A COSMIC-RAY-DOMINATED INTERSTELLAR MEDIUM IN ULTRA LUMINOUS INFRARED GALAXIES: NEW INITIAL CONDITIONS FOR STAR FORMATION

For the UV-shielded, and mostly subsonic, dense gas cores deep inside molecular clouds, the heating rate is given by

$$\begin{equation} \rm \Gamma _{CR}\sim 1.5\times 10^{-24}\left(\frac{\zeta _{CR}}{10^{-17}\,s^{-1}}\right) \left(\frac{{{\it {n}}}(H_2)}{10^4\,cm^{-3}}\right)\,erg\,cm^{-3}\,s^{-1}, \end{equation} \tag{ 7 }$$

where ζ_CR(s⁻¹)∝U_CR is the CR ionization rate per H₂ molecule. The gas cooling via gas-dust interaction can be expressed as

$$\begin{equation} {\rm \Lambda _{g-d}\sim 10^{-25}\left(\frac{{{\it {n}}}(H_2)}{10^4\,cm^{-3}}\right)^2} {T^{1/2}_{\rm k}} \left({{T_{\rm {k}}-T_{\rm dust}}}\right)\,\rm erg\,cm^{-3}\,s^{-1} \end{equation} \tag{ 8 }$$

(see Tielens 2005 for derivation of both expressions). The most important line cooling of this gas phase is through rotational lines of CO and a few other molecular species whose rotational ladder energy levels reach down to E_ul/k_B ∼ (5–10) K. Following the detailed study of Goldsmith (2001), we parameterize the molecular line cooling as

$$\begin{equation} {\rm {\Lambda _{line}\sim 6\times 10^{-24}\left[\frac{{\it {n}}(H_2)}{10^4\,cm^{-3}}\right]}}^{1/2} \left(\frac{T_{\rm k}}{10\,\rm K}\right)^{\beta } \rm erg\,cm^{-3}\,s^{-1}. \end{equation} \tag{ 9 }$$

The density dependence was extracted from a fit of the parameter α in Table 2 of Goldsmith (2001), which reproduces the values of the Λ_line to within ≲20% for n(H₂) = (10⁴–10⁶) cm⁻³, which spans the density range of dense cores within GMCs. For that density range β ∼ 3.

The resulting gas temperature for the CR-heated gas can then be estimated from the equation of thermal balance

$$\begin{equation} \rm \Gamma _{CR} = \Lambda _{line}+\Lambda _{g-d}. \end{equation} \tag{ 10 }$$

Setting the dust temperature to T_dust = 0 K will yield a minimum T_k value while also allowing a simple analytic solution of Equation (10). Substituting the expressions from Equations (7)–(9) into the latter yields

$$\begin{equation} T^3 _{\rm k,10}+ \rm 0.526{{{\it {n}}}}^{3/2}_4{\it T}^{3/2} _{\rm k,10}=\rm 0.25{{{\it {n}}}}^{1/2}_4\zeta _{-17}, \end{equation} \tag{ 11 }$$

where T_k,10 = T_k/(10 K), n₄ = n(H₂)/(10⁴ cm⁻³) and ζ₋₁₇ = ζ_CR/(10⁻¹⁷ s⁻¹). An exact solution of the latter is then

$$\begin{equation} T_{\rm k,10}=\rm 0.630\big[\big({{{\it {n}}}}^{1/2}_4 \zeta _{-17}+0.276676{{{\it {n}}}}^3_4\big)^{1/2}-0.526{{{\it {n}}}}^{3/2}_4\big]^{2/3}. \end{equation} \tag{ 12 }$$

For ζ_CR,Gal = 5 × 10⁻¹⁷ s⁻¹ for the Galaxy (e.g., van der Tak & van Dishoeck 2000) and n(H₂) = 10⁴ cm⁻³, the latter yields T_k ∼ 9 K, which is typical for UV-shielded gas immersed in Galactic CR energy density (e.g., Goldsmith 2001), and is deduced by numerous observations in the Galaxy (e.g., Pineda & Bensch 2007; Bergin & Tafalla 2007). For ζ_CR,CSB = (1–4) × 10³ζ_CR,Gal expected for the ISM of extreme starbursts and n(H₂) = (10⁴–10⁶) cm⁻³ (typical densities for dense cores in the Galaxy, e.g., Bergin & Tafalla 2007), Equation (12) yields T_k ∼ (80–240) K, as the minimum possible temperature for the molecular gas in compact extreme starbursts (see Figure 1). Turbulent gas heating (e.g., Pan & Padoan 2009) that may "seep" down the molecular cloud hierarchical structures to the dense gas cores (although because they are typically subsonic, such heating is expected to be negligible), any sort of mechanical heating of the dense gas in ULIRGs (Baan et al. 2010) or warmer dust because of IR light "leaking" deep inside molecular clouds can only raise this temperature range.

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The gas cores deep inside the CR-heated regions of molecular clouds in the Galaxy and especially the highest density ones (∼(10⁵–10⁶) cm⁻³) are typically dominated by near-thermal motions (e.g., Bergin & Tafalla 2007). However, the often large turbulent linewidths found in the dense gas disks of nearby ULIRGs (Downes & Solomon 1998; Sakamoto et al. 2008; Matsushita et al. 2009) could in principle affect even dense gas core kinematics and thus their line cooling function. Following Goldsmith (2001) that any such macroscopic motions are driven mostly by self-gravity, a new effective line cooling function would be

$$\begin{equation} \rm \Lambda ^{(eff)} _{line}= \Lambda _{line}\times \left(\frac{{{\it {n}}}(H_2)}{10^3\,cm^{-3}}\right)^{1/2}, \end{equation} \tag{ 13 }$$

which roughly quantifies the effect of increased transparency (and thus cooling power) of molecular line photons. An optical depth dependence of τ∝(dV/dR)⁻¹ is adopted, where (dV/dR)_VIR = [n(H₂)/(10³ cm⁻³)]^1/2 km s⁻¹ pc⁻¹ is the velocity gradient of macroscopic motions for self-gravitating gas. The equation of thermal balance and its solution then become

$$\begin{equation} T^3_{\rm k,10}+{\rm {0.166{{{\it {n}}}}_4}} T^{3/2} _{\rm k,10} = 0.0789\zeta _{-17}, \end{equation} \tag{ 14 }$$

and

$$\begin{equation} T_{\rm k,10} = \rm 0.630\big[\big(0.02766{{{\it {n}}}}^2_4 + 0.3156\zeta _{-17}\big)^{1/2}-0.1663{{{\it {n}}}}_4\big]^{2/3}, \end{equation} \tag{ 15 }$$

(where we set again T_dust = 0 K). For ζ_CR∼(1–4) × 10³ζ_CR,Gal, the new thermal balance equation yields T_k(min) ∼ (55–115) K for the molecular gas in such environments while for more extreme CSBs with ζ_CR ∼ 10⁴ζ_CR,Gal it is T_k(min) ∼ (145–160) K. In all cases, the minimum temperatures in CRDRs remain significantly higher than the minimum T_k ∼ (8–10) K which is attained in the dark UV-shielded cores in the Galaxy (see Figure 1) where the initial conditions of star formation are set (Bergin & Tafalla 2007).

It must be noted that such high CR energy densities may also affect molecular gas chemistry, and thus the abundance of coolants such as CO. Hence, better temperature estimates of the UV-shielded dense gas cores in CRDRs can only be provided by self-consistent solutions of their thermal and chemical states. Moreover, activation of other cooling lines such as O i at 63 μm (ΔE_ul/k_B ∼ 228 K) when temperatures rise significantly above 100 K in dense gas (n(H₂) ≳ 10⁵ cm⁻³) can cap the rise of T_k(min) = F(ζ_CR) in CRDRs to ∼150 K (W.-F. Thi 2010, private communication). On the other hand, gas temperatures can be even higher than the simple estimates provided by Equation (10) because residual turbulent motion dissipation (and thus heating) remains possible in the dense gas reservoirs of ULIRGs, as they may resemble the tidally stirred dense gas in the Galactic center where turbulent heating remains important even at high densities (e.g., Stark et al. 1989; Rodríguez-Fernández et al. 2001; Güsten & Philipp 2004).

The large CR energy densities, besides significantly raising the minimum possible temperature of molecular gas in the CSBs powering ULIRGs and their high redshift counterparts, they will also dramatically raise the minimum ionization fraction. Following the treatment by McKee (1989), in UV-shielded environments with negligible photoionization and CRs as the sole cause of ISM ionization, the ionization fraction x(e) = n_e/n(H) = n_e/2n(H₂) is given by

$$\begin{eqnarray} x(e) &=& 2\times 10^{-7} \mbox{r}^{-1} _{\rm {gd}} \left(\frac{{{{\it {n}}}}_{\rm {ch}}}{2{\it {n}}(\mbox{H}_2)}\right)^{1/2} \nonumber\\ &&\times\left[\left(1+\frac{{{{\it {n}}}}_{\rm {ch}}}{8{{\it {n}}}(\mbox{H}_2)}\right)^{1/2} + \left(\frac{{{{\it {n}}}}_{\rm {ch}}}{8{{\it {n}}}(\mbox{H}_2)} \right)^{1/2}\right], \end{eqnarray} \tag{ 16 }$$

where n_ch ∼ 500 (r²_gd ζ₋₁₇) cm⁻³ is a characteristic density encapsulating the effect of CRs and ambient metallicity on the ionization balance (r_gd is the normalized gas/dust ratio r_gd = [(G/D)/100] with G/D(gas-to-dust mass) = 100 assumed for solar metallicities, e.g., Knapp & Kerr 1974; Aannestad & Purcell 1973).

For the Galaxy where ζ_CR,Gal = 5 × 10⁻¹⁷ s⁻¹, it is n_ch = 2.5 × 10³ cm⁻³ and for a typical dense core density of n(H₂) = 10⁵ cm⁻³, Equation (16) yields x(e) ∼ 2.4 × 10⁻⁸, consistent with the typical range in the Galaxy: 5 × 10⁻⁹ ≲ x(e) ≲ 1.5 × 10⁻⁷ (e.g., Langer 1985). For the much larger CR ionization rates, ζ_CR,CSB = 10³ × ζ_CR,Gal expected in the ISM of CSBs: n_ch = 2.5 × 10⁶ cm⁻³ and x(e) ∼ 4 × 10⁻⁶. The latter is 1–2 orders of magnitude larger than the typical range and ∼4 times higher than the highest value measured for dense UV-shielded cores anywhere in the Galaxy (Caselli et al. 1998). In the classic photoionization-regulated star formation scenario (McKee 1989), such high ionization fractions will be capable of turning even very dense gas in the UV-shielded/CR-ionized regions of molecular clouds subcritical (i.e., M_core/M_Φ <1, where M_Φ = 0.12Φ/G^1/2 and Φ is the magnetic field flux threading the molecular core; Mouschovias & Spitzer 1976), and thus halt their gravitational collapse until a now much slower ambipolar diffusion allows it.

In the high-extinction ISM of compact extreme starbursts, CRDRs make the dramatic change of star formation initial conditions an imperative of high-density star formation. The latter occurs irrespective of whether the bulk of the molecular gas in CRDRs of ULIRGs is dominantly heated by CRs or not as the latter will now raise the temperature minimum of pre-stellar UV-shielded dense gas cores throughout their SF volumes by a large factor. The consequences of this new imperative for star formation in extreme starburst environments remain invariant irrespective of whether gravitational instability in turbulent molecular gas (e.g., Klessen 2004; Jappsen et al. 2005; Bonnell et al. 2006) or ambipolar diffusion of magnetic field lines from dissipated dense cores followed by their gravitational collapse (e.g., Mouschovias & Spitzer 1976; McKee 1989) drive star formation in galaxies. In both schemes, the dense UV-shielded cores of molecular clouds are where the initial conditions of star formation are truly set (see Elmegreen 2007; Ballesteros-Paredes & Hartmann 2007 for recent excellent reviews). It must also be noted that this gas phase is different from the PDR-dominated gas around SF sites that dominates the global molecular line and dust continuum spectral energy distributions observed for ULIRGs and often (erroneously) used to set the star formation initial conditions in starbursts (e.g., Klessen et al. 2007).

The effects on the characteristic mass scale of young stars M^(*)_ch and thus on the stellar initial mass function (IMF; Elmegreen et al. 2008) for the ISM in CRDRs are explored in detail in a forthcoming paper (Papadopoulos et al. 2010). It is nevertheless worth pointing out that the large boost of T_k(min) in CRDRs in an (almost) extinction-free manner across a large range of densities in molecular clouds cannot but have fundamental consequences on M^(*)_ch and the emergent stellar IMF. Indicatively for a T_k(min) in UV-shielded cores boosted by a factor of 10, the Jeans mass M_J∝T^3/2_kn(H₂)^−1/2 rises by a factor of ∼32! (over an identical density range) and almost certainly raises M^(*)_ch and the characteristic mass scale of the stellar IMF. Interestingly, similar effects can also occur in AGN-induced X-ray-dominated regions (XDRs; Schleicher et al. 2010) since X-rays just as CRs (and unlike far-UV photons) can volumetrically heat large columns of molecular gas while experiencing very little extinction. The effect of X-rays on the Jeans mass of dense cores and the IMF has been recently proposed for a powerful distant QSO (Bradford et al. 2009) and it may represent a neglected but important AGN feedback factor on its circumnuclear star formation.

The molecular line diagnostics of starburst-induced CRDRs and AGN-originating XDRs can be to a large degree degenerate when galaxies host both power sources. This has been noticed in earlier comparative studies of XDRs and regions with higher U_CR values (though only up to 100 × U_CR,Gal) where only carefully chosen line ratios can distinguish between them (Meijerink et al. 2006). The still larger CR energy densities expected in CRDRs of ULIRGs will further compound these difficulties.

Provided that a powerful X-ray luminous AGN heating up the bulk of the molecular gas in its host galaxy can be somehow excluded (e.g., via hard X-ray observations), any set of molecular lines and ratios that can strongly constrain the temperature of the dense gas UV-shielded phase (n(H₂) > 10⁴ cm⁻³) in ULIRGs will be valuable. Indeed, given that in the hierarchical structures of typical molecular clouds, the dense gas regions, (1) lie well inside much larger ones that strongly attenuate far-UV light and (2) cool strongly via molecular line emission because of their high densities, any evidence for high temperatures for the dense gas phase would be an indicator of strong CR heating. In that regard, observations of high-J CO lines such as J = 6–5 and its ¹³CO isotopolog have already been proven excellent in revealing CR heated rather than UV/photoelectrically heated molecular gas in galactic nuclei (Hailey-Dunsheath et al. 2008). Irrespective of the particular set of rotational lines used as a "thermometer" of the dense gas, three general requirements must be met: (1) all lines must have high critical densities (n_crit > 10⁴ cm⁻³), (2) have widely separated E_{J + 1,J}/k_B factors, and (3) the J-corresponding lines of at least one rare isotopolog must also be observed (e.g., ¹²CO and ¹³CO or C³²S and C³⁴S, etc.). The first two requirements ensure probing of the dense gas phase while maintaining good T_k sensitivity, and the last one is necessary for reducing well-known degeneracies when modeling only transitions of the most abundant isotopolog which often have significant optical depths. Molecular lines with n_crit ≳ 10⁵ cm⁻³ (e.g., HCN, CS rotational transitions) are particularly valuable since, aside from emanating from gas well within typical pre-stellar cores, they trace a phase whose kinematic state is either dictated by self-gravity (e.g., Goldsmith 2001) or has fully dissipated to thermal motions. This constrains the line formation mechanism (and can be used to remove degeneracies of radiative transfer models, e.g., see Greve et al. 2009), but most importantly it reduces the possibility of residual mechanical heating of the dense gas (Loenen et al. 2008) that could mask as CR heating (as both can heat gas volumetrically).

A brief example of such diagnostics can be provided using a large velocity gradient (LVG) code (e.g., Richardson 1985) to compute relative line intensities for dense gas with T_k = (10–15) K, and n(H₂) = 10⁵ cm⁻³, and T_k = (100–150) K at the same density. In both cases, a gas velocity gradient due to self-gravity is assumed. For the cold gas, the CO and ¹³CO (J+1−J)/(3–2) brightness temperature ratios are: R_{(J + 1,J)/32} ≲ 0.7 for J+1 ⩾ 4 (J = 1–0, and 2–1 are not considered because they can have substantial contributions from a diffuse non-self-gravitating phase). For the warm gas R_{(J + 1,J)/32} ∼ 0.9–0.95 for J+1 = 4–6, while the corresponding ratios for ¹³CO are ∼1–1.3. Similar diagnostics but using multi-J line emission from rarer molecules (and their isotopologs) with much larger dipole moments (n_crit ≳ 10⁵ cm⁻³) such as HCN and H¹³CN can even better constrain the temperature of dense gas in galaxies. However, their much fainter emission (e.g., HCN lines are ∼5–30 times fainter than those of CO) allows their use only for the brightest nearby starburst nuclei (e.g., see Jackson et al. 1995 for an early pioneering effort), and only ALMA will enable such diagnostics for a large number of galaxies in the local and distant universe.

Strong thermo-chemical effects induced by large U_CR values and their ability to volumetrically warm large amounts of dense molecular gas can also provide a valuable diagnostic of the temperatures deep inside dense gas cores. For example, global HNC/HCN J = 1–0 brightness temperature ratios of R^(1–0)_HNC/HCN = 0.5–1.0 in galaxies can be attributed to an ensemble of PDRs but ratios R^(1–0)_HNC/HCN < 0.5 cannot and imply T_k ⩾ 100 K (Loenen et al. 2008). Such low ratios are indeed found in ULIRGs but are attributed to turbulent gas heating by SNRs in dense molecular environments (Loenen 2009) which substantially warms the gas in the absence of photons to T_k ⩾ 100 K which is necessary for the HNC+H→HCN+H reaction to proceed efficiently and convert HNC to HCN (e.g., Schilke et al. 1992). Unlike the Galaxy where turbulent gas heating has subsided in the dense subsonic gas cores that would emit in these transitions this may not be so in the ISM of ULIRGs, making CR and mechanical heating difficult to distinguish. As mentioned earlier, moving the CRDR molecular line diagnostic to tracers with ever increasing critical densities (and definitely with n_crit ≳ 10⁵ cm⁻³) may be the only way of "breaking" this degeneracy as turbulent heating is expected to become progressively weaker in higher density cores (after all star formation is expected to proceed in dense gas cores whose turbulence has fully dissipated; Larson 2005). Finally, chemical signatures that can uniquely trace the high U_CR values in CRDRs using the high ionization fractions expected for their dense gas cores are particularly valuable. Such a very sensitive probe of ζ_CR and x(e) in dense UV-shielded gas is provided by R_D = [DCO⁺]/[HCO⁺] versus R_H = [HCO⁺]/[CO] abundance ratio diagrams where the high CR energy densities and resulting ionization fractions in CRDRs would correspond to very low R_D and very high R_H values (Caselli et al. 1998). A dedicated study of these issues that explores the unique diagnostic of CRDRs is provided by Meijerink et al. (2010).

Most of the aforementioned tests will acquire additional diagnostic power once ALMA is able to conduct them at high, sub-arcsecond resolution in nearby ULIRGs. It will then be possible to discern whether very warm dense gas with high ionization fractions is localized around a point-like source, the AGN (and is thus due to XDRs), or is well distributed over the SF area of compact systems and is thus starburst-related (see Schleicher et al. 2010 for a recent such study). High-resolution imaging can also directly measure the high brightness temperatures expected for all optically thick and thermalized molecular lines (where T^(line)_ex ∼ T_k) in CRDRs. Radiative transfer models of emergent CO line emission from a self-gravitating gas phase with n(H₂) = 10⁵ cm⁻³ and T_k = (100–150) K typically yield T_b(CO) ∼ (80–140) K for J = 1–0 up to J = 7–6, and imaging at a linear resolution of ΔL ≲ 100 pc (the diameter of gaseous disks in ULIRGs) would be adequate to directly measure them (see Sakamoto et al. 2008 and Matsushita et al. 2009 for early examples). For z ≲ 0.05 (which would include a large number of ULIRGs), such linear resolutions correspond to angular resolutions of θ_b ∼ 01 (for a flat Λ-dominated cosmology with H₀ = 71 km s⁻¹ Mpc⁻¹ and Ω_m = 0.27) which would certainly be possible with ALMA.