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Modeling the H2O submillimeter emission in extragalactic sources

Recent observational studies have shown that H2O emission at (rest) submillimeter wavelengths is ubiquitous in infrared galaxies, both in the local and in the early Universe, suggestive of far-infrared pumping of H2O by dust in warm regions. In this work, models are presented that show that (i) the highest-lying H2O lines (E_{upper}>400 K) are formed in very warm (T_{dust}>~90 K) regions and require high H2O columns (N_{H2O}>~3x10^{17} cm^{-2}), while lower lying lines can be efficiently excited with T_{dust}~45-75 K and N_{H2O}~(0.5-2)x10^{17} cm^{-2}; (ii) significant collisional excitation of the lowest lying (E_{upper}<200 K) levels, which enhances the overall L_{H2O}-L_{IR} ratios, is identified in sources where the ground-state para-H2O 1_{11}-0_{00} line is detected in emission; (iii) the H2O-to-infrared (8-1000 um) luminosity ratio is expected to decrease with increasing T_{dust} for all lines with E_{upper}<~300 K, as has recently been reported in a sample of LIRGs, but increases with T_{dust} for the highest lying H2O lines (E_{upper}>400 K); (iv) we find theoretical upper limits for L_{H2O}/L_{IR} in warm environments, owing to H2O line saturation; (v) individual models are presented for two very different prototypical galaxies, the Seyfert 2 galaxy NGC 1068 and the nearest ultraluminous infrared galaxy Arp 220, showing that the excited submillimeter H2O emission is dominated by far-infrared pumping in both cases; (vi) the L_{H2O}-L_{IR} correlation previously reported in observational studies indicates depletion or exhaustion time scales, t_{dep}=Sigma_{gas}/Sigma_{SFR}, of<~12 Myr for star-forming sources where lines up to E_{upper}=300 K are detected, in agreement with the values previously found for (U)LIRGs from HCN millimeter emission...


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), H 2 O couples very well to the radiation field in warm regions that emit strongly in the far-IR. In extragalactic sources, excited lines of H 2 O 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, and with Herschel/PACS (Pilbratt et al. 2010;Poglitsch et al. 2010) in Mrk 231 , Arp 220 and NGC 4418 (González-Alfonso et al. 2012, G-A12). Modeling and analysis have demonstrated the ability of H 2 O 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 H 2 O at submillimeter (hereafter submm, λ > 200 µm) wavelengths in local sources, where the excited (i.e., nonground-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 H 2 O 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 H 2 O 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 H 2 O luminosity (L H 2 O ), taken from the 2 02 − 1 11 and 2 11 − 2 02 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 H 2 O 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 H 2 O that is slightly faster than linear with L IR in some lines (2 11 − 2 02 and 2 20 − 2 11 ) 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 H 2 (Gao & Solomon 2004a,b).
The increasing wealth of observations of H 2 O at submm wavelengths in both local and high-z sources and the correlations discovered between L H 2 O 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 H 2 O emitting regions in warm (U)LIRGs and to propose a general framework for interpreting the H 2 O submm emission in extragalactic sources.

Excitation overview
At submm wavelengths, H 2 O 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 101 1 , 75, 58, and 45 µm far-IR transitions, respectively. The ground-state line 1 has no analog pumping mechanism, so that the upper 1 11 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 opacities 2 . 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 App. 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).
If collisional excitation of the 1 11 and 2 12 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 H 2 O lines 2−6 will be boosted because these 1 11 and 2 12 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-H 2 O level 2 02 from the ground 0 00 state can significantly enhance the emission of line 2. Therefore, the H 2 O 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 (1 11 , 2 12 , and 2 02 ). Lines 7 − 8 require strong far-IR radiation density not only at 58 − 45 µm, but also at longer wavelengths, together with high H 2 O column densities (N H 2 O ) in order to significantly populate the lower backbone 3 13 and 4 14 levels.  Table 1. The o-H 2 O 3 12 − 2 21 transition is not considered due to blending with CO (10-9) (G-A10), and the far-IR 2 12 − 1 01 transition at 179.5 µm discussed in the text is also indicated in green for completeness.

Description of the models
The basic models for H 2 O 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, H 2 O 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. 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 H 2 O ortho-to-para ratio of 3. Line broadening is simulated by including a microturbulent velocity (V turb ), for which the FWHM velocity dispersion is ∆V = 1.67V turb . No systemic motions are included.

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.

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. 2008,2010,2012,2013,2014). Our values at 125 and 850 µm are κ 125 = 30 cm 2 g −1 and κ 850 = 0.7 cm 2 g −1 , in good agreement with those derived by Dunne et al. (2003). Adopting a gas-todust ratio of X = 100 by mass, and using κ 100 = 44.5 cm 2 g −1 , the column density of H nuclei is where τ 100 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 H 2 O 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 H 2 O /L IR are insensitive to β for β values above 1.5 (Sect. 4.3.3).

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 (τ 100 ), the corresponding H 2 O column density per unit of velocity interval (N H 2 O /∆V), the velocity dispersion ∆V, T gas , and the H 2 density (n H 2 ). Fiducial numbers for some of these parameters are τ 100 = 0.1, ∆V = 100 km s −1 , T gas = 150 K, and n H 2 = 3 × 10 5 cm −3 . Collisional rates with H 2 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 ratios 3 L H 2 O /L IR . We also explore models where collisions are ignored, appropriate for low-density regions (n H 2 10 4 cm −3 ), for which F i /F j only depend on T dust , τ 100 , and N H 2 O /∆V, while L H 2 O /L IR depends in addition on ∆V.
Depending on the values of the above parameters, our models can be interpreted in terms of a single source or are better 3 We denote L H 2 O as the H 2 O luminosity of a given generic H 2 O submm line, while L i is the luminosity of the H 2 O line i (numbering in Table 1). The H 2 O line fluxes F i are given in Jy km s −1 . L IR is the 8 − 1000 µm luminosity. applied to each of an ensemble of clouds within a clumpy distribution. The radius of the modeled source is where Eq. (1) has been applied. The corresponding IR luminosity can be written as L IR = 4πR 2 σT 4 dust γ, where γ(T dust , τ 100 ) ≤ 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 τ 100 = 0.1 to γ = 0.9 for T dust = 95 K and τ 100 = 1. In physical units, in L ⊙ , indicating that a model with τ 100 ∼ 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 10 11 L ⊙ (Y13). For very warm (T dust ∼ 90 K) and optically thick (τ 100 ∼ 1) sources with low average densities (n H 2 = 3 × 10 3 cm −3 ), Eq. (3) gives L IR ∼ 5 × 10 11 L ⊙ 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 ∆V in our models can be related to the velocity gradient used in escape probability methods as dV/dr ∼ ∆V/(2R), and using Eq. (2) 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 H 2 /10 5 cm −3 ) 1/2 km s −1 pc −1 (Bryant & Scoville 1996;Goldsmith 2001;Papadopoulos et al. 2007 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 which shows that, for compact and dense clumps (τ 100 = 0.1, n H 2 = 3 × 10 5 cm −3 ), ∆V ∼ 25 × (K vir /10) km s −1 and the typical observed linewidths (∼ 300 km s −1 ) are caused by the galaxy rotation pattern and velocity dispersion of clumps. In contrast, for optically thick sources with low densities (τ 100 = 1, n H 2 10 4 cm −3 ), ∆V 130 km s −1 is required for K vir 1. Instead of calculating ∆V for each model according to eq. (6), which would involve a "universal" K vir independent of the source characteristics 4 , we have used ∆V = 100 km s −1 for A&A proofs: manuscript no. watersubmm_def 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 ∆V as follows. For given T dust and τ 100 , the relative level populations, the line opacities, and thus the H 2 O line-flux ratios (F i /F j ) depend on N H 2 O /∆V, while the luminosity ratios L H 2 O /L IR are proportional to ∆V. Therefore, for any ∆V, identical results for F i /F j are obtained with the substitution while L H 2 O /L IR should be scaled as Both the line-flux ratios (F i /F j ) and the luminosity ratios L H 2 O /L IR are independent of the number of clumps (N cl ) if the model parameters (T dust , τ 100 , T gas , n H 2 , N H 2 O /∆V, and ∆V) remain the same for the cloud average. With the effective source radius defined as R eff = N 1/2 cl R, both the line and continuum luminosities scale as ∝ R 2 eff . Therefore, if the effective source size is changed and all other parameters are kept constant, a linear correlation between each L H 2 O and L IR is naturally generated, regardless of the excitation mechanism of H 2 O. (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).
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.

General results
In Fig. 3, model results are shown in which T dust is varied from 35 to 115 K, N H 2 O /∆V from 5 × 10 14 to 5 × 10 15 cm −2 /(km s −1 ), τ 100 from 0.1 to 1.0, and where collisional excitation with n H 2 = 3×10 5 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 H 2 O /L IR × (100 km s −1 /∆V) 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 H 2 O submm fluxes of lines 2 − 6 relative to those obtained ignoring collisional excitation are plotted as a function of n H 2 for T gas = 150 K.
The first conclusion that we infer from Fig. 3a1-f1 is that the relative fluxes of lines 5 − 6 generally increase with increasing T dust . These lines are pumped through the H 2 O 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 F 4 significantly above F 6 , and are thus unlike those observed in most (U)LIRGs (Y13). The two peaks in the H 2 O SLED (in lines 2 and 6) generally found in (U)LIRGs (Y13) indicate that the submm H 2 O emission essentially samples regions with T dust 45 K. Significant collisional excitation enhances line 4 relative to line 6 ( Fig. 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 , τ 100 , and N H 2 O /∆V. 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 H 2 O. 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 H 2 O /∆V 3×10 15 cm −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 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 H 2 O luminosities are well below the observed L H 2 O − L IR correlation are considered "cold". These sources are characterized by very optically thin and extended continuum emission, and/or with low N H 2 O (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 τ 100 efficiently populates level 2 02 , so that the 2/3, 2/4, 2/5, and 2/6 ratios strongly increase with increasing n H 2 (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 2 12 and 1 11 ; radiative trapping of photons emitted in the groundstate transitions increases the chance of absorption of continuum photons in the 101 and 75 µm transitions. Nevertheless, collisional excitation is negligible for high τ 100 and high T dust (Fig. 4b).

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 H 2 O 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, f 75 / f 100 . As shown in Fig. 5a, our models indeed show a steep increase in the 6/4 ratio with T dust for fixed τ 100 and N H 2 O . 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 f 75 / f 100 = 1.5 − 1.8. For the case of high τ 100 and N H 2 O /∆V, the averaged 6/4 ratio is consistent with lower T dust and f 75 / f 100 = 1 − 1.2. In general, the 6/4 ratio indicates T dust ≈ 45 − 80 K. 5 Similarly, the 6/2 ratio is also sensitive to T dust , as shown in Fig. 5b. The  (a1-f1), and for the L H 2 O /L IR ratios (for ∆V = 100 km s −1 ) as a function of T dust and L IR (assuming a source of R eff = 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 N H 2 O /∆V and τ 100 are indicated at the top of the figure. The different colors in panels a1-f1 indicate different T dust , 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 n H 2 = 3 × 10 5 cm −3 and T gas = 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 L H 2 O /L IR ratios of all lines (see text). The dashed lines in panels a2-f2 indicate the average L H 2 O /L IR ratios reported by Y13. When compared with observations, the modeled L IR 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 H 2 O submm emission traces warm regions of luminous IR galaxies (see text). 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 2 02 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 f 60 / f 100 (Y13), which should still show a correlation (though maybe less pronounced) than the expected correlation with f 75 / f 100 . As we argue in Sect. 4.3, this lack of correlation suggests that the observed far-IR f 60 / f 100 colors, and in particular the observed f 100 fluxes, are not dominated by the warm component responsible for the H 2 O 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 H 2 O 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).
The H 2 O 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 deexcitation in line 6 will be followed by a de-excitation in either line 5 or in the 3 12 − 2 21 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 F 6 /F 5 = 1.16 (Fig. 6). This is a lower limit, because in case of high N H 2 O /∆V and/or high T dust and τ 100 , absorption of line 5 emitted photons that can eventually be reemited through the 3 12 − 2 21 transition, or absorption of continuum photons in the H 2 O 4 23 − 3 12 transition, will decrease the strength of line 5 relative to line 6.
Although with significant dispersion, overall data for HII+mild AGN sources indicate F 6 /F 5 ∼ 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 F 6 /F 5 ≈ 1.6, favoring warm dust (> 55 K) and substantial columns of H 2 O and dust. This indicates that sources in both the optically thin and optically thick regimes are H 2 O emitters.
In optically thin conditions and with moderate T dust , lines 2 − 4, together with the pumping 2 20 − 1 11 101 µm transition, form a closed loop (Fig. 1) where statistical equilibrium of the level populations implies equal fluxes for the three submm lines (Fig. 3a1-c1). The rise in T dust and τ 100 , however, increases the chance of line absorption in the strong 3 22 − 2 11 transition at 90 µm, thus decreasing the flux of line 3 relative to both line 2 and 4. Consequently, the F 2 /F 3 ratio is expected to increase from ≈ 1 (for low τ 100 ) to ≈ 2 (for τ 100 ∼ 1 and N H 2 O /∆V 10 17 cm −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 (Fig. 3d-f), F 2 /F 3 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 H 2 O is predicted to increase Fig. 5. a) F 6 /F 4 (oH 2 O 3 21 − 3 12 -to-pH 2 O 2 20 − 2 11 ) and b) F 6 /F 2 (oH 2 O 3 21 − 3 12 -to-pH 2 O 2 02 − 1 11 ) line flux ratios as a function of the 75-to-100 µm far-IR color. Blue symbols: N H 2 O /∆V = 10 15 cm −2 /(km s −1 ) and τ 100 ≤ 0.1; black: N H 2 O /∆V = 5 × 10 15 cm −2 /(km s −1 ) and τ 100 ≤ 0.1; red: N H 2 O /∆V = 5 × 10 15 cm −2 /(km s −1 ) and τ 100 = 1.0; green: N H 2 O /∆V = 10 15 cm −2 /(km s −1 ) and τ 100 = 1.0; magenta: same as blue symbols but with collisional excitation included with T gas = 150 K and n H 2 = 3 × 10 5 cm −3 . The small numbers close to the symbols indicate the value of T dust . 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 H 2 O submm emission are characterized by warm dust (T dust 45 K). F 2 /F 3 but maintains F 4 /F 3 > 1 (Fig. 3a1-c1). In Mrk 231, the high F 2 /F 3 ratio and mostly the detection of lines 7 − 8 indicate very warm dust (G-A10), but the relatively low F 4 /F 3 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 τ 100 to ≈ 1.5 (Fig. 5), but Mrk 231 shows F 6 /F 2 ≈ 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 F 2 /F 3 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 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 massintensity 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 singletemperature model results on the H 2 O 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 H 2 O 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, L T IR , the contribution to the luminosity by a given T dust component i is f i = L i IR /L T IR , and the observed H 2 O-to-IR luminosity ratio is where (L H 2 O /L IR ) i are the values plotted in Fig. 3a2-f2 (for ∆V = 100 km s −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 L H 2 O /L T IR is small, even though f cold may be high. Since our

H 2 O emission and monochromatic luminosities
The H 2 O submm emission of lines 2 − 6 essentially involves two excitation processes, that of the base level (2 12 for ortho and 1 11 for para-H 2 O) 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 2 12 − 1 01 line at 179 µm (ortho) or 1 11 − 0 00 at 269 µm (para). In optically thin conditions and for fixed N H 2 O and ∆V, our models then show a linear correlation between the H 2 O luminosities L H 2 O and the product of the continuum monochromatic luminosities responsible for the excitation, L 179 × L 75 (ortho) or L 269 × L 101 (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 H 2 O is independent of L 179(269) ).  ignored, the double dependence of L H 2 O on two monochromatic luminosities implies that L H 2 O is (nearly) proportional to L IR . Our predicted SEDs indicate that, for small variations in T dust around 55 K, L 269 ∝ L 2/7 IR and L 101 ∝ L 1/2 IR . Therefore, for the para-H 2 O lines 2 − 4, L 2−4 ∝ L 0.8 IR in optically thin conditions, slightly slower than linear. For the ortho lines, L 179 ∝ L 1/3 IR and L 75 ∝ L 2/3 IR , so that L 5−6 ∝ L IR . This explains why, in Fig. 3(a2-b2), the L 2−4 /L IR ratios show a slight decrease with increasing T dust above 55 K, while L 5−6 /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).

The
In Fig. 8 we show the L H 2 O /L IR ratios (with ∆V = 100 km s −1 ; L H 2 O /L IR ∝ ∆V) for lines 2, 4, and 6 as a function of the f 25 / f 60 color. The observed f 25 / f 60 was used by Y13 to characterize the H 2 O emission and is especially relevant given that the H 2 O 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 H 2 O /L IR values are upper limits.
The first conclusion inferred from Fig. 8 is that the range of f 25 / f 60 colors measured by Y13 (between the dashed lines) matches T dust in the ranges favored by the observed H 2 O line flux ratios, that is, 50−75 K and optically thin conditions (τ 100 = 0.1) and also T dust = 60 − 95 K and τ 100 = 1.0. This indicates that the warm environments responsible for the H 2 O emission are best traced in the continuum in this wavelength range, but also that the f 25 / f 60 color alone involves degeneracy in the dominant T dust and τ 100 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 H 2 O emission in very warm sources (with detection of lines 7 − 8).
Second, it is also relevant that the L H 2 O /L IR values differ by a factor 2 between models with warm dust in the optically thin regime (T dust = 55 K, τ 100 = 0.1, N H 2 O /∆V = 10 15 cm −2 /(km s −1 )) and those with very warm dust in the optically thick regime with high H 2 O columns (T dust = 95 K, τ 100 = 1, N H 2 O /∆V = 5 ×10 15 cm −2 /(km s −1 )), potentially explaining why sources with different physical conditions show similar L H 2 O /L IR ratios (Y13).
Third, in optically thin conditions (τ 100 ∼ 0.1) and if collisional excitation is unimportant, the models with constant N H 2 O /∆V = 10 15 cm −2 /(km s −1 ) (blue symbols) predict a slow decrease in L 2−4 /L IR and a nearly constant L 5−6 /L IR with increasing f 25 / f 60 , as argued above. This behavior, however, fails to match the observed trends (Y13), as L 2−4 /L IR and L 6 /L IR decrease by factors of ∼ 2 and ∼ 3, respectively, when f 25 / f 60 increases from 0.08 to 0.15. When collisional excitation is included (magenta symbols), the L 2−4 /L IR ratios show a stronger dependence on f 25 / f 60 , but L 6 /L IR still changes only slightly with f 25 / f 60 . Therefore, optically thin models with varying T dust but constant τ 100 , N H 2 O /∆V, and ∆V cannot account for the observed L H 2 O /L IR − f 25 / f 60 trend. This indicates that, in optically thin galaxies, parameters other than T dust are systematically varied when f 25 / f 60 is increased and that optically thick sources also contribute to the observed trend: (i) Galaxies in the optically thin regime (with τ 100 < 1) are predicted to show a very steep dependence of L H 2 O /L IR on τ 100 for constant T dust and N H 2 O /(∆Vτ 100 ) (that is, for constant H 2 O abundance), with higher τ 100 impling lower f 25 / f 60 . We illustrate this point in Fig. 8 with the red squares, corresponding to fix T dust = 55 and 65 K and N H 2 O /(∆Vτ 100 ) = 5 × 10 15 cm −2 /(km s −1 ), with τ 100 ranging from 0.1 to 0.3. Therefore, we expect that the observed increase in f 25 / f 60 is not only due to an increase in T dust from source to source, but also to variations in τ 100 in the optically thin regime. Examples of galaxies in this regime are the AGNs NGC 6240 and NGC 1068 (see also App. A). (ii) In the optically thick regime (τ 100 1), galaxies are also predicted to show a relatively steep variation in L H 2 O /L IR with f 25 / f 60 due to increasing T dust (black symbols in Fig. 8) because the H 2 O 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 H 2 O /L IR ratios are not much higher than in the optically thin case even if much higher N H 2 O /∆V = 5 × 10 15 cm −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 sig- nificant contribution to L IR will additionally decrease L H 2 O /L IR (Y13).
In summary, the steep decrease in L H 2 O /L IR at f 25 / f 60 ≈ 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 f 25 / f 60 are not only due to variations in T dust but also to variations in τ 100 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 H 2 O /L IR with increasing T dust (and f 25 / f 60 ), as anticipated by Y13. To distinguish between both regimes for a given galaxy, the line ratios (specifically F 6 /F 5 , Sect. 4.2) and mostly the detection of lines 7 − 8 or the detection of high-lying H 2 O 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.

Line saturation and a theoretical upper limit to
L H 2 O /L IR Saturation of the H 2 O submm lines in optically thick (τ 100 ∼ 1) sources implies that there is an upper limit on L H 2 O /L IR × (100 km s −1 /∆V) that, in the absence of significant collisional excitation, cannot be exceeded. In Fig. 9, the L H 2 O /L IR × (100 km s −1 /∆V) ratios for lines 2, 4, and 6 are plotted as a function of τ 100 for the most favored T dust range of 55 − 95 K and N H 2 O /∆V = (1 − 5) × 10 15 cm −2 /(km s −1 ). In optically thin conditions (τ 100 0.1 for N H 2 O /∆V = 10 15 cm −2 /(km s −1 )) and without collisions, L H 2 O /L IR scales linearly (for fixed ∆V) with τ 100 because L 179(269) × L 75(101) ∝ τ 2 100 while L IR ∝ τ 100 . The curves flatten as the H 2 O lines saturate and show a maximum at τ 100 ≈ 0.5 − 1. Values of τ 100 significantly higher than unity are predicted to decrease L H 2 O /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 H 2 O submm emission is expected to arise from a region that surrounds the optically thick nuclei (see App. A). For ∆V = 100 km s −1 , the maximum attainable values of L H 2 O /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 L 2 /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 F 6 /F 2 = 2.2 ± 0.5 and F 6 /F 5 = 2.6 ± 0.8; within the uncertainties, these values are consistent with warm or very warm T dust 65 K and high N H 2 O . The H 2 O lines are most likely saturated in HFLS3 as is also indicated by the L 6 /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 H 2 O line (∼ 940 km s −1 ; see Sect 3.2). O13 reported L 2 /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 H 2 O 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 H 2 O , the above L H 2 O /L IR ratios (red dashed lines in Fig. 9) may even attain higher values, though the adopted ∆V = 100 km s −1 is too high for τ 100 < 0.3 and n H 2 = 3 × 10 5 cm −3 (Sect 3.2, eq. 6).

The correlation
The broad range in observed L IR in luminous IR galaxies with H 2 O 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 H 2 O − L IR correlations if the other parameters (T dust , τ 100 , T gas , n H 2 , N H 2 O /∆V, and ∆V) remain constant.
In Fig. 10 we show the L 2 /L IR ratio as a function of τ 100 for models with T dust = 55 K and T dust = 75 K that assume a constant H 2 O-to-dust opacity ratio, that is, N H 2 O /τ 100 = 10 18 cm −2 . According to Eq. (1), this corresponds to a constant H 2 O abundance of 7.7 × 10 −7 . Both models with ∆V = 100 km s −1 (independent of τ 100 ), and ∆V/τ 100 = 100 km s −1 (corresponding to a constant K vir = 1.3) are shown. The figure illustrates that a supralinear correlation between L H 2 O and L IR can be expected if, on average, τ 100 is an increasing function of L IR . If most sources with L IR ∼ 5 × 10 10 L ⊙ were optically thin (τ 100 ∼ 0.1), and the high-z sources with L IR ∼ 10 13 L ⊙ (O13) were mostly optically thick (τ 100 ∼ 1), one would then expect L 2 ∝ L 1.3 IR from Fig. 10. Modeled L 2 /L IR ratio as a function of τ 100 . Squares and triangles indicate T dust = 55 and 75 K, respectively. In all models, collisional excitation is included with T gas = 150 K and n H 2 = 3 × 10 4 cm −3 . N H 2 O /τ 100 = 10 18 cm −2 is adopted, corresponding to a constant H 2 O abundance of 7.7 × 10 −7 (Eq. 1). Blue symbols indicate models with ∆V = 100 km s −1 and thus with variable K vir (Eq. 5) indicated with the numbers. Green symbols show results with ∆V = 100 × τ 100 km s −1 simulating a constant value of K vir = 1.3. When compared with observations, the modeled L IR values should be considered a fraction of the observed IR luminosities (Sect. 4.3.1), and thus the modeled L 2 /L IR values are upper limits. 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.

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

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 H 2 O flux ratios reported by Y13 (their Table 2) indicate that (i) F 6 /F 2 ∼ 1.2, favoring T dust = 55 K if there is no significant collisional excitation and T dust = 75 K if the H 2 O emission arises in warm and dense gas (Fig. 5); (ii) F 6 /F 5 ∼ 1.2, consistent with the optically thin regime (Fig. 6). For these T dust , Fig. 11a shows the values of N H 2 O for ∆V = 100 km s −1 required to explain the observed L H 2 O /L IR ratios, as a function of τ 100 . Models with included or excluded collisional excitation are considered. We recall that ∆V is the velocity dispersion of the dominant structure(s) that ac- Fig. 11. a) Values of N H 2 O for ∆V = 100 km s −1 as a function of τ 100 , for T dust = 55 K (red) and 75 K (blue), required to account for the observed averaged L H 2 O /L IR 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, T gas = 150 K, and n H 2 = 5 × 10 4 − 3 × 10 5 cm −3 for T dust = 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 τ 100 = 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 L H 2 O /L IR ratios, Y13, for lines 2-6). In panels b) and d), the models indicate the values of L H 2 O /L IR for ∆V = 100 km s −1 . Lines 1, 7, and 8 are excluded from the comparison because of their low detection rates (Y13).
counts for the H 2 O emission (Sect. 3.2), and for the case of low τ 100 and relatively high densities, Eq. (6) suggests ∆V < 100 km s −1 with the consequent increase in N H 2 O (Fig. 10). The decrease in τ 100 implies the increase in N H 2 O 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 (τ 100 0.3). In Fig. 11b-e, the detailed comparison between the τ 100 = 0.1 models and the observations (Y13) is shown. Significant collisional excitation is not favored for T dust = 55 K, since it would increase F 2 relative to F 6 . In addition, these optically thin models have the drawback of overestimating F 4 /F 2 . Conversely, the T dust = 75 K models favor significant collisional excitation in order to increase F 2 relative to F 6 . The very optically thin models (τ 100 0.05) are also not favored given the very high amounts of H 2 O required to explain (with no collisional excitation) the L H 2 O /L IR ratios.
In summary, T dust = 55 − 75 K, τ 100 ∼ 0.1, and N H 2 O ∼ (0.5 − 2) × 10 17 cm −2 can explain the bulk of the H 2 O submm emission in warm star-forming galaxies ( Table 2). As shown in Fig. 8, T dust = 55 K and τ 100 = 0.1 − 0.2 predict 25-to-60 µm flux density ratios of f 25 / f 60 = (8.5 − 6.0) × 10 −2 , in agreement with the observed values for the bulk of sources (Y13), while T dust = 75 K and τ 100 = 0.1 − 0.2 predict f 25 / f 60 = 0.42 − 0.30 (close to the observed upper values). Assuming a gas-to-dust ratio of 100 by mass, τ 100 ∼ 0.1 corresponds to a column den-sity of H nuclei of N H ≈ 1.3 × 10 23 cm −2 (Eq. 1), and thus an H 2 O abundance of X H 2 O ∼ 10 −6 . To within a factor of 3 uncertainty due to the τ 100 − N H calibration, the specific values used for τ 100 and ∆V, and variations in the measurements for individual sources, this is the typical H 2 O abundance that we infer from the observed L H 2 O − L IR correlation. Molecular shocks and hot core chemistry are very likely responsible for this X H 2 O , 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 τ 100 = 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 density 7 of Σ SFR = 121 − 195 M ⊙ yr −1 kpc −2 and gas mass surface density of Σ gas = 1430 − 2860 M ⊙ 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 H 2 O 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 followup study of the relationship between L H 2 O and L HCN is required to check this point. In addition, modeling the individual sources simultaneously in the continuum and the H 2 O emission will provide further constraints on the nature of these regions.

Strong optically classified AGN sources
The general finding that the H 2 O emission is similar in starforming and strong-AGN sources (Y13) may simply indicate that the far-IR pumping of H 2 O 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 H 2 O 1 11 − 0 00 (Y13), indicating that the gas densities are higher in the circumnuclear regions of AGNs. Another difference is that the L H 2 O /L IR ratios are somewhat lower in strong AGN sources (Y13). While relatively low columns of dust and H 2 O in these sources could explain this observational result, it is also possible that high X-ray fluxes photodissociate H 2 O, reducing its abundance relative to star-forming galaxies. High abundances of H 2 O 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 H 2 O through an undepleted chemistry and pumps the excited H 2 O levels, while warm gas will further boost X H 2 O through reactions of OH with H 2 . These appear to be the ideal conditions for the presence of large quantities of H 2 O in the (circum)nuclear regions of galaxies.  Rangwala et al. 2011). Toward the far-IR optically thick nuclear region (blue symbols), the E upper < 400 K lines are expected mostly in absorption. The H 2 O emission is generated around that nuclear region, in the C extended (G-A12) component (green). Red is total. b) Resulting predicted composite PACS/SPIRE H 2 O continuumsubtracted spectrum of Arp 220, which is dominated by absorption of the continuum.