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Control of star formation by supersonic turbulence
Mordecai-Mark Mac Low
Department of Astrophysics, American Museum of Natural History,
79th Street at Central Park West, New York, NY 10024-5192, USA ∗
Ralf S. Klessen
Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany†
and UCO/Lick Observatory, University of California, Santa Cruz, CA 95064, USA
Understanding the formation of stars in galaxies is central to much of modern astrophysics. How-
ever, a quantitative prediction of the star formation rate and the initial distribution of stellar
masses remains elusive. For several decades it has been thought that the star formation process
is primarily controlled by the interplay between gravity and magnetostatic support, modulated
by neutral-ion drift (known as ambipolar diffusion in astrophysics). Recently, however, both ob-
servational and numerical work has begun to suggest that supersonic turbulent flows rather than
static magnetic fields control star formation. To some extent, this represents a return to ideas
popular before the importance of magnetic fields to the interstellar gas was fully appreciated. This
review gives a historical overview of the successes and problems of both the classical dynamical
theory, and the standard theory of magnetostatic support from both observational and theoretical
perspectives. The outline of a new theory relying on control by driven supersonic turbulence is
then presented. Numerical models demonstrate that although supersonic turbulence can provide
global support, it nevertheless produces density enhancements that allow local collapse. Inefficient,
isolated star formation is a hallmark of turbulent support, while efficient, clustered star formation
occurs in its absence. The consequences of this theory are then explored for both local star forma-
tion and galactic scale star formation. It suggests that individual star-forming cores are likely not
quasi-static objects, but dynamically collapsing. Accretion onto these objects varies depending on
the properties of the surrounding turbulent flow; numerical models agree with observations show-
ing decreasing rates. The initial mass distribution of stars may also be determined by the turbulent
flow. Molecular clouds appear to be transient objects forming and dissolving in the larger-scale
turbulent flow, or else quickly collapsing into regions of violent star formation. We suggest that
global star formation in galaxies is controlled by the same balance between gravity and turbu-
lence as small-scale star formation, although modulated by cooling and differential rotation. The
dominant driving mechanism in star-forming regions of galaxies appears to be supernovae, while
elsewhere coupling of rotation to the gas through magnetic fields or gravity may be important.
Accepted for publication in Reviews of Modern Physics
CONTENTS
I. INTRODUCTION 3
A. Overview 3
B. Turbulence 5
C. Outline 6
II. OBSERVATIONS 7
A. Composition of molecular clouds 7
B. Density and velocity structure of molecular clouds 8
C. Support of molecular clouds 10
D. Scaling relations for molecular clouds 11
E. Protostellar cores 12
1. From cores to stars 12
2. Properties of protostellar cores 13
F. The observed IMF 14
III. HISTORICAL DEVELOPMENT 17
A. Classical dynamical theory 18
B. Problems with classical theory 20
C. Standard theory of isolated star formation 22
∗Electronic address: mordecai@amnh.org
†Electronic address: rklessen@aip.de
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D. Problems with standard theory 24
1. Singular isothermal spheres 25
2. Observations of clouds and cores 26
a. Magnetic Support 26
b. Infall Motions 27
c. Density profiles 28
d. Chemical ages 29
3. Protostars and young stars 29
a. Accretion rates 29
b. Embedded Objects 30
c. Stellar Ages 30
IV. TOWARD A NEW PARADIGM 31
A. Maintenance of supersonic motions 31
B. Turbulence in self-gravitating gas 32
C. A numerical approach 32
D. Global collapse 34
E. Local collapse in globally stable regions 34
F. Effects of magnetic fields 37
G. Promotion and prevention of local collapse 40
H. The timescales of star formation 41
I. Scales of interstellar turbulence 42
J. Termination of local star formation 43
K. Outline of a new theory of star formation 43
V. LOCAL STAR FORMATION 44
A. Star formation in molecular clouds 44
B. Protostellar core models 46
C. Binary formation 48
D. Dynamical interactions in clusters 49
E. Accretion rates 50
F. Initial mass function 52
1. Models of the IMF 52
2. Turbulent fragmentation example 53
VI. GALACTIC SCALE STAR FORMATION 54
A. Formation and lifetime of molecular clouds 54
B. When is star formation efficient? 58
1. Overview 58
2. Gravitational instabilities in galactic disks 60
3. Thermal instability 61
C. Driving mechanisms 62
1. Magnetorotational instabilities 63
2. Gravitational instabilities 63
3. Protostellar outflows 64
4. Massive stars 65
a. Stellar winds 65
b. Ionizing radiation 65
c. Supernovae 66
D. Applications 67
1. Low surface brightness galaxies 67
2. Galactic disks 67
3. Globular clusters 68
4. Galactic nuclei 68
5. Primordial dwarfs 69
6. Starburst galaxies 69
VII. CONCLUSIONS 69
A. Summary 69
B. Future research problems 71
ACKNOWLEDGMENTS 73
References 73
2

I. INTRODUCTION
A. Overview
Stars are important. They are the dominant source of radiation (with competition from the cosmic microwave
background and from accretion onto black holes, which themselves probably formed from stars), and of all chemical
elements heavier than the H, He, and Li that made up the primordial gas. The Earth itself consists mainly of these
heavier elements, called metals in astronomical terminology. Metals are produced by nuclear fusion in the interior
of stars, with the heaviest elements produced during the passage of the final supernova shockwave through the most
massive stars. To reach the chemical abundances observed today in our solar system, the material had to go through
many cycles of stellar birth and death. In a literal sense, we are star dust.
Stars are also our primary source of astronomical information and, hence, are essential for our understanding of
the universe and the physical processes that govern its evolution. At optical wavelengths almost all natural light we
observe in the sky originates from stars. During day this is obvious, but it is also true at night. The Moon, the second
brightest object in the sky, reflects light from our Sun, as do the planets, while virtually every other extraterrestrial
source of visible light is a star or collection of stars. Throughout the millenia, these objects have been the observational
targets of traditional astronomy, and define the celestial landscape, the constellations.
FIG. 1. Optical image of the spiral galaxy NGC4622 observed with the Hubble Space Telescope. (Courtesy of NASA and The
Hubble Heritage Team — STScI/AURA)
When we look at the sky on a clear night, we can also note dark patches of obscuration along the band of the Milky
Way. These are clouds of dust and gas that block the light from stars further away. For roughly the last century we
have known that these clouds give birth to stars. The advent of new observational instruments made it possible to
observe astronomical objects at wavelengths ranging from γ-rays to radio frequencies. Especially useful for studying
the dark clouds are radio, sub-millimeter and far-infrared wavelengths, at which they are transparent. Observations
now show that all star formation occurring in the Milky Way is associated with the dark clouds of molecular hydrogen
and dust.
Stars are common. The mass of the Galactic disk plus bulge is about 6 × 1010 M⊙ (e.g. Dehnen & Binney 1998),
where 1M⊙ = 1.99× 1033 g is the mass of our Sun. Thus, there are of order 1012 stars in the Milky Way, assuming
standard values for the stellar mass distribution (e.g. Kroupa 2002). Stars form continuously. Roughly 10% of the
disk mass of the Milky Way is in the form of gas, which is forming stars at a rate of about 1M⊙ yr−1. Although stars
dominate the baryonic mass in the Galaxy, dark matter determines the overall mass budget: invisible material that
3

reveals its presence only by its contribution to the gravitational potential. The dark matter halo of our Galaxy is
about ten times more massive than gas and stars together. At larger scales this imbalance is even more pronouced.
Stars are estimated to make up only 0.4% of the total mass of the Universe (Lanzetta, Yahil, & Fernandez-Soto 1996),
and about 17% of the total baryonic mass (Walker et al. 1991).
Mass is the most important parameter determining the evolution of individual stars. Massive stars with high
pressures at their centers have strong nuclear fusion there, making them short-lived but very luminous, while low-
mass stars are long-lived but extremely faint. For example, a star with 5M⊙ only lives for 2.5× 107 yr, while a star
with 0.2M⊙ survives for 1.2×1013 yr, orders of magnitude longer than the current age of the universe. For comparison
the Sun with an age of 4.5 × 109 yr has reached approximately half of its life span. The relationship between mass
and luminosity is quite steep with roughly L ∝ M3.2 (Kippenhahn & Weigert 1990). During its short life a 5M⊙ star
will shine with a luminosity of 1.5× 104 L⊙, while the luminosity of an 0.2M⊙ star is only ∼ 10−3 L⊙. For reference,
the luminosity of the Sun is 1 L⊙ = 3.85× 1033erg s−1.
The light from star-forming external galaxies in the visible and blue wavebands is dominated by young, massive
stars. This is the reason why we observe beautiful spiral patterns in many disk galaxies, like NGC4622 shown in Figure
1, as spiral density waves lead to gas compression and subsequent star formation at the wave locations. Massive stars
dominate the optical emission from external galaxies. In their brief lifetimes, massive stars do not have sufficient time
to disperse in the galactic disk, so they still trace the characteristics of the instability that triggered their formation.
Hence, understanding the dynamical properties of galaxies requires an understanding of how, where, and under which
conditions stars form.
In a simple approach, galaxies can be seen as gravitational potential wells containing gas that has been able to
radiatively cool in less than the current age of the universe. In the absence of any hindrance, the gas then collapses
gravitationally to form stars on a free-fall time (Jeans 1902)
τff =
(
3π
32Gρ
)1/2
= 140 Myr
( n
0.1 cm−3
)−1/2
, (1)
where n is the number density of the gas. Interstellar gas in the Milky Way consists of one part He for every ten parts
H. The mass density ρ = µn, where we take the Galactic value for the mean mass per particle in neutral atomic gas
of µ = 2.11× 10−24 g, and G is the gravitational constant. The free-fall time τff is very short compared to the age of
the Milky Way, about 1010 yr. However, gas remains in the Galaxy and stars continue to form from gas that must
have already been cooled below its virial temperature for many billions of years. What physical processes regulate
the rate at which gas turns into stars? Another way of asking the question is, what prevented the Galactic gas from
forming stars at an extremely high rate immediately after it first cooled, and being completely used up?
Observations of the star formation history of the universe demonstrate that stars did indeed form more vigorously
in the past than today (e.g. Lilly et al. 1996, Madau et al. 1996, Baldry et al. 2002, Lanzetta et al. 2002), with as
much as 80% of star formation in the Universe being complete by redshift z = 1, less than half of the current age of
13 Gyr ago. What mechanisms allowed rapid star formation in the past, but reduce its rate today?
The clouds of gas and dust in which stars form are dense enough, and well enough protected from dissociating
UV radiation by self-shielding and dust scattering in their surface layers, for hydrogen to be mostly in molecular
form in their interior. The density and velocity structure of these molecular clouds is extremely complex and follows
hierarchical scaling relations that appear to be determined by supersonic turbulent motions (e.g. Blitz & Williams
1999). Molecular clouds are large, and their masses exceed the threshold for gravitational collapse by far when taking
only thermal pressure into account. Just like galaxies as a whole, naively speaking, they should be contracting rapidly
and forming stars at very high rate. This is generally not observed. We can define a star formation efficiency of a
region as
ǫSF = M˙∗τ/M, (2)
where M˙∗ is the star formation rate, τ is the lifetime of the region, and M is the total gas mass in the region (e.g.
Elmegreen & Efremov 1997). The star formation efficiency of molecular clouds in the solar neighborhood is estimated
to be of order of a few percent (Zuckerman & Evans 1974).
For many years it was thought that support by magnetic pressure against gravitational collapse offered the best
explanation for the slow rate of star formation. In this theory, developed by Shu (1977; and see Shu, Adams, &
Lizano 1987), Mouschovias (1976; and see Mouschovias 1991b,c), Nakano (1976), and others, interstellar magnetic
field prevents the collapse of gas clumps with insufficient mass to flux ratio, leaving dense cores in magnetohydrostatic
equilibrium. The magnetic field couples only to electrically charged ions in the gas, though, so neutral atoms can only
be supported by the field if they collide frequently with ions. The diffuse interstellar medium (ISM) with number
4

densities n ≃ 1 cm−3 (see Ferrie`re 2001 for a general review of ISM properties) remains highly ionized, enough
that neutral-ion collisional coupling is very efficient (as we discuss below in Section III.C). In dense cores, where
n > 105 cm−3, ionization fractions drop below parts per ten million. Neutral-ion collisions no longer couple the
neutrals tightly to the magnetic field, so the neutrals can diffuse through the field. This neutral-ion drift allows
gravitational collapse to proceed in the face of magnetostatic support, but on a timescale as much as an order of
magnitude longer than the free-fall time, drawing out the star formation process.
In this paper we review a body of work that suggests that magnetohydrostatic support modulated by neutral-ion
drift fails to explain the star formation rate, and indeed appears inconsistent with observations of star-forming regions.
Instead, we suggest that control of molecular cloud formation and subsequent support by supersonic turbulence is
both sufficient to explain star formation rates, and more consistent with observations. Our review focuses on how
gravitationally collapsing regions form. The recent comprehensive review by Larson (2003) goes into more detail on
the final stages of disk accretion and protostellar evolution.
B. Turbulence
At this point, we need to briefly discuss the concept of turbulence, and the differences between supersonic, com-
pressible (and magnetized) turbulence, and the more commonly studied incompressible turbulence. We mean by
turbulence, in the end, nothing more than the gas flow resulting from random motions at many scales. We further-
more will use in our discussion only the very general properties and scaling relations of turbulent flows, focusing mainly
on effects of compressibility. For a more detailed discussion of the complex statistical characteristics of turbulence,
we refer the reader to the book by Lesieur (1997).
Most studies of turbulence treat incompressible turbulence, characteristic of most terrestrial applications. Root-
mean-square (rms) velocities are subsonic, and the density remains almost constant. Dissipation of energy occurs
primarily in the smallest vortices, where the dynamical scale ℓ is shorter than the length on which viscosity acts
ℓvisc. Kolmogorov (1941a) described a heuristic theory based on dimensional analysis that captures the basic behavior
of incompressible turbulence surprisingly well, although subsequent work has refined the details substantially. He
assumed turbulence driven on a large scale L, forming eddies at that scale. These eddies interact to form slightly
smaller eddies, transferring some of their energy to the smaller scale. The smaller eddies in turn form even smaller
ones, until energy has cascaded all the way down to the dissipation scale ℓvisc.
In order to maintain a steady state, equal amounts of energy must be transferred from each scale in the cascade to
the next, and eventually dissipated, at a rate
E˙ = ηv3/L, (3)
where η is a constant determined empirically. This leads to a power-law distribution of kinetic energy E ∝ v2 ∝ k−11/3,
where k = 2π/ℓ is the wavenumber, and density does not enter because of the assumption of incompressibility. Most
of the energy remains near the driving scale, while energy drops off steeply below ℓvisc. Because of the apparently
local nature of the cascade in wavenumber space, the viscosity only determines the behavior of the energy distribution
at the bottom of the cascade below ℓvisc, while the driving only determines the behavior near the top of the cascade
at and above L. The region in between is known as the inertial range, in which energy transfers from one scale to
the next without influence from driving or viscosity. The behavior of the flow in the inertial range can be studied
regardless of the actual scale at which L and ℓvisc lie, so long as they are well separated. One statistical description
of incompressible turbulent flow, the structure functions Sp(~r) = 〈{v(~x)− v(~x+ ~r)}p〉, has been successfully modeled
by assuming that dissipation occurs in filamentary vortex tubes (She & Leveque 1994).
Gas flows in the ISM, however, vary from this idealized picture in three important ways. First, they are highly
compressible, with Mach numbers M ranging from order unity in the warm (104 K), diffuse ISM, up to as high as
50 in cold (10 K), dense molecular clouds. Second, the equation of state of the gas is very soft due to radiative
cooling, so that pressure P ∝ ργ with the polytropic index falling in the range 0.4 < γ < 1.2 as a function of density
and temperature (e.g. Spaans & Silk 2000, Ballesteros-Paredes, Va´zquez-Semadeni, & Scalo 1999, Scalo et al. 1998).
Third, the driving of the turbulence is not uniform, but rather comes from blast waves and other inhomogeneous
processes.
Supersonic flows in highly compressible gas create strong density perturbations. Early attempts to understand
turbulence in the ISM (von Weizsa¨cker 1943, 1951, Chandrasekhar 1949) were based on insights drawn from incom-
pressible turbulence. An attempt to analytically derive the density spectrum and resulting gravitational collapse
criterion was first made by Chandrasekhar (1951a,b). This work was followed up by several authors, culminating
5

in work by Sasao (1973) on density fluctuations in self-gravitating media whose interest has only been appreciated
recently. Larson (1981) qualitatively applied the basic idea of density fluctuations driven by supersonic turbulence
to the problem of star formation. Bonazzola et al. (1992) used a renormalization group technique to examine how
the slope of the turbulent velocity spectrum could influence gravitational collapse. This approach was combined with
low-resolution numerical models to derive an effective adiabatic index for subsonic compressible turbulence by Panis
& Pe´rault (1998). Adding to the complexity of the problem, the strong density inhomogeneities observed in the ISM
can be caused not only by compressible turbulence, but also by thermal phase transitions (Field, Goldsmith, & Habing
1969, McKee & Ostriker 1977, Wolfire et al. 1995) or gravitational collapse (e.g. Kim & Ostriker 2001).
In supersonic turbulence, shock waves offer additional possibilities for dissipation. Shock waves can also transfer
energy between widely separated scales, removing the local nature of the turbulent cascade typical of incompressible
turbulence. The spectrum may shift only slightly, however, as the Fourier transform of a step function representative
of a perfect shock wave is k−2. Integrating in three dimensions over an ensemble of shocks, the differential energy
spectrum E(k)dk = ρv2(k)k2dk ∝ k−2dk. This is just the compressible energy spectrum reported by Porter & Wood-
ward (1992) and Porter, Pouquet, & Woodward (1992, 1994). They also found that even in supersonic turbulence,
the shock waves do not dissipate all the energy, as rotational motions continue to contain a substantial fraction of the
kinetic energy, which is then dissipated in small vortices. Boldyrev (2002) has proposed a theory of velocity structure
function scaling based on the work of She & Leveque (1994) using the assumption that dissipation in supersonic
turbulence primarily occurs in sheet-like shocks, rather than linear filaments at the centers of vortex tubes. First
comparisons to numerical models show good agreement with this model (Boldyrev, Nordlund, & Padoan 2002a),
and it has been extended to the density structure functions by Boldyrev, Nordlund, & Padoan (2002b). Transport
properties of supersonic turbulent flows in the astrophysical context have been discussed by Avillez & Mac Low (2002)
and Klessen & Lin (2003).
The driving of interstellar turbulence is neither uniform nor homogeneous. Controversy still reigns over the most
important energy sources at different scales, but we make the argument in Section VI.C that isolated and correlated
supernovae dominate. However, it is not yet understood at what scales expanding, interacting blast waves contribute
to turbulence. Analytic estimates have been made based on the radii of the blast waves at late times (Norman &
Ferrara 1996), but never confirmed with numerical models (much less experiment). Indeed, the thickness of the blast
waves may be more important than the radii.
Finally, the interstellar gas is magnetized. Although magnetic field strengths are difficult to measure, with Zeeman
line splitting being the best quantitative method, it appears that fields within an order of magnitude of equipartition
with thermal pressure and turbulent motions are pervasive in the diffuse ISM, most likely maintained by a dynamo
driven by the motions of the interstellar gas (e.g. Ferrie`re 1992). A model for the distribution of energy and the scaling
behavior of strongly magnetized, incompressible turbulence based on the interaction of shear Alfve´n waves is given
by Goldreich & Sridhar (1995, 1997) and Ng & Bhattacharjee (1996). They found that an anisotropic Kolmogorov
spectrum k−5/3 best describes the one-dimensional (1D) energy spectrum, rather than the k−3/2 spectrum first
proposed by Iroshnikov (1963) and Kraichnan (1965). These results have been confirmed by Verma et al. (1996)
using numerical models, and by Verma (1999) using a renormalization group approach. The scaling properties of the
structure functions of such turbulence was derived from the work of She & Leveque (1994) by Mu¨ller & Biskamp
(2000; also see Biskamp & Mu¨ller 2000) by assuming that dissipation occurs in current sheets. A theory of weakly
compressible turbulence applicable in particular to small scales in the ISM has been derived by Lithwick & Goldreich
(2001), but little progress has been made towards analytic models of strongly compressible magnetohydrodynamic
(MHD) turbulence with M≫ 1. See, however, the reviews by Cho, Lazarian, & Vishniac (2002), and Cho & Lazarian
(2003). In particular, an analytic theory of the non-linear density fluctuations characteristic of such turbulence remains
lacking.
C. Outline
With the above in mind, we suggest that stellar birth is regulated by interstellar turbulence and its interplay with
gravity. Turbulence, even if strong enough to counterbalance gravity on global scales, will usually provoke collapse
on smaller scales. Supersonic turbulence establishes a complex network of interacting shocks, where converging flows
generate regions of high density. This density enhancement can be sufficient for gravitational instability. Collapse
sets in. However, the random flow that creates local density enhancements also may disperse them again. Hence,
the efficiency of star formation (eq. 2) depends strongly on the properties of the underlying turbulent velocity field,
on its driving lengthscale and strength relative to gravitational attraction. This principle holds for star formation
throughout all scales considered in this review, ranging from small star forming regions up to galaxies as a whole.
6

To lay out this picture of star formation in more detail, we first outline the observed properties of star-forming
interstellar clouds, and the distribution of stellar masses that form there in Section II. We then critically discuss
the historical development of star formation theory in Section III. We begin the section by describing the classical
dynamic theory, and then move on to the so-called standard theory, where the star formation process is controlled
by magnetic fields. After describing the theoretical and observational problems that both approaches have, we
present work in Section IV that leads us to the argument that star formation is controlled by the interplay between
gravity and supersonic turbulence. The theory is applied to individual star forming regions in Section V, where we
investigate the implications for stellar clusters, protostellar cores (the direct progenitors of individual stars), binary
stars, protostellar mass accretion, and the subsequent distribution of stellar masses. In Section VI, we discuss the
control of star formation by supersonic turbulence on galactic scales. We first examine the formation and destruction
of star-forming molecular clouds in light of models of turbulent flow. We then ask when is star formation efficient in
galaxies? We review the energetics of the possible mechanisms that generate and maintain supersonic turbulence in
the interstellar medium, and come to the conclusion that supernova explosions accompanying the death of massive
stars are the most likely agents. Then we briefly apply the theory to various types of galaxies, ranging from low surface
brightness galaxies to massive star bursts. Finally, in Section VII we summarize, and describe unsolved problems
open for future research.
II. OBSERVATIONS
All present day star formation takes place in molecular clouds (e.g. Blitz 1993, Williams, Blitz, & McKee 2000),
so it is vital to understand the properties, dynamical evolution and fragmentation of molecular clouds in order to
understand star formation. We begin this section by describing the composition (Section II.A) and density and ve-
locity structure (Section II.B) of molecular clouds. We then discuss turbulent support of clouds against gravitational
collapse (Section II.C), and introduce the observed scaling relations and their relation to the turbulent flow (Sec-
tion II.D). Finally, we describe observations of protostellar cores (Section II.E) and of the initial mass function of
stars (Section II.F).
A. Composition of molecular clouds
Molecular clouds are density enhancements in the interstellar gas dominated by molecular H2 rather than the
atomic H typical of the rest of the ISM (e.g. Ferrie`re 2001), mainly because they are opaque to the UV radiation that
elsewhere dissociates the molecules. In the plane of the Milky Way, interstellar gas has been extensively reprocessed
by stars, so the metallicity1 is close to the solar value Z⊙, while in other galaxies with lower star formation rates,
the metallicity can be as little as 10−3Z⊙. The refractory elements condense into dust grains, while others form
molecules. The properties of the dust grains change as the temperature drops within the cloud, probably due to the
freezing of volatiles such as water and ammonia (e.g. Goodman et al. 1995). This has important consequences for the
radiation transport properties and the optical depth of the clouds. The presence of heavier elements such as carbon,
nitrogen, and oxygen determines the heating and cooling processes in molecular clouds (e.g. Genzel 1991). In addition,
continuum emission from dust and emission and absorption lines emitted by molecules formed from these elements
are the main observational tracers of cloud structure, as cold molecular hydrogen is very difficult to observe. Radio
and sub-millimeter telescopes mostly concentrate on thermal continuum from dust and the rotational transition lines
of carbon, oxygen and nitrogen molecules (e.g. CO, NH3, or H2O). By now, several hundred different molecules have
been identified in the interstellar gas. An overview of the application of different molecules as tracers for different
physical conditions can be found in the reviews by van Dishoeck et al. (1993), Langer et al. (2000), van Dishoeck &
Hogerheijde (2000).
1Metallicity in astrophysics is usually defined as the fraction of heavy elements relative to hydrogen. It averages over local
variations in the abundance of the different elements caused by varying chemical enrichment histories.
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FIG. 2. Maps of the molecular gas in the Cygnus OB7 complex. (a) Large scale map of the 13CO (J = 1 − 0) emission. The
first level and the contour spacing are 0.25K. (b) Map of the same transition line of a sub-region with higher resolution (first
contour level and spacing are 0.3K). Both maps were obtained with the Bordeaux telescope. (c) 12CO (J = 1 − 0) and (d)
13CO (J = 1− 0) emission from the most transparent part of the field. (e) 13CO (J = 1− 0) and (f) C18O (J = 1− 0) emission
from the most opaque field. (g) 13CO (J = 1− 0) and (h) C18O (J = 1− 0) emission from a filamentary region with medium
density. The indicated linear sizes are given for a distance to Cygnus OB7 of 750 pc. (The figure is from Falgarone et al. 1992).
B. Density and velocity structure of molecular clouds
Emission line observations of molecular clouds reveal clumps and filaments on all scales accessible by present day
telescopes. Typical parameters of different regions in molecular clouds are listed in Table III, adopted from Cernicharo
8

(1991). The mass spectrum of clumps in molecular clouds appears to be well described by a power law, indicating
self-similarity: there is no natural mass or size scale between the lower and upper limits of the observations. The
largest molecular structures considered to be single objects are giant molecular clouds (GMCs), which have masses
of 105 to 106 M⊙, and extend over a few tens of parsecs. The smallest observed structures are protostellar cores with
masses of a few solar masses or less and sizes of <∼ 0.1 pc, and less-dense clumps of similar size. The volume filling
factor 〈n〉/n (where n is the local density, while 〈n〉 is the average density of the cloud) of dense clumps, even denser
subclumps and so on, is rather small, ranging from 10% down to 0.1% at densities of n > 105 cm−3 (e.g. McKee
1999). Star formation always occurs in the densest regions within a cloud, so only a small fraction of molecular cloud
matter is actually involved in building up stars, while the bulk of the material remains at lower densities.
The density structure of molecular clouds is best inferred from the column density of dust, which can be observed
either via its thermal emission at millimeter wavelengths in dense regions (e.g. Testi & Sargent 1998, Motte, Andre´, &
Neri 1998), or via its extinction of background stars in the infrared, if a uniform screen of background stars is present
(Lada et al. 1994, Alves, Lada, & Lada 2001). Deriving density and mass from thermal emission requires modeling
the temperature profile, which depends on optically thick radiative transfer through uncertain density distributions.
Infrared extinction, on the other hand, requires only suitable background stars. Reliance on the near-IR color excess
to measure column densities ensures a much greater dynamic range than optical extinction. This method has been
further developed by Cambre´sy et al. (2002) who use an adaptive grid to extract maximum information from non-
uniform background star fields. It turns out that the higher the column density in a region, the higher the variation
in extinction among stars behind that region (Lada et al. 1994). Padoan & Nordlund (1999) demonstrated this to be
consistent with a super-Alfve´nic turbulent flow, while Alves et al. (2001) modeled it with a single cylindrical filament
with density ρ ∝ r−2. Because turbulence forms many filaments, it is not clear that these two descriptions are
actually contradictory (Padoan, 2001, private communication), although the identification of a single filament would
then suggest that a minimum scale for the turbulence has been identified.
A more general technique is emission in optically thin spectral lines. The best candidates are 13CO and C18O,
though CO freezes out in the very densest regions (with visual extinctions above AV ≃ 10 magnitudes, see Alves,
Lada, & Lada, 1999). CO observations are therefore only sensitive to gas at relatively low densities n <∼ 105 cm−3,
and are limited in dynamic range to at most two decades of column density. The colder the gas, the lower the column
density at which lines will become optically thick. Nevertheless, the development of sensitive radio receivers in the
1980’s first made it feasible to map an entire molecular cloud region with high spatial and spectral resolution to obtain
quantitative information about the overall density structure.
The hierarchy of clumps and filaments spans all observable scales (e.g. Falgarone, Puget, & Perault 1992, Falgarone
& Phillips 1996, Wiesemeyer et al. 1997) extending down to individual protostars studied with millimeter-wavelength
interferometry (Ward-Thompson et al. 1994, Langer et al. 1995, Gueth et al. 1997, Motte et al. 1998, Testi & Sargent
1998, Ward-Thompson, Motte, & Andre´ 1999, Bacmann et al. 2000, Motte et al. 2001). This is illustrated in Figure 2,
which shows 13CO, 12CO and C18O maps of a region in the Cygnus OB7 complex at three levels of successively higher
resolution (from Falgarone et al. 1992). At each level, the molecular cloud appears clumpy and highly structured.
When observed with higher resolution, each clump breaks up into a filamentary network of smaller clumps. Unresolved
features exist even at the highest resolution. The ensemble of clumps identified in this survey covers a mass range
from about 1M⊙ up to a few 100M⊙ and densities 50 cm−3 < n(H2) < 104 cm−3. These values are typical for all
studies of cloud clump structure, with higher densities being reached primarily in protostellar cores.
The distribution of clump masses is consistent with a power law of the form
dN
dm ∝ m
α , (4)
with −1.3 < α < −1.9 in molecular line studies (Carr 1987, Stutzki & Gu¨sten 1990, Lada, Bally, & Stark 1991,
Williams, de Geus, & Blitz 1994, Onishi et al. 1996, Kramer 1998, Heithausen et al. 1998). Dust continuum studies,
which pick out the highest column density regions, find steeper values of −1.9 < α < −2.5 (Testi & Sargent 1998,
Motte et al. 1998; also see the discussion in Ossenkopf, Klessen, & Heitsch 2001), similar to the stellar mass spectrum.
The power-law mass spectrum is often interpreted as a manifestation of fractal density structure (e.g. Elmegreen &
Falgarone 1996). However, the full physical meaning remains unclear. In most studies molecular cloud clumps are
determined either by a Gaussian decomposition scheme (Stutzki & Gu¨sten 1990) or by the attempt to define (and
separate) clumps following density peaks (Williams et al. 1994). There is no one-to-one correspondence between the
identified clumps in either method, however. Furthermore, molecular clouds are only seen in projection, so one only
measures column density instead of volume density. It remains unproven that all regions of high density also have
high column density, and vice-versa. Even when velocity information is taken into account, the real 3D structure
of the cloud remains elusive. In particular, it can be demonstrated in models of interstellar turbulence that single
9

Jenkins & Shaya 1979, Bowyer et al. 1995), while at a temperature of 10 K and a density of 103 cm−3, the pressure
in a typical molecular cloud exceeds 10−12 erg cm−3. Gravitational confinement was traditionally cited to explain
the high pressures observed in GMCs (Kutner et al. 1977; Elmegreen, Lada, & Dickinson 1979; Blitz 1993; Williams
et al. 2000). Their masses certainly exceed by orders of magnitude the critical mass for gravitational stability MJ
defined by Eq. (14), computed from their average density and temperature. However, if only thermal pressure opposed
gravitational attraction, they should be collapsing and very efficiently forming stars on a free-fall timescale (Eq. 1).
That is not the case. Within molecular clouds, low-mass gas clumps appear highly transient and pressure confined
rather than being bound by self-gravity. Self-gravity appears to dominate only in the most massive individual cores,
where star formation actually is observed (Williams, Blitz, & Stark 1995; Yonekura et al. 1997; Kawamura et al. 1998;
Simon et al. 2001).
In the short lifetimes of molecular clouds (Section VI.A) they likely never reach a state of dynamical equilibrium
(Ballesteros-Paredes et al. 1999a; Elmegreen 2000). This is in contrast to the classical picture that sees molecular
clouds as long-lived equilibrium structures (Blitz & Shu 1980). The overall star formation efficiency (eq. 2) on scales
of molecular clouds as a whole is low in our Galaxy, of order of 10% or smaller (Zuckerman & Evans 1974). Only a
small fraction of molecular cloud material associated with the highest-density regions is actually forming stars. The
bulk of observed molecular cloud material is inactive, in a more tenous state between individual star forming regions.
Except on the scales of isolated protostellar cores, the observed line widths are always wider than implied by the
excitation temperature of the molecules. This is interpreted as the result of bulk motion associated with turbulence.
We will argue in this review that it is this interstellar turbulence that determines the lifetime and fate of molecular
clouds, and so their ability to collapse and form stars.
Magnetic fields have long been discussed as a stabilizing agent in molecular clouds. However, magnetic fields with
average field strength of 10µG (Verschuur 1995a,b; Troland et al. 1996, Crutcher 1999) cannot stabilize molecular
clouds as a whole. This is particularly true on scales of individual protostars, where magnetic fields appear too weak
to impede gravitational collapse in essentially all cases observed (see Section III.D). Furthermore, magnetic fields
cannot prevent turbulent velocity fields from decaying quickly (see the discussion in Section IV.A).
Molecular clouds appear to be transient features of the turbulent flow of the interstellar medium (Ballesteros-
Paredes et al. 1999a). Just as Lyman-α clouds in the intergalactic medium were shown to be transient objects formed
in the larger scale cosmological flow (Cen et al. 1994, Zhang, Anninos, & Norman 1995) rather than stable objects in
gravitational equilibrium (Rees 1986, Ikeuchi 1986), molecular clouds may never reach an equilibrium configuration.
The high pressures seen in molecular clouds can be produced by ram pressure from converging supersonic flows in
the ISM (see Section VI.A). So long as the flow persists, it confines the cloud, and supplies turbulent energy. When
the flow ends, the cloud begins to expand at its sound speed, eventually dissipating into the ISM (Va´zquez-Semadeni,
Shadmehri, & Ballesteros-Paredes 2002). Further shocks may help this process along.
D. Scaling relations for molecular clouds
Observations of molecular clouds exhibit correlations between various properties, such as clump size, velocity
dispersion, density and mass. Larson (1981) first noted, using data from several different molecular cloud surveys,
that the density ρ and the velocity dispersion σ appear to scale with the cloud size R as
ρ ∝ Rα (5)
σ ∝ Rβ , (6)
with α and β being constant scaling exponents. Many studies have been done of the scaling properties of molecular
clouds. The most commonly quoted values of the exponents are α ≈ −1.15± 0.15 and β ≈ 0.4± 0.1 (e.g. Dame et al.
1986, Myers & Goodman 1988, Falgarone et al. 1992, Fuller & Myers 1992, Wood, Myers, & Daugherty 1994, Caselli
& Myers 1995). However, the validity of these scaling relations is the subject of strong controversy and significantly
discrepant values have been reported by Carr (1987) and Loren (1989), for example.
The above standard values are often interpreted in terms of the virial theorem (Larson 1981, Caselli & Myers 1995).
If one assumes virial equilibrium, Larson’s relations (Eq.’s 5 and 6) are not independent. For α = −1, which implies
constant column density, a value of β = 0.5 suggests equipartition between self-gravity and the turbulent velocity
dispersion, such that the ratio between kinetic and potential energy is constant with Ekin/|Epot| = σ2R/(2GM) ≈ 1/2.
Note, that for any arbitrarily chosen value of the density scaling exponent α, a corresponding value of β obeying
equipartition can always be found (Va´zquez-Semadeni & Gazol 1995). Equipartition is usually interpreted as indicating
virial equilibrium in a static object. However, Ballesteros-Paredes et al. (1999b) pointed out that in a dynamic,
11

turbulent environment, the other terms of the virial equation (McKee & Zweibel 1992) can have values as large as or
larger than the internal kinetic and potential energy. In particular, the changing shape of the cloud will change its
moment of inertia, and turbulent flows will produce large fluxes of kinetic energy through the surface of the cloud.
As a result, equipartition between internal kinetic and potential energy does not necessarily imply virial equilibrium.
Kegel (1989) and Scalo (1990) proposed that the density-size relation may be a mere artifact of the limited dynamic
range in the observations, rather than reflecting a real property of interstellar clouds. In particular, in the case of
molecular line data, the observations are restricted to column densities large enough for the tracer molecule to be
shielded against photodissociating UV radiation, but small enough for the lines to remain optically thin. With limited
integration times, most CO surveys tend to select objects in an even smaller range of column densities, giving roughly
constant column density, which automatically implies ρ ∝ R−1. Surveys with longer integration times, and therefore
larger dynamic ranges, seem to exhibit an increasingly large scatter in density-size plots, as seen, for example, in the
data of Falgarone et al. (1992). Results from numerical simulations, which are free from observational bias, indicate the
same trend (Va´zquez-Semadeni, Ballesteros-Paredes, & Rodriguez 1997). Three-dimensional simulations of supersonic
turbulence (Mac Low 1999) were used by Ballesteros-Paredes & Mac Low (2002) to perform a comparison of clumps
measured in physical space to clumps observed in position-position-velocity space. They found no relation between
density and size in physical space, but a clear trend of ρ ∝ R−1 in the simulated observations, caused simply by the
tendency of clump-finding algorithms to pick out clumps with column densities close to the local peak values. Also,
for clumps within molecular clouds, the structures identified in CO often do not correspond to those derived from
higher-density tracers (see e.g. Langer et al. 1995, Bergin et al. 1997, Motte et al. 1998 for observational discussion, and
Ballesteros-Paredes & Mac Low 2002 for theoretical discussion). In summary, the existence of a physical density-size
relation appears doubtful.
The velocity-size relation appears less prone to observational artifacts. Although some measurements of molecular
clouds do not seem to exhibit this correlation (e.g. Loren 1989, or Plume et al. 1997), it does appear to be a real
property of the cloud. It is often explained using the standard (though incomplete) argument of virial equilibrium.
In supersonic turbulent flows, however, the scaling relation is a natural consequence of the characteristic energy
spectrum E(k) ∝ k−2 in an ensemble of shocks, even in the complete absence of self-gravity (Ossenkopf & Mac Low
2002, Ballesteros-Paredes & Mac Low 2002, Boldyrev, Nordlund, & Padoan 2002). Larger scales carry more energy,
leading to a relation between velocity dispersion and size that empirically reproduces the observed relation. Thus,
although the velocity-size relation probably does exist, its presence does not argue for virial equilibrium, or even
energy equipartition, but rather for the presence of a supersonic turbulent cascade.
E. Protostellar cores
1. From cores to stars
Protostellar cores are the direct precursors of stars. The transformation of cloud cores into stars can be conveniently
subdivided into four observationally motivated phases (e.g. Shu et al. 1987, Andre´ et al. 2000).
(a) The prestellar phase describes the isothermal gravitational contraction of molecular cloud cores before the
formation of the central protostar. Prestellar cores are cold and are best observed in molecular lines or dust emission.
The isothermal collapse phase ends when the inner parts reach densities of n(H2) ≈ 1010 cm−3. Then the gas and
dust become optically thick, so the heat generated by the collapse can no longer freely radiate away (e.g. Tohline
1982). The central region begins to heat up, and contraction pauses. As the temperature increases to T ≈ 2000K
molecular hydrogen begins to dissociate, absorbing energy. The core becomes unstable again and collapse sets in
anew. Most of the released gravitational energy goes into the dissociation of H2 so that the temperature rises only
slowly. This situation is similar to the first isothermal collapse phase. When all molecules in the core are dissociated,
the temperature rises sharply and pressure gradients again halt the collapse. This second hydrostatic object is the
true protostar.
(b) The cloud core then enters the class 0 phase of evolution, in which the central protostar grows in mass by the
accretion of infalling material from the outer parts of the original cloud core. Higher angular momentum material
first falls onto a disk and then gets transported inwards by viscous processes. In this phase star and disk are deeply
embedded in an envelope of gas and dust. The mass of the envelope Menv greatly exceeds the total mass M⋆ of star
and disk together. The main contribution to the total luminosity is accretion, and the system is best observed at
sub-millimeter and infrared wavelengths.
(c) At later times, powerful protostellar outflows develop that clear out the envelope along the rotational axis.
This is the class I phase during which the system is observable in infrared and optical wavebands, and for which
12

narrow line widths. These are very close to the line widths expected for thermal broadening alone and, as a result,
many of the cores appear approximately gravitationally virialized (e.g. Myers 1983). They are thought either to be
in the very early stage of gravitational collapse or to have subsonic turbulence supporting the clump. A comparison
of the line widths of cores with embedded protostellar objects (i.e. with associated IRAS sources) and the starless
cores reveals a substantial difference. Typically, cores with infrared sources exhibit broader lines, which suggests
the presence of a considerable turbulent component not present in starless cores. This may be caused by the central
protostar feeding back energy and momentum into its surrounding envelope. Molecular outflows associated with many
of the sources may be a direct indication of this process.
L1689B
FIG. 4. Radial column density profile of the prestellar core L1689B derived from combined infrared absorption and 1.3mm
continuum emission maps. Crosses show the observed values with the corresponding statistical errors, while the total uncer-
tainties in the method are indicated by the dashed lines. For comparison, the solid line denotes the best-fitting Bonnor-Ebert
sphere and the dotted line the column density profile of a singular isothermal sphere. The observed profile is well reproduced
by an unstable Bonnor-Ebert sphere with a density contrast of ∼ 50, see Bacmann et al. 2000 for a further details.
The advent of a new generation of infrared detectors and powerful receivers in the radio and sub-millimeter wave-
bands in the late 1990’s made it possible to determine the radial column density profiles of prestellar cores with
high sensitivity and resolution (e.g. Ward-Thompson et al. 1994, Andre´ et al. 1996, Motte, Andre´, and Neri 1998,
Ward-Thompson et al. 1999, Bacmann et al. 2000, Motte & Andre´ 2001). These studies show that starless cores
typically have flat inner density profiles out to radii of a few hundredths of a parsec, followed by a radial decline of
roughly ρ ∝ 1/r2 and possibly a sharp outer edge at radii 0.05–0.3 pc (e.g. Andre´ et al. 2000). This is illustrated in
Figure 4 which shows the observed column density of the starless core L1689B derived from combining mid-infrared
absorption maps with 1.3 mm dust continuum emission maps (from Bacmann et al. 2000). Similar profiles have been
derived independently from dust extinction studies (Lada et al. 1994, Alves et al. 2001). Protostellar cores often are
elongated or cometary shaped and appear to be parts of filamentary structures that connect several objects.
The various theoretical approaches to explain the observed core properties are discussed and compared in Section
V.B.
F. The observed IMF
Hydrogen-burning stars can only exist in a finite mass range
0.08<∼m<∼ 100 , (7)
where the dimensionless mass m ≡ M/(1 M⊙). Objects with m<∼0.08 do not have central temperatures and pressures
high enough for hydrogen fusion to occur. If they are larger than about ten times the mass of Jupiter, m > 0.01, they
are called brown dwarfs, or more generally substellar objects (e.g. Burrows et al. 1993, Laughlin & Bodenheimer 1993;
14

or for a review Burrows et al. 2001). Stars with m > 100, on the other hand, blow themselves apart by radiation
pressure (e.g. Phillips 1994).
It is complicated and laborious to estimate the IMF in our Galaxy empirically. The first such determination from
the solar neighborhood (Salpeter 1955) showed that the number ξ(m)dm of stars with masses in the range m to
m+ dm can be approximated by a power-law relation
ξ(m)dm ∝ m−αdm , (8)
with index α ≈ 2.35 for stars in the mass range 0.4 ≤ m ≤ 10. However, approximation of the IMF with a single
power-law is too simple. Miller & Scalo (1979) introduced a log-normal functional form, again to describe the IMF
for Galactic field stars in the vicinity of the Sun,
log10 ξ(log10 m) = A−
1
2(log10 σ)2
[
log10
( m
m0
)]2
. (9)
This analysis has been repeated and improved by Kroupa, Tout, & Gilmore (1990), who derive values
m0 = 0.23,
σ = 0.42, (10)
A = 0.1.
FIG. 5. The measured stellar mass function ξ as function of logarithmic mass log10 m in the Orion nebular cluster (upper
circles), the Pleiades (triangles connected by line), and the cluster M35 (lower circles). None of the mass functions is corrected
for unresolved multiple stellar systems. The average initial stellar mass function derived from Galactic field stars in the solar
neighborhood is shown as a line with the associated uncertainty range indicated by the hatched area. (From Kroupa 2002.)
The IMF can also be estimated, probably more directly, by studying individual young star clusters. Typical
examples are given in Figure 5 (taken from Kroupa 2002), which plots the mass function derived from star counts
in the Trapezium Cluster in Orion (Hillenbrand & Carpenter 2000), in the Pleiades (Hambly et al. 1999) and in the
cluster M35 (Barrado y Navascue´s et al. 2001).
15

it contributes to the pressure on large scales, and Chandrasekhar derived a dispersion relation similar to Eq. (12)
except for the introduction of an effective sound speed
c2s,eff = c2s + 1/3 〈v2〉 , (15)
where 〈v2〉 is the rms velocity dispersion due to turbulent motions.
Sasao (1973) noted that Chandrasekhar’s derivation neglected the effect of the inertia of the turbulent flow in
forming density enhancements while focussing only on the effective turbulent pressure. The developments through the
mid-eighties are reviewed by Scalo (1985). Both Sasao (1973) and Chandrasekhar (1951a,b) made the microturbulent
assumption that the outer scale of the turbulence is smaller than that of the turbulent clouds. However, the outer
scales of molecular cloud turbulence typically exceed or are at least comparable to the size of the system (e.g. Ossenkopf
and Mac Low, 2001), so the assumption of microturbulence is invalid. Bonazzola et al. (1987) therefore suggested a
wavelength-dependent effective sound speed c2s,eff (k) = c2s + 1/3 v2(k) for Eq. (12). In this description, the stability
of the system depends not only on the total amount of energy, but also on the wavelength distribution of the energy,
since v2(k) depends on the turbulent power spectrum. A similar approach was also adopted by Va´zquez-Semadeni
and Gazol (1995), who added Larson’s (1981) empirical scaling relations to the analysis.
An elaborate investigation of the stability of turbulent, self-gravitating gas was undertaken by Bonazzola et al.
(1992), who used renormalization group theory to derive a dispersion relation with a generalized, wavenumber-
dependent, effective sound speed and an effective kinetic viscosity that together account for turbulence at all wave-
lengths shorter than the one in question. According to their analysis, turbulence with a power spectrum steeper than
P (k) ∝ 1/k3 can support a region against collapse at large scales, and below the thermal Jeans scale, but not in
between. On the other hand, they claim that turbulence with a shallower slope, as is expected for incompressible
turbulence (Kolmogorov 1941a,b), Burgers turbulence (Lesieur 1997, p. 238), or shock dominated flows (Passot, Pou-
quet & Woodward 1988), cannot support clouds against collapse at scales larger than the thermal Jeans wavelength.
It may even be possible to describe the equilibrium state of self-gravitating gas as an inherently inhomogeneous ther-
modynamic critical point (de Vega, Sa´nchez and Combes, 1996a,b; de Vega and Sa´nchez, 2000). This may render all
applications of incompressible turbulence to the theory of star formation meaningless. In fact, it is the main goal of
this review to introduce and stress the importance of compressional effects in supersonic turbulence for determining
the outcome of star formation.
In order to do that, we need to recapitulate the development of our understanding of the star formation process
over the last few decades. We begin with the classical dynamical theory (Section III.A) and describe the problems
that it encounters in its original form (Section III.B). In particular the timescale problem lead astrophysicists to
think about the influence of magnetic fields. This line of reasoning resulted in the construction of the paradigm of
magnetically mediated star formation, which we discuss in Section III.C. However, it became clear that this so-called
“standard theory” has a variety of serious shortcomings (Section III.D). These lead to the rejuvenation of the earlier
dynamical concepts of star formation and their reconsideration in the modern framework of compressible supersonic
turbulence which we discuss in Section IV.
A. Classical dynamical theory
The classical dynamical theory focuses on the interplay between self-gravity on the one side and pressure gradients
on the other. Turbulence is taken into account, but only on microscopic scales significantly smaller than the collapse
scales. In this microturbulent regime, random gas motions yield an isotropic pressure that can be absorbed into the
equations of motion by defining an effective sound speed as in Eq. (15). The dynamical behavior of the system remains
unchanged, and we do not distinguish between the effective and thermal sound speed cs in this and the following two
sections.
Because of the importance of gravitational instability for stellar birth, Jeans’ (1902) pioneering work has triggered
numerous attempts to derive solutions to the collapse problem, both analytically and numerically. Particularly
noteworthy are the studies by Bonnor (1956) and Ebert (1957), who independently derived analytical solutions for
the equilibrium structure of spherical density perturbations in self-gravitating, isothermal, ideal gases, as well as a
criterium for gravitational collapse. See Lombardi and Bertin (2001) for a recent analysis, and studies by Schmitz
(1983, 1984, 1986, 1988) and Schmitz & Ebert (1986, 1987) for the treatment of rotation and generalized, polytropic
equations of state. It has been argued recently that this may be a good description for the density distribution
in quiescent molecular cloud cores just before they begin to collapse and form stars (Bacmann et al. 2000, Alves,
Lada, and Lada 2001). The first numerical calculations of protostellar collapse became possible in the late 1960’s
18

starlight (Hiltner 1949, 1951) that substantial magnetic fields thread the interstellar medium (Chandrasekhar & Fermi
1953a). This forced the magnetic flux problem to be addressed, but also raised the possibility that the solution to the
angular momentum problem might be found in the action of magnetic fields. The typical strength of the magnetic
field in the diffuse ISM was not known to an order of magnitude, though, with estimates ranging as high as 30 µG
from polarization (Chandrasekhar & Fermi 1953a) and synchrotron emission (e.g. Davies & Shuter 1963). Lower
values from Zeeman measurements of Hi (Troland & Heiles 1986) and from measurements of pulsar rotation and
dispersion measures (Rand & Kulkarni 1989, Rand & Lyne 1994) comparable to the modern value of around 3 µG
only gradually became accepted over the next two decades. Even now, measurements of synchrotron emission leave
open the possibility that there is a stronger disordered field in the Milky Way, although their interpretation depends
critically on the assumption of equipartition between magnetic field energy and other forms (Beck 2001).
The presence of a field, especially one much stronger than 3 µG, formed a major problem for the classical theory
of star formation. To see why, let us consider the behavior of a field in a region of isothermal, gravitational collapse
(Mestel & Spitzer 1956, Spitzer 1968). If we neglect all surface terms except thermal pressure P0 (a questionable
assumption as shown by Ballesteros-Paredes et al. 1999b, but the usual one at the time), and assume that the field, ~B
is uniform, and passes through a spherical region of average density ρ and radius R, we can write the virial equation
as (Spitzer 1968)
4πR3P0 = 3
MkBT
µ −
1
R
(
3
5
GM2 − 1
3
R4B2
)
, (17)
where M = (4/3)πR3ρ is the mass of the region, kB is Boltzmann’s constant, T is the temperature of the region, and
µ is the mean mass per particle. So long as the ionization is sufficiently high for the field to be frozen to the matter,
the flux through the cloud Φ = πR2B must remain constant. Therefore, the opposition to collapse due to magnetic
energy given by the last term on the right hand side of Eq. (17) will remain constant during collapse. If it cannot
prevent collapse at the beginning, it remains unable to do so as the field is compressed.
If we write the radius R in terms of the mass and density of the region, we can rewrite the two terms in parentheses
on the right hand side of Eq. (17) to show that gravitational attraction can only overwhelm magnetic repulsion if
M > Mcr ≡
53/2
48π2
B3
G3/2ρ2 = (4× 10
6M⊙)
( n
1 cm−3
)−2( B
3 µG
)3
, (18)
where the numerical constant is correct for a uniform sphere, and the number density n is computed with mean mass
per particle µ = 2.11× 10−24 g cm−3. Mouschovias & Spitzer (1976) noted that the critical mass can also be written
in terms of a critical mass-to-flux ratio
(M
Φ
)
cr
=
ζ
3π
(
5
G
)1/2
= 490 gG−1 cm−2, (19)
where the constant ζ = 0.53 for uniform spheres (or flattened systems, as shown by Strittmatter 1966) is used in the
final equality. Assuming a constant mass-to-flux ratio in a region results in ζ = 0.3 (Nakano & Nakamura 1978). For
a typical interstellar field of 3 µG, the critical surface density for collapse is 7 M⊙ pc−2, corresponding to a number
density of 230 cm−2 in a layer of thickness 1 pc ≈ 3.09×1018 cm. A cloud is termed subcritical if it is magnetostatically
stable and supercritical if it is not.
The very large value for the magnetic critical mass in the diffuse ISM given by Eq. (18) forms a crucial objection
to the classical theory of star formation. Even if such a large mass could be assembled, how could it fragment into
objects with stellar masses of 0.01–100 M⊙, when the critical mass should remain invariant under uniform spherical
gravitational collapse?
Two further objections to the classical theory were also prominent. First was the embarrassingly high rate of star
formation predicted by a model governed by gravitational instability, in which objects should collapse on roughly the
free-fall timescale, Eq. (1), orders of magnitude shorter than the ages of typical galaxies.
Second was the gap between the angular momentum contained in a parcel of gas participating in rotation in a
galactic disk and the much smaller angular momentum contained in stars (Spitzer 1968, Bodenheimer 1995). The
disk of the Milky Way rotates with angular velocity Ω ≃ 10−15 s−1. A uniformly collapsing cloud with initial radius
R0 formed from material with density ρ0 = 2 × 10−24 g cm−3 rotating with the disk will find its angular velocity
increasing as (R0/R)2, or as (ρ/ρ0)2/3. By the time it reaches a typical stellar density of ρ = 1 g cm−3, its angular
velocity has increased by a factor of 6× 1015, giving a rotation period of well under a second. The centrifugal force
Ω2R exceeds the gravitational force by eight orders of magnitude for solar parameters. This is unphysical, and indeed
21

low ionization fractions came to seem plausible. Nakano (1976, 1979) and Elmegreen (1979) computed the detailed
ionization balance of molecular clouds for reasonable cosmic ray ionization rates, showing that at densities greater
than 104 cm−3, the ionization fraction was roughly
x ≈ (5× 10−8)
( n
105 cm−3
)−1/2
(23)
(Elmegreen 1979), becoming constant at densities higher than 107 cm−3 or so. Below densities of 104 cm−3, the
ionization increases because of the external UV radiation field, and the gas is tightly coupled to the magnetic field.
With typical molecular cloud parameters τAD is of order 107 yr (eq. 22). The ambipolar diffusion timescale τAD
is thus about 10–20 times longer than the corresponding dynamical timescale τff of the system (e.g. McKee et al.
1993). The delay induced by waiting for ambipolar diffusion to occur was then taken as a way to explain the low star
formation rates observed in normal galaxies, as well as the long lifetimes of molecular clouds, which at that time were
thought to be about 30–100Myr (Solomon et al. 1987, Blitz & Shu 1980). See Section VI.A for arguments that they
are under 10 Myr, however.
These considerations lead to the investigation of star formation models based on ambipolar diffusion as a dominant
physical process rather than relying solely on gas dynamical collapse. In particular Shu (1977) proposed the self-
similar collapse of initially quasi-static singular isothermal spheres as the most likely description of the star formation
process. He assumed that ambipolar diffusion in a magnetically subcritical, isothermal cloud core would lead to the
build-up of a quasi-static 1/r2-density structure that contracts on timescales of order of τAD. This evolutionary phase
is denoted quasi-static because τAD ≫ τff . Ambipolar diffusion is supposed to eventually lead to the formation of a
singularity in central density, at which point the system becomes unstable and undergoes inside-out collapse. During
collapse this model assumes that magnetic fields are no longer dynamically important and they are subsequently
ignored in the original formulation of the theory. A rarefaction wave moves outward with the speed of sound, with
the cloud material behind the wave falling freely onto the core and matter ahead still being at rest.
The Shu (1977) model predicts constant mass accretion onto the central protostar at a rate M˙ = 0.975 c3s/G.
This is significantly below the values derived for Larson-Penston collapse. In the latter case the entire system is
collapsing dynamically and delivers mass to the center very efficiently, while in the former case inward mass transport
is comparatively inefficient as the cloud envelope remains at rest until reached by the rarefaction wave. The density
structure of the inside-out collapse, however, is essentially indistinguishable from the predictions of dynamical collapse.
To observationally differentiate between the two models one needs to obtain kinematical data and determine the
magnitude and spatial extent of infall motions with high accuracy. The basic predictions of inside-out collapse are
summarized in Table II. As singular isothermal spheres by definition have infinite mass, the growth of the central
protostar is taken to come to a halt when feedback processes (like bipolar outflows, stellar winds, etc.) become
important and terminate further infall.
Largely within the framework of the standard theory, numerous analytical extensions to the original inside-out
collapse model have been proposed. The stability of isothermal gas clouds with rotation, for example, has been
investigated by Schmitz (1983, 1984, 1986), Tereby, Shu, & Cassen (1984), Schmitz & Ebert (1986, 1987), Inutsuka
& Miyama (1992), Nakamura, Hanawa, & Nakano (1995), and Tsuribe & Inutsuka (1999b).
The effects of magnetic fields on the equilibrium structure of clouds and later during the collapse phase (where
they have been neglected in the original inside-out scenario) are considered by Schmitz (1987), Baureis, Ebert, &
Schmitz (1989), Tomisaka, Ikeuchi, & Nakamura (1988a,b, 1989a,b, 1990), Tomisaka (1991, 1995, 1996a,b), Galli &
Shu (1993a,b), Li & Shu (1996, 1997), Galli et al. (1999, 2001), and Shu et al. (2000). The proposed picture is that
ambipolar diffusion of initially subcritical cores that are threaded by uniform magnetic fields will lead to the build-
up of disk-like structures with constant mass-to-flux ratio. These disks are called isopedic. The mass-to-flux ratio
increases steadily with time. As it exceeds the maximum value consistent with magnetostatic equilibrium, the entire
core becomes supercritical and begins to collapse from the inside out with the mass-to-flux ratio assumed to remain
approximately constant. It can be shown (Shu & Li 1997), that for isopedic disks the forces due to magnetic tension
are just a scaled version of the disk’s self-gravity with oposite sign (i.e. obstructing gravitational collapse), and that
the magnetic pressure scales as the gas pressure (although the proportionality factor in general is spatially varying
except in special cases). These findings allow the application of many results derived for unmagnetized disks to the
magnetized regime with only slight modifications to the equations. One application of this result is that for isopedic
disks the derived mass accretion rate is just a scaled version of the original Shu (1977) rate, i.e. M˙ ≈ (1 +H0) c3s/G,
with the dimensionless parameter H0 depending on the effective mass-to-flux ratio.
However, the basic assumption of constant mass-to-flux ratio during the collapse phase appears inconsistent with
detailed numerical calculations of ambipolar diffusion processes (see Section III.D.1). In these computations the
23

1. Singular isothermal spheres
The collapse of singular isothermal spheres is the astrophysically most unlikely and unstable member of a large
family of self-similar solutions to the 1D collapse problem. Ever since the studies by Bonnor (1956) and Ebert (1957),
and by Larson (1969) and Penston (1969a) much attention in the star formation community has been focused on
finding astrophysically relevant, analytic, asymptotic solutions to this problem (see Section III.A). The standard
solution derived by Shu (1977) considers evolution of initially singular, isothermal spheres as they leave equilibrium.
His findings subsequently were extended by Hunter (1977, 1986). Whitworth & Summers (1985) demonstrated that
all solutions to the isothermal collapse problem are members of a two-parameter family with the Larson-Penston-type
solutions (collapse of spheres with uniform central density) and the Shu-type solutions (expansion-wave collapse of
singular spheres) populating extreme ends of parameter space. The solution set has been extended to include a
polytropic equation of state (Suto & Silk 1988), shocks (Tsai and Hsu 1995), and cylinder and disk-like geometries
(Inutsuka and Miyama 1992; Nakamura, Hanawa, & Nakano 1995). In addition, mathematical generalization using a
Lagrangian formulation has been proposed by Henriksen (1989, see also Henriksen, Andre´, and Bontemps 1997).
Of all proposed initial configurations for protostellar collapse, quasi-static, singular, isothermal spheres seem to be
the most difficult to realize in nature. Stable equilibria for self-gravitating, spherical, isothermal gas clouds embedded
in an external medium of given pressure are only possible up to a density contrast of ρc/ρs ≈ 14 between the
cloud center and surface. More centrally concentrated clouds can only reach unstable equilibrium states. Hence, all
evolutionary paths that could yield a central singularity lead through instability, so collapse will set in long before
a 1/r2 density profile is established at small radii r (Whitworth et al. 1996; also Silk & Suto 1988, and Hanawa &
Nakayama 1997). External perturbations also tend to break spherical symmetry in the innermost region and flatten
the overall density profile at small radii. The resulting behavior in the central region then more closely resembles the
Larson-Penston description of collapse. Similar behavior is found if outward propagating shocks are considered (Tsai
and Hsu 1995). As a consequence, the existence of physical processes that are able to produce singular, isothermal,
equilibrium spheres in nature is highly questionable.
The original proposal of ambipolar diffusion processes in magnetostatically supported gas does not yield the desired
result either. Ambipolar diffusion in magnetically supported gas clouds results in a dynamical Larson-Penston-type
collapse of the central region where magnetic support is lost, while the outer part is still hold up primarily by the field
(and develops a 1/r2 density profile). Mass is fed to the center not by an outward moving expansion wave, but by
ambipolar diffusion in the outer envelope. The proposal that singular isothermal spheres may form through ambipolar
diffusion processes in magnetically subcritical cores has been extensively studied by Mouschovias and collaborators in a
series of numerical simulations with ever increasing accuracy and astrophysical detail (Mouschovias 1991, Mouschovias
& Morton 1991, 1992a,b, Fiedler & Mouschovias 1992, 1993, Morton et al. 1994, Ciolek & Mouschovias 1993, 1994,
1995, 1996, 1998, Basu and Mouschovias 1994, 1995a,b, Desch & Mouschovias 2001; see however also Nakano 1979,
1982, 1983, Lizano & Shu 1989, or Safier et al. 1997). The numerical results indicate that the decoupling between
matter and magnetic fields occurs over several orders of magnitude in density, becoming important at n(H2) >
1010 cm−3. There is no single critical density below which matter is fully coupled to the field and above which it is
not, although ambipolar diffusion is indeed the dominant physical decoupling process (e.g. Desch & Mouschovias 2001).
As a consequence of ambipolar diffusion, initially subcritical gas clumps separate into a central nucleus that becomes
both thermally and magnetically supercritical, and an extended envelope that is still held up magnetostatically. The
central region goes into rapid collapse sweeping up much of its residual magnetic flux with it (Basu 1997).
Star formation from singular isothermal spheres is also biased against binary formation. The collapse of rotating
singular isothermal spheres very likely will result in the formation of single stars, as the central protostellar object
forms very early and rapidly increases in mass with respect to a simultaneously forming and growing rotationally
supported protostellar disk (e.g. Tsuribe & Inutsuka 1999a,b). By contrast, the collapse of cloud cores with flat inner
density profiles will deliver a much smaller fraction of mass directly into the central protostar within a free-fall time.
More matter will go first into a rotationally supported disk-like structure. These disks tend to be more massive with
respect to the central protostar in a Larson-Penston-type collapse compared to collapsing singular isothermal spheres,
they are more likely become unstable to subfragmentation, resulting in the formation of binary or higher-order stellar
systems (see the review by Bodenheimer et al. 2000). Since the majority of stars seems to form as part of binary
or higher-order system (e.g. Mathieu et al. 2000), star formation in nature appears incompatible with collapse from
strongly centrally peaked initial conditions (Whitworth et al. 1996).
25

2. Observations of clouds and cores
Before we consider the observational evidence against the standard theory of magnetically mediated star formation,
let us recapitulate its basic predictions as introduced in Section III.C. The theory predicts (a) constant accretion rates
and (b) infall motions that are confined to regions that have been passed by a rarefaction wave that moves outwards
with the speed of sound, while the parts of a core that lie further out remain static. The theory furthermore (c) relies
on the presence of magnetic field strong enough to hold up the gas in molecular cloud cores sufficiently long, so it
predicts that cores should be magnetically subcritical during most of their lifetimes. In the following we demonstrate
all these predictions appear to be contradicted by observations.
Bourke et al.
Crut
heretal.
FIG. 9. Line-of-sight magnetic field strength Blos versus column density N(Hs for various molecular cloud cores. Squares are
observations of Bourke et al. (2001) and circles are observations summarized by Crutcher (1990), Sarma et al. (2000), and
Crutcher & Troland (2000). Large symbols represent clear detections of the Zeeman effect, whereas small ones are 3σ upper
limits to the field strength. The lines drawn for the upper limits connect 3σ to 1σ limits. To guide the eye, lines of constant
flux-to-mass ratio (Φ/M)n are given, normalized to the critical value, i.e. to the inverse of equation (19). The observed line-of-
sight component Blos of the field is statistically deprojected to obtain the absolute value of the field B. The upper panel (a)
assumes spherical core geometry, while the lower panel (b) assumes a sheetlike geometry. A value of (Φ/M)n < 1 corresponds
to a magnetically supercritical core with magnetic field strengths too weak to support against gravitational contraction, while
(Φ/M)n > 1 allows magnetic support as required by the standard theory. Note that almost all observed cores are magnetically
supercritical. This is evident when assuming spherical symmetry, but even in the case of sheetlike protostellar cores the average
ratio is 〈(Φ/M)n〉 ≈ 0.4 when considering the 1σ upper limits. This is significantly lower than the critical value. The one clear
exception is RCW57, but it has two velocity components, leaving its Zeeman value in some doubt. For further discussion see
Bourke et al. (2001), from which this figure was drawn.
a. Magnetic Support
In his critical review of the standard theory of star formation, Nakano (1998) pointed out that no convincing
magnetically subcritical core had been found up to that time. Similar conclusions still hold today. All magnetic field
measurements are consistent with cores being magnetically supercritical or at most marginally critical.
26

d. Chemical ages
The chemical age of substructure in molecular clouds, as derived from observations of chemical abundances (also
see Section VI.A), is much smaller than the ambipolar diffusion time. This poses a timescale argument against
magnetically regulated star formation. The comparison of multi-molecule observations of cloud cores with time-
dependent chemical models indicates typical ages of about 105 years (see the reviews by van Dishoeck et al. 1993,
van Dishoeck & Black 1998, and Langer et al. 2000). This is orders of magnitude shorter than the timescales of up
to 107 yr required for ambipolar diffusion to become important as required by the standard model.
3. Protostars and young stars
a. Accretion rates
Observed protostellar accretion rates decline with time, in contradiction to the constant rates predicted by the
standard model. As matter falls onto the central protostar it goes through a shock and releases energy that is
radiated away giving rise to a luminosity Lacc ≈ GM⋆M˙⋆/R⋆ (Shu et al. 1987, 1993). The fact that most of the
matter first falls onto a protostellar disk, where it gets transported inwards on a viscous timescale before it is able to
accrete onto the star does not alter the expected overall luminosity by much (see e.g. Hartmann 1998).
During the early phases of protostellar collapse, while the mass Menv of the infalling envelope exceeds the mass M⋆
of the central protostar, the accretion luminosity Lacc far exceeds the intrinsic luminosity L⋆ of the young star. Hence
the observed bolometric luminosity Lbol of the object is a direct measure of the accretion rate as long as reasonable
estimates of M⋆ and R⋆ can be obtained. Determinations of bolometric temperature Tbol and luminosity Lbol therefore
should provide a fair estimate of the evolutionary stage of a protostellar core (e.g. Chen et al. 1995, Myers et al. 1998).
Scenarios in which the accretion rate decreases with time and increases with total mass of the collapsing cloud fragment
yield qualitatively better agreement with the observations than do models with constant accretion rate (Andre´ et al.
2000, see however Jayawardhana, Hartmann, & Calvet 2001, for an alternative interpretation based on environmental
conditions). A comparison of observational data with theoretical models where M˙⋆ decreases exponentially with time
is shown in Figure 10.
FIG. 11. Outflow momentum flux FCO versus envelope mass Menv, normalized to the bolometric luminosity Lbol, using the
relations Menv ∝ L0.6bol and FCOc ∝ Lbol. Protostellar cores with Menv > M⋆ are shown by open circles, and Menv > M⋆ by
filled circles (data from Bontemps et al. 1996). FCOc/Lbol is an empirical tracer for the accretion rate; the speed of light c
is invoked in order to obtain a dimensionless quantity. Menv/L0.6bol is an evolutionary indicator that decreases with time. The
abscissa therefore corresponds to a time axis, with early times at the right and later times to the left. Overlayed on the data
is a evolutionary model that assumes a flat inner density profile (for details see Henriksen et al. 1997, where the figure was
published originally).
29

A closely related method to estimate the accretion rate M˙⋆ is determination of protostellar outflow strengths
(Bontemps et al. 1996). Most embedded young protostars have powerful molecular outflows (Richer et al. 2000),
while outflow strength decreases towards later evolutionary stages. At the end of the main accretion phase, the
bolometric luminosity of protostars Lbol strongly correlates with the momentum flux FCO (e.g. Cabrit & Bertout
1992). Furthermore, FCO correlates well with Menv for all protostellar cloud cores (Bontemps et al. 1996, Hoherheijde
et al. 1998, Henning and Launhardt 1998). This result is independent of the FCO−Lbol relation and most likely results
from a progressive decrease of outflow power with time during the main accretion phase. With the linear correlation
between outflow mass loss and protostellar accretion rate (Hartigan et al. 1995) these observations therefore suggest
stellar accretion rates M˙⋆ that decrease with time. This is illustrated in Figure 11 which compares the observed
values of the normalized outflow flux and the normalized envelope mass for a sample of ∼ 40 protostellar cores with a
simplified dynamical collapse model with decreasing accretion rate M˙⋆ (Henriksen et al. 1997). The model describes
the data relatively well, as opposed to models of constant M˙⋆.
b. Embedded Objects
The fraction of protostellar cores with embedded protostellar objects is very high. Further indication that the
standard theory may need to be modified comes from estimates of the time spent by protostellar cores during various
stages of their evolution. As the standard model assumes that cloud cores in the prestellar phase evolve on ambipolar
diffusion timescales, which are an order of magnitude longer than the dynamical timescales of the later accretion
phase, one would expect a significantly larger number of starless cores than cores with embedded protostars.
For a uniform sample of protostars, the relative numbers of objects in distinct evolutionary phases roughly corre-
spond to the relative time spent in each phase. Beichman et al. (1986) used the ratio of numbers of starless cores
to the numbers of cores with embedded objects detected with the Infrared Astronomical Satellite (IRAS) and es-
timated that the duration of the prestellar phase is about equal to the time needed for a young stellar object to
completely accrete its protostellar envelope. Millimeter continuum mapping of pre-stellar cores gives similar results
(Ward-Thompson et al. 1994, 1999), leading Andre´ et al. (2000) to argue that the timespan of cores to increase their
central density n(H2) from ∼ 104 to ∼ 105 cm−3 is about the same as to go from n(H2) ≈ 105 cm−3 to the formation
of the central protostar. This clearly disagrees with standard ambipolar diffusion models (e.g. Ciolek & Mouschovias
1994), which predict a duration six times longer. Ciolek & Basu (2000) were indeed able to accurately model infall
in L1544 using an ambipolar diffusion model, but they did so by using initial conditions that were already almost
supercritical, so that very little ambipolar diffusion had to occur before dynamical collapse would set in. Ciolek &
Basu (2001) quantify the central density required to match the observations, and conclude that observed pre-stellar
cores are either already supercritical or just about to be. These observations that already in the prestellar phase the
timescales of core contraction are determined by fast dynamical processes rather than by slow ambipolar diffusion.
c. Stellar Ages
If the contraction time of individual cloud cores in the prestellar phase is determined by ambipolar diffusion, then
the age spread in a newly formed group or cluster should considerably exceed the dynamical timescale. Within a star-
forming region, high-density protostellar cores will evolve and form central protostars faster than their low-density
counterparts, so the age distribution is roughly determined by the evolution time of the lowest-density condensation.
However, the observed age spread in star clusters is very short. For example, in the Orion Trapezium cluster, a
dense cluster of a few thousand stars, it is less than 106 years (Prosser et al. 1994, Hillenbrand 1997, Hillenbrand &
Hartmann 1998), and the same holds for L1641 (Hodapp & Deane 1993). These age spreads are comparable to the
dynamical time in these clusters. Similar conclusions can be obtained for Taurus (Hartmann 2001), NGC1333 (Bally
et al. 1996, Lada et al. 1996), NGC6531 (Forbes 1996), and a variety of other clusters (see Elmegreen et al. 2000,
Palla & Stahler 1999, Hartmann 2001).
Larger regions form stars for a longer time. This correlation suggests that typical star formation times correspond
to about 2–3 turbulent crossing times in that region (Efremov & Elmegreen 1998a). This is very fast compared to the
ambipolar diffusion timescale, which is about 10 crossing times in a uniform medium with cosmic ray ionization (Shu
et al. 1987) and is even longer if stellar UV sources contribute to the ionization (Myers & Khersonsky 1995) or if the
cloud is very clumpy (Elmegreen & Combes 1992). Magnetic fields, therefore, appear unable regulate star formation
on scales of stellar clusters.
30

further evolution in a consistent way prevents the time step from becoming prohibitively small. This allows modeling
of the collapse of a large number of cores until the overall gas reservoir becomes exhausted.
ZEUS-3D, conversely, gives equal resolution in all regions, resolving shocks equally well everywhere, as well as
allowing the inclusion of magnetic fields (see Section IV.F). On the other hand, collapsing regions cannot be followed
to scales less than one or two grid zones. Once again the resolution required to follow gravitational collapse must be
considered. For a grid-based simulation, the criterion given by Truelove et al. (1997) holds. Equivalent to the SPH
resolution criterion, the mass contained in one grid zone has to be rather smaller than the local Jeans mass throughout
the computation. In the models described here, this criterion is satisfied until gravitational collapse is underway.
The computations discussed here are done on periodic cubes, with an isothermal equation of state, using up to 2563
zones (with one model at 5123 zones) or 803 SPH particles. To generate turbulent flows, Gaussian velocity fluctuations
are introduced with power only in a narrow interval k − 1 ≤ |~k| ≤ k, where k = L/λd counts the number of driving
wavelengths λd in the box (Mac Low et al. 1998). This offers a simple approximation to driving by mechanisms that
act on a single scale. To drive the turbulence, this fixed pattern is normalized to maintain constant kinetic energy
input rate E˙in = ∆E/∆t (Mac Low 1999). Self-gravity is turned on only after the turbulence reaches a state of
dynamical equilibrium.
FIG. 13. Density cubes in a model of supersonic turbulence driven on intermediate scales, with wavenumbers k = 3− 4, from
Klessen et al. (2000). Four different evolutionary stages are shown: (a) just before gravity is turned on, (b) when the first
collapsed cores are formed and have accreted M∗ = 5% of the mass, (c) when the mass in dense cores is M∗ = 25%, and (d)
when M∗ = 50%. Time is measured in units of the global system free-fall timescale τff , and dark dots indicate the locations of
protostars.
33

D. Global collapse
First, we examine the question of whether gravitational collapse can itself generate enough turbulence to prevent
further collapse. Gas dynamical SPH models initialized at rest with Gaussian density perturbations show fast collapse,
with the first collapsed objects forming in a single free-fall time (Klessen, Burkert, & Bate 1998; Klessen & Burkert
2000, 2001; Bate et al. 2002a,b). Models set up with a freely decaying turbulent velocity field behave similarly (Klessen
2000). Further accretion of gas onto collapsed objects then occurs over the next free-fall time, defining the predicted
spread of stellar ages in a freely-collapsing system. The turbulence generated by collapse (or virialization) fails to
prevent further collapse, contrary to previous suggestions (e.g. by Elmegreen 1993). Such a mechanism only works
for thermal pressure support in systems such as galaxy cluster halos where energy is lost inefficently, while turbulence
dissipates energy quite efficiently (Eq. 26).
Models of freely collapsing, magnetized gas remain to be done, but models of self-gravitating, decaying, magnetized
turbulence by Balsara, Ward-Thompson, & Crutcher (2001) using an MHD code incorporating a Riemann solver
suggest that the presence of magnetic fields does not markedly extend collapse timescales. They further show that
accretion down filaments aligned with magnetic field lines onto cores occurs readily. This allows high mass-to-flux
ratios to be maintained even at small scales, which is necessary for supercritical collapse to continue in a magnetized
medium after fragmentation occurs.
hM
J
i
turb
= 0:6
hM
J
i
turb
= 3:2
FIG. 14. Fraction Mcore of mass accreted onto protostars as function of time for different models of self-gravitating supersonic
turbulence. The models differ by driving strength and driving wavenumber, as indicated in the figure. The mass in the box
is initially unity, so the A-models (solid curves) are formally unsupported, while the B-models (dashed lines) are formally
supported. This is indicated by the effective turbulent Jeans mass 〈MJ〉turb defined at the mean density. This number has to
be compared with the total mass in the cube, which is unity. The figure shows that the efficiency of local collapse depends
on the scale and strength of turbulent driving. Time is measured in units of the global system free-fall timescale τff . Only
the model B5 which is driven strongly at scales smaller than the Jeans wavelength λJ in shock-compressed regions shows no
collapse. (From Klessen et al. 2000.)
E. Local collapse in globally stable regions
Second, we examine whether continuously driven turbulence can support against gravitational collapse. The models
of driven, self-gravitating turbulence by Klessen et al. (2000) and Heitsch et al. (2001a) described in Section IV.C show
that local collapse occurs even when the turbulent velocity field carries enough energy to counterbalance gravitational
contraction on global scales. This idea was first suggested by Hunter (1979), who used the virial theorem to make
his case. Later support for it was offered by 2D gas dynamical computations by Le´orat et al. (1990). An example of
local collapse in a globally supported cloud is given in Figure 13. A hallmark of global turbulent support is isolated,
inefficient, local collapse.
34

Local collapse in a globally stabilized cloud is not predicted by any of the analytic models for turbulent, self-
gravitating gas, as discussed in Klessen et al. (2000). The resolution to this apparent paradox lies in the requirement
that any substantial turbulent support must come from supersonic flows, as otherwise pressure support would be at
least equally important. However, supersonic flows compress the gas in shocks. In isothermal gas with density ρ
the postshock gas has density ρ′ = M2ρ, where M is the Mach number of the shock. The turbulent Jeans length
λJ ∝ ρ′−1/2 in these density enhancements, so it drops by a factor of M in isothermal shocks.
Klessen et al. (2000) demonstrated that supersonic turbulence can completely prevent collapse only when it can
support not just the average density, but also the shocked, high-density regions, as shown in Figure 14. This basic
point was earlier made by Elmegreen (1993) and Va´zquez-Semadeni et al. (1995). Two criteria must be fulfilled in
these regions. The rms velocity must be high enough, and the driving wavelength λd < λJ(ρ′) small enough. If these
two criteria are not met, the localized high-density regions collapse, although the surrounding flow remains turbulently
supported.
FIG. 15. Density cubes for (a) a model of large-scale driven turbulence and (b) a model of small-scale driven turbulence at
an evolutionary phase when M∗ = 5% of the total mass has been accumulated by protostars. Compare with Figure 13b.
Together they illustrate the influence of different driving wavelengths for otherwise identical physical parameters. Larger-scale
driving results in a more organized distribution of protostars (black dots), while smaller-scale driving results in a more random
structure. Note the different times at which M∗ = 5% is reached. (From Klessen et al. 2000.)
The length scale and strength of energy injection into the system determine the structure of the turbulent flow,
and therefore the locations at which stars are most likely to form. Large-scale driving leads to large coherent shock
structures (Figure 15a). Local collapse occurs predominantly in these filaments and layers of shocked gas (Klessen et
al. 2000). Increasing the driving scale produces larger and more massive structures that can become gravitationally
unstable. Hence, the star formation efficiency (eq. 2) increases. The same is true for weaker driving. Reducing the
turbulent kinetic energy means that more and larger volumes exceed the Jeans criterion for gravitational instability.
The more massive the unstable region is, the more stars it will form. Dense clusters or associations of stars build up,
either in the complete absence of energy input, or when small-scale turbulence is too weak to support large volumes,
or when large-scale turbulence sweeps up large masses of gas that collapse. In all of these cases star formation is high
efficiency and proceeds on a free-fall timescale (Klessen et al. 1998, Klessen & Burkert 2000, 2001).
35

amount of gas available for collapse on scales where turbulence turns from supersonic to subsonic (Va´zquez-Semadeni,
Ballesteros-Paredes, & Klessen 2003). Sufficiently strong driving on short enough scales can prevent local collapse for
arbitrarily long periods of time, but such strong driving may be rather difficult to arrange in a real molecular cloud.
If we assume that stellar driving sources have an effective wavelength close to their separation, then the condition
that driving acts on scales smaller then the Jeans wavelength in ‘typical’ shock generated gas clumps requires the
presence of an extraordinarily large number of stars evenly distributed throughout the cloud, with typical separation
0.1 pc in Taurus, or only 350 AU3 in Orion. This is not observed. Very small driving scales seem to be at odds with
the observed large-scale velocity fields in at least some molecular clouds (e.g. Ossenkopf & Mac Low 2002).
Triggering of star formation by compression has been discussed at least since the work of Elmegreen & Lada (1977)
on star formation in the gas swept up by expanding H ii regions. The turbulent compressions described here do
indeed trigger local star formation in globally supported regions. However, in the absence of the flow, global collapse
would cause more vigorous star formation. Indeed, Elmegreen & Lada (1977) noted themselves that the time for
gravitational instability to occur in the shocked, compressed layer was actually somewhat longer than the Jeans time
in the undisturbed cloud. We discuss this issue further in Section VI.D.2.
The domination of cloud structure by large-scale modes leads to the formation of stars in groups and clusters.
When stars form in groups, their velocities initially reflect the turbulent velocity field of the gas from which they
formed. However, as more and more mass accumulates in protostars, their mutual gravitational interaction becomes
increasingly important, beginning to determine the dynamical state of the system, which then behaves more and more
like a collisional N -body system, where close encounters occur frequently (see Section V.D).
F. Effects of magnetic fields
So far, we have concentrated on the effects of purely gas dynamical turbulence. How does the picture discussed
here change if we consider the presence of magnetic fields? Magnetic fields have been suggested to support molecular
clouds well enough to prevent gravitationally unstable regions from collapsing (McKee 1999), either magnetostatically
or dynamically through MHD waves.
Assuming ideal MHD, a self-gravitating cloud of mass M permeated by a uniform flux Φ is stable unless the mass-
to-flux ratio exceeds the value given by Eq. (19). Without any other mechanism of support, such as turbulence acting
along the field lines, a magnetostatically supported cloud collapses to a sheet which is then supported against further
collapse. Fiege & Pudritz (1999) found an equilibrium configuration of helical field that could support a filament,
rather than a sheet, from fragmenting and collapsing. Such configurations do not appear in numerical models of
turbulent molecular clouds, however, suggesting that reaching this stable equilibrium is difficult.
Investigation of support by MHD waves concentrates mostly on the effect of Alfve´n waves, as they (1) are not
as subject to damping as magnetosonic waves and (2) can exert a force along the mean field, as shown by Dewar
(1970) and Shu et al. (1987). This is because Alfve´n waves are transverse waves, so they cause perturbations δ ~B
perpendicular to the mean magnetic field ~B. McKee & Zweibel (1994) argue that Alfve´n waves can even lead to an
isotropic pressure, assuming that the waves are neither damped nor driven. However, in order to support a region
against self-gravity, the waves would have to propagate outwardly, because inwardly propagating waves would only
further compress the cloud. Thus, this mechanism requires a negative radial gradient in wave sources in the cloud
(Shu et al. 1987).
3One astronomical unit is the mean radius of Earths orbit around the Sun, 1AU = 1.5× 1013 cm.
37

velocity
magnetic eld
FIG. 17. Two dimensional slice through a cube of magnetostatically supported, self-gravitating turbulence driven at large scale
from Heitsch et al. (2001a). The upper panel shows velocity vectors and the lower panel magnetic field vectors. The initial
magnetic field is along the z-direction, i.e. vertically oriented in all plots presented. The field is strong enough in this case not
only to prevent the cloud from collapsing perpendicular to the field lines, but even to suppress turbulent motions in the cloud.
The turbulence barely affects the mean field. The density greyscale is given in the colorbar, in model units. The time shown
is t = 5.5tff . (From Heitsch et al. 2001a.)
It can be demonstrated (e.g. Heitsch et al. 2001a) that supersonic turbulence does not cause a magnetostatically
supported region to collapse, and vice versa, that in the absence of magnetostatic support, MHD waves cannot
completely prevent collapse, although they can retard it to some degree. The case of a subcritical region with
M < Mcr is illustrated in Figure 17. Indeed, sheets form, roughly perpendicular to the field lines. This is because
the turbulent driving can shift the sheets along the field lines without changing the mass-to-flux ratio. The sheets
do not collapse further, because the shock waves cannot sweep gas across field lines and the entire region is initially
supported magnetostatically.
38

FIG. 19. Upper panel: Core-mass accretion rates for 10 different low-resolution models (643 zones) of purely gas dynamic
turbulence with identical parameters but different realizations of the random turbulent driving. The dotted line shows a mean
accretion rate calculated from averaging over the sample. For comparison, higher-resolution runs with identical parameters but
1283 (dashed line) and 2563 (thick solid line) zones are shown as well. Lower panel: Mass accretion rates for various models with
different magnetic field strength and resolution. Common to all models is the occurrence of local collapse and star formation
regardless of the detailed choice of parameters, as long as the system is magnetostatically supercritically. (For further details
see Heitsch et al. 2001a.)
G. Promotion and prevention of local collapse
Highly compressible turbulence both promotes and prevents collapse. Its net effect is to inhibit collapse globally,
while perhaps promoting it locally. This can be seen by examining the dependence of the Jeans mass MJ ∝ ρ−1/2c3s ,
Eq. (14), on the rms turbulent velocity vrms. If we follow the classical picture that treats turbulence as an additional
pressure (Chandrasekhar 1951a,b), then we define c2s,eff = c2s +v2rms/3, giving the Jeans mass a dependence on velocity
of v3rms. However, compressible turbulence in an isothermal medium causes local density enhancements that increase
the density by M2 ∝ v2rms, adding a dependence 1/vrms. Combining these two effects, we find that
MJ ∝ v2rms (27)
for vrms ≫ cs, so that ultimately turbulence does inhibit collapse. However, there is a broad intermediate region,
especially for long wavelength driving, where local collapse occurs despite global support, as shown in Figure 15,
which can be directly compared with Figure 13b).
The total mass and lifetime of a Jeans-unstable fluctuation determine whether it will actually collapse. Roughly
speaking, the lifetime of an unstable clump is determined by the interval between two successive passing shocks:
the first creates it, while the second one, if strong enough, may disrupt the clump again (Klein, McKee & Colella
1994, Mac Low et al. 1994). If the time interval between the two shocks is sufficiently long, however, an unstable
clump can contract to high enough densities to effectively decouple from the ambient gas flow and becomes able to
survive the encounter with further shock fronts (e.g. Krebs & Hillebrandt 1983). Then it continues to accrete from
the surrounding gas, forming a dense core.
A more detailed understanding of how local collapse proceeds comes from examining the full time history of
accretion for different models (Figure 14). The cessation of strong accretion onto cores occurs long before all gas has
40

J. Termination of local star formation
It remains quite unclear what terminates stellar birth on scales of individual star forming regions, and even whether
these processes are the primary factor determining the overall efficiency of star formation in a molecular cloud (eq. 2).
Three main possibilities exist. First, feedback from the stars themselves in the form of ionizing radiation and stellar
outflows may heat and stir surrounding gas up sufficiently to prevent further collapse and accretion. Second, accretion
might peter out either when all the high density, gravitationally unstable gas in the region has been accreted in
individual stars, or after a more dynamical period of competitive accretion, leaving any remaining gas to be dispersed
by the background turbulent flow. Third, background flows may sweep through, destroying the cloud, perhaps in the
same way that it was created. Most likely the astrophysical truth lies in some combination of all three possibilities.
If a stellar cluster formed in a molecular cloud contains OB stars, then the radiation field and stellar winds from
these high-mass stars strongly influence the surrounding cloud material. The UV flux ionizes gas out beyond the
local star forming region. Ionization heats the gas, raising its Jeans mass, and possibly preventing further protostellar
mass growth or new star formation. The termination of accretion by stellar feedback has been suggested at least
since the calculations of ionization by Oort & Spitzer (1955). Whitworth (1979) and Yorke et al. (1989) computed the
destructive effects of individual blister Hii regions on molecular clouds, while in a series of papers, Franco et al. (1994),
Rodriguez-Gaspar et al. (1995), and Diaz-Miller et al. (1998) concluded that indeed the ionization from massive stars
may limit the star formation efficiency (eq. 2) of molecular clouds to about 5%. Matzner (2002) analytically modeled
the effects of ionization on molecular clouds, concluding as well that turbulence driven by Hii regions could support
and eventually destroy molecular clouds. The key question facing these models is whether Hii region expansion
couples efficiently to clumpy, inhomogeneous molecular clouds, a question probably best addressed with numerical
simulations.
Bipolar outflows are a different manifestation of protostellar feedback, and may also strongly modify the properties
of star forming regions (Norman & Silk 1980, Lada & Gautier 1982, Adams & Fatuzzo 1996). Matzner & McKee
(2000) modeled the ability of bipolar outflows to terminate low-mass star formation, finding that they can limit star
formation efficiencies to 30–50%, although they are ineffective in more massive regions. How important these processes
are compared to simple exhaustion of available reservoirs of dense gas (Klessen et al. 2000, Va´zquez-Semadeni et al.
2003) remains an important question.
The models relying on exhaustion of the reservoir of dense gas argue that only dense gas will actually collapse,
and that only a small fraction of the total available gas reaches sufficiently high densities, due to cooling (Elmegreen
& Parravano 1994, Schaye 2002), gravitational collapse and turbulent triggering (Elmegreen 2002), or both (Wada,
Meurer, & Norman 2002). This of course pushes the question of local star formation efficiency up to larger scales,
which may indeed be the correct place to ask it.
Other models focus on competitive accretion in local star formation, showing that the distribution of masses
in a single group or cluster can be well explained by assuming that star formation is fairly efficient in the dense
core, but that stars that randomly start out slightly heavier tend to fall towards the center of the core and accrete
disproportionately more gas (Bonnell et al. 1997; 2001a). These models have recently been called into question by
the observation that the stars in lower density young groups in Serpens simply have not had the time to engage in
competitive accretion, but still have a normal IMF (Olmi & Testi 2002).
Finally, star formation in dense clouds created by turbulent flows may be terminated by the same flows that created
them. Ballesteros-Pardes et al. (1999a) suggested that the coordination of star formation over large molecular clouds,
and the lack of post-T Tauri stars with ages greater than about 10Myr tightly associated with those clouds, could
be explained by their formation in a larger-scale turbulent flow. Hartmann et al. (2001) make the detailed argument
that these flows may disrupt the clouds after a relatively short time, limiting their star formation efficiency that way.
The extensive evidence for short sequences of cluster ages (e.g. Blaauw 1964, Walborn & Parker 1992, Efremov &
Elmegreen 1998b) is often attributed to sequential triggering (Elmegreen & Lada 1977) by shock fronts expanding into
existing clouds. An additional mechanism for producing such sequences may be the sequential formation of clouds
by the larger scale flow. Below, in Section VI.C we argue that field supernovae are the most likely driver for the
background turbulence, at least in the star-forming regions of galaxies. Supernovae associated with any particular
star-forming region will not be energetically important, although they may produce locally significant compressions.
K. Outline of a new theory of star formation
The support of star-forming clouds by supersonic turbulence can explain many of the same observations successfully
explained by the standard theory, while also addressing the inconsistencies between observation and the standard
43

theory described in the previous section. The key point that is new in our argument is that supersonic turbulence
produces strong density fluctuations in the interstellar gas, sweeping gas up from large regions into dense sheets and
filaments, and does so even in the presence of magnetic fields. Supersonic turbulence decays quickly, but so long as
it is maintained by input of energy from some driver it can support regions against gravitational collapse.
Such support comes at a cost, however. The very turbulent flows that support the region produce density enhance-
ments in which the Jeans mass drops as MJ ∝ ρ−1/2 (Eq. 14), and the magnetic critical mass above which magnetic
fields can no longer support against that collapse drops even faster, as Mcr ∝ ρ−2 (Eq. 18). For local collapse to actu-
ally result in the formation of stars, Jeans-unstable, shock-generated, density fluctuations must collapse to sufficiently
high densities on time scales shorter than the typical time interval between two successive shock passages. Only then
can they decouple from the ambient flow and survive subsequent shock interactions. The shorter the time between
shock passages, the less likely these fluctuations are to survive. Hence, the timescale and efficiency of protostellar
core formation depend strongly on the wavelength and strength of the driving source, and the accretion histories of
individual protostars are strongly time varying (Section V.E). Global support by supersonic turbulence thus tends to
produce local collapse and low rate star formation, exactly as seen in low-mass star formation regions characteristic
of the disks of spiral galaxies. Conversely, lack of turbulent support results in regions that collapse freely. In gas
dynamic simulations, freely collapsing gas forms a web of density enhancements in which star formation can proceed
efficiently, as seen in regions of massive star formation and starbursts.
The regulation of the star formation rate then occurs not just at the scale of individual star-forming cores through
ambipolar diffusion balancing magnetostatic support, but rather at all scales via the dynamical processes that de-
termine whether regions of gas become unstable to prompt gravitational collapse. Efficient star formation occurs in
collapsing regions; apparent inefficiency occurs when a region is turbulently supported and only small subregions get
compressed sufficiently to collapse. The star formation rate is determined by the balance between turbulent support
and local density, and is a continuous function of the strength of turbulent support for any given region. Fast and
efficient star formation is the natural behavior of gas lacking sufficient turbulent support for its local density.
Regions that are gravitationally unstable in this picture collapse quickly, on the free-fall timescale. They never
pass through a quasi-equilibrium state as envisioned by the standard model. Large-scale density enhancements such
as molecular clouds could be caused either by gravitational collapse, or by ram pressure from turbulence. If collapse
does not succeed, the same large-scale turbulence that formed molecular clouds can destroy them again.
V. LOCAL STAR FORMATION
In this section we apply the theoretical picture of Section IV to observations of individual star forming regions.
We show how the efficiency and time and length scales of star formation depend on the properties of turbulence
(Section V.A), followed by a discussion of the properties of protostellar cores (Section II.E), the immediate progenitors
of individual stars. We then speculate about the formation of binary stars (Section V.C), and stress the importance of
the dynamical interaction between protostellar cores and their competition for mass growth in dense, deeply embedded
clusters (Section V.D). This implies strongly time-varying protostellar mass accretion rates (Section V.E). Finally,
we discuss the consequences of the probabilistic processes of turbulence and stochastic mass accretion for the resulting
stellar inital mass function (Section V.F).
A. Star formation in molecular clouds
Not only does all star formation occur in molecular clouds, but all giant molecular clouds appear to form stars. At
least, all those surveyed within distances less than 3 kpc form stars (Blitz 1993, Williams et al. 2000), except possibly
the Maddalena & Thaddeus (1985) cloud (Lee, Snell, & Dickman 1996; Williams & Blitz 1998), and this last cloud
may have formed just recently.
The star formation process in molecular clouds appears to be fast. Once the collapse of a cloud region sets in,
it rapidly forms an entire cluster of stars within 106 years or less. This is indicated by the young stars associated
with star forming regions, typically T Tauri stars with ages less than 106 years (e.g. Gomez et al. 1992, Green &
Meyer 1996, Carpenter et al. 1997, Hartmann 2001), and by the small age spread in more evolved stellar clusters
(e.g. Hillenbrand 1997, Palla & Stahler 1999, 2001). Star clusters in the Milky Way also exhibit an amazing degree of
chemical homogeneity (in the case of the Pleiades, see Wilden et al. 2002), implying that the gas out of which these
stars formed must have been chemically well-mixed initially (see also Avillez & Mac Low 2002, Klessen & Lin 2003).
44

FIG. 21. Protostellar cores from a model of clustered star formation. The left side shows protostellar cores with collapsed
central objects (indicated by a black dot), the right side “starless” cores without central protostars. Cores are numbered
accoring to their peak density. Surface density contours are spaced logarithmically with two contour levels spanning one
decade, log10 ∆ρ = 0.5. The lowest contour is a factor of 10
0.5 above the mean density. (From Klessen & Burkert 2000.)
B. Protostellar core models
The observed properties of molecular cloud cores as discussed in Section II.E.2 can be compared with gas clumps
identified in numerical models of interstellar cloud turbulence. Like their observed counterparts, the model cores are
generally highly distorted and triaxial. Depending on the projection angle, they often appear extremely elongated,
being part of a filamentary structure that may connect several objects. Figure 21 plots a sample of model cores from
Klessen & Burkert (2000). Those which have already formed a protostellar object in their interior are shown on the
left, while starless cores without central protostars are shown on the right. Note the similarity to the appearance of
observed protostellar cores (Figure 3). The model clumps are clearly elongated. The ratios between the semi-major
and the semi-minor axis measured at the second contour level are typically between 2:1 and 4:1. However, there are
significant deviations from simple triaxial shapes. As a general trend, high density contour levels typically are regular
and smooth, because there the gas is mostly influenced by pressure and gravitational forces. On the other hand, the
lowest contour level samples gas that is strongly influenced by environmental effects. Hence, it appears patchy and
irregular. The location of the protostar is not necessarily identical with the center of mass of the core, especially when
it is irregularly shaped.
46

“run away” T-Tauri stars found in X-ray observation in the vicinities of star-forming molecular clouds (e.g. Sterzik
& Durison 1995, 1998, Smith et al. 1997; Klessen & Burkert 2000; or for observations e.g. Neuha¨user et al. 1995; or
Wichmann et al. 1997). However, many of these stars could not have traveled to their observed positions in their own
lifetimes if they were formed in the currently star-forming cloud It appears more likely that the observed extended
stellar population is associated with clouds that dispersed long ago.
(c) Dynamical interaction leads to mass segregation. Star clusters evolve towards equipartition. For massive
stars this means that they have on average smaller velocities than low-mass stars (in order to keep the kinetic
energy Ekin = 1/2mv2 roughly constant). Thus, massive stars sink towards the cluster center, while low-mass stars
predominantly populate large cluster radii (e.g. Kroupa 1995a,b,c). This holds already for nascent star clusters in the
embedded phase (e.g. Bonnell & Davis 1998).
(d) Dynamical interaction and competition for mass accretion lead to highly time-variable protostellar mass growth
rates. This is discussed in more detail in Section V.E.
(e) The radii of stars in the pre-main sequence contraction phase are several times larger than stellar radii on the
main sequence (for a review on pre-main sequence evolution see, e.g., Palla 2000, 2002). Stellar collisions are therefore
more likely to occur during the very early evolution of star clusters. During the embedded phase the encounter
probability is further increased by gas drag and dynamical friction.
Collisions in dense protostellar clusters have been proposed as mechanism to produce massive stars (Bonnell, Bate,
& Zinnecker 1998; Stahler, Palla, & Ho 2000). The formation of massive stars has long been considered a puzzle in
theoretical astrophysics, because 1D calculations predict for stars above ∼ 10M⊙ the radiation pressure acting on
the infalling dust grains to be strong enough to halt or even revert further mass accretion (e.g. Yorke & Kru¨gel 1977;
Wolfire & Cassinelli 1987; or Palla 2000, 2001). However, detailed 2D calculations by Yorke & Sonnhalter (2002)
demonstrate that in the more realistic scenario of mass growth via an accretion disk the radiation barrier may be
overcome. Mass can accrete from the disk onto the star along the equator while radiation is able to escape along the
polar direction. Massive stars, thus, may form via the same processes as ordinary low-mass stars (also see McKee &
Tan 2002). Collisional processes need not be invoked.
E. Accretion rates
When a gravitationally unstable gas clump collapses onto a central star, it follows the observationally well-
determined sequence described in Section II.E. In the main accretion phase (class 0), the release of gravitational
energy by accretion dominates the energy budget. Hence, protostars exhibit large IR and sub-millimeter luminosities
and drive powerful outflows. Both phenomena can be used to estimate the protostellar mass accretion rate M˙ (e.g.
Andre´ & Montmerle 1994, Bontemps et al. 1996, Henriksen, Andre´, & Bontemps 1997). These observations suggest
that M˙ varies strongly, and declines with time after the class 0 phase (Section III.D.3.a). The estimated lifetimes are
a few tens of thousands of years for the class 0 and a few hundreds of thousands of years for the class I phase.
Most models of protostellar core collapse concentrate on isolated objects, whether the models are analytic (e.g.
Larson 1969, Penston 1969a, Hunter 1977, Henriksen et al. 1997, Basu 1997) or numerical (e.g. Foster & Chevalier
1993, Tomisaka 1996, Ogino, Tomisaka, & Nakamura 1999, Wuchterl & Tscharnuter 2002). A typical accretion history
is shown in Figure 8. However, stars predominantly form in groups and clusters. Numerical studies that investigate
the effect of the cluster environment on protostellar mass accretion rates have been reported by Bonnell et al. (1990,
2001a,b), Klessen & Burkert (2000, 2001), Klessen et al. (2001), Heitsch et al. (2001a), and Klessen (2001a). These
numerical models suggest the following predictions about protostellar accretion in dense clusters:
(a) Protostellar accretion rates from turbulent fragmentation in a dense cluster environment are strongly time
variable. This is illustrated in Figure 24 for 49 randomly selected cores from the model by Klessen (2001a).
(b) The typical density profiles of gas clumps that give birth to protostars indeed exhibit a flat inner core, followed
by a density fall-off ρ ∝ r−2, and are truncated at some finite radius, which in the dense centers of clusters often is
due to tidal interaction with neighboring cores (see Section II.E and Section V.D). As a result, the modeled accretion
rates agree well with the observations. A short-lived initial phase of strong accretion occurs when the flat inner part
of the pre-stellar clump collapses, corresponding to class 0. If the cores remain isolated and unperturbed, the mass
growth rate gradually declines in time as the outer envelope accretes, giving class I. Once the truncation radius is
reached, accretion fades and the object enters class II. However, collapse does not start from rest for the density
fluctuations considered here, so accretion rates exceed the values predicted by models of isolated objects, even for
objects in the simulations far from their nearest neighbors.
(c) The mass accretion rates of protostellar cores in a dense cluster also deviate strongly from the rates of isolated
cores because of mergers and competition between cores, as discussed above (Section V.D). Mergers drastically change
50

Magellanic Cloud (Sirianni et al. 2002). Dynamical effects during the embedded phase of star cluster evolution enhance
this initial segregation even further (see Section V.D.c).
(f) Individual cores in a cluster environment form and evolve through a sequence of highly probabilistic events, so
their accretion histories differ even if they accumulate the same final mass. Accretion rates for protostars of a certain
mass can only be determined in a statistical sense. Klessen (2001a) suggests that an exponentially declining rate
with a peak value of a few 105 M⊙yr−1, a time constant in the range 0.5 to 2.5× 105 yr, and a cut-off related to gas
dispersal from the cluster offers a reasonable fit to the average protostellar mass growth in dense embedded clusters,
but with large variations.
F. Initial mass function
Knowledge of the distribution of stellar masses at birth, described by the initial mass function (IMF), is necessary to
understand many astrophysical phenomena, but no analytic derivation of the observed IMF has yet stood the test of
time. In fact, it appears likely that a fully deterministic theory for the IMF does not exist. Rather, any viable theory
must take into account the probabilistic nature of the turbulent process of star formation, which is inevitably highly
stochastic and indeterminate. We gave a brief overview of the observational constraints on the IMF in Section II.F,
and we here review models for it.
1. Models of the IMF
Existing models to explain the IMF can be divided into five major groups. In the first group feedback from the stars
themselves determines their masses. Silk (1995) suggests that stellar masses are limited by the feedback from both
ionization and protostellar outflows. Nakano, Hasegawa, & Norman (1995) describe a model in which stellar masses
are sometimes limited by the mass scales of the formative medium and sometimes by stellar feedback. Adams &
Fatuzzo (1996) apply the central limit theorem to the hypothesis that many independent physical variables contribute
to the stellar masses to derive a log-normal IMF regulated by protostellar feedback. However, for the overwhelming
majority of stars with masses M <∼ 5M⊙, protostellar feedback (i.e. winds, radiation and outflows) are unlikely to be
strong enough to halt mass accretion, as shown by detailed protostellar collapse calculations (e.g. Wuchterl & Klessen
2001, Wuchterl & Tscharnuter 2003).
In the second group of models, initial and environmental conditions determine the IMF. In this picture, the structural
properties of molecular clouds determine the mass distribution of Jeans-unstable gas clumps, and the clump properties
determine the mass of the stars that form within. If one assumes a fixed star formation efficiency for individual
clumps, there is a one-to-one correspondence between the molecular cloud structure and the final IMF. The idea that
fragmentation of clouds leads directly to the IMF dates back to Hoyle (1953) and later Larson (1973). More recently,
this concept has been extended to include the observed fractal and hierarchical structure of molecular clouds Larson
(1992, 1995). Indeed random sampling from a fractal cloud seems to be able to reproduce the basic features of the
observed IMF (Elmegreen & Mathieu 1983, Elmegreen 1997, 1999, 2000a,c, 2002). A related approach is to see the
IMF as a domain packing problem (Richtler 1994).
The hypothesis that stellar masses are determined by clump masses in molecular clouds is supported by observations
of the dust continuum emission of protostellar condensations in the Serpens, ρOphiuchi, and Orion star forming regions
(Testi & Sargent 1998; Motte et al. 1998, 2001; Johnstone et al. 2000, 2001). These protostellar cores are thought
to be in a phase immediately before a star forms in their interior. Their mass distribution resembles the stellar IMF
reasonably well, suggesting a close correspondence between protostellar clump masses and stellar masses, leaving little
room for stellar feedback processes, competitive accretion, or collisions to act to determine the stellar mass spectrum.
A third group of models relies on competitive coagulation or accretion processes to determine the IMF. This has
a long tradition and dates back to investigations by Oort (1954) and Field & Saslaw (1965), but the interest in this
concept continues to the present day (e.g. Silk & Takahashi 1979; Lejeune & Bastien 1986; Price & Podsiadlowski
1995; Murray & Lin 1996; Bonnell et al. 2001a,b; Durisen, Sterzik, & Pickett 2001). Stellar collisions require very
high stellar densities, however, for which observational evidence and theoretical mechanisms remain scarce.
Fourth, there are models that connect the supersonic turbulent motions in molecular clouds to the IMF. In particular
there are a series of attempts to find an analytical relation between the stellar mass spectrum and statistical properties
of interstellar turbulence (e.g. Larson 1981, Fleck 1982, Hunter & Fleck 1982, Elmegreen 1993, Padoan 1995, Padoan
et al. 1997, Myers 2000, Padoan & Nordlund 2002). However, properties such as the probability distribution of density
in supersonic turbulence in the absence of gravity have never successfully been shown to have a definite relationship
52

When molecular clouds were first discovered, they were thought to have lifetimes of over 100Myr (e.g. Scoville &
Hersh 1979) because of their apparent predominance in the inner galaxy. These estimates were shown to depend upon
too high a conversion factor between CO and H2 masses by Blitz & Shu (1980). They revised the estimated lifetime
down to roughly 30Myr based on the association of clouds with spiral arms, apparent ages of associated stars, and
overall star formation rate in the Galaxy.
Chemical equilibrium models of dense cores in molecular clouds (as reviewed, for example, by Irvine, Goldsmith, &
Hjalmarson 1986) showed disagreements with observed abundances in a number of molecules. These cores would take
as long as 10Myr to reach equilibrium, which could still occur in the standard model. However, Prasad, Heere, &
Tarafdar (1991) demonstrated that the abundances of the different species agreed much better with the results at times
of less than 1Myr from time-dependent models of the chemical evolution of collapsing cores (also see Section III.D.2.d).
Bergin & Langer (1997), Bergin et al. (1997), Pratap et al. (1997), and Aikawa et al. (2001) came to similar conclusions
from careful comparison of several different cloud cores to extensive chemical model networks. Aikawa et al. (2001)
and Saito et al. (2002) also studied deuterium fractionation, again finding short lifetimes.
Ballesteros-Paredes et al. (1999a) argue for a lifetime of less than 10Myr for molecular clouds as a whole. They
base their argument on the notable lack of a population of 5–20 Myr old stars in molecular clouds. Stars in the
clouds typically have ages under 3–5 Myr, judging from their position on pre-main-sequence evolutionary tracks in
a Hertzsprung-Russell diagram (D’Antona & Mazzitelli 1994; Swenson et al. 1994; with discrepancies resolved by
Stauffer, Hartmann, & Barrado y Navascues 1995). Older weak-line T Tauri stars identified by X-ray surveys with
Einstein (Walter et al. 1988) and ROSAT (Neuha¨user et al. 1995) are dispersed over a region as much as 70 pc
away from molecular gas, suggesting that they were not formed in the currently observed gas (Feigelson 1996).
Leisawitz, Bash, & Thaddeus (1989), Fukui et al. (1999) and Elmegreen (2000) have made similar arguments based
on the observation that only stellar clusters with ages under about 10Myr are associated with substantial amounts
of molecular gas in the Milky Way and the LMC.
FIG. 26. Log of number density in a cut through the galactic plane from a 3D SN-driven model of the ISM with resolution of
1.25 pc, including radiative cooling and the vertical gravitational field of the stellar disk, as described by Mac Low et al. (2003)
and Avillez & Breitschwerdt (2003). High-density, shock-confined regions are naturally produced by intersecting SN-shocks
from field SNe.
55

For these short lifetimes to be plausible, either molecule formation must proceed quickly, and therefore at high
densities, or observed molecular clouds must be formed from preexisting molecular gas, as suggested by Pringle,
Allen, & Lubow (2001). A plausible place for fast formation of H2 at high density is the shock compressed layers
naturally produced in a supernova-driven ISM, as shown in Figure 26 from simulations similar to those described in
Mac Low et al. (2003) and Avillez & Breitschwerdt (2003). Similar morphologies have been seen in many other global
simulations of the ISM, including Rosen, Bregman, & Norman (1993), Rosen & Bregman (1995), Rosen, Bregman,
& Kelson (1996), Korpi et al. (1999), Avillez (2000), and Wada & Norman (1999, 2001). Mac Low (2000) reviews
these earlier simulations. Mac Low et al. (2003) showed that pressures in the ISM are broadly distributed, with peak
pressures in cool gas (T < 103 K) as much as an order of magnitude above the average because of shock compressions
(also see Passot & Va´zquez-Semadeni 2003). This gas is swept up from ionized 104 K gas, so between cooling and
compression its density has already been raised up to two orders of magnitude from n ≈ 1 cm−3 to n ≈ 100 cm−3.
These simulations did not include a correct cooling curve below 104 K, so further cooling could not occur even if
physically appropriate, but it would be expected.
We can understand this compression quantitatively. The sound speed in the warm gas is (8.1 km s−1)(T/104 K)1/2,
taking into account the mean mass per particle µ = 2.11×10−24 g for gas 90% H and 10% He by number. The typical
velocity dispersion for this gas is 10–12 km s−1 (e.g. Dickey & Lockman 1990, Dickey, Hanson, & Helou 1990), so that
shocks with Mach numbers M = 2–3 are moderately frequent. Temperatures in these shocks reach values T ≤ 105 K,
which is close to the peak of the interstellar cooling curve (e.g. Dalgarno & McCray 1972; Raymond, Cox, & Smith
1976), so the gas cools quickly back to 104 K. The density behind an isothermal shock is ρ1 = M2ρ0, where ρ0 is
the pre-shock density, so order of magnitude density enhancements occur easily. The optically-thin radiative cooling
rate Λ(T ) drops off at 104 K as H atoms no longer radiate efficiently (Dalgarno & McCray 1972; Spaans & Norman
1997), but the radiative cooling L ∝ n2Λ(T ). Therefore density enhancements strongly increase the ability to cool.
Hennebelle & Pe´rault (1999) show that such shock compressions can trigger the isobaric thermal instability (Field et
al. 1969; Wolfire et al. 1995), reducing temperatures to of order 100K or less. Heiles (2001) observes a broad range
of temperatures for neutral hydrogen from below 100K to a few thousandK. The reduction in temperature by two
orders of magnitude from 104 K to 100K raises the density correspondingly. Combined with the initial isothermal
shock compression, this results in a total of as much as three orders of magnitude of compression. Gas that started
at densities somewhat higher than average, say at 10 cm−3, can be compressed to densities of 104 cm−3, enough to
reduce H2 formation times to a few hundred thousand years.
FIG. 27. Shock velocities Vd and pre-shock number densities n at which the cold post-shock layer is more than 8% molecular,
taken from 1D simulations by Koyama & Inutsuka (2000) that include H2 formation and dissociation, and realistic heating and
cooling functions from Wolfire et al. (1995).
Koyama & Inutsuka (2000) have demonstrated numerically that shock-confined layers do indeed quickly develop
high enough densities to form H2 in under a million years, using 1D computations including heating and cooling rates
from Wolfire et al. (1995) and H2 formation and dissociation. In Figure 27 we show the parameter space in which
56

they find H2 formation is efficient. Hartmann, Ballesteros-Paredes, and Bergin (2001) make a more general argument
for rapid H2 formation, based in part on lower-resolution, 2D simulations described by Passot, Va´zquez-Semadeni,
& Pouquet (1995) that could not fully resolve realistic densities like those of Koyama & Inutsuka (2000), but do
include larger-scale flows showing that the initial conditions for the 1D models are quite reasonable. Hartmann et
al. (2001) further argue that the self-shielding against the background UV field also required for H2 formation will
become important at approximately the same column densities required to become gravitationally unstable.
FIG. 28. Instability of radiatively cooled layer confined from left and right by strong shocks with Mach number M = 16.7,
computed in two dimensions with an adaptive mesh refinement technique by Walder & Folini (2000). White regions have
densities of 14 cm−3, while the darkest regions have densities over 104 cm−3.
Shock-confined layers were shown numerically to be unstable by Hunter et al. (1986) in the context of colliding
spherical density enhancements, and by Stevens, Blondin, & Pollack (1992) in the context of colliding stellar winds.
Vishniac (1994) demonstrated analytically that isothermal, shock-confined layers are subject to a nonlinear thin shell
instability. The physical mechanism can be seen by considering a shocked layer perturbed sinusoidally. The ram
pressure on either side of the layer acts parallel to the incoming flow, and thus at an angle to the surface of the
perturbed layer. Momentum is deposited in the layer with a component parallel to the surface, which drives material
towards extrema in the layer, causing the perturbation to grow. A numerical study by Blondin & Mark (1996) in two
dimensions demonstrated that the nonlinear thin shell instability saturates in a thick layer of transsonic turbulence
when the flows become sufficiently chaotic that the surface no longer rests at a substantial angle to the normal of the
incoming flow.
57

are situations where cooling will make the difference between gravitational stability and instability, but are those just
marginal cases or the primary driver for star formation in galaxies?
Starburst
Si
ze
High SFR Low SFR
Warm gas Toomre
unstable?
Is warm gas
compressed and
cooled?
N
Y
N
Isolated SF
(Taurus)
Local
Burst
(Orion)
Y
Can cold gas
overwhelm turbulent
support?
NY
Vrms from
SFR
LSB
galaxies,
outer disks
 from
dynamics
FIG. 29. Criteria for different regimes of star formation efficiency in galaxies. See text for further details.
In Figure 29 we outline a unified picture that depends on turbulence and cooling to control the star formation rate.
After describing the different elements of this picture, we discuss the steps that we think will be needed to move
from this cartoon to a quantitative theory of the star formation rate. The factor that determines the star formation
rate above any other is whether the gas is sufficiently dense to be gravitationally unstable without additional cooling.
Galactic dynamics, and interactions with other galaxies and the surrounding intergalactic gas determine the average
gas densities in different regions of a galaxy. The gravitational instability criterion here includes both turbulent
motions and galactic shear, as well as magnetic fields. If gravitational instability sets in at large scale, collapse will
continue so long as sufficient cooling mechanisms exist to prevent the temperature of the gas from rising (effective
adiabatic index γeff ≤ 1). Molecular clouds can form in less than 105 yr, as the gas passes through densities of
104 cm−3 or higher, as an incidental effect of the collapse. A starburst results, with stars forming efficiently in
compact clusters. The size of the gravitationally unstable region determines the size of the starburst.
If turbulent support, rather than thermal support, prevents the gas from immediately collapsing, compression-
induced cooling can become important. Supersonic turbulence compresses some fraction of the gas strongly. As
most cooling mechanisms depend on the gas density non-linearly, the compressed regions cool quickly. When these
regions reach densities of order 104 cm−3, again molecule formation occurs, allowing the gas to cool to even lower
temperatures (see Section VI.A). These cold regions then can become gravitationally unstable and collapse, if allowed
by the local turbulence. Triggering by nearby star formation events (Elmegreen & Lada 1977) represents a special
case of this mode (see Section VI.D.2). This mechanism is less efficient than prompt gravitational instability, as much
of the gas is not compressed enough to form molecules. It is, however, more efficient in regions of higher average
density. Galactic dynamics again determines the local average density, and so, in the end, the star formation efficiency
in this regime as well.
If the turbulence even in the cooled regions supports the gas against general gravitational collapse, isolated, low-rate
star formation can still occur locally in regions further compressed by the turbulence. This may describe regions of
low-mass star formation like the Taurus clouds. On the other hand, if the cooled gas begins to collapse gravitationally,
locally efficient star formation can occur. The size of the gravitationally unstable region then really determines whether
a group, OB association, or bound cluster eventually forms. Star formation in regions like Orion may result from this
branch.
59

2. Gravitational instabilities in galactic disks
Now let us consider the conditions under which gravitational instability will set in. On galactic scales, the Jeans
instability criterion for gravitational instability must be modified to include the additional support offered by the
shear coming from differential rotation, as well as the effects of magnetic fields. The gravitational potential of the
stars can also contribute to gravitational instability on large scales. Which factor determines the onset of gravitational
instability remains unknown. Five that have been proposed are the temperature of the cold phase, the surface density,
the local shear, the presence of magnetic fields, and the velocity dispersion, in different combinations.
We can heuristically derive the Toomre (1964) criterion for stability of a rotating, thin disk with uniform velocity
dispersion σ and surface density Σ using timescale arguments (Schaye 2003). First consider the Jeans criterion for
instability in a thin disk, which requires that the timescale for collapse of a perturbation of size λ
tcoll =
√
λ/GΣ (34)
be shorter than the time required for the gas to respond to the collapse, the sound crossing time
tsc = λ/cs. (35)
This implies that gravitational stability requires perturbations with size
λ < c2s/GΣ. (36)
Similarly, in a disk rotating differentially, a perturbation will spin around itself, generating centrifugal motions that can
also support against gravitational collapse. This will be effective if the collapse timescale tcoll exceeds the rotational
period trot = 2π/κ, where κ is the epicyclic frequency, so that stable perturbations have
λ > 4π2GΣ/κ2. (37)
A regime of gravitational instability occurs if there are wavelengths that lie between the regimes of pressure and
rotational support, with
c2s
GΣ < λ <
4π2GΣ
κ2 . (38)
This will occur if
Q = csκ/2πGΣ < 1, (39)
which is the Toomre criterion for gravitational instability to within a factor of two. The full criterion from a linear
analysis of the equations of motion of gas in a shearing disk gives a factor of π in the denominator (Safronov 1960,
Goldreich & Lynden-Bell 1965), while a kinetic theory approach appropriate for a collisionless stellar system gives a
factor of 3.36 (Toomre 1964).
Kennicutt (1989) and Martin & Kennicutt (2001) have demonstrated that the Toomre criterion generally can
explain the location of the edge of the star-forming disk in galaxies, although they must introduce a correction factor
α = 0.69 ± 0.2 into the left-hand-side of equation (39). Schaye (2003) notes that this factor should be corrected to
α = 0.53 to account for the use of the velocity dispersion rather than sound velocity, and the exact Toomre criterion
for a stellar rather than a gas disk.
The Toomre criterion given in Eq. (39) was derived for a pure gas disk with uniform temperature and velocity
dispersion, and no magnetic field. Relaxation of each of these assumptions modifies the criterion, and indeed each
has been argued to be the controlling factor in determining star formation thresholds by different authors.
Stars in a gas disk respond as a collisionless fluid to density perturbations large compared to their mean separation.
Jog & Solomon (1984a) computed the Toomre instability in a disk composed of gas and stars, and found it to always be
more unstable than either component considered individually. Both components contribute to the growth of density
perturbations, allowing gravitational collapse to occur more easily. Taking into account both gas (subscript g) and
stars (subscript r), instability occurs when
2πGk
(
Σr
κ2 + k2c2sr
+
Σg
κ2 + k2c2sg
)
> 1, (40)
where k = 2π/λ is the wavenumber of the perturbation considered. Jog & Solomon (1984a) and Romeo (1992)
extended this model to include the effect of the finite thickness of the disk. Elmegreen (1995) was able with some
60

The isobaric instability condition derived by Field (1965) is
(∂L
∂T
)
P
=
(∂L
∂T
)
ρ
− ρ0T0
(∂L
∂ρ
)
T
< 0, (41)
where ρ0 and T0 are the equilibrium values. Optically thin radiative cooling in the interstellar medium gives a cooling
function that can be expressed as a piecewise power law Λ ∝ ρ2T βi , where βi gives the value for a temperature range
Ti−1 < T < Ti, while photoelectric heating is independent of temperature. Isobaric instability occurs when βi < 1,
while isochoric instability only occurs with βi < 0 (e.g. Field 1965).
In interstellar gas cooling with equilibrium ionization, there are two temperature ranges subject to thermal insta-
bility (Pikel’ner 1968, Field et al. 1969). In the standard picture of the three-phase interstellar medium governed by
thermal instability (McKee & Ostriker 1977), the higher of these, with temperatures 104.5 K < T < 107 K (Raymond,
Cox, & Smith 1977), separates hot gas from the warm ionized medium. The lower range of 101.7K < T < 103.7 K
(Fig. 3a of Wolfire et al. 1995) separates the warm neutral medium from the cold neutral medium. Cooling of gas
out of ionization equilibrium has been studied in a series of papers by Spaans (1996, Spaans & Norman 1997, Spaans
& Van Dishoeck 1997, Spaans & Carollo 1998) as described by Spaans & Silk (2000). The effective adiabatic index
depends quite strongly on the details of the local chemical, dynamical and radiation environment, in addition to the
pressure and temperature of the gas. Although regions of thermal instability occur, the pressures and temperatures
may depend strongly on the details of the radiative transfer in a turbulent medium, the local chemical abundances,
and other factors.
When thermal instability occurs, it can drive strong motions that dynamically compress the gas nonlinearly. There-
after, neither the isobaric nor the isochoric instability conditions hold, and the structure of the gas is determined by
the combination of dynamics and thermodynamics (Meerson 1996, Burkert & Lin 2000, Lynden-Bell & Tout 2001,
Sa´nchez-Salcedo et al. 2002, Kritsuk & Norman 2002). Va´zquez-Semadeni, Gazol, & Scalo (2000) examined the behav-
ior of thermal instability in the presence of driven turbulence, magnetic fields, and Coriolis forces and concluded that
the structuring effect of the turbulence overwhelmed that of thermal instability in a realistic environment. Gazol et al.
(2001) and Sa´nchez-Salcedo, Va´zquez-Semadeni, & Gazol (2002) found that about half of the gas in such a turbulent
environment actually has temperatures falling in the thermally unstable region, and emphasize that a bimodal tem-
perature distribution may simply be a reflection of the gas cooling function, not a signature of a discontinuous phase
transition. Mac Low et al. (2003) examined supernova-driven turbulence and found a broad distribution of pressures,
which were more important than thermal instability in producing a broad range of densities in the interstellar gas.
Heiles (2001) confirmed the suggestions of Dickey, Salpeter, & Terzian (1978), and Mebold et al. (1982) that
substantial amounts of gas lie out of thermal equilibrium. This has provided observational support for a picture in
which turbulent flows rather than thermal instability dominates structure formation prior to gravitational collapse.
Heiles (2001) measured the temperature of gas along lines of sight through the warm and cold neutral medium by
comparing absorption and emission profiles of the Hi 21 cm fine structure line. He found that nearly half of the warm
neutral clouds measured showed temperatures that are unstable according to the application of the isobaric instability
condition, Eq. (41), to the Wolfire et al. (1995) equilibrium ionization phase diagram.
Although the heating and cooling of the gas clearly plays an important role in the star formation process, the
presence or absence of an isobaric instability may be less important than the effective adiabatic index, or similar
measures of the behavior of the gas on compression, in determining its ultimate ability to form stars.
C. Driving mechanisms
Both support against gravity and maintenance of observed motions appear to depend on continued driving of the
turbulence, which has kinetic energy density e = (1/2)ρv2rms. Mac Low (1999, 2002) estimates that the dissipation
rate for isothermal, supersonic turbulence is
e˙ ≃ −(1/2)ρv3rms/Ld = −(3× 10−27 erg cm−3 s−1)
( n
1 cm−3
)( vrms
10 kms−1
)3
( Ld
100 pc
)−1
, (42)
where Ld is the driving scale, which we have somewhat arbitrarily taken to be 100 pc (though it could well be smaller),
and we have assumed a mean mass per particle µ = 2.11× 10−24 g. The dissipation time for turbulent kinetic energy
τd = e/e˙ ≃ Ld/vrms = (9.8Myr)
( Ld
100 pc
)
( vrms
10 kms−1
)−1
, (43)
62

4. Massive stars
In active star-forming galaxies, massive stars probably dominate the driving. They could do so through ionizing
radiation and stellar winds from O stars, or clustered and field supernova explosions, predominantly from B stars no
longer associated with their parent gas. The supernovae appear likely to be most important, as we now show.
a. Stellar winds
First, we consider stellar winds. The total energy input from a line-driven stellar wind over the main-sequence
lifetime of an early O star can equal the energy from its supernova explosion, and the Wolf-Rayet wind can be even
more powerful. However, the mass-loss rate from stellar winds drops as roughly the sixth power of the star’s luminosity
if we take into account that stellar luminosity varies as the fourth power of stellar mass (Vink, de Koter & Lamers
2000), while the powerful Wolf-Rayet winds (Nugis & Lamers 2000) last only 105 years or so, so only the very most
massive stars contribute substantial energy from stellar winds. The energy from supernova explosions, on the other
hand, remains nearly constant down to the least massive star that can explode. As there are far more lower-mass
stars than massive stars, with a Salpeter IMF giving a power-law in mass of α = −2.35 (Eq. 8), supernova explosions
inevitably dominate over stellar winds after the first few million years of the lifetime of an OB association.
b. Ionizing radiation
Next, we consider ionizing radiation from OB stars. The total amount of energy contained in ionizing radiation is
vast. Abbott (1982) estimates the integrated luminosity of ionizing radiation in the disk of the Milky Way to be
e˙ = 1.5× 10−24 erg s−1 cm−3. (50)
However, only a small fraction of this total energy goes to driving interstellar motions.
Ionizing radiation contributes to interstellar turbulence in two ways. First, it ionizes the diffuse interstellar gas,
heating it to 7,000–10,000 K and adding energy to it. As this gas cools, it contracts due to thermal instabilities,
driving turbulent flows, as modeled by Kritsuk & Norman (2002a,b). They modeled the flow in a cooling instability
after a sudden increase in heating by a factor of five, and found that a flow with peak thermal energy of Eth gains a
peak kinetic energy of roughly Ekin = ηcEth, with ηc ≃ 0.07. Parravano, Hollenbach, & McKee (2003) find that the
local UV radiation field, and thus the photoelectric heating rate, increases by a factor of 2–3 due to the formation of
a nearby OB association every 100–200 Myr. However, substantial motions only lasted about 1 Myr after a heating
event in the model by Kritsuk & Norman (2002b). We can estimate the energy input from this mechanism on average
by taking the kinetic energy input from the heating event and dividing by the typical time τOB between heating
events. If we take the thermal energy to be that of n = 1 cm−3 gas at 104 K (perhaps a bit higher than typical), we
find that
e˙ = 3
2
nkTηc/τOB ≃ (5× 10−29 erg cm−3 s−1)
( n
1 cm−3
)
( T
104 K
)
( ηc
0.07
)
( τOB
100Myr
)−1
. (51)
Although comparable to some other proposed energy sources discussed here, this mechanism appears unlikely to be
as important as the supernova explosions from the same OB stars, as discussed below.
The second way that ionization drives turbulence is through driving the supersonic expansion of Hii regions after
photoionization heating raises their pressures above that of the surrounding neutral gas. Matzner (2002) computes
the momentum input from the expansion of an individual Hii region into a surrounding molecular cloud, as a function
of the cloud mass and the ionizing luminosity of the central OB association. By integrating over the Hii region
luminosity function derived by McKee & Williams (1997), he finds that the average momentum input from a Galactic
region is
〈δp〉 ≃ (260 kms−1)
( NH
1.5× 1022 cm−2
)−3/14( Mcl
106 M⊙
)1/14
〈M∗〉. (52)
The column density NH is scaled to the mean value for Galactic molecular clouds (Solomon et al. 1987), which varies
little as cloud mass Mcl changes. The mean stellar mass per cluster in the Galaxy 〈M∗〉 = 440M⊙ (Matzner 2002).
65

to collapse freely. A classic example of this is the massive star formation region NGC 3603, which locally resembles
a starburst knot, even though the Milky Way globally does not have a large star formation rate. On a smaller scale,
even the Trapezium cluster in Orion seems to have formed with efficiency of <∼ 50% (Hillenbrand & Hartmann 1998).
Triggering of star formation by compressive shocks from nearby star-forming regions (Elmegreen & Lada 1977) is
a special case of global support from turbulence leading to local collapse. Although prompt blast waves from winds
and early supernovae of OB association can compress nearby gas and induce collapse, most of the energy from that
association is released at later times as the less massive B stars explode, driving the larger-scale interstellar turbulence
that provides support against general collapse. The instances of apparent triggering seen both in linear sequences
of OB associations (e.g. Blaauw 1964) and in shells (e.g. Walborn & Parker 1992, Efremov & Elmegreen 1998b,
Kamaya 1998, Barba´ et al. 2003) may represent this prompt triggering. It seems unlikely, however, that this prompt
triggering will dominate large-scale star formation as first suggested by Gerola & Seiden (1978), and since developed
by Neukirch & Feitzinger (1988), Korchagin et al. (1995), and Nomura & Kamaya (2001). Compression due to
supersonic turbulent flows is suggested to be the main mechanism leading to stellar birth in gas-rich dwarf galaxies
without spiral density waves, such as Holmberg II (e.g Stewart et al. 2000). However, the rate compression-induced
star formation is small compared to the rates expected for global collapse, which is effectively prevented by the same
turbulent flows.
3. Globular clusters
Globular clusters may simply be the upper end of the range of normal cluster formation. Whitmore (2000) reviews
evidence showing that young clusters have a power-law distribution reaching up to globular cluster mass ranges. The
luminosity function for old globular clusters is log normal, which Fall & Zhang (2001) attribute to the evaporation of
smaller clusters by two-body relaxation, and the destruction of the largest clusters by dynamical interactions with the
background galaxy (also see Vesperini 2000, 2001). Fall & Zhang (2001) suggest that the power-law distribution of
young clusters is related to the power-law distribution of molecular cloud masses found by Harris & Pudritz (1995).
However, numerical models of gravitational collapse tend to produce mass distributions that appear more log-normal,
and are not closely related in shape to the underlying mass distributions of density peaks (Klessen 2001, Klessen et al.
2000). It remains unknown whether cluster masses are determined by the same processes as the masses of individual
collapsing objects, but the simulations do not include any physics that would limit them to one scale and not the
other. Further investigation of this question will be interesting.
4. Galactic nuclei
In galaxies with low star formation rates, the galactic nucleus is often the only region with substantial star formation
occurring. As rotation curves approach solid body in the centers of galaxies, magnetorotational instabilities die away,
leaving less turbulent support and perhaps greater opportunity for star formation. In more massive galaxies, gas is
often funneled towards the center by bars and other disk instabilities, again increasing the local density sufficiently
to overwhelm local turbulence and drive star formation.
Hunter et al. (1998) and Schaye (2003) note that central regions of galaxies have normal star formation despite
having surface densities that appear to be stable according to the Toomre criterion. This could be due to reduced
turbulence in these regions decreasing the surface density required for efficient star formation. The radial dependence
of the velocity dispersion is difficult to determine, because H i observations with sufficient velocity resolution to
measure typical turbulent linewidths of 6–12 km s−1 have rather low spatial resolution, with just a few beams across
the galaxy. Most calculations of the critical surface density therefore assume a constant value of the turbulent velocity
dispersion, which may well be incorrect (Wong & Blitz 2002).
As an alternative, or perhaps additional explanation, Kim & Ostriker (2001) point out that the magneto-Jeans
instability acts strongly in the centers of galaxies. The magnetic tension from strong magnetic fields can reduce or
eliminate the stabilizing effects from Coriolis forces in these low shear regions, effectively reducing the problem to a
2D Jeans stability problem along the field lines.
68

5. Primordial dwarfs
In the complete absence of metals, cooling becomes much more difficult. Thermal pressure supports gas that
accumulates in dark matter halos until the local Jeans mass is exceeded. The first objects that can collapse are
the ones that can cool from H2 formation through gas phase reactions. Abel, Bryan, & Norman (2000, 2002) and
Bromm, Coppi, & Larson (1999) have computed models of the collapse of these first objects. Abel et al. (2000, 2002)
used realistic cosmological initial conditions, and found that inevitably a single star formed at the highest density
peak before substantial collapse had occurred elsewhere in the galaxy. Bromm et al. (1999) used a flat-top density
perturbation that was able to fragment in many places simultaneously, due to its artificial symmetry.
Li, Klessen, & Mac Low (2003) suggest that the lack of fragmentation seen by Abel et al. (2000, 2002) may be due
to the relatively stiff equation of state of metal-free gas. Li et al. (2003) found that fragmentation of gravitationally
collapsing gas is strongly influenced by the polytropic index γ of the gas, with fragmentation continuously decreasing
from γ ∼ 0.2 to γ ∼ 1.3. The limited cooling available to primordial gas even with significant molecular fraction
may raise its polytropic index sufficiently to suppress fragmentation. Abel et al. (2000, 2002) argue that the resulting
stars are likely to have masses exceeding 100M⊙, leading to prompt supernova explosions with accompanying metal
pollution and radiative dissociation of H2.
6. Starburst galaxies
Starburst galaxies convert gas into stars at such enormous rates that the timescale to exhaust the available material
becomes short compared to the age of the universe (see the review by Sanders & Mirabel 1996). Starbursts typically
last for a few tens or hundreds of millions of years. However, they may occur several times during the life of a galaxy.
The star formation rates in starburst galaxies can be as high as 1000M⊙ yr−1 (Kennicutt 1998), some three orders
of magnitude above the current rate of the Milky Way. Starburst galaxies are rare in the local universe, but rapidly
increase in frequency at larger lookback times, suggesting that starbursts are characteristic of early galaxy evolution
at high redshifts. The strongest starbursts occur in galactic nuclei or circumnuclear regions.
However, in interacting galaxies, strong star formation is also triggered far away from the nucleus in the overlapping
regions, in spiral arms, or sometimes even in tidal tails. In these interactions a significant number of super-star clusters
form, which may be the progenitors of present-day globular clusters (Zhang & Fall 1999, Whitmore et al. 1999), or
even compact elliptical galaxies (Fellhauser & Kroupa 2002). The Antennae galaxy, the product of a major merger
of the spiral galaxies NGC 4038 and 4039, is a famous example where star formation is most intense in the overlap
region between the two galaxies (Whitmore & Schweizer 1995). Merging events always seem to be associated with the
most massive and luminous starburst galaxies, the ultraluminous IR galaxies identified by Sanders & Mirabel (1996).
Gentler minor mergers can also trigger starbursts. Such an event disturbs but does not disrupt the primary galaxy.
It recovers from the interaction without dramatic changes in its overall morphology. This could explain the origin
of lower-mass, luminous, blue, compact galaxies, which often show very little or no sign of interaction (e.g. van Zee,
Salzer, & Skillman 2001). Alternative triggers of the starburst phenomenon that have have been suggested for these
galaxies include bar instabilities in the galactic disk (Shlosman, Begelman, & Frank 1990), or the compressional effects
of multiple supernovae and winds from massive stars (e.g. Heckman, Armus, & Miley 1990), which then would lead
to a very localized burst of star formation.
Regardless of the details of the different starburst triggering mechanisms, they all focus gas into a concentrated
region quickly enough to overwhelm the local turbulence and any additional turbulence driven by newly formed
stars. Combes (2001) argues that this can only be accomplished by gravitational torques on the gas. We suggest that
starburst galaxies are just extreme examples of the continuum of star formation phenomena, with gravity overwhelming
support from turbulent gas motions on kiloparsec scales rather than the parsec scales of individual OB associations,
or the even smaller scales of low-mass star formation.
VII. CONCLUSIONS
A. Summary
The formation of stars represents the triumph of gravity over a succession of opponents. These include thermal
pressure, turbulent flows, magnetic flux, and angular momentum. For several decades, magnetic fields were thought
to dominate the resistance against gravity, with star formation occurring quasistatically as ambipolar diffusion allows
69

they are coupled to the gas, however.
We outlined the shape of the new theory in Section IV.K. Rather than relying on quasistatic evolution of magneto-
statically supported objects, it suggests that supersonic turbulence controls star formation. Inefficient, isolated star
formation is a hallmark of turbulent support, while efficient, clustered star formation occurs in its absence. When
stars form, they do so dynamically, collapsing on the local free-fall time. The initial conditions of clusters appear
largely determined by the properties of the turbulent gas, as is the rate of mass accretion onto these objects. The
balance between turbulent support and local density then determines the star formation rate. Turbulent support
is provided by some combination of supernovae and galactic rotation, along with possible contributions from other
processes. Local density is determined by galactic dynamics including galaxy interactions, along with the balance
between heating and cooling in a region. The initial mass function is at least partly determined by the initial distri-
bution of density resulting from turbulent flows, although a contribution from stellar feedback and interactions with
nearby stars cannot yet be ruled out.
We explored the implications of the control of star formation by supersonic turbulence at the scale of individual stars
and stellar clusters in Section V. We examined how turbulent fragmentation determines the star forming properties
of molecular clouds (Section V.A), and then turned to discuss protostellar cores (Section II.E), binary stars (Section
V.C), and stellar clusters (Section V.D) in particular. Strongly time-varying protostellar mass growth rates may
result as a natural consequence of competitive accretion in nascent embedded clusters (Section V.E). Turbulent
models predict protostellar mass distributions (Section V.F) that appear roughly consistent with the observed stellar
mass spectrum (Section II.F), although more work needs to be done to arrive at a full understanding of the origin of
stellar masses.
The same balance between turbulence and gravity that seems to determine the efficiency of star formation in
molecular clouds also works at galactic scales, as we discussed in Section VI. The transient nature of molecular clouds
suggests that they form and are dispersed in either of two ways. One possibility is that they form during large-scale
gravitational collapse, and are dispersed quickly thereafter by radiation and supernovae from the resulting violent
internal star formation. The other possibility is that large-scale turbulent flows in galactic disks compress and cool
gas. These same flows will continue to drive the turbulent motions observed within the clouds. Some combination of
turbulent flow, free expansion at the sound speed of the cloud, and dissociating radiation from internal star formation
will then be responsible for their destruction on a timescale of 5–10 Myr (Section VI.A).
Having considered the formation of molecular clouds from the interstellar gas, we then discuss the role of differential
rotation and thermal instability competing and cooperating with turbulence to determine the overall star formation
efficiency in Section VI.B. We examined the physical mechanisms that could drive the interstellar turbulence, focusing
on the energy available from each mechanism in Section VI.C. In star-forming regions of disks, supernovae appear to
overwhelm all other possibilities. In outer disks and low surface brightness galaxies, on the other hand, the situation
is not so clear: magnetorotational or gravitational instabilities look most likely to drive the observed flows but further
work is required on these regions. Finally, in Section VI.D, we gave examples of how this picture may apply to
different types of objects, including low surface brightness, normal, and starburst galaxies, as well as galactic nuclei
and globular clusters. We argue that efficient star formation occurs at all scales when gravity overwhelms turbulence,
with the result ranging from a single low-mass star at the very smallest scale to a starburst at the very largest scale.
B. Future research problems
Although the outline of a new theory of star formation has emerged, it is by no means complete. The ultimate
goal of a predictive, quantitative theory of the star formation rate and stellar initial mass function remains elusive. It
may be that the problem is intrinsically so complex, like terrestrial climate, that no single solution exists, but only a
series of temporary, quasi-steady states. Certainly our understanding of the details of the star formation process can
be improved, though. Eventually, coupled models capturing different scales will be necessary to follow the interaction
of the turbulent cascade with the thermodynamics, chemistry, and opacity of the gas at different densities. We can
identify several major questions that summarize the outstanding problems. As we merely want to summarize these
open issues in star formation, we refrain from giving an in-depth discussion and the associated references, which may
largely be found in the body of the review.
How can we describe turbulence driven by astrophysical processes? There is really no single driving scale, because of
the non-uniformity of explosions, and perhaps of other drivers. However, a good description of the structure around
the driving scales remains essential, as the largest perturbations lie at the largest scales in any turbulent flow. This
remains to be found. The length of the self-similar turbulent cascade also depends on the scales on which the driving
71

ACKNOWLEDGMENTS
We have benefited from long-term collaborations, discussions, and exchange of ideas and results with a large number
of people. We particularly mention (in alphabetical order) M. A. de Avillez, J. Ballesteros-Paredes, P. Bodenheimer,
A. Burkert, B. G. Elmegreen, C. Gammie, L. Hartmann, F. Heitsch, P. Kroupa, D. N. C. Lin, C. F. McKee, A˚.
Nordlund, V. Ossenkopf, E. C. Ostriker, P. Padoan, F. H. Shu, M. D. Smith, J. M. Stone, E. Va´zquez-Semadeni,
H. Zinnecker, and E. G. Zweibel. This review also benefited from two detailed anonymous reviews and extended
comments from E. Falgarone, F. Heitsch, M. K. R. Joung, A. Lazarian, M. D. Smith, and E. Va´zquez-Semadeni.
Finally, we must thank our editors V. Trimble and J. Krolik for their patience and encouragement. M-MML was
supported by CAREER grant no. AST 99-85392 from the US National Science Foundation, and by the US National
Aeronautics and Space Administration (NASA) Astrophysics Theory Program under grant no. NAG5-10103. RSK
was supported by the Emmy Noether Program of the Deutsche Forschungsgemeinschaft (grant no. KL1358/1) and by
the NASA Astrophysics Theory Program through the Center for Star Formation Studies at NASA’s Ames Research
Center, UC Berkeley, and UC Santa Cruz. Preparation of this work made extensive use of the NASA Astrophysical
Data System Abstract Service.
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TABLE I. Properties of the Larson-Penston solution of
isothermal collapse.
before core formation after core formation
(t < 0) (t > 0)
density profile ρ ∝ (r2 + r20)−1 ρ ∝ r−3/2, r → 0
(r0 → 0 as t → 0−) ρ ∝ r−2, r →∞
flattened isothermal
sphere
velocity profile v ∝ r/t as t → 0− v ∝ r−1/2, r → 0
v ≈ −3.3 cs, r →∞ v ≈ −3.3 cs, r →∞
accretion rate M˙ = 47 c3s/G
TABLE II. Properties of the Shu solution of isothermal col-
lapse.
before core formation after core formation
(t < 0) (t > 0)
density profile ρ ∝ r−2, ∀ r ρ ∝ r−3/2, r ≤ cst
singular isothermal
sphere
ρ ∝ r−2, r > cst
velocity profile v ≡ 0, ∀ r v ∝ r−1/2, r ≤ cst
v ≡ 0, r > cst
accretion rate M˙ = 0.975 c3s/G
86

TABLE III. Physical properties of interstellar clouds
GIANT MOLECULAR
CLOUD COMPLEX
MOLECULAR
CLOUD
STAR-FORMING
CLUMP
PROTOSTELLAR
COREa
Size (pc) 10− 60 2− 20 0.1− 2 <∼ 0.1
Density (n(H2)/cm3) 100− 500 102 − 104 103 − 105 > 105
Mass (M⊙) 104 − 106 102 − 104 10− 103 0.1 − 10
Line width (kms−1) 5− 15 1− 10 0.3− 3 0.1 − 0.7
Temperature (K) 7− 15 10− 30 10− 30 7− 15
Examples W51, W3, M17, Orion-
Monoceros, Taurus-
Auriga-Perseus complex
L1641, L1630, W33,
W3A, B227, L1495,
L1529
see Section II.E
a Protostellar cores in the ”prestellar” phase, i.e. before the formation of the protostar in its interior.
87