ASTR 498N Lecture
9 Radiative
transport
(online at www.astro.umd.edu/~drabin/)
The clear light of truth.
We’ve now seen the basic equations of stellar evolution, including the equation expressing the radiative transport of luminosity:
Today we’ll derive this equation and explore the
opacity that appears
in it.
Microreview
Linear aborption coefficient
Cross section
Mean free path
Mass absorption coefficient
Thermal conduction
Heat flows from hot to cold. Experimentally, for a wide range of gases,
liquids, and isotropic solids, the heat flux is found to be proportional to the
temperature gradient: , where K is the coefficient of thermal
conductivity. The microscopic basis for
conduction is the random motion of particles.
We can estimate the form of K with a simple argument.
Let be
the mean energy of a particle at x.
At any moment, roughly 1/6 of the particles are headed in each of the 6
directions ±x, ±y, ±z. Thus, per
unit time and area,
about particles cross a plane of
constant x from below, and a similar number from above. The particles crossing from below
experienced their last collision, on average, a distance l below the
plane, where l is the mean free path, and therefore carry a mean energy
across
the plane. The net energy flux in the +x
direction is
We can write this as
where the coefficient of thermal conductivity is
and is the specific heat per particle. For a monatomic ideal gas,
and
.
For a gas of neutral atoms, we can estimate the mean free path using
and
where a0
is the
Bohr radius (O&C §9.2). The
collision cross section for an ionized gas is not so obvious.
What about a photon gas? Rewrite the estimate above for conductive flux as
where u is energy density and the wavelength
averaged mass absorption coefficient (the opacity) is effectively defined by its correspondence with
the mean free path.
Finally, for blackbody radiation, u = aT4 where is the
radiation constant. Using this in du/dT above gives
In a spherical star, ,
and we
obtain the equation for radiative energy transport:
In view of our rough derivation of the coefficient of thermal conductivity, it is fortuitous that the equation comes out right, including the numerical factors. However, the approximate derivation here brings out the underlying unity of heat conduction, whether by massive particles or photons.
Rosseland mean opacity
(O&C §9.2)
As a homework problem (O&C Problem 9.16), you’ll derive from the radiative transfer equation the result
For
an isotropic radiation field, the radiation pressure is (Lecture 1) . Further assume that the radiation is
blackbody, Iλ = Bλ.
Then
Integrating over all wavelengths,
Now, if we define the Rosseland mean opacity by
we obtain
which recovers our basic result while revealing the nature of the mean opacity. Thinking in terms of transparency (1 /κλ) instead of absorption, the Rosseland mean transparency is the integral of the monochromatic transparency weighted by the temperature derivative of the Planck function. Regions, if any, of particularly high transparency (low opacity tend to dominate the integral. The weighting function de-emphasizes high and low frequencies.
Note that, when more than one source of opacity
contributes significantly to the total opacity at some frequencies, the
Rosseland mean of the total opacity is not in general equal to the sum of the
individual Rosseland mean opacities. That is, if opacity sources κ1(ν)
and κ2(ν) have Rosseland means and
,
.
Sources of opacity
(O&C §9.2)
Bound-bound transitions
The absorption (or emission) of a
photon when an electron makes an upward (or downward) transition between two
bound energy levels is the familiar process that gives rise to discrete
spectral lines. Although bound-bound
opacity is most important in cool stars and in the outermost layers of hotter
stars, it makes a nontrivial contribution to the Rosseland mean over a fairly
large region of the ρT
plane. In the Sun, the ratio of the
total opacity with lines to that without can reach 2
3. There is no convenient formula that
encapsulates bound-bound opacity.
Bound-free transitions
Bound-free absorption, also known as photoionization,
occurs when an incident photon has enough energy to liberate an electron from a
bound atomic state. If we consider the
ionization of an electron from a particular bound level, the cross section is
zero until it spikes sharply (ionization edge) at hν = χ (χ
is the ionization potential), then drops to higher energies as ν3. Thus, the total opacity due to bound-bound
transitions is a summation over sawtooth-like patterns followed by an
integration over frequency. Kramers
showed in a semi-classical approximation that
where the Gaunt factor gbf is of order unity and the mysterious guillotine factor t is typically in the range [1,100]. The inverse process to photoionization is radiative recombination.
The
negative hydrogen ion (H)
is a dominant source of bound-free opacity in the outer layers of cooler
stars. There is a bound state for a
second electron in the field of a proton, but the binding energy of the second
electron is only 0.75 eV, compared to 13.6 eV for the first electron. When hydrogen is predominantly neutral,
metals (Z > 2), which often have first ionization potentials of only
a few eV, provide most of the free electrons and therefore, through H
,
control the total opacity.
Free-free transitions
A free electron in the vicinity of an ion can accelerate and in so doing absorb or emit a photon: free-free absorption or free-free emission (also known as bremsstrahlung). Kramers came to the rescue here too with the approximate forumula
Note
that and
share the same functional dependence on ρ
and T.
Electron scattering
The cross section for electron
scattering may be calculated classically by considering that an electromagnetic
wave passing an electron has an electric field that causes the electron to
oscillate, which then tends to re-emit a photon of the same frequency in
another direction. In terms of the
classical electron radius ,
Note
that does not depend on frequency, so
.
The formula assumes a
fully-ionized, pure hydrogen-helium mixture, which is generally a good
approximation where electron scattering is important. The cross section (and opacity) must be modified for relativistic
effects at very high temperatures (
) or if the electrons are significantly
degenerate.
Electron conduction
Because normally the mean free path of a photon is much longer than the mean free path of an electron or ion, heat conduction by particles can be ignored relative to radiative diffusion. However, as we mentioned in our discussion of the Fermi-Dirac distribution, a highly degenerate electron gas becomes highly conductive because an electron on the surface of the Fermi sea cannot give up energy in a collision since all lower energy states are already occupied. Electron conduction is important in the cores of evolved stars and in white dwarfs.
Opacity
in the ρT
plane
Rosseland mean opacity graphically
