Summary of galaxy classification

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Lecture 5

• Summary of galaxy classification
• Example problems
• HW review
• Introduction to emission line spectroscopy
  review of atomic structure

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Next topic AGN

• Before we start this topic, need to review some
  basics on some of the most fundamental probes of
  AGN emission lines
• AGN emit high energy photons with energies more
  than sufficient to ionize hydrogen
• We will therefore concentrate on only this
  phase of the ISM HII regions

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Useful References

• Gaseous Nebulae
• Shu (pg 216)
• Online notes
• Osterbrock
• AGN
• CO Ch 26
• Introduction to Active Galactic Nuclei by
  Peterson(first chapter available free online see
  reading references on class webpage)
• Active Galactic Nuclei by Krolik (more advanced
  graduate-level textook)
• Quasars and Active Galactic Nuclei - An
  Introduction , by A.K.Kembhavi and J.V.Narlikar
  (slightly more advanced graduate-level textbook)

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The Interstellar Medium

• Perfect vacuum exists nowhere in space
• This stuff between stars 10-15 of visible
  mass of Galaxy
• 99 is gas, 1 dust

• So what? Why study it?

• It affects starlight, or other radiation sources

• Interesting physics in itself - probes chemical
  evolution and starformation history of all
  galaxies. For our next topic, it is very
  important

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Physics of ISM

• rmean 0.3 cm-3
• rmolecular cloud 106 cm-3
• ratmosphere 2?1018 cm-3

Thermodynamic equilibrium is not generally the
case.
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Emission-line spectroscopy from Ionized Gas

• Emission-lines are a powerful probe of the
  underlying abundances, temperature, density, and
  ionization states of AGN
• Rules governing transitions between different
  states of an atom are governed by quantum
  mechanics
• Let us review some basic atomic physics

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Spectroscopic notation

• Four quantum numbers describe any electron in an
  atom
• n principal quantum number 1, 2, 3
• l orbital angular momentum quantum number 0,
  1, n-1
• ml magnetic quantum number -l, -(l 1), ,
  0, 1, , (l 1), l ((2l1 values)
• ms spin quantum number /- ½
• No two electrons can have the same quantum
  numbers. This is the Pauli exclusion principle
  so there are 2n2 possible states for an electron
  with a given quantum number n
• We can describe location of an electron as being
  in orbitals, where for historical reasons, the
  terminology below is used

Solution of angular part of Schrodinger equation
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Spectroscopic notation
e.g., Oxygen has 8 electrons. What is its
electronic configuration? What about Neon which
has 10 electrons?
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Spectroscopic notation

• For hydrogen, we can use the Bohr model of the
  atom and find that the allowed energies for a
  hydrogen atom are
• En -13.6eV (1/n2)
• And the frequency of a transition between two
  levels is
• ? RH x (1/n2low - 1/n2high)
• RH Rydbergs constant 109677.5 cm-1 for
  hydrogen
• Similar relations work for other hydrogenic atoms
  (He II, Li III, etc )

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Spectroscopic notation

• For an atom, we dont specify the quantum numbers
  of every electron, instead we sum the individual
  l together vectorially to find the total angular
  momentum
• L ? l i
• L 0 are called S states, L 1 are called P
  states, L 3 are D states, etc!
• Similarly one sums the total electron spin
  vectorially
• S ? msi
• For an atom with an even number of electrons, S
  is an integer, with an odd number its 1/2,
  3/2, etc. S 0 and L 0 for a closed shell.
• LS coupling assumption (good for light atoms)
  implies that the total angular momentum J is just
  the (vector) sum of LS
• The magnetic quantum number M takes on values of
  J, J-1, , 0, -J-1, -J
• An atomic level is described as n2S1LJ, where
  2S1 is called the multiplicity
• (inner full shells sum to zero)
• For a term with total angular momentum J, there
  are (2J 1) separate states. Because the atom
  doesnt care how it is oriented in space, all
  these states are degenerate in energy (except in
  the presence of a magnetic or electric field)

Write down the possible term symbols for Li,
which has 3 electrons
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Example Configuration
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Selection Rules
Not all transitions between levels occur with
equal probability

• For several reasons (angular momentum must be
  conserved so since a photon takes away 1 unit of
  angular momentum), there are the following rules
  governing transitions
• ?l /- 1
• ?L-1,0,1
• ?J-1,0,1, unless J0, then ?J/-1
• ?S0
• ?M-1,0,1
• Permitted lines are lines whose where the rules
  above are obeyed and the transition probability
  is high can radiate via dipole radiation
• Forbidden lines are correspond to the case when
  the rules above are not obeyed. They can still
  occur but with low transition probability (via
  2-photon process or require quadrupole or
  octopole transitions)
• Forbidden lines are given a symbol of square
  brackets, e.g. OI

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Multiplets

• While in hydrogen the energy levels depend only
  on the quantum number n, in other atoms
  transitions arising from a specific nL term (with
  a number of values of J) to another term can give
  rise to multiplets
• One example of a multiplet is the
  OIII??4959,5007 multiplet which arises from the
  1D2 to 3P2 and 3P1 levels
• This is called fine structure
• There is also hyperfine structure due to the
  alignment of the proton spin and the electron
  spin which gives rise to the 21 cm hydrogen
  continuum line
• Also Zeeman splitting in a magnetic field (which
  breaks the degeneracy of the separate J states)

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OIII Ground state lines
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Transition Rates

• For any possible transition between two states m
  and n, there exist Einstein transition
  coefficients Amn, Bmn, and Bnm
• Amn spontaneous transition rate per unit time
  per atom
• Bmn stimulated emission coefficient (a photon
  with an energy equivalent to the transition from
  n to m can prompt stimulate an electron
  decay)
• Bnm absorption coefficient
• These coefficients are related
• gmBmn gnBnm where gm is the statistical weight
  of the level
• Amn (h?3/c2)( gn/gm) Bnm
• Transitions with high A values (108 per second)
  are permitted, if A 108 per second, then an
  electron will take, on average, 10-8 seconds to
  decay
• Also note that the larger the energy difference
  of a transition, the higher the A value as A ?3
• Forbidden transitions have low A values, A lt 1
  per second
• On Earth, the density is so high that collisional
  de-excitation dominates. You can see allowed
  transitions but forbidden lines are not typically
  seen. In the ISM, densities can be low enough
  that forbidden transitions can actually occur
  before collisional de-excitation.

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Populating Atomic Levels

• An electron may change levels for a number of
  reasons
• Photo-excitation from a lower level (very
  unimportant in the ISM)
• Collisional excitation (must overcome the
  Boltzmann factor)
• Collisional de-excitation from higher level
• Cascade from a higher level
• Recombination to an excited state
• In the ISM, where the density is extremely low,
  collisional excitations (or de-excitations) are
  only important for levels with no permitted decays

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Cross section for photo-ionization

• Hydrogen requires 13.6 eV of energy to ionize,
  which corresponds to ?912 Å, ? 3.29 x 1015 Hz.
  At this energy, the atoms cross section to
  photoionization is A06.30 x 10-18 cm2.
• a? A0 (?0/?)3
• Extreme UV light with wavelengths just shortward
  of 912 Å are absorbed very rapidly by neutral
  hydrogen in the interstellar medium
• At shorter wavelengths (say the soft x-ray)
  absorption by the ISM is much less important, the
  ISM is transparent in the hard x-ray and
  gamma-ray bands

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Cross section for photo-ionization of hydrogen
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Photo-ionization of heavy elements
Energies needed to ionize metals
Note that when hydrogen is ionized, so is oxygen!
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Recombination Coefficient

• Rate of recombination
• a ? Te-1/2

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Ionization Balance

• The rate of photo-ionizations occurring in a
  nebula is

where ? is the optical depth (i.e., the
attenuation factor), and a? is the ionization
cross-section.

• Meanwhile, the rate of recombinations is

• In a steady state nebula, the rate of
  photo-ionizations equals the rate of
  recombinations, so

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Ionization Balance

• For the case of a constant density nebula, this
  simplifies. First, multiply through by 4 ? r2

• Now integrate both sides of the equation over
  radius. If R is the radius of the ionized
  region, the

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Ionization Balance

• For a constant density nebula, integrating the
  right side of the equation is easy

• For the left-hand side, we can get rid of the
  radius by noting that the attenuation is due to
  the absorption of neutral hydrogen along the path
  of the photon. In other words,

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Ionization Balance

• By definition, at the edge of the nebula, all the
  ionizing photons have been absorbed, so the
  optical depth at that point is infinite. So

• The last integral is equal to one. Therefore, if
  we define Q(H0) as the number of photons capable
  of ionizing hydrogen, i.e.,

we get
where we have assumed Ne Np N(H). This gives
the radius of the Strömgren Sphere.
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Forbidden Lines from Ionized Gas 2-Level Atom
2
E21
1

• Downward transitions can occur through
• Spontaneous decay (rate determined by A21)
• collisional de-excitation (elastic collisions
  with electrons)

• Upward transitions can occur through
• photoionization from a lower level
• collisional excitation (inelastic collisions
  with electrons)

In equilibrium, upward transition rate from level
1 balances downard transition rate to level 1
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This means
Einstein A-coefficient
1
Collisional de-excitation coefficient
Increases with ne
Increases with n1
Collisional excitation coefficient
The collisional coefficients are related through
the Boltzmann factor
2
Where g2, g1, are the statistical weights of the
two levels (2J1)
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The collisional de-excitation coefficient is
given approximately by
Where the thermal speed of the electrons is given
by
And an elastic cross section given approximately
by
A more detailed calculation yields
Where W12 comes from quantum mechanical
calculations
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Using Equations 1 and 2, we get that the
level populations are related by
3
The limiting cases are
Boltzmann distr.
This defines the critical density for a
transition at densities above the critical
density, transitions from the upper to lower
level are more likely to occur through collisions
rather than spontaneous decay.
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How do we use this information?
The luminosity of a line (per unit volume) caused
by a downward transition from 2 -gt 1 is
Using equation 3, this becomes
At low densities
At high densities
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The transition between these two regimes occurs
at the critical density
This is a useful diagnostic - if we don't see
certain lines of an ion, it may be because the
density is too high for their radiation to occur.
This happens when the mean time between
collisions is comparable to or less than the
radiative lifetime A21-1. (This is why we dont
see forbidden lines on Earth)
For permitted transitions (electric dipole
radiation), the transitions rates occur at A
108 sec-1 and corresponding critical densities of
1015 cm-3. For forbidden lines, the A
coefficients are much smaller so the critical
densities are much lower - comparable to
densities in typical nebulae.
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Critical densities for some common transitions
Appenzeller and Östreicher 1988 (AJ 95, 45) and
Table 3.11 of Osterbrock
Many of these lines are seen in gaseous nebulae,
and the strength of the lines depends on the
density of the gas
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So, summarizing, because these lines are excited
primarily by collisions with electrons, the
observed line strengths can be used to measure
the density, temperature, and then composition of
the gas!!
To measure density
Use line ratios from lines with different
critical densities (but are close together in
energy to eliminate temperature dependence ) one
in the low-density limit (L a n2) and one in the
high-density limit (L a n).
In this example, the SII at 6716Å comes from a
4S3/2-2D5/2 transition and has a critical density
of 1.5 x 103 cm-3. The 6731Å line comes from the
4S3/2-2D3/2 transition with critical density 3.9
x 103 cm-3. Note that the line ratio is
sensitivite to density between about 100 to
10,000 cm-3.
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To measure temperature
Again use single ion (independent of abundance
and ionization) but pick lines that have large
differences in excitation potential.
The most-used example is O III with lines at
4363 and 49595007 Å.
Once these line ratios are used to constrain
density and temperature, abundances can be
derived!
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The Nature of the Ionizing Source
Since the ions observed are ionized by photons,
and the ionization potential of metals varies,
line ratios of lines from various ions can be
used to infer the spectral shape of the ionizing
radiation field. Here are the ionization
potentials (eV) for a few common ions
Since even the most massive (hottest) stars
produce very few photons beyond 50 eV, the
presence of lines from highly ionized species
indicated the presence of some source of hard
photons other than stars. What could this be?
Next topicAGN