The Critical Density and the Effective Excitation Density of Commonly Observed Molecular Dense Gas Tracers

We define the optically thin critical density without a background by solving for n_c in equation (1) when and ignoring the stimulated emission and absorption terms:

The sum over collision rates in the denominator of equation (4) has been split in two terms depending on whether the collision from level j is to an energy level lower in energy (first sum) or higher in energy (second sum) than level j. Upward collision rates are related to downward collision rates using detailed balance by

where T_k is the kinetic temperature of the collisional partner. Published collision rates (i.e., the Leiden molecular database;³ Schöier et al. 2005) quote only downward collision rates. In the two-level approximation, which is widely quoted but rarely appropriate, the sums reduce to a single term such that . As a result, will always be larger (usually by at least a factor of a few) than .

It is instructive to compare how the optically thin critical density compares to the excitation temperature. For a particular transition between levels j → k, the level population are related to the excitation temperature, T_ex, between the two levels by Boltzmann's equation

The excitation temperature (and therefore the level populations) depends on the density of colliding partners (n_c), the gas kinetic temperature of colliding partners (T_k), and the energy density of radiation fields that the molecules are exposed to. The dependence of T_ex on the density and gas kinetic temperature for the ground state transitions of ortho and para NH₃ are shown in Figure 2. The only background radiation field assumed in this example is the CMB. At low densities (n_c < 10³ cm^-3), the level populations come into equilibrium with the CMB and T_ex approaches T_cmb = 2.725 K. As the density increases, collisions become more important in determining the level populations. The excitation temperature of (1,1) centimeter wavelength inversion transition equilibrates to T_k by densities >10^5.5 cm^-3 while the submillimeter 1₀ - 0₀ rotational transition equilibrates to T_k at significantly higher densities (>10⁷ cm^-3). The general shape of the T_ex with log n_c curves is a sigmoid curve with stimulated coupling with the background radiation field setting the lower bound at low densities and collisions with colliding partners at T_k setting the upper bound. At densities where T_ex < T_k, the level populations are said to be subthermally populated. This is the situation for most of the commonly observed dense gas transitions in the ISM. When T_ex approaches T_k, the level populations are said to be thermalized.⁴

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At optical wavelengths (the high frequency limit), critical density is interpreted as the density at which an atomic forbidden line is quenched by collisions meaning that a collisional de-excitation occurs before a photon can be generated from spontaneous emission. In the optical, quenching occurs when T_ex is thermalized (T_ex → T_k). At radio wavelengths (the low frequency limit) the critical density occurs at different positions along the sigmoid T_ex curve for different molecular tracers and different transitions. In the case of the 1.3 cm (23.7 GHz) NH₃ (1,1) transition, the optically thin critical density is 2 orders of magnitude below the thermalization density (Fig. 2). Early radio astronomy observations of molecules such as OH and NH₃ occurred at centimeter wavelengths and since the critical density at these low frequencies is just above the regime where the T_ex starts to rise above equilibration with the CMB, critical density was used to indicate the density at which a transition would appear in emission. As the frequency of the transition increases, the critical density moves farther up the T_ex curve. For the 0.52 μm (572 GHz) NH₃ 1₀ - 0₀ transition, the critical density is nearly equal to the density required for thermalization. Millimeter transitions are an intermediate case; thus, neither the radio interpretation of critical density being the density at which an emission line is excited nor the optical interpretation of critical density being the density at which an emission line is quenched is appropriate.

In reality, radiative trapping is very important for many molecular transitions observed in the dense interstellar medium and cannot be ignored. The energy density is then modified using the solution to the equation of radiative transfer to include emission from the molecule at the excitation temperature describing the level populations for a transition from j → k as

where is the solid angle averaged escape fraction (see chapter 19 of Draine [2011] for a derivation). If we multiply the energy density by the conversion factor between Einstein B_jk and A_jk terms from equation (2), then we find that

In the limit that the background radiation can be ignored (), the critical density becomes

Comparing equation (9) with equation (4), we see that the effect of radiative trapping is to reduce the spontaneous transition rate by the escape fraction A_jk (see Scoville & Solomon 1974; Goldreich & Kwan 1974).

The relationship between and the line-of-sight optical depth, τ_νjk, depends on the geometry and kinematics of the region. In the case of a static, uniform density sphere

(Osterbrock 1989), while in the case of a spherical cloud with a large velocity gradient v ∝ r (the LVG or Sobolev approximation),

(Sobolev 1960, Castor 1970). For large optical depths (τ_νjk >  ∼ 5), the effective spontaneous transition rate is then given by .

This result has an important effect on the critical density by reducing its value from the optically thin value. The most commonly observed transition of molecular gas is¹²CO 1−0. The optically thin critical density at T_k = 10 K is ; however, since¹²CO 1−0 has optical depths that are typically more than a factor of 10 in molecular clouds (and usually much higher), it is easy to observe strong lines (>1 K km/s) in gas that has densities < 10² cm^-3. Optical trapping is so important for CO that it does not make sense to use the optically thin critical density.

Commonly observed lines of dense gas tracers in cores and clumps also tend to be optically thick. The upper left panel of Figure 3 shows the predicted optical depth of HCO⁺ 1−0 through 4−3 at different densities from a simple (single density, single temperature) radiative transfer model predicting that these transition are expected to be very optically thick over a wide range of densities. Observations support this prediction. For example, several hundred high-mass clumps mapped in the MALT90 Galactic survey have a median optical depth of τ = 23 for the HCO⁺ 1−0 transition (Hoq et al. 2013). Clumps observed in the Bolocam Galactic Plane Survey (BGPS) in the higher frequency transition HCO⁺ 3−2 have an average τ = 10 (Shirley et al. 2013). As a result, the analytically calculated critical densities using the optically thin approximation (eq. [4]) are systematically biased toward higher densities by an order of magnitude or more. HCO⁺ is not unique as most commonly observed dense gas tracers (i.e., HCN, CS, N₂H⁺, etc.) in the dense, cold neutral medium are typically both subthermally populated (T_ex < T_k) and optically thick (also see Plume et al. 1997; Gerner et al. 2014). In practice, to properly estimate the optical depth, a coupled radiative transfer and statistical equilibrium calculation must be performed to find the level populations upon which the optical depth depends (since τ_νjk ∝ ∫(n_k B_kj - n_j B_jk)ds along the line-of-sight s). Since the escape fraction depends nonlinearly with column density (for modest optical depths), nonlinearly with geometry, and nonlinearly with gas kinetic temperature, it is difficult to make a simple tabulation of a correction to .

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The column density of the observed molecule and the gas kinetic temperature of the colliding partner must be assumed in order to calculate n_eff. The sensitivity of n_eff to these assumptions is explored in Figure 3 for the HCO⁺ molecule. At lower column densities, a higher n_eff is required to produce a 1 K km/s line. As the column density increases by an order of magnitude, the n_eff decreases by almost exactly an order of magnitude. The effective excitation density is inversely proportional to column density

where N_ref is a reference column density. This simple scaling proportionality breaks down for n_eff ≳ 5 × 10⁶ cm^-3 and the relationship between n_eff and N becomes noticeably nonlinear (see the T_k = 10 K curve for HCO⁺ 4−3 in the bottom left panel of Fig. 3). In those cases, a new RADEX model should be run to scale tabulated results to a new column density.

The dependence of n_eff to T_k is nonlinear and more sensitive at low kinetic temperatures (T_k < 20 K) due to the inefficiency of upward collisions at lower T_k (Fig. 3). An order of magnitude higher n_eff is required to make a 1 K km/s line for T_k = 10 K than at T_k = 30 K for HCO⁺ 3−2. This sensitivity is due to the exponential dependence from upward collisional rates (eq. [5]) and becomes more pronounced when the upper energy level is significantly higher than T_k (i.e., the effect is stronger for 4−3 vs. 1−0 at T_k < 20 K because E_u/k = 42.8 K for J = 4 vs. E_u/k = 4.3 K for J = 1).

Table 1 gives n_eff for 12 dense gas tracers at T_k = 10, 15, 20, 50, and 100 K for a reference column density. Recent surveys of dense gas in the Milky Way may be used to guide our choice of the reference column density. For example, observations of several hundred clumps by the MALT90 (Meittenen et al. 2014; Hoq et al. 2013), ChaMP (Barnes et al. 2011), and BGPS (Shirley et al. 2013) surveys indicate that HCO⁺ column densities typically span from log N = 12 - 15.5. The Hoq et al. and Shirley et al. surveys find median column densities of clumps drawn from the ATLASGAL and BPGS Galactic plane surveys that are in agreement near log(med{N}) = 14.4. Miettenen's (2014) analysis of infrared dark clouds (IRDCs) indicates a significantly lower median column density of log(med{N}) = 13.1. All of these surveys use observations of H¹³CO⁺ to correct the optical depth of the HCO⁺ line. The Barnes et al. (2011) blind mapping survey of a section of the southern Galactic plane find a smaller value of log(med{N}) = 12.9, but did not observe H¹³CO⁺ to make optical depth corrections. We chose a column density that is intermediate between the IRDC, low-mass samples, and the Galactic plane surveys of log(N_ref) = 14.0. For comparison, Evans (1999) tabulate the effective density for HCO⁺ assuming log N = 13.5.

The reference column densities for each molecule in Table 1 are estimated from surveys in dense gas tracers (Suzuki et al. 1992; Zylka et al. 1992; Mangum & Wootten 1993; Plume et al. 1997; Jijina et al. 1999; Savage et al. 2002; Pirogov et al. 2003; Purcell et al. 2006; Blair et al. 2008; Rosolowsky et al. 2008; Falgarone et al. 2008; Hily-Blant et al. 2010; Barnes et al. 2011; Dunham et al. 2011; Ginsburg et al. 2011; Reiter et al. 2011; Adande et al. 2012; Benedettini et al. 2012; Wienen et al. 2012; Hoq et al. 2013; Shirley et al. 2013; Gerner et al. 2014; Miettenen et al. 2014; Yuan et al. 2014). Since these surveys contain many biases (i.e., only study massive high-mass regions or only study nearby low-mass regions, etc.), N_ref represents an educated guess at a typical value intermediate between high-mass clumps and low-mass cores. The tabulated n_eff can be easily scaled to another column density using equation (13).¹³C isotopologue column densities assume no fractionation and an isotope ratio of [¹²C]/[¹³C] = 50, appropriate for a source in the Molecular Ring (Wilson & Rood 1994). For molecules with ortho and para states, LTE statistical equilibrium was assumed (1:1 ortho to para ratio for NH₃ and 3:1 ortho to para ratio for H₂CO).⁶

The panels of Figure 4 summarize the range in and n_eff of commonly observed molecular tracers of dense gas for reference observed column densities and a gas kinetic temperature of T_k = 15 K. This temperature was chosen to be intermediate between the typical temperature of nearby low-mass cores (10 K, Jijina et al. 1999; Rosolowsky et al. 2008; Seo et al. 2015, submitted) and high-mass clumps in the Galactic plane (20 K, Dunham et al. 2011; Wienen et al. 2012). n_eff is lower than in nearly all cases by 1–2 orders of magnitude. This is partially due to the sensitivity of n_eff to line trapping in optically thick transitions. Transitions that are the most optically thick have the largest average ratios (i.e., 121 for HCO⁺ transitions and 88 for HCN transitions where the average is calculated for 1−0 through 4−3 transitions). Less optically thick species have smaller ratios (i.e., average ratios of 16 for CN and 10 for N₂H⁺). The species plotted with the smallest difference between and n_eff is the optically thin isotopologue H¹³CN with an average ratio of 1.7. Some heavier species whose transitions are likely close to optically thin listed in Table 1 also have low average ratios that are less than 3 (i.e., H₂CO, HC₃N, and CH₃CN). These results indicate that a 1 K km/s line is typically observed for transitions when in equation (1) or when the radiative rates are greater than collisional depopulation out of the upper energy level of the transition.

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There is also typically an increase in and n_eff with J_u for a particular molecule (Fig. 4). This behavior can be understood from the functional dependences on A_jk and γ_jk in equation (4) ( ∼A_jk/γ_jk). The Einstein A for electric dipole allowed transitions is , where μ_e is the molecular permanent electric dipole moment. In general, molecules with larger μ_e will trace denser gas. For instance, μ_e = 2.99 D for HCN while μ_e = 0.11 D for CO (Ebenstein & Muenter 1984; Goorvitch 1994), partially explaining why HCN is a denser gas tracer than CO (the other effect being that the optical trapping in¹²CO lines is often very large; see § 2.2). Since ν_jk ∝ J_u and the Einstein A_jk has a dependence, higher frequency transitions of a particular molecule will trace higher densities. This statement is also true for comparing molecules with similar permanent electric dipole moments but with transitions at different frequencies. As an example, the frequency of the NH₃ (μ_e = 1.47 D; Ueda & Iwahori 1986) inversion transitions (J, K = J) are nearly a factor of 4 lower than for the 1−0 transitions of HCO⁺ N₂H⁺ and HCN; thus, the spontaneous transition rate is more than 60 times smaller, lowering the critical densities of NH₃ (1,1) with respect to the 1−0 transitions of HCO⁺ N₂H⁺ and HCN. One problem with the use of becomes apparent for the NH₃ inversion transitions. Since NH₃ (1,1) through (4,4) are at nearly the same frequency (23.7 to 24.1 GHz), their are nearly identical. n_eff is sensitive to both line trapping and E_u/k of the transition resulting in an increase in n_eff for NH₃ (1,1) and (2,2) (N.B. n_eff for NH₃ (3,3) and (4,4) are undefined at T_k = 15 K as we will discuss below).

Differences in collision rates also affect the ranking of molecules with density in Figure 4. HCN is observed to have 1–2 orders of magnitude higher and n_eff than HCO⁺. Both HCO⁺ and HCN transitions have similar A_jk values, but HCO⁺ is an ion and the collisional cross section for collisions with H₂ is larger than for neutral species (see Flower [1999] vs. Dumouchel et al. [2010]). This reduces the critical density and effective excitation density of HCO⁺ relative to HCN.

The ordering of species with increasing density is different for versus n_eff among many of the molecules plotted in Figure 4. A problem with using can be seen when HCO⁺ and N₂H⁺ are compared. HCO⁺ and N₂H⁺ are both ions with similar A_jk values for transitions and similar collisional cross-sections and thus have similar ; however, it is common knowledge to observers of nearby, low-mass dense cores that N₂H⁺ is a denser gas tracer than HCO⁺. Due to chemical effects, N₂H⁺ typically has lower abundance and therefore lower column density than HCO⁺ resulting in a higher n_eff (see eq. [13]).

n_eff also does a better job than of characterizing the difference in density traced by isotopologues versus the main species. The ratio of A_jk for the 1 → 0 transition between H¹³CN and HCN is 0.92. This accounts for the slightly lower for H¹³CN versus HCN; however, we know that the isotopologues should trace higher density gas than the main species because it is more optically thin and has a lower column density (see eq. [13]). The average ratio of n_eff for H¹³CN/HCN 1 → 0 is 41 (T_k∈[10,100] K). This ratio is not exactly equal to the assumed isotope ratio and column density ratio of 50 because of differences in A_jk, collisional rates, and line trapping between the isotopologue and main species.

These examples illustrate important advantages in using n_eff rather than the critical density; however, using n_eff has a significant disadvantage in that n_eff is undefined when an integrated intensity of 1 K km/s cannot be observed. This is the case for the NH₃ (3,3) (E_u/k = 123.5 K) and higher inversion lines at low gas kinetic temperature. When E_u/k≫T_k, the upper energy state cannot attain a high enough population to result in large enough radiative rates to produce a 1 K km/s line. High J_u transitions for species such as NH₃, HCN, HCO⁺, and N₂H⁺ will be undefined at low gas kinetic temperatures (T_k ≲ 20 K).

Another problem with the standard definition of n_eff is that the more complex the spectrum of a molecule (e.g., a heavy asymmetric top molecule), the less likely a particular transition will attain enough level population to produce a 1 K km/s emission line. For instance, no transitions of HNCO have intensities greater than 1 K km/s for the observed median column density of log med{N} = 12.6 (Gerner et al. 2014). n_eff for complex organic species (i.e., CH₃CHO, CH₃OCH₃, etc.) will be undefined. One solution is to reduce the 1 K km/s criterion. Ultimately, the choice for this criterion is arbitrary, but unless all researchers agree to use the same criterion, n_eff could not be compared among different papers and for different species. The upper right panel of Figure 3 shows the nonlinear sensitivity of the integrated intensity for different transitions of HCO⁺ with density. As the integrated intensity criterion is lowered, the curves flatten with n_c indicating that a smaller decrease in the criterion results in a larger decrease in n_eff. These problems limit the utility of n_eff compared to which can always be calculated if collision rates are known.

I am grateful to the Max Planck Institut für Astronomie and Thomas Henning for hosting my Sabbatical during which this note was written. I would also like to thank John Black, Lee Mundy, Neal Evans, Henrik Beuther, Cecilia Ceccarelli, J.D. Smith, and the anonymous referee for comments and useful discussions that improved this manuscript.