Inferring the magnetic field vector in the quiet Sun. II. Interpreting results from the inversion of Stokes profiles

In a previous paper, we argued that the inversion of Stokes profiles applied to spectropolarimetric observations of the solar internetwork yield unrealistically large values of the inclination of the magnetic field vector ($\gamma$). This is because photon noise in Stokes $Q$ and $U$ are interpreted by the inversion code as valid signals, that leads to an overestimation of the transverse component $B_\perp$, thus the inclination $\gamma$. However, our study was based on the analysis of linear polarization signals that featured only uncorrelated noise. In this paper, we develop this idea further and study this effect in Stokes $Q$ and $U$ profiles that also show correlated noise. In addition, we extend our study to the three components of the magnetic field vector, as well as the magnetic filling factor $\alpha$. With this, we confirm the tendency to overestimate $\gamma$ when inverting linear polarization profiles that, although non-zero, are still below the noise level. We also establish that the overestimation occurs mainly for magnetic fields that are nearly vertical $\gamma \lesssim 20\deg$. This indicates that a reliable inference of the inclination of the magnetic field vector cannot be achieved by analyzing only Stokes $I$ and $V$. In addition, when inverting Stokes $Q$ and $U$ profiles below the noise, the inversion code retrieves a randomly uniform distribution of the azimuth of the magnetic field vector $\phi$. To avoid these problems, we propose only inverting Stokes profiles for which the linear polarization signals are sufficiently above the noise level. However, this approach is also biased because, in spite of allowing for a very accurate retrieval of the magnetic field vector from the selected Stokes profiles, it selects only profiles arising from highly inclined magnetic fields.Comment: Accepted for publication in Astronomy and Astrophysics. 14 pages. 7 color figure

Inferring the magnetic field vector in the quiet Sun. III. Disk variation of the Stokes profiles and isotropism of the magnetic field

We have studied the angular distribution of the magnetic field vector in the solar internetwork employing high-quality data (noise level $\sigma \approx 3\times 10^{-4}$ in units of the quiet-Sun intensity) at different latitudes recorded with the Hinode/SP instrument. Instead of applying traditional inversion codes of the radiative transfer equation to retrieve the magnetic field vector at each spatial point on the solar surface and studying the resulting distribution of the magnetic field vector, we surmised a theoretical distribution function of the magnetic field vector and used it to obtain the theoretical histograms of the Stokes profiles. These histograms were then compared to the observed ones. Any mismatch between them was ascribed to the theoretical distribution of the magnetic field vector, which was subsequently modified to produce a better fit to the observed histograms. With this method we find that Stokes profiles with signals above $2\times 10^{-3}$ (in units of the continuum intensity) cannot be explained by an isotropic distribution of the magnetic field vector. We also find that the differences between the histograms of the Stokes profiles observed at different latitudes cannot be explained in terms of line-of-sight effects. However, they can be explained by a distribution of the magnetic field vector that inherently varies with latitude. We note that these results are based on a series of assumptions that, although briefly discussed in this paper, need to be considered in more detail in the future.Comment: Accepted for publication in Astronomy and Astrophysics. 14 pages, 8 color figure

Modified p-modes in penumbral filaments?

Aims: The primary objective of this study is to search for and identify wave modes within a sunspot penumbra. Methods: Infrared spectropolarimetric time series data are inverted using a model comprising two atmospheric components in each spatial pixel. Fourier phase difference analysis is performed on the line-of-sight velocities retrieved from both components to determine time delays between the velocity signals. In addition, the vertical separation between the signals in the two components is calculated from the Stokes velocity response functions. Results: The inversion yields two atmospheric components, one permeated by a nearly horizontal magnetic field, the other with a less-inclined magnetic field. Time delays between the oscillations in the two components in the frequency range 2.5-4.5 mHz are combined with speeds of atmospheric wave modes to determine wave travel distances. These are compared to expected path lengths obtained from response functions of the observed spectral lines in the different atmospheric components. Fast-mode (i.e., modified p-mode) waves exhibit the best agreement with the observations when propagating toward the sunspot at an angle ~50 degrees to the vertical.Comment: 8 pages, 12 figures, accepted for publication in Astronomy & Astrophysic

The uncombed penumbra

The uncombed penumbral model explains the structure of the sunspot penumbra in terms of thick magnetic fibrils embedded in a magnetic surrounding atmosphere. This model has been successfully applied to explain the polarization signals emerging from the sunspot penumbra. Thick penumbral fibrils face some physical problems, however. In this contribution we will offer possible solutions to these shortcomings.Comment: 6 pages, 2 figures. to appear in the proceedings of the Solar Polarization Workshop I

Penumbral thermal structure below the visible surface

$Context$. The thermal structure of the penumbra below its visible surface (i.e., $\tau_5 \ge 1$) has important implications for our present understanding of sunspots and their penumbrae: their brightness and energy transport, mode conversion of magneto-acoustic waves, sunspot seismology, and so forth. $Aims$. We aim at determining the thermal stratification in the layers immediately beneath the visible surface of the penumbra: $\tau_5 \in [1,3]$ ($\approx 70-80$ km below the visible continuum-forming layer). $Methods$. We analyzed spectropolarimetric data (i.e., Stokes profiles) in three Fe \textsc{i} lines located at 1565 nm observed with the GRIS instrument attached to the 1.5-meter solar telescope GREGOR. The data are corrected for the smearing effects of wide-angle scattered light and then subjected to an inversion code for the radiative transfer equation in order to retrieve, among others, the temperature as a function of optical depth $T(\tau_5)$. $Results$. We find that the temperature gradient below the visible surface of the penumbra is smaller than in the quiet Sun. This implies that in the region $\tau_5 \ge 1$ the penumbral temperature diverges from that of the quiet Sun. The same result is obtained when focusing only on the thermal structure below the surface of bright penumbral filaments. We interpret these results as evidence of a thick penumbra, whereby the magnetopause is not located near its visible surface. In addition, we find that the temperature gradient in bright penumbral filaments is lower than in granules. This can be explained in terms of the limited expansion of a hot upflow inside a penumbral filament relative to a granular upflow, as magnetic pressure and tension forces from the surrounding penumbral magnetic field hinder an expansion like this.Comment: 5 pages; 2 figures; accepted for publication in Astronomy and Astrophysics Letter

Ranking efficient DMUs using cooperative game theory

The problem of ranking Decision Making Units (DMUs) in Data Envelopment Analysis (DEA) has been widely studied in the literature. Some of the proposed approaches use cooperative game theory as a tool to perform the ranking. In this paper, we use the Shapley value of two different cooperative games in which the players are the efficient DMUs and the characteristic function represents the increase in the discriminant power of DEA contributed by each efficient DMU. The idea is that if the efficient DMUs are not included in the modified reference sample then the efficiency score of some inefficient DMUs would be higher. The characteristic function represents, therefore, the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient units is dropped from the sample. Alternatively, the characteristic function of the cooperative game can be defined as the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient DMUs are the only efficient DMUs that are included in the sample. Since the two cooperative games proposed are dual games, their corresponding Shapley value coincide and thus lead to the same ranking. The more an ef- ficient DMU impacts the shape of the efficient frontier, the higher the increase in the efficiency scores of the inefficient DMUs its removal brings about and, hence, the higher its contribution to the overall discriminant power of the method. The proposed approach is illustrated on a number of datasets from the literature and compared with existing methods

Convective motions and net circular polarization in sunspot penumbrae

We have employed a penumbral model, that includes the Evershed flow and convective motions inside penumbral filaments, to reproduce the azimuthal variation of the net circular polarization (NCP) in sunspot penumbrae at different heliocentric angles for two different spectral lines. The theoretical net circular polarization fits the observations as satisfactorily as penumbral models based on flux-tubes. The reason for this is that the effect of convective motions on the NCP is very small compared to the effect of the Evershed flow. In addition, the NCP generated by convective upflows cancels out the NCP generated by the downflows. We have also found that, in order to fit the observed NCP, the strength of the magnetic field inside penumbral filaments must be very close to 1000 G. In particular, field-free or weak-field filaments fail to reproduce both the correct sign of the net circular polarization, as well as its dependence on the azimuthal and heliocentric angles.Comment: Accepted for publication in the Astrophysical Journal. 10 pages, 7 figures (3 in color). Uses emulatedap