4. Continuum Transitions & LTE

1. 4. 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

2. Continuum Transitions Bound-free: Extinction cross section for H and H-like transitions Kramer’s formula: n = principal quantum number of level i from which atom/ion is ionized Z = charge, gbf = Gaunt factor. s decays as 1/n3 above “edge” at n0 , zero at n < n0

3. Continuous Opacity • Show figure of H, He opacity: edges, lamda^3, etc • Negative hydrogen, other species. • Relative importance, dominant opacities

4. 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

5. Scattering Electron (Thomson) Scattering: n-independent for low energy photons: Volume extinction coefficient : aT = sTNe, High energy photons: Compton scattering. High energy electrons: Inverse-Compton scattering, both fn(n) Major source of continuous extinction in hot star atmospheres where H is ionized.

6. Rayleigh Scattering: Cross-section for Rayleigh scattering of photons with n << n0 by bound electrons with binding energy h n0 : flu and n0 characterize major bound-bound “resonance transition” of the bound electron. n4 (1/l4) dependence makes the sky blue and sunsets red.

7. Redistribution in angle or Scattering Phase Functions: Thomson and Rayleigh scattering are coherent: photon gets re-directed with same n. Re-direction has phase function ~ 1 + cos2q.

8. Local Thermodynamic Equilibrium In LTE all atomic, ionic, and molecular level populations obey Maxwellian-like Saha-Boltzmann statistics defined by the local temperature: Ni / N ~ exp(-Ei / kT). Matter in LTE Maxwell. Particles of mass m the Maxwell distribution is: for total velocity v, N = total number of particles of mass m per m3. High v tail from v2 factor. Most probable speed, vp = (2kT/m)1/2 Average speed is vav = (3kT/m)1/2.

9. Boltzmann. The Boltzmann excitation distribution is: nr,s= number of atoms per m3 in level s of ionization stage r gr,s = statistical weight of level s in stage r cr,s= excitation energy of level s in stage r, measured from ground (r, 1) cr,s - cr,s = hn for a radiative transition (r, s) to (r, t) level s “higher” (more internal energy) than level t

10. Saha. LTE population ratio between ground levels of successive ionization stages: Ne , me = electron density , mass nr+1,1 , nr,1 = populations of ground states of ionization stages r , r+1 cr = ionization energy of stage r = minimum energy to free an electron from the ground state of stage r, with cr = hnedge gr+1,1 , gr,1 = statistical weights of ground levels. Total population ratio: Partition Function:

11. Saha-Boltzmann. LTE population ratio between level i and ion state c to which it ionizes as: ni= total population of level i nc = number of ions in ionization level c cci = cr – cr,i + cr+1,c = hnedge = ionization energy from level i to state c.

12. Radiation in LTE Planck. Line source function simplifies to Bn Sn = Bn formally derived via Einstein coefficients for b-b process, but holds for all LTE or “thermal” photon processes.

13. Wien and Rayleigh-Jeans. Large n / T : Wien approx: Particle-like behaviour of photons at high energy, similar to Boltzmann Small n / T : Rayleigh-Jeans approx:

14. Stefan-Boltzman. Integration of Bn gives Stefan-Boltzman law: with

15. 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.