Download Lecture 5: The Gray Atmosphere

Lecture 5: The Gray Atmosphere
As an illustration of how the equation of radiative transfer can be used and of
how we can begin to understand stellar atmospheres in some detail through
theory, we discuss the “gray atmosphere”. This illustrates how astronomers were
able to derive the basic temperature structure of the stellar atmosphere before
anything was really known about the detailed physics of it. It is a very beautiful
and successful theory that also forms the basis of more modern approaches. For
a more detailed look than we can do in lecture, read the ApJ paper from the
Centennial Issue that is on reserve in the library.
Basic Assumptions: The most basic assumption, of course, is that the
opacity is independent of wavelength (i.e. “gray” or colorless). Therefore, the
transfer equation can be written without the nu (or lambda) subscripts and all
variables are taken to be integrated over frequency.
We also assume a plane-parallel atmosphere, infinite and uniform in two
dimensions, so that the only variable is depth into the atmosphere.
What we seek to derive is a T-tau relationship -- i.e. how does
temperature increase with optical depth. This is as far as we can go without
knowledge of the opacity, since such knowledge would be needed to move from
optical depth to physical depth. We assume LTE, so that S(tau)=B(T(tau)).
We assume “radiative equilibrium”, which means that the flux at each layer in the
atmosphere is conserved. This amounts to saying that all of the energy being
transported by the atmosphere is transported by radiation and none by
convection or conduction. A mathematical statement of this is F = const. The
amount of flux transported by the atmosphere is a “free parameter” of the
model -- i.e. we get to choose it. Given our definition of “effective temperature”
it is equivalent to choose that, since F = σTe4
Geometry: The geometry is the same as we discussed last time, but we need
to recall now that the specific intensity is a function of theta (the “polar angle”)
although not a function of phi (the “azimuthal angle”). We also note that the
optical depth (tau) will now be measured vertically into the star’s atmosphere,
and that to get to the same physical depth along an angled path will require a
larger optical depth (tau/cos(theta) = tau sec (theta)).
The book uses the notation of a subscript “v”
(careful, it looks like a “nu”!) to indicate “vertical”
optical depth. This seems unnecessarily complicated
to me. I will use the more conventional approach of
just referring to vertical optical depth as tau in this
problem.
<
θ
τ
θ
Observer
Increasing optical depth
Star’s atmosphere
τ =0
τ sec(θ)
Using our same beautiful diagram from last time, we now consider that I =
I(theta) and that along a ray at any particular angle theta, the optical depth is
given by dtau/cos(theta)
dI
cosθ
=I −S
dτ
And, since the boundary condition is that the intensity from any background
source (i.e. from the interior of the star) is negligible, the solution may be written
Isurf ace (θ) =
!
∞
B(T (τ ))e−τ secθ secθdτ
0
where we have employed the assumption of LTE that S=B(T) and we show
explicitly that T is a function of tau. So, the problem only remains to specify T(tau)
and we can calculate I(theta) at tau = 0 (i.e. the surface). This, of course, is what
we can observe and would like to compare with our theory!
To build a theory on these assumptions we begin by taking the “zeroth moment”
of the equation of radiative transfer and integrate over all solid angle:
!
!
!
dI
cosθ dΩ =
IdΩ −
SdΩ
dτ
4π
4π
4π
The first term on the right hand side is just the mean intensity (times 4 pi) and
the second term on the right is 4 pi S, since S is isotropic.
!
d
IcosθdΩ = 4π(J − S)
dτ 4π
But we recognize the integral term on the left as the Flux and by the acondition
of radiative equilibrium it is constant in the atmosphere so its derivative must be
zero. Therefore,we have: J = S = B where we have also employed the
assumption of LTE. Note that while J, S and B are all isotropic, I is not!
These relations, by themselves do not provide a T-tau relation. To proceed, we
take the first moment of the Equation of Radiative transfer, which yields:
!
!
!
2 dI
cos θ dΩ =
IcosθdΩ −
ScosθdΩ
dτ
4π
4π
4π
The first term on the right is the flux (F) and the second term is zero since S is
isotropic. Rearranging the rhs yields:
!
d
Icos2 θdΩ = F
dτ 4π
Now, recalling the definition of radiation pressure we see this is:
dPrad
c
=F
dτ
Since F is a constant, independent of tau, we can integrate this immediately to:
Prad
F
= τ + const
c
The constant can be determined from the boundary condition that at tau = 0,
the intensity field is entirely outwardly directed, so if we break the intensity field
into an inward and outward directed component I = Iin + Iout then
1
J =< I >= Iout
2
and
and
so
Prad
4πJ
2πIout
=
=
3c
3c
F = πIout
2F
F
2
const =
and Prad = (τ + )
3c
c
3
The last element we require is a relationship between Prad and J. If I were
isotropic then we know that
4π
Prad =
J
3c
The assumption that this is true even for the anisotropic radiation field in the
gray atmosphere is called the “Eddington approximation”. Its validity can be
verified for many possible anisotropies and it reproduces observations well, as
we shall see.
Putting it all together we have:
Prad
4π
σTe4
2
=
B=
(τ + )
3c
c
3
4π σT 4
σTe4
2
=
(τ + )
3c π
c
3
4
T =
4 3
Te ( τ
1
+ )
4
2
So, the theory has achieved what we sought -- a T-tau relationship! Like all good
theories it gives us some new insights, too. It tells us that the temperature at the
top of the photosphere is not really zero, but
Te
T (τ = 0) = 0.25 = 0.84Te
2
Also,
2
T = Te when τ = ( )
3
Testing the theory: comparison with limb darkening
Recall that the solution to the equation of radiative transfer in this case was:
! ∞
Isurf ace (θ) =
B(T (τ ))e−τ secθ secθdτ
0
Now that we have a T-tau relation from the theory we can calculate I(theta), but
we can also measure it for the Sun because the solar disk is resolved.
Relating the brightness at a given theta to the brightness at theta = 0 (i.e. the
center of the solar disk) we find (see book for details):
Isurf ace (θ)
2 3
= + cosθ
Isurf ace (θ = 0)
5 5
This is a pretty good fit to the data, as the next slide shows. Also, the finite
temperature of the top of the photosphere agrees pretty well with the what is
inferred from the depth of absorption lines. So -- all in all -- not a bad
description of the atmosphere of stars -- at least the Sun.