Recall....
• Opacity, κ  , can be wavelength dependent
and describes the ‘stopping power’ of the
medium to the photons
• For absorption that is uniform along the a
given path: (incident intensity is I(0) & the
path length, s, as:
I = I(0) exp(– s )
( = κ)
or:
I = I(0) exp(–)
where:  is called the (frequencydependent) optical depth of the medium,
and is unitless.
Overview of Lecture
•Equation of Radiative Transfer
•Examine what causes opacity in astronomical
spectra – in particular in stars. Also examine
wavelength dependence
•Line formation in stellar atmospheres. Limbdarkening – Source function etc.
•Stellar atmospheres: Deviations between real
and models – LTE etc.
Interpretation of the equation of
radiative transfer
• The formal solution of the radiative transfer
equation yields the observed intensity of the
radiation:
τυ
I υ  I υ (0) e -τ υ   Sυ e -τ υ dτ υ
0
• The frequency dependence of the emission and
absorption components leads to the formation
of emission and/or absorption features.
Optical depth
The overall optical depth of a batch of gas is an important
number. If tells us right away if the cloud falls into one of two
useful regimes:
optically thin:  << 1
•Chances are small that a photon will interact with particle
•Can effectively see right through the cloud
•In the optically thin regime, the amount of extinction
(absorption plus scattering) is linearly related to the amount
of material: double the amount of gas, double the extinction
 if we can measure the amount of light absorbed (or
emitted) by the gas, we can calculate exactly how much gas
there is
Optical depth
Optically thick:  >> 1
I = I(0) exp(–)  I/I(0)=
Exp(-10)=5e-5
Exp(-1)=0.37
Exp(-0.1)=0.9
Exp(-0.0001)=0.99
•Certain that a photon will interact many times with
particles before it finally escapes from the cloud
• Any photon entering the cloud will have its direction
changed many times by collisions -- which means that its
"output" direction has nothing to do with its "input"
direction.
 Cloud is opaque
•You can't see through an optically thick medium; you can
only see light emitted by the very outermost layers.
 i.e., can’t ‘see’ interior of a star – only see the
‘surface’ or the photosphere
•One convenient feature of optically thick materials: the
spectrum of the light they emit is a blackbody spectrum, or
very close to it  layers deep within a star (can assume LTE)
Sources of Opacity
•So, if the inner layers of a star can be
approximated by a BB, what are the causes of
opacity in the outer layers?
•Detailed calculation of opacity, κ, is tough and
complex problem  Major Codes Required!
•The wavelength dependence shapes the
continuous spectrum emitted by a star.
•Major influence on the temperature structure of
an atmosphere because the opacity controls how
easily energy flows at a given wavelength.
Where opacity is high, energy flow (flux) is low.
Sources of Opacity
•Opacity a function of composition, density &
temperature.
•Determined by the details of how photons
interact with particles (atoms, ions, free
electrons).
•If the opacity varies slowly with λ it determines
the stars continuous spectrum (continuum). The
dark absorption lines superimposed on the
spectrum are the result of a rapid variation of
opacity with λ.
 Can be broken down into 5 main sources
Sources of Opacity
5 primary sources of opacity:
•Bound-Bound absorption  Small, except at those discrete
wavelengths capable of producing a transition. i.e., responsible for
forming absorption lines – also can form pseudo-continuum at low R
•Bound-Free absorption  Photoionisation
has sufficient energy to ionize atom. The freed
thus this is a source of continuum opacity
e-
- occurs when photon
can have any energy,
•Free-Free absorption (Bremsstrahlung) A scattering process.
A free electron absorbs a photon, causing the speed of the electron to
increase. Can occur for a range of λ, so it is a source of continuum
opacity. Only important at high temperatures as need lots of free e-.
•Electron scattering (Thomson scattering) A photon is
scattered, but not absorbed by a free electron. A very inefficient
scattering process only really important at high temperatures where it
dominates as other as other sources decrease.
•Dust extinction  Only important for very cool stellar atmospheres
and cold interstellar medium  more important at short wavelengths, will
not treat in this course!
Examples:
•Bound-Free absorption: e.g., the Balmer jump or Balmer
decrement, the Lyman limit
i.e., Structure of the H atom  produces spectral features
Examples:
•Bound-Bound absorption: e.g., H absorption (Lyman,
Balmer, Paschen series etc.), Fe line-blanketing, molecular
etc., millions of lines for cooler gas…….
Modelled opacity in the UV due to gas at
5,000K (black) and 8,000K (red). The
opacities are due to lines, mostly HI,
FeII, SiII, NI, OI and MgII
Balmer series b-b transitions
(note the Balmer edge 
continuous, so bound-free!)
Mean Opacity
All 4 opacities can be grouped to form a mean opacity at a
specific wavelength:
If we average over all wavelengths we get the ‘Rosseland
Mean Opacity’:
•Bound-bound term
requires millions of
transitions (i.e. opacity
project)
•Bound-free term
reasonably approximated
by ~T-3.5
•Free-free term ~T-3.5 also
•Electron scattering term
simply independent of
wavelength
Contributions to mean opacity with T (at
constant density)
Sources of Opacity
Primary sources of opacity in most stellar
atmospheres are:
•Photoionisation of H- ions, but these become increasingly
ionised for stars hotter than the sun, where
photoionisation of H atoms and free-free absorption
become the main sources.
•For O stars the main source is electron scattering, and the
photoionisation of He also contributes. Also bound-bound
transitions in the UV important – actually can drive wind.
•Molecules can survive in cooler stellar atmospheres and
contribute to bound-bound and bound-free opacities. The
large numbers of molecular lines are an efficient
impediment to the flow of photons.
Optical Depth Effects in
Stellar Atmospheres:
Two examples:
1. Line Formation
2. Limb-darkening
Optical depth and spectral
line formation
• Remember that can only
‘see’ a MFP into a cloud
(or star) if it is optically
thick  so, the lower the
optical depth, the
deeper into the star we
see
• For weak lines (lower
optical depth) the
deeper the line
formation region
• For strong lines (higher
optical depth), the
shallower the line
formation region 
think of case of cloud in
earth’s atmosphere
Temperature structure of solar atmosphere
Optical depth and
spectral line formation
τ increasing
• Formation of absorption lines on the Sun
Optical depth and spectral
line formation
• Formation of absorption features can also be
understood in terms of the temperature of the local
source function decreasing towards the line centre
 Limb darkening can be understood in a similar way…
Limb Darkening
The sun  redder at the edges, also dimmer at the edges…
Limb Darkening
Can also be understood in terms of temperature
within the solar photosphere. Deeper  hotter.
Since we ‘see’ ~ 1 optical depth into atmosphere
=> can see different depths across solar disk
At centre see hotter gas
than at edges
Similar effect to line
formation earlier
Centre appears hotter,
brighter
Limb darkening!
Limb Darkening
Variation of intensity across solar disk
See notes
for further
details
Limb Darkening
Also see limb darkening in other stars, i.e., red
supergiant Betelgeuse – few pixels across!
Blackbody and thermal
radiation
Thermodynamic equilibrium
• If we assume that all the constituents of the gas in
a body are in the most probable macrostate (due
to random collisions) i.e., they are in (strict)
thermodynamic equilibrium
• The gas can then be described with a single
parameter – the temperature, T, and this same
temperature also describes the radiation field
• Since system is in equilibrium:
S  B
• This is true where collisions occur within a volume
where the state variables (specifically the
temperature) can be considered constant
– example: the deep interior of a star
Local Thermodynamic
Equilibrium
• Nearer the surface, the assumption of
thermodynamic equilibrium is only partly true:
– mean free path for photons is long, so they “see” the
stellar boundary
– mean free path for particles is short, so they can be
very close to the boundary and yet still act as if they
are in equilibrium i.e., they still obey the MaxwellBoltzmann distribution
• Away from the boundary, where the mean free
path for photons << thermal scale height, local
thermodynamic equilibrium (LTE) is satisfied and
we have: S  B


Local Thermodynamic
Equilibrium
• The assumption of LTE is appropriate if
collisional processes amongst particles
dominate the competing photoprocesses,
or are in equilibrium with them at a
common matter and radiation temperature
• For gaseous nebulae, interplanetary,
interstellar or intergalactic media non-LTE
processes are important:
– gas is optically thin
– photoexcitation is important
– Particle density low (few interactions)
Radiation and equilibrium
• For a perfect absorber of radiation, the emitted
radiation is described by the Planck equation for
blackbody radiation at the gas temperature, T:
2hυ3
1
S  B (T ) 
υ
υ
c 2 e h kT  1
• This equation is important for the derivation of
stellar atmospheric properties. In general
terms, when material is in thermodynamic
equilibrium it is in mechanical, thermal, and
chemical equilibrium.
Source function and Kirchoff’s
Laws
• Given:
τυ
I υ  I υ (0) e -τ υ   Sυ e -τ υ dτ υ
0
• This equation can be simply integrated to
give:
 ( total )
   

I  I (0) e  S  e


υ
υ

 I (0) e  
υ
υ

 0
  (total) 

 S 1  e 

υ


• Two important limits: τ << 1 and τ >> 1
Source function and Kirchoff’s
Laws
• For τ << 1 we can simplify:
  (total) 

I  I (0) e  S 1  e 

υ υ
υ


~ I (0)  S 1  (1   (total))
υ
υ

 I (0)  S  (total)
υ
υ
 


• So the emission increases with path
length (recall that optical depth = σ s)
• i.e., emission lines from solar corona at
eclipse (so background source)
Source function and Kirchoff’s
Laws
• For τ >> 1 we can simplify:
I  I (0) e
υ υ
~S
 
  (total) 

 S 1  e 

υ


υ
• So the emission has a constant value
Question: how far into the source do
we see?
Hint: think about the definition of Sυ
Kirchoff’s Laws
• Bunsen, Kirchoff (1859)
• The three basic types of spectra:
– continuum
– emission
– absorption
Think about the application of radiation transfer
to these cases (hint: identify source and absorber)
Thermalisation
Blackbody spectra
Recall the shape of the blackbody curve – this is
the limiting emission that an optically thick
medium will reach for that temperature
Thermalisation
• Consider a uniform slab of gas of
thickness L and temperature T that
radiates like a blackbody, with an
absorption coefficient συ which is small
everywhere except at a strong line of
frequency υ0
• Compare the emitted intensity in the
line relative to the neighbouring
continuum for different limiting optical
thicknesses of the slab
Thermalisation
• For a gas in TE we have:
  (total) 

I  S (T ) 1  e 

υ
υ


• and, for frequencies which are similar:
 0 L
1 e

I 1  e L
I0
• we now have three interesting cases,
depending on the balance of the optical
depths (absorption cross-sections)
Thermalisation
 0 L
1 e

I 1  e L
I0
• Case I: medium optically thin for all
frequencies –


 1
I

I 0
0
• Case II: line is optically thick, but continuum
isn’t I 0
1

 1
I   L
• Case III: medium is optically thick at all
frequencies -
I 0
I
1
Approach to thermalisation
Blackbody curve
~0
 large
 small
 very large
Approach to thermalisation – line and continuum changes
Scattering
Scattering
• Scattering may be either:
– frequency dependent
• e.g., line scattering
– frequency independent
• e.g., scattering by free electrons
• If scattering is independent of frequency it
is said to be “grey”
Example: Electron scattering
• For electron scattering we have:
es = Th x column density
where: Th is the Thomson cross-section:
Th = 6.652 x 10–25 cm2
• Note: The optical depth (amount of scattering) is
directly proportional to the number of electrons along
the line-of-sight
Model Stellar
Atmospheres…
…some general points…
Model Atmospheres
• Model atmospheres are the key to interpreting
observations of real stars
• By definition, most of stellar photons we receive are
from ‘photosphere’ ( optical depth 2/3 at 500nm)
• Need to model in order to compare with observations
• Initial model constructed on the basis of observations &
known physical laws.
• Then modified and improved iteratively until good match
achieved. Can then infer certain properties of a star:
temperature, surface gravity, radius, chemical
composition, rate of rotation, etc as well as the
thermodynamic properties of the atmosphere itself.
Model Atmospheres
A number of simplifications usually
necessary!:
Plane-parallel geometry  making all physical
variables a function of only one space coordinate
Hydrostatic Equilibrium  no large scale
accelerations in photosphere, comparable
to surface gravity, no dynamical significant mass loss
No fine structures  such as granulation, starspots
Magnetic fields are excluded
Stars as Black Bodies?
Thermal Equilibrium?
Basic condition for the BB as emitting source  negligible
fraction of radiation escapes!
Below the lower photosphere optical depth to the surface is
high enough to prevent escape of most photons. They are
reabsorbed close to where they were emitted thermodynamic equilibrium - & radiation laws of BB apply.
However, a star cannot be in perfect thermodynamic
equilibrium! That would imply no net outflow of energy!
Higher layers deviate increasingly from BB as this leakage
becomes more significant. There is a continuous transition
from near perfect local thermodynamic equilibrium (LTE)
deep in the photosphere to complete nonequilibrium
(non-LTE) high in the atmosphere.
Stars as Black Bodies?
Thermal Equilibrium?
Thermodynamic Equilibrium is applied to relatively small
volumes of the model photosphere - volumes with dimensions
of order unity in optical depth LTE
The photosphere may be characterized by one physical
temperature at each depth.
(L)TE means atoms, electrons & photons interact enough that
the energy is distributed equally among all possible forms
(kinetic, radiant, excitation etc), and the following theories
can be used to understand physical processes:
•Distribution of photon energies: Planck Law (Black-Body
Relation)
• Distribution of kinetic energies: Maxwell-Boltzmann
Relation
• Distribution among excitation levels: Boltzmann Equation
• Distribution among ionization states: Saha Equation
So one temperature can be used to describe the gas locally
Stars as Black Bodies?
Thermal Equilibrium?
•So LTE usually assumed  works for non-extreme
conditions
•Often poorly describes very hot stars (strong
radiation field) and very extended stars (low
densities, i.e., red giants)
•Sometimes LTE works for some spectral features,
but not for other features in the same star
(different lines formed in different regions of
photosphere)
•Generally models can re-produce stellar spectra
very well….
Notes for lectures 3+4:
http://www.maths.tcd.ie/~ccrowley/
Astro_spec_lecture_3_4.ppt
http://www.maths.tcd.ie/~ccrowley/Astro_s
pec_notes_3_4.doc