ORTHO-H₂ AND THE AGE OF INTERSTELLAR DARK CLOUDS

The low-mass star formation process is relatively well understood, however its first step details remain uncertain, especially regarding the question of time and age. Still much debated questions are how long it takes for a cloud to form from the diffuse atomic medium, how long a molecular cloud lives, and how long it takes to form a condensation that will evolve into a prestellar core that subsequently collapses to form a protostar. Today, much debate exists upon the lifetime of clouds, e.g., Hartmann et al. (2001) defending a short lifetime (one to a few million years) while Tassis & Mouschovias (2004) and Mouschovias et al. (2006) claim a typical age of 10 million years, but most arguments are either of limited statistical significance or model dependent (unknown magnetic field strength, small-scale turbulence rapid dissipation, etc.). Clues are needed and while the usual chemical age modeling is not satisfying (problem of unknown initial conditions), we present here a simple constraint based on basic chemistry: the absence of DCO⁺ in dark cloud envelopes can only be explained if the clouds are young enough and we propose to determine an upper limit for this age which is as much independent from the initial conditions as possible.

Dark clouds contain dust grains embedded in interstellar gas, itself mostly molecular as these clouds are self-shielded against our Galaxy's UV background. These dark clouds are cold (typically 10 K, with cores as low as 7 K; Pagani et al. 2003, 2005), and deep inside, where the density reaches a few 10⁴ cm⁻³, the heavy species freeze out onto grains to form ices. Both these ices and the gas are subject to a rich chemistry. Recently, strong deuteration (deuterium enhancement with respect to hydrogen carriers: DCO⁺ versus HCO⁺, HDCO and D₂CO versus H₂CO, etc.) has been recognized as a useful tool to better understand both the chemistry itself and the star formation process (Bergin & Tafalla 2007; Ceccarelli et al. 2007).

H₂ is the main hydrogen reservoir in dark clouds, and similarly, HD is the main deuterium reservoir. Its relative abundance to H₂ is ∼3 × 10⁻⁵ (Hébrard 2006; Linsky et al. 2006). Watson (1976) was the first to invoke chemical deuterium enrichment via the important reaction

$$\begin{equation} {\rm H_3^+ + HD \rightarrow H_2D^+ + H_2} + \Delta E. \end{equation} \tag{ 1 }$$

ΔE = 232 K when all species (reactants and products) are in their ground state. In cold dark clouds, the forward reaction is strongly favorable, creating H₂D⁺/H⁺₃ ratios much larger than the original HD/H₂ ratio. If CO is abundant (X[CO] ≈ 1.5 × 10⁻⁴, where X[] denotes the relative abundance of a species with respect to H₂), H⁺₃ reacts five times more often with CO than with HD, and forms HCO⁺, strongly decreasing the abundance of H₂D⁺.

Once H₂D⁺ is formed, it can be enriched in deuterium by reacting with HD again to form D₂H⁺ and D⁺₃ as first recognized by Roberts et al. (2003):

$$\begin{eqnarray} &&\rm H_2D^+ + HD \rightarrow D_2H^+ + H_2 + 187 \,{\rm K} \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&\rm D_2H^+ + HD \rightarrow D_3^+ + H_2 + 234 \,{\rm K}. \end{eqnarray} \tag{ 3 }$$

These two reactions are again favorable in the forward direction at low temperatures because of their exothermicity. Then, these H⁺₃ isotopologues can easily transfer their deuterons to other species:

$$\begin{eqnarray} &&\rm CO + H_2D^+ \rightarrow \frac{2}{3}HCO^+ + \frac{1}{3}DCO^+ + \frac{1}{3}H_2 + \frac{2}{3}HD \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} &&\rm CO + D_2H^+ \rightarrow \frac{1}{3}HCO^+ + \frac{2}{3}DCO^+ + \frac{1}{3}HD + \frac{2}{3}D_2 \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &&\rm CO + D_3^+ \rightarrow DCO^+ + D_2 \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&\rm N_2 + H_2D^+ \rightarrow \frac{2}{3}N_2H^+ + \frac{1}{3}N_2D^+ + \frac{1}{3}H_2 + \frac{2}{3}HD \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} &&\rm N_2 + D_2H^+ \rightarrow \frac{1}{3}N_2H^+ + \frac{2}{3}N_2D^+ + \frac{1}{3}HD + \frac{2}{3}D_2 \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} &&\rm N_2 + D_3^+ \rightarrow N_2D^+ + D_2. \end{eqnarray} \tag{ 9 }$$

When CO is abundant, H₂D⁺ is much less abundant as noted above and also reacts with CO preferentially, quenching all the other deuteration paths. CO also reacts with species like N₂H⁺ and N₂D⁺ to destroy them:

$$\begin{eqnarray} &&\rm N_2H^+ + CO \rightarrow HCO^+ + N_2 \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} &&\rm N_2D^+ + CO \rightarrow DCO^+ + N_2. \end{eqnarray} \tag{ 11 }$$

The strong correlation between CO freezeout onto grains and deuteration (e.g., Crapsi et al. 2005) confirms the chemical models (e.g., Roberts et al. 2003) as sketched above and has led to the conclusion that CO freezeout was necessary for strong deuteration to occur. There is however one exception to this statement: the deuteration of CO itself can form DCO⁺. In first approximation, if H⁺₃ is mainly destroyed by CO, X[H₂D⁺] varies like X[CO]⁻¹, whereas the production of DCO⁺ is proportional to X[CO] and X[H₂D⁺] (Reaction (4)). Thus, the dependency upon X[CO] cancels out and X[DCO⁺] should remain approximately constant. Figure 1 shows the abundance of X[DCO⁺], X[H⁺₃ isotopologues], and X[N₂D⁺] as a function of X[CO] in a steady-state chemical model. The X[H⁺₃ isotopologues] is the sum of the abundances of all three deuterated H⁺₃ isotopologues, which represent altogether the main deuteration partners of species like CO and N₂. The model is the same as in Pagani et al. (2009) but CO now runs from an undepleted relative abundance of 1.5 × 10⁻⁴ down to a depletion by a factor of 300 (X[CO] = 5 × 10⁻⁷), for a cloud density of 10⁴ cm⁻³ at 10 K. When X[CO] starts to drop, X[H⁺₃ isotopologues] increases faster than proportional (indicated by the −1 slope) because the D₂H⁺ abundance depends both on the destruction of H⁺₃ by CO and on the destruction of H₂D⁺ by CO, and the D⁺₃ abundance depends also on the destruction of D₂H⁺ by CO. This explains that at the beginning, X[DCO⁺] increases: the deuteration capabilities increase faster than X[CO] decreases. While X[CO] keeps decreasing, the deuteration capabilities reach saturation and X[DCO⁺] becomes proportional to X[CO] (traced by the +1 slope). X[N₂D⁺], a representative gas-phase deuterated species, follows the deuteration capabilities of the cloud, as expected. It is clear that DCO⁺ has a behavior different from the other deuterated species, with an abundance varying only by ±50% while the CO abundance drops by a factor of 30, and most importantly by reaching its maximum abundance when CO is high and deuteration capabilities are down. Therefore, one would expect DCO⁺ to be present everywhere in the cloud. It is not the case. Indeed, DCO⁺ is, as the other deuterated species, only detected in the depleted prestellar cores on extents similar to NH₃ (Butner et al. 1995; Juvela et al. 2002) and nowhere else. This can only be explained if a high H₂ ortho:para ratio (OPR) is present. Indeed the H₂ OPR is an important controller of the cold cloud deuteration chemistry, as already discussed in Pineau des Forêts et al. (1991), Flower et al. (2006), and Pagani et al. (1992, 2009).

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H₂ has two different possible nuclear spin states due to the proton spin of $\frac{1}{2}$: parallel (ortho, I = 1, weight 2I + 1 = 3) and anti-parallel (para, I = 0, weight = 1). The ortho state corresponds to the odd rotational levels (J = 1, 3, 5,...), the para state to the even levels. The first ortho state (J = 1) is 170 K above the ground (J = 0) para state. The ortho–para exchange is possible neither via radiative processes, nor via inelastic collisions. It can occur in the gas phase and on the surface of dust grains (Le Bourlot 2000). The conversion rate on a solid surface is however highly uncertain (Watanabe et al. 2010; Chehrouri et al. 2011) and it is most probable that this is mainly realized by gas-phase chemical reactions with H⁺ or H⁺₃ ions (Dalgarno et al. 1973; Flower & Watt 1984; Le Bourlot 1991; Flower et al. 2006). The importance of ortho-H₂ in the deuteration process was first studied by Pineau des Forêts et al. (1991) and Pagani et al. (1992), and later on, Flower et al. (2006) revealed its importance as a deuteration regulator. However, they considered the consequences of the OPR evolution only for the prestellar core formation itself. In a study of L183 main prestellar core, Pagani et al. (2009) noted that the H₂ OPR evolution timescale required by the observed N₂D⁺/N₂H⁺ ratio was on the order of the free-fall time. They also noted that the OPR had to drop below 0.01 for N₂D⁺ to become detectable.

The H₂ OPR value is not easily accessible. H₂ does not emit in cold dark clouds and is seen only in shocked regions, especially protostar outflows, or in absorption in UV in low-extinction regions. The energy release from the H₂ formation reaction, assumed to be 1.5 eV in the equipartition hypothesis, is large enough to populate many levels of both ortho and para species: it is hence generally accepted, though not certain, that the OPR is 3:1 when H₂ forms on grains. Once released in the gas phase, H₂ tends to relax toward its thermal equilibrium state but never reaches it because at trace abundance levels, ortho-H₂ destruction is compensated by fresh H₂ formed onto grains from residual H and from destruction of ions such as H⁺₃ and H₂D⁺ that can release an ortho-H₂ (Le Bourlot 1991; Flower et al. 2006). It is therefore necessary to model the chemical evolution of the H₂ OPR to understand the evolution of dark clouds. Once H₂ is formed, it can react with H⁺ and H⁺₃ to exchange states (Le Bourlot 1991; Flower et al. 2006). For example,

$$\begin{equation} {\rm o\hbox{-} H_2 + H^+ \rightarrow p\hbox{-} H_2 + H^+} + \Delta E. \end{equation} \tag{ 12 }$$

ΔE = 170 K is the energy released as H₂ goes from J = 1 to J = 0 levels. The reverse endothermic reaction is thus difficult to obtain in a cloud at 10 K and ortho-H₂ slowly decreases to low abundance levels (≈10⁻³ to 10⁻⁴). Similarly, in Reaction (1), except for HD, all the species have ortho and para states and ΔE therefore depends in fact on the considered initial and final states. If only para states are involved in the products, ΔE is maximal (ΔE = 232 K with para-H⁺₃ and 265 K with ortho-H⁺₃, HD being always in its ground state, J = 0), but if both products are in their ortho ground state while the reactants are in their lowest state, para-H⁺₃ ground state and HD (J = 0), ΔE becomes negative and the reaction is slightly endothermic (H₂D⁺ lowest ortho state is 87 K above the para ground state, and the total required energy is 25 K). Similarly, while the reverse reaction is impossible in a 10 K cloud with both para reactants, it becomes rapid with ortho-H₂D⁺ and ortho-H₂ even at 10 K, as this channel is slightly exothermic (ΔE = 25 K) to produce para-H⁺₃. A few such channels are opened with ortho-H₂ allowing the efficient destruction of the H⁺₃ deuterated isotopologues. Ortho-H₂ is thus an important chemical poison to deuteration in dark clouds. Its abundance is therefore critical to control the deuteration enhancement (Flower et al. 2006; Pagani et al. 2009). Because DCO⁺ is not observed everywhere in the clouds outside the depleted cores where it is detected (Butner et al. 1995; Juvela et al. 2002) while its abundance should be maximal, this sets a minimum level for the abundance of ortho-H₂. In turn, this minimum level gives rise to an upper limit to the age of the cloud, as ortho-H₂ abundance must eventually decrease to low levels as implied by Reaction (12), and in order to explain the deuteration enhancements seen in the depleted cores, which, in some cold cores, can become huge (amplification up to 10¹²; Lis et al. 2002; van der Tak et al. 2002; Parise et al. 2004).

To address these questions, we used the deuteration network from Roueff et al. (2005, 2007) in which we have included the ortho–para chemistry as described in Pagani et al. (2009), based itself on the work by Walmsley et al. (2004), Flower et al. (2006), and the new rate coefficients of Hugo et al. (2009). We have also included analytical approximations of the spin-state-dependent dissociative recombination rates of H⁺₃ isotopologues from the tables in Pagani et al. (2009). The deuteration network includes all simple species (up to six atoms) based on C, N, O, and S with up to five deuterium substitutions (CD⁺₅). For those reactions with a small endothermicity (E/k_B < 600 K), we have corrected the endothermicity when the reaction with ortho-H₂ was involved (e.g., CD + H₂ → CH + HD is endothermic with para-H₂ and exothermic with ortho-H₂). For all reactions with rare species producing H₂ (e.g., HCN⁺ + H → CN⁺ + H₂), we have always assumed that H₂ is released in its para form, the most favorable energetically (similarly, only ortho-D₂ is considered in similar reactions with deuterated species). This does not noticeably accelerate the decay of ortho-H₂. We checked that after, e.g., 10⁵ yr the para-H₂ production rate is 99.9% dominated by reactions of ortho-H₂ with H⁺₃ and H⁺. Our model is not aimed at reproducing accurately a given cloud, but it allows to follow the evolution of the H₂ OPR in a molecule-rich gas. We have computed the chemical evolution of a cloud of constant density (n(H₂) = 10⁴ cm⁻³), constant temperature (10 K), average grain radius of 0.1 μm, and cosmic ionization rate of 1 × 10⁻¹⁷ s⁻¹, which represent the usual conditions met in starless dark clouds. We have started with all species being atomic except H which is considered to be already entirely converted to H₂ and D to HD, and we have considered different OPR starting ratios from 3 down to 3 × 10⁻³ to account for the possibility of H₂ production OPR below 3. Figure 2 shows the evolution of DCO⁺ and ortho-H₂ with time. We have traced the detectability limit of DCO⁺ for a cloud with an H₂ column density of 10²² cm⁻², considering that a DCO⁺ J:1→0 line with a width of 0.5 km s⁻¹ and an intensity of 0.1 K would be easily detectable with present-day radio telescopes. With the RADEX non-LTE radiative transfer model (van der Tak et al. 2007), we find that a DCO⁺ column density of ∼2 × 10¹¹ cm⁻² is detectable, equivalent to a relative abundance to H₂ of ∼2 × 10⁻¹¹. Figure 2 shows that for any initial H₂ OPR below 0.1, DCO⁺ should be detected throughout the cloud in less than 3 × 10⁵ yr. For an initial OPR = 0.1, the DCO⁺ becomes marginally detectable around ∼5 × 10⁵ yr, disappears, and finally becomes clearly detectable after 2 Myr. For OPR > 0.1, it takes 3–6 million yr to become detectable, i.e., when the H₂ OPR drops below ∼0.03. Compared to the H₂ OPR drop from 3 to ∼0.01 in depleted cores which takes less than 2 × 10⁵ yr to happen (Pagani et al. 2009), the difference is twofold: as density increases during the prestellar core formation, (1) the chemistry accelerates, and (2) depletion sets in and H⁺ is no more destroyed by the other species which have now disappeared from the gas phase (H⁺₃ also becomes more abundant but to a lesser extent). H⁺ can even become the dominant ion. The higher abundance of H⁺ and H⁺₃ and the accelerated chemistry provoke the rapid decline of ortho-H₂.

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On the same figure, we also trace the abundance of the sum of the deuterated H⁺₃ isotopologues in the case H₂ OPR = 3. It shows that under 1 Myr, deuteration is dominated by the carbon chemistry (CH₂D⁺, C₂HD⁺, etc.) which is not dependent upon the ortho-H₂ abundance due to exothermicities close to 400 K. It is only beyond 1 Myr that the H⁺₃ deuteration network takes over when ortho-H₂ starts to be negligible. If ortho-H2 is low since the beginning, then the H⁺₃ isotopologue contribution to the abundance of DCO⁺ is important from the start which explains the disappearance of the dip between 1 and ∼3 Myr.

The model is also sensitive to two other initial conditions: the abundance of metals and the cosmic ray ionization rate. Metals, especially alkali and alkaline earth metals like Ca, Na, and K, and others like Fe, have a direct influence on the electronic equilibrium of the chemical model which in turn changes the dissociative recombination efficiency for DCO⁺ and therefore its abundance. For standard ζ (1 × 10⁻¹⁷ s⁻¹) and H₂ OPR (3), we have varied their abundance from 3.4 × 10⁻⁸ to 1.3 × 10⁻⁷. For the lowest abundance (50% of the default one), DCO⁺ becomes detectable much earlier (2 × 10⁵ yr) than in the standard case, drops below the detectability limit after 1 Myr, and is detectable again after 4 Myr. For the highest abundance (twice the default one), DCO⁺ is detectable after 7.5 Myr. Concerning the sensitivity of the model to the cosmic ray ionization rate (Figure 3): if the rate increases, the production of H⁺ and H⁺₃ increases, therefore accelerating the decay of ortho-H₂ and increasing the abundance of HCO⁺ and DCO⁺. For ζ = 3 × 10⁻¹⁷ s⁻¹, DCO⁺ becomes detectable in less than 10⁵ years. Values of ζ in between 1 and 3 × 10⁻¹⁷ s⁻¹ would bring our model in agreement with some recent estimates of cloud lifetime (Enoch et al. 2008; Hatchell & Fuller 2008). Conversely, for ζ = 3 × 10⁻¹⁸ s⁻¹, the chemistry is considerably slowed down, DCO⁺ becomes detectable only after 28 Myr, and steady state is reached only after 47 Myr. It must be noted however that such a low ionization rate seems unlikely following Padovani et al. (2009).

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Ortho-H₂ abundance is most probably the main limiting parameter for DCO⁺ production. The absence of DCO⁺ outside depleted cores, while its abundance is expected to be maximum for CO abundances close to 10⁻⁴, is due to a high ortho-H₂ abundance in dark clouds (OPR ⩾ 0.1) maintained long enough. Such high abundance can survive thanks to low H⁺ and H⁺₃ abundances, remaining low via charge or proton exchanges with many neutral species. However, ortho-H₂ eventually disappears. This sets an upper limit to the age of DCO⁺-quiet dark clouds of 3–6 million yr after H₂ has formed for the most probable initial conditions. Higher cosmic ray ionization rates would increase the abundance of H⁺ and H⁺₃ and can thus only shorten the maximal possible age of the clouds to values which are too small while lower rates would tend to relax the age constraint but seem very unlikely. This result strengthens the results of Hartmann et al. (2001), Enoch et al. (2008), and Hatchell & Fuller (2008), and is not compatible with those of Tassis & Mouschovias (2004) and Mouschovias et al. (2006). From the chemical point of view, cold dark clouds seem to be really short-lived.

We are grateful to Patrick Hennebelle for fruitful discussions. E.R. acknowledges the support of the ANR No. 09-BLAN-020901.