ANALYTICAL STAR FORMATION RATE FROM GRAVOTURBULENT FRAGMENTATION

According to various simulations of hydrodynamic or MHD supersonic turbulence, the density PDF is well represented in both cases by a lognormal form

$$\begin{eqnarray} {\cal P}(\delta) &=& {1 \over \sqrt{2 \pi \sigma _0^2}} \exp \left(- { (\delta - \bar{\delta })^2 \over 2 \sigma _0 ^2} \right), \; \delta = \ln (\rho / \rho _0)\nonumber \\ \bar{\delta }&=&-\sigma _0^2/2 \;, \; \sigma _0^2=\ln (1 + b^2 {\cal M}^2), \end{eqnarray} \tag{ 1 }$$

where ${\cal M}$ is the Mach number and b ≃ 0.5 (Federrath et al. 2010).

The essence of the KM analysis is to assume that there is a critical density, ρ_crit, above which star formation is occurring. Then, the SFR (Equation (20) of KM) is simply obtained by estimating the fraction of gas with density larger than ρ_crit,

$$\begin{eqnarray} &&{\rm SFR}_{\scriptsize\textit{ff}} = \epsilon {\tau _{\scriptsize\textit{ff}}^0 \over \tau _{{\scriptsize\textit{ff}},cr} \phi _t} \int ^\infty _{\ln \widetilde{\rho }_{{\rm crit}}} \widetilde{\rho } {\cal P}({\delta }) d\widetilde{\delta }, \end{eqnarray} \tag{ 2 }$$

with ${\widetilde{\rho }} = \rho / \rho _0$. KM further assume that $\tau _{{\scriptsize\textit{ff}},cr} \simeq \tau _{\scriptsize\textit{ff}}^0$.

In this expression, is the (supposedly mass-independent) efficiency with which the mass within the collapsing prestellar cores is converted into stars. Calculations (e.g., Matzner & McKee 2000; Ciardi & Hennebelle 2010) as well as observations (e.g., André et al. 2010) suggest that ≃ 0.3–0.5. The parameter ϕ_t corresponds to the time needed for a self-gravitating fluctuation to be replenished. KM estimate it to be of the order of a few, in agreement with the analysis we propose in the Appendix.

In KM, ρ_crit is determined from the condition that the corresponding Jeans length must be equal to the sonic length. Their underlying assumption is that turbulent support will be too efficient to enable star formation at scales larger than the sonic length. This yields $\widetilde{\rho }_{{\rm crit}, {\rm KM}} = (\phi _x \lambda _{J0}/\lambda _s)^2$, where ϕ_x is a coefficient of order unity, λ_J0 is the Jeans length at the mean cloud density, and λ_s is the sonic length.

The expression obtained by PN is similar to the KM one, stated by Equation (2), except that they do not assume $\tau _{\scriptsize\textit{ff}}^0 = \tau _{{\scriptsize\textit{ff}},{\rm cr}}$, but instead $\tau _{{\scriptsize\textit{ff}},{\rm cr}}/ \tau _{\scriptsize\textit{ff}}^0 = \sqrt{\tilde{\rho }_{{\rm crit}}}$, as indeed comes out from the integral in Equation (2). They consider that both and ϕ_t are equal to 1, except in the magnetized case where they argue that ≃ 0.5 (which appears to be the main reason for the reduced SFR in the magnetized case). With Equations (1) and (2), this yields (Equation (30) of PN)

$$\begin{eqnarray} &&{\rm SFR}_{\scriptsize\textit{ff}} = {\epsilon \over 2 \phi _t} \widetilde{\rho }_{{\rm crit}}^{1/2} \left[ 1 + {\rm erf} \left({ \sigma _0^2 -2 \ln (\widetilde{\rho }_{{\rm crit}}) \over 2^{3/2} \sigma _0} \right) \right]. \end{eqnarray} \tag{ 3 }$$

The main difference with the KM model, however, resides in the choice of ρ_crit. In PN, this latter is obtained by requiring that the corresponding Jeans length be equal to the typical thickness of the shocked layer, inferred by combining isothermal shock jump conditions and a turbulent velocity scaling v∝l^0.5. This yields $\widetilde{\rho }_{{\rm crit}, {\rm PN}} \simeq 0.067\, \theta ^{-2} \alpha _{{\rm vir}} {\cal M}^2$, where θ ≈ 0.35 is the ratio of the cloud size over the turbulent integral scale and α_vir is the virial parameter, α_vir = 2E_kin/E_grav = 5V²₀/(πGρ₀L²_c), where V₀ is the rms velocity within the cloud, representative of the level of turbulent versus gravitational energy in the cloud (Equations (8) and (9) of PN).

In the HC theory of star formation (see HC08, HC09) prestellar cores are the outcome of initial density fluctuations that isolate themselves from the surrounding medium under the action of gravity. These fluctuations are determined by identifying in the cloud's random field of density fluctuations the structures of mass M which at scale R are gravitationally unstable, according to the virial theorem. This condition defines a scale-dependent (log)-density threshold, δ^c_R = ln (ρ_c(R)/ρ₀), or equivalently a scale-dependent Jeans mass, M^c_R

$$\begin{eqnarray} &&M_R^c = a_J^{2/3} \left({ (C_s)^2 \over G } R + {V_0^2 \over 3\, G } \left({R \over 1 {\rm pc}}\right)^{2\eta } R \right), \end{eqnarray} \tag{ 4 }$$

where C_s is the sound speed, G is the gravitational constant, a_J is a constant of order unity, while V₀ and η ≃ 0.4 determine the rms velocity:

$$\begin{eqnarray} &&\big\langle V_{\rm rms}^2\big\rangle = V_0^2 \times \left({R \over 1 {\rm pc}} \right) ^{2 \eta }. \end{eqnarray} \tag{ 5 }$$

A fluctuation of scale R will be replenished within a typical crossing time τ_R, and will thus be replenished a number of time equal to $\tau ^0 _{\scriptsize\textit{ff}} / (\phi _t \tau _{R,{\scriptsize\textit{ff}}})$, where $\tau _{R,{\scriptsize\textit{ff}}}=\tau _R/\phi _t$ is the freefall time at scale R (see the Appendix), i.e., at density ρ_R ∼ M_R/R³. Including this condition into the HC formalism yields, after some algebra, for the number-density mass spectrum of gravitationally bound structures, ${\cal N} (M)=d(N/V)/dM$:

$$\begin{eqnarray} &&{\cal N}(M_R) \simeq { {\rho _0} \over M_R} {dR \over dM_R} \times \left(-{d \delta _R \over dR} e^{\delta _R} \left({\tau ^0 _{\scriptsize\textit{ff}} \over \tau _R}\right) {\cal P}(\delta _R) \right). \end{eqnarray} \tag{ 6 }$$

Apart for the time ratio ${\tau ^0 _{\scriptsize\textit{ff}} / \tau _R}$, this equation is similar to Equation (33) of HC08 and Equation (27) of HC09. According to this definition, SFR_ff is thus given by the integral of the mass spectrum specified by Equation (6):

$$\begin{eqnarray} &&{\rm SFR}_{\scriptsize\textit{ff}} =- {\epsilon \over \phi _t} \int _0 ^{M_{{\rm cut}}} { dM \over M} {dR \over dM} \, { d \delta _R \over dR} {\tau ^0_{\scriptsize\textit{ff}} \over \tau _{R,{\scriptsize\textit{ff}}}} e^{\delta _R} {\cal P}(\delta _R). \end{eqnarray} \tag{ 7 }$$

According to Equation (4), M_cut corresponds to the mass associated with the largest size fluctuations that can turn unstable in the cloud, y_cut = 2R/L_c. We verified that, as long as y_cut is not too small, the results depend only weakly on its value (P. Hennebelle & G. Chabrier 2012, in preparation). In the following, we will pick y_cut ≈ 0.1 as our fiducial value.

A few remarks are worth discussing at this stage. First, in the present theory there is no explicitly introduced critical scale or density for star formation, as we sum up over all gravitationally unstable cores, irrespective of their scale or density. This is achieved through the multi-scale analysis expressed by Equations (6) and (7). Indeed, turbulence is by essence a multi-scale phenomenon and introducing a critical scale does not appear clearly justified. Indeed, a piece of fluid, even if dominated by turbulence, can still collapse if it is self-gravitating.

Another essential difference with the KM and PN theories is that they rely on a unique characteristic collapsing time (the mean cloud freefall time in KM and the critical density freefall time in PN). This can be seen from Equations (2) and (3), where the term $\widetilde{\rho }^{1/2}$, taken at $\widetilde{\rho }_{{\rm crit}}$, lies outside the integral. In contrast, in our theory (see Equation (7)), the freefall density-dependence of each collapsing structure is properly accounted for, as the freefall time consistently varies with mass M and scale R, $\tau _{R,{\scriptsize\textit{ff}}}\propto \rho _R^{-1/2}$.

Even though we stress that the SFR cannot be properly determined by a simple integral of the density PDF, because such an integral, unlike Equation (7), does not take into account the spatial distribution of the gas, we suggest the following simplified but more consistent expression, which retains the collapsing time density-dependence, instead of Equations (2) and (3):

$$\begin{eqnarray} \nonumber {\rm SFR}_{\scriptsize\textit{ff}}^{{\rm simp}} &=& \epsilon \int ^\infty _{\delta _{{\rm crit}}} {\tau _{\scriptsize\textit{ff}}^0 \over \tau _{\scriptsize\textit{ff}}(\rho) \phi _t} \widetilde{\rho } {\cal P}(\delta) d{\delta } = {\epsilon \over \phi _t} \int ^\infty _{\delta _{{\rm crit}}} \widetilde{\rho }^{3/2} {\cal P}(\delta) d\delta \\ &=& {\epsilon \over 2 \phi _t} \exp \big(3 \sigma _0^2 / 8\big) \left[ 1 + {\rm erf} \left({ \sigma _0^2- \ln (\widetilde{\rho }_{{\rm crit}}) \over 2^{1/2} \sigma _0 } \right) \right]. \end{eqnarray} \tag{ 8 }$$

This SFR is larger than the ones given by Equations (2) and (3), as shown in the next section.

We consider different choices for ρ_crit. When using ρ_{crit, KM} or ρ_{crit, PN}, we refer to the corresponding SFR as "multi-freefall KM or PN," respectively. We also consider another value for ρ_crit, obtained by simply requiring that the Jeans length at this density is equal to y_cutL_c. This corresponds to the assumption that only fluctuations smaller than a given cloud size fraction can collapse. We simply refer to this model as "multi-freefall." It is easy to check that Equation (8) depends only weakly on y_cut, except when y_cut → 0.

As in KM and PN, we define the cloud properties by α_vir and ${\cal M}$. Figure 1 displays ${\rm SFR}_{\scriptsize\textit{ff}}^0$, which corresponds to SFR_ff for = 1 and ϕ_t = 1, obtained with different formalisms, for various values of the virial parameter, and for three typical Mach numbers, namely, ${\cal M}=16$, 9, and 4.

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Both the HC and the multi-freefall models are larger by a factor of ∼2–3 than PN and by at least an order of magnitude than KM. This stems from the fact that, when taking into account the density-dependence of the structure collapsing times, (1) dense regions collapse fast and (2) fluctuations denser than ρ_crit have a smaller freefall time than the ones at ρ_crit, globally increasing the value of ${\rm SFR}_{\scriptsize\textit{ff}}^0$. When such a density-dependence is properly accounted for, all SFR determinations are in better agreement. Nevertheless, some differences persist between the various multi-freefall models. This stems from the choice of ρ_crit. Indeed, the choice of $\widetilde{\rho }_{{\rm crit}}$ in the KM and PN theories (see Sections 2.1 and 2.2) yields $y_{{\rm cut}}\approx {\cal M}^{-2}$ and thus corresponds to very small values of y_cut, implying that only small Jeans masses (or conversely only very dense structures) are taken into account in these models.

Two interesting trends can be inferred from Figure 1. First, increasing the virial parameter leads to a decrease of the SFR, with a severe reduction above some typical value of α_vir, which decreases with decreasing Mach number. This naturally arises from the fact that, as α_vir increases, the increasing contribution of kinetic energy over potential energy prevents gravitational collapse and thus inhibits star formation, a point already noticed by KM and PN. Second, the SFR increases, although modestly, with the Mach number. This is because increasing the Mach number extends the core mass function into the low-mass domain (see PN and HC08) and, as small-scale structures have shorter freefall times, this increases the number of collapsing small cores, thus the SFR. This positive dependence of the SFR upon ${\cal M}$ is in agreement with the results of PN but contrasts with the ones of KM (see their Equation (30)), as seen in the figure. Such a decreasing dependence of the SFR with increasing Mach in the KM theory clearly stems from the fact that the $\widetilde{\rho }_{{\rm crit}}^{1/2}$ term (see Equation (3)) is lacking in their Equation (20).

Star-forming giant molecular clouds in the Milky Way have masses of 10³ ≲ M_c/M_☉ ≲ 3 × 10⁶, with ${\cal M}\approx 4$–30 and α_vir ≈ 0.3–3. The observed SFR per cloud freefall time lies in the range $0.03\lesssim {\rm SFR}_{\scriptsize\textit{ff}}\lesssim 0.3$, with a mean value $\langle {\rm SFR}_{\scriptsize\textit{ff}}\rangle \approx 0.16$ (Murray 2011; although see Feldmann & Gnedin 2011 for caution). Evans et al. (2009) and Heiderman et al. (2010) find SFRs in the range of ≈0.02–0.12 for nearby molecular clouds and of ≈0.03–0.5 for massive star-forming dense clumps, yielding a mean value of ≈0.1, about an order of magnitude larger than the values predicted by KM. Krumholz & Tan (2007, Figure 5) report lower values at low density. At high density (≳ 10⁴ cm⁻³), however, two of their three data (ONC and CS(5–4)) are compatible with the aforementioned mean values.⁴ According to our calculations (see Figure 1), for such cloud/clump characteristics, ${\rm SFR}_{\scriptsize\textit{ff}}^0$ is predicted to lie within the range of ≈0.3–3. As discussed in Section 2.1 and in the Appendix, the effective SFR is ${\rm SFR}_{\scriptsize\textit{ff}} = (\epsilon / \phi)\times {\rm SFR}_{\scriptsize\textit{ff}}^0$, where /ϕ_t ≈ 0.1–0.2. Therefore, according to the present calculations, theories based on a multi-freefall formalism yield SFRs per freefall time in typical molecular clumps in the range ${\rm SFR}_{\scriptsize\textit{ff}}\approx$ 0.03–0.6, going from low-dense clouds to the densest clumps, consistent with the observed values.

Figure 2 displays the SFR per unit area, ${\dot{\Sigma }}_\star = {\rm SFR}_{\scriptsize\textit{ff}}\times \Sigma _g/\tau ^0_{\scriptsize\textit{ff}}$, with Σ_g = L_cρ₀ = M_c/πL²_c, as a function of cloud surface densities, Σ_g, for four typical cloud sizes, L_c = 1, 4, 10, and 40 pc. The clouds are assumed to follow Larson's (1981) relations and thus have velocities given by Equation (5) and densities n₀ × (L_c/1pc)^−0.7, where n₀ = 10² to 10⁴ cm⁻³, yielding cloud masses M_c = 200–10⁶ M_☉. From these values, $\cal M$ and α_vir can be consistently determined. In Figure 2, we have taken /ϕ_t = 0.1.⁵

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Lada et al. (2010) and Heiderman et al. (2010) data for clouds and massive clumps are shown for comparison. Clearly, almost all theories exhibit a direct correlation between the density of star formation and the gas density and, when including the /ϕ_t factor, reproduce well the observational results, except possibly for the densest clumps, where the SFRs are about a factor of ∼5 larger. The large spread of the data precludes a clear distinction between the theories at this stage, apart from the KM one which clearly lies well below most of the data points.

Interestingly, both the exact HC and "multi-freefall" calculations predict a drastic drop in the SFR below Σ_c ≈ 110–120  M_☉ pc⁻² and a change of slope in the ${\dot{\Sigma }}_\star \propto {\Sigma }_g^{N}$ relation, with N ≈ 4.8 and N ≈ 1.6, similar to the Kennicutt–Schmidt relation (N_KS ≈ 1.4), respectively, below and above Σ_c, in very good agreement with the observations (Heiderman et al. 2010). This density corresponds to a visual extinction A_V ≈ 6 (A_K ≈ 1). A similar density threshold for significant dense core population has been identified in several surveys (Onishi et al. 1998; Johnstone et al. 2000; Kirk et al. 2006; Enoch et al. 2007; Lada et al. 2010; André et al. 2010). Various authors (e.g., Johnstone et al. 2000; Kirk et al. 2006; Heiderman et al. 2010) have suggested that the origin of such a density threshold is related to magnetic fields, which cannot support the gas against gravitational collapse above some density. The present calculations, however, show that there is no need to invoke magnetic field support or MHD shock conditions to get such a threshold although, as mentioned in the Appendix, magnetic fields may contribute dynamically by reducing the value of ϕ_t. The threshold simply stems from the fact that at the corresponding density, the size of the clumps becomes comparable to the Jeans length and thus the amount of gas appropriate to form stars drops drastically (see HC09, Figure 8). The relative similarity between the various "multi-freefall" predictions clearly indicates that, besides the /ϕ_t factor, the key physical quantity which determines the SFR is the density-dependence of the freefall time of the collapsing overdense regions induced by turbulence. An alternative possibility as recently suggested by Krumholz et al. (2011) is to assume that the SFR is simply a constant factor times the cloud's or galaxy's volume density over mean freefall time, although a clear physical explanation is lacking at this stage.