Master equation theory applied to the redistribution of polarized radiation in the weak radiation field limit

Abstract

Context. In previous papers of this series, we presented a formalism able to account for both statistical equilibrium of a multilevel atom and coherent and incoherent scatterings (partial redistribution). Aims. This paper provides theoretical expressions of the redistribution function for the two-term atom. This redistribution function includes both coherent (RII) and incoherent (RIII) scattering contributions with their branching ratios. Methods. The expressions were derived by applying the formalism outlined above. The statistical equilibrium equation for the atomic density matrix is first formally solved in the case of the two-term atom with unpolarized and infinitely sharp lower levels. Then the redistribution function is derived by substituting this solution for the expression of the emissivity. Results. Expressions are provided for both magnetic and non-magnetic cases. Atomic fine structure is taken into account. Expressions are also separately provided under zero and non-zero hyperfine structure. Conclusions. Redistribution functions are widely used in radiative transfer codes. In our formulation, collisional transitions between Zeeman sublevels within an atomic level (depolarizing collisions effect) are taken into account when possible (i.e., in the non-magnetic case). However, the need for a formal solution of the statistical equilibrium as a preliminary step prevents us from taking into account collisional transfers between the levels of the upper term. Accounting for these collisional transfers could be done via a numerical solution of the statistical equilibrium equation system.

Importance of Angle-dependent Partial Frequency Redistribution in Hyperfine Structure Transitions Under Incomplete Paschen-Back Effect Regime

“…1(b) and 2(b)). We recall that, traditionally the loops in the polarization diagram (namely, a plot of Q/I versus U/I for a given wavelength and for a range of magnetic field strength or orientation values) are identified to be due to the level-crossings in the incomplete PBE regime (see e.g., Bommier 1980;Landi Degl'Innocenti & Landolfi 2004, see also Sowmya et al 2014). When a given curve in the polarization diagram forms a loop the Q/I and U/I values tend towards the non-magnetic value.…”

“…The collisional PRD matrix for a two-term atom without HFS and in the incomplete PBE regime is given in Equation (A.1) of Bommier (2017Bommier ( , 2018. By using the following quantum number replacement L → J; J → F ; S → I; J * → F * in Equation (A.1) of Bommier (2018), we obtain the collisional PRD matrix for a two-level atom with HFS.…”

“…Subsequently McKenna (1985) and Faurobert (1987Faurobert ( , 1988 considered the effects of type-II 2 and type-III 3 AD-PRD functions of Hummer (1962) on linear polarization profiles of resonance lines. The case of weak field Hanle effect with AD-PRD matrices of Bommier (1997, given by the so-called Approximation-II) was considered by Nagendra et al (2002), while the case of scattering in arbitrary fields was considered by Sampoorna et al (2008Sampoorna et al ( , 2017. The above cited papers solved the transfer equation in the Stokes vector basis.…”

“…and the corresponding transfer solutions were presented in Anusha & Nagendra (2012). The decomposition technique of Anusha & Nagendra (2011) is particularly useful for handling the Approximation-II of Bommier (1997) for weak field Hanle effect, and for any geometry. The usefulness of their technique for the planar geometry is presented in Supriya et al (2013a).…”

Master equation theory applied to the redistribution of polarized radiation in the weak radiation field limit

“…This is indeed a multilevel problem and the solution was enabled following Bommier (2016a). This two-term redistribution matrix is given in Bommier (2017), with a corrigendum in Bommier (2018, thanks to Ernest Alsina Ballester).…”