Chapter 8 – Continuous Absorption Physical Processes Definitions Sources of Opacity –Hydrogen bf and ff –H - –He –Scattering

Physical Processes 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. Scattering – deflection of a photon from its original path by a particle, without changing wavelength –Rayleigh scattering if the photon’s wavelength is greater than the particle’s resonant wavelength. (Varies as -4 ) –Thomson scattering if the photon’s wavelength is much less than the particle’s resonant wavelength. (Independent of wavelength) –Electron scattering is Thomson scattering off an electron Photodissociation may occur for molecules

Electron Scattering vs. Free-Free Transition Electron 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.

Why Can’t an 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)

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

Monochromatic Absorption Coefficient Recall d  =   dx. We need to calculate , the absorption coefficient per gram of material First calculate the atomic absorption coefficient  (per absorbing atom or ion) Multiply by number of absorbing atoms or ions per gram of stellar material (this depends on temperature and pressure)

Bound-Bound Transitions These produce spectral lines At high temperatures (as in a stellar interior) these may often be neglected. But even at T~10 6 K, the line absorption coefficient can exceed the continuous absorption coefficient at some densities

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 (g bf – not the statistical weight!) For the nth bound level below the continuum and < n where  0 = x 10 –26 for in Angstroms

Converting to the MASS Absorption Coefficient Multiply by the number of neutral hydrogen atoms per gram in each excitation state n Back to Boltzman and Saha! g n =2n 2 is the statistical weight u 0 (T)=2 is the partition function

Class Investigation Compare  bf at =5000A and level T=Teff for the two models provided Recall that and k=1.38x10 -16, a 0 =1x And Use the hydrogen ionization chart from your homework.

Free-Free Absorption from H I Much less than bf 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:

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?

Opacity from the H - Ion Only one known bound state for bound-free absorption eV binding energy So < hc/h = 16,500A 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)

More H - Bound-Free Opacity Per atom absorption coefficient for H- can be parameterized as a polynomial in : Peaks at 8500A

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  3 Small at optical wavelengths Comparable to H - bf at 1.6 microns Increases to the infrared

He absorption 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)

Electron Scattering Thomson scattering: Independent of wavelength In hot stars (O and early B) where hydrogen dominates, then P e ~0.5P g, and  (e) is independent of pressure In cool stars, e- scattering is small compared to other absorbers for main sequence star but is more important for higher luminosity stars

Rayleigh Scattering Generally can be neglected But – since it depends on  4 it is important as a UV opacity source in cool stars with molecules in their atmospheres. H 2 can be an important scattering agent

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 -, C 2 -, H 2 0 -, CH 3, TiO are important in late and/or very late stars

Molecular Hydrogen Opacity H 2 is more common than H in stars cooler than mid-M spectral type (think brown dwarfs!!) H 2 does not absorb in the visible spectrum H 2 + does, but is less than 10% of H - in the optical H 2 + is a significant absorber in the UV H 2 - ff absorption in the IR

Opacity vs. Spectral Type Main Sequence Supergiants

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

Sources of Opacity for Teff=4500 Log g = 1.5

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  in AMU and k=1.38x How does  e compare to  Rosseland