Evaluating the reliability of a simple method to map the magnetic field azimuth in the solar chromosphere
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Evaluating the reliability of a simple method to mapthe magnetic field azimuth in the solar chromosphere
Jan Jurˇc´ak, Jiˇr´ı ˇStˇep´an, and Javier Trujillo Bueno Astronomical Institute of the Czech Academy of SciencesFriˇcova 298, 25165 Ondˇrejov, Czech Republic Instituto de Astrof´ısica de CanariasE-38205 La Laguna, Tenerife, Spain Departamento de Astrof´ısica, Facultad de F´ısica, Universidad de La LagunaE-38206 La Laguna, Tenerife, Spain Consejo Superior de Investigaciones Cient´ıficas, Spain (Received XXXX; Revised XXXX; Accepted XXXX)
Submitted to ApJABSTRACTThe Zeeman effect is of limited utility for probing the magnetism of the quiet solar chromosphere.The Hanle effect in some spectral lines is sensitive to such magnetism, but the interpretation of thescattering polarization signals requires taking into account that the chromospheric plasma is highlyinhomogeneous and dynamic (i.e., that the magnetic field is not the only cause of symmetry breaking).Here we investigate the reliability of a well-known formula for mapping the azimuth of chromosphericmagnetic fields directly from the scattering polarization observed in the Ca II II Keywords:
Sun: chromosphere — techniques: polarimetric — methods: data analysis INTRODUCTIONWith the upcoming new generation of solar tele-scopes, like DKIST (presently in the commissioningphase, Rimmele et al. 2020) and EST (presently in thepreparatory phase, Jurˇc´ak et al. 2019), there is an ur-gent need for suitable methods to infer the magneticfield information from the unprecedented spectropolari-metric data that these telescopes will provide. In par-ticular, reliable diagnostic methods are important for
Corresponding author: Jan Jurˇc´[email protected] the solar chromosphere, a highly inhomogeneous anddynamic atmospheric region where there are multipleeffects that significantly complicate the development ofreliable inversion methods. The present paper studiesquantitatively the possibility of obtaining realistic mapsof the magnetic field azimuth in the solar chromospheredirectly from the scattering polarization observed in theCa II I ), linear po-larization ( Q and U ) and circular polarization ( V ) inspectral lines –that is, we need to measure the Stokesprofiles I ( λ ), Q ( λ ), U ( λ ) and V ( λ ) as as function ofwavelength (e.g., Stenflo 1994; del Toro Iniesta 2003). Jurˇc´ak et al.
While the linear polarization of the radiation is mostcommonly described in terms of the Stokes parameters Q and U , an equivalent description is provided by thetotal linear polarization, P L = p Q + U , and the az-imuth χ of the linear polarization orientation with re-spect to a suitable axis. The simple relation betweenthese two representations is Q = P L cos 2 χ , (1) U = P L sin 2 χ , (2)which implies that χ = 12 arctan UQ + χ , (3)where χ = 0 if Q >
U > χ = 180 ◦ if Q >
U <
0, and χ = 90 ◦ if Q < χ B in the observer’s reference frame is constantalong the line of sight (LOS). Under such circumstances,the observed orientation of the linear polarization vec-tor is either parallel or perpendicular to the projection ofthe magnetic field vector onto the plane of the sky (e.g.,Section 13.5 in Landi Degl’Innocenti & Landolfi 2004).Then, it follows from Eqs. (1–3) thattan 2 χ B = U ( λ ) /Q ( λ ) . (4)In summary, if a spectral line is in the saturation regimeof the Hanle effect and χ B is constant along the LOS,Eq. (4) provides a well-known recipe to determine themagnetic field azimuth from the orientation of the ob-served linear polarization.In chromospheric lines the atomic excitation is typ-ically dominated by radiative transitions. As a re-sult, scattering processes and the Hanle effect are of-ten the main cause of the spectral line radiation emerg-ing from weak-field regions. Although the scatteringpolarization in most chromospheric lines is sensitive to magnetic strengths in the gauss range (e.g., the Ca i line at 4227 ˚A ), there are a few chromospheric spec-tral lines that are effectively in the saturation regimeof the Hanle effect. In this regime, which occurs whenthe Zeeman splitting in frequency units is much largerthan the inverse lifetime of the relevant atomic levels,the linear polarization that results from scattering pro-cesses is sensitive only to the orientation of the mag-netic field, but not to its strength (e.g., Section 13.5 inLandi Degl’Innocenti & Landolfi 2004). For such spec-tral lines, like the forbidden lines of the solar corona(e.g., Judge 2007), Eq. (4) holds if the magnetic fieldazimuth is constant along the LOS.Interestingly, the well-known Ca II D / (Manso Sainz & Trujillo Bueno 2003a,b, 2010). Itis generally believed that such magnetic fields areomnipresent in the quiet solar chromosphere (e.g.,Bianda et al. 1998) and this is also the case in the 3Dmodel of Carlsson et al. (2016) we have selected forthis investigation. Other authors have also indicatedthat such conditions can be expected for the quiet so-lar chromosphere (e.g., see the left panel of Figure 7 inCarlin & Asensio Ramos 2015).The idea of applying Eq. (4) to map the azimuth ofsolar magnetic fields from the forward scattering po-larization observed in a spectral line that is close tothe saturation regime of the Hanle effect is not new.Collados et al. (2003) applied it to map the magneticfield azimuth of solar coronal filaments from forward-scattering polarization observations in the He i ii II eliability of a method to map the chromospheric magnetic field azimuth Figure 1.
The two left panels show the calculated linear polarization signals integrated over the inner core of the Ca II χ U/Q determined from the calculated
Q/I and
U/I signals, and the map of the azimuth χ sim of the magnetic field vector at the corrugated surface within the 3D modelwhere the line-center optical depth is unity in the Ca II | χ U/Q − χ sim | < ◦ . (see ˇStˇep´an & Trujillo Bueno 2016). The same appliesto other chromospheric lines (e.g., Jaume Bestard et al.2021).The aim of the present paper is to investigate the re-liability of Eq. (4) for determining the azimuth of thechromospheric magnetic field from the scattering polar-ization observed in the Ca II Q and U signals that result from full 3D radiative transfercalculations in the 3D model of Carlsson et al. (2016).It is important to emphasize the following points:(1) We take fully into account the effects of 3D non-LTE radiative transfer on the polarization of the emer-gent spectral line radiation, including the impact of themodel’s macroscopic velocity gradients. This is impor-tant because, as mentioned above, in full 3D radiativetransfer the scattering polarization signals are stronglyaffected by the model’s horizontal inhomogeneities andby the effects of spatial gradients in the three vectorialcomponents of the plasma’s macroscopic velocity (e.g.,the review by ˇStˇep´an 2015). Neglecting these effectswould not be suitable to study the quantitative impactof the instrumental effects on the inferred magnetic fieldazimuth, and this is precisely the main goal of this pa-per.(2) The calculated polarization signals are caused bythe combined action of scattering processes and theHanle and Zeeman effects.(3) The azimuth of the model’s magnetic field is notexactly constant along the LOS.(4) The strength of the model’s magnetic field if noteverywhere sufficiently strong so as to guarantee thatthe Ca II Q/I and
U/I profiles calculated byˇStˇep´an & Trujillo Bueno (2016) in the 3D snapshotmodel of Carlsson et al. (2016) include all the key phys-ical ingredients needed to reach solid conclusions con-cerning the reliability of the basic formula of Eq. (4)for mapping the azimuth of the magnetic field in quietregions of the solar chromosphere. ANALYSIS OF THE THEORETICAL DATAWe use a 3D snapshot model of the solar atmosphereresulting from a radiative-magnetohydrodynamics simu-lation of an enhanced network region (see Carlsson et al.2016). We use snapshot 385 of the time-dependent sim-ulation. The disk-center field of view covered by this3D model is 32 . ′′ × . ′′
6. The model has a grid size ofapproximately 48 km along the horizontal directions.The Stokes profiles of the emergent radiationin the Ca II (ˇStˇep´an & Trujillo Bueno2013), taking into account scattering processesand the Hanle and Zeeman effects. Figure 1of ˇStˇep´an & Trujillo Bueno (2016) and Fig. 4 inJurˇc´ak et al. (2018) provide information on the physicalconditions of the 3D model atmosphere at the corru-gated surface where the optical depth is unity at the The public version of the PORTA radiative transfer code can befound at https://gitlab.com/polmag/PORTA.
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Figure 2.
Maps of the magnetic field azimuth taking into account different telescope apertures (d) and noise-to-signal (N/S)ratios. The pixel size used corresponds to the critical sampling of the theoretical spatial resolution.
Figure 3.
Histograms of the differences between the mag-netic field azimuth χ U/Q inferred from the
U/Q ratio (seeEq. (4)) and the actual azimuth χ sim of the model’s mag-netic field at the atmospheric heights where the optical depthis unity at the center of the Ca II − (blue), 5 × − (orange), and10 − (green). The thin vertical lines mark where we havea cumulative probability of 50% for each telescope and N/Sratio. center of the Ca II Q/I and
U/I profiles. The mid-dle panel gives the map of the magnetic field azimuth obtained using Equation (4), after restricting the result-ing χ U/Q values between 0 ◦ and 90 ◦ due to the azimuthambiguities that follow from Eq. (4) . The next pan-els show the actual values of the model’s magnetic fieldazimuth ( χ sim ) at the corrugated surface of line-centeroptical depth unity in the vertical direction and a prob-ability density function (PDF) that quantifies the accu-racy of the inferred magnetic field azimuth.In order to evaluate the reliability of the inferredmagnetic field azimuth, χ U/Q , we sum the probabil-ities within the red dashed lines in the PDF panelof Fig. 1; i.e., we calculate the total probability that | χ U/Q − χ sim | < ◦ . To investigate the reliability of χ U/Q depending on different observing setups, we de-grade the theoretical data considering various N/S ratiosand telescope apertures. To this end, we: (1) convolvethe calculated Stokes profiles with the spatial PSF of For a detailed discussion of the magnetic field ambiguities re-sulting from the inference of the polarization observed in spec-tral lines we refer the reader to sections 1.9 and 11.7 ofLandi Degl’Innocenti & Landolfi (2004) eliability of a method to map the chromospheric magnetic field azimuth σ ranging from 5 × − to 10 − .The Q and U signals used in Equation 4 result fromaveraging those corresponding to the nine wavelengthpoints located around the minimum of the Stokes I pro-file. This method is applicable also when the individual Q and U signals are dominated by noise. We did notmodify the 12 m˚A wavelength sampling of the theoret-ical Stokes profiles, which corresponds to a spectral res-olution R ∼
350 000. Averaging nine wavelength pointseffectively decreases the noise level by a factor three.This might not be an option if the spectrograph usedhas lower spectral resolution. In such a case, such lowN/S ratios would be achieved with the same exposuretime because the spectrograph automatically integratesthe Stokes signals in wavelength. Note that the spa-tial grid size of the 3D model atmosphere is comparableto the diffraction limit of a telescope with an entrancepupil of 160 cm, and that such spatial resolution is sig-nificantly worse than the diffraction limit of the DKISTand EST telescopes. RELIABILITY OF THE AZIMUTHDETERMINATIONFigure 2 shows the resulting maps of the inferred χ U/Q magnetic field azimuth for different telescope diameters(columns) and N/S ratios (rows). There is a clear trendof improvement with increasing telescope aperture anddecreasing N/S ratio. The lower right χ U/Q map isnearly identical to the map of the model’s χ sim azimuthshown in Fig. 1.In order to compare the inferred χ U/Q maps with the χ sim ones we show in Fig. 3 the histograms of the | χ U/Q − χ sim | differences for each of the maps shown in Fig. 2.Note that a random distribution of χ U/Q would producea straight line at 2.22% in Fig. 3. Figure 4 shows thecumulative probability of | χ U/Q − χ sim | < ◦ for a rangeof N/S ratios and telescopes with diameters d ≤
300 cm.The different columns of Fig. 2 illustrate the impor-tance of the telescope diameter. With d = 50 cm wealready have a spatial resolution that captures reason-ably well the global structure of the magnetic field (thepixel size is roughly three times larger that the model’sgrid size). With increasing telescope aperture, we donot gain a significant improvement in the reliability of χ U/Q . The data with the lowest N/S ratio of 10 − showsthe highest probability of | χ U/Q − χ sim | < ◦ for tele- scope diameters larger than ∼
150 cm, for which thespatial resolution is comparable to the model’s grid size(Fig. 4). For such telescopes the cumulative probabil-ity is around 53%, decreasing only to 51% and 42% fortelescope diameters of 50 cm and 10 cm, respectively.Clearly, the N/S ratio has an important impact onthe reliability of the inferred magnetic field azimuth (seeFigs 2 and 4). For telescopes with d = 200 cm the cumu-lative probability value drops from 53% for a N/S ratioof 10 − to 45% and 39% for N/S values of 5 × − and10 − , respectively. For larger N/S ratios we approach acumulative probability value of 23%, which is equivalentto a random distribution of the χ U/Q azimuths. Theseresults indicate that to some extent we can sacrifice thespatial resolution in order to decrease the noise level inour simulation of measured Q and U signals. SUMMARY AND CONCLUSIONSWe have investigated the reliability of a simple for-mula for obtaining maps of the magnetic field azimuthfrom the linear polarization signals of the Ca II II II Q/I and
U/I signals are sufficiently realistic so as to argue that wecan use them to reach reasonable conclusions regardingthe reliability of the applied method for inferring themagnetic field azimuth from real observations. Giventhat 3D radiative transfer calculations are computation-ally costly, in this investigation we have used a single
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Figure 4.
The probability of | χ U/Q − χ sim | < ◦ for different telescope diameters and N/S levels. The white points indicatethe cases considered in Figs. 2 and 3. The colored isosurface is interpolated between the grid points that correspond to theblack-curves crossings.
3D snapshot model (i.e., we have not accounted for theimpact of the temporal evolution of the solar chromo-spheric plasma). Nevertheless, with the new generationof large-aperture solar telescopes and spectropolarime-ters the exposure time needed to detect the Ca II χ U/Q ) we use the cumulative probabil-ity that the obtained χ U/Q values are within 10 ◦ of themodel’s magnetic field azimuth at the heights where theline-center optical depth is unity in the Ca II χ sim ). For the sake of simplicity, we have presentedour analysis for the forward-scattering geometry of thedisk-centre line of sight.Our results show that the azimuth estimation is verysensitive to the noise level of the spectropolarimetric ob-servation and that, to some extent, we can sacrifice thespatial resolution to improve the polarimetric sensitiv- ity. However, this conclusion has been reached usingStokes profiles from radiative transfer calculations in a3D model atmosphere having a relatively large grid size,which is comparable to the spatial resolution of tele-scopes with a diameter ∼
150 cm. Moreover, the sim-ulation box contains a relatively simple magnetic fieldconfiguration, while the magnetic structure of the realsolar chromosphere is probably more complex. A recentinvestigation contrasting the calculated Stokes profiles(ˇStˇep´an & Trujillo Bueno 2016) with disk-center spec-tropolarimetric observations of the Ca II II eliability of a method to map the chromospheric magnetic field azimuth d = 4 m) the exposure time needed to achievea given N/S ratio is a factor 16 lower than with a 1 mtelescope. Assume we want to achieve a N/S ratio of10 − at the core of the Ca II R = 40 000 (which roughly correspondsto the spectral resolution of averaging nine wavelengthpoints, as in our study), and a spatial sampling of 0 . ′′ Bianda, M., Solanki, S. K., & Stenflo, J. O. 1998, A&A,331, 760Carlin, E. S. 2013, PhD thesis, Generation and Transfer ofPolarized Radiation in Hydrodynamical Models of theSolar Chromosphere, Universidad de La Laguna,arXiv:1402.1567. https://arxiv.org/abs/1402.1567Carlin, E. S. 2015, in Polarimetry, ed. K. N. Nagendra,S. Bagnulo, R. Centeno, & M. Jes´us Mart´ınez Gonz´alez,Vol. 305, 146–153, doi: 10.1017/S1743921315004676Carlin, E. S., & Asensio Ramos, A. 2015, ApJ, 801, 16,doi: 10.1088/0004-637X/801/1/16Carlsson, M., Hansteen, V. H., Gudiksen, B. V., Leenaarts,J., & De Pontieu, B. 2016, A&A, 585, A4,doi: 10.1051/0004-6361/201527226Collados, M., Trujillo Bueno, J., & Asensio Ramos, A.2003, in Astronomical Society of the Pacific ConferenceSeries, Vol. 307, Solar Polarization, ed. J. Trujillo-Bueno& J. Sanchez Almeida, 468 del Toro Iniesta, J. C. 2003, Introduction toSpectropolarimetry (Introduction to Spectropolarimetry,by Jose Carlos del Toro Iniesta, pp. 244. ISBN0521818273. Cambridge, UK: Cambridge UniversityPress, April 2003.)Harvey, J. W., & Solis Team. 2020, IAU Symposium, 354,42, doi: 10.1017/S1743921320000125Jaume Bestard, J., Trujillo Bueno, J., ˇStˇep´an, J., & delPino Alem´an, T. 2021, arXiv e-prints, arXiv:2101.04421.https://arxiv.org/abs/2101.04421Judge, P. G. 2007, ApJ, 662, 677, doi: 10.1086/515433Jurˇc´ak, J., Collados, M., Leenaarts, J., van Noort, M., &Schlichenmaier, R. 2019, Advances in Space Research, 63,1389, doi: 10.1016/j.asr.2018.06.034Jurˇc´ak, J., ˇStˇep´an, J., Trujillo Bueno, J., & Bianda, M.2018, A&A, 619, A60, doi: 10.1051/0004-6361/201732265Landi Degl’Innocenti, E., & Landolfi, M. 2004, Polarizationin Spectral Lines, Vol. 307,doi: 10.1007/978-1-4020-2415-3