Modeling ionized gas in low-metallicity environments: the Local Group dwarf galaxy IC10

Our objective is to investigate the physical properties of the ionised gas of the low-metallicity dwarf galaxy, IC 10, at various spatial scales: from individual HII regions to the entire galaxy scale and examine whether diagnostics for integrated measurements introduce bias in the results. We modeled the ionised gas combining the mid- and far-infrared fine-structure cooling lines observed with Spitzer/IRS and Herschel/PACS, with the photoionisation code Cloudy. The free parameters of the models are the age of the stellar cluster, the density and the ionisation parameter of the ionised gas as well as the depth of the cloud. The latter is used to investigate the leakage of the ionising photons from the analysed regions of IC 10. We investigate HII regions in the main star-forming body, on scales of ~25 pc, three in the main star-forming region in the center of the galaxy and two on the first arc. We then consider larger sizes on the scale of ~200 pc. We find that most clumps have nearly identical properties, density ~10$^{2.} $ - 10$^{2.6}$ cm$^{-3}$, ionisation parameter between 10$^{-2.2}$ and 10$^{-1.6}$ and age of the stellar cluster ~5.5 Myr. All of them are matter-bounded regions, allowing ionising photons to leak. The relatively uniform physical properties of the clumps suggest a common origin for their star formation activity, which could be related to the feedback from stellar winds or supernovae of a previous generation of stars. The properties derived for ~200 pc size"zones"have similar properties as the HII regions they encompass, but with the larger regions tending to be more radiation-bounded. Finally, we investigate the fraction of [CII] 157.7 {\mu}m, [SiII] 34.8 {\mu}m, and [FeII] 25.9 {\mu}m, emission arising from the ionised gas phase and we find that most of the emission originates from the neutral gas, not from the ionised gas.


Introduction
The interstellar medium (ISM) plays a key role in understanding star formation (SF) process, as it is at the same time the reservoir of gas and dust and the repository of stellar ejecta, enriched by the elements produced by nucleosynthesis in massive stars. While it is still difficult to assess the details of the ISM properties at high redshifts, galaxies in the Local Group offer the opportunity to study several 'chemically-young', i.e. metal-poor, dwarf galaxies. Metal-poor dwarf galaxies are not genuinely young (i.e., they are at least older than ∼1 Gyr) and consequently they cannot be directly compared to high-redshift galaxies. Nevertheless, these unevolved nearby dwarf galaxies remain the best laboratories to examine SF and physical conditions in the metal-poor ISM. Some irregular dwarf galaxies harbour super star clusters (e.g. 30 Doradus in the Large Magellanic Cloud, Hunter 1999) and some of them host Wolf-Rayet (WR) stars (e.g. IC 10, Massey & Holmes 2002), hinting at intense SF activity within the last 10 Myr. The combination of a young massive stellar population which produces hard ultraviolet (UV) photons and a dust-poor ISM (more transparent) can result in a hard radiation field extending over galaxy-wide scales. Observations are consistent with the above picture, with ionised gas tracers detected throughout these galaxies (e.g. Kawada et al. 2011;Cormier et al. 2015).
The gas tracers, i.e. cooling lines, provide access to the gas properties of the ISM, such as elemental abundances (e.g. Garnett 1990;Stasińska 2007;Kewley et al. 2010), temperature and density, and reveal the nature of the heating mechanism of the gas (UV, X-rays, shocks, etc.; e.g. Baldwin et al. 1981; Article number, page 1 of 48 arXiv:1810.03633v1 [astro-ph.GA] 8 Oct 2018 A&A proofs: manuscript no. IC10 Osterbrock & Ferland 2005;Kaufman et al. 2006;Dimaratos et al. 2015;Lee et al. 2016;Lebouteiller et al. 2017). The midand far-infrared (MIR and FIR) lines -the focus of this paperare less affected by the dust and gas attenuation compared to optical lines, and are therefore potentially ideal tracers of the ISM parameters deeper into star forming clouds.
The Herschel Dwarf Galaxy Survey (DGS, Madden et al. 2013) enabled the observations of many such cooling lines in some of the most metal-poor dwarf galaxies in the nearby Universe. It provided FIR and submillimetre (submm) photometric and spectroscopic observations of 48 low-metallicity dwarf galaxies. The spectroscopic data from the Herschel telescope (Pilbratt et al. 2010), together with Spitzer (Werner et al. 2004) MIR spectroscopic data, provide ∼20 MIR and FIR fine-structure cooling lines to study the ionised, atomic and molecular gas in dwarf galaxies. Analysis on global galaxywide scales of the DGS by Cormier et al. (2012; shows that the physical properties of the ISM of these galaxies are different compared to those of the more metal-rich (starburst, spiral, IR-bright) galaxies. The Cormier et al. (2015) study highlights, for example, that the brightest FIR line for these galaxies on global scales, is the [OIII] 88.4 µm line, while it is [CII] 157.7 µm in more metal-rich galaxies. The emission of the [OIII] 88.4 µm line requires photons with energy greater than 35 eV, thus the strength of [OIII] 88.4 µm emission is indicative of the presence of a significant Lyman continuum flux, and the extent of [OIII] 88.4 µm emission is a consequence of the low dust abundance.
An important step is now to connect integrated galaxy scale analyses to resolved region studies (e.g. Chevance et al. 2016;Lee et al. 2016;Fahrion et al. 2017), by examining the ISM properties of a nearby galaxy that can be resolved and fully mapped. Performing an analysis at different spatial scales can help to develop a more consistent picture of the ISM characteristics. The challenge is to take into account the variations in signal-to-noise ratio of the diagnostic tracers and their spatial coverage, compelling us to select specific combinations of tracers at the different scales. Since the modeling strategy depends on what tracers are available, it is important to extract the best possible constraints on the physical properties as a function of the spatial scales used and identify the potential corresponding biases.
The proximity of the dwarf irregular galaxy IC 10 (∼715 kpc; Sakai et al. 1999;Sanna et al. 2009;Kim et al. 2009) makes it an opportune object to investigate the relation between the local properties of the different gas phases and the larger scales characteristics in a low metallicity environment and to understand which combinations of emission lines provide useful diagnostics as a function of the spatial scale considered. Moreover, IC 10 has been mapped in all of the tracers available with Spitzer and Herschel, providing larger data cubes than currently available in the optical. Combinations of ionic and atomic tracers should provide the constraints necessary to build a self-consistent model of the ISM and thus assess the physical properties (e.g., density, filling factor) of the different ISM phases.
In the present paper we perform a study of the ionised gas: the dense H ii regions as well as the diffuse ionised gas of IC 10. The best solutions in this study will then set a reference model of the ionised gas distribution and characteristics that will be used to further model the associated neutral and molecular gas components, in a subsequent paper. This paper is organized as follows. First, we present an overview of IC10 (Section 2) and the dataset used in this study (Section 3). Next, in Sections 4 we describe our modeling strat- egy. Modeling the MIR and FIR line emission we determine the physical properties of the ionised gas in the H ii regions as well as at larger scales (Section 5). Finally, we discuss the results obtained (Section 6) and we summarize our conclusions in Section 7.

Overview of IC10
IC 10 is an irregular dwarf galaxy, with a metallicity of 12+log(O/H) = 8.26 (Garnett 1990;Lequeux et al. 1979;Richer et al. 2001;Magrini & Gonçalves 2009), ≈2.7 times lower than the solar metallicity (12+log(O/H) = 8.69; Asplund et al. 2009) and between that of the Small and Large Magellanic Clouds. It was identified as an external object by Mayall (1935) and as a member of the Local Group by Hubble (1936). The estimated distance is uncertain because of the location of this galaxy close to the Galactic plane (b = -3.3 • ). The distance has been determined to be from 500 kpc to 3 Mpc (Sandage & Tammann 1974;Sakai et al. 1999;Borissova et al. 2000;Hunter 2001). In this paper we adopt 715 kpc (Kim et al. 2009;Lim & Lee 2015), the distance calculated by Kim et al. (2009) using the tip of the red giant branch (TRGB) method. This distance is close to 700 kpc, a value frequently adopted in the recent literature (e.g. Heesen et al. 2018). At this distance 1 ≡ 3.5 pc. Thus, IC 10 is close enough for its SF regions to be resolved and far enough to be fully mapped.
IC 10 consists of a main body and several star forming arcs (Fig. 1). These components are sitting in an extended and complex H i envelope whose diameter is 7 times larger than the optical diameter (Huchtmeier 1979;Wilcots & Miller 1998;Ashley et al. 2014). H i holes are prominent throughout the body of IC 10, the origin of which has been interpreted to be cumulative effects of stellar winds (Wilcots & Miller 1998;Nidever et al. 2013). The velocity field of the ionised gas closely matches the HI velocity field (Thurow & Wilcots 2005). Supernova remnants and winds from WR stars have been identified as the main driver of the gas kinematics (Thurow & Wilcots 2005). The hard radiation from the WR stars, together with photons leaking from the H ii regions, are the sources of such extended ionised gas emission (Hidalgo-Gámez 2005).
Several parameters suggest that IC 10 is undergoing a starburst phase. The star formation rate is ∼0.2 M yr −1 and Hodge & Lee (1990) discovered a large number (144) of H ii regions, which is relatively high for a galaxy with stellar mass of 4×10 8 M . The starburst nature of the galaxy is confirmed by the stellar component of the galaxy. The well-studied stellar population of IC 10 highlights young stellar clusters (≤10 Myr) located in the Hα emitting regions (Sanna et al. 2009;Yin et al. 2010;Vacca et al. 2007), while the old clusters are distributed over a wide area of the disk (Lim & Lee 2015). Moreover, WR stars abound in IC10 (e.g. Massey & Holmes 2002) 1 . IC 10 shows high SF activity considering the low molecular gas surface density, determined from CO observation (e.g., Wilson & Reid 1991;Leroy et al. 2006), although a large quantity of H 2 (100 times the mass of molecular gas present in the CO core) has been inferred from the CO-dark gas tracer [CII] 158 µm (Madden et al. 1997). Finally, the gas-to-dust mass (G/D) ratio is estimated to be 240 -475 (Rémy-Ruyer 2013; G/D ∼160 for the Milky Way), lower than that expected based on a linear relation between gas-to-dust and metallicity (∼ 10 3 ; Rémy-Ruyer et al. 2014).

Data
IC 10 has been observed with the Herschel/PACS (Photodetector Array Camera and Spectrometer; Poglitsch et al. 2010) instrument, as part of the DGS sample, and it has also been observed in the MIR with the IRS (Infrared Spectrograph; Houck et al. 2004) onboard Spitzer, providing access to a wide range of gas tracers. In this section, we present the data used in our study.
3.1. Spitzer/IRS spectroscopy IC 10 was mapped with the low-resolution modules (R = λ/∆λ ≈ 60 − 127) of the IRS with the Spitzer Space Telescope: two Short-Low (SL; λ= 5.7 -14.5 µm) and two Long-Low (LL; λ= 14 -38 µm). The SL map consists of 8×58 slit positions while the LL map consists of 2×20 slit positions. Five additional pointings were observed toward bright knots with the high-resolution modules (R ≈ 600) Short-High (SH; λ= 9.9 -19.6 µm) and Long-High (LH; λ= 18.7 -37.2 µm). Figure 2 shows the area covered by the low-resolution modules and the positions of the high-resolution pointings. Table 1 provides observational parameters. The integrated rest-frame spectrum (∆v = 348) together with the modeled spectral energy distribution (SED) for one of the selected zones (Sec. 4.1) of size ∼ (174 pc × 226 pc) in the center of IC10 is shown for illustration in Figure 3 while a zoom on individual lines is displayed in Figure 4.
We refer to  for the general principles on the reduction of the IRS data and map reconstruction, while here we present an updated reduction. The data was first reduced with CUBISM (Smith et al. 2007) and the bad pixels in the detector were identified and ignored using the backtracking tools before projecting and exporting the datacube. The SL and LL data cubes were projected independently with different map 1 Crowther et al. (2003) identified in total 25 WR, 14 WC (WR star whose spectrum is dominated by lines of carbon) and 11 WN (WR star whose spectrum is dominated by lines of nitrogen), which is an unusually high WC/WN for a metal-poor galaxy. Observed WC/WN as a function of the metallicity for different galaxies of the Local Group is shown in Massey & Holmes (2002) and in Crowther et al. (2003). parameters (different sampling), moreover the SL maps exhibit gaps, which can result in some discrepancy in the continuum flux level for a given spatial region. The amplitude of this discrepancy was estimated by calculating the stitching factor between the SL and LL continuum for the overlap wavelength window around 14 -15 µm. The ratio is close to one with a standard deviation σ st of about 15% per pixel of 4 . A Monte-Carlo method was used to produce the spectral maps and estimate the associated uncertainties. For each of 100 realisations we have done the following: 1. The gaps in the SL map were filled using a b-spline fit in the direction perpendicular to the slits. 2. A median filtering was applied in the dispersion direction to accommodate the lower spatial resolution due to the incomplete sampling in that direction. 3. Each plane was convolved to a resolution of either 4 (SL) or 12 (SL and LL). 4. Each plane was resampled to a pixel size of either 4 (SL) or 12 (SL and LL). 5. Each spectral line and the continuum of a plane were simultaneously fitted with a Gaussian and a second order polynomial, respectively.
The final line flux and associated uncertainties were calculated using the median and median absolute deviation of the line flux distribution among the 100 outcomes. In Figure 5 we show the line maps prior to convolution and resampling (points 3 and 4), and Table 2 reports the general properties of those lines. The individual high-resolution pointings (Appendix, Fig. A.1) were retrieved from the CASSIS spectral database (Lebouteiller et al. 2011;. For some pointings, the emission is extended, and, as a consequence, the SH and LH spectra do not align because both modules have different slit sizes (4.7 × 11.3 for SH, 11.1 × 22.3 for LH). For such cases, we used the wavelength-dependent extended source flux calibration available in CASSIS, and applied a scaling factor to match SH A&A proofs: manuscript no. IC10 to LH in the overlapping wavelength range of 19.0 − 19.5 µm. Some pointings are dominated by a point source, and in such cases the SH and LH spectra, after using a point-source flux calibration (a factor between 1.5 to 4.5), aligned fairly well. The final fluxes obtained with the high-resolution modules for each pointing are presented in the Table A.1.
The line fluxes are not corrected for possible attenuation due to the silicate absorption bands at ≈10 and ≈20 µm (Galliano et al. 2017). The [ArIII] and [SIV] lines, located at 9.0 and 10.5 µm respectively are the most likely to be attenuated by the silicate dust (the silicate absorption band at ≈ 20 µm is comparatively weaker). The silicate absorption optical depth at 10 µm in IC 10 is about τ ≈ 0.2 on average across the main body, with peaks around 0.4 toward the clumps . From τ we calculated the extinction A 10µm < 1.086 × 0.4 ∼ 0.4 mag. This value is an upper limit, since our line measurements correspond to spatial scales larger than those used for which the silicate absorption optical depth peaks were determined. If we wish to calculate the attenuation of the [ArIII] and [SIV] lines, we need to assume a geometry, i.e., either "screen" (assuming that the gas is located behind the dust) or "mixed" (assuming the gas is mixed with the dust). I ν /I inc ν = e −τ is the solution of the transfer of radiation through dust, where I ν is the observed intensity and I inc ν is the incident intensity. For a homogeneous mixture of gas and dust, the equation becomes (Mathis 1972). If we assume that the gas and dust are well mixed, the silicate absorption measured toward IC 10 clumps corresponds to an attenuation of less than 50% with lower values expected for the large zones (τ ≈ 0.2 on average across the main body, while 50% is calculated assuming τ ≈ 0.4). Overall, we consider that the infrared lines are little affected by extinction, thus we do not correct the line emission for it.  Table 2 reports the general properties of those emission lines.

Herschel/PACS spectroscopy
The PACS array consists of 5 × 5 spatial pixels covering a total field of view 47 × 47 . The velocity resolution is ∼ 90 km s −1 at 60 µm, ∼ 125 km s −1 at 90 µm and ∼ 295 km s −1 at 120 µm (PACS Observer's Manual 2013). The observations have been done in unchopped mode, in which an offset position is observed before and after the IC 10 observation. The data cubes have been reduced with Herschel Interactive Processing Environment (HIPE, Ott 2010) v12.0.0 and then processed with PAC-Sman  for the line fit and map construction. To estimate the uncertainty on the fit parameters, a Monte-Carlo approach was used. Details of the observations, the reduction of the PACS data and uncertainties can be found in Cormier et al. (2015).

Ancillary data: Hα
Another tracer of the ionised gas is the optical line Hα. IC 10 has been observed in Hα with the 1.8 m Perkins Telescope at Lowell Observatory at a resolution of 2.3 (Hunter & Elmegreen 2004). The data is calibrated but not extinction-corrected. The map was convolved to the Spitzer/IRS SL resolution (3.7 ) using a gaussian kernels. In this way the Hα image is consistent with the highest resolution available for the MIR line maps. Since Hα suffers from significant extinction from Galactic dust along the line of sight toward IC 10 (Galactic latitude of IC 10 ∼3.3 • ) and from dust internal to IC 10, we do not use Hα as a constraint in the analysis. Instead, we will use the model unattenuated predictions for Hα and the observed value in order to estimate the extinction a posteriori (see Section 6.3).

Morphology
The maps of MIR and FIR emission lines ( Figure 5) show bright compact clumps distributed in the main star-forming region and in two arcs. The emission of these lines peak in the same clumps as Hα .
The MIR fine-structure lines [ArIII] 8.9 µm, [SIV] 10.5 µm and [NeIII] 15.5 µm have high critical density, ≥ 5 × 10 4 cm −3 (Table 2), and high ionisation potential, 27.6 eV, 34.7 eV and 41 eV respectively. These lines are good diagnostics of the dense ionised gas, i.e. the younger H ii regions. Other MIR tracers, such as [ArII] 6.9 µm and [NeII] 12.8 µm, in addition to the compact clumps also show prominent extended emission, which likely arises in relatively more diffuse medium. The corresponding ions exist for lower energies, thus they trace better the outer shell of H ii regions as well as relatively diffuse low-ionisation gas Dimaratos et al. 2015). The S 2+ ion with ionisation potential of 23.3 eV has been observed at two wavelengths, 18.7 µm and 33.5 µm, which have critical densities for collisions with electrons of 2 × 10 4 cm −3 and 7 × 10 3 cm −3 , respectively. Hence, we have access to the useful density tracer of the ionised gas, [SIII] 33.5 µm/[SIII] 18.7 µm (Osterbrock & Ferland 2005).
The FIR fine-structure lines [OIII] 88.4 µm and [NII] 121.9 µm are characterised by much lower critical densities than the MIR tracers. Both lines arise from more diffuse ionised gas (Table 2). In particular, the extended [OIII] 88.4 µm emission, which has an ionisation potential of 35.1 eV and a critical density 500 cm −3 , suggests a high filling factor of diffuse ionised gas. The low-metallicity ISM seems to be very porous, allowing hard photons, such as those creating the O 2+ ion, to penetrate over large distances, even extending to galaxy scales (e.g. Cormier et al. 2015).
Our study focuses on the physical conditions of the compact, dense H ii regions as well as the diffuse, ionised gas of IC 10. Thus, we will use the lines from all species that trace mostly the ionised gas: N + , O 2+ , Ne + , Ne 2+ , S 2+ , S 3+ , Ar + , and Ar 2+ . We will not use in our models species with ionisation potentials lower than 13.6 eV since they may be ionised in regions where hydrogen remains neutral; these include C + , Si + , or Fe + and the corresponding prominent emission lines in the infrared, [CII]

Various spatial scales
Three spatial scales have been selected to analyse the ionised gas properties: clumps, zones and body.

Clumps
The smallest spatial scale accessible is limited by the spatial resolution of the instruments. The best resolution is achieved with the Spitzer/IRS SL observations (4 , corresponding to 14 pc at A&A proofs: manuscript no. IC10 the distance of IC 10). With this resolution we can disentangle the clumps from the more extended gas component. Although many clumps could, in principle, be investigated, we focused the analysis on the brightest clumps of the galaxy in MIR and FIR lines. We identified five such clumps in the body of IC 10: three in the central main star-forming region (M#1, M#2 and M#3), and two in the first star-forming arc (A1#1 and A1#2). Figure 6 shows the clump locations.
We used a 2D gaussian fit to disentangle the clumps and an underlying background component, and to calculate the integrated flux of each of them. For all tracers, the minimum Gaussian width was set by the resolution of the instrument, σ 2 min = (FWHM/2.35) 2 . The flux uncertainties were calculated using a Monte-Carlo method. Each map was perturbed 100 times based on the line fit uncertainties and for each iteration the integrated flux was calculated. Since the spatial resolution of SL and LL models are different, we treated these data differently. For tracers with spatial resolution better than 12 (SL-IRS maps: [ArII] 6.9 µm, [ArIII] 8.9 µm, [SIV] 10.5 µm and [NeII] 12.8 µm; PACS maps: [OIII] 88.4 µm and [NII] 121.9 µm), some degree of freedom on the position, (maximum) size, and asymmetry was allowed. The asymmetry of the Gaussian, described by the parameter R ab = σ a /σ b (with σ a and σ b measured in the two axes of the Gaussian), was constrained to be between 0.5 and 2. For the tracers with low spatial resolution (> 12 ; [SIII] 18.7 µm, [SIII] 33.5 µm and [NeIII] 15.5 µm) the clumps are not spatially resolved. In fact, we computed intrinsic dimensions of the clumps that were similar to or lower than 12 . Hence we fixed the width of the Gaussian to the resolution of the maps and the reference positions were fixed based on the Hα peaks (convolved to the Spitzer resolution).
For the analysis at this scale (∼25 pc) we use the absolute line fluxes of the SL tracers ( 12.8 µm (or their combination) can be used in principle to trace some model parameters, we decide not to use them because they involve lines from different modules/instruments and, most importantly, because our strategy relies on absolute fluxes whenever possible. The final fluxes are presented in Table 3.

Zones
We selected large areas of the galaxy (few hundreds pc), that are expected to exhibit different local physical conditions (Figure 6). This allows us to consider integrated emission to improve the signal-to-noise ratio (S/N) of faint areas. Some zones enclose the clumps described above (M and A1 ) and structures F. L. Polles: Modeling the ionised gas in IC10 visible in Hα (Arc2 (A2), Central (C), Diffuse2 (D2) and Dif-fuse3 (D3)) while the Diffuse1 (D1) zone was selected in order to examine the most diffuse phase possible. A single spectrum was produced by stacking spectra in the cube for all pixels of the zone. Line fluxes and errors were propagated through a Monte-Carlo simulation. For each Monte-Carlo iteration a zone spectrum was produced and one set of line fluxes was calculated. The fluxes and the associated errors were calculated using the median and the standard deviation of the flux distribution. For each PACS line, instead, we simply summed the line flux over the pixels in each zone, and propagated errors accordingly. The HIPE pipeline does not provide uncertainties on the measurements. Instead, empirical errors were calculated from the dispersion of the data cloud at every wavelength bin, and the line flux and errors were then inferred from each projected pixel (see . Table 4 presents the fluxes and errors measured for each zone. Figure 3 shows as an example the set of lines corresponding to the integrated M zone of ∼(174 pc × 226 pc). By increasing the spatial scale from clumps to zones, we are able to use in the analysis the absolute fluxes of all of the tracers, including the IRS LL tracers ([NeIII] 15.5 µm [SIII] 18.7 µm and [SIII] 33.5 µm).

Body (B)
The largest scale corresponds to most of the star-forming body of the galaxy, encompassing a region of 545 pc × 743 pc and includes most of the zones examined individually and described above. It is the largest region available in IC 10 with all of the Spitzer spectroscopic observations. The integrated fluxes of this area were calculated with the same method used for the other zones. We omit the PACS tracers from the analysis of the Body region, because the observations do not cover the full map.

Cloudy setup
To model the ionised gas, we use the spectral synthesis code Cloudy c13.03 (Ferland et al. 2013). Cloudy is a 1D spectral synthesis code which computes the physical and chemical structure and predicts the resulting spectrum of a region of gas and dust exposed to an ionising radiation field. Our calculation is a constant pressure model 2 . The code requires the following input parameters: 1. the shape of the source spectrum of the radiation field striking the cloud. We only consider a radiation field from a stel-2 Cloudy can compute the propagation of the radiation assuming constant density or constant pressure. This is important when the model is computed beyond the ionised phase. In constant pressure setup the density is adjusted such that the total pressure is constant throughout the cloud. Total pressure includes thermal pressure, turbulent, ram, and magnetic pressures and radiation pressure, both from the stellar continuum and internally generated light. lar population as a source, using the assumption of an instantaneous burst (age of the burst, t burst , which is varied); 2. the intensity of the input radiation field. In our case we use the dimensionless ionisation parameter (U, which is varied); 3. the hydrogen density at the illuminated face of the cloud (n H , which is varied); 4. the chemical composition and grain properties (which are fixed); 5. the cloud depth (which is varied).
The model parameters are described in detail below.

Shape of the source spectrum
We use the spectral synthesis code Starburst99 (Leitherer et al. 2010) to create the stellar ionizing continuum that serves as input for the Cloudy models. Specifically, we choose a Salpeter initial mass function (α = 2.35) with an upper mass limit, M up , of 100 M as done in López-Sánchez et al. (2011) and Padova asymptotic giant branch tracks with a metallicity of 0.008. We assume a single-burst star formation event. A continuous star formation rate may be envisaged, but the shape of the UV spectrum below 912 Å (i.e., for energies above 13.6 eV) is almost age-independent in this case, making this scenario difficult to test based on ionised gas tracers alone.
Several papers have studies the stellar population of IC10 (e.g. Sanna et al. 2009;Yin et al. 2010;Vacca et al. 2007). Motivated by these studies, the age of the cluster is varied within the range of 2.5 to 7 Myr (steps of 0.1 Myr between 2.5 to 3.5 Myr and between 5 to 6 Myr, and steps of 0.5 Myr between 3.5 to 5 Myr and between 6 to 7 Myr 3 ).

Intensity of the radiation field
We use the ionisation parameter U, to constrain the source brightness. U is the dimensionless ratio of the hydrogen-ionizing photons to total hydrogen density, characterising the intensity of the radiation field: where r 0 is the distance from the source to the inner edge of the cloud, n H is the total hydrogen density, c is the speed of light, Q(H) is the number of hydrogen-ionising photons emitted by the central source and Φ(H) is the surface flux of ionising photons. In our model, log U ranges from -4 to -1 with steps of 0.2.

Hydrogen density
The hydrogen density at the illuminated face of the cloud of our model grid covers the range of 10 -10 4 cm −3 with steps of 0.2 dex. In pressure equilibrium, which is the case of our models, the density in the H ii region stays almost constant, since the temperature remains stable.

Chemical composition
We set the elemental abundances of oxygen, nitrogen, neon, argon and sulphur to observed values, based on Magrini & Gonçalves (2009) and López-Sánchez et al. (2011). We adopt F. L. Polles: Modeling the ionised gas in IC10   (Figure 6). The dust grain properties of IC 10 are not well known. We chose to use the grain properties of the Small Magellanic Cloud. We adopt the grain size distribution presented in Weingartner & Draine (2001) which consists of graphites and silicates. The total dust abundance is scaled to the metallicity of IC 10.

Cloud depth and stopping criterion
We are interested in the ionised gas properties. The maximum depth into the model cloud that we consider corresponds to the ionisation front. The ionisation front is defined as the depth where the hydrogen ionisation fraction H + /H drops below 1%. Calculating the model until the ionisation front is equivalent to calculating a radiation-bounded H ii region, i.e. a cloud which is optically thick to ionising radiation from which no photons with energies above 13.6 eV will escape. Models that are stopped at lower depth correspond to matter-bounded H ii regions (sometimes called density-bounded) from which a significant fraction of the ionising photons escape. Figure 7 shows an example of the total hydrogen density and the ionisation fraction (n e /n H ) as a function of the depth. IC 10 shows significantly extended ionised gas emission and the regions investigated in this work are small compared to the extent of the ionised gas emission. Thus it is possible that a large fraction of the ionising photons escapes from the individual regions. In order to simulate matter-bounded as well as radiationbounded regions, we treat the depth of the cloud as a free parameter. In this study the normalised depth into the cloud is expressed as d/d IF where d is the depth at which the calculation is stopped and d IF is the ionisation front depth. If the region is radiation-bounded, the observed lines will be better reproduced at the ionisation front (cloud depth of 1). Otherwise, if the H ii region is matter-bounded, a better model can be found with a smaller depth. The depth at which the calculation is stopped is an important parameter for those lines with relatively low ionisation potentials, such as [ArII] 6.9 µm and [NII] 121.9 µm with ionisation potentials of 15.7 and 14.5 eV, respectively. This is because these lines are predominantly emitted in the outer parts of the H ii region (see Fig. 8). If some of the observed regions indeed have a significant escape fraction, the matter-bounded models will better reproduce the faintness of those lines. It should be kept in mind that the derived value of the depth corresponds to the depth for a single modeled cloud. In reality, multiple H ii regions may contribute to the observed emission lines. In that case depth represents an averaged property, which is related to the fraction of matter-vs. radiation-bounded clouds.

Optimization
In order to derive the physical properties of the ionised gas, we compare the fluxes calculated (Section 4.1) with the grid of models presented in Section 4.2.
We want to compare the observed line luminosities to the values predicted by Cloudy (similar to the method used by, e.g., Cormier et al. (2012); Dimaratos et al. (2015)). Since our Cloudy calculations are performed using intensities (erg cm −2 s −1 ; Section 4.2) we convert predicted intensities to absolute luminosities, which requires a scaling factor 4 , defined as the ratio 4 The scaling factor implies a conversion of intensities to absolute luminosities for the model and it is a free parameter because the absolute between the observed line luminosity, O, and the modeled line intensity, M. Instead of using a reference line that would be used to normalize the models to the observations, s global is an additional parameter in our models. In this way there is not a priori selection of the best line for the normalization. The correlated uncertainties of the tracers are accounted for by building a covariance matrix between all tracers. For each model, the goodness of the fit is calculated with the χ 2 as: where X is a N-dimensional vector (N is the total number of the observed lines used as constraints). For each line j, X j = O j − (M j * s global ). O j is the observed emission of line j, M j is the model-predicted emission and s global is the global scaling factor described above. V is the covariance matrix (N × N) with V i j = ρ i j σ i σ j , where σ i and σ j are the uncertainties related to the lines i and j, respectively, and ρ i j is the correlation coefficient between σ i and σ j . The uncertainties taken into account to calculate the matrix are the calibration uncertainties (5% for IRS and 12% for PACS), the uncertainties on the line fit (Table 3 and Table 4), the uncertainties on the elemental abundances (Table 5), and the uncertainties due to the SL/LL stitching (15% for IRS LL; Section 3.1). We populate the covariance terms (uncertainties and correlation coefficient) using a Monte Carlo simulation with 10 6 iterations. The best model is found by minimizing the χ 2 value. In order to understand the relationship and possible degeneracy between free parameters, we calculate the probability density function (PDF), which, for each physical parameter p = (p 1 , ..., p n ), is calculated as: where the sum is performed over all of the j-models for which the free parameter p is equal to p1, p2,...,p n . The PDF is normalised by PDF N bin , where N bin is the number of the bins. With this normalisation, the PDFs can be easily compared even if they have a different number of bins.
We compute the PDFs of each free parameter individually as well as the 2D PDFs for each pair of parameters, in order to highlight potential degeneracies between parameters in the parameter space. We also compute the histograms of each parameter for the best models only. The best models are those with a χ 2 value lower than χ 2 min + ∆χ 2 , where ∆χ 2 = 5.89 is the 1σ confidence interval with five free parameters (Press et al. 1992), i.e., n H , U, t burst , depth and s. luminosities of the ionizing sources are not known. Ideally, we would like to use the observed line-to-bolometric luminosity ratio as a constraint in the models, but this proved difficult since we only had access to the infrared luminosity with the Herschel and Spitzer photometry measurements and not to the bolometric luminosity. We should convert the infrared luminosity to the bolometric luminosity, but since we are studying resolved star-forming regions and zones we do not have enough informations to provide an infrared-to-bolometric conversion factor. F. L. Polles: Modeling the ionised gas in IC10

Results
The best model solutions for each clump are listed in Table 6. Most clumps have nearly identical properties except A1#1, which has lower density and higher ionisation parameter, but with larger error bars. The density, n H , of the clumps is well constrained, with values between 100 and 400 cm −3 . As a separate approach we use the theoretical ratio of [SIII] 33.5 µm/[SIII] 18.7 µm as a tracer of the density. The high spectral resolution IRS pointings give us access to the [SIII] line ratio at higher resolution (e.g. Dudik et al. 2007). Since the high-resolution observations are pointings with relatively small apertures, there is no spatial information. Hence, we can use these observations (Appendix Table A.1) only for clumps that coincide with the positions of the high-resolution pointings: M#3, A1#1, and A1#2. Using the calculation of a 2level system, as opposed to a full model, we find densities of 10 2.1 , 10 2.5 and 10 2.5 cm −3 , respectively. These results are compatible, within the uncertainties, with the values that we found with the model grids.
The physical depth is lower than 1 (between 0.55 and 0.90), which can be interpreted either as one cloud around the stellar cluster being matter-bounded, or some fraction of the clouds around the stellar cluster being matter-bounded and some fraction being radiation-bounded. However, this parameter shows large uncertainties. The depth is mostly constrained by [ArII] 6.9 µm and [NII] 121.9 µm, and both lines have large error bars compared to most of the other tracers. For all of the clumps, the [ArII] 6.9 µm S/N is overall lower than 2 while [NII] 121.9 µm is not only faint but it also carries a relatively large calibration uncertainty (12%) because it is observed with PACS, unlike most of the other lines observed with IRS (5% calibration uncertainty).
The derived age of the burst and the ionisation parameter are rather homogeneous across the galaxy, with ages ranging from 5.3 to 5.7 Myr and ionisation parameters from 10 −2.2 and 10 −1.6 . These two parameters also show large uncertainties.
We calculate the PDFs and the histograms of the best models to estimate uncertainties. Figure 9 shows the results for the clump M#1. The PDFs and the histograms of the other clumps are shown in Figures B.1, B.2, B.3 and B.4. Only the density seems to be tightly constrained by our tracers. All the other parameters do not have a well-defined solution. For a given clump, many models have similar χ 2 values, close to the minimum χ 2 . Hence the PDFs are almost flat and the histograms include many models, providing parameter values distributed over the entire parameter range used in the model grid. Thus, we cannot provide a useful error estimate. The large uncertainties on the ionisation parameter and the starburst age are related to the degeneracy between these two parameters, which can be clearly seen in the 2D PDF parameters. A model with a young stellar population combined with a low ionisation parameter produces a similar spectrum as a model with older starburst and a high ionisation parameter (Morisset et al. 2016).

Constraining the age range
The PDFs of t burst show two peaks at ≈3 and ≈5.5 Myr. Since we do not have enough information on the age of the stellar population, we explored the effect of restricting the starburst age to either one of the two peaks, in order to witness the effects on the PDFs of the other parameters. Figure 10 shows the results for the clump M#1. The figure shows the PDFs when constraining t burst between 2.8 and 3.4 Myr (i.e., the WR stage), and the PDFs for constraining t burst between 5.2 and 5.8 Myr (i.e., the typical H ii region age for an instantaneous burst hypothesis; Section 4.2). The comparisons for the other clumps are shown in Figures B.5, B.6, B.7 and B.8. As expected the PDFs for the ionisation parameter change. This is due to the well known degeneracy described above. The PDFs for the physical depth also change, which can be easily understood since the ionisation parameter, the starburst age, and the physical depth all depend on the relative intensity of high-vs. low-ionisation tracers. In the case of the physical depth, this is because higher ionisation species are located closer to the ionizing sources. The density parameter, instead, is not affected by the range of starburst age.
For all of the clumps, the starburst age around 5.5 Myr is preferred by the models (higher probability), which is consistent with the model solution for the zones (Section 5.2). The subset of the models with an age around 5.5 Myr (Figures 10) corresponds to relatively low physical depth parameter for the clumps (0.7 -0.9), U between 10 −2.5 and 10 −1 , and density consistently around ≥ 250 cm −3 , except for the clump A1#1 with a lower density around 100 cm −3 . Hence, these experiments show that well constrained solutions can be found if the degeneracy between U, the starburst age, and the physical depth parameters can be lifted, for instance by forcing the age around 5.5 Myr.
The best age solution equal to 5.5 Myr for all of the clumps (as well as for the zones) is probably driven by the detection of appreciable amounts of highly-ionised species such as S 3+ together with the hypothesis of an instantaneous burst. Such highly-ionised species are produced by energetic photons from short-lived stars ( 6 Myr). Therefore, for ages much larger than ≈ 6 Myr, we should not be able to detect [SIV] (or [NeIII]). A continuous SF could also be considered as an alternative to the instantaneous burst, but with our tracers we cannot investigate this case (Section 4.2.1).
The H ii regions that we have analyzed are all distributed along the edge of a large H i and Hα hole. This hole is probably the result of the combined effect of stellar winds or supernovae over several Myrs. Wilcots & Miller (1998) calculated that the expanding bubble shell ought to become gravitationally unstable within 10 7 yr, implying that a second SF event, after the one that created the hole, is likely to be triggered. Hence, the young stellar population that we found with our modeling could be the second SF event triggered by stellar winds of the first stellar generation. An additional support of this scenario is given by the identification of several giant molecular clouds (GMCs) around the H i bubbles throughout IC 10 (Leroy et al. 2006). This is reminiscent of studies of Milky Way GMCs, which are formed at the overlapping interface between several Galactic H i supershells produced by previous episodes of stellar feedback, such as stellar winds or supernovae (Dawson et al. 2015;Inutsuka et al. 2015). Hence, the age found is consistent with a scenario of overlapping interfaces between several H i supershells that can trigger new episodes of SF and explain the location of GMCs in IC 10. However, this scheme is in contradiction with others studies that found that stellar feedback are not the dominant trigger of molecular cloud formation (e.g. Dawson et al. 2013). Another scenario that could explain this young stellar population is the interaction between the extended H i envelope and the galaxy. The gas infalling from the large reservoir may have driven the present star formation in the galaxy (Wilcots & Miller 1998).

Comparison with previous optical studies
The properties of the H ii regions have been already investigated in previous studies using optical lines. It is interesting to compare our results with previous results. However, the comparison between our results and those obtained with optical spectroscopy should be regarded with caution. In fact, infrared lines probe deeper within the galaxy or within individual clouds than optical lines, possibly reaching regions with different physical conditions (Section 6.3).
López-Sánchez et al. (2011) analysed a region that corresponds to M#1. For the whole area, they derived densities between 100 and 400 cm −3 and stellar population ∼ 3.3 Myr. This bright H ii region has been studied also by Arkhipova et al. (2011), which investigated the properties of the brightest H ii regions of IC 10. Combining optical line ratios with Cloudy models, they found densities between 30 to 200 cm −3 , stellar age between 2.5 to 5 Myr and an ionisation parameter between 10 −3.6 to 10 −2.5 .
Both optical studies found a stellar age ∼3 Myr. These solutions for stellar ages correspond to one of our PDF peaks we find for stellar age (Figures 9, and from B.1 to B.4). However, it does not correspond to our best solutions which have older stellar ages (∼5.5 Myr), higher U (10 −2.2 -10 −1 ) and are matterbounded clouds. The smaller ages derived in the optical study also have consequences in that they derive lower U due to the stellar age-U degeneracy already discussed (Section 5.1.2). A younger stellar age would require a lower U value. Regarding the density, our results are in agreement with the densities derived by López-Sánchez et al. (2011) and with the highest values obtained by Arkhipova et al. (2011). Table 7 summarizes the model solutions with 1σ uncertainties for the zones and the Body. Note that the last column of Table 7 provides the reduced χ 2 for those zones that have more constraints than the number of free parameters (which was not the case for the clump analysis). Overall, the physical depth parameter is larger for the zones, between 0.8 to 1, compared to the individual clumps (0.7 -0.9). As the volume increases, a larger fraction of ionizing photons is absorbed by the gas and the ionised gas component becomes globally almost radiationbounded. Interestingly, the depth of the largest area analyzed, B, is clearly below 1 and peaks at 0.85, which implies that even on the scale of the entire body, a significant fraction of ionizing radiation may escape. One would need the observations of the MIR and FIR ionised gas lines over the entire IC 10 galaxy scale to infer if ionizing photons manage to escape the galaxy outside its large H i halo extension. The stellar age we determine is almost the same for most of the zones, ≈5.5 Myr, and it coincides with the most likely age of the stellar population of the clumps. Similar U and n H are found for the integrated M zone and the star-forming clumps M#1, M#2 and M#3 which are the brightest components in M (Table 6 and 7). The same conclusion is reached for the arc zone A1 and the clumps A1#1 and A1#2, and for the zone B dominated by the bright regions associated with SF. This suggests that the properties estimated at the ∼ 200 pc scale (zones) are dominated by the brightest compact clumps (∼ 25 pc scales).

Model results for zones and the Body
The results concerning the ionisation parameter and the density highlight important differences between the regions in IC 10. Indeed, the zone M has log U = -1.4 and n H = 10 2.2 cm −3 , while the zone A2 has log U = -2.4 and n H = 10 2.0 cm −3 . These differences could be due to more stars or to the fact that SF might be embedded in denser and more compact regions.
As for the clump analysis, we show in Figure 12 the PDFs and histograms of the best models (defined by the minimum χ 2 ) of the zone M. Each parameter shows a well-defined peak in the PDF and the histograms of the best models occupy narrow ranges, implying that reliable values have been determined. This shows that with a suite of well-detected lines from different species with a range of critical densities and ionisation potentials, and with an adequate method for finding the best model, one can derive several of the physical parameters at once and break the degeneracies, in particular between the starburst age and the ionisation parameter. Even the depth parameter is well constrained (≈ 0.80) due to the robust detection of [ArII] when integrating over the zone M.
The PDFs and the histograms for the other zones are presented in the Appendix (Figures C.1 to C.7). One can see that depending on the S/N of the tracers used, it becomes impossible to constrain some parameters. This is particularly evident for the physical depth, due to lack of sufficient S/N of [ArII] as well as the density, since the [SIII] 33.5 µm line also has low S/N. The zone C is a star-forming region fainter than zones M or A1 and it was not observed in the [OIII] 88.4 µm line. The ionisation parameter in the zone C is lower than in zone M and the best model of the zone C prefers high depth values (> 0.85), however the large uncertainty of the [ArII] 6.9 µm line prevents a reliable determination of the physical depth for this zone. The zone D1 is the region with the lowest surface brightness and [ArII] 6.9 µm, [SIV] 10.5 µm, and [SIII] 18.7 µm were not detected. There were not enough constraints to solve for the free parameters. The low S/N of [ArII] 6.9 µm has a direct impact on the depth determination, while the low S/N of [SIII] 18.7 µm has a direct impact on the density determination. Even the starburst age or the ionisation parameter cannot be determined, which is in part due to the non-detection of [SIV] 10.5 µm. For zone B (the main body of the galaxy) all of the tracers are available except [OIII] 88.4 µm, whose spatial coverage over the main body is incomplete. Results are shown in Figure C.7. The parameters are well constrained, with a density around 100 cm −3 , a large ionisation parameter (log U> -2) and depth around 0.85.

Comparison between observations and best predicted emission
We compare the observed and predicted absolute line fluxes to quantify how observations are reproduced by our model. Ta IC10   Table 7. Results of the best-model parameters with the relative bounds of 1σ uncertainties, the absolute minimum χ 2 and the number of line ratios available to constrain the solution, for each zone and Body areas.  µm and [NII] 121.9 µm originate from more diffuse ionised gas due to their low critical densities (5×10 2 cm −3 and 3×10 2 cm −3 , respectively). The best models tend to fit the dense component because most of the lines that we use to constrain the solutions arise from a dense component and those tracers of the diffuse ionised gas have a larger calibration uncertainty. They thus carry less weight than the other lines in the calculation of the χ 2 . Unfortunately we do not have enough tracers to constrain a model that combines both components. We also notice some systematic biases: [ArII] 6.9 µm and [ArIII] 8.9 µm are always overestimated by the models by a factor of about two. This result may indicate that the assumed argon elemental abundance in the models is too high, which seems reasonable considering the large uncertainty on the argon abundance (Table 5).
We compute the PDFs of the constraints (observation/model) also for the zones. Figure 13 shows the PDFs for the zone M. Although the model parameters are well constrained, we can see that some lines, such as [NeII] and [OIII] are under-predicted by a factor of 3 -4. Thus we argue that these two lines are likely to arise in the relatively diffuse ionised gas ([NeII] because Ne + exists for a large range of relatively low photon energies: 22-41eV).
Summarising, for the clumps as well as for the zones, most of the lines are reproduced within a factor of 2. The cases for which the predicted fluxes are several factors away from the observed value would need to be investigated, but we tentatively describe the discrepancy due to the mix of the diffuse and dense ionised gas. While the denser phase related to the young clusters seems to be well constrained by a model with a single component at any spatial scale, such a model misses and ignores other phases that may be important.

Porosity of the ISM
Our models consider a single central ionizing source fully surrounded by a ionised gas cloud (Section 4.2). In reality, multiple  clouds may contribute to the observed emission-lines, potentially not covering all of the lines of sight from the ionizing source (i.e., some radiation may escape). These clouds may be a combination of matter-and radiation-bounded clouds. Therefore, as for other parameters, the physical depth parameter represents some kind of average property. Still, our results for the clump depth, between 0.75 to 0.90, imply that a significant fraction of ionizing photons is able to escape the clumps. Such results are compatible with the investigation performed by Hidalgo-Gámez (2005), that shows that the photons leaking from H ii regions, in addition to the ionisation provided by the WR stars, are responsable for the extended diffuse ionised gas in IC 10. This has implication for the PDRs, which are the neutral zones bordering only the radiationbounded clouds. Thus, the estimated fraction of escaping ionizing photons may be used to infer a covering factor of radiationbounded clouds around the ionizing source, which, in turn, sets important constraints when comparing PDR model predictions with the observations. The depth parameter is larger when considering larger zones. However, some zones still require a depth parameter less than 1. This implies that the larger zones are getting close to a radiation-bounded case, but not fully. The fact that the ISM is porous over large spatial scales is related to the abundance of dust in the galaxy, and therefore to the metallicity. Several studies show that the ISM at low metallicity is relatively more porous than more metalrich environments (e.g. Madden et al. 2006;Kawada et al. 2011;Cormier et al. 2015). IC 10 shows extended H i structures, and it is likely that the fraction of ionizing photons escaping IC 10 it is close to zero when considering the entire galaxy. Direct measurements of the Lyman continuum in external galaxies suggest a small fraction of escaping ionizing photons (e.g. Cowie et al. 2009;Bridge et al. 2010;Leitet et al. 2013).
The importance of the ISM porosity at low metallicity is also suggested by the extended spatial distribution of [OIII] 88.4 µm line, which traces the diffuse ionised gas. It is shown to extend over large spatial scales in nearby giant H ii regions in the Magellanic Clouds Kawada et al. 2011). IC 10 also shows an extended [OIII] 88.4 µm line emission throughout the main body of the galaxy ( Figure 5). Another observational tracer of the porosity is the ratio [OIII]/[CII], that mostly measures the amount of PDR vs the amount of low density ionised gas. It is observed that [OIII] 88.4 µm is the brightest infrared line in low-metallicity galaxies (while [CII] is relatively brighter in more metal-rich environments, Cormier et al. 2015). In IC 10 the [OIII]/[CII] ratio varies from 0.6 toward the more diffuse regions up to 4.5 toward the brightest H ii regions. These characteristics of [OIII] 88.4 µm line emission in low metallicity environments suggest a high filling factor of diffuse highly-ionised gas due to the lower abundance of dust which, in turn, can cause enhanced disruption of the natal molecular cloud resulting in a lower covering factor of dense clouds around young stellar clusters.
The depth parameter and the [OIII]/[CII] ratio are two signs of the ISM porosity. The way we have introduced the depth parameter in the model suggests a promising avenue for quantifying the ISM porosity. Understanding how the porosity changes as a function of metallicity requires applying this method developed for IC 10 to other spatially-resolved objects

Origin of [CII], [FeII] & [SiII]
The C + , Fe + and Si + ions have ionisation potentials lower than that of hydrogen (11.3 eV, 7.9 eV and 8.2 eV, respectively), while the potential of the next ionisation stage is above 13.6 eV. Therefore, the origin of these lines is ambiguous. They may arise from the neutral and from the ionised gas. In the neutral gas, the collision partners are mostly H 0 , H 2 , and free electrons (coming from the ionisation of species with ionisation potentials below 13.6 eV or from cosmic ray and soft X-ray photoionisation of H). In the ionised gas, the collision partners are e − coming from the ionisation of H by Lyman Continuum photons. Observationally, [CII] 157.7 µm is found to be a strong emission line originating in the surface layers of PDRs illuminated by the radiation field of massive stars (e.g. Negishi et al. 2001), but it can also arise from the ionised gas phase (Madden et al. 1993;Abel et al. 2005;Cormier et al. 2012).
For 34.8 µm emission over the best model solution is at the low end cutoff of the PDF, preventing a definite conclusion on the origin of the observed [SiII] for this zone. However, as discussed in Section 5.2, there is no satisfactory solution for the zone D1 so the "best" solution considered should be regarded with caution. Hence, using the model solutions to compare to the observed [CII], [SiII], and [FeII], we find that these lines seem to originate either in the PDR or in a potentially diffuse ionised gas component (which is not accounted for in the models).
Alternatively [NII] 121.9 µm can be used to estimate the [CII] 157.7 µm coming from the ionised gas. With an ionisation potential of 14.5 eV it originates only in the ionised phase. Knowing the elemental abundance and knowing the ionisation fractions C + /C and N + /N, the theoretical ratio [CII]/[NII] 122µm is then a function of gas density and temperature (Heiles 1994;Abel 2006;Oberst et al. 2006). Temperature plays a minor role and the ratio mostly depends on density. We have calculated the observed [CII]/[NII] pixel-by-pixel (for each pixel with good S/N) and we have found that only 1/10 of the observed [CII] can be explained to originate in the ionised gas, for any density. The difference is due to the presence of [CII] in the neutral gas.
We conclude that [CII] does not arise from H ii regions nor from a diffuse ionised gas phase but it arises from neutral gas/PDRs. [CII] can therefore be used with little or no correction for the future PDR modeling effort.

Extinction
IC 10 lies close to the Galactic plane (Galactic latitude b = -3.3 • ), which implies a significant foreground reddening in addition to the internal reddening.
Assuming the models are well constrained by the suite of infrared lines, we can compare the predicted intrinsic Hα emission to the observed value (not corrected for extinction; Section 3) and infer the extinction. The relation between the observed and the emitted Hα, assuming screen extinction, is given by the rela-tion: Hα obs = Hα mod e −τ (Hα) (4) where Hα obs and Hα mod are the observed and the predicted emission, respectively, and τ(Hα) is the optical depth, τ(Hα) = C×k(Hα). The k(Hα) is the opacity curve used in Cloudy (grain properties of the Small Magellanic Cloud from Weingartner & Draine 2001) and C is the reddening factor. Knowing C (thus τ(Hα)), then the extinction is A Hα = 1.086 τ(Hα). Finally, using the reddening relation A Hα = 3.40 5 E(B − V) , we calculate the reddening. The values obtained with this method are presented in Table 10. We estimate for the clumps E(B-V) between 1.6 to 2.0 mag. For an independent estimate, we also calculated the extinction using the HI recombination line Huα 12.3 µm observed with the Spitzer/IRS high-resolution pointings (Table A.1). From Hummer & Storey (1987) we estimated the theoretical ratio Hα/Huα expected for case B recombination, assuming a temperature 10 000 K and density of 100 cm −3 , resulting in a value of ≈294. We thus compute τ(Hα) using the relation: Hα obs Huα obs = Hα pred Huα pred × 10 −0.434 6 (τ(Hα)−τ(Huα)) where both τ(Hα) and τ(Huα) are estimated using the opacity curve of Cloudy. Then we calculate the E(B-V) values, following the procedure described before. The values are reported in Table 10. We estimate E(B-V) ∼2 mag, i.e., quite similar to the values obtained from the Cloudy model predictions. Both methods provide high extinction values compared to previous works. This is not surprising since in our calculations we are using MIR and FIR lines to predict the emitted unreddened Hα (either the metallic species used for the Cloudy predictions or the HI recombination line Huα), which allow us to probe deeper into dusty regions. However, the fact that we estimate consistently large values, even from clumps that are located far away from each other, suggests that the extinction is due to a uniform absorption component rather than from the star-forming region. Previous studies report reddening values between 0.7 mag to 1.0 mag, depending on the method used. Using Cepheid variable stars, for example, Sakai et al. (1999) calculated E(B-V) = 1.16 mag, while Massey & Armandroff (1995) estimated E(B-V) = 0.75 -0.80 mag based on WR stars and the blue stellar population. Table 10 summarizes the extinction values from the literature 7 . These measurements correspond to various regions and various spatial scales, so the comparison with our model estimates toward the clumps is not trivial. Borissova et al. (2000) investigated several regions used in our study (Table 10). The values we find with our methods are similar to those measured by Borissova et al. (2000) using the NIR/optical line ratio Brγ/Hα. Both studies used IR/optical line ratio to estimate E(B-V), probing deeper into the dust than the methods based on optical observations alone.
The measurement of the visual extinction seen by the gas enables the potential use of many optical lines as constraints to the models. These new constraints complement the infrared tracers, by giving access to lines of different species, with different critical densities. Some remaining degeneracies between physical parameters may be broken using this combination.

Summary
We have presented the Spitzer/IRS and Herschel/PACS spectroscopic observations of the infrared cooling lines tracing the ionised gas in the nearby irregular dwarf galaxy IC 10. The proximity of this galaxy allows us to investigate the multi-phase ISM on different spatial scales. We have focused our investigation on the nature of ionizing sources as well as the physical properties of the ionised gas, based on the observational constraints and the Cloudy modeling solutions. The main results are the following: -We have modeled the brightest H ii regions in MIR and FIR emission lines. Three of these regions are located on the main star-forming regions of the galaxy (M) and two regions on the first arc (A1). We found t burst between 5.3 and 5.7 Myr, U between 10 −2.2 and 10 −1 , density between 10 2 to 10 2.6 cm −3 and depth between 0.55 to 0.9. Thus, the physical properties of the clumps (t burst , log U and n H ) are quite uniform, suggesting a common origin for their SF activity. The origin of the ionizing sources in the H ii regions analyzed in this study, could be related to the feedback from stellar winds or supernovae of a previous generation of stars. -Using [ArII] 6.9 µm and [NII] 121.9 µm we determined that the clumps are matter-bounded clouds, with a significant fraction of ionizing photons escaping the nebula. Solutions for larger spatial scales suggest that the clouds are almost radiation-bounded. The matter-bounded nature at almost any spatial scale indicates that the ISM is quite porous, which is possibly due to the low metallicity of the environment. -In the case of the clumps, the method seems to fail in constraining well the model parameters and we had to include additional constraints ([SIII] line ratio or fixing the age of the starburst) to narrow down the range of parameters. However, when all of the tracers are available and the S/N is sufficient, the method provides satisfactory solutions with reasonable reduced χ 2 values, with few biases and with the possibility of breaking degeneracies between parameters. -We have estimated the total extinction in the clumps with two different methods: 1) modeling solution, comparing the observed Hα with that predicted by the model, and 2) using the ratio of Hα(3-2)/Huα(7-6). The values obtained with the two methods are similar, with E(B-V) between 1.6 to 2 mag. -Modeling larger areas (∼200 pc; zones), which were expected to exhibit different local physical conditions, we found that overall, the physical depth parameter is larger for the zones, between 0.8 to 1, than for the clumps. The estimated stellar age is almost the same for most of the zones, ∼5.5 Myr, the results are in agreement with the stellar age determined for the clumps. Instead, the results concerning the ionisation parameter and the density highlight differences between the regions in IC 10. -Modeling the zones with a single component model, we found that similar physical conditions are obtained for various spatial scales as long as bright components dominate the emission in the integrated zone. -The comparison between the observations and the best predicted emission reveals that at all of the spatial scales analysed, a single component model is not enough to reproduce all of the tracers. Indeed, another diffuse ionised gas component may be necessary. -We have investigated the origin of the [CII] 157.7 µm, [FeII] 25.9 µm and [SiII] 34.8 µm emission to determine the fraction of the emission arising from the ionised gas phase. We found that most of the emission of the [CII], [FeII], and [SiII] arises in the PDR component.
A&A proofs: manuscript no. IC10 Fig. 9. Results for the clump M#1. Top: PDFs for n H , log U, t burst (Age), and physical depth. The blue histograms show the PDF for each single parameter. The 2D PDFs are shown as density plots (black ∼1, to white ∼0) at the intersection between two parameters. The blue horizontal and vertical lines indicate the parameter value corresponding to the best model solution, with the corresponding parameter value given in blue. Bottom: Histograms of parameter distributions for the best models (see text for details).
Article number, page 18 of 48 F. L. Polles: Modeling the ionised gas in IC10 Fig. 10. PDFs for the clump M#1. The blue histograms show the PDFs from Figure 9, with the best model (χ 2 min ) shown by the vertical blue line and the best model parameter values given by the blue number on the top left corner. The PDFs overplotted in red are for the subset of the models with t burst (Age) constrained between 2.8-3.4 Myr (top) and between 5.2-5.8 Myr (bottom). For each subset PDF, the vertical red line and the red number on the right corner indicate the best model (χ 2 min,sel ) in that particular subset.