\section{Introduction}

The blue compact dwarf emission-line galaxy I\,Zw\,18 is famous for
being the most metal poor galaxy known so far. Its oxygen abundance
is about 2\% the solar value, as first shown by Searle and Sargent
(1972), and then confirmed by many studies (e.g. Lequeux et al. 1979,
French 1980, Kinman \& Davidson 1981, Pagel et al. 1992, Legrand et
al. 1997, Izotov et al. 1997b, V\'{\i}lchez
\& Iglesias-P\'{a}ramo 1998). Because of this,  I\,Zw\,18 has played
an essential role in the determination of the primordial helium mass
fraction.  Also, due to its
extreme properties, I\,Zw\,18 has been a choice target for studies of
star formation history in blue compact galaxies (Dufour \& Hester
1990, Hunter \& Thronson 1995, Dufour et al. 1996, De Mello et al.
1998, Aloisi et al. 1999), of the elemental enrichment in dwarf
galaxies (Kunth \& Sargent 1986, Kunth et al. 1995) and of the
interplay between star formation and the interstellar medium (Martin
1996, van Zee et al. 1998). An important clue is the distribution of
the oxygen abundance inside the H~{\sc ii}\ regions (Skillman \& Kennicutt
1993, V\'{\i}lchez \& Iglesias-P\'{a}ramo 1998)
and in the neutral gas
(Kunth et al. 1994, Pettini \& Lipman 1995, van Zee et al. 1998).
Another clue is the carbon and nitrogen abundance (Garnett et al.
1997, V\'{\i}lchez \& Iglesias-P\'{a}ramo 1998, Izotov \& Thuan 1999).
On the whole, there is a
general consent about an intense and recent
burst of star formation in I\,Zw\,18  - which provides the ionizing
photons - following previous star formation episodes. How exactly has
the gas been enriched with metals during the course of the evolution
of I\,Zw\,18  remains to be better understood.

Much of our understanding (or speculations) on the chemical evolution
of I\,Zw\,18  (and other galaxies in general) relies on the
confidence
placed in the chemical abundances derived from the lines emitted in
the H~{\sc ii}\ regions. These are generally obtained using standard,
empirical,
methods which have been worked out years ago, and rely on the theory
of line emission in
photoionized gases. Photoionization models are, most of the time,
used merely as a guide to evaluate
the temperature of the low excitation regions once the characteristic
temperature of the high excitation zones has been obtained through
the [O~{\sc iii}] $\lambda$4363/5007\ ratio. They
also serve to provide formulae for estimating the correction factors
for
the unseen ionic species of a given element.

Direct fitting of
the observed emission  line spectrum by tailored photoionization
models provides more accurate
abundances only if all the relevant line ratios are perfectly
reproduced by the model
(which is rarely the case in model fitting history) and if the
model reproducing all the observational constraints is unique.

One virtue of model fitting, though, is that it permits to check
whether the assumptions used in abundance determinations are correct
for a given object. For example, there is the long standing debate
whether so-called ``electron temperature fluctuations'' (see e.g. Mathis 1995,
Peimbert 1996, Stasi\'{n}ska 1998) are present
in H~{\sc ii}\ regions to a
sufficient level so as to significantly affect elemental abundance
determinations. If a photoionization model is not able to reproduce
all the temperature sensitive line ratios, the energy balance is not
well understood, and one may question the
validity of abundance determinations. Also, photoionization models
are a potential tool (see e.g. Esteban et al. 1993,  Garc\'{\i}a-Vargas
1996, Stasi\'nska \& Schaer\-er 1997, Crowther et al.\ 1999)
to uncover the spectral distribution of
the ionizing radiation field, thus providing information on the
ionizing stars, their evolutionary status and the structure of their
atmospheres.

These two points are a strong motivation for a photoionization model
analysis of I\,Zw\,18. There have already been a few such attempts in
the past (Dufour et al. 1988, Campbell 1990, Stevenson et al. 1993).
None of those models were, however, able to reproduce the He~{\sc ii} $\lambda$4686\
line,
known to exist in I\,Zw\,18 since the work of French (1980). The
reason
is that, in those models, the spectral distribution of the ionizing
radiation was that of a single
star whose radiation field was interpolated from a grid of
plane-parallel, LTE model atmospheres for massive stars.
Recently,  Wolf-Rayet stars have been identified in I\,Zw\,18 through
the characteristic bump they produce at 4650~\AA\ (Izotov et al. 1997a,
Legrand et al. 1997).
Spherically expanding non-LTE model atmospheres for hot Wolf-Rayet
stars
with sufficiently low wind densities (Schmutz et al. 1992) do predict
an
output of radiation above the He~{\sc ii}\ ionization edge, which might, at
least
qualitatively, provide a natural explanation for the narrow He~{\sc ii} $\lambda$4686\
line
observed in I\,Zw\,18. Schaerer (1996) has, for the first time,
synthesized the broad (stellar) and narrow (nebular) He~{\sc ii}\ features
in young starbursts using the Geneva stellar evolution tracks and
appropriate stellar model atmospheres. He then extended his
computations to the metallicity of I\,Zw\,18 (De Mello et al. 1998).

In this paper, we use the emergent radiation field from the synthetic
starburst model presented in De Mello et al. (1998) to construct
photoionization models of I\,Zw\,18. One
of the objectives is to see whether this more realistic ionizing
radiation field permits, at the same time, to solve the electron
temperature problem encountered in previous studies. Former
photoionization models predicted too low a [O~{\sc iii}] $\lambda$4363/5007\ ratio, unless
specific geometries were adopted (Dufour et al. 1988, Campbell 1990),
which
later turned out to be incompatible with Hubble Space Telescope
({\em HST}) images.
The synthetic starburst model we use is based on spherically
expanding non-LTE stellar atmosphere models for main sequence stars
(Schaerer \& de Koter 1997) and for Wolf-Rayet stars (Schmutz et al.\
1992). These models have a greater heating power than the LTE model
atmospheres of same effective temperature (see Fig.\ 3;
also Schaerer \& de Koter)

The progression of the paper is as follows. In Section 2, we  discuss
in more detail the photoionization models proposed previously for
I\,Zw\,18 and show in what respect they are not consistent with
recent
observations. In Section 3, we present our own model fitting
methodology, including a description of the computational tools. In
Section 4, we describe the models we have built for I\,Zw\,18,
and discuss the effects of the assumptions involved in the computations.
Our main results are summarized in Section 5.

\section{Previous photoionization models of I\,Zw\,18}

The first attempt to produce a photoionization model for I\,Zw\,18 is
that of Dufour et al. (1988). Their observational constraints were
provided by spectra obtained in an aperture of 2.5\arcsec $\times$
6\arcsec  of the NW region
combined with IUE observations yielding
essentially the C~{\sc iii}] $\lambda$1909\ line. Using Shields's photoionization code
NEBULA, they modelled the object as an ionization bounded sphere of
constant density $n$ = 100~cm$^{-3}$\ and adjustable volume filling
factor
$\epsilon$ so as to reproduce the observed [O~{\sc iii}] $\lambda$5007/[O~{\sc ii}] $\lambda$3727\ ratio. The
ionizing radiation was provided by a central source of radiation,
represented by the LTE model atmospheres of Hummer \& Mihalas (1970),
modified to take into account the low metallicity of I\,Zw\,18.
Discarding the
He~{\sc ii}\ problem, they obtained a model that was reasonably successful
except that it had an O$^{++}$\ temperature, T(O$^{++}$), marginally smaller
than observed
(17200~K) compared to the value of 18100 (+1100, -1000)~K
derived directly from their observed [O~{\sc iii}] $\lambda$4363/5007\ = (2.79 $\pm$
0.35) $\times$ 10$^{-2}$ (the errors quoted being 2$\sigma$)
\footnote{A summary of various measurements of [O~{\sc iii}] $\lambda$4363/5007\ and corresponding
electron temperatures is shown in Fig.\ 2.}.
This model was obtained for
an effective temperature of 45000~K. These authors showed that,
because of the
dominant role played by Ly$\alpha$ cooling in I\,Zw\,18, it was
impossible, for
the adopted geometry, to produce a model with
noticeably higher T(O$^{++}$), by varying the free parameters at hand.
Even increasing the effective temperature did not raise
T(O$^{++}$) appreciably, because then the ionization parameter had to be
lowered in order to maintain [O~{\sc iii}] $\lambda$5007/[O~{\sc ii}] $\lambda$3727\ at its observed level, and
this resulted in a greater
H$^{0}$\ abundance, thus enhancing  Ly$\alpha$ excitation.  Dufour et al.
then proposed a
composite model, in which the [O~{\sc iii}]\ line would be mainly produced
around high temperature stars (T$_{eff}$\ $>$ 38000~K) and the [O~{\sc ii}]\ line
would be mainly emitted around stars of lower T$_{eff}$\ ($<$ 37000~K).
Alternatively, one could have, around a star of T$_{eff}$\ $<$ 45000~K, a
high ionization component emitting most of the [O~{\sc iii}]\ and a low
ionization component emitting most of the [O~{\sc ii}].  Since then, the
{\em HST} images (Hunter \& Thronson 1995, Meurer et al.
1995, Dufour et al. 1996,
De Mello et al. 1998) have revealed that the NW region appears
like a shell of ionized gas about 5\arcsec\ in diameter, encircling a dense
star cluster. Thus the geometries proposed by Dufour et al. (1988),
although quite reasonable a priori, do not seem to apply to the case
under study.

Campbell (1990), using Ferland's photoionization code CLOUDY,
constructed
models to fit the spectral observations of Lequeux et
al. (1979) obtained through a slit of 3\arcsec.8 $\times$ 12\arcsec.4. These
observations were giving a [O~{\sc iii}] $\lambda$4363/5007\ ratio of (3.75 $\pm$
0.35) $\times$ 10$^{-2}$. With a
constant density, ionization bounded spherical model and a LTE Kurucz
stellar atmosphere with metallicity 1/10 solar, in which the
adjustable parameters were O/H, T$_{eff}$, $n$ and $\epsilon$, Campbell
obtained
a best fit model that had [O~{\sc iii}] $\lambda$4363/5007\ = 3.07~10$^{-2}$, i.e. much lower
than the
value she aimed at reproducing. She then proposed a density gradient
model, in which the inner regions had a density $n$
$>$~10$^{5}$~cm$^{-3}$, so as
to induce collisional deexcitation of [O~{\sc iii}] $\lambda$5007. Applying standard
abundance derivation techniques to this model yields an oxygen
abundance higher by 70\% than the input value. This led Campbell to
conclude that I~Zw18 was not as oxygen poor as previously thought.
The density gradient model of Campbell (1990) can be checked
directly using the density sensitive [Ar~{\sc iv}] $\lambda$4741/4713\ ratio. The only
observations giving this line ratio are those of Legrand et al.
(1997), and they indicate a density in the Ar$^{+++}$\ region lower than
100~cm$^{-3}$ . Direct images with the {\em HST} do not support Campbell's
density gradient model either, since, as stated above,
 the appearance of the H~{\sc ii}\ region
is that of a shell surrounding the excitation stars.

Stevenson et al. (1993), using a more recent version of CLOUDY,
constructed a spherical, ionization bounded constant
density photoionization model to fit their own data. They used as an
input an extrapolation of the Kurucz LTE model atmospheres. Their
modelling procedure was very similar to that of Campbell (1990) for
her constant density model. Their best fit model had O/H =
1.90~10$^{-5}$
and returned [O~{\sc iii}] $\lambda$4363/5007\ = 2.79~10$^{-2}$, to be compared to their
observed value of (3.21 $\pm$ 0.42) $\times$ 10$^{-2}$.

What complicates the discussion of the three studies above is that
they use different codes with probably different atomic data, and
they aim at fitting different sets of observations. Nevertheless, it
is clear that all those
models have difficulties in reproducing the high
[O~{\sc iii}] $\lambda$4363/5007\ observed. They have other weak points, as noted by their
authors. For example, Dufour et al. (1988) and Stevenson et al. (1993)
comment on the unsatisfactory fitting of the sulfur lines. However,
the atomic data concerning sulfur are far less well established than
those concerning oxygen, therefore the discrepancies are not
necessarily meaningful. Besides, it is not surprising that, with a
simple density structure, one does not reproduce perfectly at the
same time the [O~{\sc iii}]/[O~{\sc ii}]\ and [S~{\sc iii}]/[S~{\sc ii}]\ ratios.

The most important defect shared by the three models just discussed
is that they predict no He~{\sc ii} $\lambda$4686\ emission. This is simply due to the
fact that they used an inadequate input stellar radiation field.
With the presently available stellar population synthesis models for
the exciting stars of giant H~{\sc ii}\ regions which make use of more
realistic model atmospheres (Schaerer \& Vacca 1998), and especially
models that are relevant for the Wolf-Rayet stages of massive stars,
it is interesting to reanalyze the problem. Using simple photon
counting arguments, De Mello at al. (1998) have already shown that a
starburst with a Salpeter initial mass function and an upper mass
limit of 150~M$_{\odot}$\ could reproduce the equivalent width
of
the Wolf-Rayet features and of the narrow He~{\sc ii} $\lambda$4686\ emission line in
I\,Zw\,18.

It is therefore interesting, using the emergent radiation field from
such a synthetic stellar population,
to see whether one can better reproduce the [O~{\sc iii}] $\lambda$4363/5007\
ratio
observed in I\,Zw\,18, with a model that is more compatible with the
density structure constrained by the {\em HST} images.

\section{Our model fitting methodology }

\subsection{Computational tools and input parameters}

As in the previous studies, we concentrate on the so-called NW
component, seen in the top of Fig.\ 1, which shows the WFPC2
H$\alpha$\ image of the {\em HST} (cf.\ Fig.\ 1 of De Mello et al.\ 1998).
Throughout the paper, we
adopt a distance to I~Zw~18 of 10 Mpc, assuming $H_{o}$ =
75 km~s$^{-1}$~Mpc$^{-1}$, as in many studies (Hunter \&
Thronson 1995, Martin 1996, van Zee et al. 1998)
\footnote{Izotov et al. (1999) have submitted a paper suggesting
a distance of 20 Mpc to I Zw 18. Should this be the case,
the conclusions of our paper that are linked to the ionization structure
and the temperature of the nebula would hardly be changed. The
total mass of the ionized gas would be larger, roughly by a factor 2$^{3}$.}.

\subsubsection{The stellar population}

We use the same model for the stellar population as described in De
Mello et al. (1998). It is provided by a evolutionary population
synthesis code using stellar tracks computed with the Geneva code at
the appropriate metallicity (1/50~Z$_{\odot}$). The stellar atmospheres
used are spherically expanding non-LTE models for WR stars
(Schmutz et al.\ 1992) and O stars ({\em CoStar} models at $Z=0.004$,
Schaerer \& de Koter 1997), and Kurucz models at [Fe/H]=-1.5 for
the remainder. More details can be found in
De Mello et al. (1998) and Schaerer \& Vacca (1998).  We assume an
instantaneous burst of star formation, with an upper mass limit of
150~M$_{\odot}$\ and a lower mass limit of 0.8 ~M$_{\odot}$.
Since all observational quantities considered here depend
only on the properties of massive stars, the choice for
$M_{\rm low}$ has no influence for the results of this paper.
It merely serves as an absolute normalisation.
The total initial mass of the stars is adjusted in such a way
that, at a distance of 10~Mpc, the flux at 3327~\AA\ is equal to
1.7~10$^{-15}$ ~erg~s$^{-1}$~cm$^{-2}$~\AA$^{-1}$,
the value measured in the flux calibrated WFPC2 F336W image of De Mello
et al.\ (1998)  within a circle of 2.5\arcsec\ radius centered on the
NW region (see Fig.\ 1).
This flux is dominated by the latest generation of
stars in I\,Zw\,18, so that our normalization is hardly sensitive
to the previous star formation history in the NW region of I\,Zw\,18.
It yields a total stellar
mass of 8.7~10$^{4}$~M$_{\odot}$, at a distance of 10~Mpc.

Actually, most of the flux comes from a region much smaller in size,
and our photoionization modelling is made with the ionizing cluster
located at the center of the nebula and assuming that its spatial
extension is negligible.

We consider that the observed ultraviolet flux is only negligibly
affected by extinction
\footnote{A direct fitting of the ultraviolet stellar continuum
by population synthesis models, of which we became aware after the
paper had been submitted,
yields  C(\ifmmode {\rm H}\beta \else H$\beta$\fi) $<$ 0.06 (Mas-Hesse \& Kunth 1999).}. For an extinction C(\ifmmode {\rm H}\beta \else H$\beta$\fi)
of 0.04, such as
estimated by Izotov \& Thuan (1998), the corrected flux would be only
about 10\% larger, which is insignificant in our problem. Other
observers give values of C(\ifmmode {\rm H}\beta \else H$\beta$\fi) ranging between 0. and 0.2. If C(\ifmmode {\rm H}\beta \else H$\beta$\fi)
were as large as 0.20, as estimated by  V\'{\i}lchez \& Iglesias-P\'{a}ramo
(1998) and some other observers, the true stellar flux would be a
factor two higher. However, all the determinations of C(\ifmmode {\rm H}\beta \else H$\beta$\fi), except
the one by Izotov \& Thuan (1998), do not take into account the
underlying stellar absorption at \ifmmode {\rm H}\beta \else H$\beta$\fi\, and therefore overestimate the
reddening.
A further cause of overestimation of C(\ifmmode {\rm H}\beta \else H$\beta$\fi), which applies
also to the work of Izotov \& Thuan (1998), is that the intrinsic
H$\alpha$/\ifmmode {\rm H}\beta \else H$\beta$\fi\ ratio is assumed to be the case B recombination value, while
collisional excitation of H$\alpha$\ is not negligible in the case of
I\,Zw\,18 as noted by Davidson \& Kinman (1985). We will come back
to this below.

\subsubsection{The nebula}

The photoionization computations are performed with the code PHOTO
using the atomic data listed in Stasi\'{n}ska \& Leitherer (1996). The
code assumes spherical geometry, with a central ionizing source. The
diffuse radiation is treated assuming that all the photons are
emitted outwards in a solid angle of 2$\pi$, and the transfer of the
resonant photons of hydrogen and helium is computed with
the same outward only approximation, but multiplying the photo-absorption
cross-section by an appropriate factor to account for the increased path
length due to scattering (Adams 1975).

The nebular abundances used in the computations are those we derived
from the spectra of Izotov and Thuan (1998) for the NW component of
I\,Zw\,18,
with the same atomic data as used in the photoionization code. For
helium, however, we adopted the abundance derived by Izotov \& Thuan
(1998) for the SE component, as stellar absorption contaminates the
neutral helium lines in the NW component. The nominal value of the
temperature derived from [O~{\sc iii}] $\lambda$4363/5007\ is 19800~K. This value was used to
compute the ionic abundances of all the ions except O$^{+}$, N$^{+}$\ and
S$^{+}$,
for which a value of 15000~K was adopted (this is the typical value
returned by  our photoionization models for I\,Zw\,18). The electron
density deduced from [S~{\sc ii}] $\lambda$6731/6717\ is 140~cm$^{-3}$, and this density was
adopted in
the computation of the ionic abundances of all species.  The
ionization correction factors to compute the total element abundances
were those of Kingsburgh \& Barlow (1994), which are based on
photoionization models of planetary nebulae and are also suitable for
H~{\sc ii}\ regions. They give slightly smaller oxygen abundances (by a few
\%) than the traditional ionization correction factors which assume
that the O$^{+++}$\ region is coextensive with the He$^{++}$\ (we did not
iterate on the ionization correction factors after our photoionization
model analysis since this would have not changed any of the conclusions
drawn in this paper). The carbon
abundance used
in the computations follows from the C/O ratio derived by Garnett et
al. (1997) from {\em HST} observations of I\,Zw\,18. The abundances of the
elements not constrained by the observations (Mg, Si) and (Cl, Fe)
have
been fixed to  10$^{-7}$ and 10$^{-8}$ respectively. Table 1 presents
the abundance set
used in all the computations presented in the paper. As already noted
by previous authors, at the metallicity of I\,Zw\,18, the heavy
elements
(i.e. all the elements except hydrogen and helium) play a secondary
role in the thermal balance. Their role in the absorption of ionizing
photons is completely negligible. Any change of abundances, even that
of helium, compatible with the observed intensities of the strong
lines, will result in a very small change in the electron
temperature, and we have checked that the effect they will induce in
the predicted spectra are small compared to the effects discussed
below.

\begin{table}
\caption{Input abundances relative to hydrogen for the models of
I\,Zw\,18 (by number).}
\begin{flushleft}
\begin{tabular}{rr}
\hline
He & 7.60 10$^{-2}$\\
C & 3.03 10$^{-6}$ \\
N & 3.89 10$^{-7}$ \\
O & 1.32 10$^{-5}$ \\
Ne & 2.28 10$^{-6}$ \\
S & 3.72 10$^{-7}$ \\
Ar & 9.13 10$^{-8}$ \\
\hline
\end{tabular}
\end{flushleft}
\end{table}

We do not include dust in the computations. While it is known that,
in general, dust mixed with the ionized gas may absorb some of the
ionizing photons, and contribute to the energy balance of the gas by
photoelectric heating and collisional cooling (e.g. Baldwin et al.
1991, Borkowski \& Harrington 1991, Shields \& Kennicutt 1995), the expected effect in
I\,Zw\,18 is negligible, since the dust-to-gas ratio is believed to
be small at such metallicities (cf.\ Lisenfeld \& Ferrara 1998).

The case of I\,Zw\,18 is thus very interesting for photoionization
modelling, since due to the very low metallicity of this object, the
number of unconstrained relevant parameters is minimal.

\subsection{Fitting the observational constraints}

In judging the value of our photoionization models, we do not
follow the common procedure of producing a table of intensities
relative to \ifmmode {\rm H}\beta \else H$\beta$\fi\ to be compared to the observations. A good
photoionization model is not only one which reproduces the observed
line ratios within the uncertainties. It must also satisfy other
criteria, like being compatible with what is known from the
distribution of the ionized gas, and what is known of the ionizing
stars themselves. On the other hand, many line ratios are not at all
indicative of the quality of a photoionization model. For example,
obviously, two lines arising from the same atomic level like [O~{\sc iii}] $\lambda$5007\
and [O~{\sc iii}] $\lambda$4959 have intensity ratios that depend
only on the respective transition probabilities. In H~{\sc ii}\ regions,
the
ratio of hydrogen Balmer lines (if case B applies) is little
dependent on the physical conditions in the ionized gas, and this is
why it can be used to determine the reddening. The ratios of the
intensities of neutral helium lines do depend somewhat on the
electron density distribution and on selective absorption by dust of
pseudo-resonant photons, (Clegg \& Harrington 1989, Kingdon \& Ferland
1995), and
these are introduced in photoionization models. In the
case of the NW component of I\,Zw\,18, the observed neutral helium lines
are
affected by absorption from stars or interstellar sodium
(Izotov \& Thuan 1998,
 V\'{\i}lchez \& Iglesias-P\'{a}ramo 1998), and cannot be easily used as
constraints for
photoionization models.

Generally speaking, once line ratios indicative of the
electron temperature (like [O~{\sc iii}] $\lambda$4363/5007,  [N~{\sc ii}] $\lambda$5755/6584), of the electron density
(like
[S~{\sc ii}] $\lambda$6731/6717, [Ar~{\sc iv}] $\lambda$4741/4713) and of the global ionization structure (like
[O~{\sc iii}] $\lambda$5007/[O~{\sc ii}] $\lambda$3727\
or [S~{\sc iii}] $\lambda$9532/[S~{\sc ii}] $\lambda$6725) have been fitted, the ratios of all the strong
lines with respect to \ifmmode {\rm H}\beta \else H$\beta$\fi\ are necessarily reproduced by a
photoionization model whose input abundances were obtained from the
observations. The only condition is that the atomic data to derive
the abundances and to compute the models should be the same. Problems
may
arise only if the empirical ionization factors are different from the
ones given by the model, or if there is insufficient information on
the distribution of the electron temperature or density inside the
nebula (in the case of I\,Zw\,18 no direct information is available
on the temperature in the low ionization zone, but we adopted
 a value inspired by the models).
Therefore, intensity ratios such
as [O~{\sc iii}] $\lambda$5007/\ifmmode {\rm H}\beta \else H$\beta$\fi, [Ne~{\sc iii}] $\lambda$3869/\ifmmode {\rm H}\beta \else H$\beta$\fi, [N~{\sc ii}] $\lambda$6584/\ifmmode {\rm H}\beta \else H$\beta$\fi,
[Ar~{\sc iii}] $\lambda$7135/\ifmmode {\rm H}\beta \else H$\beta$\fi\ or C~{\sc iii}] $\lambda$1909/\ifmmode {\rm H}\beta \else H$\beta$\fi\ are not a measure of the quality of the
photoionization model.

To judge whether a photoionization model is acceptable, one must work
with outputs that are significantly affected by the physical
processes on which the photoionization model is based, i.e.
the transfer of the ionizing radiation, the processes determining the
ionization equilibrium of the various atomic species and the thermal
balance of the gas. Table 2 lists the quantities that can be used in
the case of I\,Zw\,18, given the observational information we have on
the
object. The value of the \ifmmode {\rm H}\beta \else H$\beta$\fi\ flux is derived from the H$\alpha$\ flux
 measured in a circle of radius
$\theta$=2.5\arcsec\ (shown in Fig. 1),
assuming C(\ifmmode {\rm H}\beta \else H$\beta$\fi) = 0. The line
ratios He~{\sc ii} $\lambda$4686/\ifmmode {\rm H}\beta \else H$\beta$\fi, [O~{\sc iii}] $\lambda$4363/5007, [S~{\sc ii}] $\lambda$6731/6717, [O~{\sc iii}] $\lambda$5007/[O~{\sc ii}] $\lambda$3727, [S~{\sc iii}] $\lambda$6312/[S~{\sc ii}] $\lambda$6725\ and [O~{\sc i}] $\lambda$6300/\ifmmode {\rm H}\beta \else H$\beta$\fi\
are the values observed by Izotov \& Thuan (1998) in the
rectangular aperture  whose approximate position is shown in Fig. 1.
It is important to define in
advance the tolerance we accept for the difference between our model
predictions and the observations. This must take into account both
the uncertainty in the observational data, the fact that the spectra
were taken through an aperture not encompassing the whole nebula, and
the fact that the nebula does not have a perfect, spherical symmetric
structure. This latter aspect is, of course, difficult to  quantify,
and the numbers given in Column 3 of Table 2 are to be regarded
rather as guidelines. In Column 4, we indicate which is the dominant
factor determining the adopted tolerance : the signal-to-noise, or
the geometry. For example, such ratios as He~{\sc ii}/\ifmmode {\rm H}\beta \else H$\beta$\fi, [O~{\sc iii}]/[O~{\sc ii}], [S~{\sc iii}]/[S~{\sc ii}]\ or
[O~{\sc i}]/\ifmmode {\rm H}\beta \else H$\beta$\fi\ are obviously more dependent on geometrical effects than [O~{\sc iii}] $\lambda$4363/5007.
 Note that, even for that ratio,
the tolerance given in Table 2 is larger than the
uncertainty quoted by Izotov \& Thuan (1998). The reason is that the
many observations of the NW component of I\,Zw\,18 made over the years,
with
different telescopes, detectors, and apertures, yield distinct values
for this ratio, as shown in Fig. 2. In view of this figure, a
tolerance of 10\% with respect with the [O~{\sc iii}] $\lambda$4363/5007\ ratio measured by
Izotov
\& Thuan (1998) seems reasonable. The status of the [S~{\sc ii}] $\lambda$6731/6717\ ratio
is somewhat different. It indicates the average electron density in
the zone emitting [S~{\sc ii}]. This is very close to an input parameter,
since photoionization models are built with a given density
structure. However, because the density deduced from [S~{\sc ii}] $\lambda$6731/6717\ is not
the
total hydrogen density but the electron density
in the region emitting [S~{\sc ii}], and because the
density is not necessarily uniform, it is important to check that the
model returns an [S~{\sc ii}] $\lambda$6731/6717\ value that is compatible with the
observations.
For the total \ifmmode {\rm H}\beta \else H$\beta$\fi\ flux, we accept models giving
 F(\ifmmode {\rm H}\beta \else H$\beta$\fi) larger than the observed value, on account
of the fact that the coverage factor of the ionizing source by the nebula
may be smaller than one.

\setcounter{table}{1}

\begin{table*}[htb]
\caption{Observables that a photoionization model of I\,Zw\,18 should
fit (references in Sect.\ 3.2). }
\begin{flushleft}
\begin{tabular}{llllll}
\hline
Quantity & Value & Tolerance & Major source of uncertainty & Symbol in Figs.\ 4--6 \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
F(\ifmmode {\rm H}\beta \else H$\beta$\fi) [erg~cm$^{-2}$~s$^{-1}$] & 4.0~10$^{-14}$ & + 0.5 dex & geometry
(see text) & circle\\
angular radius $\theta$ [arc sec ] & 2.5 & $\pm$ 0.08 dex & geometry & cross \\
He~{\sc ii} $\lambda$4686/\ifmmode {\rm H}\beta \else H$\beta$\fi  & 0.034 &  $\pm$ 0.2 dex &
geometry (see text) & square \\
[O~{\sc iii}] $\lambda$4363/5007\ & 3.28~10$^{-2}$ & $\pm$ 0.04 dex & S/N & open triangle\\
[S~{\sc ii}] $\lambda$6731/6717\ & 1.3 & $\pm$ 0.04 dex & S/N & diamond \\
[O~{\sc iii}] $\lambda$5007/[O~{\sc ii}] $\lambda$3727\ & 6.82 & $\pm$ 0.1 dex & geometry & filled triangle \\
[O~{\sc i}] $\lambda$6300/\ifmmode {\rm H}\beta \else H$\beta$\fi\ & 0.007  & $\pm$ 0.3 dex & geometry & plus \\
[S~{\sc iii}] $\lambda$6312/[S~{\sc ii}] $\lambda$6725\ & 0.173 & $\pm$ 0.2 dex & geometry & asterisk \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{flushleft}
\end{table*}

Thus, in the following, we compute photoionization models with the
input parameters as defined in Section 3.1, and see how they
compare with the constraints specified in Table 2. We will not
examine the effects of varying the elemental abundances, since, as
mentioned above, they are negligible in our problem. Uncertainties in
the measured stellar flux have only a small impact on our models, and
are therefore not discussed here.
Similarly, we discard the effects of an error in the distance
$d$ to I~Zw~18. These are not crucial on the ionization structure of a
model designed to fit the observed flux at 3327\AA, since
the mean ionization parameter varies
roughly like $d^{2/3}$. What we mainly want to see is
whether, with our present knowledge, we can satisfactorily explain
the observed properties of I\,Zw\,18. As will be seen, the gas
density
distribution plays an important role.

\section{Climbing the ladder of sophistication}

\subsection{The ionizing radiation field }

Before turning to proper photoionization modelling, it is worthwhile
examining the gross properties of the ionizing radiation field of the
synthetic stellar population model we are using, and compare it to
single star model atmospheres. Two quantities are particularly
relevant. One is  $Q({\rm{He^{+}}})$/$Q({\rm{H^{0}}})$, the ratio of the number of photons above
54.4 and 13.6~eV emitted by the ionizing source.
This ratio allows one to estimate the He~{\sc ii} $\lambda$4686/\ifmmode {\rm H}\beta \else H$\beta$\fi\ that would be observed in
a
surrounding nebula, by using simple conservation arguments leading to
the
formula: He~{\sc ii} $\lambda$4686/\ifmmode {\rm H}\beta \else H$\beta$\fi\ = 2.14 $Q({\rm{He^{+}}})$/$Q({\rm{H^{0}}})$\ (taking the case B
recombination coefficients given in Osterbrock 1989). As is known,
this expression is
valid if the nebula is ionization bounded and the observations
pertain to the whole volume. It is less commonly realized that it
also assumes the
average temperature in the He$^{++}$\ region to be the same as in the
entire H~{\sc ii}\ region. This is may be far from true, as will be shown
below, so a
correction
should account for that. Another assumption is that all the photons above
54.4~eV are absorbed by He$^{+}$\
ions. This is not what happens in objects with a low ionization
parameter. There, the residual neutral hydrogen particles are
sufficiently numerous to compete with He$^{+}$. In such a case, the
expression above gives an upper limit to the nebular He~{\sc ii} $\lambda$4686/\ifmmode {\rm H}\beta \else H$\beta$\fi\
ratio. In spite of these difficulties, $Q({\rm{He^{+}}})$/$Q({\rm{H^{0}}})$\ remains a useful
quantity to estimate the intensity of the nebular He~{\sc ii} $\lambda$4686\ line.
Fig. 3a shows the variation of $Q({\rm{He^{+}}})$/$Q({\rm{H^{0}}})$\ as a function of
starburst age for the synthetic model population we are considering.
As already stated in De Mello et al. (1998), the strength of the
He~{\sc ii}\ nebular line in I\,Zw\,18 indicates a starburst between 2.9
and 3.2~Myr.

Another important ratio is $Q({\rm{He^{0}}})$/$Q({\rm{H^{0}}})$, sometimes referred to as the
hardness parameter of the ionizing radiation field. It provides a
qualitative
measure of the heating power of the stars. We have represented this
quantity in Fig. 3c. We see that, as the
starburst ages, its heating power gradually declines, and shows only
a very mild bump at ages around 3~Myr, where the Wolf-Rayet stars are
present.
As we will show below this modest increase of the heating power
is not sufficient to explain the high electron temperature observed
in I\,Zw\,18.

For comparison, we show in Figs.\ 3b and d respectively, the values
of $Q({\rm{He^{+}}})$/$Q({\rm{H^{0}}})$\ and $Q({\rm{He^{0}}})$/$Q({\rm{H^{0}}})$\ as a function of the stellar effective
temperature for the LTE model atmospheres of Kurucz (1991),
the {\em CoStar} model atmospheres corresponding to main sequence
stars (Schaerer \& de Koter 1997) and for the model atmospheres for
Wolf-Rayet stars of Schmutz et al.\ (1992).
The {\em CoStar} models show an increased He$^{+}$\ ionizing flux
compared to Kurucz models which have a negligible flux even for
very low metallicities ([Fe/H]$=-1.5$). The reasons for this
difference have been discussed in Schaerer \& de Koter (1997).
In addition to the $T_{\rm eff}$\ dependence,  $Q({\rm{He^{+}}})$\ from WR models depend
strongly on the wind density. He$^{+}$\ ionizing photons are only
predicted by models with sufficiently thin winds (cf.\ Schmutz et al.\
1992).
Figure 3d shows the increase of the hardness of the radiation field,
at a given $T_{\rm eff}$,
between the spherically expanding non-LTE models for O and WR stars
and the traditional Kurucz models (see discussion in Schaerer \& de
Koter 1997). This provides a greater heating power which, as will
be shown later, is however still insufficient to explain the
observations.

\subsection{I\,Zw\,18 as a uniform sphere}

We start with the easiest and most commonly used geometry in
photoionization modelling: a  sphere uniformly filled with
gas at constant density, occupying a fraction $\epsilon$ of the whole
nebular volume. The free parameters of the models are then only the
age of the starburst, the gas density and the filling factor. Each
model is computed starting from the center, and the computations are
stopped either when the [O~{\sc iii}]/[O~{\sc ii}]\ ratio has reached the observed
value given in Table 2, or when the gas becomes neutral. In other
words, we examine also models that are not ionization bounded, in
contrast to previous studies.

Figure 4 shows our diagnostic diagram for a series of models having a
density $n$= 100~cm$^{-3}$\ and a filling factor $\epsilon$=0.01. The left
panel shows the
computed values of log F(\ifmmode {\rm H}\beta \else H$\beta$\fi) + 15 (open circles), log He~{\sc ii} $\lambda$4686/\ifmmode {\rm H}\beta \else H$\beta$\fi\ + 2
(squares), angular radius $\theta$ (crosses),
[O~{\sc iii}] $\lambda$4363/5007\ $\times$ 100 (open triangles), [S~{\sc ii}] $\lambda$6731/6717\ (diamonds),
log ([O~{\sc iii}]/[O~{\sc ii}])
(black triangles), log ([S~{\sc iii}]/[S~{\sc ii}]) (asterisks) and log [O~{\sc i}]/\ifmmode {\rm H}\beta \else H$\beta$\fi\ +3
(plus)  as a
function of the starburst age. The black circles correspond to the
value of log F(\ifmmode {\rm H}\beta \else H$\beta$\fi) + 15  that the nebula would have if it were
ionization bounded. Thus, by comparing the positions of an open
circle and a black circle, at a given abscissa, one can immediately
see whether
the model is density bounded and how much diffuse H$\alpha$\ or \ifmmode {\rm H}\beta \else H$\beta$\fi\ emission
 is expected to be emitted outside the main body of the nebula.
 In the right panel, the observed
values are represented on the same vertical scale and with the same
symbols as the model predictions. The tolerances listed in Table
2 are represented as vertical error bars (the horizontal displacement
of the symbols has no particular meaning). We readily see that the
age of the starburst is
important only for the He~{\sc ii} $\lambda$4686\ line, the other quantities varying very
little for ages 2.7--3.4 Myr. Therefore, for the following
runs of models, we adopt an age of 3.1~Myr. In principle, one
can always adjust the age for the model to reproduce the
observed He~{\sc ii} $\lambda$4686/\ifmmode {\rm H}\beta \else H$\beta$\fi\ ratio exactly.

Figure 5 shows the same sort of diagnostic diagram as Fig. 4 for a
series of models with $n$ = 100~cm$^{-3}$\ and varying filling factor.
For a
filling factor around 0.1 or larger, with the adopted electron density,
the model is ionization bounded, and its [O~{\sc iii}]/[O~{\sc ii}]\ is larger than
observed. For filling factors smaller than that, the gas distribution
is more extended, so that the general ionization level drops. The
observed [O~{\sc iii}]/[O~{\sc ii}]\ can then only be reproduced for a density
bounded
model. In such a case, the \ifmmode {\rm H}\beta \else H$\beta$\fi\ radiation actually produced by the
nebula is smaller
than if all the ionizing photons were absorbed in the nebula.
A filling factor of 0.002 -- 0.05 gives values of [O~{\sc iii}]/[O~{\sc ii}], F(\ifmmode {\rm H}\beta \else H$\beta$\fi)
and $\theta$  in agreement with the observations.
 But such models give [O~{\sc iii}] $\lambda$4363/5007\ too small
compared with the observations, and [O~{\sc i}]/\ifmmode {\rm H}\beta \else H$\beta$\fi\ below the observed
value by nearly two orders of magnitude. It is interesting, though,
to understand the qualitative behavior of these line ratios as
$\epsilon$ decreases. [O~{\sc i}]/\ifmmode {\rm H}\beta \else H$\beta$\fi\ decreases because the model becomes
more
and more density bounded in order to reproduce the observed
[O~{\sc iii}]/[O~{\sc ii}]\, and levels off at $\epsilon$ = 0.1, because the
ionization
parameter of the model is then so small that the [O~{\sc i}]\ is gradually
emitted by residual neutral oxygen in the main body of the
nebula and not in the outskirts. [O~{\sc iii}] $\lambda$4363/5007\ decreases as $\epsilon$
decreases, because of the increasing proportion of L$\alpha$ cooling
as the ionization parameter drops.

One can build other series of models with different values of $n$
that are still compatible with the observed [S~{\sc ii}] $\lambda$6731/6717. Qualitatively,
their behavior is
the same and no acceptable solution is found.

Interestingly, Fig. 5 shows that models  with $\epsilon$  $\ge$ 0.1
have [O~{\sc iii}] $\lambda$4363/5007\ marginally compatible with the observations ( [O~{\sc iii}] $\lambda$4363/5007\ = 3.03
10$^{-2}$ for $\epsilon$ = 0.1), but such models have too large
[O~{\sc iii}]/[O~{\sc ii}] ($>$ 10 compared of the observed value 6.8) and too small angular
radius ($<$ 1.6\arcsec\ instead of the observed value 2.5\arcsec). Note,
by the way, that such models, being optically thick, return a rather
large [O~{\sc i}]/\ifmmode {\rm H}\beta \else H$\beta$\fi, actually close to the observed value, and a
[S~{\sc iii}]/[S~{\sc ii}]\ compatible with the observations. However, we do not
use [S~{\sc iii}]/[S~{\sc ii}]\ as a primary criterion to judge the validity of a model,
since experience with photoionization modelling of planetary nebulae shows
that it is difficult to reproduce at the same time the sulfur and the
oxygen ionization structure of a given object, and, in principle, one
expects the atomic data for oxygen to be more reliable than those for
sulfur. The strongest argument against models with $\epsilon > 0.1$ is
their angular size, which is definitely too small compared with the
observations. This remains true even when considering a reasonable error
on the distance since, with the condition that we impose on the flux at
3327~\AA\ to be preserved,  the angular radius of a model goes roughly
like $d^{1/3}$. This illustrates the importance of taking into account
other parameters in addition to line ratios to accept or reject a
photoionization model.

\subsection{I~ Zw~18 as a spherical shell }

The series of models presented above had mainly a pedagogical
value, but they are obviously
incompatible with the observed morphology in H$\alpha$. The next step is
to consider a model consisting of a hollow, spherical shell of
constant density, similar to the one constructed by  Garc\'{\i}a-Vargas et al. (1997)
for NGC\,7714 for example. In such a case, there is an additional free parameter,
$R_{in}$, the radius of the inner boundary of the shell. It is fixed,
more or less, by the appearance of the H$\alpha$\ image. Figure 6 shows a
diagnostic diagram for a series of models with $R_{in}$ = 2.25
10$^{20}$~cm (corresponding to an angular radius of 1.5\arcsec), $n$ =
100~cm$^{-3}$\, and varying $\epsilon$. The qualitative behavior is
similar to
that seen for the uniform sphere models presented in Fig. 1, but the
[O~{\sc iii}] $\lambda$4363/5007\ ratio is now even lower (it never exceeds 2.5~10$^{-2}$ in
this series). This is because of the enhanced role
of
L$\alpha$ cooling, which is strong in all the parts of the nebula,
while for the full sphere model, in the zone close to the star, the
ionization parameter is very high and consequently the population of
neutral hydrogen very small.

Apart from the [O~{\sc iii}] $\lambda$4363/5007\ problem, models with $\epsilon$ = 0.002 --
0.05 are satisfactory as concerns the main diagnostics ([O~{\sc iii}]/[O~{\sc ii}],
F(\ifmmode {\rm H}\beta \else H$\beta$\fi) and $\theta$). The models become progressively
density bounded
towards smaller values of $\epsilon$ $<$ 0.02, meaning that there is
a leakage of ionizing photons. From Fig. 6, one sees that
these photons are enough to produce an H$\alpha$\ emission in an extended diffuse
envelope that is at least comparable in strength to the total
emission from the dense shell. This is in agreement with Dufour \&
Hester's (1990) ground-based observation of extended H$\alpha$\ emission
surrounding the main body of star formation.

\subsection{Other geometries}
Closer inspection of the {\em HST} H$\alpha$\ image shows that the gas emission
is incompatible with a spherical bubble.
This is illustrated in Fig.\ 7, where the observed cumulative
surface brightness profile within radius $r$ (dashed line) is compared
to the expected profiles for constant density spherical shells of
various inner radii (solid lines).
The theoretical profiles are
obtained assuming that the temperature is uniform
in the gas, but taking into account a reasonable temperature gradient
in the model hardly changes the picture. Clearly, the observed
profile is not compatible with a spherical density distribution. The
column density of emitting matter in the central zone of the image is
too small. One must either have an incomplete shell
with some matter stripped off from the poles, or even a more extreme
morphology like an diffuse axisymmetric
body with a dense ringlike equator seen face on. Such geometries are
actually common among planetary nebulae (Corradi \& Schwartz 1995)
and nebulae surrounding luminous blue variables (Nota et al. 1995),
being probably the result of the interaction of an aspherical
stellar wind from the central stars (Mellema 1995, Frank et al. 1998)
and are also suggested to exist in superbubbles and supergiant shells
and to give rise to the blow-out phenomenon (Mac Low et al. 1989,
Wang \& Helfand 1991, Tenorio-Tagle et al. 1997, Oey \& Smedley 1998,
 Martin 1998).

Does the consideration of such a geometry help in solving the [O~{\sc iii}] $\lambda$4363/5007\
problem? In the case of a spherical shell with some matter lacking at
the poles, our
computations overestimate the role of the diffuse ionizing radiation,
which is supposed to come from a complete shell. Since the
heating power of the diffuse ionizing radiation is smaller than that
of the stellar radiation, one may be underestimating the electron
temperature. As a test, we have run a model switching off completely
the diffuse radiation, so as to maximize the electron temperature.
The effect was to increase the O$^{++}$\ temperature by only 200~K. In
the case of a ring with some diffuse matter seen in projection inside
the ring, the gas lying close to the stars would be
at a higher electron temperature than the matter of the ring, and one
expects that an adequate combination of the parameters describing the
gas distribution interior to and inside of the ring
might reproduce the [O~{\sc iii}] $\lambda$4363/5007\ measured in the
aperture shown in Fig. 1. However, we notice that the region of high
[O~{\sc iii}] $\lambda$4363/5007\ is much larger than that. It extends over almost 20\arcsec\ (
 V\'{\i}lchez \& Iglesias-P\'{a}ramo 1998) and [O~{\sc iii}] $\lambda$4363/5007\ is
neither correlated with [O~{\sc iii}]/[O~{\sc ii}]\ nor with the H$\alpha$\ surface
brightness. Therefore, one cannot explain the high [O~{\sc iii}] $\lambda$4363\ by the
emitting matter being close to the ionizing stars.
In passing we note that the observations of  V\'{\i}lchez \&
Iglesias-P\'{a}ramo (1998) show the nebular He~{\sc ii} $\lambda$4686\ emission to be
extended as well (although not as much as [O~{\sc iii}] $\lambda$4363). If this emission is
due to photoionization by the central star cluster, as modelled in
this paper, this means that the H$\alpha$\ ring is porous, since He~{\sc ii} $\lambda$4686\
emission necessarily comes from a region separated from the stars by
only a small amount of intervening matter.

In summary, we must conclude that, if we take into account all the
available information on the structure of I\,Zw\,18, we are not able
to
explain the high  observed [O~{\sc iii}] $\lambda$4363/5007\ ratio with our photoionization
models. In our best models, [O~{\sc iii}] $\lambda$4363/5007\ is below the nominal value of
Izotov \& Thuan (1998) by 25 - 35\%.

\subsection{Back to the model assumptions}

Can our lack of success be attributed to an improper description of
the stellar radiation field? After all, we know little about the
validity of stellar model atmospheres in the Lyman continuum
(see discussion in Schaerer 1998).
Direct measurements of the EUV flux of early B stars revealed an
important EUV excess (up to $\sim$ 1.5 dex) with respect to plane
parallel model atmospheres (Cassinelli et al.\ 1995, 1996), whose
origin has been discussed by Najarro et al.\ (1996), Schaerer \& de
Koter (1997) and Aufdenberg et al.\ (1998).
For O stars a similar excess in the {\em total} Lyman continuum
output is, however, excluded from considerations of their bolometric
luminosity and measurements of H~{\sc ii}\ region luminosities
(Oey \& Kennicutt 1997, Schaerer 1998).
The hardness of the radiation field, which is crucial for the heating
of the nebula, is more difficult to test.
Some constraints can be obtained by comparing the line emission of
nebulae surrounding hot stars with the results of photoionization models
(Esteban et al.\ 1993,  Pe\~{n}a et al.\ 1998, Crowther et al.\ 1999),
but this is a difficult task, considering that
the nebular emission depends also on its geometry.
Although the hardness predicted by  the {\em CoStar} O stars models
permits to build grids of photoionization models that seem to explain the
 observations of Galactic and LMC H~{\sc ii}\ regions
(Stasi\'nska \& Schaerer 1997), the constraints are not sufficient
to prove or disprove the models.

To check the effect of a harder radiation field, we have run a series of models
where the radiation field above 24.6~eV was arbitrarily multiplied by
a factor 3 (raising the value of $Q({\rm{He^{0}}})$/$Q({\rm{H^{0}}})$\ from  0.33 to 0.59,
corresponding to $T_{\rm eff}$\ from $\ga$ 40000 K to $\sim$ 100\,000~K or
a blackbody of the same temperature (cf.\ Fig.\ 3d).
This drastic hardening of the radiation field resulted in an increase
of [O~{\sc iii}] $\lambda$4363/5007\ of only 10\%. It is only by assuming a blackbody radiation
of 300\,000~K (which has $Q({\rm{He^{0}}})$/$Q({\rm{H^{0}}})$\ = 0.9) that one approaches the
observed [O~{\sc iii}] $\lambda$4363/5007. A model similar to those presented in Fig. 6 but
with such a radiation field gives [O~{\sc iii}] $\lambda$4363/5007\ =3.15 10$^{-2}$. But is has
a He~{\sc ii} $\lambda$4686/\ifmmode {\rm H}\beta \else H$\beta$\fi\ ratio of 0.53, which is completely ruled out by the
observations. Of course, a blackbody is probably not the best
representation for the spectral energy distribution of the radiation
emitted by a very hot body, but in order to explain the emission line
spectrum of I~Zw~18 by stars, one has to assume an ionizing radiation
strongly enhanced at energies between 20 -- 54 eV, but not above 54.4
eV, compared to the model of De Mello et al.\ (1998).
If this is realistic cannot be said at the present time.

We have also checked the effect of heating by additional X-rays that
would be emitted due to the interaction of the stellar winds with
ambient matter (see Martin 1996 for such X-ray observations),
 by simply adding a bremsstrahlung spectrum at T = 10$^{6}$
or  T = 10$^{7}$ K to the radiation from the starburst model. As
expected, the effect on [O~{\sc iii}] $\lambda$4363/5007\ was negligible, since the X-rays are
mostly absorbed in the low ionization regions (they do raise the
temperature in the O$^{+}$\ zone to T$_{e}$ $\simeq$ 16\,000~K).

As already commented by previous authors, changing the elemental
abundances does not improve the situation. Actually, even by putting
all the elements heavier than helium to nearly zero abundance,
[O~{\sc iii}] $\lambda$4363/5007\ is
raised by only 7\%. Varying the helium abundance in reasonable limits
does not change the problem.

The neglect of dust is not expected to be responsible for the
too low [O~{\sc iii}] $\lambda$4363/5007\ we find in our models for I~Zw~18.
Gas heating by photoemission from
grains can contribute by as much as 20\% to the electron thermal
balance when the dust-to-gas ratio is similar to that found in the
Orion nebula. But, as discussed by Baldwin et al. (1991), it is
effective close to the ionizing source where dust provides most of
the opacity. Besides, the proportion of dust in I\,Zw\,18 is expected
to be small, given the low metallicity. Extrapolating the relation found by
Lisenfeld \& Ferrara (1998) between the dust-to-gas ratio and the
oxygen abundances in dwarf galaxies to the metallicity of I\,Zw\,18
yields a dust-to-gass mass ratio 2 to 3 orders of magnitudes smaller
than in the solar vicinity.

It remains to examine the ingredients of the photoionization code. We
first stress that the [O~{\sc iii}] $\lambda$4363/5007\ problem in I\,Zw\,18 has been
encountered
by various authors using different codes, even if the observational
material and the points of emphasis in the discussion were not the
same in all the studies.
As a further test, we compared the same model for I~Zw~18 run by CLOUDY 90
and by PHOTO and the difference in the predicted value of [O~{\sc iii}] $\lambda$4363/5007\ was
only 5\%.
One must therefore incriminate
something that is wrongly treated in the same way in many codes.
One possibility that comes to mind is the diffuse ionizing radiation,
which is treated by  some kind of outward only approximation in all
codes used to model I~Zw~18.
However,  we do not expect
 that an accurate treatment of the diffuse ionizing radiation
would solve the problem.
Indeed, comparison of codes that treat more
accurately the ionizing radiation with those using an outward only
approximation shows only a negligible difference in the electron
temperature (Ferland et al.\ 1996).
Besides, as we have shown, even quenching the diffuse ionizing
radiation does not solve the problem.

Finally, one can also question the
atomic data. The most relevant ones here are those governing the
emissivities of the observed [O~{\sc iii}]\ lines and the H~{\sc i}\  collision
strengths. The magnitude of the discrepancy we wish to solve would
require modifications of the collision strengths or transition
probabilities for [O~{\sc iii}]\ of about 25\%.
This is much larger than the expected uncertainties and the differences
between
the results of different computations for this ion (see discussion in
Lennon and Burke 1994 and  Galavis et al. 1997). Concerning L$\alpha$
excitation, dividing the collision strength by a factor 2 (which is
far above any  conceivable uncertainty, see e.g. Aggarwal et al.
1991) modifies [O~{\sc iii}] $\lambda$4363/5007\ only by 2\% because L$\alpha$ acts like a
thermostat.

We are therefore left with the conclusion that the [O~{\sc iii}] $\lambda$4363/5007\ ratio
cannot be explained in the framework of photoionization models alone.

\subsection{Condensations and filaments}

Another failure of our photoionization models is that
they predict too low [O~{\sc i}]/\ifmmode {\rm H}\beta \else H$\beta$\fi\ compared to observations. This is a
general feature of photoionization models and it is often taken as
one of the reasons to invoke the presence of shocks. However, it is
well known that the presence of small condensations of filaments
enhances the [O~{\sc i}]/\ifmmode {\rm H}\beta \else H$\beta$\fi\ ratio, by reducing the local ionization
parameter. Another possibility is to have an intervening filament
located at a large distance from the ionizing source, whose projected
surface on the aperture of the observations would be small.

In order to see under what conditions such models can quantitatively
account for
the observed [O~{\sc i}]/\ifmmode {\rm H}\beta \else H$\beta$\fi\ ratio, we have mimicked such a situation by
computing a series of ionization bounded photoionization
models for filaments of different densities located at various distances
from the exciting stars.
For simplicity, we assumed that there is no intervening matter
between the filaments and the ionizing source.
The models were actually computed
for complete shells. The radiation coming from a filament
can be simply obtained
by multiplying the flux computed in the model by the
covering factor $f$ by
which the filament is covering the source.

\begin{table*}
\caption{Line intensities relative to \ifmmode {\rm H}\beta \else H$\beta$\fi\ for models of filaments.
}
\begin{flushleft}
\begin{tabular}{l|l|lll|ll}
\hline
\noalign{\smallskip}
 &     Main body & \multicolumn{3}{l|}{Filaments of various $n$} &
 \multicolumn{2}{l}{Filaments at various distances} \\
\hline
\noalign{\smallskip}
$n$ [cm$^{-3}$]  & 10$^2$   $^a$   &   10$^4$    &  10$^5$  &  10$^6$ & 10$^2$ &  10$^2$   \\
$\theta_{\rm in}$ [\arcsec] &  &       1.5   &   1.5    &   1.5   &   20.  &  100.    \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
F(\ifmmode {\rm H}\beta \else H$\beta$\fi)$^b$
 & 2.02e-13   &   3.33e-13  & 3.32e-13 & 3.35e-13  & 3.41e-13 &
3.47e-13 \\
[O~{\sc i}] $\lambda$6300\   & 1.04e-4    &    5.28e-2  & 1.57e-1  & 3.83e-1   & 6.90e-2  &
2.81e-1  \\
[O~{\sc ii}] $\lambda$3727\  & 2.15e-1    &    2.41e-1  & 5.39e-2  & 1.19e-2   & 4.71e-1  &
2.85e-1  \\
[O~{\sc iii}] $\lambda$4363\   & 3.46e-2    &    3.63e-5  & 1.12e-6  & 1.04e-7   & 1.20e-5  &
1.74e-7  \\
[O~{\sc iii}] $\lambda$5007\    & 1.42       &    3.76e-3  & 1.41e-4  & 1.25e-5   & 1.41e-3  &
3.08e-5  \\
[S~{\sc ii}]\ $\lambda$6717 & 5.58e-4    &    5.20e-2  & 5.23e-2  & 5.27e-2   & 1.87e-1  &
5.70e-1  \\
[S~{\sc ii}]\ $\lambda$6731.   & 4.19e-3    &    8.04e-2  & 9.16e-2  & 1.03e-1   & 1.35e-1  &
3.96e-1  \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\small
$^a$ The model for the main body is density bounded (see text)}
\noalign{\small
$^b$ The total \ifmmode {\rm H}\beta \else H$\beta$\fi\ flux is given in  erg s$^{-1}$ for a covering factor $f$=1. }
\end{tabular}
\end{flushleft}
\end{table*}

Table 3 presents the ratios with respect to \ifmmode {\rm H}\beta \else H$\beta$\fi\ of the [O~{\sc i}] $\lambda$6300, [O~{\sc ii}] $\lambda$3727,
[O~{\sc iii}] $\lambda$5007, [S~{\sc ii}] $\lambda$6717 and [S~{\sc ii}] $\lambda$6731 for
these models. The first column of the table corresponds to a
photoionization model for the main body of the nebula with
$R_{in}$ = 2.25 10$^{20}$ cm (corresponding to
$\theta_{\rm in}=$ 1.5\arcsec), $n$=100~cm$^{-3}$, and $\epsilon$=0.01,
and  density bounded so
as to obtain the observed [O~{\sc iii}]/[O~{\sc ii}]\ (this is one of the models of
Fig. 6). It can be readily seen that, for an intervening filament of
density 10$^{2}$~cm$^{-3}$\ located at a distance of 500~pc from the star
cluster, or for condensations of density 10$^{6}$~cm$^{-3}$, one can
reproduce the observed [O~{\sc i}]/\ifmmode {\rm H}\beta \else H$\beta$\fi\ line ratio without
strongly affecting the remaining lines, not even the density
sensitive [S~{\sc ii}] $\lambda$6731/6717\ ratio, if one assumes a covering factor
$f$ of about 0.1.

This explanation may appear somewhat speculative.
 However, one must
be aware that the morphology of the ionized gas in I\,Zw\,18 shows
filaments in the main shell as well as further out and this has been
amply discussed in the literature (Hunter \& Thronson 1995, Dufour
et al. 1995, Martin 1996). Our point is that, even in the
framework of pure photoionization models, if one accounts for a
density structure suggested directly by the observations, the
strength of the [O~{\sc i}]\ line can be easily understood.

We note that the condensations or filaments that produce [O~{\sc i}]\  are
optically thick, and therefore their
neutral counterpart should be seen in H~{\sc i}. Unfortunately, the
available H~{\sc i}\ maps of I\,Zw\,18 (van Zee et al. 1998) do not have
sufficient spatial resolution to reveal such filaments. But these
authors show that the peak emission in H~{\sc i}\ and H$\alpha$\ coincide (in
their paper,
the peak emission in H$\alpha$\ actually refers to the whole NW component), and
find that the entire optical system, including the diffuse emission,
is embedded in a diffuse, irregular and clumpy neutral cloud. In such
a situation, it is very likely that some clumps or filaments,
situated in front of the main nebula, and having a small covering
factor, produce the observed [O~{\sc i}]\ line.

By using photoionization
models to explain the observed [O~{\sc i}]/\ifmmode {\rm H}\beta \else H$\beta$\fi\ ratio, one can deduce
the presence of density fluctuations, even if those are not seen
directly.  Such density fluctuations could then be, more directly,
inferred from density sensitive ratios of [Fe~{\sc ii}]  lines, such
as seen
and analyzed in the Orion nebula by Bautista \& Pradhan (1995).

\subsection{A few properties of I\,Zw\,18 deduced from models}

Although we have not built a completely satisfactory photoionization
model reproducing all the relevant observed properties of I\,Zw\,18,
we
are not too far from it. We describe below some properties of the
best models that may be of interest for further empirical studies of
this object.

Tables 4 and 5 present the mean ionic fractions and mean ionic
temperatures
(as defined in Stasi\'{n}ska 1990) for the different elements
considered, in the case of the best fit models with a uniform sphere
and a spherical shell respectively, with $n$=100~cm$^{-3}$, and $\epsilon$=0.01.

It can be seen that, while both geometries yield similar ionic
fractions for the main ionization stages, the relative populations of
the highly charged
trace ions are very different. In the case of the uniform sphere,
the proportion of O$^{+++}$, for example, is twice as large as in
the shell model. Also, the characteristic temperature of ions with
high ionization potential are much higher in the case of the filled
sphere, for reasons commented on earlier. As a result, the
OIV]1394/\ifmmode {\rm H}\beta \else H$\beta$\fi\ ratio is 9.7~10$^{-3}$ in the first case and 8.6~10$^{-4}$
in the second. This line is too weak to be measured in I\,Zw\,18, of
course, but it is useful to keep this example in mind for the study
or more metal rich objects.

It is interesting to point out that, in the case of the uniform
sphere, the total flux in the He~{\sc ii} $\lambda$4686\ line is smaller than in the case
of the shell model (3.1~10$^{-15}$  vs. 3.3~10$^{-15}$ erg~cm$^{-2}$~s$^{-1}$)
 despite the
fact that the ionic fractions of He$^{++}$\ are similar. This is
because the He$^{++}$\  region is at a much higher temperature (25\,000~K
versus 18\,000~K).

Tables 4 and 5 can be used for estimating the ionization correction
factors for I\,Zw\,18.
Caution should, however, be applied regarding especially
the ionization structure predicted for elements of the third row of
Mendeleev's table.
Experience in photoionization modelling of
planetary nebulae, where the observational constraints are larger,
shows that the ionization structure of these elements is rarely
satisfactorily reproduced with simple models (Howard et al. 1997, Pe\~na
et al., 1998).

The total ionized mass in the NW component is relatively well determined,
since we know
the total number of ionizing photons, the radius of the emitting
region and have
an estimate of the
mean ionization parameter through the observed [O~{\sc iii}]/[O~{\sc ii}]. Indeed,
at a given $Q({\rm{H^{0}}})$,
for a constant density sphere with a filling factor $\epsilon$,
$n^{2} \epsilon$ is proportional to the cube of the radius,
 while $n \epsilon^{2}$ is proportional to the cube of the ionization
 parameter. Of course, we have just made the point that the NW
 component
 of I\,Zw\,18 is not a sphere. Nevertheless, we have an order of
 magnitude estimate, which is of 3.~10$^{5}$~M$_{\odot}$\ at $d$ = 10~Mpc (this
 estimate varies like $d^{3}$).

Finally, it is important to stress, as already mentioned above, that
H$\alpha$\ is partly excited by collisions. In all our models for the main
body of the nebula,  H$\alpha$/\ifmmode {\rm H}\beta \else H$\beta$\fi\ lies between 2.95 and 3,
while the case B recombination value is 2.7. This means that the
reddening of I\,Zw\,18 is smaller than the value obtained using the
case B
recombination value at the temperature derived from [O~{\sc iii}] $\lambda$4363/5007. If we
take the observations of Izotov \& Thuan (1998), who also correct for
underlying stellar absorption, we obtain C(\ifmmode {\rm H}\beta \else H$\beta$\fi)=0.

\section{Summary and conclusions}

We have built photoionization models for the NW component of I~Zw~18
using the radiation field from a starburst population
synthesis at appropriate
metallicity (De Mello et al. 1998) that is consistent with the
Wolf-Rayet signatures seen in the spectra of I~Zw~18. The aim was to
see whether, with a nebular density structure compatible with recent
{\em HST} images, it was possible to explain the high
 [O~{\sc iii}] $\lambda$4363/5007\ ratio seen in this object, commonly interpreted as
 indicative of electron temperature $\simeq$ 20\,000K.

For our photoionization analysis we have focused on properties which
are relevant and crucial model predictions.
For the observational constraints we have not only used line ratios, but
also other properties such as the integrated stellar flux at 3327~\AA\
and the observed angular radius of the emitting region as seen by the
{\em HST}. Care has also been taken to include tolerances on model
properties which may be affected by deviations from various simple
geometries.

We have found that [O~{\sc iii}] $\lambda$4363/5007\ cannot be explained by pure photoionization
models, which yield too low an electron temperature. We have
considered the effects due to departure from spherical symmetry
indicated by the {\em HST} images.
Indeed these show that
the NW component of I~Zw~18 is neither a uniform
sphere nor a spherical shell, but rather a bipolar structure with a
dense equatorial ring seen pole on.

We have discussed the consequences that an inaccurate description of the
stellar ionizing radiation
field might have on [O~{\sc iii}] $\lambda$4363/5007\, as well as additional photoionization
by X-rays.
Finally, we have considered possible errors in the atomic data.

All these trials were far from solving the electron temperature
problem, raising the [O~{\sc iii}] $\lambda$4363/5007\ ratio by only a few percent while the
discrepancy with the observations is on the 30\% level.
Such a discrepancy means that we are missing a heating source whose
power may be of the same magnitude as that  of the stellar ionizing
photons. It is also possible that
the unknown energy source is not so powerful, but acts in such a way
that small quantities of gas are emitting at very high temperatures,
thus boosting the [O~{\sc iii}] $\lambda$4363\ line. Shocks are of course one of the
options (Peimbert et al. 1991, Martin 1996, 1997), as well as conductive heating
at the interface of an X-ray plasma with optically visible gas (Maciejewski et al. 1996).
Such ideas need to be examined quantitatively, and applied to the
case of I~Zw~18, which we shall attempt in a future work.

What are the consequences of our failure in understanding the energy
budget on the abundance determinations in I~Zw~18? It depends on how
 the electron temperature is distributed in the O$^{++}$ zone.
 As emphasized by Peimbert (1967, 1996) over the years (see also
 Mathis et al. 1998), the existence of even small zones at very high
  temperatures will boost  the lines with high excitation threshold
  like [O~{\sc iii}] $\lambda$4363, so that the temperature derived from [O~{\sc iii}] $\lambda$4363/5007\ will
  overestimate the average temperature of the regions emitting [O~{\sc iii}] $\lambda$5007.
  Consequently, the true O/H ratio will be larger than the one derived
  by the standard methods. The C/O ratio, on the other hand,  will be
  smaller than derived empirically, because the ultraviolet C~{\sc iii}] $\lambda$1909\
  line will be extremely enhanced in the high temperature regions.
  Such a possibility was invoked by Garnett et al. (1997) to explain
  the high C/O found in I~Zw~18 compared to other metal-poor
  irregular galaxies. Presently, however, too little is known both
  theoretically and observationally to estimate quantitatively this
effect,
  and it is not excluded that the abundances derived so far may be
  correct within 30\%.
  Obviously, high spatial resolution mapping of the [O~{\sc iii}] $\lambda$4363/5007\ ratio in
  I~Zw~18 would be valuable to track the origin of the high [O~{\sc iii}] $\lambda$4363\
  seen in spectra integrated over a surface of about 10~\arcsec $^{2}$.

Beside demonstrating the existence of a heating problem in I~Zw~18,
our photoionization model analysis led to several other results.

The intensity of the nebular He~{\sc ii} $\lambda$4686\ line can be reproduced with a
detailed photoionization
model having as an input the stellar radiation field that is consistent
the observed Wolf-Rayet features in I~Zw~18.
This confirms the results of De Mello et al.  (1998) based on simple
Case B recombination theory.

By fitting the observed [O~{\sc iii}]/[O~{\sc ii}]\ ratio and the angular size of the NW
component with a model where the stellar radiation flux was adjusted
to the observed value, we were able to show that the H~{\sc ii}\ region is
not ionization bounded and about half of the
ionizing photons are leaking out of it, sufficient to explain the extended
diffuse H$\alpha$\ emission observed in I~Zw~18.

While the [O~{\sc i}]\ emission is not reproduced in simple models, it can
easily be accounted for by condensations or by intervening filaments
on the line of sight. There is no need to invoke shocks to excite the
[O~{\sc i}]\ line, although shocks are probably involved in the creation of
the filaments, as suggested by Dopita (1997) in the context of
planetary nebulae.

The intrinsic H$\alpha$/\ifmmode {\rm H}\beta \else H$\beta$\fi\ ratio is significantly affected by collisional
 excitation: our photoionization models give a value of 3.0, to be compared
  to the case
 B  recombination value of 2.75 used in most observational papers.
 Consequently, the reddening is smaller than usually estimated, with
 C(\ifmmode {\rm H}\beta \else H$\beta$\fi) practically equal to 0.

Our models can be used to give ionization correction factors
appropriate for I~Zw~18 for more accurate abundance determinations.
However, the largest uncertainty in the abundances of C, N, O and Ne
ultimately lies in the unsolved temperature problem.

It would be, of course, of great interest to find out whether other
galaxies share with I~Zw~18 this [O~{\sc iii}] $\lambda$4363/5007\ problem.
There are at least two other cases which would deserve a more
thorough analysis.
One is the starburst galaxy NGC 7714, for which
published photoionization models (Garc\'{\i}a-Vargas et al. 1997)
also give [O~{\sc iii}] $\lambda$4363/5007\
smaller than observed. However, it still needs to be demonstrated
that this problem remains when modifying the model assumptions
 (e.g. the gas density distribution, possible heating of the
gas by photolelectric effect on dust particles etc...).
In the case of NGC 2363, the photoionization models of
Luridiana et al. (1999) that were built using the oxygen abundances
derived directly from the observations yielded a [O~{\sc iii}] $\lambda$4363/5007\ ratio marginally
compatible with the observations. These authors further argued that,
due to the presence of large spatial temperature fluctuations,
the true gas metallicity  in this object is higher than derived by
empirical methods. In such a case, the [O~{\sc iii}] $\lambda$4363/5007\ ratio becomes even more
discrepant.

It might well be that additional
heating sources exist in giant H~{\sc ii}\ regions, giving rise to such
large temperature variations and and enhancing the [O~{\sc iii}] $\lambda$4363\ emission.
As mentioned above, such
a scenario needs to be worked out quantitatively.

Further detailed observational and theoretical studies of individual objects
would be helpful,
since we have shown that with insufficient observational constraints,
high [O~{\sc iii}] $\lambda$4363/5007\ may actually be produced by photoionization models.
The effort is worthwhile, since it
would have implications both on our understanding of the energetics of
starburst galaxies and on our confidence in abundance derivations.

\begin{acknowledgements}
This project was partly supported by the ``GdR Galaxies''.
DS acknowledges a grant from the Swiss National Foundation of Scientific
Research.
We thank Duilia De Mello for providing the {\em HST} images and
Jean-Fran\c{c}ois Le Borgne for help with IRAF.
During the course of this work, we benefited from conversations
with Jose V\'{\i}lchez, Rosa  Gonz\'{a}lez-Delgado, Enrique P\'{e}rez,
Yurij Izotov, Trinh Xuan Thuan. Thanks are due to Valentina Luridiana,
Crystal Martin and Claus Leitherer for reading the manuscript.
\end{acknowledgements}


\section{Introduction}
Massive binary stars provide vital insights to our understanding of massive-star evolution. This is primarily due to the accuracy with which their masses can be determined: an essential ingredient for understanding a wide range of stellar properties. Because of the additional constraints that can be placed on their age and evolution, these stars provide information on initial masses, chemical mixing and mass-loss \citep{moffat_binaries,selma}. In a broader context, they can then act as crucial calibration points for models of both stellar atmospheres and evolution.

From the catalogue of bright stars in the Magellanic Clouds by \citet{feast}, R139 has a $V$-band magnitude of $\sim$12, making it one of the brightest objects in the 30 Doradus nebula\footnote{Other aliases of R139 include: Brey~86 \citep{brey}, Parker~952 \citep{parker}, BAT99-107 \citep{batt}, and Selman~2 \citep{selman}.}. \citet{walborn_and_b} noted R139 as potentially one of the most massive stars in 30 Doradus, urging more detailed study.

Multi-epoch spectroscopy of R139 was obtained as part of the campaign by \citet{moffat}. His mean radial velocity from observations in 1982 showed an offset of $\sim$100\,kms$^{-1}$ compared to the mean velocity from earlier data. He noted R139 as a single-lined binary, with a tentative period of
52.7\,d adopted from a number of possible fits to the data.  \citet{schnurr} presented spectroscopy of R139 from three observing seasons (spanning 2001 to 2003). While noting that the system displayed `slightly variable radial velocity', it was concluded that R139 was single, citing the relatively large uncertainties in Moffat's past work for the conflicting scenarios.

R139 has now been observed as part of the VLT-FLAMES Tarantula Survey\footnote{Observations obtained at the European Southern Observatory Very Large Telescope in programme 182.D-0222.} (VFTS), an ESO Large Programme which has obtained multi-epoch spectroscopy of over 800 massive stars in 30~Doradus. A full overview of the survey is given by Evans et al. (Paper~I, 2011), here we report on the discovery of R139 as a massive double-lined binary.

\section{Observations}\label{observations}
The observations of R139 are summarised in Table \ref{tab_1}. The primary dataset of the VTFS has been obtained with the Giraffe spectrograph using the `Medusa' fibre-fed mode of the FLAMES instrument on the Very Large Telescope (VLT). Details of the reductions and observational strategy can be found in Paper~I, in which R139 is catalogued as object VTFS\,527.

After the initial detection of binarity made from the FLAMES data, follow-up observations have been obtained on the 6.5m Magellan Clay Telescope with the MagE instrument, with X-Shooter on the VLT and also with FEROS on the MPG/ESO 2.2m telescope at La Silla\footnote{ FEROS observations obtained as part of programme 086.D-0997}. All these follow-up observations provide coverage across the entire visible spectrum, with a typical signal-to-noise ratio of order 150 - although the FEROS data must be degraded to $R\sim 9000$ to achieve this.

\begin{table}[t]
\caption{Observational epochs for R139 detailing; the instrument used, the resolving power ($R$), the date and the time elapsed since the first epoch ($\Delta $ HJD).
\label{tab_1} }
\tiny
\begin{center}
\begin{tabular}{llccr}
\hline\hline
Epoch & Instrument\,/\,Setting & $R$  & HJD +2400000 & $\Delta $ HJD \\

\hline
1 & FLAMES\,/\,LR02 &  7,000  & \phantom{it}54748.7769\tablefootmark{a} & - \\
2 & FLAMES\,/\,LR02 &   & 54748.8228 & 0.046\\
3 & FLAMES\,/\,LR02 & & 54749.7233 & 0.946\\
4 & FLAMES\,/\,HR15N & 16,000  & 54749.7726 & 0.996\\
5 & FLAMES\,/\,LR03 &  8,500  & 54755.6882 & 6.911\\
6 & FLAMES\,/\,HR15N &    & 54810.6697 & 61.893\\
7 & FLAMES\,/\,LR03 &   & 54810.7374 & 61.961\\
8 & FLAMES\,/\,LR03 &  & 54810.7808 & 62.004\\
9 & FLAMES\,/\,LR03 &  & 54810.8357 & 62.059 \\
10 & FLAMES\,/\,LR02 &   & 54837.6332 & 88.856\\
11 & FLAMES\,/\,LR02 &  & 54868.5569 & 119.780\\
12 & FLAMES\,/\,LR02 &  & 55112.8129 & 364.036\\
\hdashline
13 & MagE \,/\,1'' slit &  4,500 & 55527.8017 & 779.025\\
\hdashline
14 & X-Shooter\,/\,0.5'' slit & \phantom{it}9,100\tablefootmark{b} & 55580.5477  & 831.771\\
15 & X-Shooter &  & 55584.7076 & 835.931\\
16 & X-Shooter  &  & 55588.6922  & 839.915\\
\hdashline
17 & FEROS & 48,000 & 55604.6909  & 855.936 \\
18 & FEROS  &  & 55605.5613 & 856.739\\

\hline
\vspace{-0.6cm}
\end{tabular}
\end{center}
\tablefoottext{a}{The FLAMES observations were comprised of back-to-back observations so only the mid-exposure HJD is given.}\ \newline
\tablefoottext{b}{The resolving power quoted for X-shooter is that of the UV-arm only, which overlaps the LR02-03 regions of the FLAMES data.}
\vspace{-0.3cm}
\end{table}

\section{Results}\label{results}
\subsection{Binary identification and spectral classification}
\citet{feast} described the spectrum of R139 as exhibiting `O-type absorption plus weak W emission'. \citet{walborn_and_b} argued that R139 has very strong Of emission features, classifying it as O7 Iafp, while Schnurr et al. (2008b) adopted the spectral type WN9h::a. This ambiguity has likely arisen because the emission features are the superposition from two similar stars. When observed at lower resolution, this would have falsely suggested enhanced emission or broadening of the lines.

The increased resolution and time-sampling of the new data have revealed that many of R139's prominent emission and absorption features separate into two distinct and similar components. This is best illustrated by contrasting epochs 11 and 16 in Figure~\ref{spectra}, which show the minimum and maximum observed separations respectively. The epochs which display well-separated components have allowed a precise classification of both components. The system consists of a more massive and more luminous primary, which is an O6.5\,Iafc supergiant, and a slightly less luminous O6\,Iaf companion. These spectral types are determined from visual inspection of the He~\2 / He~\1 absorption line ratios between $\lambda$4200 / $\lambda$4026 and also between $\lambda$4542 / $\lambda$4471, the latter of which can be seen in Figure~\ref{spectra}.
\begin{figure}[!b]
\centering
\includegraphics[width=9.0cm]{r139_fig1.pdf}
\caption{Normalised spectra of R139 at epochs of minimum (\#11) and maximum (\#16) observed separation. The He~\1 $\lambda$4471 line profile in epoch \#16 suffers from a slight nebular over-subtraction but does not prevent clear identification of the components.
\label{spectra}}
\vspace{-0.4cm}
\end{figure}

The Ifc classification of the primary arises from emission of C~\3 $\lambda$4647 and  $\lambda$4650-4652. This relatively rare feature led to the recent introduction of the Ofc category in the morphological framework used to classify O-type spectra \citep{walborn_ofc}. Figure~\ref{carbon} clearly illustrates that the N~\3 emission lines separate into two distinct components, whereas the two C~\3 emission lines show no evidence of separation, but merely shift between epochs in the same sense as the primary. The Ofc feature has previously been associated with the O5 spectral type \citep{walborn_ofc}; this therefore, is an interesting example of the phenomenon in a later-type star, albeit in the LMC.

Where possible, the relative shift of the components was identified through the C~\3 emission and also He~\1 $\lambda$4922 emission, which are only present in the primary. In the LR02 observations, the Si~\4 $\lambda$4116 emission line was used as the main diagnostic for the relative shift of the components. The Struve-Sahade effect \citep[e.g.][]{linder} can most likely be neglected in this system given its relatively long period.

\begin{figure}[t]
\centering
\includegraphics[width=8.cm,height=4.6cm]{r139_fig2.pdf}
\caption{Normalised spectra showing the complex N~\3 and C~\3 emission region. Both of the N~\3 emission lines and the He~\2 $\lambda$4686 emission are comprised of two un-equal components, the stronger of which is associated with the primary. The C~\3 emission lines show no separation, but exhibit radial velocity shifts in the same direction as the primary.
\label{carbon}}

\end{figure}

\subsection{Radial velocity analysis and lower mass limits}
A global $\chi^2$ fitting approach has been used to determine the radial velocity shifts of the different epochs. This technique fits double Gaussian profiles to a number of lines: He~\1 $\lambda$4026, Si~\4 $\lambda$4116, He~\2 $\lambda$4200, S~\4 $\lambda$4485, S~\4 $\lambda$4505, He~\2 $\lambda$4542, He~\2 $\lambda$ 4686 and He~\1 $\lambda$4922. The fitting is performed simultaneously on all the observations, which ensures that consistent profile shapes are used, including at conjunction. This approach improves the disentangling of the contribution from each star for the data sets with limited phase coverage, but ignores the possibility of line profile variations. The formal errors on the measurements for each component are a few km\,s$^{-1}$.

The mass ratio of the system can be found from the ratios of the primary and secondary radial velocities, and is independent of any other assumptions about the orbit \citep{rauw2000}. For R139 the mass ratio is found to be: $M_1 / M_2 = 1.20 \, \pm \, 0.05$.

Period searches based on Fourier analysis of the measured radial velocity shifts were performed using the methods of \cite{gosset}: the dominant signal indicated a period of 153.9 days. The corresponding orbital solution, which has an rms uncertainty of 6.2\,km\,s$^{-1}$, is shown in Figure \ref{orbit} \citep[based on methods of][]{sana06stars}. The corresponding lower mass limits for the stars are: $M_1 \sin^3{i} = 78 \pm 8\,M_{\sun}$ and $M_2 \sin^3{i} = 66 \pm 7\,M_{\sun}$. Other orbital parameters are listed in Table \ref{orb_params}. These are sensitive to the behaviour of the system around periastron: further observations of this stage in the orbit would allow confirmation of these results.

\begin{table}[b]
\caption{The parameters associated with the best-fit orbital solution. The errors quoted are the formal errors on the best-fit from the Fourier analysis, and therefore may not be fully representative of the uncertainty in the parameter values.
\label{orb_params} }
\tiny
\begin{center}
\begin{tabular*}{0.45\textwidth}{@{\extracolsep{\fill}}ll}
\hline\hline

Property & Best-fit value \\

\hline

Period, P  & 153.9  $\pm$   0.1\,days\\
Eccentricity, e &  0.46       $\pm$   0.02\\
Argument of periastron, $\omega$  &  106.9       $\pm$   5.0\,deg\\
Date of $\phi=0$ (HJD - 2450000), T$_0 $ &    6035.9    $\pm$   1.3\,days\\
Maximum velocity of primary, K1 & 107.8       $\pm$   3.8\,km\,s$^{-1}$ \\
Maximum velocity of secondary, K2 &   127.0       $\pm$   4.5\,km\,s$^{-1}$\\
Projected semi-major axis for primary, a$_1 \sin{i}$ &  290.6       $\pm$  10.8\,$R_{\sun}$\\
Projected semi-major axis for secondary, a$_2 \sin{i}$ &  342.3       $\pm$  12.8\,$R_{\sun}$\\

\hline
\vspace{-0.6cm}
\end{tabular*}
\end{center}
\end{table}

\subsection{The luminosity of R139}
To estimate the luminosity of the system, model atmospheres were calculated with CMFGEN \citep{hillier}, adopting abundances from \cite{asplund} and scaling them appropriately for the LMC. These were used to constrain the effective temperature ($T_{\rm eff}$) consistent with:
1) the absence of N~\4 $\lambda$4058 emission;
2) the presence of N~\3 $\lambda$4640 emission;
3) the intensity of the He~\1 $\lambda$4471 absorption.
This gives an estimated $T_{\rm eff}$ for both components of $34 \pm 2$\,kK.

The luminosity was then determined by matching optical and infrared photometry from \cite{selman} and 2MASS \citep{skrutskie}. For this purpose, the visual extinction $A_V=R\times E(B-V)$ was determined for each model, based on the relation $R = 1.12 \times E(V-K)/E(B-V) + 0.02$ from \citet{fitzpatrick}. The resulting luminosity for the composite system is $\log(L/L_{\odot})=6.4\pm 0.1$ (with $R$ in the range 3.4--3.9).

\begin{figure}[t]
\begin{center}
\includegraphics[width=7.5cm]{r139_fig3.pdf}
\caption{The best-fit orbital solution from the measured radial velocities of the components, indicating a 153.9 day orbit. The closed circles denote the velocities of the primary and the open circles the secondary.}\label{orbit}
\end{center}
\vspace{-0.4cm}
\end{figure}

\section{Photometric Variability}\label{fts}
R139 was identified as showing slight photometric variability by \cite{feit}. They observed a 0.3 magnitude dimming in the $V$ band over a 25 day period. However, this was from only three observations taken with a wide (18'') aperture.

An active component of the VTFS is photometric follow-up with the 2\,m Faulkes Telescope South, which has been used to monitor seven fields in the 30~Dor region. The default mode of the camera is 2\,$\times$\,2 binning of the CCD pixels, giving an effective pixel-scale on the sky of 0\farcs278.

We have 54 V-band epochs for the relevant field, spanning an 18 month period starting in January 2009. The Faulkes data are reduced automatically following observations, but are not calibrated photometrically. Given the crowding in this field, we used {\sc apphot} in {\sc iraf} to obtain instrumental magnitudes of R139 from aperture photometry. Five `check' stars of similar brightness were selected from the frames for comparison: R133, R137, R138, Mk\,11, and R146. From these, differential residuals ($\Delta V$) were calculated for R139 compared to the mean magnitude of the check star for each epoch. The deviation was found to be consistent with that calculated between the check stars themselves, indicating that R139 shows no photometric variability. These results are shown in Figure \ref{phot}, where the observations have been phased to the 153.9\,day orbit.

If the inclination ({\it i}) of the system was $90^{\circ}$, the maximum duration of eclipses near apastron and periastron has been calculated to be 7.9 and 2.9\,days respectively. From the sampling of our photometric data, it is unlikely that such events would have gone undetected, see Figure \ref{phot}. However, if the inclination is lower ($80^{\circ}\!\!\lesssim\!i\! \lesssim \!\!86^{\circ}$), there is no eclipse near apastron and the periastron eclipse is shorter. Consequently, an intensive photometric observing campaign is required near to periastron to conclusively determine if there is any evidence for an eclipse.

\begin{figure}[t]
\begin{center}
\includegraphics[width=8.0cm,height=5.1cm]{r139_fig4.pdf}
\caption{Differential V-band residuals for R139 compared to the mean of the five check stars, phased to the 153.8\,day orbit. Open circles denote the five epochs when the seeing was in excess of 2\farcs0. The dotted lines indicate the expected mid-point of any possible eclipses: $\phi = 0.39$ and 0.98 for the eclipses near apastron and periastron respectively.
}\label{phot}
\end{center}
\vspace{-0.4cm}
\end{figure}

\section{X-rays}\label{xrays}
R139 was detected by \cite{zwart02} as an X-ray source in the 30~Dor field observed with the Advanced CCD Imaging Spectrometer on the \emph{Chandra X-Ray Observatory}. Further analysis of the \emph{Chandra} data was carried out by \citet{townsley} and more recently by \citet{guerrero}. These studies found R139 to have a relatively low X-ray luminosity compared with other W-R stars in the region. \citet{guerrero} also considered data from the R\"ontgen Satellite (\emph{ROSAT}), but it did not detect R139 owing to its lower sensitivity.

The X-ray luminosity and the bolometric luminosity of massive O stars are linked by the relationship $L_X \approx 10^{-6.9}L_{bol}$ \citep{sana06X}. Therefore, with a luminosity of $\log(L/L_{\odot})=6.4$ for the combined system, an X-ray luminosity of $1.2 \times 10^{33} \, \rm{erg\,s^{-1}}$ would be expected. This is slightly lower than Guerrero's result of $2.7 \times 10^{33} \, \rm{erg\,s^{-1}}$ in the 0.5 - 7.0keV range - even considering the possible 25\% error in the detected count rate. This slight excess emission might be associated with X-rays generated through the interaction of the system's stellar winds. There is no evidence, however, for phase-dependent line profile variations, which would have also suggested colliding winds.

\section{Discussion}\label{discussion}
\emph{Previous observations:} In order to compare our result with the earlier work of \citet{schnurr}, who found an insignificant radial velocity shift, our LR02 observations were degraded to the same resolving power as Schnurr's ({\it R}\,$\sim1000$) and a number of lines were fit with a single Gaussian function. The radial velocity variation was found to be only 10.3\,km\,s$^{-1}$, while the FWHM of the profiles varied by around 40\,km\,s$^{-1}$. This suggests that, even if the system had been observed near periastron, it would have been difficult to confirm its binary nature.

\vspace{0.2cm}
\noindent{\emph{Comparison systems:}\
Some binary systems have been identified where both components are more massive than those of R139: NGC-3603-A1, is a system comprised of a 116M$_{\sun }$ primary and a 89M$_{\sun }$ star \citep{schnurr_binary}, and also WR20a, an 83M$_{\sun }$ and 82M$_{\sun }$ system \citep{wr20_rauw,wr20_bonanos}. However, there are not many systems with a pair of massive evolved O-stars. Closer analogs are the Cyg OB2-B17 system \citep{vanessa}, where the component stars are O7 and O9 supergiants and Cyg OB2-\#5 where one of the stars is an O6-7 supergiant \citep{rauw99}. It would appear that neither of these systems contain stars as massive as those predicted here. Consequently, it can be argued that R139 is the most massive O supergiant binary system yet discovered. }

\begin{figure}[!t]
\centering
\includegraphics[width=7.5cm,height=4.8cm]{r139_fig5.pdf}
\caption{Hertzsprung-Russell diagram showing mass estimates derived from luminosity fits to evolutionary tracks (Brott et al. 2011; Friedrich et al. in prep.).~The red circle indicates the best fit for the total luminosity of the R139 system, while the blue squares show the fit for the luminosity of the two components based on the mass ratio and assuming the objects are coeval. From these it is possible to infer the initial masses, see text for further details.
\label{hrd}}
\vspace{-0.3cm}
\end{figure}

\vspace{0.2cm}
\noindent{\emph{Evolutionary masses:}\
Figure \ref{hrd} shows how the R139 system compares to evolutionary tracks from Friedrich et al. (in prep.), computed analogously to the models of \cite{ines}. The figure shows that the total luminosity and $T_{\rm eff}$ of the system equals that of a single star with an initial mass above 125 $M_{\sun}$.}

Assuming R139 consists of coeval stars with a mass ratio of 1.2,  the estimates for the current masses are $75 \pm 14 M_{\sun}$ for the (cooler) primary and $62 \pm 11 M_{\sun}$ for the (hotter) secondary.  These values have been derived using a $\chi^2$ method to fit the combined luminosity of the stars against that quoted for the system.  The effective temperatures of the stars were fitted against the CMFGEN-derived temperatures of $34 \pm 2$\,kK. Interestingly, these estimated masses are in close agreement with the lower-mass limits from the orbital solution. This implies that the system has a high inclination and supports the need for additional photometric observations.

In these models an initial equatorial velocity of 110 km\,s$^{-1}$ was adopted in agreement with the current observed $v \sin i$. The effect of rotation on the evolutionary tracks is very limited for initial rotation rates up to about 300\,km\,s$^{-1}$\citep{ines}. Nevertheless, these tracks are sensitive to uncertain physical processes such as internal mixing and mass loss. The errors on the mass estimates represent the formal 1-sigma confidence limits of the $\chi^2$ fit and do not include systematic uncertainties in the model physics.

The best fit corresponds to an age of 2 - 2.5\,Myr and implies that both stars have significantly evolved off the zero-age main sequence. As the stars are assumed to be coeval, the substantial mass ratio implies a large difference in temperature between the components (see Fig. \ref{hrd}). This is surprising given the similar spectral types; \citet{martins} predict a temperature difference nearer to 1\,kK for a 0.5 variation in spectral types. This discrepancy may well reflect our still limited understanding of the physics of the most massive stars, illustrating the potential of massive binaries as tools to evaluate our models.

\vspace{0.4cm}
\noindent{The high quality, time-sampled VFTS observations have revealed that R139 is a binary system. The data suggest that it is the most massive evolved O-star binary system yet discovered: a result which further observations around periastron would help to confirm. As demonstrated here, such a massive system has already presented challenges for theoretical models to reproduce its observed properties and it will likely provide a crucial test for evolutionary and atmospheric models in the future.
}

\acknowledgements{Thanks to the referee, Anthony Moffat, for their constructive comments. SdM acknowledges NASA Hubble Fellowship grant HST-HF-51270.01-A awarded by STScI, operated by AURA for NASA, contract NAS 5-26555. AZB acknowledges support from the European Commission\,FP7 under the Marie Curie International Reintegration Grant PIRG04-GA-2008-239335. RHB acknowledges partial support from DIULS Project PR09101.}

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