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Massive star formation in 100,000 years from turbulent and pressurized molecular clouds

Massive stars (with mass m_*>8 solar masses) are fundamental to the evolution of galaxies, because they produce heavy elements, inject energy into the interstellar medium, and possibly regulate the star formation rate. The individual star formation time, t_*f, determines the accretion rate of the star; the value of the former quantity is currently uncertain by many orders of magnitude, leading to other astrophysical questions. For example, the variation of t_*f with stellar mass dictates whether massive stars can form simultaneously with low-mass stars in clusters. Here we show that t_*f is determined by conditions in the star's natal cloud, and is typically ~10^5 yr. The corresponding mass accretion rate depends on the pressure within the cloud - which we relate to the gas surface density - and on both the instantaneous and final stellar masses. Characteristic accretion rates are sufficient to overcome radiation pressure from ~100 solar mass protostars, while simultaneously driving intense bipolar gas outflows. The weak dependence of t_*f on the final mass of the star allows high- and low-mass star formation to occur nearly simultaneously in clusters.

are far greater than those associated with regions of low-mass star formation, but are comparable to the central surface density in the Orion Nebula Cluster 9 , which is about 1 g cm −2 . These regions have very high mean pressures P : where k B is Boltzmann's constant and f gas is the fraction of the cloud's mass that is in gas, as opposed to stars. They have power-law density profiles 10 ρ ∝ r −kρ , with k ρ ≃ 1.5 ± 0.5.
Molecular clouds are observed to be inhomogeneous on a wide range of scales, and numerical simulations show that this is a natural outcome of supersonic turbulence 11 . Dense, self-gravitating inhomogeneities that are destined to become stars are termed cores. We assume that the distribution of core masses determines the resulting distribution of stellar masses (the initial mass function) and we take this distribution as given.
Because of the disruptive effects of protostellar outflows, only a fraction ǫ core of the total core mass M core ends up in the star: m * f = ǫ core M core , where m * f is the final stellar mass (for binary and other multiple star systems, m * f is the total mass of stars in the system). To estimate ǫ core for massive stars, we assume that their outflows are scaled versions of the outflows from low-mass stars, which is qualitatively consistent with observation 12 . For low-mass stars, protostellar outflows typically eject 25-75% of the core mass 13 . A similar analysis applied to high-mass stars yields comparable results (J.C.T. & C.F.M., in prep.), so we adopt ǫ core = 0.5 as a typical value.
Our fundamental assumption is that star-forming clumps and the cores embedded within them are each part of a self-similar, self-gravitating turbulent structure that is virialized-that is, in approximate hydrostatic equilibrium-on all scales above the thermal Jeans mass. The density and pressure are then power laws in radius (for the pressure, P ∝ r −kP ), so that the cloud is a polytrope with P ∝ ρ γp . In be the effective sound speed; if the pressure is dominated by turbulent motions, then c is proportional to the line width. Molecular clouds and cloud cores satisfy a line width-size relation in which c increases outwards 15 , corresponding to γ p < 1. The equation of hydrostatic equilibrium gives M = k P c 2 r/G for the mass inside r, and ρ = Ac 2 /(2πGr 2 ) for the density at r, where A = γ p (4 − 3γ p )/(2 − γ p ) 2 . It is then immediately possible to determine the properties of a core in terms of the pressure at its surface and the mass of the star that will form within it (see Fig. 1). The radius of a core is then r s = 0.074(m * f /30 M ⊙ ) 1/2 Σ −1/2 pc; recall that the typical clump observed in ref. 7 has a radius of 0.5 pc, which is set by the angular resolution of those observations. The mean density in a core isn H = 1.0 × 10 6 (m * f /30 M ⊙ ) −1/2 Σ 3/2 cm −3 , and the r.m.s. velocity dispersion is 1.65(m * f /30 M ⊙ ) 1/4 Σ 1/4 km s −1 . The thermal sound speed at temperature T is 0.3(T /30 K) 1/2 km s −1 , and so a high-mass core is dominated by supersonic turbulence and should be very clumpy.
We now consider the timescale for a star to form in such a core. On dimensional grounds, we expect the protostellar accretion rate to be 16 : so long as radiation pressure is not important. Here m * is the instantaneous stellar mass, is the free-fall time and φ * is a dimensionless constant of order unity. This equation could be violated in the sense that φ * ≫ 1 only in the unlikely case that the star forms from a coherent spherical implosion; if the star formation is triggered by an external increase in pressure, φ * could be increased somewhat, but deviations from spherical symmetry in the triggering impulse and in the protostellar core will generally prevent φ * from becoming too large. It could be violated in the opposite sense that φ * ≪ 1 if the core is magnetically dominated, so that collapse could not begin until the magnetic field diffused out of the core.
However, magnetic fields are not observed to be dominant in molecular cores 17 . Equation (2) implies that the accretion rate, and thus the star formation time t * f ∝ m * /ṁ * , depends weakly on the properties of the ambient medium,ṁ * ∝ ρ 1/2 .
We now show that if the collapse is spherical and self-similar, then φ * is quite close to unity provided that radiation pressure does not disrupt the flow. Although we assume the collapse is spherical far from the star, it will naturally proceed via a disk close to the star owing to the angular momentum of the accreting material; we assume this does not limit the flow of matter onto the star as otherwise the disk would become very massive and gravitationally unstable. Shielding by the disk reduces the importance of radiation pressure on the accretion flow, and allows the formation of massive stars provided the accretion rate is sufficiently large 18 . Under the assumption of a polytropic structure in hydrostatic equilibrium, equation (2) implies: where the value of ρ entering t ff is evaluated at the radius in the initial cloud that just encloses the gas that goes into the star when its mass is m * . For the isothermal case (γ p = 1) and for ǫ core = 1, this reduces to the classic result of Shu 19 show that the accretion ratė m * is proportional to t 3−3γp , so that for γ p < 1, as observed, the accretion rate accelerates 2,4 . As discussed in ref. 2, termination of the accretion breaks the self-similarity once the expansion wave encloses m * f , which occurs at t ∼ 0.45t * f . Thereafter, the relationṁ * ∝ t 3−3γp becomes approximate, but the approximation should be reasonable good 2 . The star-formation time is then given by: Evaluating φ * (see Fig. 1), we conclude thatṁ * ≃ m * /t ff to within a factor 1.5 for spherical cores in which the effective sound speed increases outward.
We can express the accretion rate in terms of the mean pressure of the clump in which the star forms, P (eq. 1), and the final mass of the star, m * f , where φ P ≡ P s /P is the ratio of the core's surface pressure to the mean pressure in the clump and j ≡ 3(2 − k ρ )/[2(3 − k ρ )]; for k ρ = 3/2, j = 1/2. This typical accretion rate is large because the core is embedded in a high-pressure environment, so that in hydrostatic equilibrium it has a high density and short free- Let P s = φ P P be the surface pressure of a core. The properties at the surface of the core are given by r s = [AGm 2 * f /(2πk 2 P ǫ 2 core P s )] 1/4 , ρ s = [Ak 2 P ǫ 2 core P 3 s /(2πG 3 m 2 * f )] 1/4 , and c s = [2πG 3 m 2 * f P s /(Ak 2 P ǫ 2 core )] 1/8 . We anticipate that the overall star formation efficiency in the clump will be relatively high, so in equation (1) we adopt f gas = 2/3 as a fiducial value. We estimate φ P = P s /P ≃ 2 (C.F.M. & J.C.T., in prep.), and we set α vir equal to unity 7 . We take k ρ = 1.5 as a typical value, corresponding to γ p = 2/3, k P = 1 and A = 3/4. Following equation (3), we evaluate where m 0 is tabulated in their paper. Over the entire range of γ p and k ρ relevant to molecular clouds 48k ρ to within about 1%. The star formation time decreases from 3.5t ff to 1.5t ff as γ p varies from 0 to 1. The variation of t * f andṁ * f relative to the k ρ = 1.5 case is also shown. Note that the singular polytropic model in hydrostatic equilibrium breaks down for k ρ = 1, γ p = 0, since then the pressure gradient vanishes (k P = 0).