% Draft notes - presently for internal consumption only
%
%
% dhc - 7 June 06 - based on some fitting experiments Asif and I have begun
%
% testing the tradeoffs between mass-loss rate reduction and porosity
% (isotropic) on x-ray line profiles; using Chandra HETGS observation
% of zeta Pup as the test-bed
%
% presently focusing on Fe XVII 15.014 MEG only 
%
% v2 - included Ne X 12.13
%
% v3 (3Aug06) - after talking with Asif, Stan, and Maurice; now including O VIII 18.97
% and formatted/written more like a paper
%
% v4 (22Sep06) - more complete in scope; many smaller changes
%


\documentclass[preprint, 12pt]{aastex} 
\usepackage{amsmath,amssymb}

%\usepackage{emulateapj5}

\newcommand{\chandra}{\textit{Chandra}}
\newcommand{\xmm}{\textit{XMM}}
\newcommand{\xspec}{\textit{XSPEC}}
\newcommand{\zpup}{$\zeta$~Pup}
\newcommand{\zori}{$\zeta$~Ori}
\newcommand{\dori}{$\delta$~Ori}
\newcommand{\tori}{$\theta^1$~Ori C}
\newcommand{\tsco}{$\tau$~Sco}
\newcommand{\bcru}{$\beta$~Cru}
\newcommand{\gcas}{$\gamma$~Cas}
\newcommand{\taustar}{\ensuremath{\tau_*}}
\newcommand{\Rstar}{\ensuremath{\rm R_*}}
\newcommand{\hinf}{\ensuremath{h_{\infty}}}
\newcommand{\hprime}{\ensuremath{h\accent}}
\newcommand{\Rmin}{\ensuremath{R_{\rm min}}}
\newcommand{\umax}{\ensuremath{u_{\rm max}}}
\newcommand{\lamlab}{\ensuremath{\lambda_{\rm lab}}}
\newcommand{\q}{\ensuremath{q}}
\newcommand{\hetgs}{HETGS}
\newcommand{\meg}{MEG}
\newcommand{\etal}{et al.}
\newcommand{\vinf}{\ensuremath{v_{\infty}}}
\newcommand{\Teff}{\ensuremath{T_{\rm eff}}}
\newcommand{\lx}{\ensuremath{L_{\rm x}}}
\newcommand{\lya}{${\rm Ly}_{\alpha}$}
\newcommand{\Ro}{$R_\mathrm{o}$}
\newcommand{\Rone}{\ensuremath{R_\mathrm{1}}}
\newcommand{\uo}{u_\mathrm{o}}
\newcommand{\msunyr}{\ensuremath{\mathrm{M}_\sun \mathrm{yr}^{-1}}}
\newcommand{\mdot}{\ensuremath{\dot{M}}}
\newcommand{\kms}{\ensuremath{{\rm km s}^{-1}}}
\newcommand{\ie}{{\it i.e.\;}}


\shorttitle{Porosity vs.\ M-dot tradeoffs constrained by $\zeta$ Pup line profiles}
\shortauthors{Cohen, Leutenegger, ud-Doula, \& Owocki}

\begin{document}

\title{Quantifying mass-loss rate reduction and wind porosity using
  \chandra\ \hetgs\ emission line profiles of $\zeta$ Puppis}

\author{David H. Cohen\altaffilmark{1}, Maurice A. Leutenegger\altaffilmark{3}, Asif
  ud-Doula\altaffilmark{1,2}, Stanley P. Owocki\altaffilmark{2}}

%\email{cohen@astro.swarthmore.edu; asif@bartol.udel.edu, }

\altaffiltext{1}{Swarthmore College Department of Physics and
  Astronomy, 500 College Ave., Swarthmore PA 19081}

\altaffiltext{2}{Bartol Research Institute, University of Delaware,
  217 Sharp Laboratory, Newark DE 19716}

\altaffiltext{3}{Columbia University, Department of Physics and
  Columbia Astrophysics Laboratory, 550 W.  120$^{\rm th}$ St., New
  York NY 10027}

\begin{abstract}

  We explore the joint effects of mass-loss rate reduction and wind
  porosity on symmetrizing X-ray emission line profiles, using the
  \chandra\ grating spectrum ({\it Co-authors: we may add \xmm\ data
    to the analysis}) of $\zeta$ Puppis as a test case.  We focus on
  three relatively high signal-to-noise, representative lines --
  \ion{Ne}{10} at 12.13 \AA, \ion{Fe}{17} at 15.01 \AA, and \ion{O}{8}
  at 18.97 \AA\ ({\it Co-authors: we can add more lines to the
    analysis}) -- and show that when the data are of high enough
  quality, small values of wind optical depth, \taustar, and porosity
  length, $h$, are preferred over porous models with larger optical
  depths.  We quantitatively explore the trade-offs between \taustar\
  and \hinf\ -- the terminal velocity porosity length -- in those
  cases where models with high porosity and high wind optical depth
  provide acceptable fits.  We find that $\hinf \approx 5$ \Rstar\
  before the \taustar\ values implied by the literature mass-loss rate
  can be accommodated.
% Furthermore, reasonable fits with high
% porosity models can only be achieved if the radial dependence of the
% X-ray filling factor falls off sharply with radius ($q \approx 1$),
% in order to surpess the emission at the line wings that tends to be
% present in porous models, but is not present in the low optical
% depth models or the data.  
  We also find that porosity has little effect on the line-profile
  fits for $\hinf \lesssim 1$ \Rstar.  {\it Co-authors: A sentence or
    two about introducing the {\bf non}-isotropic porosity model in
    the context of the porosity length formalism, and the results of
    fits with this model.}  By both examining the porosity length
  ({\it Co-authors: we haven't done this yet, and it may prove to be
    too problematic and/or not worth doing}) and by synthesizing line
  profiles from the output of two-dimensional numerical simulations of
  the wind instability, we show that physically realistic clumpy and
  porous wind structure does not have large enough porosity lengths to
  significantly affect the emergent X-ray line profile shapes. We also
  explore the overall level and wavelength dependence of the atomic
  opacity via detailed modeling and show that relatively flat
  opacities between 0.5 and 1 keV are easily obtained in a smooth
  wind.


\end{abstract}


\keywords{line: profiles --- stars: early-type --- stars: mass loss
  --- stars: winds, outflow --- stars: individual (\zpup) --- X-rays:
  stars}



\section{Introduction}

X-ray emission line profiles provide very useful constraints on the
physical properties of the hot, X-ray emitting plasma above the
surfaces of O stars.  The observed large line widths are evidence that
the hot plasma resides in the stars' radiation-driven winds, with the
specific profile shapes providing information about the distribution
of line-of-sight velocity in the hot plasma.  Attenuation via
continuum opacity in the cool, bulk wind is manifest as a blue-shifted
line peak and a red-skewed asymmetry, because the red-shifted photons
arise in the back hemisphere of the wind, and are therefore subject to
more attenuation, passing through a larger wind column, than the
blue-shifted photons emitted from the front hemisphere
\citep{MacFarlane1991,oc2001}.  Thus the observed profile shapes also
provide information about the wind optical depth, which in turn
jointly constrains the effective wind opacity and the spatial
distribution of the X-ray emitting plasma.

The roughly ten O stars that have been observed with the \chandra\ and
\xmm\ gratings typically\footnote{There are special exceptions --
  \tsco\ (which is a B0 star) and, to a lesser extent, \tori\ -- with
  quite narrow, but still resolved ($v_{HWHM} \approx 200$ \kms),
  X-ray emission lines.  These special cases seem to involve magnetic
  fields, if not dynamo activity.} show wind-broadening, but less of
the asymmetry and skewness expected from the effects of wind
attenuation.  These results have generally been interpreted as
evidence for smaller mass-loss rates, in accord with other, recent
indications from the analysis of UV, FUV, H-alpha, and radio
observations of O star winds ({\it refs to be inserted}).  However,
recent work describing the effects of porosity on X-ray line profiles
suggest that porosity's symmetrizing effects can explain what
previously had been interpreted as reduced continuum opacity (and,
ultimately, attributed to reduced mass-loss rates), though only by
invoking very large inter-clump spacing \citep{ofh2006}.  This raises
two qeustions: (1) How do line-profile fits with and without porosity
compare quantitatively?  And (2) To the extent that porous models
provide acceptable fits, are the large derived porosity lengths
physically reasonable?

To address these two issues we first fit several of the stronger lines
in the \chandra\ HETGS spectrum of \zpup\ with a parameterized wind
model that includes porosity effects.  By doing this, we can address,
quantitatively, how well porous models fit the data and what the joint
constraints are on the parameters that describe porosity and wind
optical depth. Specifically, we can determine the values of the
porosity length (defined in \S2.2) that are required to fit the data,
assuming that the literature mass-loss rates are correct.  We address
the second issue above by analyzing two-dimensional
radiation-hydrodynamics simulations of the line-driven instability.
We analyze the wind structure produced in these simulations both by
characterizing the porosity length ({\it Co-authors: This hasn't been
  done yet}) and by synthesizing X-ray line profiles using the clumped
structure from the simulations.

In \S2 we describe the data and the basic, isotropic porosity model we
use for the initial analysis. In \S 3 we present the fits to the data,
with emphasis on the joint constraints on the wind optical depth and
the porosity length, assuming isotropic porosity (\ie\ spherical
clumps). In \S 4 we introduce an extension of our basic model, which
accounts for non-isotropic porosity in the context of the
porosity-length formalism.  This model is equivalent to the compressed
shells, or ``pancakes'' that have already been used to describe the
X-ray line profiles of several O stars \citep{ofh2006}. We also
present fits to the \zpup\ \chandra\ data using this new,
non-isotropic model in this section. In \S5 we analyze the numerical
simulations of an O star wind in order to investigate the extent to
which the porosity requirements derived from fitting data in the
previous sections are realistic. In \S6 we present detailed
calculations of the wind opacity, which are needed to interpret the
derived wind opacity values in terms of mass-loss rates, as well as to
assess the implications of the apparent grayness of the effective
opacity.  We discuss the results and summarize the conclusions in \S
7.


\section{The Data and the Model}


\subsection{The Data}


In this paper, we report primarily on fits to the \ion{Fe}{17}
emission line at $\lambda = 15.014$ \AA\ in the \chandra\ \hetgs\
spectrum of \zpup\ (MEG $m = \pm 1$ orders; the HEG spectrum has
negligible flux in this line). {\it Co-authors: Again, note that we
  can expand the scope of the data involved -- both to more lines in
  the \chandra\ spectrum (including HEG too, for lines with enough
  counts, though 15.01 and 18.97 do not have enough counts in the HEG
  to warrant inclusion of those data for those lines) and to the long
  \xmm\ RGS spectrum of \zpup.}  This line is representative -
\citet{kco2003} fit the profile with a spherically symmetric wind
model with attenuation \citep{oc2001}, and found a best-fit $\taustar
= 1.0$.  We also report on fits to two other strong lines in the MEG
spectrum of \zpup\ -- the Lyman-alpha lines of neon and oxygen, at
12.13 \AA\ and 18.97 \AA, respectively.  {\it Maurice: I can change
  the quoted wavelengths to the nearest m\AA, throughout the paper.}

Although this data set has been discussed and analyzed in several
previous papers \citep{Cassinelli2001,kco2003,Leutenegger2006}, we
re-reduced the data using recent versions of the CIAO software package
(v.\ 3.3) and the CALDB calibration files (v.\ 3.2.2), starting from
the level 1 events table. We saw nothing unusual in the data.  The
grating spectra we extracted appear very similar, if not identical to,
those reported by previous authors.  The effective exposure time of
the observation is 67,636 s, with a total of 17,448 counts in the MEG
first order spectra ($+1$ and $-1$ combined).



\subsection{The Model}


Detailed numerical simulations of the line-driven instability (LDI),
performed under the assumption of one-dimensional spherical symmetry
due to computational constraints, show that thin compressed shells of
wind material form quite easily in the wind above about half a stellar
radius \citep{co1994,Feldmeier1997}.  Rapidly accelerated, rarefied
wind material slams into these slow moving, dense shells, producing
shock-heated plasma at temperatures of several million degrees, and
associated X-ray emission.  It is reasonable to assume that the cold,
compressed shells break up on some lateral scale, although this effect
obviously cannot be produced in one-dimensional
simulations\footnote{The first two-dimensional simulations of the LDI
  do indeed show structures on a rather small scale, both laterally
  and radially \citep{do2003}}.  The effects of such ``fragmented
shells'' have been studied in a series of papers
\citep{foh2003,ofh2004} and recently this model has been compared to
\chandra\ spectra of several O stars \citep{ofh2005,ofh2006}.

A more general treatment of the effect of wind clumping and the
associated porosity on X-ray line profiles demonstrated that the key
parameter is the porosity length, $h \equiv \ell/f$, where $\ell$ is
the characteristic clump scale, $f \equiv \ell^3/L^3$ is the volume
filling factor of the clumps, and  $L$ is the inter-clump spacing
\citep{oc2006}. The porosity length can be thought of as the photon
mean free path as it traverses the interclump medium, at least for
optically thick clumps. 
%Two key points of this analysis are that (1)
%the porosity effect depends on the clump scale, which is not the case
%for density-squared processes, such as free-free emission, which are
%sensitive only to the clump density contrast and not the scale, and
%(2) that individual clumps must be optically thick for the porosity
%effect to be significant. And t
The key quantitative conclusion is that the porosity length must be of
order the local radius, $r$, before X-ray line profiles can be
significantly affected.  This conclusion is consistent with the
conclusions drawn by \citet{ofh2006} from the comparison of their more
specialized model with data.  One goal of the current paper is to put
this result on a firm quantitative and statistical footing, via the
use of real data and the statistical analysis of model fits with
adjustable parameters.

The line-profile model we use is the standard \citet{oc2001}
four-parameter (\taustar, \umax ($\equiv \Rstar/\Rmin$),
normalization, and \q) model, modified for the effects of isotropic
porosity, using the effective opacity treatment of \citet{oc2006},
according to eq.\ (8) in that paper. However, rather than assuming
that the porosity length is proportional to the local radius, as
$h=h'r$, we assume that the clumps follow the wind velocity law (taken
to be $\beta=1$) -- the so-called ``stretch'' porosity, or ``beta''
porosity, model \citep{stan_notes}.  The local value of the porosity
length thus scales as $h=\hinf(1-\Rstar/r)$.  This radial distribution
of porosity length is reasonable if the clumps are embedded in the
wind and move with a similar velocity to law to that of the bulk wind.
Fortunately, the expression for the effective optical depth is
analytically integrable with the beta porosity law.  {\it Co-authors:
  Should we lay out the mathematics here?  Substituting the beta
  porosity law into equation 8 and deriving/showing the governing
  equation for the profile (analogous to equation 11 in \citet{oc2006}
  -- from the beta/stretch version of equation 9 combined with
  equation 10)?}  This parameterization also has the benefit of
allowing for a direct comparison to the parameter, $n_o$, which
\citet{ofh2006} use to characterize the frequency at which clumps are
released at the base of the stellar wind.

\subsection{The Fitting Procedure}

This wind profile with porosity model is implemented as a custom model
in \xspec\ (v.\ 11.3.1).  It has five parameters described above, with
the two most relevant ones being the terminal porosity length, \hinf,
and the fiducial wind optical depth, \taustar.  The other three
parameters are the normalization (photons s$^{-1}$ cm$^{-2}$), \umax,
which parameterizes the onset radius of X-ray emission, and $q$, which
is the power-law index for the assumed radial dependence of the X-ray
emitting plasma's filling factor.  These last four parameters are
identical to those used by \citet{oc2001}. The effect of porosity on
the line profiles is completely accounted for by the single new
parameter, \hinf.  We performed all of the fitting in \xspec, and
included a continuum component in all model fitting. The continuum was
forced to be flat, but the height of the continuum was allowed to be a
free parameter.  We typically included a continuum region on either
side of the line that was about as large as the wavelength range
covered by the line itself, although we occasionally excluded nearby
wavelength regions if they were contaminated by emission from other
lines.  This was the case for the \ion{Fe}{17} line at 15.014 \AA, for
which we fit data on the range $14.85~{\rm \AA} < \lambda < 15.13~{\rm
  \AA}$, which excludes the continuum on the red side of the line, as
it is contaminated by the broadened emission of another \ion{Fe}{17}
line at 15.26 \AA.  Following this procedure, we had 110 bins in the
combined $m = \pm 1$ MEG spectra.  We used the C statistic
\citep{Cash1979} as the fit statistic for all the fitting described in
this paper, as the small number of counts in the extreme wings and the
continuum invalidate the ${\chi}^2$ statistic and its assumption of
Gaussian-distributed errors.


\section{Fitting the Isotropic Porosity Model to the Data}

The first fit we performed was the most general: all five parameters
(\hinf, \taustar, \umax, normalization, and $q$) were free to vary, as
was the normalization of the continuum.  The best-fit model parameters
are in good agreement with the results of \citet{kco2003}, including
$\taustar=1.1$ (compare to their 1.0), and has the best-fit terminal
porosity length value of $\hinf = 0.00$.  The best-fit parameters are
summarized in Table \ref{tab:best-fit} (quoted errors on the fit
parameters are 68\% confidence limits, throughout).  The best-fit
model is shown, superimposed on the data in Fig.\ \ref{fig:best-fit}.
The fit is very good.  In fact, the very low C statistic value implies
a rejection probability of only 3\%.  This is determined from fitting
an ensemble of Monte Carlo simulated datasets and comparing the C
statistic from the fit to the data to the distribution of C statistics
generated from the Monte Carlo simulations (via the {\it goodness}
command in \xspec).

{\it Co-authors: We should think about what it means that the fit is
  formally so good.  Perhaps the model is more detailed or complex
  than it needs to be considering the quality (S/N) of the data.  We
  could eliminate one or more of the free parameters.  But we really
  have no {\it a priori} way to decide which ones are reasonable to
  eliminate (and what values we'd fix them at).  In fact, if we were
  to eliminate one parameter, I would say the most reasonable thing
  would be to fix $\q=0$.  However, this won't ``improve'' the Monte
  Carlo probabilities (i.e. they'll still be too good/low).  And
  fixing $\q=0$ would have implications for high-\taustar\ fits, as we
  will see below, where we show that the high-\taustar\ fits require
  $q \approx 1$.  Maybe the thing to do is to start off this section
  fixing $q = 0$.  Then we can show that high-\taustar,\hinf\ fits are
  really quite bad, with the strong blue wing, as is seen in Fig.\
  \ref{fig:q0}, that doesn't fit the data.  And only at that point
  would we allow that \q\ could be a free parameter.  So, we'd show
  that in order to get decent fits with models with high porosity (but
  still not nearly as good as models without porosity) you have to
  ``artificially'' crank up \q.}

Given this best-fit model, we can quantify the uncertainties on the
derived model parameters via the usual ``$\Delta$C statistic'' method.
This allows us to see how much larger than zero \hinf\ can be before
the model fit is significantly worse than the best-fit model.  It will
also enable us to see how much higher the derived \taustar\ value can
be.  We show the joint constraints on \taustar\ and \hinf\ in Fig.\
\ref{fig:conf_contours}. It can be seen from this figure that at the
68\% confidence level, \hinf\ can be almost as large as 1.7 \Rstar.
But interestingly, even if \hinf\ is this large, \taustar\ does not
have to increase by very much ($\taustar \approx 3$ if $\hinf = 1.7$
\Rstar\ compared to the $\taustar \approx 15$ expected from the
literature \mdot\ and $\kappa$ values\footnote{{\it Co-authors: A
    somewhat detailed calculation of the wavelength-dependent atomic
    opacity in the wind is pending.}}).

Next, we look at a model with a high \taustar\ value; something
consistent with the literature mass-loss rate.  Without a specific,
detailed calculation of the X-ray opacity of the wind, we are guided
by previously published opacities and related quatities for this and
other stars' winds. We choose $\taustar = 15$ as the value
representative of a smooth wind and the literature mass-loss rate.
{\it Co-authors: I will explicitly show this calculation -- and the
  implicit assumption for the value of $\kappa$ -- when I get a
  chance.}

In Fig.\ \ref{fig:high_taustar} we show the best-fit model with
\taustar\ fixed at 15, and all the other parameters free.  The best
fit model has $\hinf = 5.5$ \Rstar. The lower limit on \hinf\ is 4.5
\Rstar.  So, a rather large value of the porosity length is required
if the mass-loss rate is not reduced\footnote{\citet{ofh2006} find a
  value of $n_{\rm o} = 2.6 \times 10^{-5}$ s$^{-1}$ for \zpup.  Using
  their values for \vinf\ and \Rstar, this value of $n_{\rm o}$
  corresponds to $\hinf=6.4$ \Rstar. {\it Co-authors: Note that this
    value is now $1.7 \times 10^{-4}$ in the recently accepted version
    of \citep{ofh2006}, corresponding to $\hinf = 1$ \Rstar. It is
    fixed, now, not fit. Should we also quote this original value from
    their submitted manuscript that represents their fit-by-eye?}}.
The model parameters of this fit are summarized in Table
\ref{tab:high_taustar}.  Note that \Rmin\ did not change very much,
but \q\ did.  The best-fit value is $\q = 0.82$.  This high \q\ value
will de-emphasize the wings of the line by weighting the inner wind
more.  Note also that a bump or flattening out of the profile can be
seen near line center in this model.  The model systematically
overpredicts the flux in this part of the profile. {\it Co-authors: It
  should be noted that the C statistic (like the chi-square statistic)
  is unaffected by correlations in the model-data deviations; by long
  runs of bins that all either over- or under-predict the level seen
  in the data.  Some rank-order statistic (like the K-S statistic)
  might be more sensitive in this regard.}

Note that this model is significantly worse than the best-fit model
shown in Fig.\ \ref{fig:best-fit} -- the C statistic is 98 compared to
86 (implying $>99\%$ significance).  {\it Co-authors: We will likely
  be refitting the data with a background spectrum included. This will
  not change the results but will change the raw values of the C
  statistic.} However, because the quality of the best fit is so good,
the significantly worse fit still has a reasonable goodness of fit.
So, although we cannot rule out this high-\taustar, high-\hinf\ model,
the low-\taustar, low-\hinf\ model is {\it preferred} -- with high
statistical significance -- over the high-\taustar, high-\hinf\ model.
Because the high-\taustar, high-\hinf\ model requires a large increase
in \q, and because one might reasonably choose to fix $\q=0$ in order
to keep the number of free model parameters small, it is instructive
to look at a high-\taustar, high-\hinf\ model with $\q=0$. The
best-fit model with $\taustar=15$ and $\q=0$ fixed is shown in Fig.\
\ref{fig:q0} and described in Table \ref{tab:q0}.  Indeed this fit is
significantly worse than the others we have shown.  This seems to be
due to an excess of emission on the blue wing, which is a universal
characteristic of high-\taustar, high-\hinf\ models (and which is
suppressed by larger \q\ values, as large \q\ gives extra weight to
the inner, slow-moving portions of the wind).

To assess the level of degeneracy among the model parameters, we
compare three of the models we have discussed above, without
convolving them with the instrumental response.  This comparison is
shown in Fig.\ \ref{fig:three_models}.  It can be seen from this
figure that the models produce line profiles with quite similar,
though not identical, morphologies.  However, as we have just shown,
the systematic differences in these morphologies can, in fact, be
distinguished with data of the quality of the \ion{Fe}{17} 15.014 \AA\
line in the \zpup\ MEG spectrum.

Finally, the high value of \hinf\ found in the high-\taustar\
($\taustar=15$) model may be partly due to the nature of the stretch
model, as parameterized by \hinf.  So, we also fit a model that
parameterizes the radial dependence of the porosity length as $h =
h'r$, which was the assumption used by \citet{oc2006}.  Indeed, we
find when we do this that, again, a model without porosity is
preferred to one with porosity.  Furthermore, when we force $\taustar
= 15$, we find a high value of $h'$.  Quantitatively, we find that
$h'$ must be between 1.0 and 1.5; so, roughly four times smaller than
the corresponding \hinf.  This is sensible, as the inner, optically
thick, portion of the wind is where porosity will have its largest
effect.  And in this region, the wind speed is several times below the
terminal velocity, so to get the same porosity length in the inner
wind, $h'$ must be smaller than \hinf\ by a factor of a few. Profiles
calculated using these two different assumptions regarding the
porosity-length distribution are shown in Fig.\ \ref{fig:stan_suite}.
{\it Co-authors: we could eliminate this discussion/paragraph
  completely; what do you think? I'm inclined to keep in this brief
  discussion because the $h'$ model was used in \citet{oc2006}, and
  the \hinf\ parameterization was not discussed there at all.}

The results we have reported on here, for \ion{Fe}{17} at 15.014 \AA,
are representative of what we find for other lines, though generally
the uncertainties associated with fitting other lines in the \chandra\
spectrum are greater, as the signal-to-noise of the 15.014 \AA\ line
is relatively high.  However, we performed a comparable suite of fits
to the \ion{Ne}{10} \lya\ line at 12.134 \AA.  Two of these fits -- a
global fit with all parameters free and one in which we fix $\taustar
= 15$ -- along with the confidence contours in the \taustar-\hinf\
plane are shown in Fig.\ \ref{fig:NeX} and the fit parameters are
listed in Table \ref{tab:NeX}.  While high-\taustar, high-\hinf\
models are excluded at the 90\% confidence level for this line too,
the fits they provide are not as bad as the comparable fits to the
\ion{Fe}{17} 15.014 \AA\ line.  We also did the same for the oxygen
\lya\ line at 18.97 \AA.  This line has a lower signal-to-noise than
the neon \lya\ line, and consequently the bounds that can be placed on
the model parameters are less stringent.  Here the high wind optical
depth model cannot be ruled out at all, although as can be seen in
Fig.\ \ref{fig:OVIII}, the same correlation between \taustar\ and
\hinf\ is seen in the fits to this line too.  {\it Co-authors: I
  haven't included a table for the fits to this line, because the
  model parameters are so poorly constrained.  Also, should we state
  the wavelength range over which we fit the data along with the total
  number of bins, for these two lines, as we did for the \ion{Fe}{17}
  line?}


\section{Including Non-isotropic Porosity in Models of the X-ray Emission Line Profiles}

{\it Co-authors: Should this be an appendix instead?  I think we'll
  want to discuss fits of the non-isotropic model to the data and
  perhaps compare isotropic and non-isotropic models directly.  So, at
  this point, I feel we should include this within the body of the
  paper.}

\subsection{Extending the Model to Include Non-isotropic Porosity}

{\it Co-authors: We will lay out some of the formalism in \citep{stan_notes}... }

\subsection{Fits to the \chandra\ Spectrum}

{\it Co-authors: Maurice has incorporated this model into {\it windprof} and we will do a series of fits like those described in the previous section...}

\section{Porosity Structure in Two-Dimensional Numerical Simulations of Line-Driven Winds}

Now that we have information about the quantititive requirements for
porosity to have a significant effect on X-ray line profiles, we are
ready to assess the likelihood that real winds have such large
porosity lengths. The simplest assumption is that the line-driven
instability, which is thought to lead to the shock heating and
X-ray production in the first place, is also responsible for
generating the clumping that could lead to porosity.  It is also
possible, however, that other mechanisms -- perhaps photospheric
disturbances, causing propagating density enchancements in the wind
which, when modulated by rotation, lead to discrete absorption
components (DACs) and co-rotating interaction regions (CIRs); or
perhaps magnetic fields seeding wind structure -- dominate the
clumping at larger spatial scales.

To address the clumpy and porous structure generated by the LDI, we
have examined state-of-the-art, two-dimensional (2D) radiation
hydrodynamics simulations, as described in \citet{do2003}. Multiple
snapshots from the simulations described in that paper were mapped
onto a three dimensional grid, in order to provide a realistic
distribution of cold wind material.  The X-ray emitting plasmas was
assumed to be smoothly distributed above some minimum radius and to
have an emissivity proportional to the ambient density squared.  This
scenario closely mirrors the assumptions in the \citet{oc2001} model,
with $\q = 0$. Note that the 2D simulations are isothermal, so X-ray
emission cannot be directly computed from them in a straightforward
way.  And in any case, the method we adopt here allows for a clearer
comparison to the line profiles expected from a corresponding smooth
wind.

{\it Co-authors - but especially Luc and Stan: I'm unclear about a
  couple of things -- reading Luc's email of May 21, 2006 to Stan --
  the description of how the different phi-slices were or weren't
  populated is a bit confusing to me: 12 degree wide wedges... but
  only 9 of them?  And then your statement about ``not actually
  us(ing) different 2D hydro inputs in other phi angles'' is not so
  clear to me, either.  Sorry for being obtuse.  Maybe a useful way to
  address my confusion would be for you, Luc (and/or Stan) to comment
  on/modify the text I've written above.  Another question I have is
  how you determined the opacity of the wind material (or, of
  individual clumps).  What did you use for the atomic opacity? OK,
  wait, I assume that you fiddled the opacity to give the three values
  of \taustar.  Is that correct? Finally, why do the solid lines in
  the figure look a little bumpy too?}

In Fig.\ \ref{fig:sims} we show a series of line profiles calculated
from the simulations, as described above.  Each panel has a different
onset radius of X-rays ($R_o$) ({\it Co-authors: we need to use
  consistent notation: $R_o$ or \Rmin?}) and shows line profiles for
three values of wind optical depth (\taustar), calculated using the
clumped wind structure from the simulations and also from a
corresponding smooth wind.  The profiles that include clumping show a
low level of random noise, as would be expected from the numerical
simulation generated structure, but show very little, if any,
systematic effects on the overall shapes.  Examining the simulation
output itself (figure 1 in \citet{do2003}), this is not surprising, as
the clumpy structure is quite small scale, yielding porosity lengths
that are significantly less than a stellar radius. Note that this is
quite different than the larger interclump spacing seen in
one-dimensional simulations \citep{Feldmeier1997}, probably because in
one dimension, clumps (shells, really) are prone to collide and merge
whereas in higher-dimensional simulations the shell fragments, or
clumps, are able to move past each other without interacting or
interacting via shearing, rather than merging.  It seems likely
that instabilities (e.g.\ Rayleigh-Taylor) also cause larger
structures to fragment on small lateral scales. Finally, note that the
small lateral scales of wind clumps (as distinct from, e.g., CIRs)
have been noted in other studies, both theoretical \citep{do2002} and
observational \citep{lm1999}.

{\it Co-authors: To what extent is it (a) possible and (b) desirable
  to compute porosity lengths directly from these same -- or similar
  -- simulations?  Identifying discrete clumps, assessing their length
  scales and assessing interclump spacings might not be so
  straightforward.  We could, alternatively, eyeball the simulation
  outputs and qualitatively/semi-quantitatively characterize these
  values.}

\section{Wind Opacity}

{\it Co-authors: I have started using Joe MacFarlane's Prismspect code
  to model ionization and opacity (using his ATBASE atomic structure
  models).  I have yet to include x-ray photoionization in the models,
  among other things, but I think it's desirable to include this in
  this paper.  Not only is it necessary to have opacities to connect
  $\taustar$ to mass-loss rates, but the wavelength-dependence of
  these opacities has also become an issue with the opacities shown by
  \citet{ofh2006} and their discussion of gray opacity pointing to
  optically thick clumping.}

\section{Discussion and Conclusions}

\vspace{1in}


%\acknowledgements

%DHC acknowledges NASA contract AR5-6003X to Swarthmore College through
%the Chandra X-ray Center.




% BIBLIOGRAPHY 

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% TABLES



% TABLE 1: Best-fit model


\begin{deluxetable}{lc}
\tablewidth{0pt}
\tablecaption{Fitted parameters for the best global model -- all parameters free}

\tablehead{
 \colhead{parameter} & 
 \colhead{value\tablenotemark{a}} 
}

\startdata

\q & $-0.02^{+.13}_{-.02}$ \\
\taustar & $1.1^{+0.7}_{-0.1}$ \\
\hinf & $0.00^{+1.7}_{-0.0}$ \\
\Rmin & $1.53^{+.11}_{-.11}$ \\
C & 82.95 \\
rej.\ prob.\tablenotemark{b} & 3\% \\ 

\enddata

\tablewidth{400pt}

\tablenotetext{a}{Uncertainties quoted here and in other tables are
  the 68\% confidence limits on a given parameter, calculated using
  the $\Delta{\rm C} = 1$ criterion while letting all the other
  parameters vary freely.}  

\tablenotetext{b}{The rejection probability represents the percentage
  of Monte Carlo simulated datasets which gave a C statistic value
  lower than that produced by the actual data.}

\label{tab:best-fit}

\end{deluxetable}  



% TABLE 2: high_taustar


\begin{deluxetable}{lc}
\tablewidth{0pt}
\tablecaption{Parameters for a model with $\taustar=15$ -- all other parameters, including \hinf, free}

\tablehead{
 \colhead{parameter} & 
 \colhead{value} 
}

\startdata

\q & $0.82^{+.88}_{-.42}$ \\
\taustar & $15$ \\
\hinf & $5.5^{+1.2}_{-1.0}$ \\
\Rmin & $1.73^{+.23}_{-.06}$ \\
C & 98.44 \\
rej.\ prob. & 10\% \\ 

\enddata

\label{tab:high_taustar}

\end{deluxetable}  



% TABLE 3: high_taustar, q=0 fixed


\begin{deluxetable}{lc}
\tablewidth{0pt}
\tablecaption{Parameters for a model with $\taustar=15$ and $\q=0$ -- all other parameters, including \hinf, free}

\tablehead{
 \colhead{parameter} & 
 \colhead{value} 
}

\startdata

\q & $0$ \\
\taustar & $15$ \\
\hinf & $6.7^{+0.9}_{-1.2}$ \\
\Rmin & $1.56^{+.08}_{-.07}$ \\
C & 102.44 \\
rej.\ prob. & 21\% \\ 

\enddata

\label{tab:q0}

\end{deluxetable}  



% TABLE 4: stretch porosity model (all free, and then taustar=15
% fixed) fit to the Ne X line at 12.134 A)


\begin{deluxetable}{lcc}
  \tablewidth{0pt} \tablecaption{Parameters for a porosity
    model fit to the \ion{Ne}{10} line at 12.134 \AA: the right-most
    column lists the fit parameters for a model with \taustar\ fixed
    at $\taustar=15$}

\tablehead{
 \colhead{parameter} & 
 \colhead{value} &   \colhead{value}
\\ 
\colhead{} & \colhead{(all parameters free)} & \colhead{($\taustar=15$)}
}


\startdata

\q & $0.16^{+.24}_{-.19}$ & $0.45^{+.23}_{-.15}$   \\
\taustar & $1.89^{+3.52}_{-0.38}$ & $15$ \\
\hinf & $0.37^{+2.43}_{-0.34}$ & $4.10^{+.70}_{-.50}$ \\
\Rmin & $1.48^{+.12}_{-.11}$ & $1.62^{+.10}_{-.08}$ \\
C & 79.40 & 84.40 \\
rej.\ prob. & 14\% & 21\% \\ 

\enddata

\label{tab:NeX}

\end{deluxetable}  






% FIGURES 

\newpage



% FIGURE 1: BEST-FIT, q FREE

\begin{figure}
\begin{center}
\includegraphics[angle=90,scale=0.60]{figs/FeXVII_1501/fexvii_1501_windprof_best_q_thawed.ps}
\end{center}
\caption{The global best-fit wind profile model for the \ion{Fe}{17}
  line ($\lamlab = 15.014$ \AA) measured in the MEG first order
  (negative and positive orders co-added).  In this model, the free
  parameters were  \hinf, \taustar, \umax, \q, and the normalization of
  the line profile, as well as the normalization of the power-law
  continuum model (for which we fixed the power-law index at
  $\alpha=2$). The fit is formally good. (fexvii\_1501\_windprof\_best\_q\_thawed.ps) }
\label{fig:best-fit}
\end{figure}


% FIGURE 2: BEST-FIT, q FREE  - CONFIDENCE REGIONS

\begin{figure}
\begin{center}
\includegraphics[angle=90,scale=0.60]{figs/FeXVII_1501/fexvii_1501_hinf_tau.ps}
\end{center}
\caption{The 68\% and 90\% confidence regions for two parameters of
  interest (${\Delta}C=2.30$, $4.61$).  The calculated grid is 20 by
  20, and for each of the 400 models, the other parameters (\q, \umax,
  normalization, and the normalization of the power-law continuum)
  were free to vary until a best-fit model for those values of
  \taustar\ and \hinf\ was found. The best-fit value on the grid is
  indicated with an asterisk (the fit shown in Fig.\
  \ref{fig:best-fit}). The X indicates the best-fit \hinf\ value for
  the fit where $\taustar = 15$ was fixed (the fit shown in Fig.\
  \ref{fig:high_taustar}).  (fexvii\_1501\_hinf\_tau.ps) }
\label{fig:conf_contours}
\end{figure}


% FIGURE 3: high_taustar

\begin{figure}
\begin{center}
  \includegraphics[angle=90,scale=0.60]{figs/FeXVII_1501/fexvii_1501_windprof_best_tau_15.ps}
\end{center}
\caption{This fit has \taustar\ fixed at 15, with all the other
  parameters being free. The terminal porosity length has a value of
  $\hinf=5.5$ \Rstar.  The fit is significantly worse than the
  best-fit model shown in Fig.\ \ref{fig:best-fit}. The model seems to
  have trouble producing the observed profile shape on the red side of
  the line, especially.  (fexvii\_1501\_windprof\_best\_tau\_15.ps)}
\label{fig:high_taustar}
\end{figure}


% FIGURE 4: high_taustar, q=0 fixed

\begin{figure}
\begin{center}
  \includegraphics[angle=90,scale=0.60]{figs/FeXVII_1501/fexvii_1501_windprof_best_q_fixed_hinf_thawed.ps}
\end{center}
\caption{This fit has \taustar\ fixed at 15 and \q\ fixed at 0, with
  all the other parameters being free. Note the bump on the blue wing
  of the model in addition to the systematic over-prediction of flux
  on the red side of the profile, as is also seen in the fit shown in
  Fig.\ \ref{fig:high_taustar}.  The fit is significantly worse than
  the best-fit $\taustar=15$, \q\ free model shown in Fig.\
  \ref{fig:high_taustar}.  
  (fexvii\_1501\_windprof\_best\_q\_fixed\_hinf\_thawed.ps)}
\label{fig:q0}
\end{figure}


% FIGURE 5: COMPARISON OF THREE UNCONVOLVED MODELS

\begin{figure}
\begin{center}
  \includegraphics[angle=0,scale=1.6]{figs/FeXVII_1501/best_fit_zpup.ps}
\end{center}
\caption{Three of the models we fit to the 15.014 \AA\ line, shown
  with perfect resolution: the global best fit, the model with
  $\taustar=15$, and the model with $\taustar=15$ and $\q=0$. {\it
    Co-authors: The image quality of this figure is fine when it's
    viewed in ghostview.  Don't know what's going on with the
    embedding in LaTeX, but I'll get it figured out.} (best\_fit.eps)}
\label{fig:three_models}
\end{figure}


% FIGURE 6: SUITE OF POROSITY (LINEAR V. STRETCH) MODELS

\begin{figure}
\begin{center}
  \includegraphics[angle=0,scale=0.65]{figs/stan_suite.eps}
\end{center}
\caption{Suite of models comparing the linear treatment of porosity
  (left-most column) to the stretch (beta) treatment (second-from-left
  column) for a series of values of $h'$ and \hinf, respectively.  In
  each panel, different values of \taustar\ are shown.  All models
  assume $\Rmin=1.5 \Rstar$ and $\q=0$. {\it Co-authors: The third
    column shows non-isotropic porosity models, while the fourth
    column shows ``shell'' models, where each ``clump'' subtends
    4$\pi$ steradians.  Stan - or I - can remake this figure when we
    decide exactly what we want to show -- maybe a reduced version of
    the second and third columns.} (stan\_suite.eps)}
\label{fig:stan_suite}
\end{figure}


% FIGURE 7: Ne X 

\begin{figure}
\begin{center}
  \includegraphics[angle=90,scale=0.38]{figs/NeX_1213/nex_1213_best_fit.ps}
  \includegraphics[angle=90,scale=0.34]{figs/NeX_1213/NeX_1213_hinf_tau.ps}
  \includegraphics[angle=90,scale=0.38]{figs/NeX_1213/nex_1213_best_tau_15.ps}
\end{center}
\caption{Best-fit porosity model -- with all parameters free
  -- fit to the \ion{Ne}{10} Lyman-alpha line at 12.134 \AA\ (top).
  We also show the 68\% and 90\% confidence limits for this fit
  (middle) and a fit to the same line with a porosity model
  for which we fixed $\taustar=15$ (bottom). In the middle panel, the
  best global fit (shown in the top panel) is indicated by the
  asterisk, while the best-fit high optical depth model (shown in the
  bottom panel) is indicated by the X.  (nex\_1213\_best\_fit.ps;
  NeX\_1213\_hinf\_tau.ps; nex\_1213\_best\_tau\_15.ps)}
\label{fig:NeX}
\end{figure}


% FIGURE 8: O VIII 

\begin{figure}
\begin{center}
  \includegraphics[angle=90,scale=0.40]{figs/OVIII_1897/oviii_1897_best_hinf_zero.ps}
  \includegraphics[angle=90,scale=0.37]{figs/OVIII_1897/oviii_1897_hinf.ps}
  \includegraphics[angle=90,scale=0.40]{figs/OVIII_1897/oviii_1897_tau15_prime_zero.ps}
\end{center}
\caption{Best-fit porosity model -- with all parameters free -- fit to
  the \ion{Ne}{10} Lyman-alpha line at 12.134 \AA\ (top).  We also
  show the 68\% and 90\% confidence limits for this fit (middle) and a
  fit to the same line with a porosity model for which we fixed
  $\taustar=15$ (bottom).  (nex\_1213\_best\_fit.ps;
  NeX\_1213\_hinf\_tau.ps; nex\_1213\_best\_tau\_15.ps)}
\label{fig:OVIII}
\end{figure}




% FIGURE 9: Luc's sims

\begin{figure}
\begin{center}
  \includegraphics[angle=0,scale=0.75]{figs/xray_porous_q0.ps}
\end{center}
\caption{Line profiles calculated from the wind structure generated in
  2D hydro simulations.  The emission assumes an \citet{oc2001} model
  with $\q = 0$ and the onset radius, $R_o = 1.1, 1.5, 2.0, 3.0$
  \Rstar, as indicated in each panel.  Within each panel, we show
  profiles for three assumptions of the wind optical depth, given by
  $\taustar = 1, 2, 5$. The numerically calculated profiles are
  indicated by the dashed lines, while the corresponding smooth-wind
  profiles are indicated by the solid lines.  (xray\_porous\_q0.ps)}
\label{fig:sims}
\end{figure}




\end{document}


