Faurobert decay of the atom then leads to linear polarization of the re-emitted radiation eld. This so-called scattering polarization is analoguous to

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1 The Last Total Solar Eclipse of the Millennium in Turkey ASP Conference Series, Vol , 1999 W. C. Livingston, and A. Ozguc, eds. Hanle eect of weak solar magnetic elds M. Faurobert Observatoire de la C^ote d'azur. Departement G.D. Cassini (CNRS/UMR 659). BP 49. F Nice Cedex 4. France Abstract. The Hanle eect provides a diagnostic tool for weak magnetic elds, which do not give rise to a measurable Zeeman eect, such as turbulent eldsormagnetic canopies in the chromosphere. The lines which are sensitive to the Hanle eect are formed under non-lte conditions by scattering of photons. After a brief description of the physical mechanism at hand, I will present some approximate expressions for the linear polarization of such lines in the presence of a weak magnetic eld and show how the Hanle eect may be used for the diagnostics of magnetic canopies. 1. Introduction The diagnostics of strong photospheric magnetic elds is mainly derived from measurements of the Zeeman polarization in spectral lines due to Zeeman splitting of the atomic levels. The sensitivity of this method is limited to magnetic elds stronger than a few hundred Gauss, for which Zeeman splitting is not too small as compared to the line Doppler width. Weaker elds, or mixed polarity elds do not give rise to measurable Zeeman eect. In the photosphere, there is more and more evidence that weak magnetic elds are present outside uxtubes, in the form of turbulent elds or intranetwork elds. The Hanle eect is an alternative tool for a direct detection of those elds (Faurobert-Scholl et al Steno et al. 1997). In the chromosphere, the expansion of uxtubes leads to a decrease of the magnetic eld strength and to bending of the eld lines. Such magnetic congurations of the canopy type are dicult to detect by their Zeeman eect. However their Hanle eect has been observed on lines such as the CaI 47 A and the SrII 4078A resonance lines (Bianda, Solanki & Steno 1998a Bianda, Steno & Solanki 1998b). More generally, there is a need for direct diagnostics of the magnetic eld in higher regions of the solar atmosphere, such as the high chromosphere or the transition region, where the magnetic eld is much weaker than in the photospheric layer. We now present briey the Hanle eect. Let us rst consider a spectral line formed in the absence of any magnetic eld by scattering of photons, i.e. by absorption immediately followed by re-emission of radiation. If the absorbed radiation eld is anisotropic, then the (degenerate) Zeeman sub-levels of the excited state have dierent populations and there are phase relationship among them. This phenomenon is referred to as atomic polarization. The radiative 1

2 Faurobert decay of the atom then leads to linear polarization of the re-emitted radiation eld. This so-called scattering polarization is analoguous to Rayleigh polarization. In the presence of a weak magnetic eld, if the Larmor frequency is of the order of the inverse of the upper-level radiative decay time, there is a small splitting of the Zeeman-sublevels, but they still partially overlap. Their phase relationships relax with time, this leads to a decrease of the linear polarization of the re-emitted radiation together with a rotation of its polarization plane. This is the so-called Hanle eect. Its eciency domain is dened by the condition that the Larmor frequency is of the order of the inverse life-time of the atomic levels. For lines in the visible domain this corresponds to magnetic elds on the order of a few Gauss to one hundred Gauss. This eect has already been used to determine the magnetic eld in solar prominences. A combination of He D3 and H emission lines was used (Leroy, Bommier & Sahal-Brechot 1984). Here we are interested in the diagnostics of magnetic elds in the solar atmosphere by means of the Hanle eect on absorption lines. Scattering polarization has been observed in the solar spectrum outside active regions by Steno, Keller, & Gandorfer (1998). They used a new generation very sensitive polarimeter, ZIMPOL, at Kitt Peak. They have discovered that the linear polarization spectrum is as rich as the intensity spectrum, but with completely dierent features, which still mostly remain to be interpreted. The more strongly polarized lines are strong resonance lines such as the CaI 47 A and SrII 4078 A, that we will consider in the following. Those two lines have also been observed at IRSOL (Locarno) by Bianda et al. (1998a b), the Hanle eect was detected as a rotation of the line core polarization which does not aect the wings. For such strong lines, formed under non-lte conditions, the interpretation of the observed polarization requires in principle the solution of a polarized radiative transfer equation which takes into account multiple scattering in the presence of a weak magnetic eld. We shall show that some approximate expressions may be used to get a rst order estimate of the line core polarization. We shall then derive a diagnostic method for magnetic canopies by making use of the observations in the CaI and SrII lines.. Approximate expressions for the line polarization in the presence of Hanle eect In the Stokes parameter formalism the radiation eld is described by a 3- component vector I ~ = (I Q U) y. Because of the presence of the magnetic eld the radiation eld is not axially-symetrical, it depends on the 4 variables ( '), as usual =cos where is the colatitude of the line of sight, ' is its azimuth, and are respectively the line optical depth and the frequency. One can show that in the presence of a weak magnetic eld the absorption matrix is scalar, and that the absorption coecient is the same as in the nonmagnetic case. The transfer equation for I ~ is d~ I d =(()+)(~ I ; ~ S) (1)

3 Hanle eect of weak solar magnetic elds 3 where is the absorption prole in the line and is the ratio of the line to the continuous absorption coecients, ~ S is a 3-component source vector which is the sum of a scattering term and of a thermal creation term Z ~S( ') =(1; ") Z d 0 ()d 4 ^P B ( ' 0 ' 0 ) ~ I( 0 ' 0 )+" ~ B () where ^PB is the Hanle phase matrix and B is the Planck function. Equation (??) may be solved numerically (see Nagendra, Frisch, & Faurobert- Scholl 1998, hereafter NFF, and Nagendra et al. 1999). However, quite simple approximate expressions for the line Stokes parameters may be derived. They allow to get some insight in the most important features of the Hanle eect and to investigate its diagnostics possibilities. These approximations are obtained from an azimuthal Fourier expansion of the radiation eld and a factorization of the source function, as described in Faurobert-Scholl (1991) and NFF. This leads to I( ') ' J I ( x ) (3) p 1 Q( ') ' W p 3(1 ; ) M + p q 3 1 ; [M 3 cos ' ; M 4 sin '] ; p ) 3 (1 + )[M 5 cos ' + M 6 sin '] J Q ( x ) (4) p q U( ') ' W ; 3(1 ; )[M 4 cos ' + M 3 sin '] + p 3 [M6 cos ' ; M 5 sin ']o J Q ( x ): (5) with x ' =(), W is a coecient which depends on the quantum numbers of the upper and lower levels of the transition. Here ' is the dierence in azimuth between the line of sight and the magnetic eld vector. J I which is essentially the source function for the intensity, is not aected by the magnetic eld, whereas J Q is a component of the polarized source function. It is not strongly aected by the magnetic eld, as shown in NFF. The main eects of the magnetic eld on the line Stokes parameters are thus described by the terms enclosed by curly braces. They depend on the direction of the line of sight and on the M i coecients which are given by M = 1 ; 3SB B 1+ r B 3 M 3 = ; [1 ; 3 B SB] C B BS B 1+B [1 ; 6 B SB]

4 4 Faurobert r 3 M 4 = ; S B B 1+B [1 ; 3 B SB] r 3 M 5 = S B B [1 ; 3 B 1+B CB] r 3 M 6 = S B C 3B 3 B (1 + B )(1 + 4 B In these expressions B is the Hanle eect parameter ): (6) B = Lg J A (7) where L = eb=4mc is the Larmor frequency of the electron in the magnetic eld, g J is the Lande factoroftheupperlevel and A is the sum of the radiative, inelastic and depolarizing collision rates (see e.g. Bommier 1996, Eq. (3)), and C B =cos B S B =sin B (8) where B is the colatitude, or inclinaison with respect to the vertical direction, of the magnetic eld. In the absence of a magnetic eld, or if the magnetic eld is vertical, M = 1, the other M i coecients vanish. We see from Eq. (??) that the intensity prole is almost not changed by the magnetic eld. The ratio U=Q is at rst order independent of W and of radiative transfer eects, because the source term J Q is eliminated out of this ratio. It is related to the Hanle rotation angle H by tg( H ) = U=Q. The Hanle depolarization is dened by depol =1; p Q + U q Q (9) 0 where Q 0 is the value of the Stokes parameter Q in the absence of magnetic eld. As the source term J Q is only weakly aected by the magnetic eld (see NFF), the Hanle depolarization does not depend, at rst order, on radiative transfer eects. Its dependence on the magnetic eld is mainly given by the expressions in curly braces in Eqs. (??)-(??). Let us now showhow the Hanle eect may be used to make the diagnostics of magnetic canopies in the low chromosphere. 3. Diagnostics of magnetic canopies In quiet regions of the Sun, the photospheric uxtubes expand on short scales and merge together. This gives rise to a magnetic layer where the magnetic eld is weak and almost horizontal. Such a magnetic conguration should produce a detectable Hanle eect on spectral lines which are formed in the low chromosphere. Actually, the rotation of the polarization plane has been detected in two strong resonance lines, namely the CaI 47 A and the SrII 4078 A lines which are formed in this region (see Bianda et al. 1998a b).

5 Hanle eect of weak solar magnetic elds 5 Figure 1. Largest value of the Hanle depolarization (dened in (??)) as a function of the Hanle parameter (dened in (??)). The dierent curves are obtained for dierent values of the inclination of the magnetic eld: asterisk : B = 90, diamond : B = 60, triangle : B =45, square : B =30. For the CaI line (respectively SrII line) the Hanle parameter B is of order unity for magnetic elds of the order of 5 Gauss (respectively 1 Gauss). For agiven magnetic eld, the ratio of the Hanle parameters in the two lines is B (CaI) =0:4: (10) B (SrII) The linear polarization in those two lines has been recorded using the polarimeter at IRSOL in Locarno. Several measurements were made for a given limb-distance, they show variable Hanle rotations and depolarizations. Here we interprete these observations by the fact that for dierent points on the solar disk, the azimuth of the line of sight varies with respect to the azimuth of the magnetic eld, whereas we assume that the magnetic intensity and inclination remain constant for all the observation points at a given limb-distance. This is consistent with a magnetic conguration of the canopy type all over the low chromosphere. Figure 1 shows the largest depolarization derived from (??) and (??) when ' varies for lines of sight at = 0:1, as a function of the Hanle parameter B, and various values of the inclination angle B. We observe the well known saturation of the Hanle depolarization when B becomes larger than about 3. The saturation value depends on the inclinaison of the magnetic eld. The maximum depolarization may be determined by systematic measurements of the linear polarization in the line for a given limb-distance all around the solar disk. We see on Fig. (1) that a given maximum depolarization observed in one

6 6 Faurobert line corresponds to several possible magnetic congurations. As an example, if the maximum depolarization is 0.7 in the SrII line (we take this value from the observations recorded by Bianda et al. 1998b), this corresponds either to an almost horizontal magnetic eld, with B ' 1, or to an inclined eld with a larger intensity, B =45 and B '. Measurements made in a second line, with a dierent sensitivity to the Hanle eect, may allow to distinguish between these two congurations. If the CaI line is used, then the horizontal conguration with B (SrII) = 1 should produce a maximum depolarization of 0.35 (because for the CaI line B ' 0:4), whereas the inclined conguration with B (SrII) = would produce a maximum depolarization of 0.45 ( B (CaI) ' 0:8) in the CaI line. 4. Conclusion This contribution is a rst attempt in the new area of Hanle diagnostics. We applied it to absorption lines formed in the low chromosphere. But the Hanle eect should also be quite well suited for the determination of weak elds in the higher chromosphere and in the transition region, where emission or absorption lines in the ultra-violet part of the spectrum are formed by scattering of photons. This is a completely unexplored eld because of the present lack of sensitive polarimeters operating in the UV domain. References Bianda, M., Solanki, S. K., & Steno, J.O. 1998a, A&A, 331, 760 Bianda, M., Steno, J.O., & Solanki S.K. 1998b, A&A, 337, 565 Bommier, V. 1996, Solar Phys., 164, 9 Faurobert-Scholl, M. 1991, A&A, 46, 469 Faurobert-Scholl, M., Feautrier, N., Machefert, F., Petrovay, K., & Spieledel, A. 1995, A&A, 98, 89 Leroy, J.L., Bommier, V., & Sahal-Brechot, S. 1984, A&A, 131, 33 Nagendra, K.N, Frisch, H., & Faurobert-Scholl, M. 1998, A&A, 33, 610 Nagendra, K.N., Paletou, F., Frisch, H., & Faurobert-Scholl, M. 1999, in Solar Polarization, Proc. nd SPW, eds K.N. Nagendra and J.O. Steno (Kluwer, Dordrecht), 17 Steno, J.O., Bianda, M., Keller, C.U., & Solanki, S.K. 1997, A&A, Steno, J.O., Keller, C.U., & Gandorfer, A. 1998, A&A, 39, 319

Vector Magnetic Field Diagnostics using Hanle Effect

Meeting on Science with Planned and Upcoming Solar Facilities in the Country, Bangalore, India, November 2-3, 2011 Vector Magnetic Field Diagnostics using Hanle Effect M. Sampoorna Indian Institute of