1. Chapter 8 – Continuous Absorption
Physical Processes
Definitions
Sources of Opacity
Hydrogen bf and ff H- H2 He
Scattering
How does kn affect the spectrum?
More continuous absorption, less continuum light at that wavelength
More continuous absorption, lines must form in shallower layers, at lower optical depth
Need kn to determine T(t) relation

2. Many physical processes contribute to opacity
Bound-Bound Transitions – absorption or emission of radiation from electrons moving between bound energy levels.
Bound-Free Transitions – the energy of the higher level electron state lies in the continuum or is unbound.
Free-Free Transitions – change the motion of an electron from one free state to another.
Electron Scattering – deflection of a photon from its original path by a particle, without changing its wavelength
Rayleigh scattering – photons scatter off bound electrons. (Varies as l-4)
Thomson scattering –photons scatter off free electrons (Independent of wavelength)
Photodissociation may occur for molecules

3. What can various particles do?
Free electrons – Thomson scattering
Atoms and Ions –
Bound-bound transitions
Bound-free transitions
Free-free transitions
Molecules –
BB, BF, FF transitions
Photodissociation
Most continuous opacity is due to hydrogen in one form or another

4. Monochromatic Absorption Coefficient
Recall dtn = knrdx. We need to calculate kn, the absorption coefficient per gram of material
First calculate the atomic absorption coefficient an (per absorbing atom or ion)
Multiply by number of absorbing atoms or ions per gram of stellar material (this depends on temperature and pressure)
MOSTLY HYDROGEN

5. Bound-Bound Transitions
Bound-bound transitions produce spectral lines
At high temperatures (as in a stellar interior) these may often be neglected.
But even at T~106K, the line absorption coefficient can exceed the continuous absorption coefficient at some densities
Remember the hydrogen atom:
R is the Rydberg Constant, R = 1.1 x 10-3 Å-1
As m > ∞, the transition approaches a bound-free condition. For photons of higher energy, the hydrogen atom is ionized

6. Bound Free Transitions
An expression for the bound-free coefficient was derived by Kramers (1923) using classical physics.
A quantum mechanical correction was introduced by Gaunt (1930), known as the Gaunt factor (gbf is not the statistical weight!)
(for the nth bound level below the continuum and l < ln)
where a0 = x 10–26 for l in angstroms and gbf is of order 1
The atomic absorption coefficient abf(H) has units of cm2 per neutral H atom

7. Must also consider level populations
Back to Boltzman and Saha!
gn = 2n2 is the statistical weight
u0(T) = 2 is the partition function
So, the abs. coef. per neutral H atom is (summing over all levels n):

8. One more step
Terms with n > n0+2 can be replaced with an integral (according to Unsöld)
Plus a little manipulation, gives
This is the absorption coefficient per neutral hydrogen atom
Here, I is the ionization potential, NOT the intensity!

10. Model Flux Distributions
Sharp edges are the result of sudden drop in bound-free opacities due to ionization

11. Free-Free Absorption from H I
Much less than bound free absorption
Kramers (1923) + Gaunt (1930) again
Absorption coefficient depends on the speed of the electron (slower electrons are more likely to absorb a photon because their encounters with H atoms take longer)
Adopt a Maxwell-Boltzman distribution for the speed of electrons
Again multiply by the number of neutral hydrogen atoms:

12. Opacity from Neutral Hydrogen
Neutral hydrogen (bf and ff) is the dominant source of opacity in stars of B, A, and F spectral type
Discussion Questions:
Why is neutral hydrogen not a dominant source of opacity in O stars:
Why not in G, K, and M stars?

13. Opacity from the H- Ion Bound–free and free-free
Only one known bound state for bound-free absorption
0.754 eV binding energy
So l < 16,500A = 1.65 microns
Requires a source of free electrons (ionized metals)
Major source of opacity in the Sun’s photosphere
Not a source of opacity at higher temperatures because H- becomes too ionized (average e- energy too high)

14. More H- Bound-Free Opacity
Per atom absorption coefficient for H- can be parameterized as a polynomial in l:
Units of cm2 per neutral hydrogen atom

15. H- Bound-Free Absorption Coefficient
Two theoretical calculations
Important in the optical and near infrared
Peaks at 8500Å

16. H- Free-Free Absorption Coefficient
The free-free H- absorption coefficient depends on the speed of the electron
Possible because of the imperfect shielding of the hydrogen nucleus by one electron
Proportional to l3
Small at optical wavelengths
Comparable to H- bf at 1.6 microns
Increases to the infrared

17. H- Free Free Absorption Coefficient
H- ff is important in the infrared
combining H- bf and ff gives an opacity minimum at 1.6 microns
H- ff parameterized as
the f’s are functions of logl and q is 5040/T
Units are cm2 per neutral H atom

18. Molecular H2, H2+, H2- Opacities
H2 is more common than H in stars cooler than mid-M spectral type (think brown dwarfs!!)
Recall that these are important in L and T dwarfs! Also in cool white dwarfs…
Not important in optical region (H2+ less than 10% of H- in the optical)
H2 in the infrared
H2+ in the UV,
H2- has no stable bound state, but ff absorption is important in cooler stars

19. Collision induced opacity of molecular hydrogen
Linsky/JILA
Collision induced opacity of molecular hydrogen
H2 has no dipole moment - no rotation or vibration-rotation spectrum
Collisions with (H2, He, H) can induce transient dipole moments
Fundamental VR band at 4162 cm-1 (2.4 microns).
First overtone VR band at 8089 cm-1 (1.2 microns).
Second overtone VR band at cm-1 (0.2 microns).
Collisions are fast - individual spectral lines broad and overlap
H2CIO is important for computing the temperature structure of brown dwarfs because it is a near-continuous opacity source that fills in the opacity gaps between the molecular absorption lines.

20. Helium Absorption
He in hot stars only, O and early B stars – c1=19.7eV, I1=24.6 eV, I2=54.4 eV
He I absorption mimics H
He II also mimics H, but x4 in energy, ¼ in l
Bound-free He- absorption is negligible (excitation potential of 19 eV!)
Free-free He- can be important in cool stars in the IR
BF and FF absorption by He is important in the hottest stars (O and early B)

21. Electron Scattering vs. Free-Free Transition
Electron scattering (Thomson scattering) – the path of the photon is altered, but not the energy
Free-Free transition – the electron emits or absorbs a photon. A free-free transition can only occur in the presence of an associated nucleus. An electron in free space cannot gain the energy of a photon.

22. Why Can’t a Lone Electron Absorb a Photon?
Consider an electron at rest that is encountered by a photon, and let it absorb the photon….
Conservation of momentum says
Conservation of energy says
Combining these equations gives
So v=0 (the photon isn’t absorbed) or v=c (not allowed)

23. Electron Scattering
Thomson scattering (photons scatters off a free electron, no change in l, just direction):
Independent of wavelength
In hot stars (O and early B) where hydrogen is ionized (Pe~0.5Pg), k(e)/Pe is small unless Pe is small
In cool stars, e- scattering is small compared to other absorbers for main sequence star but is more important for higher luminosity stars

24. Rayleigh Scattering
Photons scatter off bound electrons (varies as l-4)
Generally can be neglected
But – since it depends on l-4, it is important as a UV opacity source in cool stars with molecules in their atmospheres.
H2 can be an important scattering agent

25. Other Sources
Metals: C, Si, Al, Mg, Fe produce bound-free opacity in the UV
Line Opacity: Combined effect of millions of weak lines
Detailed tabulation of lines
Opacity distribution functions
Statistical sampling of the absorption
Molecules: CN-, C2-, H20- , CH3, TiO are important in late and/or very late stars

26. Sources of Opacity for Teff=4500 Log g = 1.5

27. Opacity Sources at 5143K

28. Opacity at 6429 K

29. Opacity at 7715 K

30. Opacity at K

31. Opacity vs. Spectral Type
Main Sequence
Supergiants

32. Dominant Opacity vs. Spectra Type
Low
Electron scattering
(H and He are too
highly ionized)
Low pressure –
less H-, lower
opacity
Electron Pressure
He+ He
Neutral H H- H-
High
(high pressure forces more H-)
O B A F G K M

33. Class Exercise – Electron Scattering
Estimate the absorption coefficient for electron scattering for the models provided at a level where T=Teff
Recall that
and
with m in AMU and k=1.38x10-16
How does ke compare to kRosseland

34. Class Investigation
Compare kbf at l=5000A and level T=Teff for the two models provided
Recall that
and k=1.38x10-16, a0 =1x10-26
And
Use the hydrogen ionization chart from your homework.