MOLECULAR GAS IN EXTREME STAR-FORMING ENVIRONMENTS: THE STARBURSTS Arp 220 AND NGC 6240 AS CASE STUDIES

At ν>230 GHz, the A3 receiver on the JCMT may present a calibration problem that has to do with the fact that the sideband ratio in this double sideband (DSB) receiver is not unity but can vary as a function of the local oscillator (LO) frequency. This may have affected our HCN 3–2 measurements, which we now reduce separately for each sideband, and intend to correct properly before coadding.

In the standard calibration, the raw normalized spectrum is scaled by T_cal = (1 + G_is)ΔT, where G_is = G_i/G_s (=1 for the DSB mode) is the ratio of the image and signal sideband gains, and ΔT depends on the mean atmospheric and telescope cabin temperatures, the telescope transmission efficiency, and the line-of-sight optical depth (see Equation (A12) of Kutner & Ulich 1981). The first step for reducing spectra for which G_is ≠ 1 is to divide the spectra by 2 in order to undo the standard online calibration. Next, the lower sideband (LSB) and upper sideband (USB) tuned spectra were averaged and corrected separately using the G_is(ν_LO) dependence given on the JCMT Web site,⁶ which for the HCN 3–2 line yielded G_is = 0.81. Finally, the resulting corrected spectra were coadded.

In Arp 220, high-resolution interferometric observations have revealed the double-horn profiles of the ¹²CO 1–0 and 2–1 lines (Scoville et al. 1991) to originate from two nuclei embedded within a circumnuclear ring or disk (Scoville et al. 1997; Downes & Solomon 1998). We see the same line profile in our ¹²CO 2–1 spectrum (Figure 1(a)), with a strong emission peak at ≃−200 km s⁻¹ (relative to V_LSR = 5454 km s⁻¹) and a less-intense peak at ≃+50 km s⁻¹. These two peaks correspond, respectively, to the western, blueshifted nucleus and to the eastern, redshifted nucleus in the molecular disk (Downes & Solomon 1998). Our ¹²CO 3–2 line, single-dish spectrum shows a double-horn profile similar to the lower transitions (Lisenfeld et al. 1996; Mauersberger et al. 1999).

One subtle difference exists however, and is confirmed by our own ¹²CO 3–2 spectrum (Figure 1(b)). The relative strength of the peak at ≃+50 km s⁻¹ is greater in the 3–2 line than in the 1–0 and 2–1 spectra. In the 3–2 spectrum, the western component is only slightly stronger than the eastern. Recent interferometric observations with the Submillimeter Array (SMA) showed that the CO 3–2 and the 890-μm dust emission is indeed strongest in the western nucleus (Sakamoto et al. 2008).

The ¹³CO lines observed in Arp 220 with the IRAM 30 m telescope and the JCMT are shown in Figures 1(c) and (d), respectively. Although the ¹³CO lines seem somewhat narrower than their ¹²CO counterparts, this may be simply an effect of the low signal-to-noise ratio (S/N) in the line wings and the uncertain baseline level. In general, all the ¹²CO and ¹³CO lines seem to have very similar line profiles, within the available S/N. Both the ¹³CO 1–0 and 2–1 spectra observed with the IRAM 30 m telescope show tentative evidence of double peaks, consistent with the ¹²CO lines. In the 1–0 spectrum, both peaks are equally strong, whereas the peak at ≃−200 km s⁻¹ is stronger in the 2–1 spectrum. This also appears to be the case in the 2–1 and 3–2 spectra observed with the JCMT (Figure 1(d)), although we note the 3–2 spectrum in particular is very noisy. The fact that the ¹³CO 2–1 spectra obtained with the two telescopes are in agreement in terms of line shape as well as flux density gives us confidence that our observations are able to robustly discern the line shapes of even faint lines such as ¹³CO 2–1.

The HCN 1–0, 3–2, and 4–3 spectra observed in Arp 220 are shown in Figures 1(e)–(g), respectively. The HCN 1–0 spectrum shows a clear double-horn profile similar to ¹²CO 3–2, with equally strong peaks. The HCN 3–2 and 4–3 line profiles are both very broad (FWHM ≃ 540–590 km s⁻¹) and share the same overall shape. They both show evidence of double peaks with the ≃+50 km s⁻¹ component being more prominent. The shape is unlike the HCN(1–0) spectrum—as well as the ¹²CO 2–1 and 3–2 spectra—where the ≃−200 km s⁻¹ component is brighter.

The CS lines—of which only CS 3–2 has been previously reported for Arp 220—trace a much overlapping excitation range with that of HCN (see Table 6), and are shown in Figures 1(h)–(k). All CS lines have double-horn profiles, with the ≃+50 km s⁻¹ peak stronger in the CS 3–2, 5–4, and 7–6 spectra, while the two peaks are almost the same strength in the 2–1 spectrum. This mimics the observed evolution from low- to high-J in the HCN line profiles. We note, however, the apparent asymmetry of the CS 3–2 line may be an artifact of the limited receiver bandwidth. In determining the baseline, we forced the right edge of this spectrum to zero, so our line-flux estimate for this line is likely an underestimate.

Finally, in the last column of Figure 1, we show the HCO⁺ 4–3, HNC 1–0, and the blended C¹⁸O 1–0 and HNCO 5_0,5–4_0,4 lines. Similarly to the HCN and CS lines, the HCO⁺ spectrum peaks at ≃+50 km s⁻¹, and shows some faint emission at ≃−200 km s⁻¹, which corresponds to the stronger ¹²CO feature. In this respect, the HCO⁺ spectrum looks similar to CS 7–6, where the western component is almost completely quenched. In stark contrast to this, the double-horn spectrum of HNC 1–0 shows a remarkably strong emission peak at ≃−200 km s⁻¹, suggesting a substantial fraction of the HNC is concentrated in the western nucleus. The blended C¹⁸O 1–0 and HNCO 5_0,5–4_0,4 spectrum is shown in Figure 1(n). In order to determine separate parameters for C¹⁸O 1–0 and HNCO 5_0,5–4_0,4, two Gaussian components with the same width and fixed separation were fitted. From the resulting intensities we calculated integrated line fluxes and luminosities (Table 3).

It means that the double-horn structure seen in the low-J¹²CO spectra is also discernible in the high-J spectra of HCN, CS, and HCO⁺, albeit with the relative strengths of the two peaks reversed. In fact, a rather clear trend is apparent: the ≃−200 km s⁻¹ peak is stronger in the ¹²CO 2–1 spectrum, the peaks are about equally strong in the ¹²CO 3–2 and HCN 1–0 profiles, and the ≃+50 km s⁻¹ peak is stronger in the HCN 3–2 and 4–3 spectra. Thus, as we climb the J-ladder and probe higher and higher critical densities, the strength of the emission shifts from the western to the eastern nucleus. This picture is confirmed by the CS 7–6, and HCO⁺ 4–3 spectra which have even higher critical densities than the HCN 1–0 line and are completely dominated by the ≃+50 km s⁻¹ component with very little or no emission at all at ≃−200 km s⁻¹.

An important point to make here is that it is only due to the high-density gas tracers such as HCO⁺, HCN, and CS that we are able to conclude that the eastern nucleus harbors most of the dense gas. Without those, we would have been inclined to conclude, based on the CO 3–2 spectrum (Figure 1(b)) and its distribution (Sakamoto et al. 2008), that most of the dense gas was contained in the western component.

Given the modest (yet real) differences of the relative line strengths across the velocity range of Arp 220, and for reasons of simplicity and straightforward comparison to NGC 6240, we decided to use the line intensities average over its entire profile and the associated line ratios in our subsequent analysis (Section 4).

The ¹²CO 2–1 and 3–2 spectra of NGC 6240 are shown in Figures 2(a) and (b), respectively. Unlike Arp 220, the ¹²CO 3–2 and 2–1 spectra in NGC 6240 have almost identical shapes, with spectra having a single-peaked profile, skewed toward the blue (≃−200 km s⁻¹ relative to v_LSR = 7359 km s⁻¹). This is consistent with high-resolution interferometric observations of low-J  ¹²CO lines in the inner regions of NGC 6240, which suggest more than half of the molecular gas resides in a central structure dominated by turbulent flows, rather than systematic orbital motion (Tacconi et al. 1999). The ¹³CO 2–1 spectra obtained with the JCMT and IRAM 30 m are shown in Figures 2(c) and (d), respectively, and both appear similar in shape to their ¹²CO counterparts. Unfortunately, we cannot compare the ¹³CO 3–2 spectrum (Figure 2(d)) as no obvious detection of this transition was made. We are able, however, to set a sensitive upper limit on its integrated flux (Table 3).

The HCN 1–0, 3–2, and 4–3 transitions observed in NGC 6240 are shown in Figures 2(e)–(g). Although the HCN 1–0 line has previously been detected (Solomon et al. 1992a; Tacconi et al. 1999; Nakanishi et al. 2005), this is the first time the 3–2 and 4–3 transitions have been detected in this galaxy. All three HCN lines peak at ≃−200 km s⁻¹ and are consistent with the CO line profiles.

Unlike Arp 220, where several CS lines were observed, only the 7–6 line was observed in NGC 6240 due to limited observing time. The spectrum is shown in Figure 2(h). This is the only detection of CS in NGC 6240 to date and the line profile is consistent with those of CO and HCN.

Finally, the HCO⁺ 4–3 spectrum observed in NGC 6240 is shown in Figure 2(i). The HCO⁺ 4–3 line profile is in good agreement with the 1–0 line (Nakanishi et al. 2005) and is consistent with a single peak at ≃−200 km s⁻¹.

In summary, all of the transitions detected in NGC 6240 exhibit a similar, Gaussian-like velocity profile, peaking at about the same velocity (≃−200 km s⁻¹), in contrast to Arp 220 where all the lines not only show a double-peaked profile but a strong evolution in the dominant peak is also discernible. The similar velocity profile over such a large range of densities probed in NGC 6240 signifies a kinematical state of the molecular gas where the various phases remain well mixed down to much smaller scales than in Arp 220 where the denser phases seem settled to two counter-rotating dense gas disks, enveloped by a much more diffuse CO-bright gas phase. Subarcsec interferometric imaging of the HCN, CS, and HCO⁺ imaging of these systems is valuable in confirming such a picture, a possible result of the different merger state of these two strongly interacting systems.

A major goal of this study was to use our observations as well as results from the literature to compile a large, identical molecular line catalog for Arp 220 and NGC 6240. Thus the literature was scanned for all relevant molecular line measurements and for each transition the weighted average of every independent flux measurement was calculated. In cases where a measurement deviated significantly from the mean for obvious reasons (limited bandwidth, poor conditions, etc.), it was discarded from the average. Note that often in the literature line measurements are given in (main beam or antenna) temperature units and only rarely in Janskys. In those cases, we used the aperture and/or main-beam efficiencies quoted in the paper or, if those were not given, the average values for the particular telescope (typically tabulated in the observing manual). The final tally of lines is given in Table 3 along with their corresponding fluxes and global line luminosities. Global line ratios were calculated as the velocity/area averaged brightness temperature ratios (equivalent to taking the ratio of the line luminosities) and are listed in Tables 4 and 5.

Table 4. CO Line Ratios for Arp 220 and NGC 6240

Galaxy	12CO(2–1)	12CO(3–2)	12CO(1–0)	12CO(2–1)	12CO(3–2)	C18O(1–0)
	12CO(1–0)	12CO(1–0)	13CO(1–0)	13CO(2–1)	13CO(3–2)	13CO(1–0)
	r21	r32	$\mathcal{R}_{10}$	$\mathcal{R}_{21}$	$\mathcal{R}_{32}$	$\mathcal{R}^{(18)}_{10}$
Arp 220	0.7 ± 0.1	1.0 ± 0.1	43 ± 10	18 ± 3	8 ± 2	1.0 ± 0.3
NGC 6240	1.2 ± 0.2	1.1 ± 0.2	45 ± 15	52 ± 12	⩾32	⋅⋅⋅

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Our observations of ¹²CO 2–1 in both Arp 220 and NGC 6240 yield somewhat higher line fluxes than previous single-dish and interferometry observations listed in Table 3, although, we note that Wiedner et al. (2002) using the JCMT measured an even larger line flux in Arp 220 than ours. A careful scrutiny of the individual subscans in our data and of the calibration on the dates the observations were made revealed nothing out of the ordinary. In fact, given that each source was observed on different nights, yet both have ∼40% higher line fluxes than previously published values, suggest that the offset may well be real. A close inspection of previously published ¹²CO 2–1 spectra reveals they all suffer from either limited bandwidth (the case for single-dish measurements), and/or the sources have been overresolved spatially (in the case of interferometric observations). Because only few observations favor the higher ¹²CO 2–1 fluxes in Arp 220 and NGC 6240, however, we conservatively adopt the mean flux value of all available measurements for our analysis. The HCN 4–3 line in Arp 220 was first observed by Wiedner et al. (2002), who found a line flux only half as large as ours. Here, we adopt our own measurement since our spectrum has a higher S/N than that of Wiedner et al. (2002), and the latter used aperture efficiencies which deviate significantly from the standard values given for JCMT.

In NGC 6240 we find a HCN 1–0 flux that is half the previously published single-dish measurement (Solomon et al. 1992a) but is in excellent agreement with the interferometric measurement by Nakanishi et al. (2005). Because our data were obtained over the course of 2 days in excellent weather conditions and because a careful reduction of the data from each day separately yielded consistent results, we find the lower values more likely. In the case of Arp 220, our estimate of the HCN 1–0 line flux is about 35% larger than that of Solomon et al. (1992a), which is just within the errors. Recent, independent observations of the HCN 1–0 line in Arp 220 and NGC 6240 with the IRAM 30 m telescope yield line fluxes in agreement with our measurements (J. Graciá-Carpio 2006, private communication), suggesting the HCN 1–0 line fluxes given by Solomon et al. (1992a) may have been affected by systematic calibration uncertainties. Nonetheless, for the line ratio analysis in this paper, we conservatively adopt the weighted average of all three HCN 1–0 measurements in these two systems.

We end this section by noting that Table 3 constitutes the most extensive molecular line catalog available for any (U)LIRG. In terms of extragalactic objects in general, it is matched only by data published for the much less extreme starburst galaxies M 82 and NGC 253 (e.g., Weiß et al. 2005; Seaquist et al. 2006; Bayet et al. 2004; Bradford et al. 2003). However, unlike M 82 and NGC 253, where only the ¹²CO 1–0 and 2–1 transitions have been observed over their entire extent (e.g., Weiß et al. 2005), our study picks up the global emission from Arp 220 and NGC 6240 for all transitions, allowing for an unbiased study of the properties of their entire molecular gas reservoir using the corresponding millimeter/submillimeter line ratios. This mirrors observations of the millimeter/submillimeter lines observed in high-z starbursts where much fewer lines are available, and thus cannot yield meaningful constraints on the conditions or the mass per gas phase for their molecular gas reservoirs.

The emergence of ¹³CO transitions as the first ones marking a strong deviation from any single gas phase reproducing the relative strengths of the ¹²CO transitions is further highlighted in the case of Arp 220 by the observed strong C¹⁸O J = 1–0 line emission. There the C¹⁸O/¹³CO 1–0 line ratio of 1.0 ± 0.3 is the highest such ratio obtained over an entire galaxy—by comparison, Papadopoulos et al. (1996) measure such high values toward only a few of the brightest giant molecular clouds (GMCs) in NGC 1068. Assuming the low abundance of [C¹⁸O/¹³CO] = 0.15 found for the Milky Way (Langer & Penzias 1993) such high C¹⁸O/¹³CO brightness ratios argue for much of the ¹³CO J = 1–0 emission emerging from a phase where $\rm \tau _{10}(^{13}CO)> 1$, sharply deviating from the optical thin transitions deduced from the LVG of only the ¹²CO, ¹³CO transitions described in the previous section. A much denser gas phase can naturally yield the significantly higher optical depths needed to explain the high C¹⁸O/¹³CO J = 1–0 ratio.

In this section, we use the many transitions of HCN, CS, and HCO⁺ in our catalog—probing a wide range of critical densities and excitation temperatures in a redundant manner (see Table 6)—to constrain the properties of the dense gas phase.

In order to model the radiative transfer of these lines, we constructed a modified version of the original LVG code. The modifications were straightforward since HCN, CS, and HCO⁺ are simple linear molecules such as CO. For HCN, CS, and HCO⁺, the collision rate constants provided by Green & Thaddeus (1974), Turner et al. (1992), and Flower (1999) were adopted. We explored a similar density range as we did for the CO lines, but limited the temperature range to T_d ⩽ T_k ⩽ 120 K. The lower limit set by the dust temperature (T_d), reflects the fact that the dust and gas are thermally decoupled, with the FUV-induced photoelectric and turbulent gas heating (and its cooling via the spectral line rather than the continuum emission) setting T_k ≳ T_d. Only in the densest, most FUV-shielded and quiescent regions of the GMCs can we expect T_k ≃ T_d. The upper limit is the typical equilibrium temperature of purely atomic, C⁺-cooled HI in the cold neutral medium (e.g., Wolfire et al. 2003). Based on detailed fits to their FIR/submillimeter SEDs (Dopita et al. 2005), we adopted a dust temperature of ∼40 K for both Arp 220 and NGC 6240. Since the molecules have different abundances, the range in Λ is also different for a common velocity gradient (dv/dr). In our own Galaxy, typical abundances are [HCN/H₂] ∼ 2 × 10⁻⁸ (Bergin et al. 1996; Lahuis & van Dishoeck 2000), [CS/H₂] ∼ 10⁻⁹ (Paglione et al. 1995; Shirley et al. 2003), and [HCO⁺/H₂] = 8 × 10⁻⁹ (Jansen 1995). In order to accommodate the substantial uncertainties associated with the abundances, the range of Λ used was 10⁻¹⁰–10⁻⁷, 10⁻¹¹–10⁻⁸, and 10⁻¹¹–10⁻⁸(km s⁻¹pc⁻¹)⁻¹ for HCN, CS, and HCO⁺, respectively. These correspond to velocity gradients in the range ∼0.2–800 km s⁻¹ pc⁻¹ for the abundances quoted above.

In the case of Arp 220, the observed HCN ratios are consistent with LVG solutions in the range n(H₂) ≃ (0.3 − 1) × 10⁶ cm⁻³, and Λ_HCN ≃ (0.3 − 1) × 10⁻⁹(km s⁻¹pc⁻¹)⁻¹, for which τ_HCN(1-0) ≃ 3 − 7. From the fit of the CS lines we find n(H₂) ≃ (0.3 − 3) × 10⁶ cm⁻³, and Λ_CS ≃ (0.01 − 1) × 10⁻⁸(km s⁻¹pc⁻¹)⁻¹. Finally, the subthermal HCO⁺ ratios allow for solutions in the range n(H₂) ≃ (0.3 − 1) × 10⁴ cm⁻³, and $\Lambda _{{{\rm HCO^+}}}\simeq (0.3{-}1)\times 10^{-8}\, \left(\mbox{km\,s}^{-1}\mbox{pc}^{-1} \right)^{-1}$. All the aforementioned solutions are for T_k ⩾ 40 K and their details are shown in Table 7. There it can be seen that while HCN, CS, and HCO⁺ all trace what in broad terms can be characterized as dense gas (i.e., n(H₂) ≳ 10⁴ cm⁻³), with the density range of the CS and HCN solutions being very similar. Moreover we find the CS and HCN emission tracing gas with densities ∼100 times higher than that traced by the HCO⁺ lines. This simply reflects the high HCN, CS and the low HCO⁺ line ratios measured in Arp 220, further reinforced by the ∼ five times lower critical densities of the HCO⁺ transitions than, e.g., those of HCN at the same rotational level (see Table 6). These results are certainly consistent with Galactic surveys of CS lines, which have shown that they are unmistakable markers of very dense, high-mass star-forming cores (Plume et al. 1997; Shirley et al. 2003).

Table 7. LVG Solution Ranges, see Section 4.2 for Details

Galaxy	Tk/[K]	n(H2)/cm−3	Λ/$\rm (km\,s^{-1}\,pc^{-1})^{-1}$	χ2†ν	Kvir
Arp 220:
HCO+
	10–40	(1 − 3) × 104	(0.3 − 1) × 10−8	2.1–2.8	0.4–0.7
	45 − 95	1 × 104	0.3 × 10−8	2.1	1.2
	100 − 120	0.3 × 104	1 × 10−8	2.0 − 2.1	0.7
HCN
	10–25	(0.3–1) × 106	(0.3–1) × 10−7	0.5–1.3	0.02–0.17
	30–40	1 × 106	0.3 × 10−9	0.7 − 0.8	3.1
	45 − 120	0.3 × 106	0.1 × 10−8	0.7 − 0.8	1.7
CS
	10–30	10 × 106	(0.001 − 0.3) × 10−8	0.2–0.5	0.005 − 1.5
	35–40	3 × 106	0.01 × 10−8	0.2–0.3	0.3
	45 − 80	1 × 106	0.1 × 10−8	0.2 − 0.3	0.05
	85 − 120	0.3 × 106	(0.3 − 1) × 10−8	0.3	0.009 − 0.03
NGC 6240:
HCO+
	10–25	(1 − 3) × 104	(0.3 − 1) × 10−8	1.9 − 2.7	0.4–0.7
	30 − 55	1 × 104	0.3 × 10−8	1.8 − 1.9	1.2
	60 − 120	0.3 × 104	1 × 10−8	1.6 − 1.7	0.7
HCN
	10–20	(0.1 − 1) × 106	(0.3 − 1) × 10−7	1.3 − 1.7	0.03 − 0.1
	25–30	0.3 × 106	0.1 × 10−8	1.3 − 1.4	1.7
	35 − 55	0.1 × 106	1 × 10−8	1.3 − 1.5	0.3
	60 − 120	0.3 × 106	0.03 × 10−8	1.4 − 1.5	5.5

Notes. The most likely solutions have been highlighted in bold. † the chi-squared fit was calculated as $\chi ^{2}_{\nu } = \Sigma_i \frac{1}{\sigma _i}^{2} (R_{\rm obs} - R_{\rm model})^2$, where R_obs and R_model are the observed and modeled line ratios, respectively, and σ_i is the associated error.

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For NGC 6240, the HCN LVG fits converge mostly over n(H₂) ≃ (1 − 3) × 10⁵ cm⁻³, T_k ⩾ 40 K, and Λ_HCN ≃ 3 × 10⁻¹⁰ − 1 × 10⁻⁸(km s⁻¹pc⁻¹)⁻¹, with an optically thick HCN J = 1–0 line (τ_HCN(1-0) ≃ 3 − 29). From the HCO⁺ lines we find solutions in the range n(H₂) ≃ (0.3 − 1) × 10⁴ cm⁻³, T_k ⩾ 40 K, and $\Lambda _{{{\rm HCO^+}}}\simeq (0.3{-}1)\times 10^{-8}\, \left(\mbox{km\,s}^{-1}\mbox{pc}^{-1} \right)^{-1}$. Thus in NGC 6240, as in Arp 220, HCN is tracing gas which is significantly denser than that traced by HCO⁺, over a similar range of temperatures.

In an attempt to "break" the significant degeneracy that still exists in the LVG solutions for the dense gas we make use of the fact that at least in the Galaxy, most of the dense star-forming gas is found in self-gravitating cores (Shirley et al. 2003). In the cases of Arp 220 and NGC 6240, we use this to try and limit the allowed (n(H₂), T_k, Λ_X)-parameter space by examining the ratio of the velocity gradient inferred from the LVG fits to that expected for virialized cores, namely

$$\begin{equation} K_{\rm vir} = \frac{(dv/dr)_{{{\rm LVG}}}}{(dv/dr)_{\rm vir}} \sim 1.54 \frac{[{\rm X}/{\rm H}_2]}{\sqrt{\alpha } \Lambda _{X}} \left(\frac{n({\rm H}_2)}{10^3\,\mbox{cm}^{-3}} \right) ^{-1/2}, \end{equation} \tag{ 5 }$$

where [X/H₂] is the abundance ratio of the given molecule X (see Papadopoulos & Seaquist 1999, or Goldsmith 2001 for a derivation), and α ∼ 1 − 2.5 is a constant whose value depends on the assumed density profile of a typical cloud (Bryant & Scoville 1996).

For virialized gas we expect K_vir ∼ 1 to within a factor of a few (due to uncertainties in cloud geometry, density profile, and abundance assumed to deduce the observed dv/dr from Λ_X). LVG solutions that correspond to K_vir >> 1 indicate nonvirial, unbound motions and are ruled out for dense star-forming gas along with those that have K_vir ≪ 1 (gas motions cannot be slower than those dictated by its self-gravity). For Arp 220 and Table 7 we see that for both HCO⁺ and HCN, LVG solutions with T_k ⩾ 40 K have K_vir ≃ 1, and thus these conditions do not single out any particular range of solutions. For CS however, we find K_vir ≪ 1 for most solutions except for T_k = 35–40 K, $\rm n({\rm H}_2)=3\times 10^6$ cm⁻³ and $\rm \Lambda _{{{\rm CS}}}= 10^{-10}\,(km\,s^{-1}\,pc^{-1})^{-1}$. This suggests that the true CS abundance may be significantly higher than assumed (by at least an order of magnitude). It should be noted, however, that other studies have found evidence for CS abundances close to 10⁻⁸ in Galactic star-forming cores (Hatchell & van der Tak 2003). Adopting this value would result in K_vir values closer to unity. Finally for NGC 6240 all solutions that fit the HCO⁺ ratios are consistent with K_vir ∼ 1, while for HCN this is the case for T_k = 40 − 55 K and n(H₂) = 10⁵ cm⁻³ and Λ_HCN = 10⁻⁸(km s⁻¹ pc⁻¹)⁻¹ (see Table 7).

4.2.1. HNC versus HCN

In both galaxies the low HCO⁺ line ratios result in LVG solutions with densities mostly ∼10–100 times lower than those deduced from the analysis of the HCN ratios (Table 7). Thus if only HCO⁺ line observations were used, they would imply a bulk dense gas phase dominated by less extreme densities than is actually the case, especially in Arp 220. The several HCO⁺ and HCN transitions observed, and the independent analysis afforded by the CS transitions (for Arp 220), make it unlikely that the aforementioned conclusions can be attributed to enhanced HCN abundances (and thus thermalization of its transitions at lower densities because of radiative trapping) caused by XDRs, or simply to IR-pumping of HCN J = 1–0. Thus recent claims that HCO⁺ is a better tracer of the dense gas in (U)LIRGs than HCN (Graciá-Carpio et al. 2006), do not seem to hold in the cases of Arp 220 and NGC 6240, and a more comprehensive approach is needed to decide the case (Papadopoulos 2007).

In this section, we make use of the constraints on the dense gas in Arp 220 and NGC 6240 obtained in Section 4.2 to derive the total mass of the dense gas in these two systems. We do this using two different methods, the results of which are summarized in Table 8. The first one assumes the dense gas reservoir to be fully reducible to an ensemble of self-gravitating units whose molecular line emission does not suffer any serious cloud–cloud shielding. Then it can be shown (e.g., Dickman et al. 1986) that the total mass of, e.g., an HCN-luminous dense gas phase can be found from

$$\begin{equation} M_{{{\rm dense}}}({\rm H}_2) = \alpha _{{{\rm HCN}}} L^{\prime }_{{{\rm HCN}}}, \end{equation} \tag{ 6 }$$

where $\alpha _{{{\rm HCN}}} \simeq 2.1 \sqrt{n({\rm H}_2)}/T_{b,{{\rm HCN(1-0)}}}$ is a conversion factor (e.g., Radford et al. 1991a). We insert the range of n(H₂) and T_b,HCN(1-0) values inferred from the LVG solutions to the HCN line ratios, which for Arp 220 are n(H₂) = 0.3 × 10⁶ cm⁻³ and T_b,HCN(1-0) = 34 − 62 K, and for NGC 6240 n(H₂) = (0.1 − 0.3) × 10⁶ cm⁻³ and T_b,HCN(1-0) = 31 − 40 K. From these values we find α_HCN ≃ (19 − 34) M_☉ (K km s⁻¹ pc²)⁻¹ and α_HCN ≃ (17 − 37) M_☉ (K km s⁻¹ pc²)⁻¹ for Arp 220 and NGC 6240, which in turn yields dense gas mass traced by HCN of M_dense(H₂) ≃ (1.8 − 4.2) × 10¹⁰ M_☉  and ≃(1.0 − 2.8) × 10¹⁰ M_☉  for these two systems respectively.

In a similar fashion, the analysis of HCO⁺ and CS line ratios with their corresponding T_b and n(H₂) values yields corresponding conversion factors which, along with the luminosities of the lowest HCO⁺ and CS transitions available, yield the dense gas mass traced by each species (Table 8). The constraints of an underlying density-size cloud hierarchy (see Section 4.5) demand $M({\rm H}_2)_{{{\rm HCN/CS}}}\le {\it M}_{{{\rm HCO^+}}}$ (since the HCN, CS lines are tracing a denser gas than those of HCO⁺). Thus from Table 8 we obtain M_dense(H₂) ∼ 1.5 × 10¹⁰ M_☉ (Arp 220) and M_dense(H₂) ∼ (1 − 2) × 10¹⁰ M_☉ (NGC 6240) as the best dense gas-mass estimates conforming to the aforementioned inequality. Systematic biases can be introduced by (1) the lack of a pressure-term in Equation (6) correcting for molecular gas overlying the dense gas regions, (2) by the existence of substantial stellar mass even within the dense gas phase. In both of these cases the true conversion factor and the deduced gas masses will be lower (see Bryant & Scoville 1996; Downes & Solomon 1998 for the appropriate formalism).

Finally, given the significant number of HCN transition available for both systems, we can estimate the dense gas mass by assuming the bulk of the HCN molecules are in rotational states J ⩽ 6. Then

$$\begin{eqnarray} M_{{{\rm dense}}}({\rm H}_2) &=& [{\rm H}_2/{\rm HCN}] \left[ N_1 + N_2 + N_3 + N_4 + N_5\right.\nonumber\\ && \left.+\ N_6\right] \mu m_{\mbox{{H$_2$}}}, \end{eqnarray} \tag{ 7 }$$

where N_J is the total number of HCN molecules in state J and μ = 1.36 accounts for He. The HCN population numbers are calculated from

$$\begin{equation} N_J = \frac{8\pi k \nu ^2_{J,J-1}}{h c^3 A_{J,J-1}} \beta ^{-1}_{J,J-1} L^{\prime }_{{{\rm HCN({\it J},{\it J}-1)}}}, \end{equation} \tag{ 8 }$$

where A_J,J−1 and $\beta _{J,J-1} = (1-e^{-\tau _{J,J-1}})/\tau _{J,J-1}$ are the Einstein coefficient and the escape probability for the J → J − 1 transition, respectively. Unlike the first method this approach depends directly on the assumed $\rm [HCN/H_2]$ abundance ratio while it is in principle independent of any assumptions regarding the velocity field of the dense gas phase (i.e., it remains valid for a nonvirialized, unbound gas phase), as long as the HCN line emission remains effectively optically thin (i.e., any large τ's emerge within small gas "cells"). The dense gas masses estimated with the second method (see Table 8) bracket the best estimates yielded by the first one (constrained by the density-mass hierarchy assumption), though more tightly for Arp 220 than NGC 6240.

The two methods overlap over the same range of M(H₂)_dense ∼ (1–2) × 10¹⁰ M_☉ for both galaxies, which is comparable to the total molecular gas mass estimated independently from models of their interferometrically imaged CO emission (Downes & Solomon 1998), or their CI J = 1–0 fluxes (Papadopoulos & Greve 2004). Thus the bulk of the molecular ISM in these two extreme starbursts is in a dense state with $\it n(H_2)\sim (10^5$–10⁶) cm⁻³.

It is worth comparing this mass to the fundamental upper limit set by the dynamical mass in these objects. For Arp 220, the latter was estimated to be M_dyn ≃ 4 × 10¹⁰ M_☉  (adopted to the cosmology used here) within a radius of 1.4 kpc, based on resolved observations of CO (Downes & Solomon 1998). For NGC 6240 M_dyn ≃ (0.7 − 1.6) × 10¹⁰ M_☉  within a radius of ∼500 pc (Tacconi et al. 1999; Bryant & Scoville 1999). The CO-emitting region in NGC 6240 extends significantly beyond that (Tacconi et al. 1999) and thus a significant amount of molecular gas may reside beyond the inner 500 pc. Assuming a constant rotation curve, we obtain an upper limit of M_dyn = 6 × 10¹⁰ M_☉  within R = 1.4 kpc for NGC 6240. Since any HCN/CS/HCO⁺ bright region will be contained within the CO-bright regions we expect M_dyn ⩾ M_CO ⩾ M_dense in both systems. Thus the range of M(H₂)_dense ∼ (1 − 2) × 10¹⁰ M_☉ corresponds to ∼20%–50% of the dynamical mass in Arp 220, and ≳15% up to 100% for NGC 6240.

In this section, we attempt to go a step beyond using HCO⁺, HCN, and CS lines as isolated estimators of the dense gas mass by assuming their emission emerging from an underlying density-size hierarchy, found for Galactic molecular clouds by Larson (1981), and verified by subsequent high-resolution multitransition studies. The origin of the density-size and line-width-size power laws revealed for GMCs in the Galaxy may lie in the characteristics of supersonic turbulence and its global driving mechanisms (Heyer & Brunt 2004) when present on self-gravitating structures. As such they may remain similar even in the extreme ISM conditions present in ULIRGs.

A rigorous approach to test this in extragalactic environments (where molecular clouds cannot be viewed with the same level of detail) using the global molecular line emission of various species would require incorporating such power laws in the radiative transfer model used to interpret the emergent molecular line emission. In our case, we opt for a simpler approach in which the denser HCN-bright gas is simply assumed to be "nested" within the less-dense regions traced by the HCO⁺ transitions.

We can rudimentary test for this by deriving the velocity-averaged area-filling factor of the HCN-bright relative to the HCO⁺-bright gas phase: $f_{{{\rm HCN,HCO^+}}}$, which ought to be <1. This can be found from

$$\begin{equation} \frac{L^{\prime }_{{{\rm HCN}}}}{L^{\prime }_{{{\rm HCO^+}}}}= f _{{{\rm HCN,HCO^+}}} \frac{T_{b,{{\rm HCN}}}}{T_{b,{{\rm HCO^+}}}}. \end{equation} \tag{ 9 }$$

For the LVG solution ranges above $\rm T_k=40\,K$ (Table 7) and the J = 1–0 transition of both species, the observed line luminosities and LVG-deduced intrinsic brightness temperatures yield: $0.2 \lesssim f_{{{\rm HCN,HCO^+}}} \lesssim 1.0$ (Arp 220) and $0.1 \lesssim f_{{{\rm HCN,HCO^+}}} \lesssim 0.3$ (NGC 6240). For an underlying density-size power law $\rm \langle n(H_2)\rangle \propto R^{-\alpha }$, we set $\alpha = 2 {\rm log}\left(n_{{{\rm HCO^+}}}/n_{{{\rm HCN}}}\right)/{\rm log}f_{{{\rm HCN,HCO^+}}}$ as an approximation (n_HCN, $n_{{{\rm HCO^+}}}$ are the mean volume densities deduced from the LVG fits of the HCN and HCO⁺ line ratios). For the range of densities corresponding to LVG fits of the HCN and HCO⁺ ratios (for T_k ⩾ 40 K), we find α ≳ 2, steeper than the value of α ∼ 1 typical for Galactic molecular clouds. Using the HCO⁺ and HCN J = 4–3 transitions instead yields $0.09 \lesssim f_{{{\rm HCN,HCO^+}}} \lesssim 0.6$ (Arp 220), and $0.03 \lesssim f_{{{\rm HCN,HCO^+}}} \lesssim 0.2$ (NGC6240). These smaller values are expected for emission from the two lines bracketing the entire range critical densities in our line inventory (Table 6), and an underlying density-size ISM hierarchical structure. However, even for these values of $f_{{{\rm HCN,HCO^+}}}$ the corresponding density ranges yield α ≳ 1.3, still steeper than a Larson-type power law. Such a steepening could in principle be the result of extreme tidal stripping of GMCs and then merging of the surviving densest clumps to superdense structures, yet another signature of vastly different average molecular gas properties in extreme starbursts than those in more quiescent environments.

The observed ¹²CO line emission will have contributions from both the dense and diffuse gas and the observed global line ratio of any given $J+1\longrightarrow J$ line with respect to ¹²CO 1–0 can therefore be written as

$$\begin{equation} r_{J+1,J} = \frac{r^{(A)}_{J+1,J} + C^{(12)}_{10} r^{(B)}_{J+1,J}}{1 + C^{(12)}_{10}}, \end{equation} \tag{ 10 }$$

where r^(A)_J+1,J is the line ratio for the diffuse phase (denoted by A), r^(B)_J+1,J the one for the dense phase (denoted by B), and the contrast factor, C⁽¹²⁾₁₀ is discussed below. It is seen from Equation (10) that in order to constrain the properties of the diffuse gas, one has to determine and subtract the contribution from the dense gas phase. Fortunately, this can now be done with the constraints put on the dense phase properties in Section 4.2. The contrast factor, which determines the relative emission contribution of the two gas phases, is given by C⁽¹²⁾₁₀ = f_BA(T^(B)_b,10/T^(A)_b,10), where f_BA is the velocity-integrated relative filling factor between the two phases and T^A,B_b,10 are the velocity- and area-integrated line brightness temperatures of ¹²CO 1–0. In order to determine this quantity, we make the reasonable assumption that the HCN emission is dominated by the dense gas phase and, as a result, we can write the global (observed) HCN(1–0)/¹²CO(1–0) line ratio as

$$\begin{equation} \frac{T_{b,{{\rm HCN}}}}{T_{b,{{\rm CO}}}} = \frac{C^{(12)}_{10}}{1+{C^{(12)}_{10}}} \frac{T^{(B)}_{b,{{\rm HCN}}}}{T^{(B)}_{b,{{\rm CO}}}}. \end{equation} \tag{ 11 }$$

We then estimate C⁽¹²⁾₁₀ using the observed value of T_b,HCN/T_b,CO and the T^(B)_b,HCN/T^(B)_b,CO values obtained from the LVG fits of the dense gas phase. The observed global line ratio for Arp 220 is T_b,HCN/T_b,CO = 0.18 ± 0.03, and T^(B)_b,HCN/T^(B)_b,CO = 0.5 − 0.8 as estimated from the dense gas solution range. In the case of NGC 6240, the same two line ratios are 0.08 ± 0.01 and 0.4 − 0.8. Folding in the uncertainties on the observed global ratios and allowing for the range in the ratio predicted for the dense gas, we find C⁽¹²⁾₁₀ ∼ 1.5 − 3.0 (Arp 220) and C⁽¹²⁾₁₀ ∼ 1.3 − 2.8 (NGC 6240).

With C⁽¹²⁾₁₀ determined, we can then disentangle the contributions to the ¹²CO line ratios from the diffuse and dense phases using Equation (10). If we assume the $\mathcal{R}$ ratios dominated by the diffuse phase, the resulting ¹²CO and ¹³CO emission-line ratios for Arp 220 are consistent with n(H₂) ≃ (1 − 3) × 10² cm ⁻³, T_k ⩾ 40 K, and Λ_CO ≃ (3 − 10) × 10⁻⁵(km s⁻¹ pc⁻¹)⁻¹. For NGC 6240, the LVG model converges on n(H₂) ≃ (1 − 3) × 10³ cm ⁻³ T_k ⩾ 40 K, and Λ_CO ≃ (1 − 3) × 10⁻⁶(km s⁻¹ pc⁻¹)⁻¹. All solutions correspond to nonvirialized K_vir ≳ 5 values (assuming a Galactic abundance [CO/H₂]=10⁻⁴) and have moderate ¹²CO J = 1–0 optical depths (τ₁₀ ∼ 1). This seems to be the typical diffuse gas phase found in such systems (e.g., Aalto et al. 1995; Downes & Solomon 1998) which, while dominating their low-J CO line emission, it does not contain much of their total molecular gas mass.