Multilevel PRD problem: convergence of the calculation of the Second Solar Spectrum of the Na I D lines

International audienceThe calculation is an application of the multilevel PRD theory built by Bommier (2016, A&A, 591, A59), by applying the atomic density matrix formalism. The application code to the solar Na I D lines was described in Bommier (2016, A&A, 591, A60). However, an error and an imprecision were remaining, which have now been corrected for. The results, theoretical linear polarisation profiles of the Na I D lines observed 4.1 arcsec inside the limb, will be presented. They qualitatively agree with the observed ones, more with the ones of Bommier & Molodij, 2002, A&A, 381, 241, and of Trujillo Bueno et al., 2001, ASP Conf. Ser., 236, 141, who both display purely antisymmetric Na I D1 profiles, rather than with the one by Stenflo & Keller, 1997, A&A, 321, 927, who display a net linear polarisation in Na I D1. A better agreement would be obtained if the dimension number of the atmosphere model was increased. Indeed, the present atmosphere model is 1D, which remains probably highly insufficient for describing these lines, which are formed in the low chromosphere. Emphasis will be put on the convergence of the lambda-iteration in this calculation. It will be shown that although the Stokes parameters I and Q are far from being each converged within the 166 iteration steps of the calculation, simplifications occur by forming their ratio Q/I, which is the emerging linear polarisation rate and the main result of the calculation, leading to sufficient convergence for this ratio in 166 steps. It will be shown that the convergence of the emerging polarisation Q/I is of the Riemann series type for exponent larger than unity (unity is the case of the harmonic series, which does not converge)