1. I = I(0) exp(– s ) ( = κ)
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 (frequency-dependent) optical depth of the medium, and is unitless.
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2. 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. Limb-darkening – Source function etc.
Stellar atmospheres: Deviations between real and models – LTE etc.
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3. Interpretation of the equation of radiative transfer
The formal solution of the radiative transfer equation yields the observed intensity of the radiation:
The frequency dependence of the emission and absorption components leads to the formation of emission and/or absorption features.
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4. Optical depth optically thin:  << 1
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
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5. 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( )=0.99
Optically thick:  >> 1
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)
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6. 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.
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7. 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
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8. 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 - occurs when photon has sufficient energy to ionize atom. The freed e- can have any energy, thus this is a source of continuum opacity
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!
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9. 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
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10. 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!)
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11. 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)
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12. 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.
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13. Sources of Opacity
Main opacity sources in the sun (from Gray, also in Rutten notes)
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14. Optical Depth Effects in Stellar Atmospheres: Two examples: 1
Optical Depth Effects in Stellar Atmospheres: Two examples: 1. Line Formation 2. Limb-darkening
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15. 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
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16. Optical depth and spectral line formation
τ increasing
Formation of absorption lines on the Sun
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17. 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…
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18. Limb Darkening
The sun  redder at the edges, also dimmer at the edges…
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19. 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!
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20. Limb Darkening Variation of intensity across solar disk
See notes for further details
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21. Limb Darkening
Also see limb darkening in other stars, i.e., red supergiant Betelgeuse – few pixels across!
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22. Blackbody and thermal radiation
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23. 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:
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
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24. 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 Maxwell-Boltzmann distribution
Away from the boundary, where the mean free path for photons << thermal scale height, local thermodynamic equilibrium (LTE) is satisfied and we have:
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25. 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)
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26. Blackbody and thermal radiation
Note that we need to draw a distinction between:
Thermal radiation – for which
and
Blackbody radiation, for which:
Blackbody radiation is only emitted for large optical depths
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27. 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:
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.
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28. Source function and Kirchoff’s Laws
Given:
This equation can be simply integrated to give:
Two important limits: τ << 1 and τ >> 1

29. Source function and Kirchoff’s Laws
For τ << 1 we can simplify:
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)

30. Source function and Kirchoff’s Laws
For τ >> 1 we can simplify:
So the emission has a constant value
Question: how far into the source do we see?
Hint: think about the definition of Sυ

31. 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)

32. Thermalisation
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33. Blackbody spectra
Recall the shape of the blackbody curve – this is the limiting emission that an optically thick medium will reach for that temperature

34. 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

35. Thermalisation For a gas in TE we have:
and, for frequencies which are similar:
we now have three interesting cases, depending on the balance of the optical depths (absorption cross-sections)

36. Thermalisation Case I: medium optically thin for all frequencies –
Case II: line is optically thick, but continuum isn’t -
Case III: medium is optically thick at all frequencies -

37. Approach to thermalisation
Blackbody curve
 ~ 0
 small
 large
 very large
Approach to thermalisation – line and continuum changes

38. Scattering
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39. 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”
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40. Example: Electron scattering
For electron scattering we have:
where: Th is the Thomson cross-section:
Note: The optical depth (amount of scattering) is directly proportional to the number of electrons along the line-of-sight
es = Th x column density
Th = x 10–25 cm2
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41. Model Stellar Atmospheres… …some general points…
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42. 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.
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43. 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
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44. 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.
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45. 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
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46. 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….
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47. Notes for lectures 3+4:
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