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This article provides a comprehensive overview of continuum transitions and sources of opacity in LTE (Local Thermodynamic Equilibrium). It explores topics such as bound-free and free-free transitions, scattering processes like Thomson and Rayleigh scattering, and the principles of LTE and statistical equilibrium. The article also discusses the validity of LTE assumptions and the importance of considering non-LTE effects in certain scenarios.
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5. Continuum Transitions & LTE • Rutten: 2.4, 2.5 • Continuum transitions • Sources of opacity? • LTE: Maxwell, Saha, Boltzmann • LTE: Planck, Stefan-Boltzmann • Statistical weights, partition functions, etc • NLTE, Statistical equilibrium
Continuum Transitions Bound-free: For bound-free transitions the extinction cross section for hydrogen and hydrogen-like transitions is given by Kramer’s formula: With n the principal quantum number of level i from which the atom or ion is ionized. Z the ion charge and gbf the Gaunt factor, a quantum mechanical correction of order 1. The cross section decays as 1/n3 above the threshold (“edge”) frequency n0 and is zero below it.
Continuous Opacity • Show figure of H, He opacity: edges, lamda^3, etc • Negative hydrogen, other species. • Relative importance, dominant opacities
Free-free: Free-free transitions have Sn = Bn when the Maxwell velocity distribution holds, “thermal Bremsstrahlung.” The volume extinction coefficient is: with Z the ion charge, Ne, Nion the electron and ion densities, gff the appropriate Gaunt correction factor
Scattering Electron (Thomson) Scattering: Thomson scattering of photons by free electrons has a frequency independent cross-section for low energy photons: The corresponding volume extinction coefficient is aT = sTNe, with Ne the electron density. For high energy photons we have Compton scattering. For high energy electrons we have inverse-Compton scattering, which are frequency dependent. Thomson scattering is the major source of continuous extinction in the atmospheres of hot stars where hydrogen is ionized.
Rayleigh Scattering: The cross-section for Rayleigh scattering of photons with n << n0 by bound electrons with binding energy h n0 is: where the oscillator strength flu and frequency n0 characterize the major bound-bound “resonance transition” of the bound electron. For example the Ly a transition in neutral hydrogen of a weighted sum over all Lyman lines. The n4 (1/l4) dependence makes the sky blue and sunsets red.
Redistribution in angle or Scattering Phase Functions: Thomson and Rayleigh scattering are coherent: the photon gets re-directed, but keeps the same n. The re-direction has the phase function ~ 1 + cos2q. This is the angular shape that the scattered photons have.
Local Thermodynamic Equilibrium In LTE all atomic, ionic, and molecular level populations are given by Maxwellian-like Saha-Boltzmann statistics defined by the local temperature: Ni / N ~ exp(-Ei / kT). Matter in LTE Maxwell. For particles of mass m the Maxwell distribution is: for total velocity v, N the total number of particles of mass m per m3. There is a high velocity tail due to the v2 factor. The peak of the distribution gives the most probable speed, vp = (2kT/m)1/2, the average speed is vav = (3kT/m)1/2.
Boltzmann. The Boltzmann excitation distribution is: with nr,sthe number of atoms per m3 in level s of ionization stage r; gr,s the statistical weight of level s in stage r; cr,sthe excitation energy of level s in stage r, measured from the ground level (r, 1) of stage r and cr,s - cr,s = hn for a radiative transition between levels (r, s) and (r, t) with level s “higher” (more internal energy) than level t.
Saha. The Saha ionization distribution for the population ratio between the ground levels of succesive ionization stages is: with Ne and me the electron density and mass; nr+1,1 and nr,1 the population densities of the two ground states of the successive ionization stages r and r+1. cr the ionization energy of stage r, the minimum energy needed to free an electron from the ground state of stage r, with cr = hnedge. gr+1,1 and gr,1 are the statistical weights of the two ground levels. Total population ratio: Partition Function:
Saha-Boltzmann. Combination of the two distributions gives the LTE population ratio between level i and ion state c to which it ionizes as: with nithe total population density of level i, nc the number of ions in ionization level c and cci = cr – cr,i + cr+1,c = hnedge the ionization energy from level i to state c.
Radiation in LTE Planck. In LTE the line source function simplifies to the Planck Function: This equality Sn = Bn is formally derived through the Einstein coefficients for bound-bound process, but holds for all LTE or “thermal” photon processes.
Wien and Rayleigh-Jeans. For large n / T we have the Wien approx: expressing the particle-like behaviour of photons at high energy being similar to the Boltzmann distribution. For small n / T we get the Rayleigh-Jeans approx:
Stefan-Boltzman. Integration of Bn gives the Stefan-Boltzman law: with
LTE Validity and NLTE • Where is it reasonable to assume LTE? • Paragraph about NLTE: radiation field determines level populations, not just local temperature. • Statistical equilibrium.
