Chin. J. Astron. Astrophys. Vol. 4 (2004), No. 4, 365–376 Chinese Journal of ( http://www.chjaa.org or http://chjaa.bao.ac.cn ) Astronomy and Astrophysics Calibration of Vector Magnetogram with the Nonlinear Least-squares Fitting Technique ∗ Jiang-Tao Su and Hong-Qi Zhang National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012; [email protected] Received 2003 November 16; accepted 2004 April 30 Abstract To acquire Stokes profiles from observations of a simple sunspot with the Video Vector Magnetograph at Huairou Solar Observing Station (HSOS), we scanned the FeI λ5324.19 A˚ line over the wavelength interval from 150 mA˚ redward of the line center to 150 mA˚ blueward, in steps of 10 mA.˚ With the technique of analytic inversion of Stokes profiles via nonlinear least-squares, we present the cali- bration coefficients for the HSOS vector magnetic magnetogram. We obtained the theoretical calibration error with linear expressions derived from the Unno-Becker equation under weak-field approximation. Key words: Sun: activity — Sun: magnetic fields — sunspots 1 INTRODUCTION We use the data of vector magnetic fields obtained by the imaging vector magnetograph on the Solar Magnetic Field Telescope of the Huairou Solar Observing Station (HSOS) of the National Astronomical Observatories of China. The data are normally in the form of the Stokes polarization images in portions of the spectral line obtained using a birefringent filter (1/8 A˚ bandpass). The working line is FeI λ5324.19 A˚ for the photospheric magnetic field measurements. The equivalent width of the line is 0.33 A˚ and the Land´efactor g = 1.5. For measurement of the longitudinal field, the filter is tuned to –0.075 A˚ of the line center, which is the optimum place for the coefficient of calibration with both moderate sensitivity and linearity (Wang, Ai & Deng 1996). Ai et al. (1982) pointed out that the linearity was best when calibrating the transverse field at ±0.10 ∼ ±0.11 A˚ from the line center. However, for the well-known reason that cross-talk effects are weakest at the line center, the measurement of transverse field is taken at the line center. To reconstruct the vector magnetogram, we use the linear relations between the magnetic field and the Stokes parameters I, Q, U and V established under the condition of weak-field approximation, V Q2 U 21/4 B = C ,B = C + , (1) k k I ⊥ ⊥ I I where Ck and C⊥ are the calibration coefficients for the longitudinal and transverse magne- tograms, respectively. The theoretical calibration for HSOS vector magnetogram was first made ∗ Supported by the National Natural Science Foundation of China. 366 J. T. Su & H. Q. Zhang by Ai et al. (1982) using the early solar atmosphere models given by Allen (1973). The results indicated that the calibration coefficients were sensitive to the atmosphere models. For routine observations, an empirical calibration is used by adopting the standard line profile observed at Kitt Peak, instead of solving the Unno-Beckers equation. The calibration coefficients Ck 3 3 and C⊥ are 10.00 × 10 and 9.73 × 10 G for the longitudinal and transverse magnetic fields, respectively (Wang et al. 1996). Wang, Ai & Deng (1996) used two methods to calibrate the longitudinal magnetograms, an empirical calibration, a velocity calibration and Ck were found to be 8.88 × 103 G with the former and 9.60 × 103 G with the latter. However, their measure- ments were all restricted to solar quiet regions. In this paper, our observations are made on a sunspot. The Stokes profiles were obtained by scanning each wavelength of the FeIλ5324.19 A˚ line with the vector magnetograph system on HSOS. Such an effort was first made with the Mashell Space Flight center system for observations of a sunspot (Balasubramaniam & West 1991). We try to obtain the calibration coefficients for the longitudinal and transverse fields by recovering the vector magnetic field parameters and other physical parameters with the nonlinear least- squares technique of analytic inversion of the Stokes profiles. 2 OBSERVATIONS AND DATA REDUCTION The observational data were obtained on 2003 October 23, with the HSOS vector magne- tograph for a relative simple sunspot in the super active region NOAA 10484 located at N04 E12.4. The observations can be divided into two categories: spectral scan data and standard deviation data. The spectral scan data were Stokes images extending from –150 mA˚ in the blue wing to +150 mA˚ in the red wing of the FeI λ5324.19 A˚ line at steps of 10 mA.˚ First the V/I image was acquired; next the Q/I image, followed by the U/I image. The time required to obtain a set of Stokes images was about 1 minute for each spectral location. The standard deviation data were Stokes images observed at ±60 mA˚ from the line center and 10 sets of such data were obtained. Each image was a 256-frame integration. The observations were carried out from 01:00 to 02:20 UT. The cross-talk corrections are shown in Fig. 1, over the entire field of view. A linear fit of the data is sufficient. A scatter plot of the standard deviation in the polarization signals as a function of the polarization intensity is shown in Fig. 2 for the sunspot penumbra. The Stokes intensities Q, U, and V at each of the 31 filter positions were registered with the data set taken at 75 mA˚ to the blue of the line center. We also present three vector magnetograms in Fig. 3: the top one was obtained by super- posing the azimuths observed at 120 mA˚ in the blue wing (thin line) on those observed at the line center (thick line) and the figure was corrected for cross-talk effects; the middle one and the bottom one were obtained by superposing the azimuths observed at 120 mA˚ (thin line) in the blue wing on those observed at 120 mA˚ in the red wing (thick line) without and with cross-talk corrections, respectively. The three plots were superposed on the line of sight magnetogram observed at –75 mA˚ from the line center. Figure 3b shows that the red and blue transverse magnetograms differ obviously in both azimuth and magnitude. After the cross-talk corrections, the differences between the two mag- netograms become trivial, as shown in Fig. 3c. For some pixels of the inner umbra, however, a magnitude difference still remains. This is ascribed mainly to the asymmetric character of the FeIλ5324.19 A˚ line in the two wings when observed in the umbra (Wallace et al.). So Vector Magnetic Field Calibration 367 we omit some data of the inner umbra in the following analysis. The azimuth differences (φ − φ ) of the first panel were mainly caused by magneto-optical effects while the 0.00A˚ −0.12A˚ differences (φ − φ ) of the third panel were mainly caused by cross-talk effects. The 0.12A˚ −0.12A˚ mean value is 14.1◦ for the former and 7.3◦ for the latter. These results may serve as an evidence that measurements of the transverse fields should be made at the far wings of the spectral line even though the cross-talk effects there are more serious. Fig. 1 Polarization cross-talk introduced by the polarimeter and sample scatter plots derived from one set of ±60 mA˚ data. Scatter plot of Stokes (Qb − Qr)/I vs. (Vb − Vr)/I is shown on (a) plot and scatter plot for (Ub − Ur)/I vs. (Vb − Vr)/I is shown at (b) plot. The solid lines in two plots represent the linear fits to the data. Fig. 2 Scatter plots of the standard deviation in the linear and circular polarization as a function of the polarization intensity for the penumbra. The solid lines in the figure represent the linear fits. The standard deviation is symmetric about zero for the negative polarization. 368 J. T. Su & H. Q. Zhang Fig. 3 Three bird-foot vector magnetograms of a simple sunspot of NOAA 10484 on 2003 October 23, the field of view is 84×84 arcsec. North is up and west is to the right. (a) The azimuths observed at 120 mA˚ to the blue of the line center (thin line) superposed on the azimuths from observations at the line center (thick line) and the cross-talk corrections were applied. (b) The azimuths observed at ±120 mA˚ from the line center. The thin line is for the blue wing and the thick one for the red wing. (c) Similar to the middle panel, but with the cross-talk corrections applied. 3 ANALYTICAL SOLUTION OF STOKES PROFILES The analytical solutions of the Stokes profiles as derived by Landofi & Landi Degl’Innocenti (1982) are uB I = B + 1 {(1 + η )[(1 + η )2 + ρ2 + ρ2 + ρ2 ]}, 0 ∆ I I Q U V uB Q = − 1 [(1 + η )2η + (1 + η )(η ρ − η ρ ) + ρ R], ∆ I Q I V U U V Q uB U = − 1 [(1 + η )2η + (1 + η )(η ρ − η ρ ) + ρ R], ∆ I U I Q V V Q U uB V = − 1 [(1 + η )2η + ρ R], (2) ∆ I V V where 2 2 2 2 2 2 2 2 2 I = (1 + ηI ) [(1 + ηI ) − ηQ − ηU − ηV + ρQ + ρU + ρV ] − R , R = ηQρQ + ηU ρU + ηV ρV . (3) Here µ = cos θ, θ is the angle between the line of sight and the normal to the solar surface. The absorption profiles ηI,Q,U,V and the dispersion profiles ρQ,U,V are given by 2 2 ηI = 0.5[ηp sin ψ + 0.5(ηb + ηr)(1 + cos ψ)], 2 ηQ = 0.5[ηp − 0.5(ηb + ηr)] sin ψ cos 2φ, 2 ηU = 0.5[ηp − 0.5(ηb + ηr)] sin ψ sin 2φ, ηV = 0.5(ηr − ηb) cos φ, 2 ρQ = 0.5[ρp − 0.5(ρb + ρr)] sin ψ cos 2φ, 2 ρU = 0.5[ρp − 0.5(ρb + ρr)] sin ψ sin 2φ, ρV = 0.5(ρr − ρb) cos φ, (4) Vector Magnetic Field Calibration 369 where ψ is the inclination and φ is the azimuth of the magnetic field vector.