Letter

Nature 438, 332-334 (17 November 2005) | doi:10.1038/nature04280; Received 12 August 2005; Accepted 29 September 2005

The formation of stars by gravitational collapse rather than competitive accretion

Mark R. Krumholz1, Christopher F. McKee2,3 & Richard I. Klein3,4

  1. Astrophysics Department, Princeton University, Princeton, New Jersey 08544, USA
  2. Physics Department,
  3. Astronomy Department, UC Berkeley, Berkeley, California 94720, USA
  4. Lawrence Livermore National Laboratory, Livermore, California 94550, USA

Correspondence to: Mark R. Krumholz1 Correspondence and requests for materials should be addressed to M.R.K. (Email: krumholz@astro.princeton.edu).

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There are two dominant models of how stars form. Under gravitational collapse, star-forming molecular clumps, of typically hundreds to thousands of solar masses (Mcircle dot), fragment into gaseous cores that subsequently collapse to make individual stars or small multiple systems1, 2, 3. In contrast, competitive accretion theory suggests that at birth all stars are much smaller than the typical stellar mass (approx0.5Mcircle dot), and that final stellar masses are determined by the subsequent accretion of unbound gas from the clump4, 5, 6, 7, 8. Competitive accretion models interpret brown dwarfs and free-floating planets as protostars ejected from star-forming clumps before they have accreted much mass; key predictions of this model are that such objects should lack disks, have high velocity dispersions, form more frequently in denser clumps9, 10, 11, and that the mean stellar mass should vary within the Galaxy8. Here we derive the rate of competitive accretion as a function of the star-forming environment, based partly on simulation12, and determine in what types of environments competitive accretion can occur. We show that no observed star-forming region can undergo significant competitive accretion, and that the simulations that show competitive accretion do so because the assumed properties differ from those determined by observation. Our result shows that stars form by gravitational collapse, and explains why observations have failed to confirm predictions of the competitive accretion model.

In both theories, a star initially forms when a gravitationally bound gas core collapses. The crucial distinction between them is their prediction for what happens subsequently. In gravitational collapse, after a protostar has consumed or expelled all the gas in its initial core, it may continue accreting from its parent clump. However, it will not accrete enough to change its mass substantially13, 14. In contrast, competitive accretion requires that the amount accreted after the initial core is consumed be substantially larger than the protostellar mass. We define Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com as the fractional change in mass that a protostar of mass m* undergoes each dynamical time tdyn of its parent clump, starting after the initial core has been consumed. Gravitational collapse holds that fmless double1, whereas competitive accretion requires fmdouble greater than1.

We consider a protostar embedded in a molecular clump of mass M and mass-weighted one-dimensional velocity dispersion sigma. Competitive accretion theories usually begin with seed protostars of mass m* approximately 0.1Mcircle dot (refs 4, 5, 6–7), so we adopt this as a typical value. We consider two possible geometries: spherical clumps of radius R and filaments of radius R and length L, where Ldouble greater thanR. These extremes bracket real star-forming clumps, which have a range of aspect ratios. The virial mass for (spherical, filamentary) clumps is:

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

and the virial parameter is alphavir equivalent to Mvir/M (refs 15, 16). The dynamical time is tdyn equivalent to R/sigma.

First, we suppose that the gas that the protostar is accreting is not collected into bound structures on scales smaller than the entire clump. Because the gas is unbound, we may neglect its self-gravity and treat this as a problem of a non-self-gravitating gas accreting onto a point particle. This process is Bondi–Hoyle accretion in a turbulent medium, which gives an accretion rate12:

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

where Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com is the mean density in the clump. The factor phiBH represents the effects of turbulence, which we estimate in terms of parameters sigma, m* and R in the Supplementary Information12. From equation (2) and the definitions of the virial parameter and the dynamical time, we find that accretion of unbound gas gives:

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

for a (spherical, filamentary) star-forming region. From this result, we can immediately see that competitive accretion is most effective in low-mass clumps with virial parameters much smaller than unity.

Tables 1 and 2 show a broad sample of observed star-forming regions. None of them have a value of fm–BH near unity, which is inconsistent with competitive accretion and in agreement with gravitational collapse. We note that the Bondi–Hoyle rate is an upper limit on the accretion. If the stars are sufficiently close-packed, their tidal radii will be smaller than their Bondi–Hoyle radii, and the accretion rate will be lower5. Also, radiation pressure will halt Bondi–Hoyle accretion onto stars larger than approx10Mcircle dot (ref. 17).



The second possible way that a star could gain mass is by capturing and accreting other gravitationally bound cores. We can analyse this process by some simple approximations. First, when a star collides with a core it begins accreting gas from it, causing a drag force18. If drag dissipates enough energy, the two become bound. We can therefore compute a critical velocity below which any collision will lead to a capture and above which it will not. Second, cores and stars should inherit the velocity dispersion of the gas from which they form, so we assume they have maxwellian velocity distributions with dispersion sigma. The true functional form may be different, but this will only affect our estimates by a factor of order unity. Third, we neglect the range of core sizes, and assume that all cores have a generic radius Rco and mass Mco. Competitive accretion requires Mco less than or equal to m*, so we take Mco = m*, which gives the highest possible capture rate. Finally, we make use of an important observational result: cores within a molecular clump have roughly the same surface density as the clump itself19, that is, Sigma = M(piR2, 2RL)-1 for (spherical, filamentary) clumps. This enables us to compute the escape velocity from the surface of a core in terms of the properties of the clump:

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

With these approximations, it is straightforward to compute the amount of mass that a protostar can expect to gain by capturing other cores. In the Supplementary Information, we show that it is:

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

where phico is the fraction of the parent clump mass that is in bound cores and uesc equivalent to vesc/sigma. Surveys generally find core mass fractions of phico approximately 0.1 (refs 20, 21–22), so we adopt this as a typical value, giving the numerical values of fm–cap shown in Table 2. As with fm–BH, all the estimated values are well below unity.

If we let fm = fm–BH + fm–cap, then we can use our simple models to determine where in parameter space a star-forming clump must fall to have fm greater than or equal to 1. For simplicity, we consider a spherical clump with fixed phiBH = 5 and phico = 0.1 (typical values for observed regions), and a seed protostar of mass m* = 0.1. In this case, both fm–BH and fm–cap are functions of Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com alone, and we find fm greater than or equal to 1 for Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com. The functional dependence is more complex if we include filamentary regions and allow phiBH and phico to vary, but the qualitative result is unchanged. Observed star-forming regions have alphavir approximately 1 and M approximately 102 - 104Mcircle dot (ref. 23), which produces fmless double1. No known star-forming region has Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com small enough for competitive accretion to work. Thus, the cores from which stars form must contain all the mass they will ever have, which is the gravitational collapse model.

Our simple estimate of fm is consistent with simulations of competitive accretion as well, and explains why competitive accretion works in the simulations. All competitive accretion simulations have virial parameters alphavirless double1. In some cases the simulations start in this condition5, 6, 24, 25, with alphavir approximately 0.01 as a typical choice. In other cases, the virial parameter is initially of order unity, but as turbulence decays in the simulation it decreases to less double1 in roughly a crossing time7, 9, 10, 26. Once competitive accretion gets going, these simulations reach alphavirless double1 as well. In addition, many of the simulations consider star-forming clumps of masses considerably smaller than the approx5,000Mcircle dot typical of most galactic star formation23, with Mless than or similar to100Mcircle dot not uncommon. Consequently, the simulations have Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com, which explains why they find competitive accretion to be important. Note that simulations where turbulence decays will have phiBH approximately 1, rather than the typical value of phiBH = 5 we have used for real regions, but this does not substantially modify our conclusions.

Three other aspects of the simulations increase even further their estimate of fm. First, the Bondi–Hoyle radius of a 0.1Mcircle dot seed protostar in a typical clump is only 5 au (astronomical units), a smaller scale than any of the competitive accretion simulations resolve. This under-resolution may enhance accretion12. Second, small virial parameters lead most of the mass to collapse to stars, giving phico approximately 0.5–1 after a dynamical time, and also tend to make the cloud fragment into smaller pieces, lowering M. Third, rapid collapse leaves no time for large cores to assemble. For example, one simulation of a approx1,000Mcircle dot clump produces no cores larger than 1Mcircle dot (ref. 7), inconsistent with observations that find numerous cores more massive than this in similar regions22, 27. With no large cores, large stars can form only via competitive accretion.

Thus, our results are consistent with the simulations, but they show that the simulations are not modelling realistic star-forming clumps. One might argue that all clumps do enter a phase with alphavirless double1 that occurs rapidly and has therefore never been observed, but that most stars are formed during this collapse phase. In this scenario, though, protostars associated with observed star-forming regions should have systematically lower masses than the field star population, because they were formed before the collapse phase in which competitive accretion might occur. We would expect to see a systematic variation in mean stellar mass with age in young clusters, corresponding to cluster evolution into a state more and more favourable to competitive accretion. We do not observe this.

We hypothesize that the primary problem with the simulations—the reason they evolve to alphavirless double1—is that they omit feedback from star formation. Recent observations of protostellar outflow cavities show that outflows inject enough energy to sustain the turbulence and prevent the virial parameter from declining to values much less than unity28. Another possible problem in the simulations is that they simulate isolated clumps containing too little material. Real clumps are embedded in molecular clouds, and large-scale turbulent motions in the clouds may cascade down to the clump scale and prevent the turbulence from decaying. A third possibility is that turbulence decays too quickly in the simulations because they do not include magnetic fields and their initial velocity fields, unlike in real clumps, are balanced rather than imbalanced between left- and right-propagating modes29.

One implication of our work is that brown dwarfs need not have been ejected from their natal clump, so their velocity dispersions should be at most slightly greater than those of stars, and their frequency need not change as a function of clump density. This also removes a discrepancy between observations showing that brown dwarfs have disks11 and theoretical models of their origins. We also conclude that the mean stellar mass need not vary from one star-forming region to another as competitive accretion predicts, removing a discrepancy between theory8 and observations that have thus far failed to find any substantial variation in typical stellar mass with the star-forming environment. In the gravitational collapse scenario, the mean stellar mass may be roughly constant in the Galaxy, but may vary with the background radiation field in starburst regions and in the early Universe3.

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Supplementary Information

Supplementary information accompanies this paper.

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Acknowledgements

We thank R. T. Fisher for discussions and P. Padoan for comments. This work was supported by grants from NASA through the Hubble Fellowship, GSRP and ATP programmes, by the NSF, and by the US DOE through the Lawrence Livermore National Laboratory. Computer simulations for this work were performed at the San Diego Supercomputer Center (supported by the NSF), the National Energy Research Scientific Computer Center (supported by the US DOE), and Lawrence Livermore National Laboratory (supported by the US DOE). M.R.K. is a Hubble Fellow.

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Competing interests statement

The authors declare no competing financial interests.

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