A Forth‐and‐Back Implicit Λ‐Iteration

Faurobert

2012

A&A

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 C 2 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 C 2 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 C 2 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.

Section: Solving the Radiative Transfer Problem In Spherical Geometrymentioning

confidence: 99%

Modeling scattering polarization in molecular solar lines in spherical geometry

Faurobert

2012

A&A

“…Saad, 2003). For the solution of the 1D NLTE RT problem this idea was implemented in two different ways by Trujillo Bueno & Fabiani Bendicho (1995) and Atanacković-Vukmanović et al (1997). In the paper by Trujillo Bueno & Fabiani Bendicho (1995) standard approximate Λ operator approach with three-point algorithm to set up short characteristics of the second order is used.…”

Section: Gauss-seidel-type Iterationmentioning

confidence: 99%

Accelerating NLTE radiative transfer by means of the Forth-and-Back Implicit Lambda Iteration: A two-level atom line formation in 2D Cartesian coordinates

Advances in Space Research

Atanackovic

2014

State-of-the-art methods in multidimensional NLTE radiative transfer are based on the use of local approximate lambda operator within either Jacobi or Gauss-Seidel iterative schemes. Here we propose another approach to the solution of 2D NLTE RT problems, Forth-and-Back Implicit Lambda Iteration (FBILI), developed earlier for 1D geometry. In order to present the method and examine its convergence properties we use the well-known instance of the two-level atom line formation with complete frequency redistribution. In the formal solution of the RT equation we employ short characteristics with two-point algorithm. Using an implicit representation of the source function in the computation of specific intensities, we compute and store the coefficients of the linear relations J = a + bS between the mean intensity J and the corresponding source function S. The use of iteration factors in the 'local' coefficients of these implicit relations in two 'inward' directions, along with the update of the source function in other two, 'outward', directions leads to four times faster solution than the Jacobi's one. Moreover, the update made in all four consecutive sweeps of the grid leads to an acceleration by a factor of 6-7 compared to the Jacobi iterative scheme.

“…by doing Λ iteration for the polarization). For the first step we use "forth-and-back" implicit lambda iteration method by Atanacković-Vukmanović et al (1997). A further simplification for this part is that solar sphericity is only important for rays that do not intersect the solar disk, so one can use a plane-parallel model.…”

Section: Polarized Radiative Transfermentioning

confidence: 99%

Hanle diagnostics of weak solar magnetic fields:

Faurobert

2012

A&A

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 C 2 and MgH molecules with different sensitivities to the Hanle effect. We consider six C 2 -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 C 2 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 C 2 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.