1. Lecture 5 Summary of galaxy classification Example problems HW review
Introduction to emission line spectroscopy: review of atomic structure

2. 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

3. Useful References Gaseous Nebulae: AGN: Shu (pg 216) Online notes
Osterbrock
AGN:
C&O 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)

4. 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

5. Physics of ISM Many “phases” of ISM
rmean ~ 0.3 cm-3
rmolecular cloud ~ 106 cm-3
ratmosphere ~ 21018 cm-3
Many “phases” of ISM
Thermodynamic equilibrium is not generally the case.

6. 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

7. 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 ((2l+1 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 radial part of Schrodinger equation
Solution of angular part of Schrodinger equation l= 1 2 3 4
Letter used s p d f g

8. Spectroscopic notation
From the above selection rules, if n=1, for example, l can go from 0 to n-1 which is 0. Because of the Pauli exclusion principle, no two electrons can have identical quantum numbers, so only 2 electrons in an “s-shell”.
For n=2, can have l=0,1 and then ml=-1,0,1. If l=0, ml=0, this is an s-state, can have only two electrons. For l=1, can have ml=-1,0,1 and ms=+/-1/2 so can have a total of 6 electrons in the “p shell”, etc. This is how you “build up” atomic structure.
e.g., Oxygen has 8 electrons. What is its electronic configuration? What about Neon which has 10 electrons?

9. 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 = Rydberg’s constant = cm-1 for hydrogen
Similar relations work for other hydrogenic atoms (He II, Li III, etc …)

11. Spectroscopic notation
For an atom, we don’t 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 it’s +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 L+S
The magnetic quantum number M takes on values of J, J-1, …, 0, …-J-1, -J
An atomic level is described as: n2S+1LJ, where 2S+1 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 doesn’t 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

12. Example Configuration:

13. 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 J=0, then J=+/-1
S=0
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]

14. 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)

15. [OIII] Ground state lines

16. 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 < 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.

17. 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

18. 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 atom’s cross section to photoionization is A0=6.30 x 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

19. Cross section for photo-ionization of hydrogen

20. Photo-ionization of heavy elements
Energies needed to ionize metals:
Element II
III IV V
Hydrogen
13.6eV
Helium
24.6
54.4
Carbon
11.3
24.4
47.9
64.5
Nitrogen
14.5
29.6
47.4
77.7
Oxygen
13.6
35.1
54.9
77.4
Neon
21.6
41.0
63.5
97.1
Note that when hydrogen is ionized,
so is oxygen!

21. Recombination Coefficient
Rate of recombination
a  Te-1/2

22. 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

23. 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

24. 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,

25. 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.

27. Forbidden Lines from Ionized Gas: “2-Level Atom”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

28. 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 (=2J+1)

29. 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

30. The limiting cases are:
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.

31. How do we use this information?
The luminosity of a line (per unit volume) caused by a downward transition from 2 -> 1 is:
Using equation [3], this becomes:
At low densities:
At high densities:

32. 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 don’t 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.

33. 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

34. 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.

35. The most-used example is [O III] with lines at 4363 and 4959+5007 Å.
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 Å.
Once these line ratios are used to constrain density and temperature, abundances can be derived!

36. Next topic:AGN 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 topic:AGN