1
Monday, October 3: Stellar Atmospheres
There exist entire books written about stellar atmospheres; I will only give
a brief sketch of the simplest approximations used in studying stellar at
mospheres.
In particular, I want to discuss that most useful quantity, the
Rosseland mean opacity
.
1
A photon of frequency
ν
travels an infinitesimal distance
ds
inside a star.
In that distance, it can be scattered or absorbed. For simplicity, I will assume
that the scattering is coherent, and that the energy of an absorbed photon re
emerges as thermal radiation. The radiative transfer equation for scattering
is
dI
ν
ds
(scattering) =

σ
ν
I
ν
+
σ
ν
J
ν
,
(1)
where
σ
ν
is the scattering coefficient and
J
ν
is the angleaveraged specific
intensity.
2
The radiative transfer equation for absorption is
dI
ν
ds
(absorption) =

α
ν
I
ν
+
α
ν
S
ν
,
(2)
where
α
ν
is the absorption coefficient, and the source function
S
ν
is equal to
the Planck function
B
ν
(
T
) if, as we’ve assumed, the energy of the absorbed
photons is converted to thermal radiation.
Combining the effects of absorption and scattering, we find
dI
ν
ds
=

(
α
ν
+
σ
ν
)
I
ν
+ (
α
ν
B
ν
+
σ
ν
J
ν
)
.
(3)
This equation can be converted to a simpler form
dI
ν
ds
=

(
α
ν
+
σ
ν
)(
I
ν

S
ν
)
(4)
if we define a source function
S
ν
≡
α
ν
B
ν
+
σ
ν
J
ν
α
ν
+
σ
ν
,
(5)
1
A search in astroph, for instance, reveals an ongoing conflict between the Rosseland
mean opacity computed by the Opacity Project and that computed by the OPAL group.
Why do these people care so intensely about the Rosseland mean opacity?
I hope to
explain in this lecture.
2
Note that if the specific intensity
I
ν
is isotropic to begin with, coherent scattering
doesn’t affect it.
1
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which is just the mean of the Planck function and the angleaveraged specific
intensity, weighted by the absorption coefficient and the scattering coefficient,
respectively.
In general, solving the radiative transfer equation is a tedious chore. How
ever, in studying stellar atmospheres, some simplifying assumptions can be
made. The outer layers of a star are relatively low in density, so radiation is
the only significant method of transporting energy.
3
Moreover, gradients in
temperature and specific intensity are much greater in the radial direction
than in the transverse directions.
Thus, a stellar atmosphere can usually
be well approximated as a
plane parallel
system, in which properties of the
atmosphere, such as
T
,
σ
ν
, and
α
ν
depend only on the vertical coordinate
z
.
4
In the plane parallel approximation, the specific intensity
I
ν
depends only
on
z
and on the angle
θ
between the direction of the light ray and the
z
axis:
when the light travels straight upward, in the same direction that
z
increases,
θ
= 0; when the light travels straight down,
θ
=
π
. In traversing a thin layer
of the atmosphere, of vertical thickness
dz
, the distance traveled by light is
ds
=
dz
cos
θ
,
(6)
implying (
ds
)
2
≥
(
dz
)
2
. The radiative transfer equation in a plane parallel
atmosphere thus takes the form
cos
θ
∂I
ν
(
z, θ
)
∂z
=

[
α
ν
(
z
) +
σ
ν
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 Fall '05
 RYDEN
 Polarization, Magnetic Field, electric field strength

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