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Chapter 8 – Continuous Absorption • Physical Processes • Definitions • Sources of Opacity – – – – – Hydrogen bf and ff HH2 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 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 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 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 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: 1 1 R 2 2 l n m 1 R is the Rydberg Constant, R = 1.1 x 10-3 Å-1 As m > ∞, the transition approaches a boundfree condition. For photons of higher energy, the hydrogen atom is ionized 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!) 3 2 6 Rg a g l 32 e bf 0 bf a bf (l , n) 3 5 3 n5 3 3 h nn (for the nth bound level below the continuum and l < ln) • where a0 = 1.044 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 Must also consider level populations • Back to Boltzman and Saha! Nn g n kT e N u0 (T ) • 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): an Nn l k ( H bf ) a n 3 g bf e kT N n0 n0 n 3 One more step • Terms with n > n0+2 can be replaced with an integral (according to Unsöld) • Plus a little manipulation, gives n0 2 gbf log e 3 3 I k (Hbf ) a 0 l 3 10 (10 10 ) 2I n0 n • This is the absorption coefficient per neutral hydrogen atom • Here, I is the ionization potential, NOT the intensity! Model Flux Distributions • Sharp edges are the result of sudden drop in bound-free opacities due to ionization 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: log e I 3 k ( H ff ) a 0 l g f 2I 10 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 • 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) More H- Bound-Free Opacity • Per atom absorption coefficient for H- can be parameterized as a polynomial in l: abf a0 a1l a2l ... 2 N (H ) 5040 log log Pe I 2.5 log T 0.1248 N (H ) T 5 2 k (Hbf ) 4.158x10 abf Pe 10 10 0.754 • Units of cm2 per neutral hydrogen atom H- Bound-Free Absorption Coefficient • Two theoretical calculations • Important in the optical and near infrared • Peaks at 8500Å 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 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 k ff (H ) 10 Pe10 ff 26 f0 f1 log f2 log2 • the f’s are functions of logl and is 5040/T • Units are cm2 per neutral H atom 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 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 11786 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. Helium Absorption • He in hot stars only, O and early B stars – 1=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) 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. 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 hn mv c m0 2 v v 1 2 c hn m0 c 2 (m m0 )c 2 m0 c 2 • Combining these equations gives 1 ( v ) 2 (1 v ) 2 c c • So v=0 (the photon isn’t absorbed) or v=c (not allowed) Electron Scattering • Thomson scattering (photons scatters off a free electron, no change in l, just direction): 8 e2 2 a e ( 2 ) 6.654x1025 cm2e 1 3 mc Ne Pe k (e ) a (e ) a (e ) r PH • 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 Rayleigh Scattering • Photons scatter off bound electrons (varies as l-4) • Generally can be neglected • But – since it depends on l4, it is important as a UV opacity source in cool stars with molecules in their atmospheres. • H2 can be an important scattering agent Other Sources • Metals: C, Si, Al, Mg, Fe produce boundfree 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 100% 80% H(BF) 60% H2+ 40% Mg+Al+Si 20% H- Optical Depth 10 1 0.1 0.01 0% 0 Fraction of Opacity Sources of Opacity for Teff=4500 Log g = 1.5 Opacity Sources at 5143K Opacity at 6429 K Opacity at 7715 K Opacity at 11600 K Dominant Opacity vs. Spectra Type Low Electron scattering (H and He are too highly ionized) He+ He Low pressure – less H-, lower opacity Neutral H H- H- High (high pressure forces more H-) O B A F G K M