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Continuous Opacity Sources

Continuous Opacity Sources. Continuous Opacity Sources. Principal Sources: Bound-Bound Transitions Bound-Free Free-Free (Bremstralung) Electron Scattering (Thompson & Compton) Molecular Transitions We consider only H or H-Like cases

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Continuous Opacity Sources

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  1. Continuous Opacity Sources

  2. Continuous Opacity Sources • Principal Sources: • Bound-Bound Transitions • Bound-Free • Free-Free (Bremstralung) • Electron Scattering (Thompson & Compton) • Molecular Transitions • We consider only H or H-Like cases • We will do this classically (and correct to the QM result) • Stars are mostly H (in one form or another). Continuous Opacity

  3. Dominant Opacity Sources Continuous Opacity

  4. Caveats and Details • At High Temperatures: (1-e-hν/kT) → 0 so all of the bb, bf, and ff sources go to 0! • Electron scattering takes over (is always there and may be important). • Free-Free is not the same as electron scattering: Conservation of momentum says a photon cannot be absorbed by a free particle! • In principal we start with a QM description of the photon - electron interaction which yields the cross section for absorption/scattering of the photon of energy hν, call the cross section ai(ν). Continuous Opacity

  5. The Opacity Is • The opacity (g/cm2) for the process is Κi(ν) = niai(ν)/ρ • ni is the number density (#/cm3) of the operant particles • The subscript i denotes a process/opacity • The total opacity is • Κtotal = ∑Κi(ν) Continuous Opacity

  6. Compton Scattering • This process is important only for high energy photons as the maximum change is 0.024Å. • Reference: Eisberg -- Fundamentals of Modern Physics p. 81 ff. Continuous Opacity

  7. Electron Scattering Conditions • Collision of an electron and a photon • Energy and Momentum Must be conserved • In stellar atmospheres during photon-electron collisions the wavelength of the photon is increased (assume the Eelectron < Photon (hν)) • At 4000Å: hν = 4.966(10-12) ergs • ½ mv2 < 5(10-12) ergs ==> v < 108 cm/s • At 5000K vRMS = 6.7(105) cm/s • At 100000K vRMS = 3(106) cm/s Continuous Opacity

  8. Thompson Scattering Classical Electron Scattering • Reference: Marion - Classical Electromagnetic Radiation p. 272 ff • Low Energy Process: v << c • Energy Absorbed from the EM field is • a = acceleration of the electron: a = eE/me and E is the magnitude of the electric field. Continuous Opacity

  9. Thompson Scattering • Field Energy Density = <E2/4π> (Time Average) • Energy Flux Per Electron = c <E2/4π> • Now Take the Time Average of (#): • But that has to be the energy flux per electron times the cross section which is σT c <E2/4π>. Continuous Opacity

  10. The Thompson Cross Section At High Temperatures this breaks down: T ≥ 109 K Continuous Opacity

  11. Electron Scattering Opacity • Κe = σTNe/ρ • There is no frequency dependence! • Scattering off other ions is unimportant: • Cross Section Goes as (1/m)2 so for ions (1/AmH)2 while for electrons it goes as (1/me)2 • Ratio: Ions/Electrons = (me/AmH)2 = (1/A(1840))2 < 10-6 Continuous Opacity

  12. Rayleigh Scattering: σR • Scattering of a low energy photon by a bound electron. • Classically: Rayleigh scattering occurs when a photon of energy less than the atomic energy spacing is absorbed. • The electron then oscillates about the unperturbed energy level (harmonically). • The electron reradiates the same photon but remains in the same energy state. Continuous Opacity

  13. The Rayleigh Cross-Section • The cross section is: σ = σT/(1-(ν0/ν)2)2 • Where hν is the photon energy • hν0 is the restoring force for the oscillator • When ν << ν0: σR = σT (λ0/λ)4 • Now since ν << ν0 we have E << kT • Which implies T ~ 1000 K for this process. Continuous Opacity

  14. Free-Free Opacities Absorption Events • Bremstralung • Electron moving in the field of an ion of charge Ze emits or absorbs a photon: • Acceleration in field produces a photon of hν • De-acceleration in field consumes a photon of hν • Consider the Emission Process • Initial Electron Velocity: v′ • Final Electron Velocity: v • Conservation of Energy Yields • ½ me v2 + hν = ½ me v′2 Continuous Opacity

  15. Energy Considerations • Energy Absorbed: dE/dt = 2/3 (e2/c3) a2 where a = eE/me (a is the acceleration) • Most energy is absorbed during the time t  b / v′ when the electron is close to the ion • b is called the impact parameter • b is the distance of closest approach • Acceleration is ~ Ze2/meb2 • Eabs  2/3 (e2/c3) (Ze2/meb2)2 (b / v′) •  2/3 ((Z2e6)/(me2 c3 b3 v′)) Continuous Opacity

  16. Frequency Dependence Expand dE/dT in a Fourier Series • Greatest contribution is when 2πνt  1 during time t  b / v′ • Therefore 2πν b / v′ = 1 or 2πν = v′ / b • The energy emitted per electron per ion in the frequency range d is • dqν = 2πb db Eabs Why? 2πb db is an area! • dqν = energy emitted per electron per ion per unit frequency Continuous Opacity

  17. Bremstralung Energy • Total Energy Emitted: ninev′f(v′)dv′dqν • ni = Ion number density • ne = electron number density (note that the electron flux is nev′f(v′)dv′) • The reverse process defines the Bremstralung absorption coefficient aν giving the absorption per ion per electron of velocity v from the radiation field. • In TE: Photon Energy Density = Uνp = (4π/c) Bν(T) Continuous Opacity

  18. The Absorption Coefficient • Net energy absorbed must be the product of the photon flux (photon energy = hν) cUνpdν and ninef(v)dvaν and (1-e-hν/kT) or • cUνpdν ninef(v)dvaν (1-e-hν/kT) • But in TE that must be equal to the emission: • cUνpdν ninef(v)dvaν (1-e-hν/kT) = nine v′f(v′)dv′ dqν • aν = π/3 (Z2e6)/(hc me2ν3 v) • This is off by 4/3 from the exact classical result. Continuous Opacity

  19. The Bremstralung Opacity • V is v in the previous equations • This reduces to κff(ν) = 4/3 nine (2π/3mekT)1/2 ((Z2e6)/(hc meν3)) gff(ν) • gff(ν) is the mean Gaunt factor and the result has been corrected to the exact classical result. Continuous Opacity

  20. Bound Free Opacities Transition from a Bound State to Continuum or Visa Versa • This process differs from the free-free case due to the discrete nature of one of the states • Nth Discrete State: • Then the electron capture/ionization process must satisfy: • ½ me v2 - En = hν Continuous Opacity

  21. Bound Free • V is the velocity of the ejected or absorbed electron. • Semiclassical treatment of electron capture • Electron Initial Energy: ½ mev2 and is positive • The energy decreases in the electric field as it accelerates seeing ion of charge Ze • Q: Why does it loose energy as it accelerates? • A: It radiates it away • The energy loss per captured electron may be estimated as: • dqν = 2πbdbEabs = (8π2/3) ((Z2e6)/(me2 c3 v′2dν)) Continuous Opacity

  22. Cross Section • The cross section for emission of photons into frequency interval dν is defined by • hν dσν = dqν = hν (dσν/dν) dν • The final state is discrete so define σcn as the cross section for capture into state n (energy En) characterized by n in the range (n, n+dn) so that dσν = σcndn. Then • hν (dσν/dν) = hνσcn dn/dν • Solve for σcn and use En = -IHZ2/n2 to get dn/dν • Thus: σcn = ((2IHZ2/n2)/(h2νn3))(dqν/dν) = (32/3) π4 ((Z2e10)/(mec3h4v2νn3)) Continuous Opacity

  23. Photoionization The reverse process is related by detailed balance • Let σνn = photoionization cross section • The number of photons absorbed of energy hν with the emission of electrons of energy ½mev2 - En from the nth atomic state is • (cUνp/hν) dνσνn Nn (1-e-hν/kT) • Nn is the number of atoms in state n • The reverse process: the number of electrons with initial velocities v captured per second into state n is Niσcn ne v f(v) dv Continuous Opacity

  24. State Population • The Boltzmann Equation for the system is: • where g1 = 2 and gn = 2n2 (H-like ions) and the Saha equation is • Detailed Balance says: • (cUνp/hν)dνσνnNn(1-e-hν/kT) = Niσcnnevf(v)dv Continuous Opacity

  25. The Bound Free Coefficient • We assume • Maxwellian distribution of speeds • Boltzmann and Saha Equations • σνn = (gi/gn) (mevc/hν)2σcn • We assume the Z-1 electrons in the atom do not participate, set gi = 1, and correct for QM then Continuous Opacity

  26. Limits and Conditions • Photon must have hν > En • σ = 0 for hν < En (= IHZ2/n2) • Recombination • Levels coupled to ν by bound free processes have • En for n > n* where n* = (IHZ2/hν)1/2 • Total Bound Free Opacity: Continuous Opacity

  27. Other Opacities Continuous Opacity

  28. The Rosseland Mean Opacity Continuous Opacity

  29. More Spectroscopic Notation H Like Only • Electron and proton have • Spin ½ • Angular momentum /2 and each has a corresponding magnetic moment • The magnetic moment of the electron interacts with both the orbital magnetic moment of the atom (spin-orbit) and the spin of the proton (spin-spin). • Spin-orbit: fine structure (multiplets) • Spin-spin: hyperfine structure Continuous Opacity

  30. Notation • Principal Quantum Number n • Orbital Quantum Number l: l ≤ n - 1 • S P D F • l = 0 1 2 3 • Total angular momentum of a state specified by l= l [(l+1)]½ • Spin angular momentum (s=1/2): [s(s+1)]½ = 3  /2 • Orbital and spin angular momentum interacts to produce a total angular momentum quantum number j = l ± 1/2: S States have j = 1/2, P states have j = 1/2, 3/2; and D has j = 3/2, 5/2 Continuous Opacity

  31. H Like Notation Continuous Opacity

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