TIMESCALES FOR LOW-MASS STAR FORMATION IN EXTRAGALACTIC ENVIRONMENTS: IMPLICATIONS FOR THE STELLAR INITIAL MASS FUNCTION

Low-mass star formation essentially occurs by the collapse of a molecular cloud under its own weight accompanied by the dissipation of gravitational potential energy. In any situation where the star formation is not externally triggered by a violent event, for example the passage of a shock, the actual collapse time will be somewhat longer than the free-fall time due to turbulence within the cloud. In this work, we use the free-fall time as a useful underestimate of the typical collapse time for a molecular cloud undergoing quiescent star formation.

We can relate the free-fall time of a collapsing core to the number density of hydrogen nuclei as follows:

$$\begin{equation} t_{{\rm ff}}=\sqrt{\frac{3\pi }{16G\rho }}=0.75\times 10^8/{(n_H)}^{\frac{1}{2}}\,{\rm yr}, \end{equation} \tag{ 1 }$$

where n_H = (n(H) + 2n(H₂)) is the local number density of hydrogen nuclei in cm⁻³ (Dyson & Williams 1997).

In order for a cloud of gas to continue to collapse and form a star, the cloud must continue to lose energy, otherwise, the high temperatures and densities within the cloud would halt collapse before the formation of a star could occur. Energy loss occurs via different atomic and molecular transitions, while at high densities, collisions between hot gas and cooler dust grains can also be efficient in cooling the gas. The rate at which energy is lost depends on the cooling function, Λ_tot/erg cm⁻³ s⁻¹, which in turn depends on the abundance of various different molecular coolants. The total cooling time is then given by

$$\begin{equation} t_{\rm cool}=\frac{\frac{5}{2}n_HkT}{\xi \Lambda _{\rm tot}}\;\ {\rm yr,} \end{equation} \tag{ 2 }$$

where k is the Boltzmann constant, T is the gas temperature, and ξ is the metallicity in units of solar metallicity. The quantity $\frac{5}{2}n_HkT$ represents the total energy, both thermal and internal, of the system and ξΛ_tot is the total interstellar cooling rate.

Note that our chemical models described in Section 3 do not include the effects of removing molecular coolants from the gas phase by freeze-out. If freeze-out occurs, the cooling timescale will be underestimated in regions of parameter space where there is significant ice formation.

A detailed calculation of the rate per unit volume at which a species freezes out can be found in Rawlings et al. (1992). This quantity depends on the relative abundance of dust grains compared to hydrogen nuclei, the physical properties of the grain as well as the species whose freeze-out is being considered, and electrostatic effects to take into account the fact that ionic species will freeze-out at a different rate to neutral species. Due to the presence of more than one grain population in most systems, the grain properties need to be averaged over the grain size distribution. Rawlings et al. (1992) relate this size distribution to the depletion coefficient, D, that depends on the abundance of very small grains and on the metallicity.

Taking these assumptions into consideration, the freeze-out timescale can be derived from the rate of freeze-out given by Rawlings et al. (1992):

$$\begin{equation} t_{fo}=\frac{0.9\times 10^6{(m_{X}/28)}^\frac{1}{2}}{(T/10)^\frac{1}{2}(n_H/10^{4})D} {\rm \;yr,} \end{equation} \tag{ 3 }$$

where m_X is the molecular mass of the species whose freeze-out is being considered and T is the gas temperature.

Desorption is the process by which molecules in ices on the surface of dust grains are returned to the ISM. The mechanism through which heat is generated for this desorption is still debated (Roberts et al. 2007). In this work, we consider only one mechanism for desorption, namely exothermic reactions occurring on the grain surface. More specifically, the liberation of energy from H₂ formation on dust grains results in a proportion of the frozen molecules on the dust grain being returned to the gas phase.

We calculate the desorption rate of CO, which is the most abundant molecular species besides hydrogen. This rate is

$$\begin{equation} \frac{d n(CO)}{dt}\bigg| _{\rm des}=3\times 10^{-17}nn_{H}\xi \gamma \sqrt{\frac{T}{100}}. \end{equation} \tag{ 4 }$$

The rate depends on the metallicity that scales linearly with the dust to gas ratio, the abundance of both molecular and atomic hydrogen, the temperature, T, and the number of CO molecules desorbed for every H₂ formed, γ. The coefficient 3 × 10⁻¹⁷ represents the canonical H₂ formation rate in the Milky Way.

From this expression, we can calculate a typical timescale associated with desorption

$$\begin{equation} t_{\rm des}=\frac{1\times 10^9}{\big[n_Hn(H)(T/100)^\frac{1}{2}\gamma \xi \big]} {\rm \;yr,} \end{equation} \tag{ 5 }$$

where n_H is the local number density of hydrogen nuclei and n(H) is the column density of hydrogen atoms only. The number of CO molecules desorbed per H₂ formation, γ, can be constrained by considering a prestellar core in the Milky Way like L1689B where significant depletion of CO molecules has been directly observed (Redman et al. 2002). Assuming for such an object with n_H ∼ 10⁴–10⁵ cm⁻³ that the timescale for freeze-out is less than half the timescale for desorption gives an upper limit of ∼0.25 on γ. We use γ = 0.2 throughout this work and assume that this number does not change in high-redshift systems. This is plausible as we expect the microscopic properties of the dust in high-redshift galaxies to be fairly similar to those in the Milky Way. Both the freeze-out and desorption timescales depend on these microscopic properties.

Magnetic fields may play an important role in supporting dense molecular clouds and many of the fragments within them. The magnetic field lines are coupled directly to the ions and indirectly to the neutral matter through frequent ion-neutral collisions. At low ionizations, the frequency of the ion–neutral collisions becomes less regular and the field lines decouple from the neutral matter thereby allowing the matter to drift in response to the gravitational potential. This process is known as ambipolar diffusion and plays an important role in determining the rate of quiescent star formation.

The characteristic timescale associated with this process of decoupling the field from the neutral matter will depend on the relative drift velocities of the ions and neutral particles. Hartquist & Williams (1989) define the ambipolar diffusion timescale as follows:

$$\begin{equation} t_{\rm amb}=4\times 10^5(x_i/10^{-8})\,\ {\rm yr.} \end{equation} \tag{ 6 }$$

This is a function only of the fractional ionization, x_i, within the cloud and is consistent with the more detailed treatment of Mouschovias (1979). However, in the earlier paper, the ambipolar diffusion timescale is shown to be a function of position within the cloud. Therefore, variations in the degree of ionization as a function of cloud depth become important. For the purposes of our rather crude calculations, we neglect this positional dependence. Equation (6) therefore corresponds to the Mouschovias (1979) timescale at an intermediate position toward the interior of the core. It should be noted that this timescale is only relevant for magnetically subcritical cores.

The chemical timescale is the time taken to create molecular coolants from the initially atomic gas assuming that nearly all of the hydrogen is molecular. In other words, it is the time taken to ionize sufficient hydrogen to allow complete chemical conversion of C and O to molecular form. Cosmic rays are responsible for ionizing the hydrogen. The relative abundances of carbon and oxygen will scale with the metallicity, ξ. The chemical timescale is

$$\begin{equation} t_{\rm chem}\simeq \frac{3\xi \big(n_{{\rm C}}^{\rm tot}+n_{{\rm O}}^{\rm tot}\big)}{n({H}_2)\zeta }= \frac{5\times 10^6\xi }{\zeta /1\times 10^{-17}} {\rm \;yr.} \end{equation} \tag{ 7 }$$

The chemistry is important for providing the coolant molecules and therefore must not be too long if star formation is to occur. The analytical expression for the chemical timescale represented by Equation (7) depends only on the metallicity and cosmic ray ionization rate and is rather crude. The chemical timescale is less relevant in cores than in the parent cloud that may be collapsing from a diffuse to a dark state as there is not much change in the chemistry once cores begin to form.

We vary the hydrogen nuclei number density between 100 cm⁻³ and 10⁶ cm⁻³ in this study and look at the effect of this variation on the low-mass star-formation timescales. The results are summarized in Figure 1.

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As expected, the free-fall time decreases with increasing number density converging to ∼10⁶ yr at 10⁴ cm⁻³. At this density, all the different timescales converge to roughly the same order of magnitude. Thus, both freeze-out and desorption proceed with desorption a slightly slower process so that molecular ices do form. Desorption maintains a population of molecular coolants in the gas phase. The free-fall time is comparable to the cooling time, thermally induced collapse occurs, and stars are able to form. The ambipolar diffusion timescale is comparable to the dynamical free-fall time and star formation can occur in magnetically subcritical clouds. We note that these conditions are appropriate for a Milky Way type galaxy and therefore our approach indicates as expected that low-mass star formation is reasonably efficient in Milky Way type systems.

However, at very high densities, the free-fall time is shorter than the cooling time and the cores will heat up during the collapse that is then arrested.

The freeze-out timescale and desorption timescale are comparable and both decrease with increasing density at the same rate suggesting that molecular ices are able to form at all densities between 100 cm⁻³ and 10⁶ cm⁻³. This also implies that the cooling time is systematically underestimated as ice formation, which is not included in our chemical model, removes molecular coolants from the gas phase at all densities.

Looking deeper into the cloud, i.e., going from A_v ≃ 3 to A_v ≃ 10, does not seem to affect significantly the freeze-out or desorption timescales. However, the cooling time is slightly larger at higher visual extinctions as the cooling function decreases with increasing A_v. This is probably due to increased radiation trapping in the cooling lines.

The ambipolar diffusion timescale is much longer than the other timescales at very low densities. This suggests as expected that star formation cannot occur in clouds with n_H < 10³ cm⁻³ that are magnetically subcritical as the field is very slow to decouple.

The metallicity, ξ, is another important physical property that will affect low-mass star formation. This parameter is particularly important for modeling high-redshift systems as the primordial gas from which these objects formed is likely to be less chemically enriched than the interstellar medium in the Milky Way in some cases, and will therefore have lower metallicities. As we are interested in low-metallicity objects that formed early in the history of the universe, we consider metallicities as low as 10⁻⁴ξ_☉. We also consider metallicities enhanced above solar values to 3ξ_☉. In our model, we assume that the elemental abundances of all metals scale linearly with metallicity. We also assume that the dust to gas mass ratio scales linearly with metallicity. The results from this study are summarized in Figure 2.

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As expected, the cooling time falls with increasing metallicity due to the increased abundance of molecular coolants. We can see that at metallicities below ξ = 0.1ξ_☉, the cooling time is much longer than the free-fall time and collapse and star formation are likely to be impeded by gas pressure. The freeze-out time also decreases with increasing metallicity due to the presence of more dust grains but the desorption time is independent of metallicity. At low metallicities, freeze-out is slower than desorption so ice formation is inhibited. Temperatures however remain high due to the long cooling timescales.

The ambipolar diffusion timescale remains fairly insensitive to the metallicity. The fractional ionization is much higher at low visual extinctions so the ambipolar diffusion timescale falls with increasing depth into the cloud. Note that the visual extinction scales linearly with metallicity so high visual extinctions in low-metallicity clouds will correspond to larger physical distances within the cloud. Hence, the calculations for A_v ≃ 10 are probably unfeasible for low-metallicity clouds. We have considered outputs at A_v ≃ 7.5 and A_v ≃ 9.5 for ξ = 10⁻⁴ξ_☉ and ξ = 10⁻³ξ_☉, respectively. As the cloud is likely to be optically thick even at these lower depths, the chemistry is unlikely to be very different to the A_v ≃ 10 case.

The chemical timescale increases linearly with metallicity as expected. Low-metallicity systems therefore convert a significant fraction of available elements into molecular coolants.

Although little is known about the origin of extragalactic cosmic rays, they are very important in ionizing hydrogen and driving the chemistry during star formation. The level of ionization within a dark cloud may determine whether or not the cloud is stable against gravitational collapse if magnetic fields are present.

An estimate for the cosmic ray ionization rate in the Galaxy is ∼10⁻¹⁷ s⁻¹ (Hartquist et al. 1978). In this study, we vary the cosmic ray ionization rate between 10⁻¹⁸ s⁻¹ and 10⁻¹³ s⁻¹ in order to mimic both quiescent star-forming systems such as those in our own galaxy as well as starburst regions and active galactic nuclei (AGNs) that are expected to have cosmic ray fluxes of the order of 10⁻¹⁵ s⁻¹ to 10⁻¹³ s⁻¹ (Meijerink et al. 2006; Suchkov et al. 1993; Bradford et al. 2003). The results of varying the cosmic ray ionization rate are summarized in Figure 3.

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As the cosmic ray ionization rate is increased, the cooling time falls and at ζ ∼ 10⁻¹⁶ s⁻¹ it drops below the free-fall time. Cores cool more and more quickly and are therefore able to collapse to form stars. The chemical timescale is small at high values of the cosmic ray ionization rate explaining the low cooling times as molecular coolants are quick to form in such circumstances.

The freeze-out time is greater than the desorption time and the propensity for ice formation is reduced thereby allowing molecular coolants to remain in the gas phase to effectively cool the collapsing gas. Desorption is therefore unlikely to be a significant process. However, formally, the desorption time becomes shorter as expected with increasing cosmic ray ionization rate as more and more hydrogen molecules are ionized thereby increasing the abundance of atomic hydrogen as well as the temperature of the gas. If ices could be formed, they would therefore be quick to evaporate off the dust grains once again replenishing the gas phase with molecular coolants and encouraging star formation.

The ambipolar diffusion timescale becomes very large as the fractional ionization increases and the magnetic field is slow to be released from the collapsing core. Based on the results of Crutcher et al. (1987) and Athreya et al. (1998), the magnetic field strength, B, of molecular clouds varies between 1 μG and 100 μG for densities between 1 cm⁻³ and 10⁵ cm⁻³. This field strength, B scales with the density as B ∝ (n_H)^κ where $\frac{1}{3}\le \kappa \le \frac{1}{2}$ (Mouschovias 1976) is consistent with observations. In this study, based on various observations (Crutcher et al. 1987; Bourke et al. 2001; Beck et al. 2005; Kim et al. 1990) we consider κ ∼ 0.4. We can calculate the ratio of the magnetic pressure and thermal pressure, α, using

$$\begin{equation} \alpha =\frac{B^2}{8\pi n_H k T}. \end{equation} \tag{ 8 }$$

For α ≫ 1 the magnetic pressure dominates and for α ≪ 1 the thermal pressure dominates. For high cosmic ray fluxes and at a constant density of n_H = 10⁵ cm⁻³, we compute α ∼ 0.4. This suggests that the magnetic pressure may in some cases be sufficient to halt the collapse but in other cases, depending on the temperature of the core and the exact value of κ, low-mass star formation can proceed unimpeded.

The FUV radiation field is a significant source of heating of neutral interstellar gas. As explained in Section 3, the radiation field is taken to be the standard interstellar radiation field (Draine 1978) multiplied by a factor of χ where χ = 1.7 corresponds to a flux of 1.6 × 10⁻³ erg s⁻¹ cm⁻². The FUV radiation field strength is linked to the number of massive stars in the Galaxy and given the presence of many massive stars at high redshift (Abel et al. 2002; Bromm et al. 1999) is likely to be large in these systems. In this study, we vary χ between 1.7 and 1.7 × 10⁴ and study the effect on our star-formation timescales. The results are summarized in Figure 4.

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At A_v ≃ 10, the FUV radiation is sufficiently attenuated to not affect the chemistry within the cloud and all our timescales therefore remain fairly constant with FUV flux strength. At lower visual extinctions, we find that the cooling time is almost always smaller than the free-fall time and collapse and star formation are therefore encouraged. The freeze-out time is also almost always smaller than the desorption timescale except at very high FUV radiation field strengths. This suggests that the formation of icy mantles on dust grains is possible even at FUV radiation field strengths of χ = 1.7 × 10³ at A_v ∼ 3. The desorption time and freeze-out time are fairly comparable at all radiation field strengths so any icy mantles deposited on the dust grain are quick to evaporate. At very high FUV flux strengths, the desorption time is shorter than the freeze-out time and ice formation is inhibited. Both these processes mean that the ISM is always rich in molecular coolants at high FUV radiation field strengths and collapse is encouraged.

The ambipolar diffusion timescale is very large for high radiation field strengths and at low A_v. This once again suggests that magnetically supercritical cores will be stable against collapse. However, at high A_v where the radiation field cannot penetrate the cloud, the ambipolar diffusion timescale is comparable to the dynamical timescale and the magnetical criticality is less significant.

As we are linking both the FUV radiation field strengths and the cosmic ray ionization rate with massive stars in this rather simplistic analysis, it is not surprising that both parameters have similar effects on the star-formation timescales. We conclude that the presence of massive stars resulting in an elevated FUV flux strength serves to encourage quiescent low-mass star formation in every possible way at low A_v provided the residual magnetic pressure implied by the long ambipolar diffusion timescale does not halt collapse. We compute the ratio of the magnetic to thermal pressure for χ = 1.7 × 10⁴ to be α ∼ 0.5 and conclude that the magnetic pressure may in some cases be sufficient to halt collapse.

Bayet et al. (2008) have identified the Cloverleaf (z ≃ 2.6) and APM08279 (z ≃ 3.9) as possible models for high-redshift sources. Although the physical properties of these sources are not well constrained, molecular detections suggest that they are actively forming stars (Lutz et al. 2007; Gao & Solomon 2004; Wagg et al. 2005). We expect such objects to have a range of different metallicities with galaxies with significant observed dust obscuration such as HCM6A (z = 6.56; Chary et al. 2005) likely to be more chemically enriched. The presence of many massive stars in these systems will also result in high FUV radiation field strengths and cosmic ray ionization rates. Collapse is likely to occur at moderately high densities due to the lack of available molecular coolants compared to a Milky Way type object.

In order to model such a system, we calculate the star-formation timescales for three different models—(I) n = 10⁴ cm⁻³, ξ = 0.05ξ_☉, ζ = 10⁻¹³ s⁻¹, and χ = 1.7 × 10⁴; (II) n = 10⁵ cm⁻³, ξ = 0.05ξ_☉, ζ = 10⁻¹⁴ s⁻¹, and χ = 1.7 × 10³; and (III) n = 10⁵ cm⁻³, ξ = ξ_☉, ζ = 10⁻¹⁴ s⁻¹, and χ = 1.7 × 10³—at depths corresponding to A_v ∼ 3 and A_v ∼ 10. As previously stated, the presence of many massive stars in high-redshift systems justifies our choice of elevated FUV radiation field strengths when modeling such systems. In order to mimic starburst galaxies with high cosmic ray fluxes such as M82 (Suchkov et al. 1993) and NGC253 (Bradford et al. 2003) at high redshift, we adopt high cosmic ray ionization rates. Meijerink et al. (2006) estimate a cosmic ray ionization rate of 5 × 10⁻¹⁵ s⁻¹ for a star-formation rate of ∼100 M_☉ yr⁻¹ and we expect high-redshift galaxies with higher star-formation rates to have elevated cosmic ray fluxes. Finally, there is evidence that high-redshift galaxies exist at a range of metallicities with galaxies with significant observed dust obscuration likely to be more chemically enriched and we therefore consider both solar and subsolar metallicities. Our results are summarized in Table 1.

We can see that for the first model, the free-fall time is comparable to the cooling time and cores can collapse to form stars even at depths as low as A_v ∼ 3. The formation of icy mantles is likely to be inhibited in such systems as the freeze-out time is very large compared to the desorption time.

The second model has slightly lower cosmic ray fluxes and FUV radiation field strengths but is considered to be at a higher density. In this case, the free-fall timescale is shorter than the cooling time at low A_v and cores can collapse to form stars, but at A_v ∼ 10 the cooling time is larger and the propensity for low-mass star formation is reduced. Once again, the formation of icy mantles is inhibited in such a system.

The third model has also elevated FUV radiation field strengths and cosmic ray fluxes but is now considered at solar metallicity. The cooling time is now significantly lower than the free-fall time at all depths and low-mass star formation is unimpeded by thermal pressure while ice formation still remains inhibited.

In all three cases, the ambipolar diffusion timescale is very large so the residual field may be able to halt collapse. Using the temperatures and densities of these model runs, we calculate the thermal and magnetic pressures for each of the models in the same way as in Section 4.3 and their ratio, α. In both the subsolar metallicity models, we compute α ∼ 0.05 so the thermal pressure dominates over the magnetic pressure. The cooling times are small compared to the dynamical free-fall time, particularly at low A_v and low-mass star formation is encouraged. The third model at solar metallicity has α ∼ 0.6 suggesting that in this case the magnetic pressure may be sufficient to halt collapse. We conclude that recent evidence for a top-heavy IMF is only supported by this study if high-redshift galaxies are considered to be at solar metallicities in which case the magnetic pressure may halt the collapse and formation of low-mass stars.