M. Faurobert

Center-to-limb variation of scattering polarization in molecular solar lines: Observations and modeling

We present center-to-limb measurements of the scattering polarization observed in molecular lines of C 2 and MgH in the spectral range between 515.7 and 516.1 nm, together with a radiative transfer model for the formation of these lines. The observations were performed in July 2000 with THEMIS inside the south polar limb. We were able to measure the polarization at distances between 1 arcsec and 50 arcsec from the solar limb. The lines appear as very weak absorption features in the intensity spectrum but their linear polarization clearly dominates in the polarization spectrum. We introduce here a simple radiative transfer model which allows to interpret the observed center-to-limb variations of both the intensity and linear polarization. The basic assumption is that molecular lines are formed higher in the photosphere than the continuous photospheric radiation. Molecules are thus illuminated by the polarized continuum photospheric radiation field. We account for a possible Hanle eect due to weak unresolved magnetic fields but we neglect depolarizing collisions. The model depends on four parameters which are determined by fitting both the intensity and polarization in 9 molecular lines of the observed spectral domain. Making use of the dierential Hanle eect in the dierent lines of C2 we show that the C2 lines are aected by the Hanle eect due to a weak unresolved magnetic field. Its mean strength is on the order of 15 Gauss in the upper photosphere and increases to values on the order of 50 Gauss at larger depths. These results are in good agreement with those derived previously (Faurobert et al. 2001) from the linear polarization of the SrI 460.7 nm line which was observed simultaneously. Such a weak field has almost no eect on the MgH lines.

Vertical structure of sunspots from THEMIS observations

We have analysed two-dimensional spectro-polarimetric data taken with the MSDP observing mode of THEMIS in the Na I D1 line to investigate the height variation of the magnetic eld in sunspot umbrae. From the Zeeman-induced circular polarization measured at individual MSDP channels within the line prole, maps of the longitudinal magnetic eld have been computed. A method based on Response Functions has been developed to estimate the depth in the atmosphere at which the Zeeman measurements are originated, thus providing the line-of-sight eld at dierent altitudes in the photosphere. The magnetogram corresponding to the deepest level has served as a boundary condition to perform the potential eld extrapolation into the corona. We have found that the spatial distribution of vertical eld gradient contours predicted from extrapolation is in qualitatively good agreement with that inferred from observations. Quantitatively, however, the longitudinal eld gradients obtained with both methods dier about one order of magnitude, being larger for observations. The origin of this discrepancy has been discussed with respect to possible observation biases, as well as to idealizations used for eld extrapolation. This is a crucial problem to be addressed in future work, and may have important implications for the physics of how the magnetic eld evolves through sunspots and how the flux is distributed in the corona.

Hanle effect in the solar Ba II D2 line: a diagnostic tool for chromospheric weak magnetic fields

Context. The physics of the solar chromosphere depends in a crucial way on its magnetic structure. However there are presently very few direct magnetic field diagnostics available for this region. Aims. Here we investigate the diagnostic potential of the Hanle effect on the Ba ii D2 line resonance polarization for the determination of weak chromospheric turbulent magnetic fields. Methods. The line formation is described with a non-LTE polarized radiative transfer model taking into account partial frequency redistribution with an equivalent two-level atom approximation, in the presence of depolarizing collisions and the Hanle effect. We investigate the line sensitivity to temperature variations in the low chromosphere and to elastic collision with hydrogen atoms. We compare center-to-limb variations of the intensity and linear polarization profiles observed at THEMIS in 2007 to our numerical results. Results. We show that the line resonance polarization is very strongly affected by partial frequency redistribution effects both in the line central peak and in the wings. Complete frequency redistribution cannot reproduce the polarization observed in the line wings. The line is weakly sensitive to temperature differences between warm and cold components of the chromosphere. The effects of elastic collisions with hydrogen atoms and of alignment transfer due to multi-level coupling with the metastable 2 D5/2 levels have been studied in a recent paper showing that they depolarize the 2 P3/2 level of the line. In the conditions where the line is formed we estimate the amount of depolarization due to this mechanism as a factor of 0.7 to 0.65. If we first neglect this effect and determine the turbulent magnetic field strength required to account for the observed line polarization, we obtain values between 20 G and 30 G. We show that this overestimates the magnetic strength by a factor between 1.7 and 2. Applying these correction factors to our previous estimates, we find that the turbulent magnetic field strength is between 10 G and 18 G. Conclusions. Because of its low sensitivity to temperature variations, the solar Ba ii D2 line appears as a very good candidate for the diagnosis of weak magnetic fields in the low chromosphere (z ≥ 900 km) by means of the Hanle effect.

Measuring line formation depths by cross-spectral analysis Numerical simulations for the 630 nm Fe I line pair

Context. Numerical three-dimensional simulations of the solar photosphere have progressed greatly in the last 15 years. Observational tests are needed to independently verify the realism of these simulations. Aims. We aim to measure the perspective shift between monochromatic images at different wavelengths taken away from disk center. We investigate the feasibility of our method by applying it to simulated spectra of the Fe i line pair at 630.15 and 630.25 nm calculated from several snapshots of a three-dimensional simulation of solar magneto-convection. Methods. We present a method to determine line formation depths from spectroscopic observations without relying on assumptions about an atmospheric model. Our method is based on the measurement of a perspective shift, which is detected as a linear phase term in the cross-spectrum of the images. In principle this detection is independent of the spatial resolution of the observations, and provides a valuable test for numerical simulations of the solar photosphere. Results. To obtain accurate formation heights we need to correct spectra for convective Doppler shifts, and we need to accumulate successive phase shifts between images in nearby wavelengths, rather than compare images from the continuum and core directly. The comparison of images provides large dissimilarities, which result from the temperature contrast inversion in the granulation with height. We verify that the cross-spectrum phase of the simulated images shows the expected linear behavior with spatial frequency when considering two close enough wavelengths in a spectral line profile. This linear behavior is however only obtained at small spatial frequencies, i.e. for large granular structures. Derived line formation heights of the two lines range from 239 and 287 km above the continuum formation height for the 630.15 nm line, and from 138 to 201 km for the 630.25 nm line, with significant variation between snapshots. Formation height estimates from optical depth unity give on average 319 km and 244 km respectively. Conclusions. Our numerical tests validate measurements of line formation depths from cross-spectra between images at different wavelengths and stress the value of measuring the phase of the cross-spectra as an important test for numerical simulations.

Direct measurement of the formation height difference of the 630 nm Fe I solar lines

Context. Spectral lines formed over a limited height range in either a stellar or planetary atmosphere provide us with information about the physical conditions within this height range. In this context, an important quantity is the so-called line formation depth. It is usually determined from numerical calculation of the atmospheric opacity in the line of interest and then converted into geometrical depth by using atmospheric models. Aims. We develop a radically different approach, which allows us to measure directly line formation depths from spectroscopic observations without relying on assumptions about an atmospheric model. This method requires spatially resolved observations, which up to now have been available only for solar or planetary studies. We apply this method to images of the solar granulation. Methods. The method was presented and tested numerically in previous papers. It is based on the measurement of the perspective shift between images at different wavelengths, formed at different heights, when they are observed away from disk center. Because of the Fourier transform properties, this shift gives rise to a deterministic linear phase term in the cross spectrum of the images. Results. The method is applied to observations of solar quiet regions performed with the SOT spectropolarimeter on HINODE in the Fe I line pair at 630.15 and 630.25 nm. We derive the difference in formation heights between the two lines and its center-to-limb variations. We show that the high sensitivity of the measurements allows us to detect variations in the line formation heights between magnetized and non-magnetized regions of the solar atmosphere. Conclusions. Our results are the first direct measurements of line formation depths in the solar photosphere. Cross spectral analysis provides us with a new observable quantity, which may be measured with an accuracy well bellow the spatial resolution of the observations. We recall that the Fe I line pair at 630.15 and 630.25 nm is often used to determine solar magnetic fields by spectropolarimetric observations and inversion methods. The difference in the line formation heights that we measure should be taken into account in the inversion procedures.

Differential speckle interferometry: in-depth analysis of the solar photosphere

Aims.We present the results of an experiment performed at the solar telescope THEMIS in 2002 to measure the depth over which the solar granulation extends in the photosphere. Methods: Observations made in the 523.3 nm and 557.6 nm photospheric non-magnetic iron lines were correlated with images in the continuum using spectrograms. The difference in depth between the different levels in the photosphere is projected into a difference of position along the slit of the spectrograph, using a perspective effect similar to the well-known Wilson effect for sunspots. This requires measuring displacements, ones much smaller than the telescope resolution. This is made possible by using a differential speckle interferometric technique, cross-correlating images taken in the continuum and the line. The method is not adapted to following displacements of structures in the core of strong lines, due to their difference in shapes with the structures observed in the continuum. In this case, a sequential cross-spectrum method is developed to cross-correlate images taken at close wavelengths. Results: The raw results are surprising: displacements measured in the blue and the red wings of a line have opposite signs! North and South observations, however, clearly show the expected behavior attributed to a perspective effect. After a description of the observations, we give a first interpretation of the results. The main part of the observed displacement comes from the effect of unresolved Doppler shifts produced by horizontal velocities in the solar photosphere. The perspective effect we seek appears as a second-order term; we find that its amplitude is 2 or 3 times larger than predicted by theoretical 1D models. In the core of strong lines we detect a contrast inversion that also shows up in the cross-correlation function as an anti-correlation peak at line center. Conclusions: .This first use of the differential speckle interferometry technique on the Sun is quite promising for 3D studies at high spatial resolution. Further observations with very good image quality are needed to take advantage of this new technique. THEMIS is operated on the Island of Tenerife by CNRS-CNR in the Spanish Observatorio del Teide of the Instituto de Astrofisica de Canarias.

Modeling scattering polarization in molecular solar lines in spherical geometry

Context. The atmosphere of the Sun is permeated by a vast amount of magnetic flux that remains invisible in magnetograms based on the Zeeman effect. A model-independent way of measuring weak hidden magnetic fields makes use of the differential Hanle effect on the scattering polarization of molecular lines with different sensitivities to magnetic fields. Aims. The observed line scattering polarization steeply increases at the solar limb. Here we are interested in interpreting observations performed at the solar limb, where plane-parallel semi-infinite geometry is not valid. The main reason is that the sphericity of the atmosphere means that the line-of-sight optical path intersects only a finite part of the solar atmosphere. In this paper we revisit the modeling of scattering polarization in two molecular lines of C2 and MgH in the spectral range from 515.60 nm to 516.20 nm, where observations performed both inside and above the solar limb are available. Methods. The solar atmosphere is described by a one-dimensional, spherically symmetric medium following either the FALC or the FALX quiet Sun model. Both the line and background continuum scattering polarizations are computed by means of the “along-theray” approach. We assume a two-level atom formalism for the line source function, and we compute the molecule number densities and line opacities assuming LTE. We estimate the elastic and inelastic collision rates by fitting the line intensity and linear polarization in several couples of lines of the Second Solar Spectrum Atlas. Results. The limb variations of scattering polarization, both in the lines and in the continuum, are strongly modified when the sphericity of the solar atmosphere is accounted for. We show that the line polarization goes through a maximum at 0.4 �� above the limb, for both MgH and C2 lines. The contribution of the line rapidly goes to zero at a larger limb distance, but continuum polarization keeps increasing. The maximum polarization rates have an amplitude of 2% to 2.5% when the FALC model is used, which agrees with previous observations, whereas the FALX model leads to much higher rates. We then investigate the Hanle effect of microturbulent magnetic fields on the C2 line linear polarization. We show that polarization observed close to the limb would provide valuable diagnostics of weak magnetic fields in the region of the temperature minimum.

Solar-cycle variations of the internetwork magnetic field

Context. The quiet Sun exhibits a rich and complex magnetic structuring that is still not fully resolved or understood. Aims. We intend to contribute to the debate about the origin of the internetwork magnetic fields and whether or not they are related to the global solar dynamo. Methods. We analyzed center-to-limb polarization measurements obtained with the SOT/SP spectropolarimeter onboard the Hinode satellite outside active regions in 2007 and 2013, that is, at a minimum and a maximum of the solar cycle, respectively. We examined 10 �� × 10 �� maps of the unsigned circular and linear polarization in the FeI 630.25 nm line in regions located away from network elements. The maps were corrected for bias and focus variations between the two data sets. Then we applied a Fourier spectral analysis to examine wether the spatial structuring of the internetwork magnetic fields shows significant differences between the minimum and maximum of the cycle. Results. Neither the mean values of the unsigned circular and linear polarizations in the selected 10 �� × 10 �� maps nor their spatial fluctuation power spectra show significant center-to-limb variations. For the unsigned circular polarization the power of the spatial fluctuations is lower in 2013 than in 2007, but the spectral slope is unchanged. The linear polarization spectra show no significant differences in 2013 and 2007, but the spectrum of 2013 is more strongly affected by noise. Conclusions. The small-scale magnetic structuring in the internetwork is different in our 2013 and 2007 data. Surprisingly, we find a lower spatial fluctuation power at the solar maximum in the internetwork magnetic structuring. This indicates some complex interactions between the small-scale magnetic structures in the quiet Sun and the global dynamo, as predicted by recent numerical simulations. This result has to be confirmed by further statistical studies with larger data sets.

Hanle diagnostics of weak solar magnetic fields: - Inversion of scattering polarization in C2 and MgH molecular lines

Context. The quiet Sun magnetism has been intensively investigated in recent years by various observational techniques. But several issues, such as the question of the isotropy and of the energy density spectrum of the mixed polarity turbulent magnetic fields, are still under debate. Aims. Here we present an inversion method that allows us to constrain the depth-dependence of the magnetic field strength. We use the center-to-limb variations of linear scattering polarization measured in molecular lines of C2 and MgH molecules with different sensitivities to the Hanle effect. We consider six C2-triplets and one MgH line in the spectral range between 515.7 nm and 516.1 nm observed with the THEMIS Telescope. Methods. One of the delicate problems with Hanle diagnostics is to disentangle the effects of elastic depolarizing collisions from the depolarization due to the Hanle effect of the magnetic field. By making use of the different sensitivities of the molecular lines in our spectral range to microturbulent magnetic fields and, by using a non-LTE radiative transfer modeling of the line formation, we are able to determine both the depolarizing collision cross-section and the magnetic strength. We use a standard 1D quiet Sun atmospheric model and we invert the full set of center-to-limb polarization rates measured at line centers, with a depth-dependent magnetic field described by three free parameters. The depolarizing collision cross-section is also treated as a free parameter. A downhill simplex method is used to find the best-fitting values for the collisional and magnetic strength parameters. Results. For the elastic depolarizing collisions cross-section for the C2 lines we obtain α (2) = 1.6 ± 0.4 × 10 −9 cm 3 s −1 , which is within an order of magnitude of the value previously obtained for MgH lines from a differential Hanle effect analysis. The observational constraints provided by the MgH and C2 line polarization give access to the altitude range between z = 200 km and z = 400 km above the base of the photosphere. We find that the turbulent magnetic field strength decreases from 95 Gauss at the altitude z = 200 km to 5 Gauss at z = 400 km. Conclusions. The turbulent magnetic field strength that we derive from the Hanle effect shows a strong vertical gradient in the upper photosphere. We point out that this behavior may explain why very different turbulent magnetic field strengths have been inferred from the interpretation of Hanle depolarization when using different lines formed at different altitudes. We notice that the presence of a strong depth gradient is not compatible with the assumption of isotropy of the turbulent field.

New view on exoplanet transits - Transit of Venus described using three-dimensional solar atmosphere STAGGER-grid simulations

Context. An important benchmark for current observational techniques and theoretical modeling of exoplanet atmospheres is the transit of Venus (ToV). Stellar activity and, in particular, convection-related surface structures, potentially cause fluctuations that can affect the transit light curves. Surface convection simulations can help interpreting the ToV as well as other transits outside our solar system. Aims. We used the realistic three-dimensional (3D) radiative hydrodynamical (RHD) simulation of the Sun from the Stagger-grid and synthetic images computed with the radiative transfer code Optim3D to predict the transit of Venus (ToV) in 2004 that was observed by the satellite ACRIMSAT. Methods. We computed intensity maps from the RHD simulation of the Sun and produced a synthetic stellar disk image as an observer would see, accounting for the center-to-limb variations. The contribution of the solar granulation was considered during the ToV. We computed the light curve and compared it to the ACRIMSAT observations as well as to light curves obtained with solar surface representations carried out using radial profiles with different limb-darkening laws. We also applied the same spherical tile imaging method as used for RHD simulation to the observations of center-to-limb solar granulation with Hinode. Results. We explain ACRIMSAT observations of 2004 ToV and show that the granulation pattern causes fluctuations in the transit light curve. We compared different limb-darkening models to the RHD simulation and evaluated the contribution of the granulation to the ToV. We showed that the granulation pattern can partially explain the observed discrepancies between models and data. Moreover, we found that the overall agreement between real and RHD solar granulation is good, either in terms of depth or ingress/egress slopes of the transit curve. This confirms that the limb-darkening and granulation pattern simulated in 3D RHD of the Sun represent well what is imaged by Hinode. In the end, we found that the contribution of the Venusean aureole during ToV is ∼10 −6 times less intense than the solar photosphere, and thus, accurate measurements of this phenomena are extremely challenging. Conclusions. The prospects for planet detection and characterization with transiting methods are excellent with access to large a amount of data for stars. Being able to consistently explain the data of 2004 ToV is a new step forward for 3D RHD simulations, which are becoming essential for detecting and characterizing exoplanets. They show that granulation has to be considered as an intrinsic uncertainty (as a result of stellar variability) on precise measurements of exoplanet transits of, most likely, planets with small diameters. In this context, it is mandatory to obtain a comprehensive knowledge of the host star, including a detailed study of the stellar surface convection.