Solar Magnetometry

1. Solar Magnetometry J. Sánchez Almeida Instituto de Astrofísica de Canarias Magnetometry: set of techniques and procedures to determine the physical properties of a magnetized plasma(magnetic field and more ...)

2. Main Constraints: • No in-situ measurements are possible; inferences have to be based on interpreting properties of the light. • Interpretation not straightforward. The resolution elements of the observations are far larger than the magnetic structures (or sub-structure) Needed Tools: • Radiative transfer for polarized light • Instrumentation: telescopes and polarimeters • Inversion techniques (interpreting the polarization through many simplifying assumptions)

3. Purpose: • To give an overview of all ingredients that must be considered, and to illustrate the techniques with examples taken form recent research. • It is not a review since part of the techniques used at present are not covered (not even mentioned). Explicitly • Devoted to the magnetometry of the photosphere. • No proxi-magnetometry (jargon for magnetic field measurements which are no based on polarization) • No extrapolations of photospheric magnetic fields to the Corona) • No in-situ measurements (solar wind)

4. Summary – Index (1): Radiative Transfer for Polarized Radiation. • Stokes parameters, Jones parameters, Mueller matrixes and Jones matrixes • Equation of radiative transfer for polarized light • Zeeman effect • Selected properties of the Stokes profiles, ME solutions, etc. Instrumentation: • Polarimeters, including magnetographs • Instrumental Polarization Inversion Techniques: • General ingredients • Examples, including the magnetograph equation Examples of Solar Magnetometry: • Kitt Peak Synoptic maps • Line ratio method • Broad Band Circular Polarization of Sunspots • Quiet Sun Magnetic fields

5. Summary – Index (2): Advanced Solar magnetometry. • Hanle effect based magnetometry • Magnetometry based on lines with hyperfine structure • He 1083nm chromospheric magnetometry • Polarimeters on board Hinode goto end

7. Radiative Transfer for Polarized Light Stokes parameters, Jones parameters, Mueller Matrixes and Jones Matrixes • The light emitted by a point source is a plane wave • Monochromatic implies that the EM fields describe elliptical motions in a plane • The plane is quasi-perpendicular to the direction of propagation • Quasi monochromatic implies that the ellipse changes shape with time

8. y x Quasi-monochromatic means that the ellipse change with time

9. t 1/t ex(t) time (t) Frequency (1/t) w/2p 2p/w = 10-15 s, in the visible (5000 A) t : coherency time, for which the ellipse keeps a shape • t = 10-8 s, electric dipole transition in the visible • t = 5 x 10-10 s, (multimode) He-Ne Laser • t = 5 x 10-10 s, high resolution spectra (Dl/l=200000) Integration time of the measuremengts: 1 s (<< t << 2p/w), ellipse changes shape some 108 -109 times during the measurement

10. Jones Vector, complex amplitude of the electric field in the plane perpendicular to the Line-of-Sight (LOS). It completely describes the radiation field, including its polarization. Consider the effect of an optical system on the light. It just transfoms Most known optical systems are linear (from a polarizer sheet to a magnetized atmosphere) Jones Matrix (Complex 2x2 matrix)

11. The polarization of the light can be determined using intensity detectors (CCDs, photomultipliers, etc.) plus linear optical systems. (T: integration time of the measurement)

12. Stokes Parameters, that completely characterize the properties of the light from an observational point of view is the complex conjugate of describes the properties of the optical system

13. (Some) Properties of the Stokes Parameters • Two beams with the same Stokes parameters cannot be distinguished • Which kind of polarization is coded in each Stokes parameter? • The Stokes parameters of a beam the combines two independent beams is the sum of the Stokes parameters of the two beams • Any polarization can be decomposed as the incoherent superposition of two fully polarized beams with opposite polarization states • A global change of phase of the EM field does not modify the Stokes parameters

14. (Some) Properties of the Linear Optical Systems • Only seven parameters characterize the change of polarization produced by any optical system. A Jones matrix is characterized by 4 complex numbers (8 parameters) minus an irrelevant global phase. • The modification of the Stokes parameters produced by one of these systems is linear Stokes vector Mueller Matrix

15. The Mueller matrix contain redundant information. It has 16 elements, but only seven of them are independent. The relationships bewteen the elements are not trivial, though. • The Mueller matrix becomes very simple if the optical element is weakly polarizing, i.e., if with then

16. - Mueller Matrix for an optical system producing selective absorption Stokes Vector de type of absorbed light Change of amplitude produced by the selective OS

17. Example: linear polarizer transmitting the vibrations in the x-axis Then for unpolarized input light one ends up with

18. - Mueller Matrixfor an optical system producingselective retardance Stokes Vector de type of polarization that is retarded Change of phase produced by the selective OS

19. - The Mueller matrix of a series of optical systems is the product of the individual matrixes. The order does matter if the chain is formed by weakly polarizing optical systems, then the order of the different elements is irrelevant

20. S+DS S line-of-sight observer Dz layer of atmosphere Equation of Radiative Transfer for Polarized Light Emission produced by the layer Mueller matrix of i-th process changing the polarization

21. change of amplitude change of phase Stokes vector of the selective absorption + retardance

23. Emission term ? Simple assuming emitted radiation field is in LTE (Local Thermodynamic Equilibrium). In TE and with B the Planck function then

24. Radiative transfer equation for polarized light in any atmosphere whose emission is produced in LTE

25. Example: linear polarizer transmitting the vibrations in the x-axis There is just one i which absorbs and no emission (B=0)

26. Typical Mueller matrix of a linear polarizer

27. Zeeman Effect Purpose: work out the h´s and r´s in the absorption matrix in the case of amagnetized atmosphere Work out contributions to the change of polarization due to: 1) Spectral line absorption Assumptions: • Electric dipole transitions • Hydrogen-like atoms • Linear Zeeman effect 2) Continuum absorption

28. Spectral line absorption The wave function characterizing eigenstate of theses Hydrogen-like atoms can be written down as where M is the magnetic quantum number and E is the energy of the level. The electric dipole of the corresponding distribution of charges will be

29. When you have a transition between states b (initial) and f (final), the wave function is a linear combination of the two states constant over the period of the wave

30. Which leads to the selection rules for E-dipole transitions each one associated with a polarization

31. observer z x There are only three types of polarization We are interested in the projection in the plane perpendicular to the line of sight (x-y plane) y y a) For DM=0 x

32. y x 1 b) For DM=Mb- Mf=+1 b) For DM=Mb- Mf=-1 y x 1

33. B=0 B=B0 w0 w0 w w w0+Dw w0+Dw w0 w0 If the atom is in a magnetized atmosphere, the energy of each Zeeman sublevel is different, which produces a change of resonance frequency of the transitions between sublevels depending on DM, Dw Associated to each transition there is a absorption profile plus a retardance profile

34. y x DM=+1 x y w0+Dw w0+Dw w0 w0 In short: for an electric dipole atomic transition, only three kinds of polarizations can be absorbed. They just depend on DM (with M the difference of magnetic quantum numbers between the lower and the upper levels) DM=0 observer y x y DM=-1 x absorption retardance

35. Continuum Absorption Although, no details will be given, it is not difficult to show that the continuum absorption has a characteristic polarization for selective absorption of the order of (Kemp 1970), • For the solar magnetic fields (1kG magnetic field strengths), the continuum absorption is unpolarizedunless you measure degrees of polarization of the order of 10-5. • In white dwarfs, B ~ 106 G, leading to large continuum polarization (~ 1%)

36. Radiative Transfer Equation in a Magnetized Atmosphere The equation is generated considering the four types of polarization that are possible same for ´s with replacing ´s with ´s

37. observer y x Unno-Rachkovsky Equations

38. Zeeman triplet

39. general Zeeman pattern

40. effect of a change of macroscopic velocity

41. effect of a change of magnetic field strength

42. weak magnetic field strength regime

44. Selected Properties of the Stokes Profiles Stokes Profiles representation of the four stokes parameters as a function of wavelength within a spectral line

45. Stokes Profiles

46. 1.- Symmetry with respect to the central (laboratory) wavelength of the spectral line. If the macroscopic velocity is constant along the atmosphere, then I() = I(- ) Q() = Q(- ) U() = U(- ) V() = -V(- )   wavelength - laboratory wavelength of the spectral line corrected by the macroscopic velocity No proof given, but it follows from the symmetry properties of the ´s and ´sof the absorption matrix these symmetries disappear  the velocity varies within the resolution elements (asymmetries of the Stokes profiles)

47. Symmetries and asymmetries Stokes Profiles

48. 2.- Weak Magnetic Field Approximation, the width of the absorption and retardance coefficients of the various Zeeman components are much smaller than their Zeeman splittings if    is the Zeeman splitting of a Zeeman triplet, and D is the width of the line, it can be shown that (e.g., Landi + Landi 1973) then to first order in ( / D)

49. (a) Since there is no polarization at the bottom of the atmosphere (b)

50. I-V I+V 2 cos  B  I+V and I-V follow to equations that are identical to the equation for unpolarized light except that the absorption is shifted by  cos  B If the longitudinal component of the magnetic field is constant then cos  B is constant and I+V and I-V are identical except for a shift