\documentclass[journal]{rmaa}

\usepackage[latin1]{inputenc}


\title{Kinematics from spectral lines for AGN outflows based on time-independent radiation-driven wind theory}


\author{
J. M. Ram\'irez\affil{Leibniz-Institut f\"{u}r Astrophysik Potsdam, Germany}}


\shortauthor{Ram\'irez}

\shorttitle{Kinematics for AGN outflows}


\fulladdresses{
\item J. M. Ram\'irez: Leibniz-Institut f\"{u}r Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany (jramirez@aip.de).}


\listofauthors{J.~M.~Ram\'irez}

\indexauthor{Ram\'irez, J. M.}


\SetYear{2011}
\ReceivedDate{2011 January 17}
\AcceptedDate{2011 August 5}


\resumen{Construimos un modelo de fotoionizaci\'on dependiente de la velocidad para el absorbedor tibio de NGC 3783. Adoptando formas funcionales 
de la velocidad del flujo y su densidad del n\'umero de part\'iculas con el radio, apropiadas para un viento acelerado por radiaci\'on, calculamos 
el nivel de ionizaci\'on, la temperatura, el corrimiento Doppler y las profundidades \'opticas de las l\'ineas como funci\'on de la distancia. El 
modelo reproduce la relaci\'on observada entre la ionizaci\'on del gas y el corrimiento del centroide de las l\'ineas de absorci\'on en NGC~3783. 
Este espectro requiere de la presencia de dos flujos: uno altamente ionizado responsable de las alas azules de las l\'ineas de alto grado de 
ionizaci\'on y al mismo tiempo las alas rojas de las l\'ineas de menor grado de ionizaci\'on; y uno de baja ionizaci\'on que produce las alas azules 
de las l\'ineas de ox\'igeno.}


\abstract{We build a bulk velocity-dependent photoionization model of the warm absorber of the Seyfert 1 galaxy NGC 3783. By adopting functional forms 
for the velocity of the flow and its particle density with radius, appropriate for radiation-driven winds, we compute the ionization, the temperature, 
the line Doppler shift, and the line optical depths as a function of distance. The model reproduces the observed relationship between the gas ionization 
and the velocity shift of the absorption line centroids in the X-ray spectrum of NGC 3783. The distribution of asymmetry seen in this spectrum requires 
the presence of two outflows: a higher ionization component responsible for the blue wings of the high ionization lines and the red wings of the low 
ionization oxygen lines, and a lower ionization flow that produces the blue wings of the oxygen lines.\bigskip}


\addkeyword{galaxies: Seyfert}
\addkeyword{quasars: absorption lines}
\addkeyword{quasars: individual (NGC 3783)}



\begin{document}


\maketitle


\section{Introduction}

X-ray (0.5--10~keV) {\it Chandra} spectra of type-I active galactic nuclei (AGN) often show the presence of absorption lines coming from H- and He-like 
ions of O, Ne, Mg, Si, S as well as from \ion{Fe}{xvii}-\ion{Fe}{xxiii} L-shell transitions. These lines are near the region of the bound-free 
absorption edges of \ion{O}{vii} and \ion{O}{viii} at $\sim$0.8~keV, which are the hallmark of warm absorbers \citep{george1998a,komossa1999a}. In 
Seyfert 1 galaxies these absorption lines can be narrow (Seyfert 1 NALs) with full width at half maximum (FWHM) spanning $\sim$100--500~km~s$^{-1}$ 
\citep{kaspi2000a}, or broad (Seyfert 1 BALs) with FWHM $\sim$500--2000~km~s$^{-1}$ \citep{kaspi2001a}. The troughs of the lines are blue-shifted 
relative to rest frame of the host galaxy with velocities that go from several hundred to a few thousand km~s$^{-1}$ \citep{yaqoob2003a}, implying the 
presence of outflows that cover a wide range of velocities and ionization stages \citep{kaastra2000a,kaspi2002a,krolik2001a,rozanska2006a}.

In Seyfert galaxies and quasars, absorption spectra show a range of ionization stages that arise from ionization parameters that span 
$\xi$$\sim$10--1000~erg~cm~s$^{-1}$ \citep{kaspi2002a}. Simultaneous UV and X-ray observations may enhance this range even further \citep{shields1997a,gabel2005a}. 
The location of these absorbing material is uncertain. Models of X-ray absorbers in AGN place them in a wide range of distances from the central source, 
from winds originated at the accretion disk \citep{murray1995a,elvis2000a}, out to the dusty ($\sim$1~pc) torus \citep{krolik2001a} and beyond the 
narrow-line region \citep[e.g.,][]{ogle2000a}. 

In the Seyfert galaxy NGC~3783 the 900~ks {\it Chandra} spectrum of \citet{kaspi2002a} allows the precise measurements of radial velocities and widths 
of the lines. It is seen that the velocity shift of lines from \ion{Fe}{xxiii}-\ion{Mg}{xii} covers a range of $\sim$60--600~km~s$^{-1}$ while the 
low-ionized lines \ion{Si}{xiii}-\ion{O}{vii} cover velocities $\sim$500--1000~km~s$^{-1}$ (see Figure~6 in Ram\'irez, Bautista, \& Kallman 2005). The average 
velocity of the warm absorber outflow of NGC~3783 is around $\sim$500~km~s$^{-1}$. The spectrum also reveals that the line profiles are asymmetric 
\citep{kaspi2002a}, in such a way that approximately 90\% of the lines have extended blue wings. Such asymmetries were quantified by Ram\'irez et al. 
(2005). In terms of ionization, mostly high ionization species are seen in the short-wavelength portion of the spectrum $\sim$4--12~\AA. Here, resonant 
lines from \ion{Fe}{xxiii}, \ion{Fe}{xxii}, \ion{Fe}{xxi}, \ion{S}{xvi}, \ion{S}{xv}, \ion{Si}{xiv}, and \ion{Mg}{xii} cover ionization parameters 
$\xi$ from $\sim$630 to $\sim$150~erg~cm~s$^{-1}$. The longer wavelengths of the spectrum, $\sim$10--20~\AA, are dominated by lower ionization lines from 
\ion{Si}{xiii}, \ion{Fe}{xviii}, \ion{Ne}{x}, \ion{Mg}{xi}, \ion{Fe}{xvii}, \ion{Ne}{ix}, \ion{O}{viii}, and \ion{O}{vii} that span ionization parameters 
$\xi$ from $\sim$125~erg~cm~s$^{-1}$ down to $\sim$8~erg~cm~s$^{-1}$. 

Thus, photoionization modeling shows that the observed spectrum is hardly explained by a single ionization parameter. Rather two or three components 
are needed, and this has been the subject of well-detailed studies \citep{krongold2003a,netzer2003a,krongold2005a}. In these papers two/three components 
in pressure equilibrium are enough to account for all the charge states seen in the spectra. At the start of these simulations, the ionization parameter 
is not kept constant throughout the cloud but made to vary in a self-consistent way. On the other hand, \citet{steenbrugge2003a} used a nearly continuous 
distribution of ionization parameters that spans three orders of magnitude. In reality both approaches do not need to be contradictory. We just have to 
look at the possibility of introducing species between the components proposed by \citet{krongold2003a,netzer2003a,krongold2005a}; and this immediately 
leads us to the discussion of whether the material absorbing is clumpy or not. And this is in part the goal of the present work. We present both 
possibilities, continuous {\it vs} clumpy flows, with their physical implications.

\vspace{0.05cm}

Although there is no evident correlation between the ionization of the absorption species and the velocity shifts of lines \citep{kaspi2002a}, 
\citet{ramirez2005a} suggested the possibility for such relationship to exist, as the lines are consistent with being originated from an outward  flowing 
wind. In that work a wind velocity law was adopted (Castor, Abbott, \& Klein 1975), with an ionization fraction that goes as a power-law of the ionization 
parameter $q\sim \xi^{\eta}$ with $\eta>0$ for all $\xi$ and all lines. Further, $q\sim \xi^{\eta}$ and an optical depth of the form proposed by the SEI 
method from Lamers, Cerruti-Sola, \& Perinotto (1987) were used to fit the lines in the spectrum and to study the possible relationship between the profiles 
and the expansion velocity of the flow.

\vspace{0.05cm}

Here, in this theoretical approach no analytical dependence of $q$ with $\xi$ is assumed, but it is computed self-consistently using the photoionization 
code {\sc xstar} with modifications of the optical depth of the lines to work with an expanding outflow.

\vspace{0.05cm}

It is the aim of the present paper to further study the wind scenario for the warm absorber of NGC~3783 by modeling a photoionized wind flow and trying 
to reproduce the main features in the spectrum in terms of both equivalent widths and line profiles. The model consists of a spherically symmetric gas 
flow with velocity increasing with radius according to  the wind velocity law of \citet{castor1975a}. The microphysics of the plasma is then solved in 
detail along the wind, and radiative transfer is treated for the flowing plasma including Doppler shift effects on the emerging spectrum. This kind of 
approach has been used previously in studies of the ionization and thermal properties of O stars by \citet{drew1989a}, on cataclysmic variables by 
\citet{drew1985a} and in an evaluation of absorption line profiles from winds in AGN by \citet{drew1982a}.

\vspace{0.05cm}

Self-consistent hydrodynamic modeling of AGNs has been performed in the past \citep[see e.g.,][]{murray1995a} who qualitatively predict the same 
$\beta$-velocity law, where $v(r)\sim (1-r_0/r)^{\beta}$, as adopted here. However, these models did not include a detailed treatment of the thermal 
and spectral processes that lead to synthetic spectra.

\vspace{0.05cm}

The present paper is organized as follows: In \S~\ref{signi1} we discuss the statistical significance of the X-ray lines under study. Later in \S~\ref{meth}, 
we present the theoretical method we have used in this work. In \S~\ref{results1} we present the main results of our work. We present the final physical 
solution in \S~\ref{twoouts1}. We summarize in \S~\ref{summ1}.


%%%%% FIGURA 1

\begin{figure}[!t]
\centering
\includegraphics[angle=-90,width=0.99\columnwidth]{jramirezfig1.eps}
\caption{Statistical significance of the shifting of the X-ray lines found in the 
spectrum of NGC~3783. Each box is made of five-number summaries: the smallest 
observation (sample minimum, which are the thin bars at the extreme left of each 
box), a lower quartile (Q1, which is the left thick border of each box), the median 
(Q2, black thick vertical line), the upper quartile (Q3, the right thick border of 
each box) and largest observation (sample maximun, which are the thin bars at 
the extreme right of each box).}
\label{boxplot1}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Significance of the shifting in the X-ray Lines}
\label{signi1}

Since our analysis searches for possible correlation between the ionization state of the ions forming the absorption lines observed in the X-ray 
spectrum of this object, and the Doppler velocity shift of those lines, we first study the statistical significance of this possible shifting. 
A convenient way to do this is through the separation of the ions into groups to search for the velocity shifting. We take all the ions we find 
in Table~\ref{tblasim1} (see Table~3 below). The line's centroids are taken from \citet{kaspi2002a}. The ionization fraction curves for the 
classification are taken from \citet{kallman2001a,bautista2001a}. Afterwards we collect them into three groups\footnote{We group and generate the 
figure using the statistical package R (\url{http://www.r-project.org}).}: Group 1, representative of all ions of low ionization with ionization 
parameter $0<\xi({\rm erg~cm~s^{-1}})<10$; Group 2, $10<\xi({\rm erg~cm~s^{-1}}) <250$, representative of intermediate ionization state; and Group 
3, $250<\xi({\rm erg~cm~s^{-1}})<650$. A graphical way to represent this grouping scheme is shown in Figure~\ref{boxplot1}. Each box is made of 
five-number summaries: the smallest observation (sample minimum, which is the thin bar at the extreme left of each box); a lower quartile (Q1), 
which is the left thick border of each box; the median (Q2), the black thick vertical line; upper quartile (Q3), the right thick border of each 
box and largest observation (sample maximum) which is the thin bar at the extreme right of each box. The first conclusion is that all three Q2s 
are different. It is clear that Groups 1 and 3 are different at $\gtrsim$1$\sigma$. There is overlap between Groups 1 and 2 (possibly due to limited 
resolution of the instrument), but Groups 2 and 3 are different at $\gtrsim$1$\sigma$. We continue our analysis bearing in mind these overlaps in 
the groups.


\section{Method}
\label{meth}

Let us consider a radiation source with $L$$\sim$$10^{44}-10^{47}$~erg~s$^{-1}$, arising from a supermassive ($\sim$$10^8~M_{\odot}$) black hole (BH). 
Material $\sim$0.1--1 pc from the BH only needs to absorb a small fraction of this energy to be accelerated to few thousand km~s$^{-1}$ in Seyfert 
galaxies and up to 0.1--0.2~$c$ in high redshift Quasars (Arav, Li, \& Begelman 1994; Ram\'irez 2008; Saez, Chartas, \& Brandt 2009; Chartas et al. 
2009). By conservation of mass the number density of hydrogen can be written, assuming spherical symmetry, as
\begin{equation}
n_H(r)= \frac{\dot{M}}{4\pi r^2 v(r)\mu m_H}, 
\label{den1}
\end{equation}
where $\dot{M}$ is the mass-loss rate, $v(r)$ is the outflow speed at radius $r$, $\mu$ is the mean atomic weight per hydrogen atom and $m_H$ is 
the hydrogen mass.

We adopted a velocity law $v(r)$ compatible with the predictions of the radiatively driven wind theory \citep{castor1975a}. The velocity law has 
two fundamental roles. The first is to shift the frequency of the absorbing lines according to the Doppler effect. The second is to dilute the gas 
density, affecting radiative transfer across the gas  and consequently the ionization and thermal state of the gas. The velocity law varies with 
distance as
\begin{equation}
w(x)=w_0+(1-w_0)\left(1-\frac{1}{x}\right)^\beta.
\label{vel1}
\end{equation}
Here, $w$ is the velocity normalized to the terminal velocity of the wind $v_{\infty}$, $w_0$ is the velocity in the base of the wind and $x$ is the 
distance normalized to the radius of the central core $r/r_0$. The parameter $\beta$ is the quantity governing the slope of the velocity with the 
distance and its {\it ad hoc} value depends on the type of radiative force acting on the wind. Analysis of hot stars suggests that 0.5~$\lesssim \beta \lesssim$~1 
\citep{lamers1987a}. The evaluation of radiation-driven wind in AGNs of \citet{drew1982a}, and more recent dynamical calculations  (e.g., Murray 
et al. 1995; Proga, Stone, \& Kallman 2000), suggest that in AGNs 0.5~$\lesssim \beta \lesssim$~2. We have computed models using velocity laws 
$\beta=0.5,1,1.5$ and 2, representing a range from fast ($\beta=0.5$) to slow winds ($\beta=2$). We found that fast winds were not able to simultaneously 
cover the range in velocity and ionization state observed in the spectrum of NGC~3783, needed for the goal of this work, and that the best-fit 
velocity law was $\beta=2$. This is why for the rest of this work we use $\beta=2$. Then, we rewrite the number density in terms of the velocity law as, 
\begin{equation}
n_H(x)=n_0x^{-2}w^{-1},
\label{den2}
\end{equation}
where $n_0=\left(\frac{\dot{M}}{4\pi \mu m_H} r_0^{-2} v_{\infty}^{-1}\right)$. The absorbing line frequencies are shifted according to the Doppler relation,
\begin{equation}
w=\frac{c}{v_\infty} \left(1-\frac{\lambda}{\lambda_0}\right),
\label{dop1}
\end{equation} 
where $\lambda_0$ is rest wavelength. 
 
Our models are based on clouds illuminated by a point-like X-ray source. The input parameters are the source spectrum, the gas  composition, the gas 
density $n_H(x)$, and the outflow velocity $w(x)$, where $x$  is the position of each slab inside the cloud normalized to the radius of the most exposed 
face to the source, $r_0$. The source spectrum is described by the spectral luminosity $L_{\epsilon}=Lf_{\epsilon}$, where $L$ is the integrated luminosity 
from 1 to 1000~Ryd, and $\int_{1}^{1000{\rm Ryd}} f_{\epsilon} d\epsilon=1$. This spectral function is taken to be a power law $f_{\epsilon}\sim \epsilon^{-\alpha}$, 
with $\alpha=1$. The gas consists of the following elements, H, He, C, N, O, Ne, Mg, Si, S, Ar, Ca and Fe. We use solar abundances of Grevesse, Noels, 
\& Sauval (1996), in all our models.

Thermal and statistical equilibrium in our models are computed with the code {\sc xstar} \citep{kallman2001a,bautista2001a}. The code includes all 
relevant atomic processes and computes the equilibrium temperature and optical depths of the most prominent X-ray and UV lines identified in AGN spectra. 

We consider two types of models, the single absorber model (SA) and the multicomponent model (MC). In both cases the absorption line profiles depend 
simultaneously upon the ionization and the kinematics of the absorbing gas.


\subsection{The Single absorber Model}
\label{sm}

This model consists of a single extended cloud directly in the line of sight between the observer and the central source. Such a cloud flows away from 
the central source and towards the observed. For this model we make use of the Sobolev approximation for computing the line optical depths \citep{castor1975a},
\begin{equation}
\tau_{\nu}(r)= \frac{\pi e^2} {mc}f\lambda_0 n_i(r) \left(\frac{dv}{dr}\right ) ^{-1},
\label{opt1b}
\end{equation}
where $f$ is the absorption oscillator strength, $\lambda_0$ (in cm) is the laboratory wavelength of the transition, $n_i$ (in cm$^{-3}$) is the number 
density of the absorbing ion, and $dv/dr$ is the velocity gradient in the wind. This gives us the relation between the outflow state and the radiation 
field. This is different from the calculation of the optical depth in the static case,  which is directly proportional to the column density.  

Inside the cloud we use a one-step forward differencing formula for the radiation transfer \citep{kallman1982a}
\begin{equation}
L_{\nu}(r+\Delta r)= L_{\nu}(r) e^{-\tau_{\nu}(r)} +4\pi r^2j_{\nu}(r) \frac{1-e^{-\tau_{\nu}(r)}} {\kappa_{\nu}(r)},
\label{tra1}
\end{equation}
where $j_{\nu}(r)$ is the emission coefficient at radius $r$.

Under ionization equilibrium conditions the state of the gas depends just upon the shape of the ionizing spectrum and the ionization parameter $\xi$, that
 we define as in Tarter, Tucker, \& Salpeter (1969)
\begin{equation}
\xi(r)=\frac{4\pi F(r)}{n_H(r)},
\label{xi}
\end{equation}
where $F(r)$ is the total ionizing flux,
\begin{equation}
F(r)=\frac{1}{4\pi r^2}  \int_{1}^{1000 {\rm Ryd}} L_{\nu}(r)d\nu,
\end{equation}
with $L_{\nu}(r)$ given by equation~(\ref{tra1}). The requirement that $\xi$ spans various orders of magnitude, as observed, yields a cloud geometrically 
thick throughout most of the spectrum, i.e. a {\it photoionization bounded} cloud. For instance, a luminosity $L\sim 10^{44}$~erg~s$^{-1}$ and 
$\Delta \xi = \xi_2-\xi_1=\frac{L}{n} (\frac{1}{r^2 _2} -\frac{1}{r^2 _1})=10^3$ yield $n\sim10^{6}$ cm$^{-3}$ and the column density of the absorbing material 
is $N_H\sim n \Delta R \sim 9.7 \times 10^{24}$~cm$^{-2}$. This is a large value, such that even lines with moderately small oscillator strengths become 
saturated in the emergent spectrum, unless the metal abundances are reduced by several orders of magnitude with respect to solar. 


%%%%% FIGURA 2

\begin{figure}[!t]
\centering
\includegraphics[angle=-90,width=0.99\columnwidth]{jramirezfig2.eps}
\caption{Optical depth ($\tau$) static case vs outflow model for the line 
\ion{Ne}{x} $\lambda$12.134 (see text).}
\label{nex1}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In Figure~\ref{nex1} we compare the computed optical depth for the \ion{Ne}{x} $\lambda$12.134 line in a stationary nebula with $L=10^{44}$~erg~s$^{-1}$, 
and constant density $n= 10^{6}$~cm$^{-3}$ with a flow with $n(x)=3.3\times 10^{6}x^{-0.5}w^{-1}$, a wind velocity function with $\beta$= 2, and 
$v_{\infty}=1000$~km~s$^{-1}$. In both cases we adopt a turbulence velocity of 200~km~s$^{-1}$ and solar abundances. In the stationary case, $\tau$ is 
proportional the column density of the absorbing material and can reach very large values. In the outflow model, $\tau$ peaks at $\log \xi \sim 1.3$ 
($v\sim 890$~km~s$^{-1}$) reaching $\sim$5000, that is, nearly two orders of magnitude smaller than in the stationary cloud for the same $\xi$. Such high 
optical depths are common to other lines in the X-ray band, like \ion{O}{viii} in the 14--20~\AA\ and the line \ion{O}{viii} $\lambda 18.969$. In such 
cases lines appear saturated in the absorption spectrum in contrast with observation. In \citet{ramirez2005a} we fitted the integrated optical depths of 
resonant lines in the spectrum of NGC~3783 and found $T_\mathrm{tot}\lesssim 1$ for most of the lines. We illustrate in Figure~\ref{nex2} the consequences 
of taking this model for the reproduction of the profile of the line \ion{Ne}{x} $\lambda$12.134. In order to have a non-saturated line (as observed), 
we had to reduce the abundance to $\sim$1\% solar, which has no physical motivation. We do not go further with this model, and present the clumpy 
(multicomponent) scenario in the next section.


%%%%% FIGURA 3

\begin{figure}[!t]
\centering
\includegraphics[angle=-90,width=0.99\columnwidth]{jramirezfig3.eps}
\caption{Profile of the line \ion{Ne}{x}~$\lambda 12.134$ formed by taking 
$F_{\lambda}(v)=F_c\times \exp[-\tau_{\lambda} (v)]$ as function of the velocity. 
The rectangle profile is made by a cloud with solar abundances (open circles). 
The profile with an extended red wing and a sharp blue wing is made by a cloud 
with 1\% solar abundances (open square).}
\label{nex2}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{The Multicomponent Model}
\label{mc}

Now, we examine the scenario in which the absorption profile seen in the X-ray and UV spectra of AGNs are made up of multiple components. The main difference 
between this model calculations and previous ones by other authors \citep[e.g.,][]{kaspi2002a,krongold2003a,rozanska2006a}, is that in our model all the 
components are linked by a velocity law and a gas density distribution.

Each absorber is specified by an ionization parameter $\xi$, a column density $N_H$, an absorption covering factor, a gas density $n_H(r)$ and an outflow 
velocity $v(r)$ in order to shift the absorption lines according to the Doppler effect. Once the ionic fraction is calculated from the ionization equations, 
and the ionic levels computed, the opacity in each frequency bin is
\begin{equation}
\kappa_{\nu}\rho= \frac{\pi e^2}{m_ec}f_l n_l \left[ 1- \frac{n_u g_l}{n_l g_u} \right ] \phi (\Delta \nu),
\label{k1}
\end{equation}
where $\kappa_{\nu}$ is the opacity at the frequency $\nu$, $\rho=\mu m_H n_H$ is the mass density, $f_l$ is the absorption oscillator strength,  $n_l$, $n_u$ 
(in $cm^{-3}$), $g_l$, $g_u$ are the number density and the statistical weight of the lower and upper levels of the transition, respectively. We allow for the 
lines to have a finite width characterized by the line profile $\phi (\Delta \nu)$, with a width which is the greater one between the thermal and the turbulent 
motions. In all our models the turbulence velocity is assumed to be 200~km~s$^{-1}$. The optical depth of a line in each component is
\begin{equation}
\tau_{\nu}=\int_{r_1}^{r_2} \kappa_{\nu}(r) \rho(r) dr,
\end{equation}
where $r_2$ and $r_1$ are the limits of the cloud.

It is important to highlight a special difference between the way we use the ionization parameter in the MC model and the way it is used in the SA model. 
Because the clouds intervening in this model are optically thin, the ionization parameter at the most exposed face of the cloud remains essentially constant 
through the cloud, i.e. 
\begin{equation}
\xi(r)=\frac{4\pi F(r)}{n_H(r)},
\label{xi1}
\end{equation}
where $F(r)$ is the total ionizing flux at the radius $r$.

When two or more outflows are put together they are assumed to be distributed such that the observed spectrum is the result of the addition of all components. 
Thus, the radiative flux in each bin of frequency for the composite spectrum is
\begin{equation}
F_{\nu}=\sum_{i=1}^{m} f_{\nu}(x_i),
\label{tautot123}
\end{equation}
where $m$ is the number of absorbing clouds in the line-of-sight, $f_{\nu}(x_i)$ is the flux resulting from the pass of the continuum radiation through the 
absorbing cloud $i$, and $x_i$ is the normalized spatial radius of the cloud.

In \citet{ramirez2005a}, it is shown that in order to fit the line profiles in NGC~3783 a geometry for the gas distribution different from spherical is 
required. Such a deviation from spherical geometry has two effects. In the normalized notation the number density is
\begin{equation}
n_H(x)=n_0 x^{-2+\kappa}w^{-1},
\label{den3}
\end{equation}
where $0 \leq \kappa \leq 2$. A positive value of $\kappa$ implies that the gas flow dilutes more slowly than in a free spherical expansion, i.e. that there 
are sources of gas embedded in the flow, or that the flow is confined. A negative value corresponds to sinks of gas in the flow, or expansion of an initially 
confined flow in a flaring geometry. 

Secondly, we  allow the radiation flux to have a form
\begin{equation}
F=F_0
\times x^{-2-p},
\label{f}
\end{equation}
where $p$ is an index to mimic a deviation of flux from the pure geometrical dilution case. This is expected if the medium between clouds has a significant 
optical depth (if $p$ is positive) or if there are sources of radiation  embedded in the flow (if $p$ is negative).  


\section{Assumption about the number density}
\label{assumden1}

Before going into the presentation of the results we would like to highlight important differences between the underlying physical/geometrical 
motivation of the present work and previous ones. After reviewing the single (geometrically thick) model and the clumpy (geometrically thin) 
model, we favored the latter, and require for the production of the line profile found in the spectrum of Seyfert galaxies that $\Delta R/R \ll 1$ 
(also based on results by \citet{gabel2005a} from UV data).

Based on UV CIII$^{*}$ density constraints \citep{gabel2005a}, the electron density for the absorber could be $n_e\approx 10^4$~cm$^{-3}$. With 
this (using equation~\ref{xi1} and $n_H\approx n_e/1.2$), the distance between the absorber and the central source is $R\approx 25$~pc 
($\approx$10$^{20}$~cm). As in \citet{gabel2005a}, the low ionization species (XLI in that paper) seems to share the same kinematics with the UV 
absorber. Using a typical luminosity $L$ for this object of $\approx$10$^{44}$~erg~s$^{-1}$, $n_H \approx 10^4$~cm$^{-3}$, and an ionization parameter 
of $\xi \approx 1$, $R \approx 10^{20}$~cm; similar to that computed by \citet{gabel2005a}; but if we take $n_H \approx 10^{11}$~cm$^{-3}$, as is 
required for $\Delta R \ll R$, then $R \approx 10^{16.5}$~cm. This is not in contradiction with \citet{krongold2005a}, where they set limits on 
the density and location of the absorber; with $n_e>10^{4}$~cm$^{-3}$, and $D<5.7$~pc. However, is clearly different from the computation made by 
\citet{netzer2003a}, of $n_e < 5 \times 10^{4}$~cm$^{-3}$ $D > 3.2$ pc for the $\log(U_\mathrm{ox})=-2.4$ component, $n_e < 10^{5}$~cm$^{-3}$ $D > 0.63$~pc 
for the $\log(U_\mathrm{ox})=-1.2$ component, and $n_e < 2.5 \times 10^{5}$~cm$^{-3}$ $D > 0.18$~pc for the $\log(U_\mathrm{ox})=-0.6$ component. It is 
clear that some of the differences in the estimations can be due to differences in the nature of the physics behind the estimations. In 
\citet{krongold2005a} (and also \citealt{reeves2004a}, for the Fe K shell), the fundamental assumption behind the computation of $n_e$ is that the 
absorber responds instantaneously to changes of flux of the ionizing source, while in \citet{netzer2003a} the estimations are based on temperature 
derived models, average recombination rates and no response to continuum variations on timescales of 10 days. And, for our purposes, this basically 
translates into differences between the two proposed physical mechanisms: thermally accelerated winds, which in principle have to consider a sublimination 
radius for the material not being evaporated, and radiatively accelerated winds with origin possibly in the accretion disk of subparsec scale; 
both are competitor theories in the explanation of the origen of the warm-absorber outflows.


%%%%% FIGURA 4

\begin{figure}[!t]
\centering
\includegraphics[angle=-90,width=0.99\columnwidth]{jramirezfig4.eps}
\caption{Variation of the main kinematic and ionization variables for model A (see text). 
Upper left: Velocity law $w(x)$ for the model. Upper right: The ionizing radiative flux, 
decaying  as $\propto r^{-2}$ (dashed line) and as $\propto r^{-2-p}$ (dotted line). 
Lower left: Variation of the density as function of $x$. Lower right: Variation of the 
ionization parameter ($\xi$) with the normalized velocity.}
\label{fmodelA}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%% TABLA 1

\begin{table}[!t]
\vspace{0.1cm}
\centering
\setlength{\tabnotewidth}{0.99\columnwidth}
\tablecols{2}
\setlength{\tabcolsep}{3\tabcolsep}
\caption{Composition and parameter values of the kinematic model A\tabnotemark{a}}
\label{tbl1solar}
\begin{tabular}{cc}
\toprule
Element & Relative Abundance \\
\midrule
H     & 1.0 \\
He    & 0.1 \\
C     & 0.3540E-03 \\
N     & 0.9330E-04 \\
O     & 0.7410E-03 \\
Ne    & 0.1200E-03 \\
Mg    & 0.3800E-04 \\
Si    & 0.3550E-04 \\
S     & 0.2140E-04 \\
Ar    & 0.3310E-05 \\
Ca    & 0.2290E-05 \\
Fe    & 0.3160E-04 \\
\bottomrule
\tabnotetext{a}{In this model $w_0=0.4$, $v_{\infty}=900$~km~s$^{-1}$, $\log _{10} r_0=15.75$ [cm], 
$v_\mathrm{turb}=200$~km~s$^{-1}$ and $N_H=5\times 10^{20}$~cm$^{-2}$.}
\end{tabular}
\vspace{0.3cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 5

\begin{figure}[!t]
\centering
\includegraphics[angle=-90,width=0.99\columnwidth]{jramirezf5.eps}
\caption{Measured vs model predicted velocities for the group of unblended lines given 
in Table~2 of \citet{ramirez2005a}. The measurements are taken from \citet{kaspi2002a}. 
The solid line is the best line after a linear regression for model A.} 
\label{mvelA}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\vspace{0.15cm}

\section{Results and Discussions}
\label{results1}

In Figure~\ref{fmodelA} we show the variation of the different variables governing the kinematics and the ionization structure of one of our models 
(model~A). This is a wind with a velocity slope $\beta=2$, $\kappa=1.5$ and $p=1.5$. The number of kinematic components $m$ is 11. The assumed abundance 
is solar, as is given in Table~\ref{tbl1solar}, and each component has a column density $N_H=5\times 10^{20}$~cm$^{-2}$. In this model the variation 
in the ionization parameter is $\log \xi=3.5-(-0.65)$ [erg~cm~s$^{-1}$], where the ``launching'' ionization parameter is defined as $\log \xi_0=3.5$ 
[erg~cm~s$^{-1}$], and the variation in density $\log n_H=11.4-10.26$ [cm$^{-3}$]. We can see the variation with distance of the input parameters $w$, 
flux, $n_H$ and $\xi$, for model~A in Figure~\ref{fmodelA}. In Figure~\ref{mvelA} we plot the velocity shifts taken from the maximum line optical 
depths $\tau_\mathrm{max}$ of our model versus the line centroids measured by \citet{kaspi2002a} for NGC~3783 (see Table~3 in that paper).

We fit a linear model by robust regression using the M estimator (Huber 1964). The weights are included as the inverse of the variances, with the 
variance equal to $\Delta v=(v_\mathrm{lo}+v_\mathrm{hi})/2$, and $v_\mathrm{lo}$, and $v_\mathrm{hi}$ are the differences between centroid and lower 
and upper velocity limits of the measured lines given in Table~3 of \citet{kaspi2002a}. The best-fit slope is $0.97 \pm 0.31$ (solid line in the 
figure), and the residual standard error (rms) is 15.4 for 25 degrees of freedom (dof). The agreement is encouraging, considering  that no model 
has been suggested before to explain the possible correlation between the ionization parameter and the velocity shift seen in this Seyfert galaxy. 

Although the velocity shifts measured with respect to line minima are well explained by this model the distribution (in general) of the line profile 
shapes is not in agreement with that reported in \citet{ramirez2005a}, for instance. From that study, most lines have extended blue wings. On the 
other hand, the model does predict the correct asymmetries for a few lines from highly ionized species, but the profiles of the lower ionizations 
lines exhibit more extended red wings, unlike the observations. The discrepancy between the modeled profile of the  \ion{O}{viii} $\lambda$18.9 line 
and that observed in NGC~3783 is important. This is why we created model~B.


%%%%%% FIGURA 6

\begin{figure}[!t]
\centering
\includegraphics[angle=-90,width=0.99\columnwidth]{jramirezfig6.eps}
\caption{Variation of the main kinematic and ionization variables for model B (see text). 
Upper left: Velocity law $w(x)$ for the model. Upper right: The ionizing radiative flux, 
decaying as $\propto r^{-2}$ (dashed line) and as $\propto r^{-2-p}$ (dotted line). 
Lower left: Variation of the density as function of $x$. Lower right: Variation of the 
ionization parameter ($\xi$) with the normalized velocity.}
\label{fmodelB}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% TABLA 2

\begin{table}[!t]
\vspace{0.1cm}
\centering
\setlength{\tabnotewidth}{0.99\columnwidth}
\tablecols{2}
\setlength{\tabcolsep}{3\tabcolsep}
\caption{Composition and parameter values of the kinematic model B\tabnotemark{a}}
\label{tbl2solar}
\begin{tabular}{cc}
\toprule
Element & Relative Abundance \\
\midrule
H     & 1.0 \\
He    & 0.1 \\
C     & 0.3540E-03 \\
N     & 0.9330E-04 \\
O     & 0.7410E-03 \\
Ne    & 0.1200E-03 \\
Mg    & 0.3800E-04 \\
Si    & 0.3550E-04 \\
S     & 0.2140E-04 \\
Ar    & 0.3310E-05 \\
Ca    & 0.2290E-05 \\
Fe    & 0.3160E-04 \\
\bottomrule
\tabnotetext{a}{In this model $w_0=0.4$, $v_{\infty}=1100$~km~s$^{-1}$, $\log _{10} r_0=15.75$ [cm], 
$v_\mathrm{turb}=200$~km~s$^{-1}$ and $N_H=5\times 10^{20}$ cm$^{-2}$.}
\end{tabular}
\vspace{0.3cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In Figure~\ref{fmodelB} we present the variation of the  variables governing the kinematics and the ionization structure of another model (model B). 
Here we change slightly the parameters $\log \xi_0$, and $v_{\infty}$ (see Table~\ref{tbl2solar} for details). The wind has the parameters $\beta=2$, 
$\kappa=1.5$ and $p=1.5$. The number of kinematic components $m$ is 11. The abundances and the column density are as in model A. In this model the 
variations in ionization parameter and density are $\log \xi=3.00-(-1.13)$ [erg~cm~s$^{-1}$] and  $\log n_H=11.40-10.26$ [cm$^{-3}$]. We compare models 
A and B in terms of  the variation of optical depth vs. velocity of the flows of three important lines, i.e, \ion{Si}{xiv}~$\lambda 6.182$, 
\ion{Si}{xiii}~$\lambda 6.648$ and \ion{Mg}{xii}~$\lambda 7.106$. We could estimate the contribution of each cloud to the formation of the composite 
profiles of these lines. We note that model~B is superior to model~A in describing the low-velocity portion of the flow reflected by the high ionization 
lines seen in NGC~3783 (extended blue wings), but yields a worse description of the spectrum at the higher terminal velocities responsible for the center 
of the lines measured by \citet{kaspi2002a}.

Figure~\ref{mvelB} shows the fit of a linear model by robust regression using the M estimator for model B. As in model A, the weights are included as 
the inverse of the variances, with the variance equal to $\Delta v=(v_\mathrm{lo}+v_\mathrm{hi})/2$, and $v_\mathrm{lo}$, and $v_\mathrm{hi}$ are the differences 
between centroid and the lower and upper velocity limits of the measured lines given in Table~3 of \citet{kaspi2002a}. The best-fit slope is $0.72 \pm 0.20$ 
(solid line in the figure), and the residual standard error (rms) is 17.6 for 25 degrees of freedom (dof). In general, the scatter becomes worse for lines 
with terminal velocities greater than 1100~km~s$^{-1}$.


%%%%% FIGURA 7

\begin{figure}[!t]
\centering
\includegraphics[angle=-90,width=0.99\columnwidth]{jramirezf7.eps}
\caption{Measured vs model predicted velocities for the group of unblended 
lines given in Table~2 of \citet{ramirez2005a}. The measurements are taken from 
\citet{kaspi2002a}. The solid line is the best line after a linear regression for model B.}
\label{mvelB}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Modeling global properties - $\xi$ {\it vs} velocity}

One of the prime goals of this study is to examine the relationship between the kinematics and the ionization structure of the flow. This is complementary 
to earlier work \citep{krongold2003a,netzer2003a,krongold2005a} which follows the algorithm of constructing a grid of static photoionization models, with 
varying ionization parameter and column density, with the selection of this quantities which best reproduce the EW observed in the spectra, in addition to 
paying attention to the relationship between these physical parameters and the position in wavelength space of the absorption troughs.


%%%%% FIGURA 8

\begin{figure}[!t]
\centering
\includegraphics[angle=-90,width=0.99\columnwidth]{jramirezfig8.eps}
\caption{Relationship between the ionization parameter and velocity. The solid line is 
the theoretical prediction for model A. The open squares with error bars are points with 
velocities taken from \citet{kaspi2002a}, for the group of lines compared in Figure~\ref{mvelA}.}
\label{mvelxiA}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\citet{krongold2003a} fitted photoionization models to the 900~ks spectrum of NGC~3783 using two phases; one at high ionization and high temperature 
$\log~T \sim 5.98$ (HIP) and one at low ionization and temperature $\log~T \sim 4.41$ (LIP), finding  good agreement between the modeled equivalent 
width (EW) and the measured one for a set of absorption lines (see Figure~9 of \citealt{krongold2003a}). They set each of these phases at a single 
outflow velocity of $\sim$750~km~s$^{-1}$, assuming spatial coexistence of the two absorbers. The approach was similar to that from \citet{netzer2003a}, 
but in the latter they used three components in pressure equilibrium, with two kinematic components each.

One property of our model is the capability of allowing the possibility of establishing\footnote{We explicitly state {\it allow the possibility}, since the 
limited resolution of the telescope does not allow us to go beyond this possibility.} a relationship between the ionization and the kinematics of the 
gas based on a radiatively accelerated wind. Figures~\ref{mvelxiA} and \ref{mvelxiB} show this relation for models A and B, respectively. These plots 
show the predicted relationship between the observed velocities and the ionization parameter. While absorption from highly ionized ions originates from 
low-to-intermediate (200--600~km~s$^{-1}$) velocities, lines from lower ionization stages are formed at intermediate-to-high velocities (600--1000~km~s$^{-1}$). 


%%%%% FIGURA 9

\begin{figure}[!t]
\centering
\includegraphics[angle=-90,width=0.99\columnwidth]{jramirezfig9.eps}
\caption{Relationship between the ionization parameter and velocity. The solid line is 
the theoretical prediction for model B. The open squares with error bars are points with 
velocities taken from \citet{kaspi2002a}, for the group of lines compared in Figure~\ref{mvelB}.}
\label{mvelxiB}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 10

\begin{figure}[!t]
\centering
\includegraphics[clip,angle=-90,width=0.97\columnwidth]{jramirezfig10.eps}
\caption{Predicted UV spectrum by model B (in the range 700--1700~\AA). These absorption 
lines are formed by the faster (or at lower ionization parameters) clouds of model B, 
i.e., $\log \xi =0.12, -0.65$. Most of these lines are found in real UV spectra of AGN.}
\label{uv700-1700}
\vspace{0.2cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% FIGURA 11

\begin{figure*}[!t]
\centering
\includegraphics[clip,angle=-90,width=0.8\textwidth]{jramirezfig11.eps}
\caption{Predicted UV spectrum by model B (in velocity space). These absorption lines 
are formed by the faster clouds (or by those at lower ionization parameters) of model B, 
i.e., $\log \xi =0.12, -0.65$. Some of the velocities displayed by these lines are 
shared by lines coming from the X-ray band. For transforming to velocity space, we have 
taken the shorter rest-wavelength of the doublet (solid line), and also the longer 
rest-wavelength of the doublet (dashed line).}
\label{uv-velocity}
\vspace{0.7cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Far-UV and UV absorbers}

The relationship between the X-ray absorption spectrum and the UV absorption spectrum is at present a subject of controversy. Kraemer, Crenshaw, \& 
Gabel (2001) and \citet{gabel2003a,gabel2005a} have analyzed de UV spectrum of NGC~3783. HST/STIS and FUSE spectra show absorption troughs from the 
low order Lyman series (i.e. Ly${\alpha}$, Ly${\beta}$, Ly${\gamma}$), \ion{C}{iv}~$\lambda \lambda 1548.2,1550.8$, \ion{N}{V}~$\lambda \lambda 1238.8,1242.8$, 
\ion{O}{vi}~$\lambda \lambda 1032,1038$. All these lines are seen in three kinematic components at $-1365$, $-548$, and $-724$~km~s$^{-1}$ (components 
1, 2 and 3 respectively). A weak fourth component is reported by \citet{gabel2003a,gabel2005a} at $-1027$~km~s$^{-1}$. Figure~\ref{uv700-1700} shows 
the spectrum predicted by model B in the range 700--1700~\AA. It is clear from this figure that lines due to the low order Lyman series, \ion{He}{ii}, 
\ion{C}{iv}, \ion{N}{v}, \ion{O}{vi}, and \ion{Ne}{viii} would be detectable in the UV band of the spectrum. Figure~\ref{uv-velocity} shows the spectra 
of several UV lines as a function of radial velocity with respect to the systemic one. At the top of the figure we present the Ly${\alpha}$ blended with the 
\ion{He}{ii}~$\lambda 1215$ line (upper left) which yields a feature centered at $\sim$$-$1300~km~s$^{-1}$. Also, an absorption feature is formed with 
the Ly${\beta}$ and \ion{He}{ii}~$\lambda 1025$ lines (upper center) with center at $\sim$$-$1200~km~s$^{-1}$, and the line Ly${\gamma}$ (upper right) 
with a velocity $\sim$$-1100$~km~s$^{-1}$. At the bottom of the figure we plot the spectrum of three important doublets (\ion{C}{iv}~$\lambda \lambda 1548.2,1550.8$, 
\ion{N}{v}~$\lambda \lambda 1238.8,1242.8$, and \ion{O}{vi}~$\lambda \lambda 1032,1038$). The solid line depicts the velocity spectra constructed 
taking the shorter wavelength of the doublet. The spectra shown as dashed lines were obtained taking the longer wavelength. One can see the similarity between 
the velocities predicted by our model and the high-velocity components (1, 4 and likely 3) seen in the UV spectrum of NGC~3783 and other Seyfert 1 
galaxies (for example NGC~5548). This is similar to the conclusion reached by \citet{gabel2005a}, based on \citet{kaspi2002a} and \citet{gabel2003a}, 
where all X-ray lines having sufficiently high resolution and S/N were found to span the radial velocities of the three UV kinematic components (see 
also next section). These are predictions which arise naturally from our model because we are modeling the wind self-consistently with a complete 
treatment of radiative processes in all wavelengths. In our models these UV features as well as the long wavelength absorption lines in the X-ray band, 
are produced by the low ionization parameter part of the flow (e.g., $\log \xi = 0.12,-1.13$  from Figure~\ref{fmodelB} lower right panel). This demonstrates 
that the same absorber can produce X-ray and UV lines with similar velocities. Figure~\ref{uv-xrays} shows the kinematic relationship between the X-ray 
and the UV absorbers. The model has been shifted (up) for clarity. Here we show the \ion{O}{viii}~$\lambda 18.969$ and the doublet 
\ion{O}{vi}~$\lambda \lambda 1032,1038$ (with respect to the shorter wavelength). For comparison we plot the histogram data of the 900~ks of NGC~3783. 
We see that our model is capable of simultaneously producing X-ray and UV absorption lines with similar velocities around $\sim$1000~km~s$^{-1}$, as 
it has been suggested from UV data \citep{gabel2003a}.


%%%%% FIGURA 12

\begin{figure}[!t]
\centering
\includegraphics[clip,angle=-90,width=0.99\columnwidth]{jramirezfig12.eps}
\caption{Possible kinematical relationship between the UV and the X-ray absorbers. 
In the figure are plotted the theoretical X-ray and UV lines 
\ion{O}{viii}~$\lambda 18.969$ and \ion{O}{vi}~$\lambda 1032$, dashed and short-dotted 
lines respectively. Also plotted is the profile of the line \ion{O}{viii}~$\lambda 18.969$ 
taken from the 900 ks spectrum of NGC~3783. All these lines are sharing velocities 
$\sim$1000~km~s$^{-1}$.}
\label{uv-xrays}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Present single wind {\it vs} multiphase wind scenario}

However, we want to highlight the most important differences between our model, consisting of a single wind governed by the laws of radiative acceleration, 
and the multiphase wind subjected to pressure equilibrium.


%%%%% TABLA 3

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\setlength{\tabnotewidth}{0.35\textwidth}
\tablecols{4}
\setlength{\tabcolsep}{1.2\tabcolsep}
\caption{Asymmetry comparison between model A and observations}
\label{tblasim1}
\centering
\small
\begin{tabular}{lccr@{$\pm$}l}
\toprule
Ion  & Line (\AA)  & $v(\tau_\mathrm{max})$/900~km~s$^{-1}$  & \multicolumn{2}{c}{\citet{ramirez2005a}\tabnotemark{a}} \\ 
\midrule
\ion{S}{xvi}    & $~4.729$  & 0.55 (R) & 0.33  & 0.1 (B)\\
\ion{S}{xv}     & $~5.039$  & 0.67 (R) & 0.23  & 0.1 (B)\\
\ion{Si}{xiii}  & $~5.681$  & 0.70 (R) & 0.35  & 0.1 (B)\\
\ion{Si}{xiv}   & $~6.182$  & 0.67 (R) & 0.23  & 0.08 (B)\\
\ion{Si}{xiii}  & $~6.648$  & 0.70 (R) & 0.43  & 0.074 (B)\\
\ion{Mg}{xii}   & $~7.106$  & 0.70 (R) & 0.32  & 0.070 (B)\\
\ion{Mg}{xi}    & $~7.473$  & 0.76 (R) & 0.74  & 0.067 (R)\\
\ion{Fe}{xxiii} & $~8.303$  & 0.55 (R) & 0.037 & 0.060 (B)\\
\ion{Mg}{xii}   & $~8.421$  & 0.70 (R) & 0.32  & 0.059 (B)\\
\ion{Mg}{xi}    & $~9.169$  & 0.76 (R) & 0.31  & 0.054 (B)\\
\ion{Ne}{x}     & $~9.708$  & 0.76 (R) & 0.45  & 0.052 (B)\\
\ion{Ne}{x}     & $10.240$  & 0.76 (R) & 0.17  & 0.048 (B)\\
\ion{Ne}{ix}    & $11.547$  & 0.82 (R) & 0.69  & 0.043 (R)\\
\ion{Fe}{xxii}  & $11.770$  & 0.55 (R) & 0.22  & 0.042 (B)\\
\ion{Ne}{x}     & $12.134$  & 0.76 (R) & 0.29  & 0.041 (B)\\
\ion{Fe}{xxi}   & $12.284$  & 0.62 (R) & 0.42  & 0.040 (B)\\
\ion{Fe}{xviii} & $14.373$  & 0.70 (R) & 0.27  & 0.035 (B)\\
\ion{Fe}{xviii} & $14.534$  & 0.70 (R) & 0.51  & 0.034 (R)\\
\ion{O}{viii}   & $14.832$  & 0.82 (R) & 0.21  & 0.033 (B)\\
\ion{Fe}{xvii}  & $15.015$  & 0.70 (R) & 0.76  & 0.033 (R)\\
\ion{O}{viii}   & $15.188$  & 0.82 (R) & 0.38  & 0.033 (B)\\
\ion{O}{viii}   & $16.006$  & 0.82 (R) & 0.26  & 0.030 (B)\\
\ion{O}{vii}    & $17.200$  & 0.90 (R) & 0.29  & 0.029 (B)\\
\ion{O}{vii}    & $17.396$  & 0.90 (R) & 0.30  & 0.029 (B)\\
\ion{O}{vii}    & $17.768$  & 0.90 (R) & 0.32  & 0.028 (B)\\
\ion{O}{vii}    & $18.627$  & 0.90 (R) & 0.32  & 0.027 (B)\\
\ion{O}{viii}   & $18.969$  & 0.82 (R) & 0.26  & 0.026 (B)\\
\bottomrule
\tabnotetext{a}{$\bar{v}/v_1$, see definition in the text.}
\end{tabular}
\vspace{0.3cm}
\end{table*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The first difference is that in our model there is a clear correlation between ionization state of the ions and velocity ($\xi-v$), and $\xi$ and 
number of particle $n_H$ ($\xi-n_H$). At the same time, because of the dependence of $v$ and $n_H$ on the spatial distance $r$, there is a dependence 
of $\xi$ on $r$. This is different from the multicomponent in pressure equilibrium model, suggested by \citet{netzer2003a}. In that model three 
components (in ionization) may coexist in the same volume of space and lie on the stable regions of the nearly vertical part of the thermal stability curve 
($\log~T~vs~\log~[U/T]$, see Figure~12 in that paper). Our absorbers are not embedded in an external medium, and they cannot coexist at exactly the 
same location, but rather follow the physical laws of radiative acceleration. In this context, \citet{gabel2005a} also find that the UV absorbers of NGC~3783 
may share some properties of the multiphase thermal wind. The UV kinematic components 1b, 2 and 3 could occupy the low-temperature base of the region 
of the thermal stability curve where a range of temperatures can coexist in pressure equilibrium. There is one weak point in the picture of inhomogeneities 
coexisting in pressure equilibrium. The low-ionization high-velocity UV absorber does not fit there, due to a pressure of a factor of 10 larger than that 
of the other component. If embedded into a more ionized-hotter material, it will eventually evaporate, unless there exists an additional confining mechanism. 
This extra confinement could be provided by magnetic pressure, requiring moderate magnetic fields ($B\approx 10^{-3}~G$), as predicted by some dynamical 
models (Emmering, Blandford, \& Shlosman 1992; de Kool \& Begelman 1995). In our model, there is no need for a confining mechanism, and if anything, the 
more ionized material is closer to the source, shielding the intense continuum radiation and preventing its evaporation. We conclude by stating that our 
model represents an additional competitor. It cannot be ruled out by comparison with the pressure equilibrium model. A more detailed comparison 
between models is beyond the scope of the present work, but could take place in the future.


\section{Two outflows}
\label{twoouts1}

So far we have been concerned with the line velocity shifts as measured with respect to the points of maximum absorption of the lines, but 
little has been said about the asymmetry of the troughs. To quantify the line asymmetries predicted by the theoretical models we have tabulated 
the fraction of the terminal velocity at which the absorption is maximum, to be compared with the ratio ($\bar{v}/v_{1}$), where $v_1$ is the 
position of the blue edge of each line, used  to quantify the asymmetry in the spectrum of NGC~3783 (see Table~3 \citealt{ramirez2005a}). We 
characterize a theoretical line as ``red'' if $v(\tau_\mathrm{max})/v_{\infty}>0.5$ and as ``blue'' if $v(\tau_\mathrm{max})/v_{\infty}<0.5$, while 
for observed lines they are grouped according to $\bar{v}/v_{1}>0.5$ and $\bar{v}/v_{1}<0.5$ (see Ram\'irez et al. 2005). In Table~\ref{tblasim1} 
we present the observed troughs. In the first and second columns are given the identifications and the wavelengths of the lines, in the third, 
the fraction $v(\tau_\mathrm{max})/v_{\infty}$ for model A, and in the fourth the classification from observed lines, as in \citet{ramirez2005a}, 
i.e., $\bar{v}/v_{1}$. We classify every trough as either (R) if it is red or (B) if it is blue. There are clear discrepancies between model~A 
and the observations. In order to improve the agreement with observations we found it necessary to create a model composed of two outflows, which 
differ by their $\log ~\xi_{0}$, the initial exposition to the source. In Table~\ref{twoflows123} we can see separately the parameters of the two 
outflows which compose model~C. The first flow, with $\log \xi_{0}=3$, which we call HIF (high ionization flow), is able to create 
the majority of the high ionization lines (\ion{S}{xvi}~$\lambda$4.729, \ion{S}{xv}~$\lambda$5.039, \ion{Fe}{xxiii}~$\lambda$8.303), with extended 
blue wings, as the flow moves outwards and acquires more velocity, in agreement with the observations. The second, which we call LIF (low ionization 
flow), characterized by $\log \xi_{0}=2$, is able to create the extended blue wing of the oxygen lines. This model has the fundamental ingredients 
that explain the asymmetry observed in the lines of NGC~3783 and, at the same time, gives us the bulk velocity of the flow.


%%%%% TABLA 4

\begin{table}[!t]
\vspace{0.1cm}
\centering
\tablecols{2}
\caption{Parameters of the two outflow model (Model C)}
\label{twoflows123}
\begin{tabular}{cc}
\toprule
Outflow HIF   &  Outflow LIF \\
\midrule
$\log \xi_0=3.0$             & $\log \xi_0=2.0$ \\
$v_{\infty}=1100$~km~s$^{-1}$  & $v_{\infty}=2200$~km~s$^{-1}$ \\
$N_H= 10^{22}$ cm$^{-2}$           & $N_H= 10^{21}$ cm$^{-2}$\\
$w_0=0.4$                    & $w_0=0.2$ \\
Solar composition            & Solar composition \\
\bottomrule
\end{tabular}
\vspace{0.3cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%% TABLA 5

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\setlength{\tabnotewidth}{0.65\textwidth}
\tablecols{4}
\setlength{\tabcolsep}{1.2\tabcolsep}
\caption{Asymmetry comparison between model C and observations}
\label{tblasim2}
\small
\begin{tabular}{lccc}
\toprule
Ion  & Line (\AA)  & $v(\tau_\mathrm{max}$[HIF])/1100~km~s$^{-1}$   & $v(\tau_\mathrm{max}$[HIF+LIF])/2200~km~s$^{-1}$ \\
\midrule
\ion{S}{xvi}    & $~4.729$   & 0.47 (B) & ~0.47 (B) \\
\ion{S}{xv}     & $~5.039$   & 0.55 (R) & ~0.55 (R) \\
\ion{Si}{xiii}  & $~5.681$   & 0.62 (R) & ~0.62 (R) \\
\ion{Si}{xiv}   & $~6.182$   & 0.55 (R) & ~0.55 (R) \\
\ion{Si}{xiii}  & $~6.648$   & 0.62 (R) & ~0.62 (R) \\
\ion{Mg}{xii}   & $~7.106$   & 0.55 (R) & ~0.55 (R) \\
\ion{Mg}{xi}    & $~7.473$   & 0.74 (R) & ~0.74 (R) \\
\ion{Fe}{xxiii} & $~8.303$   & 0.47 (B) & ~0.47 (B) \\
\ion{Mg}{xii}   & $~8.421$   & 0.55 (R) & ~0.55 (R) \\
\ion{Mg}{xi}    & $~9.169$   & 0.74 (R) & 0.37 (B)\tabnotemark{a} \\
\ion{Ne}{x}     & $~9.708$   & 0.74 (R) & ~0.74 (R)  \\
\ion{Ne}{x}     & $10.240$  & 0.74 (R)  & 0.37 (B)\tabnotemark{a} \\
\ion{Ne}{ix}    & $11.547$  & 0.74 (R)  & 0.37 (B)\tabnotemark{a} \\
\ion{Fe}{xxii}  & $11.770$  & 0.47 (B)  & ~0.47 (B) \\
\ion{Ne}{x}     & $12.134$  & 0.74 (R)  & 0.37 (B)\tabnotemark{a} \\
\ion{Fe}{xxi}   & $12.284$  & 0.47 (B)  & ~0.47 (B) \\
\ion{Fe}{xviii} & $14.373$  & 0.55 (R)  & ~0.55 (R) \\
\ion{Fe}{xviii} & $14.534$  & 0.55 (R)  & ~0.55 (R) \\
\ion{O}{viii}   & $14.832$  & 0.74 (R)  & 0.37 (B)\tabnotemark{a} \\
\ion{Fe}{xvii}  & $15.015$  & 0.62 (R)  & ~0.62 (R) \\
\ion{O}{viii}   & $15.188$  & 0.74 (R)  & 0.37 (B)\tabnotemark{a} \\
\ion{O}{viii}   & $16.006$  & 0.74 (R)  & 0.37 (B)\tabnotemark{a} \\
\ion{O}{vii}    & $17.200$  & 0.82 (R)  & 0.41 (B)\tabnotemark{a} \\
\ion{O}{vii}    & $17.396$  & 0.82 (R)  & 0.41 (B)\tabnotemark{a} \\
\ion{O}{vii}    & $17.768$  & 0.82 (R)  & 0.41 (B)\tabnotemark{a} \\
\ion{O}{vii}    & $18.627$  & 0.82 (R)  & 0.41 (B)\tabnotemark{a} \\
\ion{O}{viii}   & $18.969$  & 0.74 (R)  & 0.37 (B)\tabnotemark{a} \\
\bottomrule
\tabnotetext{a}{With the addition of the LIF, we mark those line whose symmetry change.}
\end{tabular}
\vspace{0.45cm}
\end{table*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In the third column of Table~\ref{tblasim2} we present the characterization of the lines coming from the two-outflow model. The HIF component 
is the major contributor to the formation of the high ionization lines, i.e. \ion{S}{xvi}, \ion{Fe}{xxi}, \ion{Fe}{xxii}, \ion{Fe}{xxiii}, because 
their optical depths peak close to $\log \xi_{0}$ (HIF), and  as the velocity of the flow increases their ionization fraction decreases, forming 
their blue wings. Under this picture, only a few lines are characterized as blue, while the rest are red. Some intermediate-to-low-ionization 
ions have appreciable fractions in the LIF component, which broadens the lines and yields a red appearance to the composed profile. This effect 
is illustrated by the fourth column of Table~\ref{tblasim2}.

The composite model reproduces the asymmetry of the lines. The present uncertainties in the computation of $v(\tau_\mathrm{max})$ are of the order 
of 60~km~s$^{-1}$ (or $\Delta v(\tau_\mathrm{max})/v_{\infty}=0.055$ with $v_{\infty}=1100$~km~s$^{-1}$). So the lines classified as pure red with 
$v(\tau_\mathrm{max})/v_{\infty}\gtrsim0.55$ and pure blue with $v(\tau_\mathrm{max})/v_{\infty}\lesssim 0.45$ are more reliable than those with 
$0.45 \lesssim v(\tau_\mathrm{max})/v_{\infty}\lesssim 0.55$. 

It is interesting to analyze why the \ion{O}{vii} and \ion{O}{viii} lines are blue, while the lines \ion{Mg}{xi}~$\lambda$7.473 and 
\ion{Fe}{xvii}~$\lambda$15.015 remain red. The optical depths of \ion{Mg}{xi}~$\lambda$7.473, and \ion{Fe}{xvii}~$\lambda$15.015 are below 
$10^{-4}$ at velocities greater than 1100~km~s$^{-1}$, so the LIF component does not contribute to their broadening. Another interesting example 
is the difference in classification between the lines \ion{Mg}{xi}~$\lambda$7.473 and \ion{Mg}{xi}~$\lambda$9.169. The reason is that while 
for \ion{Mg}{xi}~$\lambda$7.473 the oscillator strength is $\sim$$5\times 10^{-2}$, the f-value for the line \ion{Mg}{xi}~$\lambda$9.169 is 
almost one order of magnitude larger ($\sim$$7\times 10^{-1}$), which causes the optical depth of the latter to be significant up to velocities 
beyond $\sim$1600~km~s$^{-1}$. 

Figure~\ref{mvelC} compares the modeled velocities of the lines from the two-outflow model with observations. Robust regression using the M estimator 
gives a slope of $0.63 \pm 0.22$ (solid line in the figure), and the residual standard error (rms) is 16.6 for 25 degrees of freedom (dof).


%%%%% FIGURA 13

\begin{figure}[!t]
\centering
\includegraphics[angle=-90,width=0.99\columnwidth]{jramirezf13.eps}
\caption{Measured vs model-predicted velocities for the group of unblended 
lines given in Table~2 of \citet{ramirez2005a}. The measurements are taken from 
\citet{kaspi2002a}. The solid line is the best line after a linear regression for 
model C.}
\label{mvelC}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


A comparison between the predicted trough profiles and those observed in NGC~3783 is shown in Figure~\ref{oviii_modelC}. Here we plot the theoretical 
spectrum given by the outflows HIF and LIF in the 18.5--19.3~\AA. We have used a power law as continuum, with a spectral index of $\Gamma=1.77$, 
suitable for this AGN, and have set an extra absorber {\tt the wabs}\footnote{$N_{\rm wabs}=0.1 \times 10^{22}$~cm$^{-2}$.} model in XSPEC, and the 
composite fluxes generated by the HIF and the LIF components. To compose the \ion{O}{viii}~$\lambda$18.969 line profile we have summed the fluxes from 
both flows using equation~(\ref{tautot123}) (where $m=22$). With this approximation the profile predicted by the two-outflow model is in excellent 
agreement with observations \citep{ramirez2005a}. The red wing of the troughs in the range 0--1000~km~s$^{-1}$ is formed by the HIF, while the blue 
wing in the range $\sim$1000--2500~km~s$^{-1}$ is formed by the LIF. These values are in agreement with the values measured for this lines in the 
observed spectrum.

This is the first time that such theoretical work has been carried out to explain the asymmetry seen in absorption in the X-ray spectrum of AGNs. Other 
work has been able to explain the blue wings of UV lines, like the well-studied \ion{C}{IV}~$\lambda~1549$, for example from radiation-driven disk-wind 
models \citep{proga2003a}. So we cannot conclude that the model presented here is unique for the description of the asymmetry seen in X-ray lines of 
AGNs, until detailed comparisons are made with such models, including the X-ray line profile produced by a wind from a Keplerian accretion disk (Knigge, 
Woods, \& Drew 1995; Shlosman, Vitello, \& Mauche 1996; Proga et al. 2000; Drew \& Proga 2000). Further work is necessary for such a comparison, and we 
plan to do that in a near future.


%%%%% FIGURA 14

\begin{figure}[!t]
\centering
\includegraphics[angle=-90,width=0.99\columnwidth]{jramirezfig14.eps}
\caption{A graphical representation of the flows HIF and LIF with the line profiles 
seen in NGC~3783 (histogram). Solid line is the composite model (model C) with two 
flows. It can be seen that it was necessary to include the second low-ionization flow 
in order to reproduce the asymmetry of these lines. Otherwise the high-ionization 
flow would form a more extended red wing.}
\label{oviii_modelC}
\vspace{0.5cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Summary}
\label{summ1}

We have computed photoionized wind-flow models for the X-ray spectrum of NGC~3783. We studied single continously absorbing models, as well as multiple 
optically thin components linked through an analytic wind velocity law. We found that the single continuously absorbing model yields gas column 
densities and optical depths that are too large, unless one adopts very low ($\sim$10$^{3-4}$~cm$^{-3}$) densities and metal abundances ($\sim$0.01 solar). 
On the other hand, the multicomponent model is able to reproduce observations very well. For this model we compute ionization properties of the material 
using a velocity law compatible with a radiative wind. Our model is consistent with $\log n_0 \sim 11.35$ [cm$^{-3}$], a launching radii of 
$\log r_0 \sim 15$ [cm], and a terminal velocities of $v_{\infty} \sim 1500$~km~s$^{-1}$, which yield a mass loss rate of the order of 
$\dot{M}_\mathrm{out} \sim 1~M_{\sun}$/yr (assuming a volumetric factor $f_\mathrm{vol}=0.1$). If we assume an ionizing luminosity $L\sim 2 \times 10^{44}$~erg~s$^{-1}$  
\citep{peterson2004a}, and an accretion efficiency of $\eta=0.1$, the Eddington mass accretion rate is $\dot{M}_\mathrm{edd} \sim 0.01~M_{\sun}$/yr. This 
is consistent with the result of \citet{crenshaw2007a} for NGC~4151, and \citet{ramirez2008a} for APM~08279+5255, of $\dot{M}_\mathrm{out}/\dot{M}_\mathrm{edd} > 10$. 
However, it is different from the supposition made by \citet{goncalves2006a}, of $\dot{M}_\mathrm{out}/\dot{M}_\mathrm{edd} \le 1$, using their photoionization 
code {\sc TITAN}, for computing the single medium in pressure equilibrium.

Finally, the asymmetry seen in the lines of the X-ray spectrum of NGC~3783 required a model with two outflows. One flow, with a launching ionization 
parameter of $\log \xi_0 \sim 3$ and a column density of $N_H=10^{22}$~cm$^{-2}$, recreates the red wings of the low ionization lines from \ion{Ne}{ix} 
to \ion{O}{vii}, and the blue wings of the high ionization lines from \ion{Fe}{xxiii} to \ion{Si}{xiv}. A second flow is necessary to create the blue 
wing of the oxygen lines, which exhibit a blue character. The theoretical fitting required $\log \xi_0 \sim 2$, terminal velocities of around $\sim$2200~km~s$^{-1}$, 
and a column density of $N_H=10^{21}$~cm$^{-2}$.

Our calculations also predict a relationship between the UV and X-ray bands, as models adjusted to fit the X-ray spectrum naturally predict UV lines 
like the Lyman serie, and the \ion{O}{vi}, \ion{N}{v} and \ion{C}{iv} doublets, in apparent concordance with \citet{costantini2010a}.

\vspace{-0.15cm}

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\end{document}

