text:
\section{Introduction}\label{S1}

The solar spectrum observed close to the solar limb is linearly polarized. The polarization of the continuum, first observed by \cite{Lyot1948}, is mainly produced by scattering at neutral hydrogen (Rayleigh scattering) and free electrons (Thomson scattering). But spectral lines, linearly polarized by scattering processes, show incredibly rich and complex polarizations patterns (e.g., \citealt{StenfloTwerenboldHarvey1983,bStenfloTwerenboldHarvey1983,StenfloKellerGandorfer2000,BGandorfer2000,BGandorfer2002,BGandorfer2005}). This {\em second solar spectrum} (\citealt{Ivanov1991,StenfloKeller1997}) has been the subject of many theoretical investigations because of its diagnostic potential for the magnetism (and thermodynamics) of the solar atmosphere; but some polarization patterns are not yet well understood (see \citealt{Trujillo2009} for a review).

Scattering line polarization is usually modeled independently of the continuum polarization. The continuum polarization is modeled as a coherent scattering process (\citealt{Debarbat1970,FluriStenflo1999,TrujilloShchukina2009}), which is a suitable approximation far from spectral lines. Thomson and Rayleigh scattering are coherent in the scatterer's frame (e.g., \citealt{BChandrasekhar1950}). However, the Doppler broadening corresponding to the thermal velocity of electrons and hydrogen atoms is several times the width of most spectral lines, which may lead to redistribution between the polarization of the spectral line and the nearby continuum (e.g., \citealt{BLandiLandolfi2004}, henceforth LL04).

The effect of non-coherent continuum scattering in radiative transfer was considered by \cite{Munch1948}, but his work did not include light polarization. He showed that the effect of the non-coherence on the intensity line spectrum is to broaden the profile and to make it shallower. The treatment of the Rayleigh and Thomson scattering was first extended to the non-coherent and polarized case by \cite{SenLee1961}, who studied some of the effects that this phenomenon may have on the emergent spectral line radiation. These initial steps were later continued by other researchers (e.g., \citealt{NagendraRangarajanRao1993,Rangarajan1999}) who studied the problem of non-coherent electron scattering and partial frequency redistribution on the polarization of resonance lines, pointing out the significance of electron scattering redistribution in the far wings of the line polarization profile. This result has been recently confirmed by \cite{SupriyaNagendraSampoornaRavindra2012} after solving the same type of problem through the application of more efficient numerical radiative transfer methods.

In this paper we treat the radiation transfer problem of resonance line polarization taking into account its interaction with non-coherent scattering in the continuum. We treat the Rayleigh and Thomson redistribution as angle independent (angle averaged redistribution), and the line emission and absorption using the two-level atom model with unpolarized lower level in the limit of complete frequency redistribution (CRD). To solve the relevant equations, formulated within the framework of the density matrix theory (see LL04), we develop an efficient Jacobian iterative method, which can be considered as a generalization of that proposed by \cite{TrujilloManso1999} for the CRD line transfer case. We apply this numerical method to solve the radiation transfer problem in a Milne-Eddington atmosphere and in a stratified model atmosphere with a temperature minimum and a chromospheric temperature rise. We study the effects of the non-coherence of the continuum scattering on intrinsically unpolarizable (transition between upper and lower levels with angular momentum $J_{u} = J_{\ell} = 1/2$) and polarizable ($J_{u}=1$ and $J_{\ell}=0$) lines. In particular, we show the possibility of generating ``emission" fractional linear polarization features (i.e., with larger polarization than in the adjacent continuum) in the core of intrinsically unpolarizable spectral lines.

\section{Formulation of the problem}

    We consider resonance line polarization (assuming the CRD and two-level atom model without stimulated emission) in the presence of a polarized continuum in a plane-parallel, static and non-magnetic atmosphere. Due to the symmetry of the problem, the radiation field is rotationally invariant with respect to the vertical direction (which we choose to be the $z$ axis) and it is thus linearly polarized along a direction either parallel or perpendicular to the projected limb. Using the reference system for polarization of Fig. \ref{FIGAxis} the radiation field is characterized by just the Stokes parameters $I$ and $Q$. Assuming that the lower level of the transition is unpolarized (either its total angular momentum is $J_{\ell} = 0$ or $1/2$, or collisions dominate its excitation), the absorption process is isotropic and the radiative transfer equations for $I$ and $Q$
at frequency $\nu$ and propagation direction $\vec{\Omega}$ are
\begin{figure}[tb]
\centering
\includegraphics[width=8.6cm]{FIG1.eps}
 \caption{Reference system for polarization. $\theta$ and $\chi$ are the polar and the azimutal angles of the ray under consideration, respectively. $\vec{\Omega}$ is the propagation direction, $\vec{e_{1}}$ is perpendicular to $\vec{\Omega}$ and is on the meridian plane, and $\vec{e_{2}}$ is perpendicular to $\vec{\Omega}$ and $\vec{e_{1}}$. In all the equations, the direction of positive Stokes $Q$ is taken along $\vec{e_{1}}$, i.e., perpendicular to the projected limb.}
\label{FIGAxis}
\end{figure}
\begin{subequations}\label{EQRT}\begin{align}
  \frac{d I}{d \tau} = & I - S_{I} , \label{EQRTI} \\
  \frac{d Q}{d \tau} = & Q - S_{Q} , \label{EQRTQ}
\end{align}\end{subequations}
where $d\tau = -\chi d\ell$ is the element of optical distance (where $\ell$ is the geometrical distance measured along the ray direction), $\chi = \chi_{l}\phi\left(x\right) + \chi_{c}$ is the total absorption coefficient; $\chi_{l}$ and $\chi_{c} = \kappa + \sigma$ are the integrated line and total continuum absorption coefficients, respectively; $\kappa$ and $\sigma$ are the thermal and scattering continuum absorption coefficients; $\phi\left(x\right)$ is the line absorption profile, and $x=\left(\nu - \nu_{0}\right)/\Delta\nu_{D}$ is the frequency separation from the resonance frequency $\nu_{0}$ in units of the Doppler width $\Delta\nu_{D}$. $S_{I}$ and $S_{Q}$ are the source functions, which for a two-level atom with polarized continuum are
\begin{subequations}\label{EQS}\begin{align}
S_{I}\left(x\right)  = &  r_{x} S^{l}_{I} + \left(1 - r_{x} \right) S_{I}^{c}\left(x\right)
, \label{EQSI} \\
S_{Q}\left(x\right) = & r_{x} S^{l}_{Q} + \left(1 - r_{x} \right) S_{Q}^{c}\left(x\right)
, \label{EQSQ}
\end{align}\end{subequations}
where $r_{x} = \chi_{l} \phi\left(x\right) / \left(\chi_{l} \phi\left(x\right) + \chi_{c}\right)$.
The line source functions are expressed in terms of the excitation state of the upper level of the transition. In this case, due to symmetry, the only non-zero spherical components are $\rho^{0}_{0}$ ($\sqrt{2J_{u} +1}$ times the total population) and $\rho^{2}_{0}$ (alignment coefficient) of the density matrix (\citealt{BBlum1981}) of the upper level, and the line source functions are (e.g., \citealt{TrujilloManso1999})
\begin{subequations}\label{EQSline}\begin{align}
S_{I}^{l} = & \frac{2 h \nu^{3}}{c^{2}}\frac{2 J_{\ell} + 1}{\sqrt{2 J_{u} + 1}} \left[\rho^{0}_{0} + \frac{w^{\left(2\right)}_{J_{u} J_{\ell}}}{2 \sqrt{2}}\left(3 \mu^{2} - 1\right) \rho^{2}_{0}\right]
, \label{EQSIline} \\
S_{Q}^{l} = & \frac{2 h \nu^{3}}{c^{2}}\frac{2 J_{\ell} + 1}{\sqrt{2 J_{u} + 1}} \frac{3 w^{\left(2\right)}_{J_{u} J_{\ell}}}{2 \sqrt{2}}\left(\mu^{2} - 1\right) \rho^{2}_{0}
, \label{EQSQline}
\end{align}\end{subequations}
where $w^{(2)}_{J_{u}J_{\ell}}$ is a numerical coefficient which depends on the total angular momentum of the levels involved in the transition (Table 10.1 in LL04; e.g., $w^{\left(2\right)}_{1 0} = 1$, $w^{\left(2\right)}_{\frac{1}{2} \frac{1}{2}} = 0$). $\mu = \cos{\theta}$, where $\theta$ is the angle of the line of sight (LOS) to $z$ (see Fig. \ref{FIGAxis}).

The density matrix elements are obtained from the following statistical equilibrium equations (\citealt{TrujilloManso1999}):
\begin{subequations}\label{EQrho}\begin{align}
\frac{2 h \nu^{3}}{c^{2}}\frac{2 J_{\ell} + 1}{\sqrt{2 J_{u} + 1}}\rho^{0}_{0} = & \left(1 - \epsilon\right)\bar{J}_{0}^{0} + \epsilon B_{\nu}
, \label{EQrho00} \\
\frac{2 h \nu^{3}}{c^{2}}\frac{2 J_{\ell} + 1}{\sqrt{2 J_{u} + 1}}\rho^{2}_{0} = & \frac{1-\epsilon}{1 + \left(1-\epsilon\right)\delta^{\left(2\right)}}w^{\left(2\right)}_{J_{u}J_{\ell}}\bar{J}_{0}^{2}
, \label{EQrho20}
\end{align}\end{subequations}
where $B_{\nu}$ is the Planck function, $\epsilon = C_{u\ell}/(A_{u\ell}+C_{u\ell})$ is the collisional destruction probability due to inelastic collisions ($C_{u\ell}$ and $A_{u\ell}$ are the collisional de-excitation rate and Einstein coefficient for spontaneous emission, respectively) and $\delta^{\left(2\right)}=D^{\left(2\right)}/A_{u\ell}$ ($D^{\left(2\right)}$ is the depolarizing rate of the level due to elastic collisions with neutral hydrogen).

The radiation field tensors in Eqs. \eqref{EQrho} are given by
\begin{subequations}\label{EQJbar}\begin{align}
\bar{J}^{0}_{0} = & \int dx \, \phi\left(x\right) J^{0}_{0}\left(x\right)
,\label{EQJ00bar} \\
\bar{J}^{2}_{0} = & \int dx \, \phi\left(x\right) J^{2}_{0}\left(x\right)
, \label{EQJ20bar}\end{align}\end{subequations}
where $J^{0}_{0}\left(x\right)$ and $J^{2}_{0}\left(x\right)$ are the frequency-dependent radiation field tensors defined as (LL04)
\begin{subequations}\label{EQJx}\begin{align}
J^{0}_{0}\left(x\right)  = & \frac{1}{2}\int_{-1}^{1} d \mu' I\left(x, \mu'\right)
,\label{EQJ00x} \\ \begin{split}
J^{2}_{0}\left(x\right)  = & \frac{1}{4 \sqrt{2}}\int_{-1}^{1} d \mu' \Big[\left(3 \mu'^{2} - 1\right) I\left(x, \mu'\right)  \\  & + 3 \left(\mu'^{2} - 1\right) Q\left(x, \mu'\right)\Big]
.\label{EQJ20x}
\end{split}\end{align}\end{subequations}

The source functions for the background continuum in Eqs. \eqref{EQS}, taking into account thermal emission and scattering, can be expressed as (e.g., \citealt{MansoTrujillo2011})
\begin{subequations}\label{EQScont}\begin{align}
 S_{I}^{c}\left(x\right) = &  s B_{\nu} + \left(1 - s\right)\left[\breve{J}_{0}^{0}\left(x\right) + \frac{1}{2 \sqrt{2}} \left(3 \mu^{2} - 1\right) \breve{J}^{2}_{0}\left(x\right)\right]
, \label{EQSIcont} \\
 S_{Q}^{c}\left(x\right) = & \left(1 - s\right)\frac{3}{2 \sqrt{2}} \left(\mu^{2} - 1\right) \breve{J}^{2}_{0}\left(x\right)
, \label{EQSQcont}
\end{align}\end{subequations}
  where $s = \kappa / \chi_{c}$, with the convolved radiation field tensors
\begin{subequations}\label{EQJconv}\begin{align}
\breve{J}^{0}_{0}\left(x\right) = & \int d x' \phi_{c}\left(x,x'\right) J^{0}_{0}\left(x'\right)
, \label{EQJ00conv} \\
\breve{J}^{2}_{0}\left(x\right) = & \int d x' \phi_{c}\left(x,x'\right) J^{2}_{0}\left(x'\right)
, \label{EQJ20conv}
\end{align}\end{subequations}
where $x'$ and $x$ are the frequencies of the incident and scattered photons, respectively.  The convolution profile $\phi_{c}\left(x,x'\right)$ accounts for the frequency redistribution caused by the Doppler effect, due to the velocity distribution of the scatterers (electrons for Thomson scattering; hydrogen and helium for Rayleigh scattering).

Thomson scattering is coherent in the scatterer's reference system. We take into account the Doppler shifts due to the motions of the electrons relative to the laboratory frame by averaging over their velocity distribution, which we assume to be Maxwellian. We also take the average over the solid angle (greatly reducing the computational cost) because the angular distribution is less important than the frequency distribution (\citealt{BMihalas1978}) and the difference with the angle-dependent distribution function is small for optically thick atmospheres (\citealt{SupriyaNagendraSampoornaRavindra2012}). The final expression for the angle averaged convolution profile is (\citealt{Hummer1962,BMihalas1978})
\begin{equation}
\phi_{c}\left(x,x'\right)=\phi_{c}\left(y\right) = \frac{1}{w}\left[\frac{e^{-y^{2}}}{\sqrt{\pi}} - y\cdot {\rm erfc}\left(y\right)\right]
,\label{EQconvprof}
\end{equation}
with
\begin{equation}
y = \left|\frac{x - x'}{2 w}\right|
,\label{EQy}
\end{equation}
where $w$ is the ratio between the Doppler widths of the perturbers and the atom of interest.

Rayleigh scattering is produced in the far wings of the Lyman lines of neutral hydrogen and helium. We may consider that the scattering in the very far wings of a resonance line is essentially coherent in the scatterer rest frame (e.g., \citealt{BMihalas1978}) and the above discussion for Thomson scattering applies also to Rayleigh scattering taking into account the different value of $w$.

If we consider the simultaneous contribution of Thomson and Rayleigh scattering, $\sigma = \sigma_{\rm{T}} + \sigma_{\rm{R}}$, different source function terms appear for each convolution kernel (Thomson and Rayleigh) and convolved radiation field tensor. For simplicity, we will not write explicitly such expressions here. To avoid a lengthy expression, we consider explicitly only one of the contributions of the background continuum scattering; accounting for additional contributions is straightforward.

The source functions in Eqs. \eqref{EQS} may be expressed in a more simple and symmetric form as
\begin{subequations}\label{EQSt}\begin{align}
S_{I}\left(x\right) = & S^{0}_{0}\left(x\right) + \frac{1}{2 \sqrt{2}}\left(3\mu^{2} - 1\right) S^{2}_{0}\left(x\right)
, \label{EQSIt} \\
S_{Q}\left(x\right) = & \frac{3}{2 \sqrt{2}} \left(\mu^{2} - 1\right) S^{2}_{0}\left(x\right)
. \label{EQSQt}
\end{align}\end{subequations}
Here $S^{K}_{Q}\left(x\right)$ are the frequency-dependent source function tensors:
\begin{subequations}\label{EQSKQdef}\begin{align}
S^{0}_{0}\left(x\right) = & r_{x}S^{0}_{0} + \left(1 - r_{x}\right)S^{0}_{0}{}^{c}\left(x\right)
,\label{EQS00def} \\
S^{2}_{0}\left(x\right) = & r_{x}w^{\left(2\right)}_{J_{u}J_{\ell}} S^{2}_{0} + \left(1 - r_{x}\right)S^{2}_{0}{}^{c}\left(x\right)
,\label{EQS20def}
\end{align}\end{subequations}
where the $S^{K}_{0} = (2 h \nu_{0}^3/c^{2})(2J_{\ell} + 1)/\sqrt{(2J_{u} +1)}\rho^{K}_{0}$ tensors are given by
\begin{subequations}\label{EQSKQldef}\begin{align}
S^{0}_{0} = & \left(1 - \epsilon\right)\bar{J}_{0}^{0} + \epsilon B_{\nu}
,\label{EQS00ldef} \\
S^{2}_{0} = & \frac{1-\epsilon}{1 + \left(1-\epsilon\right)\delta^{\left(2\right)}}w^{\left(2\right)}_{J_{u}J_{\ell}}\bar{J}_{0}^{2}
,\label{EQS20ldef}
\end{align}\end{subequations}
and the continuum frequency-dependent tensors by
\begin{subequations}\label{EQSKQcdef}\begin{align}
S^{0}_{0}{}^{c}\left(x\right) = & \left(1 - s\right)\breve{J}_{0}^{0}\left(x\right) + s B_{\nu}
,\label{EQS00cdef} \\
S^{2}_{0}{}^{c}\left(x\right) = & \left(1 - s\right)\breve{J}_{0}^{2}\left(x\right)
.\label{EQS20cdef}
\end{align}\end{subequations}

\section{Numerical Method of Solution}

   Equations \eqref{EQRT} together with Eqs.\eqref{EQS}-\eqref{EQJconv} or, equivalently, \eqref{EQSt}-\eqref{EQSKQcdef}, form a coupled system of integro-differential equations which we solve numerically. We consider an iterative method of solution: if an estimate of the source functions is given, Eqs.\eqref{EQRT} can be integrated for a given set of boundary conditions; from the radiation field thus calculated we reevaluate the $S^{K}_{0}$ and $S^{K}_{0}\left(x\right)$ tensors which are in turn used to recalculate the new source functions and hence a new radiation field estimate. The formal solution integration (Sect. \ref{S31}) is based on the short-characteristics (SC) method (\citealt{KunaszAuer1988}); in order to guarantee convergence, the iterative scheme  (Sect. \ref{S32}) is a generalization of the Accelerated Lambda Iteration (\citealt{OlsonAuerBuchler1986}) developed by \cite{TrujilloManso1999}, which is based on the Jacobi method.

\subsection{Formal Solution}\label{S31}

  If the source functions are given, Eqs.\eqref{EQRT} can be integrated explicitly between two spatial points $i$ and $j$, for a given frequency and angle:
\begin{equation}
I\left(x;j\right) = I\left(x;i\right) e^{-\Delta\tau_{ij}} + \int_{0}^{\Delta\tau_{ij}} S\left(x;t\right)e^{- t} dt
, \label{EQI}
\end{equation}
  and analogously for $Q$.
  In Eq.\eqref{EQI}, $\Delta\tau_{ij}$ is the optical distance along the ray between points \emph{i} and \emph{j} at the reduced frequency $x$.

We assume that the source function varies parabolically between three consecutive points M, O and P:  O is the point where we want to calculate the Stokes parameters, while M and P are respectively the preceding and following points according to the propagation direction. Eq.\eqref{EQI} can then be rewritten as (\citealt{KunaszAuer1988})
\begin{equation}\begin{split}
I\left(\rm{O}\right) = & I\left(\rm{M}\right) e^{-\Delta\tau_{\rm{M}}} + \Psi_{\rm{M}}\left(\rm{O}\right)S_{I}\left(\rm{M}\right) \\ & + \Psi_{\rm{O}}\left(\rm{O}\right)S_{I}\left(\rm{O}\right)  + \Psi_{\rm{P}}\left(\rm{O}\right)S_{I}\left(\rm{P}\right)
, \label{EQISC}
\end{split}\end{equation}
where $I(\rm{O})$ and $I(\rm{M})$ are the intensities at points O and M, $\Delta\tau_{\rm{M}}$ is the optical distance between points M and O; $S_{I}(\rm{M})$, $S_{I}(\rm{O})$ and $S_{I}(\rm{P})$ are the values of the intensity source function at the points M, O and P, respectively, and $\Psi_{\rm{M}}(\rm{O})$, $\Psi_{\rm{O}}(\rm{O})$ and $\Psi_{\rm{P}}(\rm{O})$ are three functions that only depend on the optical distance between the local point (O in this case) and the preceding and following points (M and P in this equation).

\subsection{Iterative Scheme}\label{S32}

   Equation \eqref{EQISC} expresses the intensity at point O as a linear combination of the source function at adjacent points in the atmosphere and the intensity at a {\em previous} point M along the ray. The same scheme can in turn be applied to the {\em previous} point and repeated all the way back to the boundary where the incoming radiation is given. Therefore, the Stokes parameters at a point $i$ along a given ray in the atmosphere can be expressed as
\begin{subequations}\label{EQRTLD}\begin{align}
I\left(x,\mu;i\right) = \sum_{j = 1}^{N_{z}} \Lambda\left(x,\mu;i,j\right)S_{I}\left(x,\mu;j\right) + T_{I}\left(x,\mu;i\right)
, \label{EQRTLID} \\
Q\left(x,\mu;i\right) = \sum_{j = 1}^{N_{z}} \Lambda\left(x,\mu;i,j\right)S_{Q}\left(x,\mu;j\right) + T_{Q}\left(x,\mu;i\right)
, \label{EQRTLQD}
\end{align}\end{subequations}
where the $\Lambda\left(x,\mu;i,j\right)$ coefficients depend on the optical distances between points ``$i$'' and ``$j$'',
$T_{I,Q}\left(x,\mu;i\right)$ are the {\em transmitted} Stokes parameters from the boundary, and $N_{z}$ the number of spatial grid points.
Averaging these expressions over the angles (Eqs. \eqref{EQJx}) and taking into account the dependence of the source function components
on $S^{K}_{0}$ (Eqs. \ref{EQSt}), the radiation field tensors at a point $i$ in the atmosphere can be expressed as
\begin{subequations}\label{EQJxnum}\begin{align}\begin{split}
J^{0}_{0}\left(x;i\right) = & \sum_{j=1}^{N_{z}}\Lambda^{0}_{0}\left(x;i,j\right)S^{0}_{0}\left(x;j\right) \\ & + \sum_{j = 1}^{N_{z}}\Lambda^{0}_{2}\left(x;i,j\right)S^{2}_{0}\left(x;j\right) + T^{0}_{0}\left(x;i\right)
, \label{EQJ00xnum}\end{split} \\ \begin{split}
J^{2}_{0}\left(x;i\right) = & \sum_{j = 1}^{N_{z}}\Lambda^{2}_{0}\left(x;i,j\right)S^{0}_{0}\left(x;j\right) \\ & + \sum_{j = 1}^{N_{z}}\Lambda^{2}_{2}\left(x;i,j\right)S^{2}_{0}\left(x;j\right) + T^{2}_{0}\left(x;i\right)
.\label{EQJ20xnum}
\end{split}\end{align}\end{subequations}
The explicit expressions for $\Lambda^{K}_{K'}\left(x;i,j\right)$ and $T^{K}_{0}\left(x;i\right)$ in terms of $\Lambda\left(x,\mu;i,j\right)$ and $T_{I,Q}\left(x;i\right)$ are given in the Appendix.
It is important to emphasize that we do not need to calculate them explicity (except for the diagonal elements); they are implicitly evaluated according to the SC algorithm described in the previous section. Equations \eqref{EQJxnum} are only convenient to derive the iterative scheme, as we will now show.

Let $S^{0}_{0}{}^{\rm old}$, $S^{2}_{0}{}^{\rm old}$, $J^{0}_{0}{}^{\rm old}\left(x\right)$ and $J^{2}_{0}{}^{\rm old}\left(x\right)$ be estimates at some iterative step of the atomic and radiation field tensors, and $S^{0}_{0}{}^{\rm old}\left(x\right)$ and $S^{2}_{0}{}^{\rm old}\left(x\right)$ the corresponding frequency dependent source function tensors derived from them using Eqs. \eqref{EQSKQdef}-\eqref{EQSKQcdef}. Let $J^{0}_{0}{}^{\dagger}\left(x\right)$ and $J^{2}_{0}{}^{\dagger}\left(x\right)$ be the values of the radiation field tensors obtained through the formal solution of the radiative transfer equation (Sect. \ref{S31}) using the above-mentioned ``old" quantities ---formally, using $S^{K}_{0}{}^{\rm old}$ on the right hand side of Eqs. \eqref{EQJxnum}.

If we used $J^{K}_{0}{}^{\dagger}\left(x\right)$ to calculate the corresponding $\bar{J}^{K}_{0}{}$ and $\breve{J}^{K}_{0}\left(x\right)$ (Eqs. \eqref{EQJbar} and \eqref{EQJconv}, respectively), and then, Eqs. \eqref{EQSKQdef}-\eqref{EQSKQcdef} to obtain new estimates of $S^{K}_{0}$ and $S^{K}_{0}\left(x\right)$, we would have a generalization of the Lambda iteration scheme which is known to have very poor convergence properties (e.g., \citealt{BMihalas1978}).

In order to improve the convergence rate, let's consider Eqs. \eqref{EQJxnum}. Formally, now we shall calculate the radiation field tensors at a given point ``$i$" from the $S^{K}_{0}{}^{\rm old}\left(x; j\right)$ at all grid points $j\ne i$, and the yet unknown ``new" value $S^{K}_{0}\left(x; i\right)$ at point ``$i$". Rearranging terms:

\begin{subequations}\label{EQJxali}\begin{align}\begin{split}
J^{0}_{0}\left(x;i\right) \approx &  J^{0}_{0}{}^{\dagger}\left(x;i\right) + \Lambda^{0}_{0}\left(x;i,i\right)\delta S^{0}_{0}\left(x;i\right) \\ & + \Lambda^{0}_{2}\left(x;i,i\right)\delta S^{2}_{0}\left(x;i\right)
,\label{EQJ00xali}\end{split} \\ \begin{split}
J^{2}_{0}\left(x;i\right) \approx & J^{2}_{0}{}^{\dagger}\left(x;i\right) + \Lambda^{2}_{0}\left(x;i,i\right)\delta S^{0}_{0}\left(x;i\right) \\  & + \Lambda^{2}_{2}\left(x;i,i\right)\delta S^{2}_{0}\left(x;i\right)
,\label{EQJ20xali}
\end{split}\end{align}\end{subequations}
where
\begin{equation}
\delta S^{K}_{0}\left(x;i\right) = S^{K}_{0}\left(x;i\right) - S^{K}_{0}{}^{\rm{old}}\left(x;i\right)
. \label{EQSKQdelta}
\end{equation}

Equations \eqref{EQJxali} show how to actually compute these new radiation field tensors: $J^{K}_{0}{}^\dagger(x)$ is calculated exactly as explained in the previous paragraph; the diagonal components of the operators $\Lambda^K_{K'}(x; i.i)$ can be efficiently computed while performing the formal solution (see \citealt{AsensioTrujillo2006}); finally, the yet-to-be-obtained $S^{K}_{0}\left(x\right)$ elements are kept explicitly; the whole iterative scheme will be obtained from consistently applying these expressions for $J^{K}_{0}\left(x\right)$ and finally solving the resulting system of algebraic equations for $S^{K}_{0}\left(x\right)$. It can be demonstrated that in solar-like atmospheres the convergence rate of this iterative scheme is practically unaffected if one retains only the zeroth-order Lambda operator $\Lambda^{0}_{0}$ while putting $\Lambda^{0}_{2}=\Lambda^{2}_{0}=\Lambda^{2}_{2}=0$ in Eqs. \eqref{EQJxali} (see \citealt{TrujilloManso1999}). Therefore, we shall develop this simplified Jacobian iterative scheme in the following.

Using Eq. \eqref{EQJ00xali} we calculate the average over the line profile of the radiation field tensor:
\begin{equation}
\bar{J}^{0}_{0}\left(i\right) = \bar{J}^{0}_{0}{}^{\dagger}\left(i\right) + \int dx' \phi\left(x';i\right)\Lambda^{0}_{0}\left(x';i,i\right)\delta S^{0}_{0}\left(x';i\right)
. \label{EQMT03}
\end{equation}
Substituting this equation for the mean radiation field tensor into Eq. \eqref{EQS00ldef} for the source function $S^{0}_{0}$ and subtracting $S^{0}_{0}{}^{\rm old}\left(i\right)$, we find

\begin{equation}\begin{split}
\delta S^{0}_{0}\left(i\right) = & \left(1 - \epsilon\right)\Big[\bar{J}^{0}_{0}{}^{\dagger}\left(i\right) - \bar{J}^{0}_{0}{}^{\rm old}\left(i\right) \\ & + \int dx' \phi\left(x';i\right) \Lambda^{0}_{0}\left(x';i,i\right) \delta S^{0}_{0}\left(x';i\right)\Big]
, \label{EQMT4}
\end{split}\end{equation}
where we have made explicit the height dependence of $\phi\left(x\right)$ (the dependence of $\epsilon$, $r_{x}$ and $s$ is kept implicit). If we substitute into this equation the expression of the source function $S^{0}_{0}\left(x\right)$ of Eq. \eqref{EQS00def}, we obtain
\begin{equation}\begin{split}
\delta S^{0}_{0}\left(i\right) = & \left(1 - \epsilon\right)\Big[\bar{J}^{0}_{0}{}^{\dagger}\left(i\right) - \bar{J}^{0}_{0}{}^{\rm old}\left(i\right)  \\ & + \int dx' r_{x'} \phi\left(x';i\right) \Lambda^{0}_{0}\left(x';i,i\right) \delta S^{0}_{0}\left(i\right) \\ & + \int dx' \left(1 - r_{x'}\right) \phi\left(x';i\right) \Lambda^{0}_{0}\left(x';i,i\right) \delta S_{0}^{0}{}^{c}\left(x';i\right)\Big]
.  \label{EQMT5}
\end{split}\end{equation}
Moreover, defining
\begin{equation}
\bar{\Lambda}^{0}_{0}\left(i,i\right) = \int dx' \phi\left(x';i\right)r_{x'}\Lambda^{0}_{0}\left(x';i,i\right)
,\label{EQL00bar}
\end{equation}
taking into account that
\begin{equation}
\delta S^{0}_{0}{}^{c}\left(x;i\right) = \left(1 - s\right)\int dx' \phi_{c}\left(x,x';i\right)\delta J^{0}_{0}\left(x';i\right)
, \label{EQMT04}
\end{equation}
where
\begin{equation}
\delta J^{0}_{0}\left(x;i\right) = J^{0}_{0}\left(x;i\right) - J^{0}_{0}{}^{\rm old}\left(x;i\right)
, \label{EQMT05}
\end{equation}
noting also that
\begin{equation}\begin{split}
&\int dx' \left(1 - r_{x'}\right)\phi\left(x';i\right)\Lambda^{0}_{0}\left(x';i,i\right) \\
&\times\int dx'' \phi_{c}\left(x',x'';i\right)\delta J^{0}_{0}\left(x'';i\right) = \\
&\int dx' \delta J^{0}_{0}\left(x';i\right)  \\
&\times\int dx'' \phi\left(x'';i\right)\left(1 - r_{x''}\right)\Lambda^{0}_{0}\left(x'';i,i\right)\phi_{c}\left(x',x'';i\right)
, \label{EQMT06}
\end{split}\end{equation}
and using Eqs.\eqref{EQS00cdef} and \eqref{EQJ00conv}, we find that the correction to the line source function is
\begin{equation}\begin{split}
\delta S^{0}_{0}\left(i\right)  = & \frac{\left(1 - \epsilon\right)}{1 - \left(1 - \epsilon\right) \bar{\Lambda}^{0}_{0}\left(i,i\right)}\Big[ \bar{J}^{0}_{0}{}^{\dagger}\left(i\right) - \bar{J}^{0}_{0}{}^{\rm old}\left(i\right) \\ & + \left(1 - s\right)\int dx' \delta J^{0}_{0}\left(x';i\right) \\ & \times\int dx'' \phi\left(x'';i\right)\left(1 - r_{x''}\right)\Lambda^{0}_{0}\left(x'';i,i\right)\phi_{c}\left(x',x'';i\right)\Big]
, \label{EQMT7}
\end{split}\end{equation}
  Applying the same reasoning to the continuum source function, from Eq.\eqref{EQJ00xali}, with $\Lambda_{0}^{2} = 0$
\begin{equation}
J^{0}_{0}\left(x;i\right) = J^{0}_{0}{}^{\dagger}\left(x;i\right) + \Lambda^{0}_{0}\left(x;i,i\right)\delta S^{0}_{0}\left(x;i\right)
. \label{EQMT9}
\end{equation}
  Taking the variation of the field tensor,
\begin{equation}\begin{split}
\delta J^{0}_{0}\left(x;i\right) = & J^{0}_{0}{}^{\dagger}\left(x;i\right) - J^{0}_{0}{}^{\rm old}\left(x;i\right) \\ & + r_{x} \Lambda^{0}_{0}\left(x;i,i\right) \delta S^{0}_{0}\left(i\right) \\ & + \left(1 - r_{x}\right) \left(1 - s\right) \Lambda^{0}_{0}\left(x;i,i\right) \\ & \times\int dx'\phi_{c}\left(x,x';i\right)\delta J^{0}_{0}\left(x';i\right)
, \label{EQMT10}
\end{split}\end{equation}
  and substituting Eq.\eqref{EQMT7} into Eq. \eqref{EQMT10}, after gathering the terms in $\delta J^{0}_{0}\left(x';i\right)$, we obtain:
\begin{equation}\begin{split}
 & \int dx'\Big[ \delta\left(x-x'\right) - \left(1 - s\right)\left(1 - r_{x}\right) \phi_{c}\left(x,x';i\right)\Lambda_{0}^{0}\left(x;i,i\right) \\ & - \frac{ \Lambda^{0}_{0}\left(x;i,i\right) r_{x} \left(1 - \epsilon\right)\left(1 - s\right)}{1 - \left(1 - \epsilon\right)\bar{\Lambda}^{0}_{0}\left(i,i\right)}\\ & \times\int dx'' \phi\left(x'';i\right)\phi_{c}\left(x',x'';i\right) \left(1 - r_{x''}\right) \Lambda^{0}_{0}\left(x'';i,i\right)\Big] \\ & \times\delta J^{0}_{0}\left(x';i\right) = J^{0}_{0}{}^{\dagger}\left(x;i\right) - J^{0}_{0}{}^{\rm old}\left(x;i\right) \\ & + \frac{\Lambda^{0}_{0}\left(x;i,i\right) r_{x} \left(1 - \epsilon\right)\left(\bar{J}^{0}_{0}{}^{\dagger}\left(i\right) - \bar{J}^{0}_{0}{}^{\rm old}\left(i\right)\right)}{1 - \left(1 - \epsilon\right) \bar{\Lambda}^{0}_{0}\left(i,i\right)}
. \label{EQMT11}
\end{split}\end{equation}
  The discretization of Eq. \eqref{EQMT11} in the frequency domain gives a linear system of $N_{x}$ equations (with $N_{x}$ the number of frequency points) for $\delta J^{0}_{0}\left(x\right)$. Substitution of this solution into Eq. \eqref{EQMT7} completes the iterative scheme for $S^{0}_{0}\left(x\right)$.

  As pointed out by \cite{TrujilloManso1999} and stated above, the solution of standard resonance line polarization problems using methods based on Jacobi iteration can simply rely on the diagonal of the $\Lambda^{0}_{0}$ operator. The resulting equation for $\delta S^{2}_{0}\left(x\right)$ is thus formally equivalent to consider Lambda iteration for $S_{Q}$. However, it is crucial to note that $J^{2}_{0}\left(x\right)$ is improved at the rate of $\delta J^{0}_{0}\left(x\right)$, because the anisotropy tensor $J^{2}_{0}\left(x\right)$ is dominated by the Stokes $I$ parameter which is, in turn, basically set by the values of $S^{0}_{0}\left(x\right)$. From Eqs.\eqref{EQSQline}, \eqref{EQSQcont} and \eqref{EQrho20}
\begin{equation}\begin{split}
\delta S^{2}_{0}\left(x;i\right) = & r_{x}\frac{3w_{J_{u}J_{\ell}}^{\left(2\right)}}{2\sqrt{2}}\left(\mu^{2}-1\right)\frac{1 - \epsilon}{1 + \left(1 - \epsilon\right)\delta^{\left(2\right)}}\bar{J}^{2}_{0}{}^{\dagger}\left(i\right) \\ & + \left(1-r_{x}\right)\left(1-s\right)\frac{3}{2\sqrt{2}}\left(\mu^{2}-1\right)\breve{J}^{2}_{0}{}^{\dagger}\left(x;i\right) \\ & - S^{2}_{0}{}^{\rm{old}}\left(x;i\right)
, \label{EQS20delta}
\end{split}\end{equation}
where $\bar{J}^{2}_{0}{}^{\dagger}$ and $\breve{J}^{2}_{0}{}^{\dagger}\left(x\right)$ result from the substitution of $J^{0}_{0}{}^{\dagger}\left(x\right)$ and $J^{2}_{0}{}^{\dagger}\left(x\right)$ into Eqs.\eqref{EQJ20bar} and \eqref{EQJ20conv}.

In summary, at each iterative step we solve the system of equations \eqref{EQMT11} in order to obtain the correction of the $J^{0}_{0}\left(x\right)$ radiation field. Then, we use this result to solve Eq. \eqref{EQMT7}, which gives us the correction for the $S^{0}_{0}$ source function. Finally, Eq. \eqref{EQS20delta} gives us the correction for the $S^{2}_{0}\left(x\right)$ source function.

\subsection{Convergence}\label{S33}

  The numerical method presented in the last section makes use of Jacobi's iterative method both for the line and the continuum part. The simpler alternative of this method is using Lambda iteration for the continuum, which converges provided that the continuum opacity is weak enough with respect to that of the line. The method presented can solve both the CRD line case without continuum opacity and the coherent continuum problem without line opacity, two problems that have different convergence rates.

  In order to illustrate this property of the numerical method we show the convergence rate for three different cases: i) CRD line for a $J_{u} = 1 \rightarrow J_{l} = 0$ transition without continuum, ii) Coherent continuum without line, iii) Non-coherent continuum without line.

  For the first case, we take a Gaussian profile with $\Delta x = 0.1$ (distance between consecutive points in the frequency grid) and $\epsilon = 10^{-4}$. For the continuum cases, we take $s = 10^{-4}$ and, for the non-coherent case, $w = 11.7$ (width of the redistribution profile). We suppose an isothermal atmosphere and we solve with $\Delta z = 0.5$ (distance between consecutive points in the height grid, in units of the opacity scale height) and $N_{\mu} = 60$ Gaussian nodes for angular integration in each hemisphere. We present the corresponding convergence rates in Fig. \ref{FIGRC}.

\begin{figure}[htb]
\centering
\includegraphics[width=8.6cm]{FIG2.eps}
 \caption{Maximum relative change of $S^{0}_{0}$ (solid lines) and $S^{2}_{0}$ (dashed lines) at each iterative step for a (CRD) resonance line without continuum (black lines), and for the continuum case  without line (gray lines). We point out that the convergence rates for the coherent and non-coherent cases are indistinguishable. For the continuum case the source function is frequency dependent, but here we take a fixed frequency because the convergence rate is virtually identical for all of them.}
\label{FIGRC}
\end{figure}

  To demonstrate the virtue of the method with respect to the continuum treatment, we solve a problem where we include both a weak line and continuum, but using Lambda iteration for the continuum part. We take the same parameters used in Fig. \ref{FIGRC} and $\chi_{l}/\chi_{c} = 10$. This is a weak spectral line case, so the final rate of convergence is greatly influenced by the Lambda iteration of the continuum, that has a very poor convergence rate (see Fig. \ref{FIGRC2}).

\begin{figure}[htb]
\centering
\includegraphics[width=8.6cm]{FIG3.eps}
 \caption{Maximum relative change of $S^{0}_{0}$ (solid lines), $S^{2}_{0}$ (dashed lines) and $J^{0}_{0}\left(x\right)$ (coincident with the solid line) at each iterative step for the CRD line transfer problem with continuum using the method described in section \ref{S32} (black lines) and using Lambda iteration for the continuum part (gray lines).}
\label{FIGRC2}
\end{figure}

  The code can also use \cite{NG1974} acceleration to decrease the total computing time. To show its efficiency we solve the problem of Fig. \ref{FIGRC2} using NG acceleration of third order. The number of iterative steps needed to reach convergence is greatly reduced without increasing significantly the computing time at each iterative step (see Fig. \ref{FIGRC3}).

\begin{figure}[htb]
\centering
\includegraphics[width=8.6cm]{FIG4.eps}
 \caption{Maximum relative change of $S^{0}_{0}$ (solid lines), $S^{2}_{0}$ (dashed lines) and $J^{0}_{0}\left(x\right)$ (coincident with the solid line) at each iterative step for the CRD line transfer problem with coherent and non-coherent continuum, with (black lines) and without (gray lines) NG acceleration.}
\label{FIGRC3}
\end{figure}

\subsection{Numerical Considerations}\label{S34}

  The precision of the numerical method depends on the parameters of the discretization in space, angles and frequencies. In the figures that are shown in Section \ref{S4} we take the following discretizations. A spatial height axis from $z_{\rm min} = -16$ to $z_{\rm max}=11$ or $16$ (this is more than needed to have an optically thick atmosphere at the bottom and an optically thin surface) with $\Delta z = 0.1$ or $0.5$, with the height $z$ measured in units of the opacity scale height. We use Gaussian quadrature with $60$ nodes at each hemisphere and a frequency axis that reaches $320$ Doppler widths with $\Delta x = 0.125$ in the core and with $\Delta x$ increasing with the distance to the resonance frequency ${\nu}_{0}$ until having $\Delta x = 15$ in the far wings.

   In order to demonstrate the reliability of our radiative transfer code, we solve the radiation transfer problem in a plane-parallel homogeneous atmosphere, relying on the fact that the $\sqrt{\epsilon}$-law (\citealt{AvrettHummer1965}; generalized to the polarized case by \citealt{Ivanov1990,LandiBommier1994}) provides an exact analytical result for the solution of this problem. We solve two of the problems of section \ref{S33}: coherent scattering in the continuum without line and resonance line without continuum.

  In the far wings of the line, the spectrum can be considered frequency independent. The source function equations are thus simplified as
\begin{subequations}\label{EQScoh}\begin{align}
S_{I} = & S^{0}_{0}{}^{c} + \frac{1}{2\sqrt{2}}\left(3\mu^{2}-1\right)S^{2}_{0}{}^{c}
, \label{EQSIcoh} \\
S_{Q} = & \frac{3}{2 \sqrt{2}}\left(\mu^{2} - 1\right) S^{2}_{0}{}^{c}
, \label{EQSQcoh}
\end{align}\end{subequations}
with
\begin{subequations}\label{EQScoh2}\begin{align}
S^{0}_{0}{}^{c} = & s B_{\nu} + (1 - s)J_{0}^{0}
, \label{EQSIcoh2} \\
S^{2}_{0}{}^{c} = & (1 - s)J^{2}_{0}
. \label{EQSQcoh2}
\end{align}\end{subequations}
  The $\sqrt{\epsilon}$-law gives us the relation $\left(S^{0}_{0}{}^{c}\right)^{2} + \left(S^{2}_{0}{}^{c}\right)^{2} = s$. In Table \ref{TBLCOH} we show the relative error between the numerical result and this analytical relation; the agreement is very satisfactory.

\begin{table}[htbp]
\centering
\caption{Verification of the $\sqrt{\epsilon}$-law for Different Values of $s$}
\begin{tabular}{ccc}
\hline \hline\\[-2.5ex]
$s$ & $\left(S_{0}^{0}{}^{c}\right)^{2} + \left(S^{2}_{0}{}^{c}\right)^{2} $ & Error (\%) \\[0.5ex]
\hline
$10^{-4}$ & $1.001\cdot 10^{-4}$   & $0.1$  \\
$10^{-3}$ & $1.0006\cdot 10^{-3}$ & $0.06$   \\
$0.01$      & $0.01002$                    & $0.02$  \\
$0.1$        & $0.100004$                  & $4\cdot 10^{-3}$  \\
$0.5$        & $0.500002$                  & $3\cdot 10^{-4}$  \\
$0.9$        & $0.90000007$              & $7\cdot 10^{-6}$  \\
\hline \hline
\end{tabular}
\label{TBLCOH}
\end{table}

   In the absense of continuum the source function equations become
\begin{subequations}\label{EQSlinepure}\begin{align}
S_{I} =  &  S^{0}_{0}  + w^{\left(2\right)}_{J_{u} J_{\ell}}\frac{1}{2 \sqrt{2}}\left(3 \mu^{2} - 1\right) S^{2}_{0}
, \label{EQSIlinepure} \\
S_{Q} = & w^{\left(2\right)}_{J_{u} J_{\ell}}\frac{3}{2 \sqrt{2}}\left(\mu^{2} - 1\right) S^{2}_{0}
, \label{EQSQlinepure}
\end{align}\end{subequations}
with
\begin{subequations}\label{EQSlinepure2}\begin{align}
S^{0}_{0} = & \epsilon B_{\nu} + \left(1 - \epsilon\right)\bar{J}^{0}_{0}
, \label{EQSIlinepure2}\\
S^{2}_{0} = & w^{\left(2\right)}_{J_{u} J_{\ell}}\frac{1}{1 + \left(1 - \epsilon\right)\delta^{\left(2\right)}}\bar{J}^{2}_{0}
. \label{EQSQlinepure2}
\end{align}\end{subequations}
In Table \ref{TBLPUL} we check the $\sqrt{\epsilon}$-law for this line transfer problem with $\delta^{\left(2\right)} = 0$; the law is satisfied with good agreement.

\begin{table}[htbp]
\centering
\caption{Verification of the $\sqrt{\epsilon}$-law for Different Values of $\epsilon$.}
\begin{tabular}{ccc}
\hline \hline\\[-2.5ex]
$\epsilon$ & $\left(S_{0}^{0}\right)^{2} + \left(S^{2}_{0}\right)^{2}$ & Error (\%) \\[0.5ex]
\hline
$10^{-4}$ & $1.0005\cdot10^{-4}$ & $0.05$  \\
$10^{-3}$ & $1.0003\cdot10^{-3}$ & $0.03$   \\
$0.01$ & $0.010001$ & $0.01$  \\
$0.1$ & $0.100003$ & $3\cdot10^{-3}$  \\
$0.5$        & $0.500001$ & $3\cdot10^{-4}$  \\
$0.9$        & $0.90000006$ & $7\cdot10^{-6}$  \\
\hline \hline
\end{tabular}
\label{TBLPUL}
\end{table}

  In the next section we apply our radiative transfer code to some particular cases, where we have both line and continuum. We study some of the effects of the non-coherence of the scattering.

\section {Illustrative Examples}\label{S4}

   In this section we present some results of radiative transfer calculations in some model atmospheres. First, we make calculations in Milne-Eddington atmospheres with constant opacity ratios, because they are suitable for understanding the physics involved. Secondly, we suppose some ad-hoc variation with height of the properties of a model atmosphere with a temperature minimum and a chromospheric temperature rise. In both cases we consider line transitions with and without intrinsic polarization, the last case being quite interesting in terms of the emergent fractional polarization profile.

\subsection{Non-coherent Scattering in Milne-Eddington atmospheres}\label{S41}

   We study the interaction between a resonance line and the continuum radiation for two cases where non-coherent scattering in the continuum is taken into account or neglected. We assume a Milne-Eddington atmosphere with constant ratios between the different opacities involved. The important parameters in this model are the ratio between the opacity of the line and the continuum, $r =\chi_{l}/\chi_{c}$, and the relative weight of the thermal part to the total opacity of the continuum, $s = \kappa/\chi_{c}$.

  We assume an intrinsically unpolarizable resonance line ($J_u = J_\ell =1/2$) and an intrinsically polarizable line ($J_u = 1$, $J_\ell = 0$). In both cases we solve the radiative transfer problem for a strong line $(r = 1000)$ and for a weak line $(r = 10)$ with $\epsilon = 10^{-4}$. For the continuum redistribution width we take $w = 11.7$ (value that we choose thinking in a forthcoming application to a realistic model; in particular, this number is the ratio between the Doppler widths of Barium and Hydrogen), and different values of $s$. We use a Milne-Eddington atmosphere with slope $3/2$.

\begin{figure*}[htp]
\centering
\includegraphics[width=17.2cm]{FIG5.eps
\caption{Emergent fractional polarization close to the limb $(\mu = 0.1)$ in a Milne-Eddington atmosphere for three different values of $s = \kappa/\left(\kappa+\sigma\right)$. The left panels are for an intrinsically unpolarizable transition ($J_u = J_\ell =1/2$) and the right panels are for a polarizable transition ($J_u = 1$, $J_\ell = 0$). Black solid lines correspond to the non-coherent scattering case, while gray dashed lines are for the case of coherent scattering in the continuum. The considered line strength values ($r$) are given in the panels.}
\label{FIGME
\end{figure*}

The non-coherent continuum scattering produces changes in the shape of the emergent fractional polarization profile, as has already been demonstrated in previous works (see Sect. \ref{S1}). When we study a strong ($r = 1000$) unpolarizable line, the coherent profile gives zero polarization in the core of the line, as expected. It is interesting to note that the redistribution produced by the non-coherent scattering polarizes the core of the line, although its $Q/I$ amplitude lies always below the continuum polarization level, i.e., the line always depolarizes the continuum. Thus, aside from being wider, the fractional polarization profile also shows non-zero polarization in the line core. The same happens to the $Q/I$ profile in the case of a weak ($r = 10$) intrinsically unpolarizable line (see Fig. \ref{FIGME}, left). The change in the $Q/I$ profile is larger for weaker lines and smaller $s$ values (or, equivalently, the more important is the scattering in the continuum).

   If we consider an intrinsically polarizable line, in order to obtain a noticeable change in the fractional polarization profile due to the non-coherent scattering in the continuum, we need the scattering coefficient $\sigma$ to be dominant over the thermal absorption term (small $s$). The smaller $s$, the more the polarization profiles changes. In all the cases shown in the right panels of Fig. \ref{FIGME} the intrinsic polarization of the line is dominant in its core and the non-coherence smoothes and broadens the fractional polarization profile in the wings of the line for small enough values of $s$.

\subsection{Non-coherent Scattering in a Stratified Atmosphere with a Chromospheric Temperature Rise}\label{S42}

    We assume now a certain height variation of the parameter $s$ and of the Planck function in order to obtain a more realistic stratification in the model atmosphere. Inspired by semi-empirical models of the solar atmosphere, we choose $\sigma$ in a way such that $1-s$ tends to unity near the surface and goes to zero at the bottom of the atmosphere (see Fig. \ref{FIGVARMO}, and note that $1 - s = \sigma/\left(\kappa + \sigma\right)$). We use two models that differ in the scattering coefficient. The variation with height of the scattering coefficient in the model $1$ is larger than in the model $2$ and the value of the scattering coefficient is the same at the height where the line integrated optical depth is unity. With these atmospheric models, we solve the two-level atom line transfer problem with $\epsilon = 10^{-4}$ and a non-coherent scattering redistribution width $w=11.7$, both for an intrinsically unpolarizable line ($J_u = J_\ell =1/2$) and for a polarizable one ($J_u = 1$, $J_\ell = 0$)

\begin{figure}[htp]
\centering
\includegraphics[width=8.6cm]{FIG6.eps}
\caption{One dimensional atmospheric models. These two models have the same temperature, integrated line opacity $\chi_{l}$ and continuum thermal absorption $\kappa$ variations, but different behaviors for the continuum scattering coefficient $\sigma$. The upper panel shows the Planck function versus the integrated line optical depth. The middle panel shows the scattering coefficient $\sigma$ versus the optical depth, with the black line indicating the model $1$ and the gray line the model $2$. The bottom panel shows the quantity $1 - s = \sigma/\left(\kappa + \sigma\right)$, and we point out that it has the typical variation that can be found in semi-empirical models, such as those of \cite{FontenlaAvrettLoeser1993}.}
\label{FIGVARMO
\end{figure}

   For a strong and unpolarizable line, the coherent profile shows a strong depolarization in the core (Fig. \ref{FIGVARQI}, dashed lines). However, the non-coherent $Q/I$ profile is strongly modified because the scattering redistribution produces polarization in the core of the lines (Fig. \ref{FIGVARQI}, solid lines). For the model $1$ (see Fig. \ref{FIGVARMO}), we see that the line does not fully depolarize the continuum, but the core is polarized and the profile is wider. For the model $2$ the non-coherent scattering generates an emission $Q/I$ profile (see Fig. \ref{FIGVARQI}).

\begin{figure*}[htp]
\centering
\subfigure{\includegraphics[width=8.6cm]{FIG7a.eps}
\hspace{8pt
\subfigure{\includegraphics[width=8.6cm]{FIG7b.eps}}
\caption{Emergent fractional polarization close to the limb ($\mu = 0.1$). The left panel shows an intrinsically unpolarizable transition ($1/2\rightarrow1/2$) and the right panel shows a polarizable one ($1\rightarrow0$). Solid lines show the non-coherent scattering solution, while the dashed lines the coherent scattering one. The black and gray lines correspond to the two different models indicated in Fig. \ref{FIGVARMO}}
\label{FIGVARQI
\end{figure*}

  For a strong polarizable line, the coherent $Q/I$ profile shows the expected polarization emission in the core of the line. For the model $1$ the polarization in the core of the line does not change, and the main effect of the non-coherent scattering is the smoothing of the peaks in the wings of the line and the broadening of the $Q/I$ profile. For the model $2$, the redistribution is able to change even the polarization in the core of the line, while producing a smoother and wider $Q/I$ profile.

What we want to emphasize with Fig. \ref{FIGVARQI} is that \emph{the non-coherent scattering in the continuum can be important and, under certain conditions, there can be an emission feature in the fractional linear polarization profile even when a total depolarization is expected.}

  Finally, we study the influence of the mass of the scatterer. To this aim, we take a Milne-Eddington atmosphere with slope $3/2$ and the $\sigma$ variation of the model $2$ in Fig. \ref{FIGVARMO}. We solve the radiative transfer problem for a $J_{\ell} = J_{u} = 1/2$ transition with gaussian absorption profile, with different widths of the redistribution function (this width is inversely proportional to the square root of the mass of the scatterer).

   For small widths we approach the coherent case, where the line is depolarized. As we increase the width of the velocity redistribution profile, the linear polarization in the core of the line increases. In Fig. \ref{FIGVARWpro} we show some fractional polarization profiles for several values of the widths of the redistribution profile. If we take the center of the line as reference and we plot the fractional polarization at this frequency versus the widths of the redistribution profile, we obtain the curve shown in Fig. \ref{FIGVARW}, where we have also indicated the continuum fractional polarization level. In this figure we can see that from a given value of $w$ the line-center signal of the $Q/I$ profile lies above the continuum level and increases to an asymptotic value. From this figure we can infer that, for a given value of $s$, Thomson scattering  (whose associated width is approximately $43$ times the Doppler width of Hydrogen) produces a greater polarization than Rayleigh scattering for an intrinsically unpolarizable line.

 \begin{figure}[htb]
\centering
\includegraphics[width=8.6cm]{FIG8.eps}
\caption{Fractional polarization profiles close to the limb ($\mu = 0.1$) for a $1/2\rightarrow 1/2$ transition in a Milne-Eddington atmosphere, with the $\sigma$ and $1 - s$ variations given by the model $2$ of Fig. \ref{FIGVARMO}, for different widths of the scattering redistribution function. The dotted line shows the coherent case.}
\label{FIGVARWpro
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics[width=8.6cm]{FIG9.eps}
\caption{Line center fractional linear polarization close to the limb ($\mu = 0.1$) for a $1/2\rightarrow 1/2$ transition in a Milne-Eddington atmosphere (with $1-s$ given by the model $2$ of Fig. \ref{FIGVARMO}) versus the width of the scattering redistribution function. The gray line represents the fractional continuum polarization amplitude.}
\label{FIGVARW
\end{figure}

\section{Conclusions}

In this paper we have studied the radiative transfer problem of resonance line polarization taking into account non-coherent continuum scattering, paying particular attention to the fractional linear polarization $Q/I$ signals that can be produced around the core of intrinsically unpolarizable lines. We used the two-level atom model with CRD and angle-averaged non-coherent scattering in the continuum. To numerically solve this type of radiative transfer problem we developed a Jacobian iterative method for the line and continuum source functions, which yields a fast convergence rate even in the case of very small line strengths. The formulation of the numerical method makes it very suitable for a direct generalization to partial frequency redistribution and angle-dependent non-coherent scattering.

We have shown that, under certain conditions, the non-coherent continuum scattering can change dramatically the core spectral region of the emergent $Q/I$ profile with respect to that calculated assuming coherent continuum scattering. Interestingly, $Q/I$ polarization signals above the continuum level can be generated in the core of intrinsically unpolarizable $(1/2\rightarrow 1/2)$ lines (i.e., in spectral lines that were expected to simply depolarize the continuum polarization level). This result is of great potential interest for a better understanding of some enigmatic spectral lines of the second solar spectrum, which showed $Q/I$ line-center signals above the continuum polarization level in spite of resulting from transitions between levels that were thought to be intrinsically unpolarizable (see Stenflo et al. 2000). Of particular interest for a first application is the D$_1$ line of Ba {\sc ii} at 4934 \AA, especially because 82\% of the barium isotopes have nuclear spin $I=0$ (i.e., their D$_1$ line transition is indeed between an upper and lower level with total angular momentum $J_{u} = J_{\ell} = 1/2$). In fact, our preliminary calculations for the Ba {\sc ii} D$_1$ line (neglecting the contribution of the 18\% of barium that has hyperfine structure) suggest that under certain stellar atmospheric conditions the physical mechanism discussed in this paper can produce significant $Q/I$ emission features.

Finally, we point out that the core of strong lines with intrinsic polarization are practically not affected by the non-coherent scattering. Therefore, the effects of the non-coherent scattering in the continuum are not always relevant and depend on the spectral line under study.

\acknowledgements
Financial support by the Spanish Ministry of Economy and Competitiveness through projects \mbox{AYA2010--18029} (Solar Magnetism and Astrophysical Spectropolarimetry) and CONSOLIDER INGENIO CSD2009-00038 (Molecular Astrophysics: The Herschel and Alma Era) is gratefully acknowledged.

redpajama_set_name: RedPajamaArXiv


text:
Home James McMurtry
Avalon Theatre 40 E Dover St, Easton, Maryland 21601
"The simple fact is that James McMurtry may be the truest, fiercest songwriter of his generation..."
Stephen King, Entertainment Weekly
The son of acclaimed author Larry McMurtry (Lonesome Dove, Terms of Endearment), James grew up on a steady diet of Johnny Cash and Roy Acuff records. His first album, Too Long in the Wasteland (released in 1989), was produced by John Mellencamp and marked the beginning of a series of acclaimed projects.
Never one to rest on his laurels, McMurtry tours year round and consistently throws down unparalleled powerhouse performances. The Washington Post notes: "Much attention is paid to James McMurtry's lyrics and rightfully so: He creates a novel's worth of emotion and experience in four minutes of blisteringly stark couplets. What gets overlooked, however, is that he's an accomplished rock guitar player ... serious stuff, imparted by a singularly serious band."
Location Avalon Theatre 40 E Dover St, Easton, Maryland 21601 View Map
Avalon Foundation March shows

redpajama_set_name: RedPajamaCommonCrawl


Description: Send to your loved ones, this Thrilling Seasonal Selection Gift Basket which is an ultimate expression of love. Featuring a wide selection of assortments, this Hamper will make an ideal birthday gift as it is composed of Cupido Latte Dessert Cups (25 g), Tresor Dore Chocolate Wafer (100 g), Tresor Dore Hazelnuts Chocolate Cookies (135 g), Delfi Almonds Milk Chocolate (180 g), Bonz Hazelnut Chocolate Cookies (70 g), Lotte Pepero Stick Biscuit Chocolate (30 g), Win2 Moroll Wafer Rolls (100 g), Forum Belgian Biscuits Speculoos Choc (125 g) and Da San Yuan Mixed Fruit Candy (120 g). All these items are presented in a beautiful Festive Basket.


Amos Bronson Alcott was an American teacher, writer, philosopher, and reformer. As an educator, Alcott pioneered new ways of interacting with young students, focusing on a conversational style, and avoided traditional punishment. He hoped to perfect the human spirit and, to that end, advocated a vegan diet before the term was coined. He was also an abolitionist and an advocate for women's rights.
Born in Wolcott, Connecticut in 1799, Alcott had only minimal formal schooling before attempting a career as a traveling salesman. Worried about how the itinerant life might have a negative impact on his soul, he turned to teaching. His innovative methods, however, were controversial, and he rarely stayed in one place very long. His most well-known teaching position was at the Temple School in Boston. His experience there was turned into two books: Records of a School and Conversations with Children on the Gospels. Alcott became friends with Ralph Waldo Emerson and became a major figure in transcendentalism. His writings on behalf of that movement, however, are heavily criticized for being incoherent. Based on his ideas for human perfection, Alcott founded Fruitlands, a transcendentalist experiment in community living. The project was short-lived and failed after seven months. Alcott continued to struggle financially for most of his life. Nevertheless, he continued focusing on educational projects and opened a new school at the end of his life in 1879. He died in 1888.
Alcott married Abby May in 1830 and they eventually had four surviving children, all daughters. Their second was Louisa May, who fictionalized her experience with the family in her novel Little Women in 1868.


Adopt-A-Pet.com News < "Free Dog" and "Free Kitten" posts aren't safe! Help spread the word!
It is very common to see posts on classified sites such as Craigslist and Nextdoor with headlines such as "free dog" or "free kitten." What many people are unaware of though is that posting pets as "free to a good home" on these sites can be very dangerous. When a pet is posted as free on a site that doesn't provide any tools or guidance on selecting the right new family, it's easy for that vulnerable animal to wind up in the wrong hands.
These messages are just suggested wording – feel free to get creative and come up with your own as well! Thank you so much for helping to keep "free dog" and "free kitten" posts off of classified sites and giving pets the chance to be rehomed in a safer way by sharing Rehome!
P.S., The sweet pups in the photo above are available for adoption through Rehome! View them here!


text:
Workshop – Brazil's Integrated Education and Solidarity Economy: Opening Pathways to Income and Citizenship
by Tula Connell | Jul 31, 2013
Brazil's Integrated Education and Solidarity Economy: Opening Pathways to Income and Citizenship
• Ruth Needleman, Professor Emerita, Labor Studies Program, Indiana University, USA
• Eunice Maria Dias Wolf, Secretary of Social Development, City of Canoas, Brazil
• Mardeli de Quadros Rosa, General Coordinator, Associação de Mulheres do Multiplicar, Brazil
• Maria Regina dos Santos Braga, President, Cooperativa Vida Saudável, Brazil
David Welsh, Solidarity Center Country Program Director, Cambodia
Alex Feltham, Solidarity Center Senior Program Officer, Asia
Mardeli de Quadros Rosa (left) and Ruth Needleman described Brazilian unions' successful popular education campaign. Photo: Tula Connell
Ruth Needleman opened the workshop, which focused on a labor-run popular education project in Brazil, by stating that "the role of the union is to go beyond membership, to go into the community." At the center of that goal is understanding the importance of education.
Needleman then overviewed the program, in which the Central Unitario de Trabajadores (CUT) engaged in a far-reaching outreach project based on the concept of "integrated education," which combines traditional disciplines with experience-based learning. CUT centered its program in Quilombo, a primarily Afro-Brazilian state far from urban centers, initially training 3,000 local residents to lead the community-based program. The project, which ran from 2003 to 2006, was unique in targeting all nonunion members, and was especially successful in empowering women, many of whom were illiterate and suffered from domestic abuse.
Needleman gave an example of how the program empowered women by describing the experience of Anna, who had worked from age 12 at a shoe factory. She couldn't help her son with homework and he didn't respect her because of her lack of knowledge. So she jumped at the opportunity to get an elementary school education through the unions' integrated learning process. She went to school five nights a week for three months after working 10 hours day. Education changed everything, she said. "Now, I am somebody. Before, I was nobody and treated as a nobody."
After the initial year, the union trained 2,000 new teachers a year. Classroom activity involved computer training, investigation and analysis. Students learned traditional subjects through practical examples. For instance, they learned math by examining the price of bread as it goes up and down, or absorbed science through identifying pollution in their communities.
The process creates a community of support, and members of this community then go on to meet with government officials and other decision-makers, to push them to implement the changes identified by the community.
The project also involved training workers to market their products. In Quilombo, the union trainers found that the Afro-Brazilians planted a unique genus of rice. They tested it and found it is much more nutritious than Asian rice. In previous centuries, black slaves tucked rice grains in their hair so they would have some to plant in case they were uprooted, thus keeping that specific genus viable. When they learned how to market their rice, they went from "being people with no self worth" to people selling goods, Needleman said.
The National Metalworkers Union (SNM) took the concept of integrated education even further and re-structured elementary school education, creating an intensive, four-night a week program. Many of the participants were low-wage workers, like domestic workers.
The trade unions involved in the popular education project "made an enormous commitment to people outside the union," Needleman said. "This is very important, because most workers, including women, are outside unions."
Needleman believes CUT ended the project because it returned to a focus on membership, "and so became isolated from society." Further, the Lula government's "solidarity economy has been a tremendously important means for transforming the economy."
Eunice Maria Dias Wolf spoke briefly, describing the SNM, where she has worked for 28 years, and overseen the creation and mission of Associação de Mulheres do Multiplicar, showing images of the women involved in the association. She also noted the SNM offers a training program for women in such areas as collective bargaining and leadership training because the union wanted to ensure all local unions in the metalworkers' federation had women on board.
She ended with a quote from Simone de Beauvoir: "Only work can ensure concrete independence."
See her full presentation.
Mardeli de Quadros Rosa noted that before becoming general coordinator of Associação de Mulheres do Multiplicar, she was the former treasurer (2003-2006) of the BMBC—Biscuits, Pasta and Homemade Biscuits, a micro enterprise.
She overviewed Associação de Mulheres do Multiplicar, founded in 1997, which was created with the support of the Metalworkers. Its mission is to promote social inclusion of women to fight all forms of oppression and violence. The organization holds courses on citizenship, women's rights, women's health and has added a new course, one focused on domestic violence. It's the only organization in the state (Rio Grande do Sol) that looks after victims of domestic violence.
Maria Regina dos Santos Braga is a former student of the union education program, and now president of a cooperative that employs women seamstresses. "When I started doing this work, housework was all I knew," she said. "I am very proud to have taken the long walk these women have taken. So imagine the multiplier effect—we hope to grow and grow."
The Cooperativa Vida Saudável, a cooperative of seamstresses, recycles donated clothes and the workers customize, renovate and fix them, then sell them at subway stops. For instance, the seamstresses transform towels and turn them into handkerchiefs. Many families benefit from this income the seamstresses earn.
Santos Braga works in the Healthy Life Cooperative, which has four sections, including one where women produce cakes and bread for sale and another where they make jelly and cookies. They also cater parties and weddings. The Associação de Mulheres do Multiplicar was key for the formation of this cooperative. The association offers courses for women, and when they first start, "they have very low self-esteem," she said, and sometimes are victims of domestic violence. "We have to rescue women."
Eunice Maria Dias Wolf closed the workshop by paying tribute to an association member, Patricia Esber, who was murdered at age 32 by her husband, the victim of domestic violence.
"Cooperative sharing is intrinsic to our tradition. People engaged in popular education must have this vision," she said. "We believe it's in our capacity for teamwork, against individualism."
See the presentation of Quadros Rosa, Santos Braga and Wolf. (Portuguese)


At this time we have a selection of 43 holiday rental properties in San Pedro Alcantara. The most economical apartment can be rented for €750 per week and offers 95 m² of living space featuring a terrace of 25 m². The most expensive 5 bedroom villa requires an investment of €40,000 per week. To get more information about any of the holiday rental properties in San Pedro Alcantara submit an enquiry through the contact form or call a number you will find on the property details page.


Elegant and understated, this is the perfect clicker for almost every occasion. It makes a gorgeous septum ring and adds a touch of sparkle to your daith piercing too! Available in many different sizes so you can find the perfect fit. Get tips on finding your size here.
This clicker is made from fine silver, and is 100% nickel free, fantastic for sensitive skin. Find out more about why I use fine silver instead of sterling here.


We are located in front of the CHRISTUS Trinity Mother Frances Winnsboro Hospital in the grey brick building labeled "Medical Building 3."
U.S. Dermatology Partners Winnsboro is a community-based rural health practice in Winnsboro, TX. Our practice focuses on early detection and treatment of skin cancer and general diseases and conditions of the skin.
Led by certified nurse practitioner Roxanne Swafford, we deliver a lifetime of skin care from childhood to middle age to the golden years. We treat conditions such as acne, psoriasis and eczema to relieve or improve symptoms that limit your comfort, health and enjoyment. We provide specialized, highly effective treatments for a variety of skin cancers to restore and extend the quality of your life.
Whether you suffer from pain, irritation or embarrassment from a persistent skin condition or if you've noticed a suspicious new spot or mole, we can help. Contact us to schedule an appointment today.


Chamber After Hours - Guthrie Real Estate — Guthrie Chamber Calendar of events for Guthrie Oklahoma. These are community events and events hosted by the Guthrie Chamber of Commerce.
After Hours features an opportunity for businesses to highlight themselves by bringing other members in for a night of relaxed and casual networking. This event is held each month in a new business location and attendance is included in your membership with the Chamber. Come out and visit with other members and begin building new business relationships.


Northrup Canyon is usually a surprise for the first-time visitor. Each turn in the old road-bed access trail offers a new habitat type, as one moves into the steep-walled valley. Just off the highway, and looking out of place amid all the basalt, is a huge granite monolith relic. Apparently it was too solid and massive to yield to the geologically recent Lake Missoula Floods that scoured out the extensive basalt canyon. Dramatic panoramas of the steep cliffs are more typical of the rest of the immediate area. Next comes a shady stretch of Ponderosa pine and Douglas fir, growing out of the talus. While gradually gaining elevation, you soon encounter an extensive meadow. Within view is an old homestead, beyond which lies Northrup Lake. Spring-fed Northrup Creek winds its way back down through the lower half of the canyon. While it can be difficult to navigate along its entire length back out to the highway, it is prime riparian habitat.
Up to 100 communal roosting Bald Eagles have been counted here during the winter months. Year-round residents have included Brown Creeper, Red-breasted Nuthatch, Bewick's and Canyon Wrens, Wild Turkey, Ringed-neck Pheasant, Black-capped and Mountain Chickadees, and Downy Woodpecker. Great Horned and Long-eared Owls are reported here. Say's Phoebes and Lazuli Buntings, Western Wood-Pewees summer in the canyon.
From Route 2 northeast of Coulee City, take Route 155 north along Banks Lake to mile 18.9. Turn right to a new pay-to-park area established above the turn-off from the highway. Approaching from Grand Coulee, turn south from Route 174 and proceed to the same mile-marker, about seven miles' distance.
The canyon is prime rattlesnake habitat, so caution is advised. An interesting side trail traverses the east canyon wall following the old Almira-Bridgeport wagon road. Northern Saw-whet and Northern Pygmy-Owls frequent the larger firs along this route. This trail is closed during the November-to-March Bald Eagle winter roosting cycle. A Bald Eagle Festival is held in mid-February.


text:
The Nizam's Rubath is an accommodation building for pilgrims coming to the Mecca. It was purchased by the Nizam for the people of Hyderabad State. In 2017, a total of 3,400 pilgrims were selected for free accommodation.

Located about 2 km from the Grand Mosque of Mecca, the Rubaath now comes under the "AWQAF (Endowment) Committee" of The H.E.H. the Nizam's Charitable Trust.

History
Originally there were 42 Rubath buildings purchased in 1860 by V th Nizam, Afzal ad-Dawlah, Asaf Jah V for the people of Hyderabad State, but only a couple are left after the expansion of the grand mosque of Mecca.

With the massive expansion of Makkah's Grand Mosque, the 140-year-old Rubath today comprises only three buildings, containing 500 rooms that can accommodate a minimum of about 1200 people. The people who use the facilities of Rubaths do not have to pay any fee.

Pilgrims from the erstwhile Nizam's dominion (Hyderabad State) were accepted here. This would translate to today's Telangana, Marathwada and Hyderabad Karnataka regions who can apply for the Rubath accommodation. Since there is a great demand for accommodation during the Hajj season, the pilgrims are selected through a draw of applicants.

It is located in Ibrahim Al Khaleel Street, Al Misfalah in Mecca. It has capacity for nearly 500 people.

See also
 Nizams of Hyderabad

References

External links
Official site

Hyderabad State
Islam in India

redpajama_set_name: RedPajamaWikipedia


text:
Memorial Service for Robert West
The Working Films team is grateful for the care and support we've received from so many in the days since Robert West, our co-founder and long-time executive director passed away. Hearing and reading stories of love and appreciation for Robert – for his strategic thinking, generosity, passion for the work, and joux de vivre – expressed by friends and colleagues from around the world has made this difficult time much easier.
The memorial service celebrating Robert's life and legacy will be held on Saturday, June 22nd, at 2:00 PM, at the Cameron Museum of Art, 3201 South 17th Street, Wilmington, NC. It will be followed by a celebration that he planned at the historic firehouse at 602 South 5th Avenue, which was Robert's home and is also the headquarters of Working Films.
If you are planning or even considering coming to Wilmington for the memorial, please let us know as soon as possible by emailing or calling Andy Myers (amyers@workingfilms.org / 910-342-9000). We can help make sure you have all the information you need and your stay is comfortable. We've reserved a block of rooms for Saturday night, June 22nd at the Hilton in downtown Wilmington. You can book a room in the block by calling the Hilton directly at 910-763-5900 by Friday 6/14. The Reservation ID is WST.
For those who would like to honor Robert's life and legacy, contributions can be made to the Robert West Reel Engagement Fund, which will support an issue-driven Working Films' initiative developed and evolved under Robert's leadership. Dedicated to innovation, strategy and on-the-ground implementation, this Fund, just like Robert, will be nimble, dynamic, creative and responsive to the many urgent needs for change. Robert's obituary as printed in the Wilmington Star News is below.
Again, thank for your concern and solidarity. It is sincerely appreciated.
The Working Films' Staff and Board
Robert Kyle West
Robert Kyle West, Co-founder and long-time Executive Director of Working Films, a nationally recognized non-profit dedicated to linking non-fiction films to activism, passed away at home in Wilmington, North Carolina on June 6, 2013 from brain cancer (GBM).
Robert and his twin sister Jane West were born on April 6, 1953, in Philadelphia, PA to parents Roberta Kyle West and Joseph F. West. As a child he resided in Lansdowne, PA and Cape May Point, NJ. In 1960, Robert's father died, and in 1967 his mother, a native Virginian, moved the family to Richmond, VA. Robert attended Douglas Freeman High School and Virginia Commonwealth University where he developed what would become a life-long passion for art, history, politics, literature and film.
Early jobs in publishing and bookselling led West to the Mint Museum in Charlotte, NC, where he was curator of film and video from 1985-1999. He was most noted for founding and directing the Charlotte Film and Video Festival, drawing fiction, non-fiction, experimental filmmakers and animators from all over the U.S. and the globe. In this same period he and a team of friends also founded Charlotte's Gay and Lesbian Film Series.
Robert considered the use, curation and moderation of a public convening – especially one built around an independent film – an art form in itself. In 1999 he turned his deep belief in the power of film to build community and galvanize hearts, minds and concrete change, into the nonprofit Working Films, which he co-founded with filmmaker Judith Helfand. That year he moved to Wilmington, NC to establish his home and the headquarters of Working Films in a historic firehouse on South Fifth Avenue.
Robert and Working Films were pioneers in the field of media engagement, crafting long-term and innovative partnerships between filmmakers and some of the nation's and world's leading nonprofits and NGOs. The engagement methodologies he crafted are now part of the parlance of making, funding, distributing and actively using non-fiction films, nationally and increasingly internationally – particularly in the UK where he collaborated with BRITDOC. The working motto he loved and helped coin: STORY LEADS TO ACTION.
Robert tirelessly gave back to the field by sitting on boards including the National Alliance for Media Arts and Culture , serving on juries and nomination committees including the Paul Robeson Fund for Independent Media, the Rockefeller Media Fellowships, the Media Arts Fellowship Program of the NEA, Creative Capital, the NC Arts Council and The Independent Television Service, a production arm of PBS and serving on film festival panels including NC's Cucalorus and Full Frame, The Center for Social Media's "Making Media that Matters", Sundance and in the last few years, Sheffield (UK) and Eso-Docs (EU).
An out gay man since he was twenty-five, Robert had an unrelenting commitment to the rights and equality of all people. He truly believed that everyone has the right to "be who they are." Robert credits his uncle, Charles M. Butterworth, Jr., a Harvard trained lawyer who devoted his time and skills to the catholic worker movement, with giving him the gift of "considering a life committed to justice outside the mainstream."
Robert's surviving siblings and family members include; Fred and Hillary West of Chestnut Hill, PA and their children Kyle and Emily; Jane and Bruce Bench of Seattle, WA and their children, Robert, Jordan and Christopher; and brother John West of Santa Barbara, CA. He is predeceased by brother Joseph West, Jr. Robert also deeply loved his friends from Wilmington, Charlotte, NY and around the world. Too numerous to name, he appreciated their thoughtful care and their many messages of support throughout this year.
Robert turned the past very difficult year into a series of life-giving lessons offered to family and friends: Be Who You Are; Unplug from Technology, Take the Time to Feel the Wind and the Sun, and Be Present. It is in that spirit that we will gather on Saturday, June 22nd, at 2:00 PM, at the Cameron Museum of Art, 3201 South 17th Street, Wilmington, NC for a memorial service celebrating Robert's life and legacy. Memorials in Robert West's honor may be made to the Robert West Reel Engagement Fund at Working Films, 602 S. Fifth Ave, Wilmington, NC 28401 or workingfilms.org/robertwest. Online condolence may be sent to the family at Atlantic Cremation Service.
susan collins says:
Dear Family of Robert West,
I was so very saddened to learn about Robert's passing-on.
I can't tell you the number of times that I tried to find him
over the years! I have a lovely pastel pencil drawing of his over my desk right now, – that he did of a single geranium that I had. It's still as
lovely as when he drew it, about 39 years ago now, in Richmond.
I can see that he certainly used his many many gifts well, over the years to bless so many. He was such good company, – and so well read and enlightened! I only wish I had spoken to him before he left for his new home, – but I know that he is safe, and giving to others wherever he is, right now. What a wonderful model of kindness, wisdom, joy and love he is. Though I'm so sorry again, that he has left us, – I am forever glad that I knew him.
Janie, if you read this, please do get in touch! I tried finding you also!
I've never forgotten our car trip across the country! Please, again, do be in touch if you receive this. I am in the big apple, –NYC. And listed in the CS journal.
I send much much love, indeed,
and my prayers, condolence and love to all of Robert's family, and friends too,
Susan S. Collins
This is a magnificent tribute to Robert K.West.
If any of his family or friends see this, I would greatly
like to talk or write with any of them. I had no idea that Robert
had passed-on, – and have tried many many times to find him
over the years.
This is a beautiful memorial to him! Just tremendous.
{Not sure why my letter on this site didn't get off.
But, again, would love to hear from anyone close to him,
and or Jane too! She was a good friend. hopefully Ill be
able to locate her.


Considering Valentine's Day is only 11 days away and no one has mentioned it in this house yet, leads me to think we'll be keeping it simple this year. In fact, we keep it pretty low key every year. But this year we'll be lucky if we get to eat dinner uninterrupted and can watch Netflix without clicking the 'children' profile. When it comes to decorations and gifts I think it'll be pretty minimalist (is it ok to skip presents?) but I have made this floral candle ring to add a little romance to the proceedings… whatever they turn out to be.
If you're like me and want an easy Valentine's Day this year you can check out the tutorial over on Hunker!


Compliment your summer wardrobe or casual look with Pep. Championship winning quality through out this quality braided belt, made from a mix of premium Italian cowhide and imitation leather elastic braid. The belt tip is debossed with our famous stag logo. Made in Italy.


Apartment 303 Harbour Bridge is a bright, north-facing apartment located in the heart of South Africa's Mother City, Cape Town.
(Fibre-Optic) Wi-Fi throughout the apartment adds to the luxury this apartment offers. We are sure you will fall in love with by the end of your stay.
major banks, the new Netcare Christiaan Barnard Memorial hospital, gyms (Virgin Active is approximately 750m), museums etc. We suggest visiting the Zeitz Museum of Contemporary Art in the new Silo building, which is on route to the Waterfront.