1. NLTE polarized lines and 3D structure of magnetic fields Magnetic fields cross canopy regions, not easily investigated by extrapolations, between photosphere and chromosphere. Full knowledge of the 3D structure implies diagnostics extracted from strong NLTE lines. The data analysed below are obtained with THEMIS / MSDP and MTR in 589.6 NaI (D1) 610.27 CaI 630.2 FeI (for comparison) Fortunately, the domain of ‘’weak field ’’ approximation is more extended for such lines (smaller Lande factor, broad lines). P.Mein, N.Mein, M.Faurobert, V.Bommier, J-M.Malherbe, G.Aulanier

2. 1) D1 line and facular magnetic flux tubes Problems of filling factor, vertical gradients, MHD models Simulation of line profiles MULTI code with field free assumption, 1D model Instrumental profile included - Quiet Sun = VAL3C model - Circular polarization: I -V profile -Solid line: flux tube, dashed: quiet -Bisector for  = +/-8, 16, 24, 32 pm -Weak field assumption B//

3. 1 - 2D model flux tube compensating horizontal components of Lorentz forces Magnetic fieldDepartures from equilibrium Formation altitudes of B// for  = +/- 8, 16, 24, 32 pm B z (0,z) ~ exp(-z/h ) B z (x,z) ~ cos 2 (  x/4d(z)) d(z) by constant flux B x (x,z) by zero divergence P(x,z) compensates Lorentz horiz. comp. Vertical accelerations exceed solar gravity at high levels

4. Simulation Smoothing by seeing effects convol cos 2 (  x/4s) s=400 km B// from tube center at  = 8, 16, 24, 32 pm No smoothing by seeing effects Points at half maximum values (crosses) are in the same order as tube widths at corresponding formation altitudes wings core wings core Seeing effects ~ filling factor effects hide vertical magnetic gradients at tube center

5. Filling factors and slope-ratios of profiles flux-tube/quiet-sun Stokes V obs = f Stokes V tube Zeeman shift of I -V profile: Z obs (dI/d ) obs = f Z tube (dI/d ) tube If f << 1, from core to wings Z obs = f Z tube (dI/d ) tube / (dI/d ) QS dI/d Tube Quiet Sun Tube QS I - V Decrease of observed B// in the wings Different models for tube and quiet sun !

6. 2 - Model flux tube closer to magneto-static equilibrium B z (0,z) 2 /2  0 ~ P quiet (z) B z (x,z) ~cos 2 (  x/4d(z)) d(z) by constant flux B x (x,z) by zero divergence P(x,z) = P quiet – B z (x,z) 2 /2  0 Magnetic fieldDepartures from equilibrium Formation altitudes of B//  = +/- 8, 16, 24, 32 pm Departures from equilibrium never exceed solar gravity

7. ObservationSimulation Average of 6 magnetic structures Faculae near disk center (N17, E18) Sections for  = 8, 16, 24, 32 pm Qualitative agreement only: - tube thinner in line wings - apparent B// smaller in line wings (seeing effects) But impossible to increase the magnetic field and/or the width of the tube without excessive departures from equilibrium With seeing effects s=500 km wings core wings

8. 3 - Conglomerate of flux tubes Magnetic fieldDepartures from equilibrium

9. ObservationSimulation Seeing effects s = 700 km Better qualitative agreement (tube width) But magnetic field still too low Coronal magnetic field outside the structure? MHD models, including temperature and velocity fluctuations…? P. Mein, N. Mein,M. Faurobert, G. Aulanier and J-M. Malherbe, A&A 463, 727 (2007)

10. 2 ) Fast vector magnetic maps with THEMIS/MSDP UNNOFIT inversions NLTE line 610.27 CaI + 630.2 FeI - Examples of fast MSDP vector magnetic maps and comparison with MTR results - How to reconcile high speed and high spectral resolution by compromise with spatial resolution in MSDP data reduction - Capabilities expected from new THEMIS set-up (32 ) and EST project (40 ) - Departures between 610.3 CaI and 630.2 FeI maps Gradients along LOS? sensitivity of lines? filling factor effects?

11. Example of MSDP image (Meudon Solar Tower 2007, courtesy G. Molodij): In each channel, x and vary simultaneously along the horizontal direction

12. Compromise spatial resol / spectral resol interpolation in x, plane A, D 80 mA B, C --> E40 mA cubic interpol --> F,G20 mA

13. 610.27 CaI Profile deduced from 16 MSDP channels + interpolation x,  plane

14. THEMIS / MSDP 2006 610.3 CaI 160’’ 120’’ THEMIS / MTR 2006 630.2 FeI 70’’ UNNOFIT inversion Aug 18, NOAA 904 S13, W35

15. f B//f Bt Scatter plots Ca (MSDP) / Fe (MTR)

16. 630.2 FeI610.3 CaI 120’’ 160’’ 120’’ THEMIS/MSDP 2007 UNNOFIT inversion June 11, NOAA 10960 S05, W52

17. IQ/IU/IV/I 610.3 CaI

18. THEMIS MSDP

20. f B x f B y Similar B x and B y similar  angles

21. Bt 6103 < Bt 6302 - Gradients along line of sight ? - B t more sensitive than B// to line center, 6103 saturated NLTE line? - stray-light effects? - instrumental profile not included? - filling factor effects? - further simulations needed ….. - comparisons with MTR data (not yet reduced) Possible improvements: - Include instrumental profile - set-up 32 channels (2 cameras = effective increase of potential well) - better size of 6302 filter ! THEMIS MSDP

22. Scanning speed for targets 100’’x160’’ 9 mn

23. Weak field approximation Disk center, no rotation of B along LOS: Stokes U = 0 1 - Simple case: LTE, Milne Eddington, B vector and f independent of z V( ) ~ f B l dI/d Q( ) ~ f B t 2 d 2 I/d 2 For weak fields, line profile inversions provide only 2 quantities, f B l and f B t 2 3) Problems and plans: Gradients of B along LOS from NaD1, 610.3 Ca, 630.2 Fe, …

24. Below a given level of  2 the range of possible solutions is larger for (B transverse * f ) than for (B transverse * f ½ ) ? B t * fB t * f 1/2

25. 2 – NLTE, B function of z, f = 1, given solar model (parts of spots?) > Computation of response functions by MULTI code V( ) =  B l (z) R(,z) dz Q( ) =  B t 2 (z) R’(,z) dz ? > Formation altitudes: barycenters of response functions

26. B l = a + bz V( )=  R(,z) (a+bz) dz Choose functions with different weights at line-center and wings: S 1 =  V( ) w 1 ( ) d S 2 =  V( ) w2( ) d a, b Linear polarization: B t ??? B t 2 = a’+ b’z Q( )=  R’(,z) (a’+b’z) dz B t 2, dB t 2 /dz ??? integrations along line profile to optmize signal/noise ratio. Circular polarization: B l (0), dB l /dz > Vertical gradients: Instead of using individual points of the bisector, Examples: w 1 ( ) = +1 and -1 around line center, 0 elsewhere w 2 ( ) = +1 and -1 in line wings, 0 elsewhere > Application: comparisons between gradients from full profile of 1 line and 2 different lines

27. 3 – NLTE, B and f functions of z, given solar model (flux tubes?) Example: flux tubes, NaD1 line (section 1) V( ) =  R(,z) (a+bz) (  +  z) dz B l = a + bz f =  +  z MHD model necessary in case of weak fields (see section 1). In particular, when flux tubes are not spatially resolved, implies that gradients of f and B l are compensated (b/a ~ -  /  ) the assumption of constant flux f B l Both unknown quantities b and  are present in the coefficient of z Impossible to determine separately f and B l 4 – Possible extension of UNNOFIT to NLTE lines close to LTE ? Example: Analysis of depatures between UNNOFIT results and parameters used for synthetic profiles in case of 610.3 Ca …