1. 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): – 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. – Polarimeters, including magnetographs – Instrumental Polarization Instrumentation: Radiative Transfer for Polarized Radiation. 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): – Hanle effect based magnetometry – Magnetometry based on lines with hyperfine structure – He 1083nm chromospheric magnetometry – Polarimeters on board Hinode Advanced Solar magnetometry. goto end

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

9. time (t) e x (t) Frequency (1/t)  w/2  1/  2  /w = 10 -15 s, in the visible (5000 A)  : coherency time, for which the ellipse keeps a shape  = 10 -8 s, electric dipole transition in the visible  = 5 x 10 -10 s, (multimode) He-Ne Laser  = 5 x 10 -10 s, high resolution spectra (  Integration time of the measuremengts: 1 s (<<  << 2  /w), ellipse changes shape some 10 8 -10 9 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. (T: integration time of the measurement ) The polarization of the light can be determined using intensity detectors (CCDs, photomultipliers, etc.) plus linear optical systems.

12. is the complex conjugate of Stokes Parameters, that completely characterize the properties of the light from an observational point of view 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. linear polarizer transmitting the vibrations in the x-axis Then for unpolarized input light one ends up with

18. - Mueller Matrix for an optical system producing selective 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. Equation of Radiative Transfer for Polarized Light zz observer line-of-sight layer of atmosphere S+  S S 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. 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  ´s and  ´s in the absorption matrix in the case of a magnetized atmosphere – Electric dipole transitions – Hydrogen-like atoms – Linear Zeeman effect Work out contributions to the change of polarization due to: 2) Continuum absorption 1) Spectral line absorption Assumptions:

28. 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 Spectral line absorption

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

32. b) For  M=M b - M f =+1 b) For  M=M b - M f =-1 1 x y 1 x y

33. 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  M, w w0w0 B=0 w w0w0 B=B 0 ww Associated to each transition there is a absorption profile plus a retardance profile w0w0 w 0  w w0w0

34. In short: for an electric dipole atomic transition, only three kinds of polarizations can be absorbed. They just depend on  M (with M the difference of magnetic quantum numbers between the lower and the upper levels) x observer y x y  M=0 x y  M=-1 w0w0 w 0  w x y  M=+1 w0w0 w 0  w 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 unpolarized unless you measure degrees of polarization of the order of 10 -5. – In white dwarfs, B ~ 10 6 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. x observer y 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  ´s of 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. Since there is no polarization at the bottom of the atmosphere (a) (b)

50. 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 I-V I+V 2 cos   B

51. Magnetograph equation : the Stokes V signal is proportional to the longitudinal component of the magnetic field V cos   B 0 observer

52. The previous argumentation is based on the assumption that the Zeeman pattern is a triplet (one  component, one  + component and one  - component). If the pattern is more complex but the magnetic field is weak, one can repeat the argumentation to show that everything remains the same except that the full Zeeman pattern has to be replaced by a equivalent Zeeman triplet whose splitting is Is the so-called effective Landé factor, and it equals one for the classical Zeeman effect

53. 4.- Stokes profiles of an spatially unresolved magnetic structure (2-component magnetic atmosphere). resolution element non-magnetic magnetic filling factor, i.e., fraction of resolution element filled by magnetic fields

54. Effect on the magnetograph equation observer Magnetic flux density

55. 4.- Milne-Eddington solution of the Radiative Transfer Equation for Polarized Light (RTEPL). Assumptions: all those needed to get an analytic solution of the of the radiative transfer equations for polarized light Importance: Used for measuring magnetic field properties RTEPL: first order linear differential equation. Admits an analytic formal solution of the coefficients are constant (basic maths)

56. continuum optical depth Compact form of the RTEPL

57. Assumptions: the ratio line to continuum absorption coefficient does not depend on optical depth The source function depends linearly on continuum optical depth Broadening of the line constant (both Doppler and damping) Magnetic field vector constant with depth … all them together lead to constant absorption matrix

59. Milne-Eddington solutions of the RTEPL (e.g., Landi Degl´Innocenti, 1992) Free parameters: 1.Magnetic field strength 2.Magnetic field azimuth 3.Magnetic field inclination 4.B 0 5.B 1 6.Macroscopic velocity 7.Doppler broadening 8.Damping 9.Strength of the spectral line IDL

60. 5.- 180 o azimuth ambiguity (exact) x observer y x y These two magnetic fields produce the same polarization, therefore, one cannot distinguish them from the polarization that they generate. IDL

61. 6.- Stokes V reverses sign upon changing the sign of the magnetic field component along the line-of-sight (approximate). x observer y x y IDL

62. 7.- Q=U = 0 for longitudinal magnetic fields. V=0 for transverse magnetic fields. (Approximate.) x observer y x y IDL Q=U=0 V=0

64. Polarimeters – Modulation package – Intensity detector – Calibration package – Instrumental polarization Basic elements:

65. opticsmodulator (p j ) calibration optics telescope + optics optics Intensity detector

66. Modulation package Optical system whose Mueller matrix can be (strongly) varied upon changing a set of control parameters. Usually the last element is an optical element that fixes the polarization state of the exit beam, but this is not always the case. Example fixed linear polarizer rotating retarder ( /4) 

67. Intensity detector for example a CCD Calibration package Optical system whose exit polarization is known. It allows to determine the (linear) relationship bewteen the intensities measured by the intensity detector and the input polarization.  Example fixed linear polarizer rotating retarder ( /4)

68. Instrumental Polarization Ideally, one would like to place calibration optics in front of the optical system used to measure, including the telescope. Unfortunately, this is not possible (there are not high precision polarization optics with the size of a telescope). This causes that the solar polarization is modified (by the telescope etc.) before we can calibrate the system: instrumental polarization. It is an important effect (mostly) produced by oblique reflections (e.g. folding mirrors, and windows (stress induced birefringence of the vacuum windows)

69. GCT Obs. Teide SPh, 134, 1

70. Techniques to overcome the instrumental polarization a) carring out the analysis (the calibration) in the optical axis of the telescope (before the optical system loses axi- symmetry). Specially designed telescopes like THEMIS (Obs. Teide).

71. b) modeling (and correcting for) the Mueller matrix of the telescope. The theoretical expression for the Mueller matrixes of all individual optical elements forming the telescope are known (given the geometry the light path, complex refractive indexes of the mirrors, specific retardances of the windows, and the like). It is possible to write down a theoretical Mueller matrix than can be confronted with observations. One can use this Mueller matrix to correct the measurements

72. Instrumental Polarization: removing I  V crosstalk since at continuum wavelengths V=0

73. CCD (longitudinal) Magnetograph 2 states modulator /4-plate + linear polarizer Narrow- band color filter

74. Magnetogram : just an image of Stokes V in the wing of a spectral line.

75. Order of magnitude of the degree of polarization to be expected in the various solar magnetic structures (for a typical photospheric line used in magnetic studies):

76. Instrumental Polarization: Seeing Induced Crosstalk Important bias of any high angular resolution observation, although it is easy to explain in magnetograph observations. If the two images whose difference should render Stokes V are not taken strictly simultaneously (within a few ms, the time scale that characterizes atmospheric turbulence variations) then Stokes I  Stokes V (Lites 1987)

77. Seeing Induced Crosstalk How to solve the problem? 1.Using high frequency modulation, so that the atmosphere is frozen during a modulation cycle. (ZIMPOL like.) 2.Using simultaneous spatio-temporal modulation. Preferred technique in ground based observations. 3.Applying image restoration before demodulation. (SST approach.) 4.Going to space (e.g. Hinode), but then you have jitter from the satellite.

79. Techniques to deduce physical properties of the magnetic atmosphere upon the interpretation of the polarization that it produces. Ingredients:  model atmosphere (assumptions on the properties of atmosphere whose magnetic field will be inferred)  polarized spectral synthesis code  fitting technique (e.g.,  2 minimization techniques) All solar magnetic fields measurements (magnetometry) need, and are based on, these ingredients and assumptions. Frequently the assumptions are implicit and people tend to think that they do not exit. The inferred magnetic field depends, sometimes drastically, on the asumptions.

80. Longitudinal magnetograph It is just an image showing the degree of circular polarization in the flank of spectral line. – If the solar atmosphere where the polarization is produced has a discrete number of magnetic component – If the magnetic field of this component does not vary, neither along the line-of-sight nor across the line-of-sight – If the temperature and pressure of the atmosphere does not depend on the magnetic field – If the velocities is constant in the resolution element Model atmosphere: Synthesis Code: – Multi component atmosphere – Weak magnetic field approximation Fitting technique: – No sophistication; one observable and one free parameter

81. A calibrated magnetograph gives the longitudinal component of the magnetic flux density (mag flux per unit surface)

82. Milne-Eddington fitting technique – If the solar atmosphere where the polarization is produced has two components: one magnetic and one non-magnetic –If the magnetic field of this component does not vary, neither along the line-of-sight nor across the line-of-sight – If the line to continuum absorption coefficient ratio does not vary with height in the atmosphere – If the source function varies linearly with continuum optical depth Model atmosphere: Synthesis Code: – Milne Eddington analytic solution of the radiative transfer equations for polarized light Fitting technique: – Non-linear least squares minimization (e.g. Skumanich & Lites 1987)

83. Input model atmosphere B, , ,... new atmosphere B, , ,... giving a smaller  2 synthesis Observed I,Q,U & V  2 small enough?  2 minimization B, , ,... NO YES observed B, , 

84. Sunspot observation Skumanich & Lites 1987

85. MISMA inversion code – complex, having many different magnetic fields, velocities, temperatures, etc. Model atmosphere: Synthesis Code: – numerical solution of the radiative transfer equations for polarized light Fitting technique: – Non-linear least squares minimization

86. Observations Synthetic

88. PCA inversions (PCA: principal component analysis) Important, since they are extremely fast, and so, they are bound to become popular in the next future. For example, they may allow to process, on line, the huge data flux produced by the new synoptic magnetographs (e.g., SOLIS, see http://solis.nso.edu) It belongs to the class of Prêt-à-porter inversions as opposed to the classical Taylor-made inversions.

89. Which synthetic profiles are closest to the observed profiles? If # i are the closest ones then Pre-computed data base Prêt-à-porter inversions Obser ved I,Q,U & V

90. Eigenfaces Reconstructed faces Rees et al., 2000 # of eigenfaces used in the reconstruction Fitting technique for PCA:

91. Rees et al. (2000) Only a few eigenvalues are needed to characterize the Stokes profiles

92. Forward modeling (which is an inversion technique!!!) – Resulting from the solutions of the MHD equations under ´realistic´ solar conditions. Model atmosphere: Synthesis Code: – numerical solution of the radiative transfer equations for polarized light Fitting technique: – Not well defined (yet?) The synthetic spectra have to reproduce the observed spectra in some statistical sense.

93. Turbulent Dynamo Simulations by Cattaneo & Emonet

94. 1´´ seeing cluster analysis classification

95. The case of the large magnetic flux concentrations Observed

96. ¨A measurement process is regarded as precise if the dispersion of values is regarded as small. A measurement process is regarded as accurate if the values cluster closely about the correct value¨ (definition; e.g., Cameron 1960) Caveats to keep in mind: – The simplest the model atmosphere in which the inversion code is based, the higher the precision of the measurement (e.g., no problems of uniqueness in magnetographic observations). – However precision is not the aim of solar magnetometry; accuracy is more important since it is more difficult to achieve. – It makes no sense oversimplifying the model atmospheres to end up with magnetic field determinations that are very precise but very inaccurate.

98. Applications of the tools and techniques developed in the notes to specific problems of solar physics.

99. Understanding Real Magnetograms, e.g., Kitt Peak Synoptic Maps README_1

100. Jones et al., 1992, Solar Phys. 139, 211 README_2 Coelostat  Instrumental polarization Noise  7G

101. Line Ratio Method, or the field strength of the network magnetic concentrations The network magnetic concentrations have very low flux density (say, less than 100 G) but a large magnetic field strength similar to that of sunspots (larger than 1 kG). network This fact is known thanks to the so-called line- ratio method (Stenflo 1973) Pre-line-ratio-method situation (late 60´s and early 70´s): magnetograms of a network region taken using different spectral lines showed inconsistent results.

102. This is due to the fact that in network regions the magnetograph equation is not valid, implying network magnetic field strength of kG even though the magnetograms show a flux density of a few hundred G. Stenflo took simultaneous magnetograms in two selected lines, Fe I 5247 (g eff =2.) Fe I 5250 (g eff =3.) These two lines are almost identical if there no magnetic field in the atmoshere (same log(gf) same excitation potential, same element and ionization state), however, they have (very) different magnetic sensitivity.

103. If weak field (sub-kG): If strong field (sub-kG):

104. Line ratio obseved in network Fe I 5247 Fe I 5250 resolution element

105. Broad Band Circular Polarization of Sunspots (BBCP) Clues on the fine-scale structure of the Sunspot´s magnetic fields Observational facts: – Sunspots produce (large) Broad-Band circular polarization ( V/I  10 -3,Illing et al. 1974a,b) – It is produced by the individual spectral lines in the band-pass (i.e., it is not continuum polarization: Makita 1986) – It is maximum produced in to the so-called neutral line, where the magnetic field is supposed to be perpendicular to the line-of- sight. (Makita 1986.) – In the neutral line Stokes V is never zero but shows the cross- over effect IDL

106. Broad Band Imaging - Polarimetry

107. neutral line Sun sunspot neutral line solar limb  solar center  we

108. typical resolution element a) The BBCP is produced by gradients along the line-of-sight, i.e., the magnetic field, velocity etc. change in the sunspot over scales of less than 150 km, i.e., much smaller than the resolution element of typical observations (1” or 1000 km). Why?

109. b) it is produced by gradients of inclination along the LOS. They are present since Stokes V is never zero in the neutral line (i.e., there is no point where the magnetic field is perpendicular to the line-of-sight). SA & Lites, 1992, ApJ, 398, 359 Cross-over effect, Grigorjev and Kart, 1972, SPh, 22, 119 Stokes V

110. 150 km 750 km Resolution element c) The BBCP cannot be due to smooth well-organized vertical variations of magnetic fields inclination.

111. Sanchez Almeida (2005)

112. 150 km 750 km Resolution element The BBCP has to be due to very intermitent variations of magnetic field inclinations. This is a general feature of the magnetic fields in the penumbrae of sunspots that is inferred from the (careful) interpretation of the circular polarization that it produces, despite the fact that we do not resolve the fine-scale structuring of the magnetic field

113. Quiet Sun Magnetic Fields Cancellation of polarization signals in complex (tangled) magnetic fields Q 2 = -Q 1  Q 1 +Q 2 = Q obs = 0  V 2 = -V 1  V 1 +V 2 = V obs = 0

114. This kind of cancellation seems to take place in the quiet Sun Size of a Network cell (25000 km)

115. Turbulent Dynamo Simulations by Cattaneo & Emonet

116. Effect of insufficient angular resolution 1” seeing original

117. Variation of the Flux Density in the simulations with the angular resolution and the sensitivity of the synthetic magnetograms.

118. 1”x1” Domínguez Cerdeña et al. (03) Inter-Network Quiet Sun  angular resolution mag.  0.5”  sensitivity  20 G  VTT (obs. Teide), speckle reconstructed  Unsigned flux density  20 G

119. 1.6 G 12 G 12 G x SolarSurface = 7x10 23 Mx = solarflux@max Rabin et. al. 2001

120. How can we measure the properties of the quiet Sun magnetic fields? Need to use inversion techniques whose model atmospheres allow for the complications that the quiet Sun field has:  Different polarities in the resolution element (different magnetic field inclinations in the resolution element)  Different magnetic field strength in the resolution element  … Quite Sun fields: matter of active research

122. Techniques and methods employed in the recent literature on solar magnetometry. Used by specialist groups. Model dependent but with substantial potential. No realistic inversion techniques exist so far. – Hanle effect based magnetometry – Magnetometry based on lines with hyperfine structure – He 1083nm chromospheric magnetometry – Polarimeters on board Hinode

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

124. A weak magnetic field splits the Zeeman sublevels but … it is weaker than the natural width of the lines. The eigenstates involved in the transition are not pure states but combinations of them …Various frequencies are excited at the same time, and they add coherently. w0-ww0-w w 0  w Hanle Effect Based Magnetometry

125. In the case that two eigenstates contribute to the dipolar emergent radiation, the resulting electric dipole is. 1.Since non-monochromatic, the radiation is always partly polarized (Hanle effect is said to depolarize) 2.Modifies the state of polarization with respect to the case Δw=0 (Hanle effect rotates the plane of polarization.) 3.Purely non-LTE effect, since the integration of many atoms emitting at random times lead to the incoherent superposition of the two polarization states U 1 and U 2, and have no effect. In the coherency matrix representation,

126. Textbook case: describes linearly polarized in the x axis at t=0. unpolarized coherency time

127. We atom Sun non-magnetic scattering Hanle effect For Hanle effect to depend on the field strength (and so to be a useful tool), Hanle signals even if tangled fields

128. Faurobert et al. (2001) observed Sr I 4607Å Hanle depolarization Hanle saturation at some 50 G depolarizing collisions are critical for a proper modeling

129. general Zeeman pattern Magnetometry Based on Lines With Hyperfine

130. Magnetometry Based on Lines With Hyperfine Structure Hyperfine Structure: due to the interaction between the electron angular momentum and the nuclear angular momentum. Old theory by Landi Degl’Innocenti (1975), but recently recovered and used for actual observations by López Ariste et al. (2002, ApJ, 580, 519). What would be a single line becomes a blend of lines. They now undergo regular Zeeman effect, with their π and σ± components. Hundreds of components show up. When the HFS splitting and the Zeeman splitting become comparable, Zeeman pattern depends on the magnetic field strength (it is not the independent superposition of the Zeeman patterns of the independent components).

131. σ Landi Degl’Innocenti (1975) π López Ariste et al. (2002) Stokes V changes shape when the field is several hundred G … good diagnostic tool for hG field strengths.

132. Despite the apparent complexity, the HFS patterns present several regularities (Landi Deg’Innocenti 1975) π and σ components are normalized to one (there is no net circular polarization). When the magnetic field is weak enough, the Stokes V signal follow the weak magnetic field approximation. The centers of gravity of the π and σ components is independent of the HFS.

133. He I 1083nm Chromospheric Magnetometry The need for a simple but quantitative diagnostic of upper chromospheric magnetic fields is keenly felt (Rüedi et al. 1995, 293, 252). Popular in chromospheric magnetometry. It is a bend of 3 He I lines sharing the same lower level (19.79 ev). Optically thin. Bend modeled using ME profiles given line strengths and Zeeman splittings. Need incomplete Pashen-Back effect to carry out the calculations. Entirely formed in the chromosphere in standard 1D model atmospheres (Fontenla et al. 1993). Formed by recombination.

134. Rüedi et al. (1995) Incomplete Pashen-Back effect required for a proper analysis (Socas-Navarro et al. 2004) blend of 3 lines ME fit Creates NCP by saturation

135. Hinode, satellite ideal for polarimetry. 50 cm diffraction limited optical telescope (λ/D~0.26’’ @ 6302 Å) Polarimeters on board Hinode Open data policy! Every one is welcome to use them Launched, end of 2006 Japanese (ISAS), in cooperation with US (NASA) and Europe (PPARC, ESA). Hinode European Data Center here in Oslo.

136. SOT- SP SOT- FG SOT: Solar Optical Telescope

139. GOTO Summary -- Index: Summary

140. Selected references ref_magnetometry.pdf Exercises on solar magnetometry

142. Sutterlin et al, 1999, DOT, G-band, speckle reconstructed

143. Volume averaged in one pixel of a typical photospheric observation The cartoon shows the right scale for the horizontal and vertical smearing

144. SST, Scharmer et al. 2002 0.12 arcsec, spatial resolution 1´´ x 1´´

145. Point Source A B Observers A and B receive exactly the same signal, which is constant in the plane perpendicular to

146. Monochromatic means plane Elliptical Motion

147. Inserting monochromatic solutions of the kind into the wave equation derived from the Maxwell equations, one finds Component in the direction of Transverse component Characteristic scale for the variation of Wavelength

148. x y If J x (t) and J y (t) vary at random, then the light Unpolarized Light p is the degree of polarization> p=0 represents unpolarized light > p=1 corresponds to fully polarized light x y Monochromatic wave In general

149. x y x y x y x y x y x y

150. (because the two beams are incoherent)

151. x y Decomposition of any polarization in two fully polarized beams

152. x y The Jones vectors of these two beams are orthogonal

153. From Jones matrix to Mueller matrix

154. For any selective absorption, this set is a base of complex 2D vectors (e.g., the Jones vector) For any polarization with Jones vector The OS just changes the Jones vector as

156.  weak magnetic field approximation Zeeman shift

157. Band-pass of typical magnetogram observations

158. continuum

160. References Kemp 1970, ApJ, 162, 169, in connection with the continuumpolarization in a magnetic field Sanchez Almeida