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Combining magneto-hydrostatic constraints with Stokes profile inversions. IV. Imposing ∇ · B = 0 condition
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J.M. Borrero1, A. Pastor Yabar2, and B. Ruiz Cobo3,41- Institut f¨ur Sonnenphysik, Sch¨oneckstr. 6, D-79104, Freiburg, Germany2- Institute for Solar Physics, Department of Astronomy, Stockholm University, AlbaNova University Centre, 10691 Stockholm, Sweden3- Instituto de Astrof´ısica de Canarias, Avd. V´ıa L´actea s/n, E-38205, La Laguna, Spain4- Departamento de Astrof´ısica, Universidad de La Laguna, E-38205, La Laguna, Tenerife, SpainRecieved / AcceptedABSTRACTContext. Inferences of the magnetic field in the solar atmosphere by means of spectropolarimetric Inversions (i.e., Stokes inversion codes) yield magnetic fields that are non-solenoidal (∇ · B , 0). Because of this, results obtained by such methods are sometimes put into question. Aims. We aim to develop and implement a new technique that can retrieve magnetic fields that are simultaneously consistent with observed polarization signals and with the null divergence condition.Methods. The method used in this work strictly imposes ∇ · B = 0 by determining the vertical component of the magnetic field (Bz) from the horizontal ones (Bx, By). We implement this solenoidal inversion into the FIRTEZ Stokes inversion code and apply it to spectropolarimetric observations of a sunspot observed with the Hinode/SP instrument.Results. We show that the solenoidal inversion retrieves a vertical component of the magnetic field that is consistent with the vertical component of the magnetic field inferred from the non-solenoidal one. We demonstrate that the solenoidal inversion is capable of a better overall fitting to the observed Stokes vector than the non-solenoidal inversion. In fact, the solenoidal magnetic field fits Stokes V worse, but this is compensated by a better fit to Stokes I. We find a direct correlation between the worsening in the fit to the circular polarization profiles by the solenoidal inversion and the deviations in the inferred Bz with respect to the non-solenoidal inversion.Conclusions.In spite of being physically preferable, solenoidal magnetic fields are topologically very similar in 80% of the analyzed three-dimensional domain to the non-solenoidal fields obtained from spectropolarimetric inversions. These results support the idea that common Stokes inversion techniques fail to reproduce ∇·B = 0 mainly as a consequence of the uncertainties in the determination of the individual components of the magnetic field. In the remaining 20% of the analyzed domain, where the Bz inferred by the solenoidal and non-solenoidal inversions disagree, it remains to be proven that the solenoidal inversion is to be preferred because even though the overall fit to the Stokes parameters improves, the fit to Stokes V worsens. It is in these regions where the application of the Stokes inversion constrained by the null divergence condition can yield new insights about the topology of the magnetic field in the solar photosphere.Key words. Sun: sunspots – Sun: magnetic fields – Sun: photosphere – Magnetohydrodynamics (MHD) – Polarization

8 0.6 0.4 0.5 spine MHS spine MHS + B = 0 intraspine MHS intraspine MHS + B = 0 u m u C 0.2 log c =0.0 log c =-1.5 log c =-3.0 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0 1 2 3 4 5 z [Mm]
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5. Vertical stratication of the vertical component of the magnetic eld, Bz(z), along the two positions in Fig. 4 (see vertical white dashed lines). The red and orange colors correspond to the results along the spine from the MHS and MHS + B = 0 inversion, respectively. The shaded pink areas indicate the 3Bz condence level from the MHS inversion. Similarly, the blue and steel blue colors correspond to the results along the intraspine from the MHS and MHS + B = 0 inversion, respectively. The shaded cyan areas indicate the 3Bz condence level from the MHS inversion. the atmospheric model that best ts the observed Stokes vector. For instance, for a physical parameter denoted by index j (see Appendix B or Chapter 11.2.1 in Sanchez Almeida 1997; del Toro Iniesta 2003, respectively), 2 j = 2 F (1) j j , where the modied Hessian matrix
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1986). In the case of the spine, the MHS (non-solenoidal) inversion yields a vertical component of the magnetic eld Bz that is almost constant with height (solid red line in Fig.5), whereas the solenoidal inversion retrieves a Bz(z) that decreases by about 1000 Gauss in 500 kilometers (dashed orange line) and lies well beyond the 3 error bars from the MHS inversion. We note that the vertical gradient in this case is about dBz/dz 2 G km1. This large vertical gradient has been deemed to be at odds with the B condition for the magnetic eld (Balthasar 2018), and this is probably correct if it should be present through the entire penumbra. However, our inversion demonstrates that locally, this large gradient is not only consis
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Based on the results from the previous two examples (spine and intraspine), we subsequently aimed at determining how often the vertical component of the magnetic eld provided by the MHS + B = 0 inversion is consistent with Bz as inferred from the regular MHS (non-solenoidal) inversion. In order to investigate this, we show in Figure 6 the cumulative histogram of the grid cells at dierent log c levels as a function of Bz/Bz, where Bz is again the standard deviation in the determination of the vertical component of the magnetic eld (4) Fig. 6. Cumulative histogram of the number of pixels, shown at three dierent optical depth levels (red: log c = 0; blue: log c = 1.5; green: log c = 3), as a function of the ratio of the dierence between Bz obtained from both inversions to the standard deviation Bz. Only pixels where Bh(log c = 1.5) > 300 G are considered (see text for details).
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