Hale Collage. Spectropolarimetric Diagnostic Techniques!!!!!!!! Rebecca Centeno
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1 Hale Collage Spectropolarimetric Diagnostic Techniques!!!!!!!! Rebecca Centeno March 1-8, 2016
2 Degree of polarization The degree of polarization is defined as: p = Q2 + U 2 + V 2 I 2 In a purely absorptive medium (no emission): d I dτ = K I It can be demonstrated that: If η Q = η U = η V = 0 unpolarized light remains unpolarized.! If η I = constant, the degree of polarization, p, increases asymptotically. Absorption and dispersion processes are non-depolarizing.! Depolarizing effects can only appear through emission processes.
3 Today Symmetry properties of the Stokes profiles! Net Circular Polarization!! Longitudinal magnetograms!! Quiet Sun magnetic fields! Unsigned magnetic flux density! Effects of spatial resolution!! Shortcomings of the Zeeman effect!! Scattering polarization and the Hanle effect! Second solar spectrum
4 { Symmetry { properties of the Stokes profiles What happens to the propagation matrix when we do a! change of variable: (λ - λ0) (λ0 - λ)? K = η I η Q η U η V η Q η I ρ V ρ U η U ρ V η I ρ Q η V ρ U ρ Q η I K = η I η Q η U η V η Q η I ρ V ρ U η U ρ V η I ρ Q η V ρ U ρ Q η I Voigt profile: φ α = 1 π i S α,i H(u 0 + u B,α,i u LOS,a) central! wavelength Zeeman! shift LOS! velocity In the absence of velocity gradients, this transformation is a constant modification! of K regarding the symmetry of wavelengths with respect to (u 0 - u LOS ).
5 So the RTE transforms like this: Symmetry properties of the Stokes profiles d dτ c I(λ λ0 )=K( I(λ λ 0 ) S) d dτ c I(λ0 λ) =K ( I(λ 0 λ) S) I(λ 0 λ) =I(λ λ 0 ) Q(λ 0 λ) =Q(λ λ 0 ) U(λ 0 λ) =U(λ λ 0 ) V (λ 0 λ) = V (λ λ 0 ) Stokes I, Q and U are symmetric! with respect to their central! wavelength, while Stokes V is! antisymmetric. from Orozco Suarez et al, 2006
6 Net Circular Polarization Net circular polarization: integral of Stokes V over the wavelength! span (W) of the spectral line NCP = W V obs (λ)dλ It is zero in the absence of velocity gradients! blue - red = 0 λ
7 Net Circular Polarization B A fraction of the pixel is! non-magnetic: no contribution to! Stokes V resolution element + + = asymmetric A fraction of the pixel is magnetized, with different field strengths,! spatial extensions, plasma velocities If we assume no velocity gradients,! each Stokes V is still antisymmetric, but the sum might not. However, the total NCP = 0!
8 splitting should be negligible (e.g. Kurucz 1993); and LTE is a reasonable approximation for Fe (e.g. Shchukina & Trujillo Bueno 2000), at least for 1D model atmospheres of solar-type stars (3D NLTE studies of Fe may, however, reveal larger departures from NLTE as speculated by e.g. Nordlund 1985 and Kostik et al. 1996). By studying a large sample of neutral and ionized Fe lines with different atomic data and therefore line strengths, the solar photospheric convection at varying atmospheric layers can be probed by analysing the resulting line shifts and asymmetries. The Fe lines and their atomic data are the same as described in detail in Paper II. In particular, accurate laboratory wavelengths for the Fe i and Fe ii lines were taken from Nave et al. (1994) and Johansson (1998, private communication), respectively. The final profiles have been computed with the individual Fe abundances derived in Paper II. formation For in solar granulation. I observations, the solar FTS disk-center comparison with profiles bisectors intensity atlas by Brault &line Neckel (1987) (see also Neckel 1999) is has been used, due to its superior quality over the older Liege o atlas by Delbouille et al. (1973) in terms of wavelength calipe bration (Allende Prieto & Garcı a Lo pez 1998a,b) and contine- uum tracement. For flux profiles the solar atlas by Kurucz et d al. (1984) has been used, which is also based on FTS-spectra d with a similar spectral resolution as the disk-center atlas. The ne wavelengths for the observed profiles have been adjusted to reo- move the effects of the solar gravitational redshift (633 m s 1 ). ne All spatially averaged theoretical profiles have been convolved re with an instrumental profile to account for the finite spectral te resolution of the observed atlas. Since the atlas was acquired n with a Fourier Transform Spectrograph (FTS), the instrumental u- profile corresponds to a sinc-function with λ/ λ d in the visual rather than the normal Gaussian (e.g. Gray 1992). The additional instrumental broadening has only a minor (but Asplund et al 2000) er not entirely negligible) effect on the resulting line(from asymmetries Fig. 1. Spatially resolved profiles and bisectors for the Fe i line. 9) Line Bisectors and Convective Blueshift
9 Longitudinal magnetograms Based on the weak-field approximation: BLOS V (λ) = Cg eff B cos θ I(λ) λ Magnetograph signal as a! function of BLOS strength and! inclination angle for a spectral! line with geff = 2.5.!! From Landi Degl Innocenti & Landolfi! Polarization in spectral lines
10 Longitudinal magnetograms Only valid for weak fields (no effect on Stokes I)! Assumes magnetic field has no gradients along LOS! Only gives the component of the magnetic field along the LOS Yet line-of-sight magnetograms are by far the most popular! measurement of magnetic fields in Solar Physics. Magnetograms give us the net longitudinal component of! the magnetic field, Bz, averaged over each resolution element.
11 Quiet Sun magnetic fields HMI/SDO (IMaX - SUNRISE, courtesy of V. Martínez Pillet)
12 Unsigned magnetic flux density The unsigned magnetic flux density: φ B = A B z da A da B z = component of the magnetic field vector projected along the LOS. In general:! B z B radial
13 Effects of spatial resolution on Quiet Sun magnetic fields How do measurements of unsigned magnetic flux change with increasing spatial resolution? resolution element (from Sánchez Almeida &! Martínez González 2011)
14 Shortcomings of the Zeeman Effect The Zeeman Effect polarization signals cancel out when tangled magnetic fields are present at sub-pixel spatial scales. + = 0 Very weak magnetic fields do not produce measurable magnetic signals (when! the Zeeman splitting is much smaller than the width of the spectral line). vmac B = 100 G, θ = 0 deg
15 Shortcomings of the Zeeman Effect There is a 180 degree { azimuth ambiguity, in the plane } perpendicular to! the LOS. 2 η Q = η 0 2 η U = η { φ0 1 2 [φ +1 + φ 1 ]sin 2 θcos2φ } { φ0 1 2 [φ +1 + φ 1 ]sin 2 θsin2φ } η Anywhere but at disk center:!! The azimuth ambiguity in the LOS reference frame translates into an!! inclination (with respect to the solar radial direction) and an azimuth!! ambiguity in the local solar frame. { } When converting from LOS into a solar reference frame: Disambiguation methods are based on continuity and minimizing currents.
16 Mechanisms that produce polarization in spectral lines Anisotropy in the excitation mechanism of the atom!!! Impact polarization! Optical pumping!!! External field breaking the axis of symmetry! Electric field! Magnetic field
17 Atomic polarization 2 { η Q = η 0 2 w & w R w B 2 { φ0 1 2 [φ +1 + φ 1 ]sin 2 θcos2φ } { } } MU = 1 MU = 0 MU = -1 1/3 1/3 1/3 1/2 1/2 ML = 0 anisotropic radiation If the number of &-transitions does not compensate the number of σ-transitions per unit volume and time, we will get a linear polarization signal. Anisotropic illumination induces population imbalances between the magnetic energy sub-levels of an atom.
18 Atomic polarization y LOS Chromosphere:! non-frequent collisions! anisotropic illumination: center-to-limb variation (from Trujillo Bueno 2006)
19 Atomic polarization scattering! atoms linearly polarized! radiation unpolarized! radiation 90 degree! scattering scenario anisotropic radiation
20 Atomic polarization 2 { η Q = η 0 2 w & w R w B 2 { φ0 1 2 [φ +1 + φ 1 ]sin 2 θcos2φ } { } MU = 1 MU = 0 MU = -1 } B = 0 Stokes Q 1/3 1/3 1/3 1/2 1/2 ML = 0 Stokes U One lobe,! not three!! anisotropic radiation
21 Atomic polarization and the Hanle Effect B 0 and inclined with respect to the axis of symmetry of the radiation MU = 1 MU = 0 MU = -1 Stokes Q 1/3 1/3 1/3 1/2 1/2 ML = 0 Stokes U anisotropic radiation
22 Atomic polarization and the Hanle Effect (from Trujillo Bueno, 2006)
23 magnetic field inclined with respect to the symmetry axis of the pumping radiation Atomic polarization and the Hanle Effect field. The basic formula to estimate the magnetic field intensity, BH (measured in gauss), sufficient to produce a sizable change in the atomic level polarization results Hanlethe Effect can be sensitive very weak width magnetic fields, depending! from The equating Zeeman splitting withtothe natural (or inverse lifetime) of on thelevel spectral line (from milligauss to hectogauss). the energy under consideration: BH = /(tlife gj ), (4) where gj and tlife are, respectively, the Lande factor and the level s lifetime (in seconds), which can be to either the magnetic upper or the lower level of the It is also sensitive tangled fields at sub-pixel! chosen spectral line. scales, This formula provides a reliable estimation when radiative transitions so it doesn t cancel out as the Zeemanonly polarization! dominate completely signals would. the atomic excitation. If elastic and/or inelastic collisions are also efficient, then the critical field increases, since it turns out to be approximately given by (Trujillo Bueno, 2003a) Hanle techniques suffer from a saturation effect, so! 1magnetic + δ(1 ϵ)field strength! there is an upper limit forbthe BH, (5) 1 ϵ sensitivity. where δ = D/Aul quantifies the rate of elastic (depolarizing) collisions in units of the Einstein Aul -coefficient, and ϵ = Cul /(Cul + Aul ) is the probability that a deexcitation is caused by collisions (with It has event to be treated within the frame of! Cul the rate of inelastic collisional transitions betweentheory the upper level u and the lower level l ). The application the quantum of polarization. of this basic equation to the upper and lower levels of typical spectral lines shows that the Hanle effect may allow us to diagnose solar and stellar magnetic fields
24 The Second Solar Spectrum (Gandorfer, 2001)
25 Next time! Simple solutions to the RTE! Milne-Eddington approximation! LTE approximation!! The general inversion problem!! Spectral line inversion algorithms! Levenberg-Marquardt techniques! Principal Component Analysis
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