Presentation on theme: "Chapter 8 – Continuous Absorption"— Presentation transcript:
1Chapter 8 – Continuous Absorption Physical ProcessesDefinitionsSources of OpacityHydrogen bf and ffH-H2HeScatteringHow does kn affect the spectrum?More continuous absorption, less continuum light at that wavelengthMore continuous absorption, lines must form in shallower layers, at lower optical depthNeed kn to determine T(t) relation
2Many 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 wavelengthRayleigh 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
3What can various particles do? Free electrons – Thomson scatteringAtoms and Ions –Bound-bound transitionsBound-free transitionsFree-free transitionsMolecules –BB, BF, FF transitionsPhotodissociationMost continuous opacity is due to hydrogen in one form or another
4Monochromatic Absorption Coefficient Recall dtn = knrdx. We need to calculate kn, the absorption coefficient per gram of materialFirst 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
5Bound-Bound Transitions Bound-bound transitions produce spectral linesAt 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 densitiesRemember the hydrogen atom:R is the Rydberg Constant, R = 1.1 x 10-3 Å-1As m > ∞, the transition approaches a bound-free condition. For photons of higher energy, the hydrogen atom is ionized
6Bound 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 1The atomic absorption coefficient abf(H) has units of cm2 per neutral H atom
7Must also consider level populations Back to Boltzman and Saha!gn = 2n2 is the statistical weightu0(T) = 2 is the partition functionSo, the abs. coef. per neutral H atom is (summing over all levels n):
8One more stepTerms with n > n0+2 can be replaced with an integral (according to Unsöld)Plus a little manipulation, givesThis is the absorption coefficient per neutral hydrogen atomHere, I is the ionization potential, NOT the intensity!
10Model Flux Distributions Sharp edges are the result of sudden drop in bound-free opacities due to ionization
11Free-Free Absorption from H I Much less than bound free absorptionKramers (1923) + Gaunt (1930) againAbsorption 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 electronsAgain multiply by the number of neutral hydrogen atoms:
12Opacity from Neutral Hydrogen Neutral hydrogen (bf and ff) is the dominant source of opacity in stars of B, A, and F spectral typeDiscussion Questions:Why is neutral hydrogen not a dominant source of opacity in O stars:Why not in G, K, and M stars?
13Opacity from the H- Ion Bound–free and free-free Only one known bound state for bound-free absorption0.754 eV binding energySo l < 16,500A = 1.65 micronsRequires a source of free electrons (ionized metals)Major source of opacity in the Sun’s photosphereNot a source of opacity at higher temperatures because H- becomes too ionized (average e- energy too high)
14More 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
15H- Bound-Free Absorption Coefficient Two theoretical calculationsImportant in the optical and near infraredPeaks at 8500Å
16H- Free-Free Absorption Coefficient The free-free H- absorption coefficient depends on the speed of the electronPossible because of the imperfect shielding of the hydrogen nucleus by one electronProportional to l3Small at optical wavelengthsComparable to H- bf at 1.6 micronsIncreases to the infrared
17H- Free Free Absorption Coefficient H- ff is important in the infraredcombining H- bf and ff gives an opacity minimum at 1.6 micronsH- ff parameterized asthe f’s are functions of logl and q is 5040/TUnits are cm2 per neutral H atom
18Molecular 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 infraredH2+ in the UV,H2- has no stable bound state, but ff absorption is important in cooler stars
19Collision induced opacity of molecular hydrogen Linsky/JILACollision induced opacity of molecular hydrogenH2 has no dipole moment - no rotation or vibration-rotation spectrumCollisions with (H2, He, H) can induce transient dipole momentsFundamental 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 overlapH2CIO 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.
20Helium AbsorptionHe in hot stars only, O and early B stars – c1=19.7eV, I1=24.6 eV, I2=54.4 eVHe I absorption mimics HHe II also mimics H, but x4 in energy, ¼ in lBound-free He- absorption is negligible (excitation potential of 19 eV!)Free-free He- can be important in cool stars in the IRBF and FF absorption by He is important in the hottest stars (O and early B)
21Electron Scattering vs. Free-Free Transition Electron scattering (Thomson scattering) – the path of the photon is altered, but not the energyFree-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.
22Why 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 saysConservation of energy saysCombining these equations givesSo v=0 (the photon isn’t absorbed) or v=c (not allowed)
23Electron ScatteringThomson scattering (photons scatters off a free electron, no change in l, just direction):Independent of wavelengthIn hot stars (O and early B) where hydrogen is ionized (Pe~0.5Pg), k(e)/Pe is small unless Pe is smallIn cool stars, e- scattering is small compared to other absorbers for main sequence star but is more important for higher luminosity stars
24Rayleigh ScatteringPhotons scatter off bound electrons (varies as l-4)Generally can be neglectedBut – 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
25Other SourcesMetals: C, Si, Al, Mg, Fe produce bound-free opacity in the UVLine Opacity: Combined effect of millions of weak linesDetailed tabulation of linesOpacity distribution functionsStatistical sampling of the absorptionMolecules: CN-, C2-, H20- , CH3, TiO are important in late and/or very late stars
31Opacity vs. Spectral Type Main SequenceSupergiants
32Dominant Opacity vs. Spectra Type LowElectron scattering(H and He are toohighly ionized)Low pressure –less H-, loweropacityElectron PressureHe+HeNeutral HH-H-High(high pressure forces more H-)O B A F G K M
33Class Exercise – Electron Scattering Estimate the absorption coefficient for electron scattering for the models provided at a level where T=TeffRecall thatandwith m in AMU and k=1.38x10-16How does ke compare to kRosseland
34Class InvestigationCompare kbf at l=5000A and level T=Teff for the two models providedRecall thatand k=1.38x10-16, a0 =1x10-26AndUse the hydrogen ionization chart from your homework.