Opacity, ? ? , can be wavelength dependent and describes the

1
Recall....

• Opacity, ? ? , 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 ??)
  (?? ???)
• or
• I I(0) exp(??)
• where ?? is called the (frequency-dependent)
  optical depth of the medium, and is unitless.

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.

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.

4
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 ? ltlt 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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Optical depth
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

• Optically thick ? gtgt 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., cant 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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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, ??, 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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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 ? 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 ?.
• ? Can be broken down into 5 main sources

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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 ?, 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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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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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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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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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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Sources of Opacity
Main opacity sources in the sun (from Gray, also
in Rutten notes)
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Optical Depth Effects in Stellar
AtmospheresTwo examples1. Line Formation2.
Limb-darkening
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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 earths atmosphere

Temperature structure of solar atmosphere
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Optical depth and spectral line formation
t increasing

• Formation of absorption lines on the Sun

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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
18
Limb Darkening
The sun ? redder at the edges, also dimmer at the
edges
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Limb Darkening
Can also be understood in terms of temperature
within the solar photosphere. Deeper ? hotter.
Since we see 1 optical depth into
atmosphere gt 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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Limb Darkening
Variation of intensity across solar disk
See notes for further details
21
Limb Darkening
Also see limb darkening in other stars, i.e., red
supergiant Betelgeuse few pixels across!
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Blackbody and thermal radiation
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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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 ltlt thermal scale height, local
  thermodynamic equilibrium (LTE) is satisfied and
  we have

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)

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

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.

28
Source function and Kirchoffs Laws

• Given
• This equation can be simply integrated to give
• Two important limits t ltlt 1 and t gtgt 1

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Source function and Kirchoffs Laws

• For t ltlt 1 we can simplify
• So the emission increases with path length
  (recall that optical depth s s)
• i.e., emission lines from solar corona at eclipse
  (so background source)

30
Source function and Kirchoffs Laws

• For t gtgt 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 S?

31
Kirchoffs 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
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 s? which is small
  everywhere except at a strong line of frequency
  ?0
• 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
  isnt -
• Case III medium is optically thick at all
  frequencies -

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Approach to thermalisation
Blackbody curve
? 0
? small
? large
? very large
Approach to thermalisation line and continuum
changes
38
Scattering
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

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 6.652 x 1025 cm2
41
Model Stellar Atmospheressome general points
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.

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

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.

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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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.

47
Notes for lectures 34

• http//www.maths.tcd.ie/ccrowley/Astro_spec_lectu
  re_3_4.ppt
• http//www.maths.tcd.ie/ccrowley/Astro
  _spec_notes_3_4.doc