1 / 15

4. Continuum Transitions & LTE

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. Continuum Transitions. Bound-free:

benard
Download Presentation

4. Continuum Transitions & LTE

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related