ANALYTICAL THEORY FOR THE INITIAL MASS FUNCTION. III. TIME DEPENDENCE AND STAR FORMATION RATE

The underlying concept on which the theory developed in Papers I and II relies consists of identifying, in a random field of density fluctuations, the mass M_R which at scale R (physically speaking, R denotes the radius of the density fluctuation) is gravitationally unstable, i.e., the mass contained in regions within which gravity dominates over all sources of support. To achieve this, the first step is to determine M_R, the mass contained in regions whose density exceeds a scale-dependent density threshold, $\log \delta ^c_R$, determined by the virial condition. Since all this gas is unstable at a scale smaller than or equal to R, it is expected that it will end up in objects of mass smaller than or equal to the Jeans mass associated with the density threshold. Thus, the second step is to make M_R equal with the mass contained in the structures of mass equal to or smaller than the associated (turbulent) Jeans mass (see Equations (28) and (29) of Paper I).

This approach implicitly assumes that each part of the flow is initially assigned a specific (scale-dependent) Jeans mass, which collapses. While such an assumption is certainly reasonable in the case of cold dark matter fluctuations within the primordial universe, it is a priori not the case in a turbulent flow in which fluctuations of scale R are replenished within a few crossing times, τ_R ≃ 2R/V_rms(R), where $V_{{\rm rms}}(R)\equiv \langle \sigma _R^2\rangle ^{1/2}$ denotes the turbulent rms velocity at scale R and σ_R the velocity fluctuation over scales smaller than R. Therefore, in the case of turbulent molecular clumps, it seems necessary to take such dynamical effects into account. This is not obvious by any means, in particular for large-scale fluctuations for which, first, a long enough time is required and, second, gravity must not entirely freeze the gas motions by preventing fluid particles to escape their local Jeans masses. In this latter case, gravity will prevent the flow motions and the replenishment of the density fluctuations, which have become gravitationally unstable.

On the other hand, if the clump is accreting at a sufficiently high rate or if it contains a large fraction of gas that is not dense enough to be locally gravitationally unstable (i.e., if the Jeans length at the gas density is comparable to or larger than the size of the clump), then dissipation of large-scale turbulence leads to turbulent compression which can continuously generate density fluctuations at all scales and trigger, for the densest ones, gravitational collapse. In this case, one must take into account the fact that during the lifetime of the cloud, τ₀, the fluctuations at scale R have been replenished a number of times equal to τ₀/τ_R. This implies that Equation (31) of Paper I must be modified as

$$\begin{eqnarray} { M_{\rm tot}(R) \over V_c} &=& \int ^{\infty } _ {\delta _R^c} \bar{\rho } \exp (\delta) {\cal P}_R(\delta) \left({\tau _0 \over \tau _R}\right) d\delta \nonumber\\ &=& \int _0 ^ {M_R^c} M^{\prime } \, {\cal N} (M^{\prime }) \, P(R,M^{\prime })\, dM^{\prime }, \end{eqnarray} \tag{ 1 }$$

where $\delta =\log (\rho /{\bar{\rho }})$ denotes the (logarithm of) density fluctuation, $\delta ^c_R$ is the threshold density at scale R, ${\cal P}(\delta)$ is the density distribution (PDF),⁸ and $V_c\sim L_c^3$ denotes the clump's volume.

Apart from the term τ₀/τ_R, this equation is the same as that derived in Paper I. It uses the formulation developed by Jedamzik (1995) in the context of dark matter halos (see also Yano et al. 1996). The first equality for M_tot(R) stems from the fact that the mass contained within structures of mass $M< M_R^c$ is equal to the mass of the gas which, smoothed at scale R, has a (logarithmic) density larger than a critical threshold $\delta _R^c$. The second expression arises from the fact that the number density of structures of mass $M< M_R^c$ is ${\cal N}(M^{\prime }) P(R,M^{\prime }) dM^{\prime }$. Here, ${\cal N}(M^{\prime }) dM^{\prime }$ is the number density of structures of mass between M' and M' + dM', while P(R, M') is the probability of finding a gravitationally unstable structure of mass M' embedded inside a structure of gas which at scale R has a (logarithmic) density larger than $\delta _R^c$. P(R, M') is assumed to be equal to 1 (see Appendix D of Paper I for further justification). The time ratio τ₀/τ_R thus simply illustrates the fact that the flow fluctuations at scale R have been rejuvenated τ₀/τ_R times.

Taking the derivative of Equation (1) with respect to R, we obtain for the number-density mass spectrum of gravitationally bound structures, ${\cal N} (M)=d(N/V_c)/dM$:

$$\begin{eqnarray} {\cal N} (M_R^c) &=& { \bar{\rho } \over M_R^c} {{dR} \over {dM}_R^c} \, \left(-{d \delta _R ^c \over {dR}} \left({\tau _0 \over \tau _R}\right) \exp (\delta _R^c) {\cal P}_R(\delta _R^c)\right. \nonumber\\ &&\left. +\, \int _ {\delta _R^c}^\infty \exp (\delta) {d \over {dR}}\left[ \left({\tau _0 \over \tau _R}\right){\cal P}_R\right]\, d\delta \right). \end{eqnarray} \tag{ 2 }$$

As shown in Paper I, the first term in this expression is the most important one and dominates over the second one except when the scale R becomes comparable to the injection scale, L_i, which is basically the size of the cloud (see the interesting formulation of Hopkins 2012a, 2012b to avoid this large-scale limitation of the present formalism).

Once the expression of the critical density threshold, $\delta _R^c=\ln (\rho (R)/{\bar{\rho }})$, is specified from the virial condition, $\langle V_{{\rm rms}}^2(R)\rangle + 3C_S^2 < E_{{\rm pot}}(R)/M$ (see Section 4.2 of Paper I), the mass spectrum of the self-gravitating pieces of fluid can be inferred from

$$\begin{eqnarray} &&M_R=C_m\, \rho (R)\, R^3, \end{eqnarray} \tag{ 3 }$$

where C_m is a geometrical factor, typically of the order of 4π/3.

Before proceeding further, it is worth stressing what is exactly selected by our procedure. As seen from Equation (1), the integration is performed from 0 to $M_R^c$. This means that the smallest Jeans masses, corresponding to the densest pieces of gas, are first accounted for and removed from the available gas mass. Then the larger Jeans masses are progressively taken into account and removed. By doing so, we properly take into account the fact that there are small Jeans masses embedded into larger ones. This happens, in particular, when the PDF significantly varies locally, i.e., when turbulence is strong, which implies that in that case the scale must be large. Therefore, strictly speaking, we do not identify well-defined bound cores but rather coherent mass reservoirs in the density field that isolate themselves from the surrounding medium under the action of gravity. For sufficiently small scales, the density becomes reasonably uniform within the mass reservoir to have a clear correspondence between this latter and a well-identified "core." For large scales, which will lead eventually to the formation of massive stars, however, the one-to-one core–reservoir correspondence becomes more blurry (as the potential well associated with the mass reservoir becomes itself more shallow). In that case, the reservoir of mass out of which the most massive cores/stars will form corresponds to what is left once all the self-gravitating small-scale density fluctuations embedded in the reservoir have been properly accounted for and "taken away." In that sense, for the largest scales, and thus the most massive cores, our formalism rejoins in some sense the so-called competitive accretion process (see, e.g., Smith et al. 2009). Therefore, in principle, the turbulent fragmentation process is properly described in our formalism, apart from the fact that for the sake of simplicity, the conditional probability P(R, M) is taken to be equal to 1. What has not been taken into account so far, however, is the gravitational fragmentation that occurs during the collapse. Although such fragmentation may occur, its importance is likely to remain limited because of the impending roles of the magnetic field (Machida et al. 2005, 2008; Hennebelle & Teyssier 2008; Price & Bate 2008; Hennebelle et al. 2011) and radiation (Krumholz et al. 2007; Bate 2009; Commercon et al. 2010), particularly when both are present (Commercon et al. 2011), an issue which still needs to be properly quantified. Observations of massive cores indeed suggest that fragmentation is rather limited, with most of the mass of the core ending up in one or just a few smaller cores (Bontemps et al. 2010; Longmore et al. 2011; Bressert et al. 2010; Palau et al. 2013).

In this section, we examine the impact of time dependence on the CMF in the simple isothermal case. The barotropic case will be examined in Section 2.3. We first discuss the density PDF, then the crossing time. Finally, we derive the analytical expression and compare the results with the time-independent ones.

2.2.1. Probability Density Function

Numerical simulations (Vázquez-Semadeni 1994; Padoan et al. 1997; Passot & Vázquez-Semadeni 1998; Kritsuk et al. 2007; Federrath et al. 2008; Schmidt et al. 2009) have revealed that the turbulence-induced density distribution, ${\cal P}(\delta)$, is reasonably well described by a lognormal distribution

$$\begin{eqnarray} {\cal P}(\delta) &=& {1 \over \sqrt{2 \pi \sigma _0^2}} \exp \left(- { (\delta - \bar{\delta })^2 \over 2 \sigma _0 ^2} \right),\nonumber \\ \delta &=& \ln (\rho / \bar{\rho }), \,\, \bar{\delta }=-\sigma _0^2/2, \; \; \sigma _0^2=\ln (1 + b^2 {\cal M}^2). \end{eqnarray} \tag{ 4 }$$

In this expression, ${\cal M}$ is the characteristic cloud Mach number and b is a non-dimensional coefficient that depends on the turbulence forcing (see Federrath et al. 2010). It typically varies from 0.25 when the forcing is purely solenoidal to almost 1 when the forcing is applied only on compressible modes. Such a three-dimensional (3D) lognormal shape for density fluctuations has received observational support from its two-dimensional (2D) observed projection, namely, the power spectrum column density of molecular clouds, measured from dust extinction maps (Kainulainen et al. 2009; Brunt et al. 2010). The scale dependence of the variance of the distribution reads, in 3D (see Paper I),

$$\begin{eqnarray} &&\sigma ^2(R) = \int _{2 \pi / L_c} ^{2 \pi / R} \widetilde{\delta } (k)^2 4 \pi k^2 dk = \sigma _0^2 \left(1 - \left({R \over L_c} \right)^{n^{\prime }-3} \right), \end{eqnarray} \tag{ 5 }$$

where L_c is the cloud's size and $\widetilde{\delta } (k)^2 \propto k^{-n^{\prime }}$ is the power spectrum of log (ρ) of 3D index n'. This latter is found in isothermal, shock-dominated hydrodynamical and MHD simulations to be very similar to the index n of the velocity power spectrum, with a typical value n' ∼ n ∼ 3.8, between 11/3 (Kolmogoroff limit) and 4 (Burgers limit; see, e.g., Kritsuk et al. 2007; Federrath et al. 2010; Schmidt et al. 2009), although the exact value depends on the forcing. Then (see Paper I),

$$\begin{eqnarray} &&{\cal P}(\delta) = {1 \over \sqrt{2 \pi \sigma (R)^2}} \exp \left(- { \left[\delta + {\sigma (R)^2 \over 2} \right]^2 \over 2\, \sigma (R)^2} \right). \end{eqnarray} \tag{ 6 }$$

2.2.2. The Crossing Time, τ_R

The crossing time at scale R is given by τ_R ≡ τ_ct(R) = 2α_ctR/V_ct, where V_ct is the relevant velocity and α_ct a dimensionless coefficient of the order of a few. At large scales, V_ct is typically equal to the 1D velocity dispersion $V_{\rm rms} ^{1{\rm D}}=V_{\rm rms}/{\sqrt{3}}$, where $V_{\rm rms}\equiv V_{\rm rms} ^{3{\rm D}}$ designates the 3D velocity dispersion all along this paper, while at small scales, below the sonic length, V_ct ≃ C_s. This crossing time is the typical time that is necessary for the density field to be significantly modified at scale R, implying that a new set of fluctuations, statistically independent of the former one, has been processed.

As mentioned earlier and discussed in Papers I and II, in the present context, we select the pieces of gas which are self-gravitating, i.e., such that their internal gravitational, kinetic, and thermal energies obey the following condition: −E_grav > 2E_kin + 3P_th. At large scales, this implies that $\alpha _g {G M} / R > V_{\rm rms}^2$, where α_g is a dimensionless coefficient equal to 3/5 for a uniform density fluctuation. We thus get

$$\begin{eqnarray} &&\tau _R ={2 \alpha _{{\rm ct}} R \over V_{\rm rms} ^{1{\rm D}} } = 2 \alpha _{{\rm ct}} \sqrt{ 24 \over \pi ^2 \alpha _{g}} \tau _{{\rm ff}}(R), \end{eqnarray} \tag{ 7 }$$

where $\tau _{{\rm ff}}(R)=\sqrt{3 \pi / 32 G \rho (R) }$ is the free-fall time of the density fluctuation of density ρ(R). A similar expression is obtained below the sonic length. In the following, we thus define the crossing time of a collapsing density fluctuation of scale R as

$$\begin{eqnarray} \tau _R &=& \phi _t \tau _{{\rm ff}}(R),\ \ \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} {\rm with}\,\,\,\,\phi _t &=& 2 \alpha _{{\rm ct}} \sqrt{ 24 \over \pi ^2 \alpha _{g}}\simeq 3. \end{eqnarray} \tag{ 9 }$$

Note that Federrath & Klessen (2012) estimated ϕ_t ≃ 2 from a fit between simulations and observations.

Defining $\tau _{{\rm ff}}^0$ as the free-fall time at the clump's mean density, $\tau _{{\rm ff}}^0 =\sqrt{3 \pi / 32 G \bar{\rho }}\simeq 1.07({\mu / 2.33})^{-1/2}({\bar{n}/ 10^3\,{\rm cm}^{-3}})^{-1/2}$ Myr (where μ = 2.33 is the mean molecular weight for a cosmic H₂/He composition), we get

$$\begin{eqnarray} &&{\tau _{R}\over \tau _{{\rm ff}}^0 }= \phi _t \sqrt{{\bar{\rho }\over \rho }}. \end{eqnarray} \tag{ 10 }$$

Note that the choice of $\tau _{{\rm ff}}^0$ is not consequential at this stage as it simply modifies the value of ${\cal N}$ uniformly without affecting its shape. Combining Equations (2), (6), and (9) and dropping the second term in Equation (2), as mentioned earlier, and assuming that $\tau _{{\rm ff}}^0\simeq \tau _0$, we obtain (see Paper I)

$$\begin{eqnarray} {\cal N}(\widetilde{M}) &=& -{1 \over \phi _{t}} { \bar{\rho } \over M_J^{0} \widetilde{M}} \left({ \widetilde{M}_R^c \over \widetilde{R} ^3 } \right)^{1/2} {d \widetilde{R} \over d \widetilde{M}_R^c} {d \delta _R^c\over d\widetilde{R}} {1 \over \sqrt{2 \pi \sigma ^2} } \nonumber\\ &&\times \exp \left(-{\left(\delta _R^c\right)^2 \over 2 \sigma ^2} + {\delta _R^c \over 2} - {\sigma ^2 \over 8} \right), \end{eqnarray} \tag{ 11 }$$

where $M_R^c$ is the critical mass at scale R (see Equation (3)) while

$$\begin{eqnarray} &&\delta _R^c = \ln \left({ \widetilde{M}_R^c \over \widetilde{R}^3} \right), \end{eqnarray} \tag{ 12 }$$

where $\widetilde{R}= R / \lambda _J^0$, $\widetilde{M} = M / M_J^0$, and $M_J^0$ and $\lambda _J^0$ denote the usual mean thermal Jeans mass and Jeans length, respectively:

$$\begin{eqnarray} M_J^0&=& {a_J\over C_m}\,{ C_s^3 \over \sqrt{G^3 \bar{\rho }}}\nonumber\\ &\approx & 0.8\left({a_J\over C_m}\right) \left({T \over 10\,{\rm K}}\right)^{3/2} \left({\mu \over 2.33}\right)^{-2}\left({{\bar{n}} \over 10^4\,{\rm cm}^{-3}}\right)^{-1/2}\, {M}_\odot \nonumber\\ \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} \lambda _J^0&=& \left({\pi ^{3/2} \over C_m} \right)^{1/3}{C_s\over \sqrt{G{\bar{\rho }}}}\nonumber\\ &\approx & 0.2\left({\pi ^{3/2} \over C_m} \right)^{1/3} \left(\!{T \over 10\,{\rm K}}\!\right)^{1/2} \left({\mu \over 2.33}\right)^{-1}\left(\!{{\bar{n}} \over 10^4\,{\rm cm}^{-3}}\!\right)^{-1/2}\, {\rm pc}, \nonumber\\ \end{eqnarray} \tag{ 14 }$$

where a_J is a dimensionless geometrical factor of the order of unity. Taking the standard definition of the Jeans mass as the mass enclosed in a sphere of diameter equal to the Jeans length, one gets a_J = π^5/2/6. We now need to know $M_R^c$, and its derivative ${d M}_R^c / {dR}$, as a function of the scale, R. It is determined by the physical processes at play in the cloud, as examined in the next sections.

2.2.3. Analytical Expression

We now need to specify the density threshold, $\delta _R^c$, which is determined from the virial theorem. As shown in Paper I (Section 4.3), the condition for collapse, which simply reads $\delta >\delta _R^c$ or equivalently $M > M_R^c=M_J(R)$, yields after calculations

$$\begin{eqnarray} &&M > a_J { \left[ (C_s(\rho))^2 + \left({V_{{\rm rms}}^2 \over 3}\right) \right ]^{3/2} \over \sqrt{G^3 \bar{\rho } \exp (\delta) } }. \end{eqnarray} \tag{ 15 }$$

We assume that the effective dispersion velocity obeys the Larson's (1981) relationship:

$$\begin{eqnarray} \left\langle V_{\rm rms}^2\right\rangle &=& V_0^2 \times \left({R \over 1\ {\rm pc}} \right) ^{2 \eta },\nonumber \\ {\cal M} &\equiv & {\cal M}(R) = {\left\langle V_{\rm rms}^2\right\rangle ^{1/2}\over C_s}. \end{eqnarray} \tag{ 16 }$$

As mentioned in Paper I, the Mach number ${\cal M}$ represents "effective" values that include both the hydrodynamical and magnetic contributions, i.e., $V_{\rm rms}=\lbrace (V_{{\rm rms}})_{\rm hydro}^2+V_A^2/2\rbrace ^{1/2}$, where V_A = B/(4πρ)^1/2 denotes the Alfvén velocity. The coefficient η is related to n, the index of the velocity power spectrum, by the relation (see Equation (24) of Paper I)

$$\begin{eqnarray} &&\eta =\frac{n-3}{2}. \end{eqnarray} \tag{ 17 }$$

As mentioned earlier, 3D numerical simulations of compressible turbulence (e.g., Kritsuk et al. 2007) suggest a value n ≃ 3.8, yielding η ≃ 0.4, as is indeed found in observations.

With Equation (3), Equation (15) implies (see Paper I)

$$\begin{eqnarray} &&M > M_R^c = a_J^{2/3} \left({ C_s^2 \over G } R + {V_{\rm rms}^2 \over 3\, G }\right). \end{eqnarray} \tag{ 18 }$$

After normalization, Equation (18) becomes (see Section 5.4 of Paper I)

$$\begin{eqnarray} &&\widetilde{M}_R^c = M / M_J^0 = \widetilde{R}\, (1+ {\cal M}^2_* \widetilde{R}^{2 \eta }), \end{eqnarray} \tag{ 19 }$$

where ${\cal M}_*$ is given by

$$\begin{eqnarray} {\cal M}_* &=& { 1 \over \sqrt{3} } { V_0 \over C_s}\left({\lambda _J^0 \over 1\, {\rm pc} }\right) ^{ \eta } \nonumber\\ &\simeq & (0.8\hbox{--}1.0) \,\left({\lambda _J^0\over 0.1\,{\rm pc}}\right)^{\eta }\,\left({C_s\over 0.2\ {\rm km\ s^{-1}}} \right)^{-1}, \end{eqnarray} \tag{ 20 }$$

and thus illustrates the impact of turbulence-induced velocity fluctuations at the Jeans scale (see Paper I).

Equations (11) and (19) finally yield for the mass spectrum of gravitationally bound prestellar cores:

$$\begin{eqnarray} {\cal N} (\widetilde{M}) &=& {2\ \over \phi _{t}} {\cal N}_0 \, { 1 \over \widetilde{R}^6} \, { 1 + (1 - \eta){\cal M}^2_* \widetilde{R}^{2 \eta } \over [1 + (2 \eta + 1) {\cal M}^2_* \widetilde{R}^{2 \eta }] }\nonumber\\ && \times \left({\widetilde{M} \over \widetilde{R}^3} \right) ^{-1 - {1 \over 2 \sigma ^2} \ln (\widetilde{M} / \widetilde{R}^3) } \times {\exp (-\sigma ^2/8) \over \sqrt{2 \pi }\, \sigma }, \end{eqnarray} \tag{ 21 }$$

where ${\cal N}_0= \bar{\rho } / M_J^0$. Equation (21) is the time-dependent generalization of Equation (44) of Paper I. The time dependence appears explicitly through the factor 1/ϕ_t but also through the modification of the exponent, $-1 -{(1 / 2) \sigma ^2} \ln (\widetilde{M} / \widetilde{R}^3)$, instead of $-3/2 - {(1 / 2) \sigma ^2} \ln (\widetilde{M} / \widetilde{R}^3)$, which arises from the time correction, proportional to $\sqrt{\rho } \propto (M/R^3)^{1/2}$. As expected from the discussion in the Introduction, the time dependence thus affects the slope of the CMF. We quantify this effect in the next sections.

2.2.4. Results and Comparison with the Time-independent Model

Before comparing the time-dependent and time-independent distributions, we determine the position of the peak of the CMF. As discussed in Paper I (Section 7.1.4), the peak occurs in the thermally dominated regime, which corresponds to ${\cal M}_*=0$. The derivative of Equation (21) with respect to the mass in that case yields

$$\begin{eqnarray} &&\widetilde{M}_{\rm peak} = {M_{\rm peak}\over M_J^0} = \exp (- \sigma ^2) = { 1 \over (1 + b^2 {\cal M}^2) }, \end{eqnarray} \tag{ 22 }$$

whereas in the case of a time-independent distribution we have (Equation (46) Paper I)

$$\begin{eqnarray} &&\widetilde{M}_{\rm peak} = \exp \left(- {3 \over 4 } \sigma ^2\right) = { 1 \over (1 + b^2 {\cal M}^2) ^{3/4}}. \end{eqnarray} \tag{ 23 }$$

This implies that for a given Mach number, the peak of the distribution is shifted toward smaller masses when time dependence is taken into account. This is intuitively expected from the fact that more small-scale objects with short free-fall times are continuously produced during the clump evolution in a time-dependent collapse. It is thus important to stress that in the (both time-dependent and time-independent) HC theory of gravo-turbulent fragmentation, the peak (i.e., the characteristic) mass of the IMF depends not only on the clump's mean thermal Jeans mass but also strongly ($\propto {\cal M}^{-2}$) on the characteristic Mach number.

Another important quantity is the exponent of the high-mass tail of the distribution. Indeed, in the limit $\widetilde{M} \simeq {\cal M}_*^2 \widetilde{R}^{2 \eta +1}$, which corresponds to the turbulence-dominated regime (see Equation (19)), Equation (21) tends to a power law plus a lognormal, such as ${\cal N}\propto M^{-(1+x)}$ with

$$\begin{eqnarray} &&x = {3 \over 2 \eta +1} - 6 { 1 - \eta \over (2 \eta + 1)^2 \sigma ^2 } \ln ({\cal M}_*), \end{eqnarray} \tag{ 24 }$$

whereas in the time-independent case, this coefficient reads (see Equation (43) of Paper II)

$$\begin{eqnarray} &&x = {2 + \eta \over 2 \eta +1} - 6 { 1 - \eta \over (2 \eta + 1)^2 \sigma ^2 } \ln ({\cal M}_*). \end{eqnarray} \tag{ 25 }$$

As expected, the distribution becomes slightly steeper when the time dependence is included in the derivation of the CMF/IMF. This is due to the fact that some massive clumps fragment into smaller pieces during the cloud collapse. Typical values for x range from 1.1 to 1.5, depending on η, σ, and ${\cal M}_*$, bracketing the Salpeter value, x = 1.35.

Figure 1 displays the core mass function, dN/dlog M for η = 0.45, ${\cal M}=6$, ${\cal M}_* =\sqrt{2}$, and two typical clump densities, $\bar{n}=5000$ cm⁻³ and $\bar{n}=3 \times 10^4$ cm⁻³. Solid lines display the time-dependent results, while dotted lines display the time-independent ones. For reference, dashed lines represent the Chabrier system IMF (Chabrier 2003), shifted upward in mass along the x-axis by a factor of three to account for the observed shift between the CMF and the IMF. In the rest of the paper, we will refer to this IMF as the "shifted Chabrier IMF" (SCIMF). As mentioned above, the time-dependent distribution peaks at lower masses and has a slightly steeper high-mass slope than the time-independent one. In spite of these differences, both distributions match well the SCIMF above about the mean Jeans mass. At low masses, both distributions appear to be too narrow. As already discussed in Papers I and II, this is essentially due to the isothermal approximation for the equation of state (eos) of the gas (see Paper II and below), but also to the assumed exact correspondence between the initial core mass and the final star mass. Indeed, as shown in Chabrier & Hennebelle (2010), taking into account some (expected) dispersion between the mass of the parent core and that of the final star, i.e., between the CMF and the IMF, naturally leads to a broadening of the latter compared with the former in the low-mass regime. At last, gravity may also broaden the PDF.

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Here we compare the time-dependent and time-independent models when a barotropic eos is used. We also introduce the magnetic field in our model and investigate its impact on the mass spectrum.

2.3.1. Formalism

Again, assuming that the pieces of fluid that collapse are gravitationally dominated, Equation (15) becomes

$$\begin{eqnarray} &&M > a_J { \left[ (C_s(\rho))^2 + \left(V_{{\rm rms}}^2 \left/ 3\right) + \left(V_{a}^2 \right/ 6\right) \right]^{3/2}\over \sqrt{G^3 \bar{\rho } \exp (\delta) } }, \end{eqnarray} \tag{ 26 }$$

where V_a is the mean Alfvén speed. Based on observational and numerical results, we assume that the magnetic intensity correlates with the gas density as

$$\begin{eqnarray} &&B = B_0 \left({\rho \over \bar{\rho } } \right)^{\gamma _b}. \end{eqnarray} \tag{ 27 }$$

Typical values for B₀ and $\bar{\rho }$ are 10–20 μG and 10³ cm⁻³, respectively (Crutcher 1999). The coefficient γ_b seems to depend on density. In the diffuse gas, γ_b is typically smaller than 0.5 and around 0.1–0.3 (Troland & Heiles 1986; Hennebelle et al. 2008), while for higher densities when the gas is self-gravitating, it is of the order of 0.5 (although see Crutcher et al. 2010).

After normalization, Equation (26) becomes

$$\begin{eqnarray} &&\widetilde{M}_R^c = M / M_J^0 = \widetilde{R}\, \left({ C_s(\rho)^2 \over \left(C_s^0\right)^2 } + {\cal M}^2_* \widetilde{R}^{2 \eta } + (V_a^*)^2 \left({\rho \over \bar{\rho }} \right)^{2 \gamma _b -1} \right),\nonumber\\ \end{eqnarray} \tag{ 28 }$$

where

$$\begin{eqnarray} &&V_a^* = { 1 \over \sqrt{6} } { B_0 \over \sqrt{4 \pi \bar{\rho }} \,C_s}. \end{eqnarray} \tag{ 29 }$$

As emphasized in Paper II (see also Hennebelle 2012 for a self-consistent calculation of the thermal balance), the thermodynamics of the gas has a drastic impact on the low-mass end of the mass function. As in Paper II, we use the barotropic eos suggested by Larson (1985, see also Glover & Mac Low 2007) from observations of molecular clouds, namely,

$$\begin{eqnarray} &&T\propto \rho ^{\gamma _1-1}{\hspace{28.45274pt}} \rho <\rho ^{{\rm crit}},\nonumber \\ &&T\propto \rho ^{\gamma _2-1}{\hspace{28.45274pt}} \rho >\rho ^{{\rm crit}}, \end{eqnarray} \tag{ 30 }$$

with γ₁ ≃ 0.7 and γ₂ ≃ 1–1.1, and ρ^crit corresponds to the density above which dust grains become thermally coupled with the gas and thus dust cooling becomes the dominant cooling mechanism instead of line cooling. Observations suggest that ρ^crit ≃ 10⁻¹⁸ g cm⁻³, i.e., ${\bar{n}}^{{\rm crit}}\simeq 2.5\times 10^5$ cm⁻³. We keep the same eos as we used in Paper I, inspired from Larson (1985) and write

$$\begin{eqnarray} \nonumber C_s^2 &=& \left[ \left(\left(C_{s,1}^0\right)^2 \left({\rho \over \bar{\rho } }\right)^{\gamma _1-1}\right)^m + \left(\left(C_{s,2}^0\right)^2 \left({\rho \over \bar{\rho }}\right)^{\gamma _2-1} \right)^m \right]^{1/m} \\ &=& \left(C_{s,1}^0\right)^2 \left[ \left({\rho \over \bar{\rho } }\right)^{(\gamma _1-1)m} + \left({C_{s,2}^0\over C_{s,1}^0}\right)^{2m} \left({\rho \over \bar{\rho }}\right)^{(\gamma _2-1)m} \right]^{1/m},\nonumber\\ \end{eqnarray} \tag{ 31 }$$

where m is a real number of the order of unity. In the following, m will be equal to 3 (see Figure 4 of Paper II). Typically, γ₁ ≃ 0.7 while γ₂ ≃ 1–1.1. The critical density, ρ^crit, at which the transition between the two regimes is occurring is expected to be about 10⁻¹⁸ g cm⁻³ and we define

$$\begin{eqnarray} &&K_{\rm crit}= \left({C_{s,2}^0 \over C_{s,1}^0}\right)^2 = \left({\rho ^{{\rm crit}} \over \bar{\rho } }\right)^{\gamma _1 -\gamma _2}. \end{eqnarray} \tag{ 32 }$$

In order to get the mass spectrum from Equation (11), we need to know $M_R^c$ and ${dM}_R^c / {d R}$ as a function of R. While for the first it is not possible to get an explicit relation from Equation (28), the second can be obtained by deriving Equation (28) with respect to $\widetilde{R}$. The analytical expressions that we obtain are identical to Equation (38) of Paper II except for the extra terms related to the magnetic field. We write them explicitly for completeness:

$$\begin{eqnarray} {d \widetilde{M}_R^c \over d \widetilde{R} } &=& { B \over C}, \end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray} \nonumber B &=& D - 3 { \widetilde{M} \over \widetilde{R}^3} {d D \over d \widetilde{\rho }} + (2 \eta + 1) {\cal M}_*^2 \widetilde{R}^{2 \eta }, \\ C &=& 1 - { \widetilde{R}^ {-2}} {d D \over d \widetilde{\rho }}, \\ D &=& \left({ \widetilde{\rho } } ^{(\gamma _1-1)m} + (K_{\rm crit})^m { \widetilde{\rho } } ^{(\gamma _2-1)m} \right)^{1/m} + (V_a^*)^2 \widetilde{\rho } ^{2 \gamma _b -1}. \nonumber \end{eqnarray} \tag{ 34 }$$

2.3.2. Comparison between Time-dependent and Time-independent Mass Spectra in the Barotropic Case

When a barotropic eos is considered, the mass spectrum depends on the clump's mean density not only as a normalization factor (as for the isothermal eos) but also through the eos itself. As a fiducial value, we adopt a critical density (Equation (32)) $\bar{n}^{{\rm crit}}=2 \times 10^5$ cm⁻³, i.e., ρ^crit ≃ 10⁻¹⁸ g cm⁻³, as inferred from observations. The density and velocity dispersions are taken to follow Larson- (1981; see also Falgarone et al. 2004 where slightly different values are inferred) type relations:

$$\begin{eqnarray} \bar{n}&=& (d_0 \times 10^3 \; {\rm cm}^{-3}) \left({ R \over 1\ {\rm pc}} \right)^{-0.7}, \end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray} V_{\rm rms} &=& (u_0 \times 0.8 \; {\rm km \; s}^{-1}) \left({ R \over 1\ {\rm pc}} \right)^{\eta }. \quad\ \end{eqnarray} \tag{ 36 }$$

According to Equation (16), V₀ = u₀ × 0.8 km s⁻¹. Figure 2 (left panel) shows the mass spectrum for four clumps of size R_c = 0.5, 2, 5, and 20 pc, respectively. The value of u₀ is taken to be equal to 1 in all cases, while the top, middle, and bottom panels correspond to d₀ = 5, 4, 3, and 2, respectively. The four panels thus correspond, from top to bottom, to typical clump masses and mean densities, M_c = 183, 3564, 21995, and 355607 M_☉, $\bar{n}=8120, 2460$, 970, and 240 cm⁻³. As in Figure 1, solid lines display the time-dependent case, dotted lines the time-independent one, while the dashed line corresponds to the SCIMF. In the four cases, except possibly for the smallest clump (as expected; see Section 6.2.2 of Paper II and Section 3.2.2.1 below), the agreement with the SCIMF is very good. Importantly enough, the clump densities found to yield a good agreement with the SCIMF are smaller by a factor of about two to five than what has been found in the time-independent case for similar agreement (Figure 8 of Paper II), and thus agree fairly well with the "standard" Larson density normalization values, d₀ ≃ 3. This is a direct consequence of the peak position being shifted toward smaller masses, as discussed earlier. Another noticeable improvement is the width of the distribution, which is larger than for the isothermal cases presented in Figure 1. As discussed in Paper II, this improvement stems from the larger compressibility of the gas for a softer than isothermal eos, promoting small-scale collapsing structures, and from the strong dependence of the peak position on γ (see Figure 3 of Paper II), which becomes even more acute in the time-dependent CMF, whose peak is shifted toward smaller masses compared with the time-independent solution.

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2.3.3. Influence of the Magnetic Field on the Core Mass Function

To study the influence of the magnetic field on the mass spectrum, we define

$$\begin{eqnarray} &&V_a = V_a^0 {B_0^{{\rm ref}} \over \sqrt{4 \pi \bar{\rho } }} \left({ \bar{n} \over 1000 \, {\rm cm}^{-3}} \right)^{\gamma _b}, \end{eqnarray} \tag{ 37 }$$

where $B_0^{{\rm ref}}$ is a reference magnetic intensity equal to 10 μG, $\bar{n}$ is the cloud mean particle density, and $\bar{\rho }=m_p \bar{n}$. We consider four cases. The first one is purely hydrodynamical and serves as a reference. The second one uses γ_b = 0.3 and $V_a^0=1$. The third and fourth ones use γ_b = 0.1 and $V_a^0=1$ or $V_a^0=3$, respectively. We primarily investigate values of γ_b smaller than 0.5 because our analysis consists of analyzing the density fluctuations generated by supersonic turbulence and identifying the ones that are self-gravitating and that subsequently will be amplified by gravity. Therefore, at the stage where the analysis is performed, the correlation between magnetic intensity and density is the result of turbulent processes rather than gravity, implying B∝ρ^0.1–0.3. Note that as emphasized in Hennebelle & Chabrier (2008), the case γ_b = 0.5 is formally equivalent to the isothermal case and all the magnetic results can be obtained by a simple renormalization of the rms velocity and Mach number (see Equation (16)).

The impact of the magnetic field on the mass spectrum can be inferred from a comparison between the various magnetized cases and the hydrodynamical one (Figure 2, right panel). When $V_a^0=1$, γ_b = 0.3 (dot dashed line), the mass spectrum is essentially shifted toward larger masses with respect to the purely hydrodynamical case. This stems from the magnetic support, as expressed by Equation (18), which adds up to the thermal pressure. Indeed, for this value of γ_b, the magnetic pressure density dependence, P_mag∝ρ^0.6, is very similar to the thermal pressure density dependence, in particular, because the Larson eos has an effective exponent at low density which is equal to 0.7.

When γ_b = 0.1, the impact of the magnetic field is different. For $V_a^0=1$ (dotted line), the departure from the hydrodynamical case (solid line) is only marginal. This is because the magnetic support increases very slowly with density; thus, at high density (which corresponds to low-mass cores), the magnetic support is less important than the thermal one. At high masses, on the other hand, the support is dominated by turbulence. For $V_a^0=3$ (dashed line), the distribution tends to be shallower at high masses. As discussed in Paper II, the exponent of the mass distribution at high masses, ${\cal N}(M)\propto M^{-(1+x)}$, when only pressure terms are included (${\cal M}_\star =0$), is given by (see Equation (32) of Paper II) 1 + x = (9 − 6γ)/(4 − 3γ), so that if γ < 0.2 the slope becomes shallower than the Salpeter value.

Overall, for a typical magnetic field intensity in the diffuse (low-density) gas of the order of B ≃ 10 μG, and a density dependence such that γ_b < 0.3, the mass spectrum is not very different from the hydrodynamical case. For γ_b = 0.3, the CMF still resembles the SCIMF but is shifted toward larger masses, a direct consequence of the magnetic support, which shifts the characteristic Jeans mass to larger values. When the magnetic field intensity is significantly larger than the aforementioned value (e.g., 30 μG, like in the considered example), however, the mass spectrum becomes too shallow at large masses and starts departing significantly from a Chabrier/Salpeter IMF in this regime. Therefore, one of the predictions of our theory is that diffuse magnetized environments with magnetic field intensities largely exceeding about 10 μG should have high-mass IMF tails shallower than the Salpeter value.

Following Krumholz & McKee (2005, KMK), we define the dimensionless star formation rate per free-fall time, SFR_ff, as the fraction of clump mass converted into stars per clump mean free-fall time, i.e., the free-fall time defined at the clump mean density $\bar{\rho }$:

$$\begin{eqnarray} &&{\rm SFR}_{{\rm ff}} = {\dot{M}_*\over M_c} \, \tau _{{\rm ff}}^0, \end{eqnarray} \tag{ 38 }$$

where $\dot{M}_*$ denotes the total SFR arising from a clump of mass M_c and volume $V_c\simeq R_c^3$. The star formation efficiency (SFE) is defined as the global mass fraction of a clump converted into stars during the lifetime τ₀ of the clump, which can last a few free-fall times (e.g., Murray 2011):

$$\begin{eqnarray} &&{\rm SFE} = {{M}_*\over M_c}={\rm SFR}_{{\rm ff}} \times \left({\tau _0\over \tau _{{\rm ff}}^0}\right). \end{eqnarray} \tag{ 39 }$$

According to these definitions, SFR_ff is thus given by the integral of the mass spectrum specified by Equation (2) as

$$\begin{eqnarray} \nonumber {\rm SFR}_{{\rm ff}} &=& \epsilon \, \int _0 ^{M_{{\rm cut}}} { M {\cal N} (M) {dM} \over \bar{\rho }} \\ &=& \epsilon \int _0 ^{M_{{\rm cut}}} {dM} {{dR} \over {dM}} \, \left(-{d \delta _R \over {dR}} {\tau ^0_{{\rm ff}} \over \tau _R} \exp (\delta _R) {\cal P}_R(\delta _R)\right.\nonumber\\ &&\left. +\, \int _ {\delta _R}^\infty {d \over {dR}} \left[{\cal P}_R \left({\tau ^0_{{\rm ff}} \over \tau _R}\right)\right]\, e^{\delta _R} d\delta \right). \end{eqnarray} \tag{ 40 }$$

In this expression, is the (supposedly uniform) efficiency with which the mass within the collapsing prestellar cores is converted into stars.⁹ Indeed, during the collapse, a substantial fraction of the mass initially within the core is blown away by outflows and jets. Calculations (e.g., Matzner & McKee 2000; Ciardi & Hennebelle 2010), as well as observations, suggest that

$$\begin{eqnarray} &&\epsilon \simeq 0.3\hbox{--}0.5. \end{eqnarray} \tag{ 41 }$$

Note that Federrath & Klessen (2012) estimated ≃ 0.3–0.7 by comparing numerical simulations with observations. In Equation (40), M_cut corresponds to the mass of the largest star that can possibly form in the cloud. Equivalently, according to the mass–scale relation given by Equations (19) or (28), one can define y_cut = R/R_c, which denotes the largest size fluctuations that can turn unstable in the cloud. The value of M_cut is crucial in setting SFR_ff. Indeed, as emphasized in Paper I (Section 5.1.3), the integral $V_c \int _0 ^ \infty M {\cal N}(M) {dM}$ is equal to the mass of the system itself because integrating up to infinity implies that all pieces of gas, including the very diffuse ones, are Jeans unstable. Therefore, restricting the integration to a finite value implies that any piece of fluid that is not dense enough, i.e., whose density corresponds to a Jeans length larger than a significant fraction of the cloud's size, is excluded from the mass spectrum of collapsing structures. The immediate consequence of such a truncation of the integral is obviously to reduce the SFR.

If R_c is the clump's radius, more generally its characteristic scale, then the question is thus to determine up to which fraction of this scale or up to which fraction of the clump mass, the integration should be performed? A similar question concerns the behavior of the density power spectrum whose scale dependence is characterized by σ²(R) (Equation (5)). While in the inertial domain, it is well established that the power spectrum of log (ρ) exhibits a power-law behavior (Beresnyak et al. 2005; Federrath et al. 2010; Audit & Hennebelle 2010), but this may not be the case when the scale R is approaching the clump size R_c. This question is directly related to the very definition of the clump itself and how it connects to the surrounding medium. Unfortunately, these questions are far from being settled. Finally, as discussed in Paper I, the second term of the right-hand side in Equation (40) is important only when y_cut → 1. Therefore, given all the uncertainties when approaching this limit and because the influence of this second term remains limited, we elected to drop it in the rest of the calculations. In that case, after proper normalization, Equations (10), (12), (11), and (40) yield

$$\begin{eqnarray} {\rm SFR}_{{\rm ff}}^0 &=& - \int _0 ^{\widetilde{M}_{{\rm cut}}} \left({ \widetilde{M}_R^c \over \widetilde{R} ^3 } \right) {d \widetilde{R} \over d \widetilde{M}_R^c} {d \delta _R^c\over {d\widetilde{R}}} {1 \over \sqrt{2 \pi \sigma ^2} } \nonumber\\ &&\times \exp \left(-{\left(\delta _R^c\right)^2 \over 2 \sigma ^2} - {\sigma ^2 \over 8} \right) d \widetilde{M}, \end{eqnarray} \tag{ 42 }$$

with

$$\begin{eqnarray} &&{\rm SFR}_{{\rm ff}} = \left({\epsilon \over \phi _{t}}\right)\times {\rm SFR}_{{\rm ff}}^0. \end{eqnarray} \tag{ 43 }$$

For the case of an isothermal eos, for which the mass–size relationship is given by Equation (19), this yields

$$\begin{eqnarray} {\rm SFR}_{{\rm ff}}^0 &=& -2 \int _0 ^{\widetilde{M}_{{\rm cut}}} \, { 1 \over \widetilde{R}^3} \, { 1 + (1 - \eta){\cal M}^2_* \widetilde{R}^{2 \eta } \over [1 + (2 \eta + 1) {\cal M}^2_* \widetilde{R}^{2 \eta }] } \nonumber\\ &&\times \left({\widetilde{M} \over \widetilde{R}^3} \right) ^{-{1 \over 2 \sigma ^2} \ln ({\widetilde{M} \over \widetilde{R}^3}) } \times {e^{(-{\sigma ^2\over 8}) }\over \sqrt{2 \pi }\,\sigma }\, d \widetilde{M}.\nonumber \\ \end{eqnarray} \tag{ 44 }$$

Conversely, the SFR can also be derived from Equation (40) (still dropping the second term) as

$$\begin{eqnarray} \nonumber {\rm SFR}_{{\rm ff}}^0 &=& \int ^\infty _{\widetilde{\delta }_{{\rm cut}}} {\tau _{{\rm ff}}^0 \over \tau _{R}(\delta _R) } e^{\delta _R} {\cal P}(\delta _R)\, d\delta _R \nonumber \\ &=& \int ^\infty _{\widetilde{\rho }_{{\rm cut}}} \widetilde{\rho }^{1/2} {\cal P}(\widetilde{\rho })\, d\widetilde{\rho } \nonumber \\ &=& {1 \over 2 } e^{({ 3 \sigma ^2 \over 8 })} \left[ 1 + {\rm erf} \left({ \sigma ^2- \ln (\widetilde{\rho }_{{\rm cut}}) \over 2^{1/2} \sigma } \right) \right]. \end{eqnarray} \tag{ 45 }$$

We stress that this is possible only because the second term in Equation (40) has been dropped. The value of $\widetilde{\rho }_{{\rm cut}}$ in Equation (45) can be derived from the collapse condition derived in our theory (see Equation (29) of Paper I), namely, $\widetilde{\rho }=e^{\delta _R}>e^{\delta _R^c}$, or similarly from the mass–size relations given by Equations (19) or (28). For the isothermal case, this simply yields

$$\begin{eqnarray} &&\widetilde{\rho }_{{\rm cut}}=\widetilde{R}_{{\rm cut}}^{-2}\,\left(1+{\cal M}_\star ^2\widetilde{R}_{{\rm cut}}^{2\eta }\right), \end{eqnarray} \tag{ 46 }$$

where $\widetilde{R}_{{\rm cut}}=(y_{{\rm cut}}R_c/\lambda _J^0)$. As clearly expressed by this equation, physically, $\widetilde{\rho }_{{\rm cut}}$ is the minimum density for which a perturbation, whose associated total (thermal + turbulent) Jeans length is equal to a maximum fraction y_cut of the clump's size, can lead to a gravitational instability. Indeed, Equation (46) can be rewritten as follows:

$$\begin{eqnarray} &&(\lambda _{J})_{{\rm cut}}={ y_{{\rm cut}}R_c\over \left[1+{\cal M}_\star ^2\,\left({y_{{\rm cut}}R_c\over \lambda _J^0}\right)^{2\eta }\right]^{1/2} }. \end{eqnarray} \tag{ 47 }$$

It is important to stress that this procedure differs not only quantitatively but also qualitatively from the ones defined in KMK and Padoan & Nordlund (2011, PN). In these author formalisms, the critical density corresponds to a new, arbitrary scale, respectively, a sonic scale or a shock scale, characteristic of a process supposed to be necessary for the collapse. In contrast, in our formalism, the only relevant scale is the size of the cloud itself. So, even though this can be expressed as a density in Equation (45), the proper way to look at it, as clearly expressed by Equation (47), is in terms of the maximum size of the fluctuations for a given cloud's size and density, which are able to grow and lead to gravitational instability. Moreover, in the KMK and PN formalisms, the critical density is proportional to ${\cal M}^2$, implying that only very dense structures will lead to star formation. In contrast, in our formalism, any structure of any density can collapse if its gravitational energy dominates over all sources of support, as long as the associated perturbation can grow and become unstable. In practice, our results do not depend significantly on the value of y_cut, except when y_cut → 0 and for low Mach numbers (see next section). Indeed, y_cut only affects the limit of the integration, which corresponds to a regime where the (lognormal) PDF is small.

As defined above, ${\rm SFR}_{{\rm ff}}^0$ represents the SFR calculated from our theory for a core-to-star mass conversion efficiency = 1 during the collapse, and for a characteristic time within which new mass reservoirs $\widetilde{M}$ of scale $\widetilde{R}$ become gravitationally unstable, i.e., new cores are produced, equal to one free-fall time at the core's density, i.e., τ_R = τ_ff(R) (ϕ_t = 1). In that sense, ${\rm SFR}_{{\rm ff}}^0$ represents the "core formation rate" per free-fall time. As already mentioned, observations and simulations point to ≃ 0.3–0.5, while our estimate of τ_R in Section 2.2.2 leads to ϕ_t ≃ 3. Consequently, we typically have

$$\begin{eqnarray} &&{{\rm SFR}_{{\rm ff}}\over {\rm SFR}_{{\rm ff}}^0 }\simeq 0.1\hbox{--}0.2. \end{eqnarray} \tag{ 48 }$$

3.2.1. Models and Assumptions

3.2.2. Star Formation Rate: Results

3.2.2.1. 

3.2.2.1. Dependence upon the clump's large-scale cutoff, size, and density. Figure 3 portrays ${\rm SFR}_{{\rm ff}}^0$, for the non-isothermal case, for b = 0.5 (left column) and b = 0.25 (right column) in Equation (4), as a function of the cutoff parameter, y_cut = R/R_c, for various clump sizes and mean densities (this means that the parameter d₀, which determines the density normalization at 1 pc in the Larson relation (Equation (35)), is constant along each line but varies from one line to another). As mentioned earlier and expected from the truncation of the integral in Equation (42), ${\rm SFR}_{{\rm ff}}^0$ is basically independent of y_cut for y_cut ≳ 0.3 but quickly drops as this parameter approaches very small values. As also expected, the strong decrease of ${\rm SFR}_{{\rm ff}}^0$ occurs at higher y_cut values for smaller, less massive clumps and the truncation becomes more drastic as density decreases. Indeed, as already noticed in Papers I and II (see Figure 8 of Paper II), the core mass spectrum is truncated both at high and small masses compared with the observationally determined CMF/IMF for small (≲ 0.5 pc) clumps, with a drastically reduced CMF for low-density clumps. This stems from two reasons: (1) the scale dependence, based on Larson's relations, of both the global and local Mach numbers, ${\cal M}$ and ${\cal M}_\star$, which enter our formalism (see Equations (16) and (20)) and are responsible for generating and stabilizing the initial density fluctuations leading eventually to prestellar cores. The smaller the clump, the smaller these values and thus the narrower the core mass spectrum (see Paper I). (2) The fact that for small clumps or low-density clumps, the Jeans scale becomes comparable to or larger than the size of the clump itself (see Equation (14)), inhibiting gravitational collapse.

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According to these results, star formation is thus predicted to occur dominantly in the largest, most massive clumps and/or in the densest (parts of) clumps, a conclusion indeed supported by observations. As seen in the figure, for densities $\bar{n} \gtrsim 2000$ cm⁻³ and for y_cut ≳ 0.1–0.3, depending on the clump's size, the SFR depends only weakly on y_cut and reaches values in the range ${\rm SFR}_{{\rm ff}}^0 \simeq 0.3$–3.0, depending on the size of the clump, for the b = 0.5 case. The fact that ${\rm SFR}_{{\rm ff}}^0$, the SFR per free-fall time, can be larger than 1 is due to its density dependence. Indeed, ${\rm SFR}_{{\rm ff}}^0$ is defined with respect to $\tau _{{\rm ff}}^0$, the free-fall time at the cloud's mean density, but fluctuations of size R whose density is larger than $\bar{\rho }$ collapse in a shorter time. As has already been discussed, the value of y_cut beyond which no star can form is ill-determined and may depend upon cloud parameters. In the rest of our calculations, we will assume y_cut ≃ 0.1–0.3 (see previous section), which means that only perturbations whose size is at most of the order of about one-tenth to one-third of the cloud's size are relevant to produce gravitationally bound prestellar cores. As just mentioned, for larger values of y_cut, the SFR remains basically unchanged.

As seen from Figure 3, low-density clumps ($\bar{n} \lesssim 1500$ cm⁻³) smaller than about ∼2 pc yield quite small or even negligible values of ${\rm SFR}_{{\rm ff}}^0$, except for values of y_cut approaching unity, a rather unlikely possibility. Phrased differently, at low density, very large clumps, but only very large clumps, still contribute appreciably to star formation. As mentioned in the previous section, this is one of the direct consequences of our theory and cannot be the case in a theory arbitrarily defining a density threshold for star formation, a point of importance when we will compare with observational determinations (Section 4). As just mentioned, star formation should thus occur dominantly in dense enough (regions of) clumps or in very large and massive clumps and to increase with clump density and clump mass/size. It is worth stressing that this strong dependence upon density or mass for efficient star formation naturally arises in our formalism and does not stem from an ad-hoc threshold condition for star formation.

Finally, the figure shows the dependence of the SFR upon the coefficient b entering Equation (4), which relates the rms Mach number to the width of the density PDF, σ₀ (see Equation (4)). Although the results are qualitatively similar, globally, ${\rm SFR}^0_{{\rm ff}}$ is smaller by a factor of a few for large values of y_cut and drops drastically below larger values of y_cut for b = 0.25 than for b = 0.5. Indeed, as shown in detail in Section 3.1 of Paper II, the mass spectrum peaks at larger masses and the density PDF gets narrower for smaller values of b because the gas is less compressible. Since the respective contributions of the turbulence forcing mechanisms (and thus the value of b) in molecular clouds are rather ill-determined and may vary from cloud to cloud, depending for instance on the presence of expanding H ii regions around massive stars or supernova explosions, this implies significant possible variations of the SFR, depending on the environment.

3.2.2.2. 

3.2.2.2. Dependence upon the virial parameter. The dependence of the SFR upon the characteristic virial parameter α_vir of a clump of radius R_c = L_c/2 and mean density ${\bar{\rho }}$, $\alpha _{{\rm vir}}=2E_K/E_G=5 V_{{\rm rms}}^2 / (\pi G {\bar{\rho }} L_c^2)$, which measures the ratio of turbulence over gravitational energy within the clump, has been examined in HC11 in the isothermal case and will be examined in the next subsection in the non-isothermal one. It is illustrated in Figure 4, which displays the values of ${\rm SFR}_{{\rm ff}}^0$ for various clump sizes/masses, for three typical Mach numbers, ${\cal M}$ = 16 (top), 9 (middle), and 4 (bottom). As a general trend, as discussed in HC11, increasing α_vir leads to a decrease of the SFR, with an abrupt reduction above some typical value of α_vir, which depends on the Mach number. This stems 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. As increasing α_vir implies decreasing the clump's mass/size, as $\alpha _{{\rm vir}}\propto L_c^{-2}\propto M_c^{-2/3}$ (at fixed Mach number and density), star formation is thus highest in the largest and most massive clumps, which correspond to the smallest values of α_vir, in agreement with the conclusion reached in the previous subsection. Giant molecular clouds (GMCs) in the Milky Way, with masses in the range ∼10³–10⁶ M_☉, are generally weakly gravitationally bound structures, with virial parameters in the range α_vir ≈ 0.3–3, with slightly decreasing α_vir with increasing cloud mass, as just noted (Heyer et al. 2009; Murray 2011). Accordingly, the rather low observed SFR values might partly stem from the fact that most molecular clouds have in general virial parameter values slightly above unity, i.e., they depart from perfect virialization, being held together partly by the confining ram pressure of turbulent flows in the interstellar medium (ISM).

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3.2.2.3. 

3.2.2.4. 

3.2.2.4. The role of magnetic field. The two bottom panels of Figure 4 show the SFR in the two magnetic cases corresponding to $V_a^0=1$, γ_b = 0.3 and $V_a^0=3$, γ_b = 0.1, respectively, adopting y_cut = 0.1.

In order to decipher the contributions of the various physical effects, we have calculated the SFR in the magnetized case, first keeping the same PDF as in the hydrodynamical case (as described by Equation (4)) but then also taking into account the fact that the magnetic field can modify the width of the density PDF, as investigated, for example, in Molina et al. (2012). In a magnetized flow, the dependence upon the Alfvénic Mach number must be taken into account. Based on their numerical simulations, Molina et al. (2012) propose an analytical relation to predict the dependence of the variance of the lognormal density distribution upon the rms Mach number in magnetized supersonic turbulent gas. The result depends on the index γ_b. For γ_b = 0, i.e., B independent of the density, they recover the hydrodynamical dependence (Equation (4)). When γ_b = 0.5, i.e., $B\propto {\sqrt{\rho }}$, they find that in the super-Alfvénic case, the variance is given by

$$\begin{eqnarray} &&\sigma _0^2 = \ln \left(1 + b^2 {\cal M}^2 {\beta \over \beta +1}\right), \end{eqnarray} \tag{ 49 }$$

where $\beta =P_{{\rm th}}/P_{{\rm mag}}=2\, C_s^2 / V_a^2$.

The basic effect of the magnetic field on the SFR, when assuming no variation of the density PDF compared with the hydrodynamical case, is illustrated by the solid lines in the bottom left and right panels, to be confronted with the SFR values displayed on the right third panel. As expected, the magnetic field tends to reduce the SFR by a factor of a few. The amplitude of this effect obviously depends on the field intensity, as explained in Section 2.3.3. For $V_a^0=1$, γ_b = 0.1 (not displayed here), the SFR is identical to the hydrodynamical case, while for $V_a^0=3$, γ_b = 0.1 (left bottom panel), it is reduced by a factor of less than 2 for ${\cal M}=16$ and about 2 for ${\cal M}=9$. Stronger values of the magnetic field obviously lead to even lower SFRs. However, the values we used are within the range of the values typically measured in molecular clouds.

The dotted lines of the bottom left and right panels show the SFR when the influence of the magnetic field on the density PDF is taken into account, as described above. With our choice of parameters, we find that at the mean density β = 0.036 for $V_a^0=3$, γ_b = 0.1 and β = 0.6 for $V_a^0=1$, γ_b = 0.3. The first value in particular significantly reduces the width of the density PDF. Clearly, the impact of the magnetic field is much more pronounced and the SFR drops by a factor of several. We stress, however, that, as discussed in Molina et al. (2012), the relation given by Equation (49) only holds for γ_b = 0.5 and does not work well at small and intermediate Mach numbers. Therefore, there are still considerable uncertainties here due to the lack of exact knowledge of the density PDF in the presence of the magnetic field and also of the precise gas density–magnetic intensity relation.

Finally, we also note that in the presence of a magnetic field, the coefficient ϕ_t could possibly be further increased, further reducing the SFR. Indeed, it is likely that the magnetic field and the velocity field of the perturbation that eventually gives rise to the gravitationally unstable perturbation must be sufficiently aligned. Otherwise, after a weak density enhancement, the magnetic pressure will probably stop the contraction.

Note that PN and Federrath & Klessen (2012) find that in their simulations, the SFR is indeed reduced by a factor two to three when the flow is significantly magnetized.

3.2.2.5. 

3.2.2.5. The role of turbulence. As in the theories of KMK, PN, and Hopkins (2012a, 2012b), turbulence is at the heart of our star formation theory. We have shown in Paper I (Section 6.3) that in the time-independent version of our IMF theory, turbulence, globally, has a negative impact on star formation. The stronger the turbulence, both at the global clump scale (as formalized by ${\cal M}$) and the local (Jeans length) scale (as formalized by ${\cal M_\star }$), the smaller the global clump mass fraction encapsulated into gravitationally bound prestellar mass reservoirs. Indeed, although increasing the Mach number increases the width of the PDF, promoting the formation of small-scale structures, it leads to a decrease in the number of collapsing structures around the peak of the CMF (see Papers I and II). As shown in the previous sections, however, star formation will keep processing during the clump's lifetime and time dependence changes the final clump's mass fraction, ending up forming collapsing cores. The impact of turbulence on ${\rm SFR}_{{\rm ff}}^0$ is illustrated in Figure 5. This figure portrays the behavior of ${\rm SFR}_{{\rm ff}}^0$ at fixed cloud density (${\bar{n}}$ = constant along each line) for different cloud sizes, as a function of V₀, and thus of ${\cal M}$ (see Equation (16)), spanning a range ${\cal M}\simeq 4$–30. The time-independent results are displayed in Figure 6. In this latter case, the SFR steadily decreases with increasing levels of turbulence, as mentioned above. As shown in Figure 5, however, including time dependence in our formalism significantly changes this behavior, with the SFR now increasing, although modestly, with the level of turbulence. The reason is twofold. First, increasing the Mach number shifts the peak of the CMF toward smaller masses (see Paper I), a behavior exacerbated when time dependence is considered, as discussed in Section 2.2.4. Since small-scale structures have shorter free-fall times (τ_R∝ρ^−1/2), this increases the number of small cores and thus globally increases the SFR (see Equation (40)). Second, massive mass reservoirs, considered to be stable in the static case, can now fragment into small-scale structures as their internal turbulent motions lead to new small-scale overdense regions, globally increasing the number of collapsing cores. This positive impact of turbulence upon the SFR is in agreement with the results of PN but contrasts with those of KMK. As demonstrated in HC11, this decreasing dependence of the SFR with increasing Mach number in the KMK theory (see their Equation (30)) stems from the missing $\widetilde{\rho }_{{\rm crit}}^{1/2}$ term in their Equation (20), which stems from the fact that KMK assume that the critical free-fall time is equal to the free-fall time of the cloud. The dependence of the SFR upon b illustrated in Figure 5 has already been addressed previously.

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The fact that in the time-dependent formalism, turbulence is found to enhance star formation seems, at first glance, to be in contradiction with earlier numerical simulations (see, e.g., MacLow & Klessen 2004). However, as discussed in HC11, the behavior we infer for the SFR is qualitatively similar to that inferred by PN (see also Federrath & Klessen 2012). In their simulations, these authors find that for a given α_vir, the SFR increases with the Mach number. The critical point here is that when turbulence is too high (i.e., α_vir is typically larger than 1), the kinetic energy will lead to an expansion of the clump, an effect which is neither included in the present analytical calculations nor in the turbulence in the box numerical simulations of PN. We thus intuitively expect a non-monotonic behavior of the SFR with turbulence. When α_vir is small, turbulence enhances star formation because of the presence of converging motions that lead to density enhancements. On the other hand, if turbulence becomes too large (α_vir ≳ 1 or so), then it triggers a fast expansion of the cloud which reduces the density and quenches star formation (note that Vázquez-Semadeni et al. 2003 do see such a non-monotonic behavior with turbulence).

3.2.2.6. 

As seen in the previous section, the SFR per free-fall time can vary by several orders of magnitude depending on the clump's conditions (mass, density, typical Mach number) but also, unsatisfactorily enough, on the values of y_cut, b, ϕ_t, and . These large uncertainties prevent a precise theoretical determination of the SFR in a cloud. Nevertheless, it is instructive to compare our determinations with observational ones. Such a comparison has partly been done in HC11 and will be extended here. The observed SFR per free-fall time in star-forming molecular clouds in the Milky Way lies in the range 0.03 ≲ SFR_ff ≲ 0.3 for clouds in the mass range 10³ ≲ M_c/M_☉ ≲ 10⁶, with ${\cal M}\approx 10$–15 and α_vir ≈ 0.3–2, with a mean value 〈SFR_ff〉 ≈ 0.16 (Murray 2011; Heyer et al. 2009). Looking at Galactic nearby molecular clouds and massive star-forming dense clumps, Heiderman et al. (2010) find SFRs in the range ≈0.02–0.12 for molecular clouds and ≈0.03–0.5 for dense clumps, yielding a mean value ≈0.1. These values are quite consistent with the ones derived above from our theory, while the typical rates obtained by KMK for similar conditions are about an order of magnitude smaller, as noted by Murray (2011) and Heiderman et al. (2010), a point we addressed in the previous section.

Traditionally, the observationally determined SFR for a given clump (or region of a clump) of mass M_c, radius R_c, lifetime τ₀, and mean dynamical time $\tau ^0_{{\rm ff}}$ is the projected SFR, i.e., the SFR per unit area:

$$\begin{eqnarray} &&{\dot{\Sigma }}_\star ={\Sigma _\star \over \tau _0}={\rm SFE}\times \left({\Sigma _g \over \tau _0}\right)={\rm SFR}_{{\rm ff}}\times \left({\Sigma _g \over \tau ^0_{{\rm ff}}}\right), \end{eqnarray} \tag{ 50 }$$

where ${\Sigma }_g=M_c/\pi R_c^2=(4/3){\bar{\rho }}R_c$ for a spherical clump is the clump gas surface density. The relationship between this projected SFR and the gas surface density is generally found to obey, within a large scatter, a power-law relationship:

$$\begin{eqnarray} &&{\dot{\Sigma }}_\star =\Sigma _{100}\,\left({ {\Sigma }_g\over 100\,{M}_\odot \,{\rm pc}^{-2} }\right)^{N}\,{M}_\odot \,{\rm yr}^{-1}\ {\rm kpc}^{-2}. \end{eqnarray} \tag{ 51 }$$

Extragalactic studies yield the well-known Kennicutt–Schmidt relation, with Σ₁₀₀ = 0.16 ± 0.6 and N = 1.4 ± 0.15.

Figure 7 displays ${\dot{\Sigma }}_\star$ predicted by Equation (42), as a function of surface density, Σ_g, for four typical clump sizes, namely, R_c = 0.5, 2, 5, and 20 pc, both in the isothermal and non-isothermal case. The impact of the uncertainties due to y_cut is shown by the two calculations with y_cut = 0.1 and 0.25, respectively. The clumps are assumed to follow Larson's (1981) relations and thus have velocities and densities given by Equations (35) and (36), with u₀ = 1 and d₀ = 0.1–5, i.e., ${\bar{n}}\approx 10$–10⁴ cm⁻³. The corresponding global and local Mach numbers entering our theory, ${\cal M}$ and ${\cal M}_\star$, are consistently derived from these values. We have taken /ϕ_t = 0.1 in the figure. Also displayed in the figure are the data of Heiderman et al. (2010), both for the class I and flat spectral energy distribution young stellar objects (YSOs) and the massive clumps, as well as the location of the observations of Gutermuth et al. (2011), as bracketed by the two large diamond areas. These latter authors determined the SFR in eight nearby molecular clouds, sampling gas surface density regions inferred from near-IR extinction mapping, ranging from ∼15 to 300 M_☉ pc⁻². Their analysis is similar to that by Heiderman et al. (2010) but, as they examined larger clouds, they placed stronger statistical constraints on the SFR versus gas surface density determinations than that study. Moreover, these authors include pre-main-sequence Class II objects in addition to Class 0, I protostars, from the Spitzer and 2MASS surveys.

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Several conclusions can be drawn from the theory–observation comparison displayed in Figure 7.

• 1.  
  All calculations reproduce well the observed values with the scatter, and exhibit a direct correlation between the density of star formation and the gas density, in agreement with the observational determinations.Clearly, it seems difficult to provide a universal relation such as Equation (51) over the whole density range, given the strong dependence, both for the slope and the normalization, of the SFR upon the properties of the clump, i.e., its mass and the nature of turbulence forcing. What seems to be more robust is that at very high density, Σ_g ≳ 300 M_☉ pc⁻², the theoretical relations seem to merge toward a strongly super-linear behavior with an exponent N ≃ 2 (an exact value is difficult to infer as it depends on the clump size) and a normalization Σ₃₀₀ ≈ 32. Star formation is thus predicted to increase basically quadratically with density, and thus to be largely dominated by the contribution of the densest (likely to be bound) clumps. Given the steeply decreasing number of clumps with increasing density, however, such dense clumps represent only a modest (≲ 20%) mass fraction of the clouds (Kainulainen et al. 2011). This slope becomes progressively steeper as density decreases, the steeper the smaller the clump, as expected from the analysis conducted in the previous section.This N ≃ 2 value for the slope is in excellent agreement with the values N = 1.67–2.67 determined by Gutermuth et al. (2011) in eight nearby molecular clouds. Lower values, including Kennicutt–Schmidt like values, N ∼ 1.4, with a star formation threshold are excluded by these authors.
• 2.  
  The dispersion in the observed SFR determinations is well explained by variations among the clump sizes, not mentioning possible variations of the (/ϕ_t) efficiency factor and of the contribution of different turbulence modes, as formalized by the factor b, between clumps.
• 3.  
  The theory, as the observations, predicts a severely decreasing SFR below a typical value Σ_g ≈ 100 M_☉ pc⁻², i.e., $N_{{\rm H_2}}\approx 5\times 10^{21}$ cm⁻², $\bar{n}\approx 2500\,(R/1\,{\rm pc})^{-1}\, {\rm cm}^{-3}$, which corresponds to a visual extinction A_V ≈ 7 (Draine 2003), except for the largest (20 pc) clumps. We recall that such a drop in the SFR naturally arises from our theory, without invoking any density threshold.
• 4.  
  There is no real density threshold in star formation. Indeed, according to our calculations, star formation is predicted to keep processing, although with a steadily decreasing rate, even at densities lower than the aforementioned value, but only at non negligible rates in the largest clumps. This was expected from the discussion in Section 3.2.2.1. In this low-density domain, our values can be compared with the observations of Gutermuth et al. (2011). As mentioned earlier, these authors examined larger clumps than previous studies and found larger SFR values than these latter at low density. The eight molecular cloud sample examined by these authors covers a range of clump sizes and surface densities L_c ∼ 4–20 pc, Σ_g ∼ 22–71 M_☉ pc⁻². In this range, they determine $\Sigma _\star /\Sigma _g^2$ in the range (3 × 10⁻⁴)–(5 × 10⁻³) pc$^2\,{M}_\odot ^{-1}$. These values can be directly confronted with the ones inferred from the two upper solid lines, which correspond, respectively, to R_c = 5 and 20 pc, in the appropriate surface density range. This yields $\Sigma _\star /\Sigma _g^2\approx 2.0\times (10^{-5}$–10⁻³) for the non-isothermal case and values larger by about a factor 10 for the isothermal case, for our fiducial value of turbulence forcing, b = 0.5. Remembering that both observational and theoretical determinations are subject to significant uncertainties, the agreement can be considered as satisfactory.

This reinforces our suggestion that the SFR smoothly decreases with decreasing gas density and that there is no real density threshold for star formation. The severe drop in the SFR, which is predicted to occur around Σ_g ≈ 100 M_☉ pc⁻² for ∼pc-size clumps, simply reflects the basic mechanisms mentioned in Section 3.2.2.1: star formation can still occur, although at significantly lower rates, in low-density clumps, provided that these latter, assuming they follow Larson's relation, are large enough to (1) significantly exceed their typical Jeans length and (2) generate enough large-scale turbulence, i.e., have large enough Mach numbers to generate a large enough spectrum of density fluctuations, and thus of prestellar cores. According to the observed clump size distribution, ${dN}_c/{dL}_c\propto L_c^{-1.9}$ (Kainulainen et al. 2011), however, such clumps are very rare. This stresses the need to observe very large areas at low-density in order to get statistically significant SFR detections.

A threshold for core formation was also ruled out in the SCUBA survey of Perseus by Hatchell et al. (2005). These authors stress the existence of submillimeter cores, identified from IRAS as likely Class I objects, down to A_V = 3, i.e., $N_{{\rm H_2}}\sim 10^{21}$ cm⁻², and demonstrate the steeply decreasing probability of finding a core with decreasing column density. This is also consistent with the analysis of André et al. (2011, Figure 3), who find a drastically decreasing but non-zero number of Class 0 prestellar core candidates below about the aforementioned A_V ∼ 7 value.

In contrast, several studies of various local clouds (Onishi et al. 1998; Johnstone et al. 2004; Enoch et al. 2007; Lada et al. 2010; André et al. 2010) do find a steep break in the SFR–gas density relationship around the aforementioned ∼100 M_☉ pc⁻² value that they identify as a density threshold for star formation. It should be stressed, however, that all these studies still find prestellar cores below this limit, suggesting that star formation continues, although with a much smaller efficiency, down to the limit of the surveys, i.e., Σ_g ≈ 10 M_☉ pc⁻². As mentioned above, a noticeable exception is the recent work by Gutermuth et al. (2011), who explore larger areas and who include pre-main-sequence Class II objects in addition to Class 0, I protostars, from the Spitzer and 2MASS surveys, and who do not find evidence for a column density threshold for star formation from 15 to 300 M_☉ pc⁻². These authors explain the different result by the very small number of Class 0, I protostars at low-gas column densities, and thus the low statistics in previous surveys, including the Spitzer-derived data of Heiderman et al. (2010). Interestingly, by comparing the SFR derived from YSO surface densities and the ones derived from far-IR luminosity, as commonly used as a proxy of the SFR, they find that the L_FIR-derived SFRs are systematically about an order of magnitude below the YSO-derived ones. This is confirmed, for instance, by the recent analysis of W40 and Serpens South with Herschel by Maury et al. (2011), who have access to Class 0 objects and find SFRs an order of magnitude higher than the typical SFRs observed for embedded infrared clusters. This suggests, if anything, that SFR determinations based only on Class 0 or Class I objects underestimate the real SFR by a fair amount. Therefore, the aforementioned identification of a cutoff in the SFR below ∼100 M_☉ pc⁻² might in fact be due to the very small number of Class 0, I protostars at low-gas column densities and the too limited statistics (too small fields of view) in the surveys. This is supported by the recent analysis of Bressert et al. (2010) who show that using only near-infrared detections and small fields of view allows identifications of young stars only in the densest parts of the clouds, a bias that can be corrected with Spitzer data and larger surveys.

As already found in the observations of Heiderman et al. (2010) and Evans et al. (2009), our SFR–gas relation, characteristic of Galaxy star-forming molecular clouds, lies above the Kennicutt–Schmidt relation by about a factor of 10 at high density. It should be remembered, however, that the Kennicutt–Schmidt relation derives from extragalactic determinations and is averaged over much larger regions (∼kpc-size) than individual molecular cloud complexes (≲ 10–100 pc). Such an average determination includes star-forming regions but also diffuse molecular gas or atomic gas that is not forming stars, leading to an overestimate of the amount of gas counted as star-forming gas (Heiderman et al. 2010). Recent SFR determinations in nearby galaxies indeed show that the measurement size scale used significantly changes the SFR–gas surface densities relation (Liu et al. 2011). Moreover, SFR in GMCs are determined from inventorying the YSOs and assuming a star formation timescale of ∼Myr while galaxy-averaged SFRs are derived from conversion of an FIR flux into a mass growth rate assuming a timescale of 10–100 Myr, naturally leading to a smaller SFR for a given average gas density. Since, as shown in the present study, the SFR depends on the clump's mean density and significantly decreases for low-density clumps, this provides a plausible explanation for the difference between GMCs and extragalactic SFR–gas quantitative determinations. In fact, a correct determination of the SFR–gas relation at extragalactic scales should take into account the density dependence of the SFR–gas relation. The same explanation applies to the characteristic value of the SFR inferred for the Milky Way, found to be about an order of magnitude smaller than in GMCs. A likely explanation is that most of the mass of the clouds that compose the Galaxy is at low column density, with only a very small fraction in condensed structures at high enough density for efficient star formation (see, e.g., Kainulainen et al. 2011). Indeed, large-scale clouds which compose the Galaxy have densities around ∼10²¹ cm⁻² (${\bar{n}}\lesssim 100$ cm⁻³). Therefore, most of the clouds in the Milky Way no longer produce stars. On the other hand, our calculations suggest that the power dependence of the SFR upon the gas surface density in the (high-density) efficient star formation regime is not drastically different for Galactic and extragalactic determinations, since our exponent at high density is N ≈ 2, whereas the one characteristic of the Kennicutt–Schmidt relation is N = 1.4 ± 0.15 (see Figure 7). Again, such a universal SFR–gas dependence naturally emerges from a picture of star formation being initiated by turbulence-induced fluctuations and the star formation rate being determined by the free-fall time of these fluctuations—not of the clump !—once gravity takes over, leading to a strong dependence of the SFR upon the gas density.

In the previous section, only the SFR dependence upon the column density was examined. In the present section, we consider the SFR dependence upon the clump's mass and upon the column and volume densities over free-fall time. The results are compared with the sample of Heiderman et al. (2010).

4.3.1. Star Formation Rate versus Clump Mass

Figure 8 portrays the SFR, ${\dot{M}}_\star$, as a function of the cloud mass, M_c, for the isothermal and non-isothermal cases. Each line corresponds to the same clump sizes as in the previous figures, i.e., R_c = 0.5, 2, 5, and 20 pc, with the normalization density d₀ increasing along the line from d₀ = 0.1 to 4.5. We note the large scatter in the predicted ${\dot{M}}_\star$ for a given clump mass, in particular, for the non-isothermal case. The data of Heiderman et al. (2010) are displayed in the figure for comparison. The theoretical determinations agree very well with the data, indicating that within our theory, the range of clump characteristics we have investigated reproduces well the observationally determined SFR values with the observed scatter. The dashed line in the figure displays the relation derived by Lada et al. (2012), ${\dot{M}}_\star = (4.6\times 10^{-8})\,({M_c/ {M}_\odot })\,{M}_\odot \ {\rm yr}^{-1}$. The present calculations show that this relation, although broadly correct, may not be an accurate representation of the real star formation process, as this latter not only depends much more steeply on the clump's mass, but also strongly depends on the clump's size.

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4.3.2. Dependence of Star Formation Rate on Free-fall Time

The values of SFRs per mean free-fall time for a sample of clumps can be directly inferred by determining the slope of the relation ${\dot{\Sigma }}_\star$ versus $({\Sigma _g / \tau ^0_{{\rm ff}}})$ (Equation (50)) or, equivalently, of the volume density relation ${\dot{n}}_\star$ versus $({n_g / \tau ^0_{{\rm ff}}})$. A universal value of the SFR would correspond to one single, constant value of the slope for the whole sample.

To confront such observational determinations with our theory, we first vary the clump's size for a fixed density normalization parameter d₀. Indeed, assuming Larson's relation for our clumps, i.e., $\rho \propto R_c^{-0.7}$, yields $\Sigma \propto R_c^{0.3}$ and $\tau _{{\rm ff}} \propto R_c^{0.35}$ for the column density and free-fall time, respectively. Therefore the column density over free-fall time, Σ/τ_ff, is almost independent of the clump's size. The corresponding ${\dot{\Sigma }}_\star$ versus $({\Sigma _g / \tau ^0_{{\rm ff}}})$ curves are thus almost parallel to the y-axis, making the dependence upon d₀ and R_c straightforward. Seven values of d₀ are computed, namely, 0.5, 1, 2, 3, 4, 5, and 10 as well as eleven sizes R_c = 0.5, 1, 2, 4, 6, 8, 10, 12, 14, 16, and 18 pc. Figure 9 displays the theoretical predictions for ${\dot{\Sigma }}_\star$ versus $({\Sigma _g / \tau ^0_{{\rm ff}}})$ as well as the observational determinations of Heiderman et al. (2010). As is obvious in the figure, and as expected from the previous subsection, the overall agreement between our predictions and the data is very good and the observational scatter is well reproduced by the explored variations of clump sizes and densities. Note that given the large spread in the data and the uncertainties discussed previously in the theory, it is difficult to assess more than a qualitative agreement.

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Interestingly, the confrontation between our results and the observed SFR values from Figure 9 leads to the conclusion that the size of star-forming clumps should decrease with increasing column density over free-fall time. Typically, the clump's size should be less than about 2 pc for log (Σ/τ_ff) ≃ 3 and less than about 0.5 pc for log (Σ/τ_ff) ≃ 3.5 (with a dispersion of a factor 2–3). To verify whether this trend is indeed real, we have plotted $\dot{\Sigma }_*$ as a function of the clump's radius (Figure 10). Indeed, Figure 10 shows that a clear trend for the highest values of ${\dot{\Sigma }}_\star$ has been obtained for the smallest clouds. The reason of this behavior is not completely clear but appears to be an important element to understanding the SFR dependence. A likely explanation is that large clouds never become dense enough, i.e., massive enough (see, e.g., Figure 8). More precisely, as massive clouds collapse, they undergo local gravitational fragmentation in dense clumps where most of the star formation occurs.

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Finally, for completeness we also display ${\dot{n}}_\star$ versus $({n_g / \tau ^0_{{\rm ff}}})$ in Figure 11 for the same cloud parameters as in Figure 9 (the various curves correspond to a fixed d₀), while Figure 12 shows the results for the same cloud parameters as in Figure 7. For the data of Heiderman et al. (2010), we derived the volume density from the surveyed area as (assuming spherical clumps of area A) ${\bar{\rho }}=(3{\sqrt{\pi }}/4)(M_c/A^{3/2})$.

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According to the present analysis, there is clearly no "universal" slope value either for the ${\dot{\Sigma }}_\star$ versus $({\Sigma _g / \tau ^0_{{\rm ff}}})$ or for ${\dot{n}}_\star$ versus $({n_g / \tau ^0_{{\rm ff}}})$ relation. Instead, these relations significantly vary with the clump's size and density. Indeed, below Σ/τ_ff ≲ 1000 M_☉ yr⁻¹ kpc⁻², there is between 1 and 2 orders of magnitude variations in the SFR. Even in dense regions corresponding to n/τ_ff > 100 cm⁻³ Myr⁻¹, the mean value of the SFR, as measured by the slope of the illustrated relations, typically varies between ∼3% and ≳ 10%. This analysis is in contrast with the claim by Krumholz et al. (2012) for a universal SFR value of about 1% from clouds to starburst galaxies. As mentioned earlier, molecular clouds are very clumpy structures with orders of magnitude of density variations. Therefore, it seems difficult to invoke an average timescale, defined at the cloud's mean density, as the relevant timescale for star formation.

According to various recent studies, in particular the ones conducted with Herschel, star formation is found to occur in filaments (André et al. 2010; Men'shchikov et al. 2010; Ward-Thompson et al. 2010; Molinari et al. 2010; Hill et al. 2011). It is well known that (unbound or bound) molecular clouds exhibit a filamentary structure, as expected from both hydrodynamical and MHD compressible (shock dominated) turbulence, even in the absence of gravity (Padoan et al. 2001; Nakamura & Li 2008; Federrath et al. 2010; see also Banerjee et al. 2009). It is thus not surprising that the locations of star-forming populations tend to follow this morphology. We recall that in our formalism, the prestellar cores, progenitors of the protostars identified as Class 0 objects, are born in turbulence-induced clumps of enhanced density and isolate themselves from the surrounding medium under the action of gravity (Hennebelle & Chabrier 2008; Chabrier & Hennebelle 2011). For the sake of simplicity, the clumps and the cores in our theory are assumed to have a spherical geometry. Nonlinear simulations of the collapse of filaments show that the resulting fragments are nearly spherical (Inutsuka & Miyama 1997). It thus seems reasonable to assume spherical collapse for the cores. The reality might be more complex for the star-forming clumps, which may have a flattened, filament-like geometry, with some of these filamentary clumps becoming themselves gravitationally unstable as they both accrete mass and dissipate turbulent energy, eventually yielding global collapse of the clump. Interestingly, the very same Herschel observations reveal that protostellar cores seem to be present only in the gravitationally bound clumps (André et al. 2010; Molinari et al. 2010; Arzoumanian et al. 2011), suggesting that star formation takes place dominantly in such clumps, prone to gravitational fragmentation. Gravitational contraction will thus cause filamentary clumps to fragment along the axis.

This possible peculiar role of filaments in star formation might affect our results by numerical factors of the order of a few but it does not fundamentally modify our theory of clump and prestellar core formation, and of the resulting CMF.¹¹ In our theory, the (scale-free) criterion for clump formation is simply set up by an arbitrary critical density threshold $\delta _c=\log (\rho _c/{\bar{\rho }})$ (see Section 3 of Paper I). The argument above thus suggests to choose for this threshold condition for the star-forming clumps—identified as the aforementioned filaments—the density at which these latter become gravitationally unstable and thus exceed a critical mass per unit length M_crit = 2〈σ〉²/G, where 〈σ〉 includes both the thermal and non-thermal velocity dispersion contributions (Ostriker 1964; Larson 1985; Inutsuka & Miyama 1992; Fiege & Pudritz 2000). For a temperature T ∼ 10 K, this corresponds to a mean column density $N_{{\rm H_2}}\simeq 10^{22}$ cm⁻², similar to the value identified in Section 4.1 for efficient star formation (point (3)), about 5–10 times the typical mean density of the surrounding cloud. This particular choice for the clump formation in our theory will not affect the resulting CMF determined by the virial condition for spherical collapse, as mentioned above.

Furthermore, a closer inspection of Table 1 and Figure 8 of Paper II (see also Figure 2) shows that small-size, low-density (≲ 3000 cm⁻³) clumps for which the CMF is very narrow and thus star formation is negligible, are unbound or only marginally bound ((E_kin + E_th) ≳ |E_grav|), whereas the ones above about this density are bound and thus prone to collapse. Therefore, our theory suggests that the clumps in which star formation is taking place dominantly should be denser than ${\bar{n}}\sim$ a few 10³ cm⁻³, i.e., $N_{{\rm H_2}}\simeq 10^{22}$ cm⁻² for pc-size clumps, in fairly good agreement with the aforementioned critical value inferred for filaments. This is consistent with the results found in Section 3.2.2.2, which show that the SFR is highest in the largest and densest clumps, with α_vir ≲ 1. Note that given the scale-free nature of the turbulent flow responsible for the clump formation, such a minimum density for efficient star formation must apply at all scales, and thus must vary accordingly with the clump's size. The aforementioned value typically applies for a 1 pc size clump and should increase as ${\sim} R_c^{-0.7}$ for smaller sizes, providing Larson's relations apply.

Therefore, a picture where star formation occurs dominantly in relatively dense, rather bound clumps with mean density ${\bar{n}}\,{\gtrsim}\, 3000\,{\rm cm}^{-3}$, in agreement with observational results, naturally emerges from our theory. The filamentary nature of the clumps does not significantly affect the results, except for possible numerical factors of the order of unity.

One of the intrinsic difficulties in inferring the mass spectrum of the self-gravitating fluctuations that may eventually lead to stars is the choice of clump parameters (density, size, velocity dispersion, etc.). As seen from our calculations, these clump properties have a strong impact on the mass spectrum. The peak position, for instance, is determined by the clumps Jeans mass, and thus mean density, temperature, and Mach number. Clumps, however, are not well-defined entities, in particular because they constantly grow in mass by accretion, so that their density and size evolve with time. Under these conditions, it is clear that the choice of a fixed set of clump parameters is a simplification. A proper investigation of the clump evolution would require us to develop a fully self-consistent model for clump formation and evolution, including growth by accretion, gravitational contraction, or turbulence dispersion as well as spatial inhomogeneity. There is no such model of clump evolution yet; we thus consider a very simple prescription, essentially inspired by observations.

As mentioned above, instead of setting a density and size, we assume that the clump is accreting at a fixed accretion rate, $\dot{M}$. Starting with a mass M(t = 0) ≡ M₀, the clump mass grows as $M_c = M_0 + \dot{M} t$. This mass entails the mass of the gas, M_g, and the mass of the stars, M_*, that formed in the clump, M_* = M_c − M_g. We further assume that the clump follows Larson-type relations as stated by Equations (35) with d₀ = 3 and u₀ = 1, which correspond to the canonical values of these relations. Then, knowing M_g(t), we can infer the gas mean density, ${\bar{n_g}}(t),$ and thus the cloud radius R_c(t). Since at a given time, t, the clump's parameters are well determined, we can calculate the SFR from our CMF/IMF theory, as stated by Equations (42) and (43). We can thus estimate the gas mass fraction converted into stars between t and t+dt as dM_*(t) = M_g × SFR_ff × dt. The total mass of stars at time t, M_*(t), is thus

$$\begin{eqnarray} &&M_*(t)=\int _0^t M_g(t) {\rm SFR}_{{\rm ff}}(t) dt. \end{eqnarray} \tag{ 52 }$$

We typically integrate from 0 to 10 Myr with time steps dt = 10⁵ yr. This procedure presents similarities with the model presented in Zamora-Avilés et al. (2012).

The accretion rate onto the molecular clumps is a difficult quantity to infer observationally and thus is not well known. Using observations of the LMC, Kawamura et al. (2009) and Fukui et al. (2009) propose that GMCs with a mass of ≃a few 10⁵ M_☉ are accreting at a rate $\dot{M}=$ (1–5) × 10⁻² M_☉ yr⁻¹. To estimate the accretion rate on smaller clumps, we simply use the Larson relations,

$$\begin{eqnarray} \nonumber \dot{M} &\simeq & {M \over \tau _c} \simeq {M \over R_c/(\sigma /\sqrt{3})} \\ & \simeq & M^{{1.3+\eta \over 2.3}} { u_0 \over \sqrt{3}} \left({4 \pi \over 3} d_0\right) ^{ 1 - \eta \over 2.3} (1 \, {\rm pc})^{ 0.7 - 3 \eta \over 2.3} \nonumber\\ &\simeq & 10^{-3} \, \left({M \over 10^4\, M_\odot } \right) ^{1.3+\eta \over 2.3} \left({\sigma \over 0.8 \, {\rm km \, s^{-1}} }\right)\nonumber\\ &&\times \left({ {\bar{n}} \over 1 \, {\rm cm}^{-3}} \right)^{1 - \eta \over 2.3}\,M_\odot \, {\rm yr}^{-1}, \nonumber \end{eqnarray} \tag{ 53 }$$

where the numerical constant has been estimated for η = 0.4. For a GMC of mass 10⁵ M_☉, this estimate yields an accretion rate $\dot{M} \simeq 5 \times 10^{-3}$ M_☉ yr⁻¹, slightly lower but nevertheless close to the estimate of Fukui et al. (2009). For a cloud of 10³ M_☉, we get 2 × 10⁻⁴ M_☉ yr⁻¹. We have thus explored three typical values for the accretion rate, namely, $\dot{M} = 10^{-2}, \, 10^{-3}$, and 10⁻⁴ M_☉ yr⁻¹, which correspond to clumps of approximately 10⁵, 10⁴, and 10³ M_☉.

Figure 13 portrays the results for $\dot{M} = 10^{-3}$ M_☉ yr⁻¹ and /ϕ_t = 0.1. The top panel shows the total mass of the clump (solid line), the mass of gas (dotted line), and the stellar mass (dashed line). While the total mass grows linearly with time, the mass of gas, which was initially almost equal to the total mass, increases less and less rapidly as star formation is occurring. At the beginning of the process, the mass of the cloud is small, and the amount of stars formed remains fairly limited. After t ≃ 4 Myr, the stellar mass starts increasing significantly and after t ≃ 8 Myr, the masses of gas and stars are roughly equal. After this time, the gas mass basically saturates, as the gas added to the cloud is rapidly converted into stars. Consequently, the size of the cloud (middle panel) increases rapidly at the beginning and much more slowly as star formation proceeds. The bottom panel displays the stellar mass spectrum at different times, namely, after 1, 4, and 7 Myr (dotted lines from bottom to top) and after t = 10 Myr (solid line); the dashed line corresponds to the SCIMF. As seen in the figure, the shape of the mass spectrum remains about the same over time but of course the integral, i.e., the total number of stars, strongly increases, a consequence of the growing size of the clump, as mentioned above.

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Figure 14 displays similar results for the same accretion rate, $\dot{M} = 10^{-3}$M_☉ yr⁻¹, but /ϕ_t = 0.3. The behavior remains qualitatively similar except that, as expected, stars form faster, since the SFR is three times larger (see Equation (43)), and thus the gas mass remains smaller by a factor of about 2 compared with the previous case. Consequently, the size of the clump is slightly smaller and less small-mass objects form because the Mach number (${\cal M}\propto R_c^\eta$) is slightly smaller (see Paper I).

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Figure 15 displays the results for $\dot{M} = 10^{-2}$ M_☉ yr⁻¹. The SFR is slightly smaller than for $\dot{M} = 10^{-3}$ M_☉ yr⁻¹ because the clump grows more rapidly and thus is less dense. Interestingly, the mass spectrum remains very similar, except for the slightly larger number of low-mass objects, a consequence of the larger Mach number. Indeed, as discussed in Paper I (Equation (47) of Paper I), there is a partial compensation between the Jeans mass and the Mach number scale dependences as a function of the cloud's size/mass so that the location of the peak of the CMF remains almost unchanged. It is interesting to compare the top panel of this figure with the top panel of Figure 5 of Vázquez-Semadeni et al. (2007), which shows the total gas+stellar masses in a simulation of colliding flows. The general behavior and even some of the details of the simulation are very similar to the results portrayed in Figure 15: (1) the total mass of the cloud in both cases is a few 10⁴ M_☉; (2) the time at which stars and gas masses are equal agree within a factor of about 2 (particularly if we choose for the beginning of the cloud formation in the simulation the time t ≃ 10 Myr, which seems more accurate than t = 0; (3) when stars start forming efficiently, the mass of the gas remains nearly constant both in the simulation and in the present model. The main differences appear in the evolution of the stellar mass, which is more sudden in the simulations and faster at the beginning than in the later phases, in contrast to the present results. Moreover, while in the model the mass of the gas always increases, it decreases in the simulation. This is likely due to gravity, whose effects on the cloud dynamics are ignored in our model.

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The last case we investigate is presented in Figure 16, which displays results for $\dot{M} = 10^{-4}$ M_☉ yr⁻¹. The mass of the clump is only about 400 M_☉. The mass spectrum peaks at about the same mass for the reason mentioned above, but the mass spectrum is much narrower with a significant deficit of both massive and low-mass objects. This stems from the fact that for such small clumps, the Jeans length becomes comparable to the size of the cloud, as discussed earlier, drastically decreasing star formation (see, e.g., Figures 2 and 8 of Paper II).

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It is now relatively well established that not only CO clumps (Heithausen et al. 1998; Kramer et al. 1998) but also infrared dark clouds (Peretto & Fuller 2010) follow a mass distribution ${\cal N}^{{\rm cl}} = {dN}^{{\rm cl}}/{dM} \propto M^{-\gamma _{{\rm cl}}}$ with γ_cl ≃ 1.7. It is important to recall that our theory predicts that this exponent, as well as the one for gravitationally bound cores when a gravitational collapse condition is taken into account, is a direct consequence of turbulence and is indeed related to the index n' of the power spectrum of log ρ by the relation γ_cl = 1 + x = 3 − n'/3 (see Paper I). Therefore, the present theory naturally takes into account the clumpy structure of the gas and in principle no further calculations are needed. In other words, for spatially close enough clumps surrounded by sufficiently dense gas, there is no need to sum up over a clump population. While strictly true as long as (1) the gas is barotropic and (2) the density PDF is lognormal, this is no longer the case in real situations. Indeed, the diffuse ISM does not follow our Equation (31), as the atomic hydrogen that fills up the Galaxy can be as warm as 10⁴ K. Therefore, for clumps that are spatially well separated, the in-between gas is not isothermal and we must sum up over a clump population. The scale at which this happens is not well determined. Numerical simulations by Vázquez-Semadeni et al. (2007), Hennebelle et al. (2008), Heitsch et al. (2008), and Banerjee et al. (2009) suggest that the warm gas is tightly mixed with the cold gas at scales of about a few parsecs.

We calculate the mass spectrum resulting from the distribution of the time-dependent clumps described in the previous section. For that purpose, we proceed as follows. We perform a series of time-dependent clump calculations, choosing values of $\dot{M}$ between 10⁻¹ and 10⁻⁵ M_☉ yr⁻¹ using logarithmic intervals. We stop the integration when M_* = M_g in the clump and we adopt for the mass of the clump the value M_cl = M_* + M_g. Then we sum up the corresponding mass spectra after multiplying by the aforementioned clump distribution ${\cal N}^{{\rm cl}} (M_{{\rm cl}}) \propto M^{-\gamma _{{\rm cl}}}$. The final core mass spectrum is given by

$$\begin{eqnarray} &&{\cal N}_{{\rm tot}} = \int _0 ^{M_{{\rm sup}}} {\cal N}_{M_c} ({M_{{\rm cl}}}) V_{cl}\, {\cal N}^{{\rm cl}} (M_{{\rm cl}}) {dM}_{{\rm cl}}, \end{eqnarray} \tag{ 54 }$$

where ${\cal N}_{M_c} ({M_{{\rm cl}}})={dn}/{dM}_c$ is the mass spectrum of self-gravitating fluctuations (i.e., cores), i.e., the CMF, for a clump of mass M_cl and volume V_cl, as given by Equation (2) (see Paper I and Paper II), and M_sup is the largest mass in our sample which corresponds to the largest accretion rate.

At this stage, it is worth stressing that since γ_cl < 2, most of the mass is contained in the most massive clumps; γ_cl = 2 then corresponds to a critical case. We have thus considered three values of γ_cl, namely 1.5, 1.7, and 2. Figure 17 displays the results.

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For γ_cl = 1.5 and γ_cl = 1.7, the mass spectra are pretty similar to the results obtained in the previous section for a single clump. The reason is that most of the mass is contained in the most massive clumps. Interestingly, varying the upper value of the accretion rate by two orders of magnitude has only a modest impact on the upper mass part of the total mass spectrum, while below about the mean Jeans mass, the low-mass part of the distribution quantitatively varies by orders of magnitude, the number of low-mass objects, in particular brown dwarfs, strongly decrease with decreasing accretion rate. Values of ${\dot{M}}_{{\rm sup}}\lesssim 10^{-3}$ M_☉ yr⁻¹ yield a strong deficit of brown dwarfs compared with the SCIMF, representative of the observed population. As mentioned above, this stems from the smaller cloud size, and thus the smaller Mach number with decreasing accretion rate, yielding a lack of overdense small-scale structures, which are the progenitors of the brown dwarfs (see Paper I and Paper II).

For γ_cl = 2, the difference with the mass spectrum obtained for a single clump is more pronounced. In particular, at intermediate mass, the distribution tends to deviate from the SCIMF. This is due to the mass of the gas being equally distributed between small and large clumps. As the CMF that results from small clumps peaks at larger masses than the CMF produced in massive clumps, this tends to create a small bump at intermediate masses (M ≃ 10 M_☉).

In this paper, we have generalized our previous formalism and developed an analytical time-dependent theory of the stellar IMF. Although not very different from the time-independent prestellar mass spectra for given sets of cloud initial conditions (temperature, density, Mach number), the time-dependent ones are not quite identical to the former ones. The most significant differences are a slightly steeper slope at large masses, which arises from the time dependence of the characteristic Jeans mass, and the position of the peak of the CMF, which is shifted toward smaller masses by a factor of the order of 2–3, depending on the initial conditions. This is a direct consequence of the small-scale structures being rejuvenated several times during the collapsing process of the larger ones. For all explored typical clump conditions, the resulting CMF, while slightly too narrow when considering an isothermal eos, agrees very well with the shifted Chabrier system IMF (i.e., the Chabrier IMF shifted by a factor of ≃ 3 to take into account the core-to-star mass conversion efficiency during the collapse) when including the thermodynamics of the gas, confirming the results obtained for the time-independent theory.

This time-dependent theory enables us to derive an expression for the SFR for molecular cloud conditions. Confrontations with previously published SFR theories show some similar trends but quantitative or even qualitative differences. As mentioned in HC11, these differences arise essentially from the two essential characteristics of our theory, namely, (1) each overdense region dominated by gravity collapses under its own density-dependent dynamical time and (2) there is no particular threshold density/scale for star formation; instead, any overdense region produced by initial density fluctuations is susceptible to collapse, no matter its degree of internal turbulence, if dominated by gravity.

We explore the dependence of the SFR upon the clump properties, namely, the size, the level of turbulence (through the virial parameter), and the magnetic field. We show that the exact value of the SFR depends significantly on these clump parameters, which, unfortunately, are ill-defined quantities, hampering precise theoretical estimates of the SFR in a clump/cloud, in particular, for magnetized clouds. We show that when time dependence is taken into account in the formation of collapsing dense cores, turbulence globally favors star formation, in contrast to the result obtained with the time-independent theory, in agreement with the results of PN. We also show that when the clump is unbound, i.e., α_vir ≳ 1, the SFR decreases drastically, as the dominant contribution of kinetic (thermal+turbulent) over gravitational energy prevents gravitational collapse of the overdense structures, inhibiting star formation. The magnetic field also reduces the SFR when it is strong enough. Its exact influence depends on the magnetic-density relation and also on its influence on the density PDF, none of these properties being known with great accuracy.

Our calculations show that SFR correlates with both the gas surface density of the surrounding region and the size of the clump, and occurs dominantly in dense and/or massive—thus bound—clumps. Such large clumps and dense regions, however, represent only a modest fraction (≲ 20%) of typical clouds. In contrast, for the smallest clumps, the SFR decreases drastically below a typical density Σ_g ≈ 100 M_☉ pc⁻², i.e., a volume density ${\bar{n}}\approx 2500\,(R_c/1\,{\rm pc})^{-1}\, {\rm cm}^{-3}$, i.e., a visual extinction A_V ≈ 7, in excellent agreement with observational determinations in Milky Way molecular clouds. For clumps below about this density, star formation is still taking place, although at a much lower rate, only in very large (≳ 10 pc) clumps, which are very rare. This stresses the need to explore large fields of view to accurately determine the SFR in low-density regions. We stress that this abrupt change in the SFR does not stem from an arbitrarily defined threshold for star formation, but naturally arises from the theory and reflects the fact that pc-size, low-density clumps barely form stars essentially because of their too large characteristic Jeans length compared with their size and their too low level of turbulence.

Interestingly enough, the aforementioned density corresponds to the density above which the cold gas in the ISM becomes essentially composed of molecular hydrogen, H₂. Indeed, simulations of compressible turbulence coupled with hydrogen chemistry (Glover & Mac Low 2007, see also Krumholz et al. 2009) suggest that (1) H₂ forms much more efficiently in (compressible) turbulent gas than in quiescent gas owing to the much shorter formation timescale and that (2) above n ∼ 300 cm⁻³, the H₂ formation rate, which scales as the square of the number density, n², becomes larger than the photodissociation rate, allowing the efficient in situ formation of molecular hydrogen. The gas becomes dominantly (entirely) molecular above about 5000 cm⁻³ (10⁴ cm⁻³) and remains "trapped" and shielded from external UV radiation in the collapsing structures. Our calculations thus support Glover & Mac Low's (2007) suggestion that dense (bound or unbound) star-forming regions drive H₂ formation, and not the opposite. Phrased differently, the formation of molecular hydrogen is not a necessary condition for star formation but instead is a consequence of efficient star formation.

Finally, we derive a fully time dependent calculation of the core mass spectrum by considering a simple time-dependent clump model. That is, instead of assuming fixed clump conditions (density, mass, size), we consider an evolving, accreting clump assuming a constant accretion rate. In order to explore the parameter space, we have conducted calculations for various accretion rates, typical of the observed determinations, assuming that the clump parameters obey the standard Larson scaling relations. The resulting core mass spectrum remains in very good agreement with the observationally derived distribution and reproduces various behaviors observed in simulations of converging flows. We also investigate the core mass spectrum that results from the contribution of a clump distribution such that dN_c/dM_c∝M^−1.7. We show that the resulting core mass spectrum is dominated by the mass spectrum of the biggest clumps because they contain most of the mass.

Our SFR model provides the general framework that includes the simplified model suggested by Lada et al. (2012), which relies on the fraction of dense (molecular) gas in the cloud and thus critically depends on a threshold density. As mentioned above (and shown in Figure 7), our theory naturally predicts a strong, roughly quadratic correlation between the SFR and the gas density, predicted to become dominantly molecular above ∼5000 cm⁻³. As shown in Section 4.3.1, such a correlation immediately implies a similar correlation with the mass of the clump/cloud itself, i.e., with the mass of molecular hydrogen above the aforementioned value. The model suggested by Lada et al. (2012) is thus a direct consequence of our general theory of the IMF+SFR. As mentioned above, however, star formation is still predicted to take place, although at a much lower rate, in large, low-density clouds, which will be composed essentially of atomic hydrogen.

However, in contrast to what has often been claimed in the literature, we show that there is no universal value of the SFR, and that this latter does not simply correlate linearly with gas density. Indeed, as mentioned above, the SFR strongly depends not only on the gas density but also on the clump mass/size, which leads to a large scatter in SFR values.

The analytical theory described in our previous and present papers, which correctly reproduces various observational constraints, as discussed in these papers, suggests the following paradigm for star formation and for the resulting CMF/IMF.

• 1.  
  Compressible (shock dominated) large-scale turbulence in the cloud, due to various possible mechanisms, such as accretion, converging cold and warm flows, or star formation itself, generates a field of density fluctuations at all scales in the cloud. The (nearly lognormal) PDF of these fluctuations is entirely determined by the characteristic (universal) log-density power spectrum index of turbulence. This PDF leads to overdense regions that correspond to the observed clump mass spectrum (see Paper I). Observations suggest that the clumps have a filamentary structure. As they accrete mass and dissipate kinetic energy, the densest clumps become gravitationally unstable above a typical density ${\bar{N}}_{{\rm H_2}}\approx 10^{22}\,{\rm cm}^{-2}$ (i.e., ${\bar{n}}\approx 10^4\,(R_c/1\,{\rm pc})^{-1}\, {\rm cm}^{-3}$ for a spherical clump) for a temperature T ∼ 10 K, triggering the fragmentation into prestellar cores. Although probably quantitatively affecting our results by numerical factors of the order of unity, this filamentary nature of the clumps does not modify the general framework of our theory, which indeed predicts a very narrow CMF below about this density for pc-size clumps, yielding a negligible number of prestellar cores.
• 2.  
  Fragmentation then introduces a scale dependence (and thus a density dependence) in the process and sets up the virial criterion for gravitational collapse (see Paper I). As demonstrated in Paper I and in Chabrier & Hennebelle (2011), we suggest that turbulence plays a crucial role in setting the massive initial mass reservoir distribution (and thus the Salpeter slope of the IMF), progenitors of massive stars, above about the mean thermal Jeans mass, at the early stages of star formation, not by providing a pressure support, in a static sense, but by dispersing the gas within these structures, which would otherwise have collapsed, until they reach a mass that we identify as their turbulent Jeans mass (see Equation (28)).
• 3.  
  Below about the typical density mentioned above, our theory predicts a drastically decreasing CMF, and thus SFR, for the typical size of the dominant clump population, essentially because the characteristic Jeans length of the clump becomes comparable to or larger than its size, stabilizing the clump against fragmentation. For a filament, the same density corresponds to the critical mass per unit length below which the clump is gravitationally stable, yielding the same conclusion. We predict, however, that star formation can still occur below this density in very large (≳ 10) pc size) clumps, large enough to significantly exceed their typical Jeans length and generate large enough turbulence levels. This seems to be supported by the observed population of Class II cores in low-density, large clumps (Gutermuth et al. 2011) and stresses the need to observe very large areas at low density in order to get statistically significant core detections. Such clumps, however, are very seldom, as inferred from the steeply decreasing clump size spectrum, and the probability of finding prestellar cores in low-density environments drops drastically, making their detection very difficult.
• 4.  
  Star formation is thus a continuous process and can occur, statistically speaking, in any density environment, as any turbulence-induced overdense region can collapse, if dominated by gravity, and eventually produce a prestellar core. There is thus no real "threshold" for star formation. This leads to a direct correspondence between the SFR and the gas (atomic + molecular) density, as observed and indeed predicted by the theory, with star formation occurring most actively in the densest regions of molecular clouds, which entails only a small (≲ 20%) fraction of their mass. However, as mentioned above, star formation is basically choked off below ${\bar{n}}\sim 1000 \, {\rm cm}^{-3}$ for the typical (∼pc) size of most clumps. Above this density, the H₂ formation rate, which increases as the square of the gas density, dominates the photodissociation rate and the gas quickly becomes dominantly and entirely molecular. This drastically decreases the photoelectric heating efficiency, causing in turn a drop in dust and gas temperature (Tielens & Hollenbach 1985; Glover & Mac Low 2007). Star formation thus promotes H₂ formation, and not the opposite.

We stress, however, that from the general point of view, star formation strongly depends on the clump characteristic properties (mass, size); therefore, there is no "universal" relation between SFR and gas density, but instead large variations depending on the clump's environment.

This paradigm, which relies on the results presented in the present and former papers, shows that an SFR determined at the early stages by turbulence-induced fluctuations provides quite a consistent picture of star formation in Milky Way molecular clouds, with star formation occurring dominantly in the densest regions of the cloud. This gravo-turbulent picture of star formation correctly predicts the observed CMF over the entire mass range from brown dwarfs to massive stars and naturally leads to an SFR versus clump masses/sizes and gas density correlations in very good agreement with observational determinations. We stress, however, that ill-defined quantities such as the size of the clump or the dynamical efficiency of core-to-star mass conversion, which may well vary from clump to clump, make the determination of the SFR uncertain by a factor of a few.