\[ \kappa_{\rm{cond}} = \frac{4 a c T^3}{3 \mathcal{D}_{\rm{e}} \rho} \]

which leads to the conductive flux,

Definition 72

\[ \mathcal{F}_{\rm{cond}} = - \frac{4 a c}{3 \kappa_{\rm{cond}} \rho} T^3 \frac{dT}{dr} \]

An approximate form for this opacity is given by

Definition 73

\[ \kappa_{\rm{cond}} \approx 4 \times 10^{-8} \frac{\mu^2_{\rm{e}}}{\mu_{\rm{I}}} Z^2_{c} \left ( \frac{T}{\rho} \right )^2 \]

whichever opacity is smaller, will be the important in determining the total opacity. In normal stars, \(\kappa_{\rm{cond}}\) is very large and conduction is negligible. The opposite is true in dense, degenerate material, such as white dwarfs.

In-Class Assignment

In-Class Assignment 7 on the Photosphere is here.