© ESO, 2014

1. Introduction

With its high dipolar moment, extremely rich spectrum, and high level spacing (in comparison to those of other molecules with low-lying transitions at millimeter wavelengths), H₂O couples very well to the radiation field in warm regions that emit strongly in the far-IR. In extragalactic sources, excited lines of H₂O at far-IR wavelengths (λ< 200 μm) were detected in absorption with the Infrared Space Telescope (ISO; Fischer et al. 1999; González-Alfonso et al. 2004, 2008), and with Herschel/PACS (Pilbratt et al. 2010; Poglitsch et al. 2010) in Mrk 231 (Fischer et al. 2010), Arp 220 and NGC 4418 (González-Alfonso et al. 2012, G-A12). Modeling and analysis have demonstrated the ability of H₂O to be efficiently excited through absorption of far-IR dust-emitted photons, thus providing a powerful method for studying the strength of the far-IR field in compact/warm regions that are not spatially resolved at far-IR wavelengths with current (or foreseen) technology.

Herschel/SPIRE (Griffin et al. 2010) has enabled the observation of H₂O at submillimeter (hereafter submm, λ > 200 μm) wavelengths in local sources, where the excited (i.e., non-ground-state) lines are invariably seen in emission. In Mrk 231, lines with E_upper up to 640 K were detected (van der Werf et al. 2010; González-Alfonso et al. 2010, hereafter G-A10), with strengths comparable to the CO lines. The H₂O lines have been also detected in other local sources (Rangwala et al. 2011; Pereira-Santaella et al. 2013), including the Seyfert 2 galaxy NGC 1068 (Spinoglio et al. 2012, S12). Furthermore, submm lines of H₂O have been detected in a dozen of high-z sources (Impellizzeri et al. 2008; Omont et al. 2011; Lis et al. 2011; van der Werf et al. 2011; Bradford et al. 2011; Combes et al. 2012; Lupu et al. 2012; Bothwell et al. 2013), even in a z = 6.34 galaxy (Riechers et al. 2013). Recently, a striking correlation has been found between the submm H₂O luminosity (L_H2O), taken from the 2₀₂−1₁₁ and 2₁₁−2₀₂ lines, and the IR luminosity (L_IR), including both local and high-z ULIRGs (Omont et al. 2013, hereafter O13). Using SPIRE spectroscopy of local IR-bright galaxies and published data from high-z sources, the linear correlations between L_H2O and L_IR for five of the strongest lines, extending over more than three orders of magnitude in IR luminosity, has recently been confirmed (Yang et al. 2013, hereafter Y13). There are hints of an increase in L_H2O that is slightly faster than linear with L_IR in some lines (2₁₁−2₀₂ and 2₂₀−2₁₁ ) and in high-z ULIRGs (O13). HCN is another key species that also shows a tight correlation with the IR luminosity, even though the excitation of the 1−0 transition is dominated by collisions with dense H₂ (Gao & Solomon 2004a,b).

The increasing wealth of observations of H₂O at submm wavelengths in both local and high-z sources and the correlations discovered between L_H2O and L_IR require a more extended analysis in parameter space than the one given in G-A10 for Mrk 231. In this work, models are presented to constrain the physical and chemical conditions in the submm H₂O emitting regions in warm (U)LIRGs and to propose a general framework for interpreting the H₂O submm emission in extragalactic sources.

2. Excitation overview

At submm wavelengths, H₂O responds to far-IR excitation by emitting photons through a cascade proccess. This is illustrated in Fig. 1, where four far-IR pumping lines (at 101, 75, 58, and 45 μm) account for the radiative excitation of the submm lines (G-A10). The line parameters are listed in Table 1, where we use the numerals 1−8 to denote the submm lines. Lines 2−4, 5−6, 7, and 8 are pumped through the 101¹, 75, 58, and 45 μm far-IR transitions, respectively.

The ground-state line 1 has no analog pumping mechanism, so that the upper 1₁₁ level can only be excited through absorption of a photon in the same transition (at 269 μm) or through a collisional event. In the absence of significant collisional excitation, and if approximate spherical symmetry holds, line 1 will give negligible absorption or emission above the continuum (regardless of line opacity) if the continuum opacity at 269 μm is low or will be detected in absorption for significant 269 μm continuum opacities². This is supported by the SPIRE spectrum of Arp 220, in which line 1 is observed in absorption (Rangwala et al. 2011) and high submm continuum opacities are inferred (González-Alfonso et al. 2004; Downes & Eckart 2007; Sakamoto et al. 2008). Collisional excitation and thus high densities and gas temperatures are then expected in sources where line 1 is detected in emission (10 sources among 176, Y13), as in NGC 1068 (S12; see also Appendix A). Line 1 can then be collisionally excited in regions where the other lines do not emit owing to weak far-IR continuum; this effect has recently been observed in the intergalactic filament in the Stephan’s Quintet (Appleton et al. 2013).

Fig. 1 Energy level diagram of H2O, showing the relevant H2O lines at submillimeter wavelengths with blue arrows, and the far-IR H2O pumping (absorption) lines with dashed-magenta arrows. The lines are numbered as listed in Table 1. The o-H2O 312−221 transition is not considered due to blending with CO (10-9) (G-A10), and the far-IR 212−101 transition at 179.5 μm discussed in the text is also indicated in green for completeness.

If collisional excitation of the 1₁₁ and 2₁₂ levels dominates over absorption of dust photons at 269 and 179 μm (i.e., in very optically thin and/or high density sources), the submm H₂O lines 2−6 will be boosted because these 1₁₁ and 2₁₂ levels are the base levels from which the 101 and 75 μm radiative pumping cycles operate (Fig. 1). In addition, in regions of low continuum opacities but warm gas, collisional excitation of the para-H₂O level 2₀₂ from the ground 0₀₀ state can significantly enhance the emission of line 2. Therefore, the H₂O submm emission depends in general on both the far-IR radiation density in the emitting region and the possible collisional excitation of the low-lying levels (1₁₁, 2₁₂, and 2₀₂). Lines 7−8 require strong far-IR radiation density not only at 58−45 μm, but also at longer wavelengths, together with high H₂O column densities (N_H2O) in order to significantly populate the lower backbone 3₁₃ and 4₁₄ levels.

3. Description of the models

The basic models for H₂O were described in G-A10 (see also references therein). Summarizing, we assume a simple spherically symmetric source with uniform physical properties (T_dust, T_gas , gas and dust densities, H₂O abundance), where gas and dust are assumed to be mixed. We only consider the far-IR radiation field generated within the modeled source, ignoring the effect of external fields (except for NGC 1068, Appendix A). The source is divided into a set of spherical shells where the statistical equilibrium level populations are calculated. The models are non-local, including line and continuum opacity effects. We assume an H₂O ortho-to-para ratio of 3. Line broadening is simulated by including a microturbulent velocity (V_turb), for which the FWHM velocity dispersion is ΔV = 1.67V_turb. No systemic motions are included.

3.1. Mass absorption coefficient of dust

The black curve in Fig. 2 shows the dust mass opacity coefficient used in the current and our past models (González-Alfonso et al. 2008, 2010, 2012, 2013, 2014). Our values at 125 and 850 μm are κ₁₂₅ = 30 cm² g^-1 and κ₈₅₀ = 0.7 cm² g^-1, in good agreement with those derived by Dunne et al. (2003). Adopting a gas-to-dust ratio of X = 100 by mass, and using κ₁₀₀ = 44.5 cm² g^-1, the column density of H nuclei is ${\mathit{N}}_{\mathrm{H}}\mathrm{=}\frac{\mathit{X}\hspace{0.17em}{\mathit{\tau }}_{\mathrm{100}}}{{\mathit{m}}_{\mathrm{H}}\hspace{0.17em}{\mathit{\kappa }}_{\mathrm{100}}}\mathrm{=}\mathrm{1.3}\mathrm{\times }{\mathrm{10}}^{\mathrm{24}}\hspace{0.17em}{\mathit{\tau }}_{\mathrm{100}}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{-2}\mathit{,}$(1)where τ₁₀₀ is the continuum optical depth at 100 μm.

For this adopted dust composition, the fit across the far-IR to submm (blue line in Fig. 2) indicates an emissivity index of β = 1.85, slightly steeper than the β = 1.5−1.6 values favored by Kóvacs et al. (2010) and Casey (2012). The H₂O excitation is sensitive to the dust emission over a range of wavelengths (from 45 to 270 μm), but we find that our results on L_H2O/L_IR are insensitive to β for β values above 1.5 (Sect. 4.3.3).

Fig. 2 Adopted mass absorption coefficient of dust as a function of wavelength. The dust emission is simulated by using a mixture of silicate and amorphous carbon grains with optical constants from Draine (1985) and Preibisch et al. (1993). As shown by the fitted blue line, the emissivity index from the far-IR to millimeter wavelengths is β = 1.85.

3.2. Model parameters

As listed in Table 2, the model parameters we have chosen to characterize the physical conditions in the emitting regions are T_dust , the continuum optical depth at 100 μm along a radial path (τ₁₀₀), the corresponding H₂O column density per unit of velocity interval (N_H2O/ΔV), the velocity dispersion ΔV, T_gas , and the H₂ density (n_H2). Fiducial numbers for some of these parameters are τ₁₀₀ = 0.1, ΔV = 100 kms^-1, T_gas = 150 K, and n_H2 = 3 × 10⁵ cm^-3. Collisional rates with H₂ were taken from Dubernet et al. (2009) and Daniel et al. (2011). Our relevant results are the line-flux ratios (F_i/F_j) and the luminosity ratios³L_H2O/L_IR. We also explore models where collisions are ignored, appropriate for low-density regions (n_H2 ≲ 10⁴ cm^-3), for which F_i/F_j only depend on T_dust , τ₁₀₀, and N_H2O/ΔV, while L_H2O/L_IR depends in addition on ΔV.

Depending on the values of the above parameters, our models can be interpreted in terms of a single source or are better applied to each of an ensemble of clouds within a clumpy distribution. The radius of the modeled source is $\mathit{R}\mathrm{=}{\mathit{N}}_{\mathrm{H}}\mathit{/}{\mathit{n}}_{\mathrm{H}}\mathrm{=}\mathrm{0.21}\mathrm{\times }\left(\frac{{\mathit{\tau }}_{\mathrm{100}}}{\mathrm{0.1}}\right)\mathrm{\times }\left(\frac{{\mathrm{10}}^{\mathrm{5}}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{-3}}{{\mathit{n}}_{{\mathrm{H}}_{\mathrm{2}}}}\right)\hspace{0.17em}\mathrm{pc}\mathit{,}$(2)where Eq. (1) has been applied. The corresponding IR luminosity can be written as ${\mathit{L}}_{\mathrm{IR}}\mathrm{=}\mathrm{4}\mathit{\pi }{\mathit{R}}^{\mathrm{2}}\mathit{\sigma }{\mathit{T}}_{\mathrm{dust}}^{\mathrm{4}}\hspace{0.17em}\mathit{\gamma }$, where γ(T_dust,τ₁₀₀) ≤ 1 accounts for the departure from a blackbody emission due to finite optical depths, ranging from γ = 0.2 for T_dust = 50 K and τ₁₀₀ = 0.1 to γ = 0.9 for T_dust = 95 K and τ₁₀₀ = 1. In physical units, ${\mathit{L}}_{\mathrm{IR}}\mathrm{=}\mathrm{1.4}\mathrm{\times }{\mathrm{10}}^{\mathrm{5}}\mathrm{\times }{\left(\frac{{\mathit{\tau }}_{\mathrm{100}}}{\mathrm{0.1}}\right)}^{\mathrm{2}}\mathrm{\times }{\left(\frac{{\mathrm{10}}^{\mathrm{5}}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{-3}}{{\mathit{n}}_{{\mathrm{H}}_{\mathrm{2}}}}\right)}^{\mathrm{2}}\mathrm{\times }{\left(\frac{{\mathit{T}}_{\mathrm{dust}}}{\mathrm{55}\hspace{0.17em}\mathrm{K}}\right)}^{\mathrm{4}}\mathrm{\times }\left(\frac{\mathit{\gamma }}{\mathrm{0.2}}\right)$(3)in L_⊙ , indicating that a model with τ₁₀₀ ~ 0.1 and moderate T_dust should be considered as one of an ensemble of clumps to account for the typically observed IR luminosities of ≳10¹¹L_⊙ (Y13). For very warm (T_dust ~ 90 K) and optically thick (τ₁₀₀ ~ 1) sources with low average densities (n_H2 = 3 × 10³ cm^-3), Eq. (3) gives L_IR ~ 5 × 10¹¹L_⊙ and the model can be applied to a significant fraction of the circumnuclear region of galaxies where the clouds may have partially lost their individuality (Downes & Solomon 1998).

The velocity dispersion ΔV in our models can be related to the velocity gradient used in escape probability methods as dV/ dr ~ ΔV/ (2R), and using Eq. (2) $\mathrm{d}\mathit{V}\mathit{/}\mathrm{d}\mathit{r}\mathrm{~}\mathrm{238}\mathrm{\times }\left(\frac{\mathrm{\Delta }\mathit{V}}{\mathrm{100}\hspace{0.17em}\mathrm{km}\hspace{0.17em}{\mathrm{s}}^{-1}}\right)\mathrm{\times }\left(\frac{\mathrm{0.1}}{{\mathit{\tau }}_{\mathrm{100}}}\right)\mathrm{\times }\left(\frac{{\mathit{n}}_{{\mathrm{H}}_{\mathrm{2}}}}{{\mathrm{10}}^{\mathrm{5}}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{-3}}\right)\hspace{0.17em}\mathrm{km}\hspace{0.17em}{\mathrm{s}}^{-1}\hspace{0.17em}\mathrm{p}{\mathrm{c}}^{-1}\mathit{.}$(4)Defining K_vir as the ratio of the velocity gradient relative to that expected in gravitational virial equilibrium, K_vir = (dV/ dr)/(dV/dr)_vir, and using (dV/dr)_vir ~ 10 × (n_H2/10⁵ cm^-3)^{1 / 2}km s^-1 pc^-1 (Bryant & Scoville 1996; Goldsmith 2001; Papadopoulos et al. 2007; Hailey-Dunsheath et al. 2012), we obtain ${\mathit{K}}_{\mathrm{vir}}\mathrm{~}\mathrm{23.8}\mathrm{\times }\left(\frac{\mathrm{\Delta }\mathit{V}}{\mathrm{100}\hspace{0.17em}\mathrm{km}\hspace{0.17em}{\mathrm{s}}^{-1}}\right)\mathrm{\times }\left(\frac{\mathrm{0.1}}{{\mathit{\tau }}_{\mathrm{100}}}\right)\mathrm{\times }{\left(\frac{{\mathit{n}}_{{\mathrm{H}}_{\mathrm{2}}}}{{\mathrm{10}}^{\mathrm{5}}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{-3}}\right)}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{·}$(5)Values of K_vir significantly above 1 and up to ~20, indicating non-virialized phases, have been inferred in luminous IR galaxies from both low- and high-J CO lines (e.g., Papadopoulos & Seaquist 1999; Papadopoulos et al. 2007; Hailey-Dunsheath et al. 2012). For clarity, the velocity dispersion is rewritten in terms of K_vir as $\mathrm{\Delta }\mathit{V}\mathrm{=}\mathrm{42}\mathrm{\times }\left(\frac{{\mathit{K}}_{\mathrm{vir}}}{\mathrm{10}}\right)\mathrm{\times }\left(\frac{{\mathit{\tau }}_{\mathrm{100}}}{\mathrm{0.1}}\right)\mathrm{\times }{\left(\frac{{\mathrm{10}}^{\mathrm{5}}\hspace{0.17em}\mathrm{c}{\mathrm{m}}^{-3}}{{\mathit{n}}_{{\mathrm{H}}_{\mathrm{2}}}}\right)}^{\mathrm{1}\mathit{/}\mathrm{2}}\hspace{0.17em}\mathrm{km}\hspace{0.17em}{\mathrm{s}}^{-1}\mathit{,}$(6)which shows that, for compact and dense clumps (τ₁₀₀ = 0.1, n_H2 = 3 × 10⁵ cm^-3), ΔV ~ 25 × (K_vir/10) kms^-1 and the typical observed linewidths (~300 kms^-1) are caused by the galaxy rotation pattern and velocity dispersion of clumps. In contrast, for optically thick sources with low densities (τ₁₀₀ = 1, n_H2 ≲ 10⁴ cm^-3), ΔV ≳ 130 km s^-1 is required for K_vir ≳ 1.

Instead of calculating ΔV for each model according to Eq. (6), which would involve a “universal” K_vir independent of the source characteristics⁴, we have used ΔV = 100 kms^-1 for comparison purposes between models (in Sect. 4.3.5 we also consider models with constant K_vir). Nevertheless, results can be easily rescaled to any other value of ΔV as follows. For given T_dust and τ₁₀₀, the relative level populations, the line opacities, and thus the H₂O line-flux ratios (F_i/F_j) depend on N_H2O/ΔV, while the luminosity ratios L_H2O/L_IR are proportional to ΔV. Therefore, for any ΔV, identical results for F_i/F_j are obtained with the substitution ${\mathit{N}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\mathrm{\to }{\mathit{N}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\mathrm{\times }\left(\frac{\mathrm{\Delta }\mathit{V}}{\mathrm{100}\hspace{0.17em}\mathrm{km}\hspace{0.17em}{\mathrm{s}}^{-1}}\right)\mathit{,}$(7)while L_H2O/L_IR should be scaled as $\frac{{\mathit{L}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}{{\mathit{L}}_{\mathrm{IR}}}\mathrm{\to }\frac{{\mathit{L}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}{{\mathit{L}}_{\mathrm{IR}}}\mathrm{\times }\left(\frac{\mathrm{\Delta }\mathit{V}}{\mathrm{100}\hspace{0.17em}\mathrm{km}\hspace{0.17em}{\mathrm{s}}^{-1}}\right)\mathrm{·}$(8)Both the line-flux ratios (F_i/F_j) and the luminosity ratios L_H2O/L_IR are independent of the number of clumps (N_cl) if the model parameters (T_dust, τ₁₀₀, T_gas , n_H2, N_H2O/ΔV, and ΔV) remain the same for the cloud average. With the effective source radius defined as ${\mathit{R}}_{\mathrm{eff}}\mathrm{=}{\mathit{N}}_{\mathrm{cl}}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathit{R}$, both the line and continuum luminosities scale as ∝${\mathit{R}}_{\mathrm{eff}}^{\mathrm{2}}$. Therefore, if the effective source size is changed and all other parameters are kept constant, a linear correlation between each L_H2O and L_IR is naturally generated, regardless of the excitation mechanism of H₂O. (For reference, however, all absolute luminosities below are given for R_eff = 100 pc.) The question, then, is what range of dust and gas parameters characterizes the sources for which the observed nearly linear correlations in lines 2−6 (O13, Y13) are observed. The detection rates of lines 1, 7, and 8 are relatively low, but the same trend is observed in the few sources where they are detected (Y13).

Fig. 3 Relevant model results for the normalized H2O SLED (a1)−f1)), and for the LH2O/LIR ratios (for ΔV = 100 kms-1) as a function of Tdust and LIR (assuming a source of Reff = 100 pc, a2)−f2)). In panels a1)–f1), model results for lines 1 to 8 (Table 1) are shown from left to right. Values for NH2O/ΔV and τ100 are indicated at the top of the figure. The different colors in panels a1)−f1) indicate different Tdust , as labeled in b1), while they indicate different lines in panels a2)−f2) (labeled in a2), see Table 1). Models with collisional excitation ignored (a)−c)), and with collisions included for nH2 = 3 × 105 cm-3 and Tgas = 150 K (d)−f)) are shown. The gray lines/symbols in panels d1)−f1) show model results that ignore radiative pumping (i.e., only collisional excitation). Collisional excitation has the overall effect of enhancing the low-lying lines (1 and 2) relative to the others and of increasing the LH2O/LIR ratios of all lines (see text). The dashed lines in panels a2)−f2) indicate the average LH2O/LIR ratios reported by Y13. When compared with observations, the modeled LIR values should be considered a fraction of the observed IR luminosities, because single temperature dust models are unable to reproduce the observed SEDs (Sect. 4.3.1); the H2O submm emission traces warm regions of luminous IR galaxies (see text).

In the following sections, the general results of our models are presented, while specific fits to two extreme sources, Arp 220 and NGC 1068, are discussed in Appendix A.

4. Model results

4.1. General results

In Fig. 3, model results are shown in which T_dust is varied from 35 to 115 K, N_H2O/ΔV from 5 × 10¹⁴ to 5 × 10¹⁵cm^-2/ (km s^-1), τ₁₀₀ from 0.1 to 1.0, and where collisional excitation with n_H2 = 3 × 10⁵ cm^-3 and T_gas = 150 K is excluded (a−c) or included (d−f). Panels a1−f1 (top) show the expected SLED normalized to line 2, and panels a2-f2 (bottom) plot the corresponding L_H2O/L_IR × (100 km s^-1/ ΔV) ratios as a function of T_dust and L_IR (for R_eff = 100 pc; all points would move horizontally for different R_eff). The effect of collisional excitation is also illustrated in Fig. 4, where the H₂O submm fluxes of lines 2−6 relative to those obtained ignoring collisional excitation are plotted as a function of n_H2 for T_gas = 150 K.

The first conclusion that we infer from Figs. 3a1-f1 is that the relative fluxes of lines 5−6 generally increase with increasing T_dust . These lines are pumped through the H₂O transition at λ ~ 75 μm (Fig. 1), thus requiring warmer dust than lines 2−4, which are pumped through absorption of 100 μm photons. The SLEDs obtained with T_dust < 45 K yield F₄ significantly above F₆, and are thus unlike those observed in most (U)LIRGs (Y13). The two peaks in the H₂O SLED (in lines 2 and 6) generally found in (U)LIRGs (Y13) indicate that the submm H₂O emission essentially samples regions with T_dust ≳ 45 K. Significant collisional excitation enhances line 4 relative to line 6 (Figs. 3d1−f1), thus aggravating the discrepancy between the T_dust < 45 K models and the observations.

Lines 7−8 provide stringent constraints on T_dust , τ₁₀₀, and N_H2O/ΔV. Since line 6 is still easily excited even with moderately warm T_dust ~ 55 K, the 8/6 and 7/6 ratios are good indicators of whether very warm dust ( > 80 K) is exciting H₂O. Sources where lines 7−8 are detected (e.g., Mrk 231, Arp 220, and APM 08279) can be considered “very warm” on these grounds, with N_H2O/ ΔV ≳ 3 × 10¹⁵cm^-2/ (km s^-1). Sources where lines 7−8 are not detected to a significant level, but where the SLED still shows a second peak in line 6, are considered “warm”, i.e. with T_dust varying between ~45 and 80 K, and N_H2O/ ΔV ~ (5−20) × 10¹⁴cm^-2/ (km s^-1).

Sources in which lines 2−6 are not detected to a significant level, that do not show a second peak in line 6, or for which the H₂O luminosities are well below the observed L_H2O−L_IR correlation are considered “cold”. These sources are characterized by very optically thin and extended continuum emission, and/or with low N_H2O (these properties likely go together). Such sources include starbursts like M82 (Y13), where the continuum is generated in PDRs and are physically very different from the properties of “very warm” sources like Mrk 231 (G-A10).

In the models that neglect collisional excitation (a1−c1), line 1 is predicted to be in absorption, transitioning to emission in warm/dense regions where it is collisionally excited (d1−f1), as previously argued. Its strength will also depend on the continuum opacity, which should be low enough to allow the line to emit above the continuum. Direct collisional excitation from the ground state in regions with warm gas but low τ₁₀₀ efficiently populates level 2₀₂, so that the 2/3, 2/4, 2/5, and 2/6 ratios strongly increase with increasing n_H2 (Fig. 4a). As advanced in Sect. 2, collisional excitation also boosts all other submm lines for moderate T_dust owing to efficient pumping of the base levels 2₁₂ and 1₁₁; radiative trapping of photons emitted in the ground-state transitions increases the chance of absorption of continuum photons in the 101 and 75 μm transitions. Nevertheless, collisional excitation is negligible for high τ₁₀₀ and high T_dust (Fig. 4b).

Fig. 4 Effect of collisional excitation on the H2O fluxes of the submm lines 2−6 as a function of nH2. The ordinates show the calculated line fluxes relative to the model that ignores collisional excitation. Tgas = 150 K is adopted in all models. a) In the case of moderate Tdust and low τ100, collisional excitation has a strong impact on the H2O fluxes at nH2 of a few × 104 cm-3, especially on line 2. b) Collisional excitation is negligible for high τ100 and very warm Tdust (note the difference in ordinate scales in a) and b)).

4.2. Predicted line ratios

In sources where lines 7 and 8 are not detected, the 6/4 flux ratio is the most direct indication of the hardness of the far-IR radiation field seen by the H₂O gas responsible for the observed emission. Since line 4 is pumped through absorption of 101 μm photons and line 6 by 75 μm photons (Fig. 1), one may expect a correlation between the 6/4 ratio and the 75-to-100 μm far-IR color, f₇₅/f₁₀₀. As shown in Fig. 5a, our models indeed show a steep increase in the 6/4 ratio with T_dust for fixed τ₁₀₀ and N_H2O. The averaged observed 6/4 ratio of ≈1.45−1.7 in strong-AGN and HII+mild-AGN sources (Y13) indicates, assuming an optically thin continuum (Fig. 5a), T_dust ≈ 55−75 K and f₇₅/f₁₀₀ = 1.5−1.8. For the case of high τ₁₀₀ and N_H2O/ΔV, the averaged 6/4 ratio is consistent with lower T_dust and f₇₅/f₁₀₀ = 1 −1.2. In general, the 6/4 ratio indicates T_dust ≈ 45−80 K⁵. Similarly, the 6/2 ratio is also sensitive to T_dust , as shown in Fig. 5b. The observed averaged 6/2 ratio of ≈1−1.2 is compatible with T_dust somewhat lower than estimated from the 6/4 ratio. This is attributable to the effects of collisional excitation of the 2₀₂ level (thus enhancing line 2 over line 6, see Fig. 4a and magenta symbols in Fig. 5b), or to the contribution to line 2 by an extended, low T_dust component.

There is, however, no observed correlation between the 6/4 ratio and f₆₀/f₁₀₀ (Y13), which should still show a correlation (though maybe less pronounced) than the expected correlation with f₇₅/f₁₀₀. As we argue in Sect. 4.3, this lack of correlation suggests that the observed far-IR f₆₀/f₁₀₀ colors, and in particular the observed f₁₀₀ fluxes, are not dominated by the warm component responsible for the H₂O emission. Indeed, current models for the continuum emission in (U)LIRGs indicate that the flux density at 100 μm is dominated by relatively cold dust components (T_dust ~ 30 K) (e.g. Dunne et al. 2003; Kóvacs et al. 2010; Casey 2012). The observed H₂O emission thus arises in warm regions whose continuum is hidden within the observed far-IR emission, but may dominate the observed SED at λ ≲ 50 μm (e.g., Casey 2012, see also Sect. 4.3.1).

Fig. 5 a)F6/F4 (oH2O 321−312-to-pH2O 220−211) and b) F6/F2 (oH2O 321−312-to-pH2O 202−111) line flux ratios as a function of the 75-to-100 μm far-IR color. Blue symbols: NH2O/ΔV = 1015cm-2/ (km s-1) and τ100 ≤ 0.1; black: NH2O/ΔV = 5 × 1015cm-2/ (km s-1) and τ100 ≤ 0.1; red: NH2O/ΔV = 5 × 1015cm-2/ (km s-1) and τ100 = 1.0; green: NH2O/ΔV = 1015cm-2/ (km s-1) and τ100 = 1.0; magenta: same as blue symbols but with collisional excitation included with Tgas = 150 K and nH2 = 3 × 105 cm-3. The small numbers close to the symbols indicate the value of Tdust . The observed averaged ratios for strong-AGN and HII+mild-AGN sources (Y13) are indicated with dashed horizontal lines, and indicate that the regions probed by the H2O submm emission are characterized by warm dust (Tdust ≳ 45 K).

Fig. 6 F6/F5 (oH2O 321−312-to-312−303) line flux ratio as a function of τ100. The small numbers on the right side of the curves indicate the values of Tdust for each curve. The H2O column density per unit of velocity interval is NH2O/ΔV = 1015cm-2/ (km s-1) (green, blue, and magenta curves) and NH2O/ΔV = 5 × 1015cm-2/ (km s-1) (red and black curves).

The H₂O lines 5 and 6 are both pumped through the 75 μm transition. Assuming that the lines are optically thin, statistical equilibrium of the level populations implies that every de-excitation in line 6 will be followed by a de-excitation in either line 5 or in the 3₁₂−2₂₁ transition (dotted arrow in Fig. 1), with relative probablities determined by the A-Einstein coefficients. In these optically thin conditions, we expect a 6/5 line flux ratio of F₆/F₅ = 1.16 (Fig. 6). This is a lower limit, because in case of high N_H2O/ ΔV and/or high T_dust and τ₁₀₀, absorption of line 5 emitted photons that can eventually be reemited through the 3₁₂−2₂₁ transition, or absorption of continuum photons in the H₂O 4₂₃−3₁₂ transition, will decrease the strength of line 5 relative to line 6.

Although with significant dispersion, overall data for HII+mild AGN sources indicate F₆/F₅ ~ 1.2 (Y13)⁶, consistent with the optically thin limit; examples of this galaxy population are NGC 1068 and NGC 6240 (Spinoglio et al. 2012; Meijerink et al. 2013). There are, however, sources like Arp 220 and Mrk 231 with F₆/F₅ ≈ 1.6, favoring warm dust ( > 55 K) and substantial columns of H₂O and dust. This indicates that sources in both the optically thin and optically thick regimes are H₂O emitters.

In optically thin conditions and with moderate T_dust , lines 2−4, together with the pumping 2₂₀−1₁₁ 101 μm transition, form a closed loop (Fig. 1) where statistical equilibrium of the level populations implies equal fluxes for the three submm lines (Figs. 3a1−c1). The rise in T_dust and τ₁₀₀, however, increases the chance of line absorption in the strong 3₂₂−2₁₁ transition at 90 μm, thus decreasing the flux of line 3 relative to both line 2 and 4. Consequently, the F₂/F₃ ratio is expected to increase from ≈1 (for low τ₁₀₀) to ≈2 (for τ₁₀₀ ~ 1 and N_H2O/ ΔV ≳ 10¹⁷cm^-2/ (km s^-1)), consistent with the relatively high values found in the warm Mrk 231 and APM 08279 (Y13). If collisional excitation is important (Figs. 3d−f), F₂/F₃ is also expected to increase because collisions mainly boost the lower lying line 2 (Fig. 4a).

One interesting caveat is, however, the behavior of the 4/3 ratio, because increasing T_dust and/or N_H2O is predicted to increase F₂/F₃ but maintains F₄/F₃ > 1 (Figs. 3a1−c1). In Mrk 231, the high F₂/F₃ ratio and mostly the detection of lines 7−8 indicate very warm dust (G-A10), but the relatively low F₄/F₃ ≲ 1 observed in the source does not match this simple scheme. The problem is exacerbated with the 6/2 ratio, which is also expected to increase with increasing T_dust and τ₁₀₀ to ≈1.5 (Fig. 5), but Mrk 231 shows F₆/F₂ ≈ 1. Nevertheless, the problem can be solved if source structure is invoked. A composite model where a very warm component accounts for the high-lying lines and a colder (dust) component enhances lines 2 −4 (with probable contribution from collisionally excited gas, as suggested by the high F₂/F₃ ratio), can give a good fit to the SLED (G-A10), although the characteristics of the “cold” component (density, extension, T_dust ) are relatively uncertain. A relatively low flux in line 4 can also be produced by absorption of continuum photons emanating from a very optically thick component, as in Arp 220 (see Appendix A).

4.3. The L_H2O – L_IR correlations

4.3.1. H₂O and the observed SED

It has long been recognized that single-temperature graybody fits to galaxy SEDs at far-IR wavelengths often underpredict the observed emission at λ< 50 μm. Therefore, multicomponent fitting, based on, for example, a two-temperature approach, a power-law mass-temperature distribution, a power-law mass-intensity distribution, or a single cold dust temperature graybody with a mid-IR power law (Dunne et al. 2003; Kóvacs et al. 2010; Dale & Helou 2002; Casey 2012), is required to match the full SED from the mid-IR to millimeter wavelengths. Our single-temperature model results on the H₂O SLED favors T_dust ≳ 45 K (Sect. 4.2), significantly warmer than the cold dust temperatures (<40 K) that account for most of the observed far-IR emission in luminous IR galaxies, indicating that the H₂O submm emission primarily probes the warm region(s) of galaxies where the mid-IR (20−50 μm) emission is generated (see footnote 5).

Relative to the total IR emission of a galaxy, ${\mathit{L}}_{\mathrm{IR}}^{\mathit{T}}$, the contribution to the luminosity by a given T_dust component i is ${\mathit{f}}_{\mathit{i}}\mathrm{=}{\mathit{L}}_{\mathrm{IR}}^{\mathit{i}}\mathit{/}{\mathit{L}}_{\mathrm{IR}}^{\mathit{T}}$, and the observed H₂O-to-IR luminosity ratio is $\frac{{\mathit{L}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}{{\mathit{L}}_{\mathrm{IR}}^{\mathit{T}}}\mathrm{=}\sum _{\mathit{i}}{\mathit{f}}_{\mathit{i}}{\left(\frac{{\mathit{L}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}{{\mathit{L}}_{\mathrm{IR}}}\right)}_{\mathit{i}}\mathrm{=}{\mathit{f}}_{\mathrm{warm}}{\left(\frac{{\mathit{L}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}{{\mathit{L}}_{\mathrm{IR}}}\right)}_{\mathrm{warm}}\mathrm{+}{\mathit{f}}_{\mathrm{cold}}{\left(\frac{{\mathit{L}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}}{{\mathit{L}}_{\mathrm{IR}}}\right)}_{\mathrm{cold}}$(9)where (L_H2O/L_IR)_i are the values plotted in Figs. 3a2−f2 (for ΔV = 100 kms^-1), and the problem is grossly simplified by considering only two “warm” and “cold” components. From the comparison of the observed average SLED (Y13) with our models, we infer that the contribution by the cold component to ${\mathit{L}}_{{\mathrm{H}}_{\mathrm{2}}\mathrm{O}}\mathit{/}{\mathit{L}}_{\mathrm{IR}}^{\mathit{T}}$ is small, even though f_cold may be high. Since our modeled L_IR emission from the warm component is thus only a fraction, f_warm, of the total IR budget, the modeled L_H2O/L_IR ratios in Figs. 3a2–f2 should be considered upper limits. The value of f_warm can only be estimated by fitting the individual SEDs.

Fig. 7 Model results showing the luminosity of the H2O line 6 (321−312) as a function of the product of the monochromatic luminosities at 75 and 179 μm. Luminosities are calculated for a source with Reff = 100 pc, NH2O/ΔV = 1015cm-2/(km s-1), and ΔV = 100 kms-1. Blue squares indicate models with τ100 = 0.1, resulting in optically thin or moderately thick H2O emission, without collisional excitation. Magenta squares show results for the same models but with collisional excitation included with Tgas = 150 K and nH2 = 3 × 105 cm-3. Green symbols indicate models with τ100 = 1.0, resulting in optically thick H2O emission. For optically thin H2O emission and without collisional excitation, the models indicate a linear correlation between LH2O and L75 × L179 (red line).

4.3.2. H₂O emission and monochromatic luminosities

The H₂O submm emission of lines 2−6 essentially involves two excitation processes, that of the base level (2₁₂ for ortho and 1₁₁ for para-H₂O) and absorption in the transitions at 75 μm (ortho) or 101 μm (para, Fig. 1). If collisional excitation is unimportant, the excitation of the base levels is also produced by absorption of dust-emitted photons in the corresponding transitions, i.e., in the 2₁₂−1₀₁ line at 179 μm (ortho) or 1₁₁−0₀₀ at 269 μm (para). In optically thin conditions and for fixed N_H2O and ΔV, our models then show a linear correlation between the H₂O luminosities L_H2O and the product of the continuum monochromatic luminosities responsible for the excitation, L₁₇₉ × L₇₅ (ortho) or L₂₆₉ × L₁₀₁ (para). This linear correlation is illustrated in Fig. 7 for line 6. The linear correlation, however, breaks down when the line becomes optically thick or when collisional excitation becomes important (in which case, L_H2O is independent of L_179(269)).

4.3.3. The L_H2O/L_IR ratios and T_dust

The above considerations are relevant for our understanding of the behavior of the modeled L_H2O/L_IR values with variations in T_dust . In the optically thin case and with collisional excitation ignored, the double dependence of L_H2O on two monochromatic luminosities implies that L_H2O is (nearly) proportional to L_IR. Our predicted SEDs indicate that, for small variations in T_dust around 55 K, ${\mathit{L}}_{\mathrm{269}}\mathrm{\propto }{\mathit{L}}_{\mathrm{IR}}^{\mathrm{2}\mathit{/}\mathrm{7}}$ and ${\mathit{L}}_{\mathrm{101}}\mathrm{\propto }{\mathit{L}}_{\mathrm{IR}}^{\mathrm{1}\mathit{/}\mathrm{2}}$. Therefore, for the para-H₂O lines 2−4, ${\mathit{L}}_{\mathrm{2}\mathrm{-}\mathrm{4}}\mathrm{\propto }{\mathit{L}}_{\mathrm{IR}}^{\mathrm{0.8}}$ in optically thin conditions, slightly slower than linear. For the ortho lines, ${\mathit{L}}_{\mathrm{179}}\mathrm{\propto }{\mathit{L}}_{\mathrm{IR}}^{\mathrm{1}\mathit{/}\mathrm{3}}$ and ${\mathit{L}}_{\mathrm{75}}\mathrm{\propto }{\mathit{L}}_{\mathrm{IR}}^{\mathrm{2}\mathit{/}\mathrm{3}}$, so that L₅₋₆ ∝ L_IR. This explains why, in Figs. 3a2−b2, the L₂₋₄/L_IR ratios show a slight decrease with increasing T_dust above 55 K, while L₅₋₆/L_IR versus T_dust attain a maximum at T_dust ≈ 55 K in optically thin models that omit collisional excitation. These results are robust against variations in the spectral index of dust down to β = 1.5 (Sect. 3.1).

Fig. 8 Modeled LH2O/LIR × (100 km s-1/ ΔV) for lines 2 (upper), 4 (middle), and 6 (lower) as a function of the f25/f60 color. The dashed vertical lines indicate the lower and upper limits for f25/f60 measured by Y13. In the upper panel, the small numbers below the squares indicate the value of Tdust , and τ100 is also indicated. Blue squares: NH2O/ ΔV = 1015cm-2/ (km s-1) and τ100 = 0.1; magenta: same as blue symbols but with collisional excitation included with Tgas = 150 K and nH2 = 3 × 105 cm-3; green: NH2O/ΔV = 1015cm-2/ (km s-1) and τ100 = 1.0; black: same as green but with NH2O/ΔV = 5 × 1015cm-2/ (km s-1). Red squares show results for fixed Tdust = 55 and 65 K with NH2O/(ΔVτ100) = 5 × 1015cm-2/ (km s-1) and τ100 = 0.1−0.3. The starred symbols indicate the positions of Arp 220 (blue), NGC 6240 (green), Mrk 231 (red), and NGC 1068 (magenta), as reported by Rangwala et al. (2011), Meijerink et al. (2013), G-A10, and Y13, respectively. When compared with observations, the modeled LIR values should be considered a fraction of the observed IR luminosities (Sect. 4.3.1). If f60 is contaminated by cold dust, the points would move to the left.

In Fig. 8 we show the L_H2O/L_IR ratios (with ΔV = 100 kms^-1; L_H2O/L_IR ∝ ΔV) for lines 2, 4, and 6 as a function of the f₂₅/f₆₀ color. The observed f₂₅/f₆₀ was used by Y13 to characterize the H₂O emission and is especially relevant given that the H₂O submm emission arises in warm regions in which the mid-IR continuum emission is not severely contaminated by cold dust. However, the continuum at λ = 60 μm may still be contaminated to some extent, in which case the data points in Fig. 8 will move toward the left. We also recall that the L_H2O/L_IR values are upper limits.

The first conclusion inferred from Fig. 8 is that the range of f₂₅/f₆₀ colors measured by Y13 (between the dashed lines) matches T_dust in the ranges favored by the observed H₂O line flux ratios, that is, 50−75 K and optically thin conditions (τ₁₀₀ = 0.1) and also T_dust = 60−95 K and τ₁₀₀ = 1.0. This indicates that the warm environments responsible for the H₂O emission are best traced in the continuum in this wavelength range, but also that the f₂₅/f₆₀ color alone involves degeneracy in the dominant T_dust and τ₁₀₀ responsible for the mid-IR continuum emission. As shown in Sect. 4.2, the first set of conditions can explain the line ratios 2−6 in warm sources (where lines 7−8 are not detected to a significant level), while the second set is required to explain the H₂O emission in very warm sources (with detection of lines 7−8).

Second, it is also relevant that the L_H2O/L_IR values differ by a factor ≲2 between models with warm dust in the optically thin regime (T_dust = 55 K, τ₁₀₀ = 0.1, N_H2O/ ΔV = 10¹⁵cm^-2/ (km s^-1)) and those with very warm dust in the optically thick regime with high H₂O columns (T_dust = 95 K, τ₁₀₀ = 1, N_H2O/ ΔV = 5 × 10¹⁵cm^-2/ (km s^-1)), potentially explaining why sources with different physical conditions show similar L_H2O/L_IR ratios (Y13).

Third, in optically thin conditions (τ₁₀₀ ~ 0.1) and if collisional excitation is unimportant, the models with constant N_H2O/ ΔV = 10¹⁵cm^-2/ (km s^-1) (blue symbols) predict a slow decrease in L₂₋₄/L_IR and a nearly constant L₅₋₆/L_IR with increasing f₂₅/f₆₀, as argued above. This behavior, however, fails to match the observed trends (Y13), as L₂₋₄/L_IR and L₆/L_IR decrease by factors of ~2 and ~3, respectively, when f₂₅/f₆₀ increases from ≲0.08 to ≳0.15. When collisional excitation is included (magenta symbols), the L₂₋₄/L_IR ratios show a stronger dependence on f₂₅/f₆₀, but L₆/L_IR still changes only slightly with f₂₅/f₆₀.

Therefore, optically thin models with varying T_dust but constant τ₁₀₀, N_H2O/ ΔV, and ΔV cannot account for the observed L_H2O/L_IR −f₂₅/f₆₀ trend. This indicates that, in optically thin galaxies, parameters other than T_dust are systematically varied when f₂₅/f₆₀ is increased and that optically thick sources also contribute to the observed trend:

• (i)

  Galaxies in the optically thin regime (with τ₁₀₀ < 1) are predicted to show a very steep dependence of L_H2O/L_IR on τ₁₀₀ for constant T_dust and N_H2O/(ΔVτ₁₀₀) (that is, for constant H₂O abundance), with higher τ₁₀₀ impling lower f₂₅/f₆₀. We illustrate this point in Fig. 8 with the red squares, corresponding to fix T_dust = 55 and 65 K and N_H2O/(ΔVτ₁₀₀) = 5 × 10¹⁵cm^-2/ (km s^-1), with τ₁₀₀ ranging from 0.1 to 0.3. Therefore, we expect that the observed increase in f₂₅/f₆₀ is not only due to an increase in T_dust from source to source, but also to variations in τ₁₀₀ in the optically thin regime. Examples of galaxies in this regime are the AGNs NGC 6240 and NGC 1068 (see also Appendix A).

• (ii)

  In the optically thick regime (τ₁₀₀ ≳ 1), galaxies are also predicted to show a relatively steep variation in L_H2O/L_IR with f₂₅/f₆₀ due to increasing T_dust (black symbols in Fig. 8) because the H₂O lines saturate and their luminosities flatten with increasing monochromatic luminosities (Fig. 7). Extreme examples of this galaxy population are Arp 220 and Mrk 231. Line saturation also implies that the L_H2O/L_IR ratios are not much higher than in the optically thin case even if much higher N_H2O/ΔV = 5 × 10¹⁵cm^-2/ (km s^-1) are present, and the corresponding ratios are consistent with the observed values to within the uncertainties in f_warm. The presence of even warmer dust ( > 100 K) with significant contribution to L_IR will additionally decrease L_H2O/L_IR (Y13).

In summary, the steep decrease in L_H2O/L_IR at f₂₅/f₆₀ ≈ 0.1−0.15 measured by Y13 is consistent with both types of galaxies (with optically thin and optically thick continuum) populating the diagram and suggests that the observed variations in f₂₅/f₆₀ are not only due to variations in T_dust but also to variations in τ₁₀₀ in the optically thin regime. At the other extreme, the optically thick (saturated) and very warm galaxies are also expected to show a decrease in L_H2O/L_IR with increasing T_dust (and f₂₅/f₆₀), as anticipated by Y13. To distinguish between both regimes for a given galaxy, the line ratios (specifically F₆/F₅, Sect. 4.2) and mostly the detection of lines 7−8 or the detection of high-lying H₂O absorption lines at far-IR wavelengths are required. The observations reported by Y13 indicate that these optically thick and warm components (diagnosed by the detection of lines 7−8) are present in at least ten sources. At least in NGC 1068 the upper limits on lines 7−8 are stringent (S12), allowing us to infer optically thin conditions.

4.3.4. Line saturation and a theoretical upper limit to L_H2O/L_IR

Fig. 9 Modeled LH2O/LIR × (100 km s-1/ ΔV) values for lines 2 (upper), 4 (middle), and 6 (lower) as a function of τ100. Collisional excitation is ignored except for the red dashed lines, where Tgas = 150 K and nH2 = 3 × 105 cm-3 are adopted. The small numbers on the right side of the upper panel indicate the value of Tdust in K, and those on the left side indicate NH2O/ΔV in cm-2/(km s-1). When compared with observations, the modeled LIR values should be considered a fraction of the observed IR luminosities (Sect. 4.3.1). For τ100 = 12 and Tdust = 95 K, line 4 is predicted to be in absorption. For fixed Tdust and τ100 ~ 1, the LH2O/LIR ratios are similar for very different NH2O, indicating line saturation.

Saturation of the H₂O submm lines in optically thick (τ₁₀₀ ~ 1) sources implies that there is an upper limit on L_H2O/L_IR × (100 km s^-1/ΔV) that, in the absence of significant collisional excitation, cannot be exceeded. In Fig. 9, the L_H2O/L_IR × (100 km s^-1/ΔV) ratios for lines 2, 4, and 6 are plotted as a function of τ₁₀₀ for the most favored T_dust range of 55−95 K and N_H2O/ΔV = (1−5) × 10¹⁵cm^-2/ (km s^-1). In optically thin conditions (τ₁₀₀ ≲ 0.1 for N_H2O/ΔV = 10¹⁵cm^-2/ (km s^-1)) and without collisions, L_H2O/L_IR scales linearly (for fixed ΔV) with τ₁₀₀ because ${\mathit{L}}_{\mathrm{179}\mathrm{\left(}\mathrm{269}\mathrm{\right)}}\mathrm{\times }{\mathit{L}}_{\mathrm{75}\mathrm{\left(}\mathrm{101}\mathrm{\right)}}\mathrm{\propto }{\mathit{\tau }}_{\mathrm{100}}^{\mathrm{2}}$ while L_IR ∝ τ₁₀₀. The curves flatten as the H₂O lines saturate and show a maximum at τ₁₀₀ ≈ 0.5−1. Values of τ₁₀₀ significantly higher than unity are predicted to decrease L_H2O/L_IR. In very optically thick components of very warm sources, the submm lines are predicted to be observed in weak emission or even in absorption, especially in line 4. Arp 220 is a case in point (Sakamoto et al. 2008), in which the H₂O submm emission is expected to arise from a region that surrounds the optically thick nuclei (see Appendix A). For ΔV = 100 kms^-1, the maximum attainable values of L_H2O/L_IR (red curves) are 3.5 × 10^-5, 4 × 10^-5, and 7 × 10^-5 for lines 2, 4, and 6, respectively, comfortably higher than the values observed in any source by Y13. Recently, a value of L₂/L_IR = (4.3 ± 1.6) × 10^-5 has been measured in the submillimeter galaxy SPT 0538-50, a gravitationally lensed dusty star-forming galaxy at z ≈ 2.8 (Bothwell et al. 2013). Although the authors do not exclude differential lensing effects, which could affect the line-to-luminosity ratios, this value is still consistent with our upper limit, suggesting strong saturation in this source. In HFLS3 at z = 6.34, Riechers et al. (2013) have measured F₆/F₂ = 2.2 ± 0.5 and F₆/F₅ = 2.6 ± 0.8; within the uncertainties, these values are consistent with warm or very warm T_dust ≳ 65 K and high N_H2O (Figs. 5, 6). The H₂O lines are most likely saturated in HFLS3 as is also indicated by the L₆/L_IR = (7.7 ± 1.3) × 10^-5 ratio, which is still consistent with the strong saturation limit for warm T_dust given the very broad linewidth of the H₂O line (~940 kms^-1; see Sect 3.2). O13 reported L₂/L_IR = (0.5−2) × 10^-5 in high-z ultra-luminous infrared galaxies, also consistent with the upper limit in Fig. 9 even for T_dust ~ 75 K when taking the broad line widths of the H₂O lines into account. Line saturation and a relatively small contribution from cold dust to the infrared emission in these extreme galaxies are implied. With collisional excitation in optically thin environments with moderate T_dust but high N_H2O, the above L_H2O/L_IR ratios (red dashed lines in Fig. 9) may even attain higher values, though the adopted ΔV = 100 kms^-1 is too high for τ₁₀₀ < 0.3 and n_H2 = 3 × 10⁵ cm^-3 (Sect 3.2, Eq. (6)).

4.3.5. The correlation

Fig. 10 Modeled L2/LIR ratio as a function of τ100. Squares and triangles indicate Tdust = 55 and 75 K, respectively. In all models, collisional excitation is included with Tgas = 150 K and nH2 = 3 × 104 cm-3. NH2O/τ100 = 1018 cm-2 is adopted, corresponding to a constant H2O abundance of 7.7 × 10-7 (Eq. (1)). Blue symbols indicate models with ΔV = 100 kms-1 and thus with variable Kvir (Eq. (5)) indicated with the numbers. Green symbols show results with ΔV = 100 × τ100 kms-1 simulating a constant value of Kvir = 1.3. When compared with observations, the modeled LIR values should be considered a fraction of the observed IR luminosities (Sect. 4.3.1), and thus the modeled L2/LIR values are upper limits.

The broad range in observed L_IR in luminous IR galaxies with H₂O emission may be attributable to varying the effective size of the emitting region. As noted in Sect. 3.2, varying R_eff (equivalent to varying the number of individual regions that contribute to L_IR or to increasing L_IR for a single source) is expected to generate linear L_H2O−L_IR correlations if the other parameters (T_dust, τ₁₀₀, T_gas , n_H2, N_H2O/ΔV, and ΔV) remain constant.

In Fig. 10 we show the L₂/L_IR ratio as a function of τ₁₀₀ for models with T_dust = 55 K and T_dust = 75 K that assume a constant H₂O-to-dust opacity ratio, that is, N_H2O/τ₁₀₀ = 10¹⁸ cm^-2. According to Eq. (1), this corresponds to a constant H₂O abundance of 7.7 × 10^-7. Both models with ΔV = 100 kms^-1 (independent of τ₁₀₀), and ΔV/τ₁₀₀ = 100 kms^-1 (corresponding to a constant K_vir = 1.3) are shown. The figure illustrates that a supralinear correlation between L_H2O and L_IR can be expected if, on average, τ₁₀₀ is an increasing function of L_IR. If most sources with L_IR ~ 5 × 10¹⁰L_⊙ were optically thin (τ₁₀₀ ~ 0.1), and the high-z sources with L_IR ~ 10¹³L_⊙ (O13) were mostly optically thick (τ₁₀₀ ~ 1), one would then expect ${\mathit{L}}_{\mathrm{2}}\mathrm{\propto }{\mathit{L}}_{\mathrm{IR}}^{\mathrm{1.3}}$ from Fig. 10, which can account for the observed supralinear correlation found by O13 and Y13. However, similar supralinear correlations would then be expected for the other submm lines 3−6.

5. Summary of the model results for optically classified starbursts and AGNs

Following the classification of sources by Y13 into optically classified star-formation-dominated galaxies with possible mild AGN contribution (HII+mild AGN sources) and optically identified strong-AGN sources, we now consider these two groups separately.

5.1. HII+mild AGN sources

We focus here on those HII+mild AGN sources where lines 2−6 are detected but lines 7−8 are undetected (that is, “warm” sources as defined in Sect. 4.1). The average H₂O flux ratios reported by Y13 (their Table 2) indicate that (i)F₆/F₂ ~ 1.2, favoring T_dust = 55 K if there is no significant collisional excitation and T_dust = 75 K if the H₂O emission arises in warm and dense gas (Fig. 5); (ii)F₆/F₅ ~ 1.2, consistent with the optically thin regime (Fig. 6). For these T_dust , Fig. 11a shows the values of N_H2O for ΔV = 100 kms^-1 required to explain the observed L_H2O/L_IR ratios, as a function of τ₁₀₀. Models with included or excluded collisional excitation are considered. We recall that ΔV is the velocity dispersion of the dominant structure(s) that accounts for the H₂O emission (Sect. 3.2), and for the case of low τ₁₀₀ and relatively high densities, Eq. (6) suggests ΔV < 100 kms^-1 with the consequent increase in N_H2O (Fig. 10).

The decrease in τ₁₀₀ implies the increase in N_H2O in optically thin conditions and when collisional excitation is unimportant. Our best fit models for the average SLED (big solid symbols) favor optically thin far-IR emission (τ₁₀₀ ≲ 0.3). In Figs. 11b−e, the detailed comparison between the τ₁₀₀ = 0.1 models and the observations (Y13) is shown. Significant collisional excitation is not favored for T_dust = 55 K, since it would increase F₂ relative to F₆. In addition, these optically thin models have the drawback of overestimating F₄/F₂. Conversely, the T_dust = 75 K models favor significant collisional excitation in order to increase F₂ relative to F₆. The very optically thin models (τ₁₀₀ ≲ 0.05) are also not favored given the very high amounts of H₂O required to explain (with no collisional excitation) the L_H2O/L_IR ratios.

Fig. 11 a) Values of NH2O for ΔV = 100 km s-1 as a function of τ100, for Tdust = 55 K (red) and 75 K (blue), required to account for the observed averaged LH2O/LIR ratios in HII-mild AGN sources (as given by Y13). Models both without (squares) and with (triangles) collisional excitation are shown. In the latter models, Tgas = 150 K, and nH2 = 5 × 104−3 × 105 cm-3 for Tdust = 55−75 K, respectively. The large filled symbols indicate the best fit models to the averaged SLED. Panels b)− e) compare in detail the four models with τ100 = 0.1 (both with and without collisions) with the observed values (black circles) in HII-mild AGN sources (both the normalized flux ratios (SLED) and the LH2O/LIR ratios, Y13, for lines 2−6). In panels b) and d), the models indicate the values of LH2O/LIR for ΔV = 100 kms-1. Lines 1, 7, and 8 are excluded from the comparison because of their low detection rates (Y13).

In summary, T_dust = 55−75 K, τ₁₀₀ ~ 0.1, and N_H2O ~ (0.5−2) × 10¹⁷ cm^-2 can explain the bulk of the H₂O submm emission in warm star-forming galaxies (Table 2). As shown in Fig. 8, T_dust = 55 K and τ₁₀₀ = 0.1−0.2 predict 25-to-60 μm flux density ratios of f₂₅/f₆₀ = (8.5−6.0) × 10^-2, in agreement with the observed values for the bulk of sources (Y13), while T_dust = 75 K and τ₁₀₀ = 0.1−0.2 predict f₂₅/f₆₀ = 0.42−0.30 (close to the observed upper values). Assuming a gas-to-dust ratio of 100 by mass, τ₁₀₀ ~ 0.1 corresponds to a column density of H nuclei of N_H ≈ 1.3 × 10²³ cm^-2 (Eq. (1)), and thus an H₂O abundance of X_H2O ~ 10^-6. To within a factor of 3 uncertainty due to the τ₁₀₀−N_H calibration, the specific values used for τ₁₀₀ and ΔV, and variations in the measurements for individual sources, this is the typical H₂O abundance that we infer from the observed L_H2O−L_IR correlation. Molecular shocks and hot core chemistry are very likely responsible for this X_H2O, which is well above the volume-averaged values inferred in Galactic dark clouds and PDRs (e.g., Bergin et al. 2000; Snell et al. 2000; Melnick & Bergin 2005; van Dishoeck et al. 2011).

Finally, we note that T_dust = 55 K and τ₁₀₀ = 0.1−0.2, and the assumption that most of the IR is powered by star formation in these sources of Y13, imply a star-formation-rate surface density⁷ of Σ_SFR = 121−195M_⊙ yr^-1 kpc^-2 and gas mass surface density of Σ_gas = 1430−2860M_⊙ pc^-2. The implied depletion or exhaustion time scale, t_dep = Σ_gas/ Σ_SFR, is ~12−15 Myr. These values lie close to the Σ_SFR−Σ_dense star-formation correlation found by García-Burillo et al. (2012) from HCN emission in (U)LIRGs with their revised HCN-M_dense conversion factors. This agreement suggests that the submm H₂O and the mm HCN emission in (U)LIRGs arise from the same regions. Sources with T_dust = 75 K would imply even shorter time scales and suggest high rates of ISM return from SNe and stellar winds. A follow-up study of the relationship between L_H2O and L_HCN is required to check this point. In addition, modeling the individual sources simultaneously in the continuum and the H₂O emission will provide further constraints on the nature of these regions.

5.2. Strong optically classified AGN sources

The general finding that the H₂O emission is similar in star-forming and strong-AGN sources (Y13) may simply indicate that the far-IR pumping of H₂O occurs regardless of whether the dust is heated via star formation or an AGN. There are, however, some differences between the two source types. Strong AGNs show a higher detection rate in H₂O 1₁₁−0₀₀ (Y13), indicating that the gas densities are higher in the circumnuclear regions of AGNs. Another difference is that the L_H2O/L_IR ratios are somewhat lower in strong AGN sources (Y13). While relatively low columns of dust and H₂O in these sources could explain this observational result, it is also possible that high X-ray fluxes photodissociate H₂O, reducing its abundance relative to star-forming galaxies. High abundances of H₂O require effective shielding from UV and X-ray photons and thus high columns of dust and gas that, in AGN-dominated galaxies, may be effectively provided by an optically thick torus probably accompanied by starburst activity. In addition, warm dust further enhances X_H2O through an undepleted chemistry and pumps the excited H₂O levels, while warm gas will further boost X_H2O through reactions of OH with H₂. These appear to be the ideal conditions for the presence of large quantities of H₂O in the (circum)nuclear regions of galaxies.

Online material

Appendix A: Two opposite, extreme cases: Arp 220 and NGC 1068

Arp 220 and NGC 1068 are prototypical sources that have been observed at essentially all wavelengths. With regard to their H₂O submm emission, these galaxies are extreme cases and deserve special consideration.

In the nearby ULIRG Arp 220, discrepancies between the observed SLED (Rangwala et al. 2011, Y13) and the single-component models of Figs. 3a1−c1 are worth noting. The observed high L₆/L_IR ≈ 2.4 × 10^-5 (Fig. 8), together with the high 6/2 ratio of ≈1.4 (Fig. A.1a), suggest T_dust ≳ 65 K and N_H2O ≳ 10¹⁷ cm^-2, consistent with detection of lines 7−8. However, high T_dust and N_H2O are mostly compatible with F₄/F₃ > 1, while the observed ratio is ≈0.7 (Fig. A.1a). As in Mrk 231, a composite model is required to account for the H₂O SLED in this galaxy.

In sources with very optically thick and very warm cores such as Arp 220 (G-A12), the increase in τ₁₀₀ above 1 decreases the submm H₂O fluxes due to the rise of submm extinction (Fig. 9). While higher T_dust generates warmer SEDs, but lowers the L_H2O/L_IR ratios for lines 2−6, the increase in τ₁₀₀ further decreases L_H2O/L_IR. This behavior suggests that the optimal environments for efficient H₂O submm line emission are regions with high far-IR radiation density but moderate extinction, i.e., those that surround the thick core(s) where the bulk of the continuum emission is generated. In contrast, the H₂O absorption at shorter wavelengths is more efficiently produced in the near-side layers of the optically thick cores, primarily if high-lying lines are involved. Absorption and emission lines are thus complementary, providing information on the source structure.

Fig. A.1 a) Proposed composite model for the H2O submm lines in Arp 220 (see G-A12), compared with the observed line fluxes (black squares, from Rangwala et al. 2011). Toward the far-IR optically thick nuclear region (blue symbols), the Eupper< 400 K lines are expected mostly in absorption. The H2O emission is generated around that nuclear region, in the Cextended (G-A12) component (green). Red is total. b) Resulting predicted composite PACS/SPIRE H2O continuum-subtracted spectrum of Arp 220, which is dominated by absorption of the continuum.

We have taken the models in G-A12 for Arp 220 to predict its submm H₂O emission. In Fig. A.1a, the blue symbols/line indicate the predicted H₂O fluxes towards the optically thick, warm nuclear region (both C_west and C_east , see G-A12), indicating that most submm lines (with the exception of lines 3, 7, and 8) are predicted in absorption. The observed H₂O submm line emission (Rangwala et al. 2011) must therefore arise in the surrounding, optically thinner region, i.e., the C_extended component, where the H₂O abundance in the inner parts (R ≲ 150 pc, where T_dust = 70−90 K) is increased relative to G-A12 (so C_extended has N_H2O = 1.3 × 10¹⁷ cm^-2 in Fig. A.1a). According to our model, the relatively low flux in line 4 is due to line absorption towards the nuclei. The main drawback of the model in Fig. A.1a is that line 7 is underestimated by a factor 2. The submm H₂O emission in Arp 220 traces a transition region between the compact optically thick cores and the extended kpc-scale disk (G-A12). The overall H₂O spectrum is, however, dominated by absorption of the continuum (Fig. A.1b).

Fig. A.2 a) Composite model for the H2O emission in NGC 1068 favored in this work. Blue/red indicate ortho/para lines, and the submm H2O 1−6 lines (Table 1) are indicated. Open squares and triangles show the contribution by a moderate-excitation (ME) and a low-excitation (LE) component, and filled symbols indicate the total emission (see text for details); in both cases the submm H2O lines 2−6 are pumped through far-IR photons emitted by dust. b) Comparison between the observed fluxes of the H2O submm lines (black squares, from Spinoglio et al. 2012) and those predicted with the composite model.

Just the opposite set of conditions characterizes the nearby Seyfert 2 galaxy NGC 1068, since the nuclear continuum emission is optically thin and collisional excitation is important (S12). All detected H₂O lines, including those in the far-IR (100−200 μm) are seen in emission, and most of them show fluxes (in erg/s/cm²) unrelated to wavelength, upper level energy (up to ≈300 K), or A-Einstein coefficient (S12). In particular, the H₂O 2₂₁−1₁₀ (108 μm) and 2₂₁−2₁₂ (180 μm) lines share the same upper level and show similar fluxes but the A-Einstein coefficient of the 108 μm transition is a factor of 8.4 higher than that of the 180 μm transition. With pure collisional excitation, the only way to account for the observed line ratios is to invoke high densities and H₂O column densities, but also a relatively low T_gas to avoid significantly populating the high-lying levels ( > 300 K). S12 found that T_gas ~ 40 K, and very high N_H2O and n_H2 can provide a reasonable fit to the SLED. However, these conditions are unrelated to the warmer gas conditions in the nuclear region of NGC 1068, as derived from the CO SLED (S12, Hailey-Dunsheath et al. 2012, hereafter H12). In addition, the observed H₂O submm SLED (Fig. A.2) is fairly similar to the SLEDs obtained in optically thin models with significant collisional excitation of the low-lying levels.

We have explored an alternative composite solution for the H₂O emission in NGC 1068 with lower densities and H₂O columns and higher T_gas , based on the far-IR pumping of the lines by an external anisotropic radiation field. In this framework, we can account for the weakness of the 108 μm line by the absorption of continuum photons, and indeed we would have to explain why this line is not observed to be even weaker than it is or in absorption. The higher lying far-IR 3₂₂−3₁₃ emission line at 156.2 μm is in this scenario pumped through absorption of continuum photons in the 3₂₂−2₁₁ line at 90 μm.

For the first component, we closely follow H12 in modeling the moderate-excitation (ME) component as an ensemble of clumps, which are described by T_dust = 55 K, τ₁₀₀ = 0.18, n_H2 = 10⁶ cm^-3, T_gas = 150 K, and N_H2O = 6.5 × 10¹⁶ cm^-2, and V_turb = 15 kms^-1 (giving K_vir ~ 10, see H12). With a mass of 7.5 × 10⁶M_⊙, this component is unable to account for the H₂O submm lines 2−6, but generates a significant fraction of the observed emission in line 1 and some far-IR lines (Fig. A.2a and panel b).

We then added another, low-excitation (LE) component, which is identified with the gas generating the low-J CO lines (Krips et al. 2011, S12) and is thus assigned a density of n_H2 = 2 × 10⁴ cm^-3. For simplicity, we also assume T_dust = 55 K, τ₁₀₀ = 0.18, and N_H2O = 6.5 × 10¹⁶ cm^-2 as for the ME, but adopt the higher V_turb of 60 kms^-1 (giving K_vir ~ 7). For the LE component, and besides the internal far-IR field described by its T_dust and τ₁₀₀, we also follow H12 in including an external field (associated with the emission from the whole region), which is described as a graybody with T_BG = 55 K and ${\mathit{\tau }}_{\mathrm{100}}^{\mathrm{BG}}\mathrm{=}\mathrm{0.05}$. The resulting mean specific intensity at 100 μm of the external field, ${\mathit{J}}_{\mathrm{ext}}^{\mathrm{100}\mathit{\mu }\mathrm{m}}$, matches the value estimated by H12 within a factor of 2 (their Eq. (1)). A crucial aspect of the present approach is that this external field is assumed to be anisotropic, that is, it does not impinge into the LE clumps on the back side (in the direction of the observer). As a result, the external field contributes to the H₂O excitation without generating absorption in the pumping far-IR lines (though some absorption is nevertheless produced by the internal field). As shown in Fig. A.2a, the LE component is expected to dominate the emission of the submm lines 2−6, as well as the emission of the majority of the far-IR lines. The required mass of the LE component is 3.5 × 10⁷M_⊙, consistent with the mass inferred from the CO lines for the CND (S12), and the IR luminosity is 2.6 × 10¹⁰L_⊙.

A key assumption of the present model is that the external radiation field does not produce absorption in the far-IR lines, as otherwise (that is, in a perfectly isotropic radiation field) the strengths of the far-IR lines would weaken, and in particular, the H₂O 2₂₁−1₁₀ line at 108 μm line would be predicted to be observed in absorption. The proposed anisotropy could be associated with the heating by the central AGN, and it seems possible as long as the source is optically thin in the far-IR. Radiative transfer in 3D would be required to check this feature. On the other hand, the external field, while having an important effect on the far-IR lines, has a secondary effect on the submm lines, which are primarily pumped by the internal (isotropic) radiation field (that is, by the dust that is mixed with H₂O). With the caveat of the assumed intrinsic radiation anisotropy in mind, we preliminary favor this model over the pure collisional one in predicting the H₂O submm fluxes and conclude that radiative pumping most likely plays an important role in exciting the H₂O in the CND of NGC 1068.

From the models for these two very different sources and the case of Mrk 231 studied previously (G-A10), we conclude that the excitation of the submm H₂O lines other than the 1₁₁−0₀₀ one is dominated by radiative pumping, though the relatively low-lying 2₀₂−1₁₁ line may still have a significant “collisional” contribution in some very warm/dense nuclear regions, and the radiative pumping may be enhanced with collisional excitation of the low-lying 1₁₁ and 2₁₂ levels. These individual cases also show that composite models to account for the full H₂O far-IR/submm spectrum in a given source may be a rather general requirement.

Acknowledgments

We are very grateful to Chentao Yang for useful discussions on the data reported in Y13. E.G.-A. is a Research Associate at the Harvard-Smithsonian Center for Astrophysics, and thanks the Spanish Ministerio de Economía y Competitividad for support under projects AYA2010-21697-C05-0 and FIS2012-39162-C06-01. Basic research in IR astronomy at NRL is funded by the US ONR; J.F. also acknowledges support from the NHSC. This research has made use of NASA’s Astrophysics Data System (ADS) and of GILDAS software (http://www.iram.fr/IRAMFR/GILDAS).

References

All Tables

All Figures

Fig. 1 Energy level diagram of H2O, showing the relevant H2O lines at submillimeter wavelengths with blue arrows, and the far-IR H2O pumping (absorption) lines with dashed-magenta arrows. The lines are numbered as listed in Table 1. The o-H2O 312−221 transition is not considered due to blending with CO (10-9) (G-A10), and the far-IR 212−101 transition at 179.5 μm discussed in the text is also indicated in green for completeness.

In the text

	Fig. 2 Adopted mass absorption coefficient of dust as a function of wavelength. The dust emission is simulated by using a mixture of silicate and amorphous carbon grains with optical constants from Draine (1985) and Preibisch et al. (1993). As shown by the fitted blue line, the emissivity index from the far-IR to millimeter wavelengths is β = 1.85.
In the text

Fig. 3 Relevant model results for the normalized H2O SLED (a1)−f1)), and for the LH2O/LIR ratios (for ΔV = 100 kms-1) as a function of Tdust and LIR (assuming a source of Reff = 100 pc, a2)−f2)). In panels a1)–f1), model results for lines 1 to 8 (Table 1) are shown from left to right. Values for NH2O/ΔV and τ100 are indicated at the top of the figure. The different colors in panels a1)−f1) indicate different Tdust , as labeled in b1), while they indicate different lines in panels a2)−f2) (labeled in a2), see Table 1). Models with collisional excitation ignored (a)−c)), and with collisions included for nH2 = 3 × 105 cm-3 and Tgas = 150 K (d)−f)) are shown. The gray lines/symbols in panels d1)−f1) show model results that ignore radiative pumping (i.e., only collisional excitation). Collisional excitation has the overall effect of enhancing the low-lying lines (1 and 2) relative to the others and of increasing the LH2O/LIR ratios of all lines (see text). The dashed lines in panels a2)−f2) indicate the average LH2O/LIR ratios reported by Y13. When compared with observations, the modeled LIR values should be considered a fraction of the observed IR luminosities, because single temperature dust models are unable to reproduce the observed SEDs (Sect. 4.3.1); the H2O submm emission traces warm regions of luminous IR galaxies (see text).

In the text

Fig. 4 Effect of collisional excitation on the H2O fluxes of the submm lines 2−6 as a function of nH2. The ordinates show the calculated line fluxes relative to the model that ignores collisional excitation. Tgas = 150 K is adopted in all models. a) In the case of moderate Tdust and low τ100, collisional excitation has a strong impact on the H2O fluxes at nH2 of a few × 104 cm-3, especially on line 2. b) Collisional excitation is negligible for high τ100 and very warm Tdust (note the difference in ordinate scales in a) and b)).

In the text

Fig. 5 a)F6/F4 (oH2O 321−312-to-pH2O 220−211) and b) F6/F2 (oH2O 321−312-to-pH2O 202−111) line flux ratios as a function of the 75-to-100 μm far-IR color. Blue symbols: NH2O/ΔV = 1015cm-2/ (km s-1) and τ100 ≤ 0.1; black: NH2O/ΔV = 5 × 1015cm-2/ (km s-1) and τ100 ≤ 0.1; red: NH2O/ΔV = 5 × 1015cm-2/ (km s-1) and τ100 = 1.0; green: NH2O/ΔV = 1015cm-2/ (km s-1) and τ100 = 1.0; magenta: same as blue symbols but with collisional excitation included with Tgas = 150 K and nH2 = 3 × 105 cm-3. The small numbers close to the symbols indicate the value of Tdust . The observed averaged ratios for strong-AGN and HII+mild-AGN sources (Y13) are indicated with dashed horizontal lines, and indicate that the regions probed by the H2O submm emission are characterized by warm dust (Tdust ≳ 45 K).

In the text

	Fig. 6 F6/F5 (oH2O 321−312-to-312−303) line flux ratio as a function of τ100. The small numbers on the right side of the curves indicate the values of Tdust for each curve. The H2O column density per unit of velocity interval is NH2O/ΔV = 1015cm-2/ (km s-1) (green, blue, and magenta curves) and NH2O/ΔV = 5 × 1015cm-2/ (km s-1) (red and black curves).
In the text

Fig. 7 Model results showing the luminosity of the H2O line 6 (321−312) as a function of the product of the monochromatic luminosities at 75 and 179 μm. Luminosities are calculated for a source with Reff = 100 pc, NH2O/ΔV = 1015cm-2/(km s-1), and ΔV = 100 kms-1. Blue squares indicate models with τ100 = 0.1, resulting in optically thin or moderately thick H2O emission, without collisional excitation. Magenta squares show results for the same models but with collisional excitation included with Tgas = 150 K and nH2 = 3 × 105 cm-3. Green symbols indicate models with τ100 = 1.0, resulting in optically thick H2O emission. For optically thin H2O emission and without collisional excitation, the models indicate a linear correlation between LH2O and L75 × L179 (red line).

In the text

Fig. 8 Modeled LH2O/LIR × (100 km s-1/ ΔV) for lines 2 (upper), 4 (middle), and 6 (lower) as a function of the f25/f60 color. The dashed vertical lines indicate the lower and upper limits for f25/f60 measured by Y13. In the upper panel, the small numbers below the squares indicate the value of Tdust , and τ100 is also indicated. Blue squares: NH2O/ ΔV = 1015cm-2/ (km s-1) and τ100 = 0.1; magenta: same as blue symbols but with collisional excitation included with Tgas = 150 K and nH2 = 3 × 105 cm-3; green: NH2O/ΔV = 1015cm-2/ (km s-1) and τ100 = 1.0; black: same as green but with NH2O/ΔV = 5 × 1015cm-2/ (km s-1). Red squares show results for fixed Tdust = 55 and 65 K with NH2O/(ΔVτ100) = 5 × 1015cm-2/ (km s-1) and τ100 = 0.1−0.3. The starred symbols indicate the positions of Arp 220 (blue), NGC 6240 (green), Mrk 231 (red), and NGC 1068 (magenta), as reported by Rangwala et al. (2011), Meijerink et al. (2013), G-A10, and Y13, respectively. When compared with observations, the modeled LIR values should be considered a fraction of the observed IR luminosities (Sect. 4.3.1). If f60 is contaminated by cold dust, the points would move to the left.

In the text

Fig. 9 Modeled LH2O/LIR × (100 km s-1/ ΔV) values for lines 2 (upper), 4 (middle), and 6 (lower) as a function of τ100. Collisional excitation is ignored except for the red dashed lines, where Tgas = 150 K and nH2 = 3 × 105 cm-3 are adopted. The small numbers on the right side of the upper panel indicate the value of Tdust in K, and those on the left side indicate NH2O/ΔV in cm-2/(km s-1). When compared with observations, the modeled LIR values should be considered a fraction of the observed IR luminosities (Sect. 4.3.1). For τ100 = 12 and Tdust = 95 K, line 4 is predicted to be in absorption. For fixed Tdust and τ100 ~ 1, the LH2O/LIR ratios are similar for very different NH2O, indicating line saturation.

In the text

Fig. 10 Modeled L2/LIR ratio as a function of τ100. Squares and triangles indicate Tdust = 55 and 75 K, respectively. In all models, collisional excitation is included with Tgas = 150 K and nH2 = 3 × 104 cm-3. NH2O/τ100 = 1018 cm-2 is adopted, corresponding to a constant H2O abundance of 7.7 × 10-7 (Eq. (1)). Blue symbols indicate models with ΔV = 100 kms-1 and thus with variable Kvir (Eq. (5)) indicated with the numbers. Green symbols show results with ΔV = 100 × τ100 kms-1 simulating a constant value of Kvir = 1.3. When compared with observations, the modeled LIR values should be considered a fraction of the observed IR luminosities (Sect. 4.3.1), and thus the modeled L2/LIR values are upper limits.

In the text

Fig. 11 a) Values of NH2O for ΔV = 100 km s-1 as a function of τ100, for Tdust = 55 K (red) and 75 K (blue), required to account for the observed averaged LH2O/LIR ratios in HII-mild AGN sources (as given by Y13). Models both without (squares) and with (triangles) collisional excitation are shown. In the latter models, Tgas = 150 K, and nH2 = 5 × 104−3 × 105 cm-3 for Tdust = 55−75 K, respectively. The large filled symbols indicate the best fit models to the averaged SLED. Panels b)− e) compare in detail the four models with τ100 = 0.1 (both with and without collisions) with the observed values (black circles) in HII-mild AGN sources (both the normalized flux ratios (SLED) and the LH2O/LIR ratios, Y13, for lines 2−6). In panels b) and d), the models indicate the values of LH2O/LIR for ΔV = 100 kms-1. Lines 1, 7, and 8 are excluded from the comparison because of their low detection rates (Y13).

In the text

Fig. A.1 a) Proposed composite model for the H2O submm lines in Arp 220 (see G-A12), compared with the observed line fluxes (black squares, from Rangwala et al. 2011). Toward the far-IR optically thick nuclear region (blue symbols), the Eupper< 400 K lines are expected mostly in absorption. The H2O emission is generated around that nuclear region, in the Cextended (G-A12) component (green). Red is total. b) Resulting predicted composite PACS/SPIRE H2O continuum-subtracted spectrum of Arp 220, which is dominated by absorption of the continuum.

In the text

Fig. A.2 a) Composite model for the H2O emission in NGC 1068 favored in this work. Blue/red indicate ortho/para lines, and the submm H2O 1−6 lines (Table 1) are indicated. Open squares and triangles show the contribution by a moderate-excitation (ME) and a low-excitation (LE) component, and filled symbols indicate the total emission (see text for details); in both cases the submm H2O lines 2−6 are pumped through far-IR photons emitted by dust. b) Comparison between the observed fluxes of the H2O submm lines (black squares, from Spinoglio et al. 2012) and those predicted with the composite model.

In the text