1. High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR) The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Research under sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer. 19 May 2005 Scattering Polarization in the Solar Atmosphere R. Casini High Altitude Observatory National Center for Atmospheric Research

2. Polarized Radiation Origin: symmetry-breaking processes of the Atom-Photon interaction (e.g., anisotropic illumination, deterministic magnetic and/or electric fields, anisotropic collisions) Zeeman effect Circular polarization Linear polarization

3. Polarized Radiation Description: 4 independent parameters –i) coherency matrix (a.k.a. polarization tensor ) –ii) Stokes parameters Jones calculus Mueller calculus

4. Polarized Radiation Operational definition of Stokes parameters For a   retarder 

5. Polarized Radiation Polarized radiation tensors Irreducible spherical tensors transformation conjugation

6. Polarized Radiation Example: Unpolarized radiation from the quiet-sun photosphere Only two non-vanishing components:

7. Atomic Polarization Gas of atoms subject to: Anisotropic and/or polarized illumination External fields Collisions 1.Atomic system not in a “pure state” 2.Population imbalances and quantum interferences between atomic levels

8. Atomic Polarization Density operator Density matrix Atomic eigenstates or some other complete basis

9. Irreducible spherical components of the density matrix If (e.g., Zeeman effect) Otherwise (e.g., Paschen-Back effect, Stark effect) Atomic Polarization transformation conjugation

10. Example: Multi-level atom 1.Population: 2.Orientation: 3.Alignment: Atomic Polarization

11. Ex. 1: Positive orientation in a level Ex. 2: Positive alignment in a level Ex. 3: Orientation and alignment in a level Atomic Polarization  Presence of net polarization in the re-emitted radiation (even in the absence of external fields)

12. Liouville’s equation Evolution equation for expectation values Time evolution of the system Perturbative expansion Atom Radiation 

13. Atom-Photon interaction to 2 nd order of perturbation Resonance Scattering 1 st order  2 nd order 

14. Restriction: Non-coherent scattering Scattering as the succession of 1 st -order absorption and re-emission Complete Re-Distribution in frequency The atom loses memory of the incident photons, and the re-emitted photons are statistically re-distributed in frequency Resonance Scattering Flat-Spectrum Approximation

15. Restriction: Non-coherent scattering Scattering as the succession of 1 st -order absorption and re-emission Two-step solution i.Determine the excitation state of the atomic system consistently with the ambient radiation field (Statistical Equilibrium Problem) ii.Calculate the scattered radiation consistently with the excitation state of the atomic system (Radiative Transfer Problem) Resonance Scattering

16. Statistical Equilibrium functions of the incident radiation

17. Radiative Transfer Absorbtion matrix Function of Stimulation matrix Function of Emission vector Function of in stationary regime

18. Resonance Scattering non-LTE of the 2 nd kind Self-consistency loop (  -iteration)

19. Difficulties 1.The Statistical Equilibrium problem grows rapidly with the complexity of the atomic system (very large sparse matrices) Possible strategy: weak-anisotropy approximation 2.The Radiative Transfer problem requires the solution of a set of 4 coupled ODEs Possible strategy: Diagonal Elements Lambda Operator (DELO) 3.No guarantee of convergence of the self-consistency loop (maybe with the exception of the simplest atomic models, with appropriate initialization) Possible strategy: ????? Resonance Scattering

20. Atom 0-1 Classical analogy in the 3D harmonic oscillator with forcing term

21. Atom 0-1

22. Atom 1-0 Hanle effect of the lower level Non-linear dependence on

23. Atom 1-0

24. Atom 1-1

26. Atomic polarization and Radiative transfer Homogeneous slab 0-1 or 1-0 w/o atomic pol. (Zeeman effect) 0-1 with atomic pol. 1-0 with atomic pol.

27. Atomic polarization and Radiative transfer Homogeneous slab   Å He I

28. Atomic polarization and Radiative transfer Homogeneous slab   Å He I Trujillo Bueno et al., Nature 415, 403 (2002)

29. Atomic polarization in Na I     F    D1D1 D2D2

30. Atomic polarization in Na I     F    D1D1 D2D2 D1D1 D2D2  Å

31. Atomic polarization in Na I     F   

32. Alignment-to-Orientation transfer     F    When quantum interferences between FS and/or HFS levels are important diagonal depolarization  coupling alignment-to-orientation

33. Atomic orientation in H I HAO Advanced Stokes Polarimeter March 2003 THEMIS heliographic telescope September 2003 Spectro-polarimetric observations of H  in solar prominences (off the limb)

34. Atomic orientation in H I Spectro-polarimetric simulations with FS and HFS THEMIS heliographic telescope September 2003 Maximum net circular polarization 1 order of magnitude too small for typical prominence fields (less than ~ 100 G)

35. Atomic orientation in H I Catalytic effect of small electric fields on H I atomic orientation  Enhanced net circular polarization in H 

36. Catalytic effect of small electric fields on H I atomic orientation HH  present also for isotropic electric fields vertical magnetic field, forward scattering Isotropic E field w/o HFS with HFS Inclinations of random-azimuth, 1 V cm -1 fields Only B Prominence B fields

37. Conclusions Spectro-polarimetric observations reveal the complexity of the atomic processes underlying resonance scattering (atomic coherences, FS and HFS effects, magnetic and electric fields, alignment-to-orientation transfer)  The local problem can already become numerically very intensive Points to focus on: a.Improve speed in the construction of the Statistical Equilibrium matrix b.Invent new strategies to accelerate convergence of the iterative scheme for atoms of arbitrary complexity and general illumination conditions