ar
X
iv
:0
70
7.
35
14
v2
[
as
tro
-p
h]
2
4 J
ul
20
07
Theory of Star Formation 1
Theory of Star Formation∗
Christopher F. McKee
Departments of Physics and Astronomy, University of California, Berkeley, CA
94720; cmckee@astro.berkeley.edu
Eve C. Ostriker
Department of Astronomy, University of Maryland, College Park, MD 20742;
ostriker@astro.umd.edu
Abstract
We review current understanding of star formation, outlining an overall theoretical frame-
work and the observations that motivate it. A conception of star formation has emerged
in which turbulence plays a dual role, both creating overdensities to initiate gravitational
contraction or collapse, and countering the effects of gravity in these overdense regions.
The key dynamical processes involved in star formation – turbulence, magnetic fields, and
self-gravity – are highly nonlinear and multidimensional. Physical arguments are used
to identify and explain the features and scalings involved in star formation, and results
from numerical simulations are used to quantify these effects. We divide star formation
into large-scale and small-scale regimes and review each in turn. Large scales range from
galaxies to giant molecular clouds (GMCs) and their substructures. Important problems
include how GMCs form and evolve, what determines the star formation rate (SFR), and
what determines the initial mass function (IMF). Small scales range from dense cores to
the protostellar systems they beget. We discuss formation of both low- and high-mass
stars, including ongoing accretion. The development of winds and outflows is increas-
ingly well understood, as are the mechanisms governing angular momentum transport in
disks. Although outstanding questions remain, the framework is now in place to build
a comprehensive theory of star formation that will be tested by the next generation of
telescopes.
CONTENTS
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
BASIC PHYSICAL PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Self-Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
MACROPHYSICS OF STAR FORMATION . . . . . . . . . . . . . . . . . . . . . 24
Physical State of GMCs, Clumps, and Cores . . . . . . . . . . . . . . . . . . . . . . . . 24
∗Posted with permission from the Annual Review of Astronomy and Astrophysics, Volume
45, c©2007 by Annual Reviews, http://www.annualreviews.org

Annu. Rev. Astron. Astrophys. 2007 45 XXX
Formation, Evolution, and Destruction of GMCs . . . . . . . . . . . . . . . . . . . . . 34
Core Mass Functions and the IMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
The Large-Scale Rate of Star Formation . . . . . . . . . . . . . . . . . . . . . . . . . . 50
MICROPHYSICS OF STAR FORMATION . . . . . . . . . . . . . . . . . . . . . . 56
Low-Mass Star Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Disks and Winds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
High-Mass Star Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
OVERVIEW OF THE STAR FORMATION PROCESS . . . . . . . . . . . . . . . 93
1 INTRODUCTION
Stars are the “atoms” of the universe, and the problem of how stars form is at
the nexus of much of contemporary astrophysics. By transforming gas into stars,
star formation determines the structure and evolution of galaxies. By tapping the
nuclear energy in the gas left over from the Big Bang, it determines the luminosity
of galaxies and, quite possibly, leads to the reionization of the universe. Most
of the elements—including those that make up the world around us—are formed
in stars. Finally, the process of star formation is inextricably tied up with the
formation and early evolution of planetary systems.
The problem of star formation can be divided into two broad categories: “mi-
crophysics” and “macrophysics”. The microphysics of star formation deals with
how individual stars (or binaries) form. Do stars of all masses acquire most of
their mass via gravitational collapse of a single dense “core”? How are the prop-
erties of a star or binary determined by the properties of the medium out of
which it forms? How does the gas that goes into a protostar lose its magnetic
flux and angular momentum? How do massive stars form in the face of intense
radiation pressure? What are the properties of the protostellar disks, jets, and
outflows associated with Young Stellar Objects (YSOs), and what governs their
dynamical evolution?
The macrophysics of star formation deals with the formation of systems of
stars, ranging from clusters to galaxies. How are giant molecular clouds (GMCs),
the loci of most star formation, themselves formed out of diffuse interstellar gas?
What processes determine the distribution of physical conditions within star-
forming regions, and why does star formation occur in only a small fraction
of the available gas? How is the rate at which stars form determined by the
properties of the natal GMC or, on a larger scale, of the interstellar medium in a
galaxy? What determines the mass distribution of forming stars, the Initial Mass
Function (IMF)? Most stars form in clusters (Lada & Lada, 2003); how do stars
form in such a dense environment and in the presence of enormous radiative and
mechanical feedback from other YSOs?
Many of these questions, particularly those related to the microphysics of star
formation, were discussed in the classic review of Shu, Adams, & Lizano (1987).
Much has changed since then. Observers have made enormous strides in char-
acterizing star formation on all scales and in determining the properties of the
medium out of which stars form. Aided by powerful computers, theorists have
been able to numerically model the complex physical and chemical processes as-
sociated with star formation in three dimensions. Perhaps most important, a
new paradigm has emerged, in which large-scale, supersonic turbulence governs
the macrophysics of star formation.
2

Theory of Star Formation 3
This review focuses on the advances made in star formation since 1987, with an
emphasis on the role of turbulence. Recent relevant reviews include those on the
physics of star formation (Larson, 2003), and on the role of supersonic turbulence
in star formation (Mac Low & Klessen, 2004; Ballesteros-Paredes et al., 2007).
The chapter by Zinnecker & Yorke in this volume of ARAA provides a different
perspective on high-mass star formation, while that by Bergin & Tafalla gives a
more detailed description of dense cores just prior to star formation. Because the
topic is vast, we must necessarily exclude a number of relevant topics from this
review: primordial star formation (see Bromm & Larson, 2004), planet formation,
astrochemistry, the detailed physics of disks and outflows, radiative transfer, and
the properties of young stellar objects.
In §2, we begin with an overview of basic physical processes and scales in-
volved in star formation, covering turbulence (§2.1), self gravity (§2.2), and mag-
netic fields (§2.3). In §3, we review macrophysics of star formation, focusing
on: the physical state of GMCs, clumps, and cores (§3.1), the formation, evolu-
tion, and destruction of GMCs (§3.2), core mass functions and the IMF (§3.3),
and the large-scale rate of star formation (§3.4). §4 reviews microphysics of star
formation, covering low-mass star formation (§4.1), disks and winds (§4.2), and
high-mass star formation (§4.3). We conclude in §5 with an overview of the star
formation process.
2 BASIC PHYSICAL PROCESSES
2.1 Turbulence
As emphasized in §1, many of the advances in the theory of star formation since
the review of Shu, Adams, & Lizano (1987) have been based on realistic evalua-
tion and incorporation of the effects of turbulence. Turbulence is in fact impor-
tant in essentially all branches of astrophysics that involve gas dynamics1, and
many communities have contributed to the recent progress in understanding and
characterizing turbulence in varying regimes. Here, we shall concentrate on the
parameter regimes of turbulence applicable within the cold ISM, and the physical
properties of these flows that appear particularly influential for controlling star
formation.
Our discussion provides an overview only; pointers will be given to excellent
recent reviews that summarize the large and growing literature on this subject.
General references include Frisch (1995), Biskamp (2003), and Falgarone & Pas-
sot (2003). A much more extensive literature survey and discussion of interstellar
turbulence, including both diffuse-ISM and dense-ISM regimes, is presented by
Elmegreen & Scalo (2004) and Scalo & Elmegreen (2004). A recent review fo-
cusing on the detailed physics of turbulent cascades in magnetized plasmas is
Schekochihin & Cowley (2005).
2.1.1 SPATIAL CORRELATIONS OF VELOCITY AND MAGNETIC
FIELDS Turbulence is defined by the Oxford English Dictionary as a state
of “violent commotion, agitation, or disturbance,” with a turbulent fluid further
defined as one “in which the velocity at any point fluctuates irregularly.” Al-
though turbulence is by definition an irregular state of motion, a central concept
1 Chandrasekhar (1949) presaged this development, in choosing the then-new theory of tur-
bulence as the topic of his Henry Norris Russell prize lecture.

4 McKee & Ostriker
is that order nevertheless persists as scale-dependent spatial correlations among
the flow variables. These correlations can be measured in many ways; common
mathematical descriptions include autocorrelation functions, structure functions,
and power spectra.
One of the most fundamental quantities, which is also one of the most in-
tuitive to understand, is the RMS velocity difference between two points sepa-
rated by a distance r. With the velocity structure function of order p defined
as Sp(r) ≡ 〈|v(x) − v(x + r)|p〉, this quantity is given as ∆v(r) ≡ [S2(r)]1/2.
The autocorrelation function of the velocity is related to the structure func-
tion: A(r) ≡ 〈v(x) · v(x + r)〉 = 〈|v|2〉 − S2(r)/2; note that the autocorrelation
with zero lag is A(0) = 〈|v|2〉 since S2(0) = 0. The power spectrum of velocity,
P (k) ≡ |v(k)|2, is the Fourier transform of the autocorrelation function. For zero
mean velocity, the velocity dispersion averaged over a volume ℓ3, σv(ℓ)2, is equal
to the power spectrum integrated with kmin = 2π/ℓ. If turbulence is isotropic
and the system in which it is observed is spatially symmetric with each dimension
≈ ℓ, then the one-dimensional velocity dispersion along a given line-of-sight (a
direct observable) will be related to the three-dimensional velocity dispersion by
σ = σv(ℓ)/
√
3. Analogous structure functions, correlation functions, and power
spectra can also be defined for the magnetic field, as well as other fluid variables
including the density (see §2.1.4). Delta-variance techniques provide similar in-
formation, and are particularly useful for reducing edge effects when making
comparisons with observational data (Bensch, Stutzki, & Ossenskopf, 2001).
For isotropic turbulence, Sp and A are functions only of r = |r|, and P is a
function only of k = |k|. The Fourier amplitude |v(k)| is then (on average) only
a function of ℓ = 2π/k, and can be denoted by v(ℓ); to emphasize that these
velocities are perturbations about a background state, the amplitude of a given
Fourier component is often written as δv(k) or δv(ℓ). When there is a large
dynamic range between the scales associated with relevant physical parameters
(see §2.1.3), correlations often take on power-law forms. If P (k) ∝ k−n for an
isotropic flow, then
v(ℓ) ∝ σv(ℓ) ∝ ∆v(ℓ) ∝ ℓq (1)
with q = (n − 3)/2. Sometimes indices n′ of one-dimensional (angle-averaged),
rather than three-dimensional, power spectra are reported; these are related by
n′ = n− 2.
The turbulent scaling relations reflect the basic physics governing the flow. The
classical theory of Kolmogorov (1941) applies to incompressible flow, i.e. one in
which the velocities are negligible compared to the thermal speed σth = (Pth/ρ)1/2
(where ρ is the density and Pth is the thermal pressure); σth is equal to the sound
speed cs = (γPth/ρ)1/2 in an isothermal (γ = 1) gas. In incompressible flows,
energy is dissipated and turbulent motions are damped only for scales smaller
than the Reynolds scale ℓν at which the viscous terms in the hydrodynamic
equations, ∼ νv(ℓ)/ℓ2, exceed the nonlinear coupling terms between scales, ∼
v(ℓ)2/ℓ; here ν is the kinematic viscosity. At scales large compared to ℓν , and
small compared to the system as a whole, the rate of specific energy transfer E˙
between scales is assumed to be conserved, and equal to the dissipation rate at the
Reynolds scale. From dimensional analysis, E˙ ∼ v(ℓ)3/ℓ, which implies n = 11/3
and q = 1/3 for the so-called “inertial range” in Kolmogorov turbulence. The
Kolmogorov theory includes the exact result that S3(ℓ) = −(4/5)E˙ℓ.
Because velocities v(ℓ) ∼ σv(ℓ) ∼ ∆v(ℓ) in molecular clouds are in general

Theory of Star Formation 5
not small compared to cs, at least for sufficently large ℓ, one cannot expect the
Kolmogorov theory to apply. In particular, some portion of the energy at a given
scale must be directly dissipated via shocks, rather than cascading conservatively
through intermediate scales until ℓν is reached. In the limit of zero pressure,
the system would consist of a network of (overlapping) shocks; this state is often
referred to as Burgers turbulence (Frisch & Bec, 2001). Since the power spectrum
corresponding to a velocity discontinuity in one dimension has P (k) ∝ k−2, an
isotropic system of shocks in three dimensions would also yield power-law scalings
for the velocity correlations, with n = 4 and q = 1/2. Note that correlations can
take on a power-law form even if there is not a conservative inertial cascade; a
large range of spatial scales with consistent physics is still required.
Turbulence in a magnetized system must differ from the unmagnetized case be-
cause of the additional wave families and nonlinear couplings involved, as well as
the additional diffusive processes – including resistive and ion-neutral drift terms
(see §2.1.3). When the magnetic field B is strong, in the sense that the Alfve´n
speed vA ≡ B/
√
4πρ satisfies vA ≫ v(ℓ), a directionality is introduced such that
the correlations of the flow variables may depend differently on r‖, r⊥, k‖, and
k⊥, the displacement and wavevector components parallel and perpendicular to
Bˆ.
For incompressible magnetohydrodynamic (MHD) turbulence, Goldreich & Srid-
har (1995) introduced the idea of a critically-balanced anisotropic cascade, in
which the nonlinear mixing time perpendicular to the magnetic field and the
propagation time along the magnetic field remain comparable for wavepackets at
all scales, so that vAk‖ ∼ v(k⊥, k‖)k⊥. Interactions between oppositely-directed
Alfve´n wavepackets travelling along magnetic fields cannot change their parallel
wavenumbers k‖ = k ·Bˆ, so that the energy transfers produced by these collisions
involve primarily k⊥; i.e. the cascade is through spatial scales ℓ⊥ = 2π/k⊥ with
v(ℓ⊥)3/ℓ⊥ ∼ constant. Combining critical balance with a perpendicular cascade
yields anisotropic power spectra (larger in the k⊥ direction); at a given level of
power, the theory predicts k‖ ∝ k2/3⊥ . Magnetic fields and velocities are predicted
to have the same power spectra.
Unfortunately, for the case of strong compressibility (cs ≪ v) and moderate or
strong magnetic fields (cs ≪ vA <∼ v), which generally applies within molecular
clouds, there is as yet no simple conceptual theory to characterize the energy
transfer between scales and to describe the spatial correlations in the velocity
and magnetic fields. On global scales, the flow may be dominated by large-
scale (magnetized) shocks which directly transfer energy from macroscopic to
microscopic degrees of freedom. Even if velocity differences are not sufficient
to induce (magnetized) shocks, for trans-sonic motions compressibility implies
strong coupling among all the MHD wave families. On the other hand, within a
sufficiently small sub-volume of a cloud (and away from shock interfaces), velocity
differences may be sufficiently subsonic that the incompressible MHD limit and
the Alfve´nic cascade approximately hold locally.
Even without direct energy transfer from large to small scales in shocks, a key
property of turbulence not captured in classical models is intermittency effects –
the strong (space-time) localization of dissipation in vortex sheets or filaments,
which can occur even with a conservative energy cascade. (Shocks in compressible
flows represent a different class of intermittent structures.) Signatures of inter-
mittency are particularly evident in departures of high-order structure function

6 McKee & Ostriker
exponents from the value p/3, and in non-Gaussian tails of velocity-increment
probability distribution functions (PDFs) (e.g. Sreenivasan & Antonia, 1997;
Lis et al., 1996). Proposed methods to account for intermittency in predicting
correlation functions for incompressible, unmagnetized turbulence have been dis-
cussed by She & Leveque (1994) and Dubrulle (1994). Boldyrev (2002) proposed
an adaptation of this framework for the compressible MHD case, but omitted
direct dissipation of large-scale modes in shocks. Research on formal turbulence
theory is quite active (see Elmegreen & Scalo, 2004 for a review of the recent
theoretical literature relevant to the ISM), although a comprehensive framework
remains elusive.
Large-scale numerical simulations afford a complementary theoretical approach
to model turbulence and to explore the spatial correlations within flows. Numer-
ical experiments can be used to test formal theoretical proposals, and to provide
controlled, quantitative means to interpret observations – within the context of
known physics – when formal theories are either nonexistant or limited in detail.
In drawing on the results of numerical experiments, it is important to ensure that
the computational techniques employed adequately capture the relevant dynam-
ical processes. For systems in which there are steep gradients of velocities and
densities, grid-based methods are more accurate in following details of the evolv-
ing flow (such as development of instabilities) than SPH methods, which have
been shown to have difficulty capturing shocks and other discontinuities (Agertz
et al., 2006).
Spatial correlations within turbulent flows have been evaluated using numerical
simulations in a variety of regimes. Overall, results are consistent with theoretical
predictions in that power-law scalings in the velocity and magnetic field power
spectra (or structure functions) are clear when there is sufficient numerical res-
olution to separate driving and dissipative scales. At resolutions of 5123 and
above, results with a variety of numerical methods show angle-averaged power-
law slopes n = 7/2−11/3 (i.e. 3.5−3.67) for incompressible (i.e. v/cs ≪ 1) MHD
flows (Mu¨ller, Biskamp, & Grappin, 2003; Haugen, Brandenburg, & Dobler, 2004;
Mu¨ller & Grappin, 2005) and n = 3.5− 4.0 for strongly compressible (v/cs >∼ 5)
flows both with (Vestuto, Ostriker, & Stone, 2003; Padoan et al., 2007) and
without (Kritsuk, Norman, & Padoan, 2006) magnetic fields. Consistent with
expectations, spectra are steeper for compressive velocity components than for
magnetic fields (and also sheared velocity components if the magnetic field is
moderate or strong), and steeper for more supersonic and/or more weakly mag-
netized models (see also Boldyrev, Nordlund, & Padoan, 2002; Padoan et al.,
2004).
When strong mean magnetic fields are present, there is clear anisotropy in the
power spectrum, generally consistent with the scaling prediction of Goldreich &
Sridhar (1995), for both incompressible and compressible MHD turbulence (Cho
& Vishniac, 2000; Maron & Goldreich, 2001; Cho, Lazarian, & Vishniac, 2002a;
Vestuto, Ostriker, & Stone, 2003; Cho & Lazarian, 2003).
In order to identify the sources of turbulence in astronomical systems, it is
also important to determine the behavior of velocity and magnetic field correla-
tions on spatial scales larger than the driving scale. Numerical simulations, both
for incompressible (Maron & Goldreich, 2001; Haugen, Brandenburg, & Dobler,
2004) and compressible MHD turbulence (Vestuto, Ostriker, & Stone, 2003),
show that the power spectra below the driving wavenumber scale are nearly flat,
n ≈ 0; that is, “inverse cascade” effects are limited. For spatially-localized forc-

Theory of Star Formation 7
ing (rather than forcing localized in k-space), Nakamura & Li (2007) also found
a break in the power spectrum, at wavelength comparable to the momentum in-
jection scale. Thus, the forcing scale for internally-driven turbulence in a system
can be inferred observationally from the peak or knee of the velocity correlation
function. If v(ℓ) continues to rise up to ℓ ∼ L, the overall scale of a system, this
implies that turbulence is either (i) externally driven, (ii) imposed in the initial
conditions when the system is formed, or (iii) driven internally to reach large
scales. Note that for systems forced at multiple scales, or both internally and
externally, breaks may be evident in the velocity correlation function (or power
spectrum).
2.1.2 TURBULENT DISSIPATION TIMESCALES Recent numerical
simulations under quite disparate physical regimes have reached remarkably sim-
ilar conclusions for the dissipation rates of turbulence. On dimensional grounds,
the specific energy dissipation rate should equal ǫU3/ℓ0, where E = U2/2 is the
total specific kinetic energy, ℓ0 is the spatial wavelength of the main energy-
containing scale (comparable to the driving scale for forced turbulence; ℓ0 ≤ L),
and ǫ is a dimensionless coefficient. For incompressible turbulence, the largest-
scale (40963 zones) incompressible, unmagnetized, driven-turbulence simulations
to date (Kaneda et al., 2003) yield a dimensionless dissipation coefficent ǫ = 0.6.
For driven incompressible MHD turbulence (at 10243 resolution), the measured
dimensionless dissipation rate ǫ ≡ (1/2)(E˙turb/Eturb)(ℓ0/U) also works out to be
ǫ = 0.6 (Haugen, Brandenburg, & Dobler, 2004). Quite comparable results also
hold for strongly-compressible (U/cs = 5) turbulence at a range of magnetiza-
tions vA/cs = 0− 10; Stone, Ostriker, & Gammie (1998) found that ǫ = 0.6− 0.7
for simulations at resolution up to 5123 zones. For decaying compressible MHD
turbulence, damping timescales are also comparable to the flow crossing time
ℓ0/v(ℓ0) on the energy-containing scale (Stone, Ostriker, & Gammie, 1998; Mac
Low et al., 1998; Mac Low, 1999; Padoan & Nordlund, 1999). Thus, although
very different physical processes are involved in turbulence dissipation under dif-
ferent circumstances, the overall damping rates summed over all available chan-
nels (including shock, reconnection, and shear structures) are nevertheless quite
comparable. Defining the turbulent dissipation timescale as tdiss = Eturb/|E˙turb|
and the flow crossing time over the main energy-containing scale as tf = ℓ0/U ,
tdiss = tf/(2ǫ). Since velocities in GMCs increase up to the largest scale, ℓ0 → d,
the cloud diameter. Assuming that on average U =
√
3σlos, the turbulent dissi-
pation time based on numerical results is therefore given by
tdiss ≈ 0.5
d
σlos
. (2)
This result is in fact consistent with the assumption of Mestel & Spitzer (1956)
that turbulence in GMCs would decay within a crossing time.
The above results apply to homogenous, isotropic turbulence, but under cer-
tain circumstances if special symmetries apply, turbulent damping rates may be
lower. One such case is for incompressible turbulence consisting of Alfve´n waves
all propagating in the same direction along the magnetic field. Note that for the
incompressibility condition ∇·v = 0 to apply, turbulent amplitudes must be quite
low (v ≪ cs). Since Alfve´n waves are exact solutions of the incompressible MHD
equations, no nonlinear interactions, and hence no turbulent cascade, can develop
if only waves with a single propagation direction are present in this case (see e.g.
Chandran, 2004 for a mathematical and physical discussion). A less extreme

8 McKee & Ostriker
situation is to have an imbalance in the flux of Alfve´n waves propagating “up-
wards” and “downwards” along a given magnetic field direction. Maron & Gol-
dreich (2001) show that in decaying incompressible MHD turbulence, the power
in both upward- and downward-propagating components decreases together until
the lesser component is depleted. Cho, Lazarian, & Vishniac (2002a) quantify
decay times of imbalanced incompressible turbulence, finding for example that
if the initial imbalance is ≈ 50% or ≈ 70%, then the time to decay to half the
initial energy is increased by a factor 1.5 or 2.3, respectively, compared to the
case of no imbalance.
For even moderate-amplitude subsonic velocities, however, Alfve´n waves couple
to other wave families and the purely Alfve´nic cascade is lost. For strongly
supersonic motions, as are present in GMCs, the mode coupling is quite strong.
As a consequence, even a single circularly polarized Alfve´n wave cannot propagate
without losses; a parametric instability known as the “decay instability” (Sagdeev
& Galeev, 1969) develops in which three daughter waves (a forward-propagating
compressive wave and two oppositely propagating Alfve´n waves when β ≪ 1)
grow at the expense of the mother wave. The initial growth rate of the instability
is γ = (0.1− 0.3) kvA when v(k)/cs = 1− 3 and β ≡ 2c2s/v2A = 0.2, and larger for
greater amplitudes and smaller β (Goldstein, 1978). The ultimate result is decay
into fully-developed turbulence (Ghosh & Goldstein, 1994; Del Zanna, Velli, &
Londrillo, 2001). Thus, for conditions that apply within GMCs, even if there were
a localized source of purely Alfve´nic waves (i.e. initially 100% imbalanced), the
power would rapidly be converted to balanced, broad-spectrum turbulence with
a short decay time. The conclusion that turbulent damping times within GMCs
are expected to be comparable to flow crossing times has important implications
for understanding evolution in star-forming regions; these will be discussed in
§§3.1 and 3.2.2.
2.1.3 PHYSICAL SCALES IN TURBULENT FLOWS In classical in-
compressible turbulence, the only physical scales that enter are the outer scale
ℓ0 at which the medium is stirred, and the inner “Reynolds” scale ℓν at which
viscous dissipation occurs. Assuming Kolmogorov scaling v(ℓ) = v(ℓ0)(ℓ/ℓ0)1/3
[for v(ℓ) ∼ σv(ℓ) ∼ ∆v(ℓ)], the dissipation scale is
ℓν
ℓ0
=
[ ν
ℓ0v(ℓ0)
]3/4
≡ Re−3/4. (3)
Here Re ≡ v(ℓ0)ℓ0/ν is the overall Reynolds number of the flow; if turbulence
increases up to the largest scales then Re = UL/ν. With ν ∼ csλmfp for λmfp the
mean free path for particle collisions, ℓν/ℓ0 ∼ (λmfp/ℓ0)3/4[v(ℓ0)/cs]−3/4. In fact,
the velocity-size scaling within GMCs has power-law index q closer to 1/2 than
1/3 on large scales, because large-scale velocities are supersonic and therefore the
compressible-turbulence results apply. Allowing for a transition from q = 1/2 to
q = 1/3 at an intermediate scale ℓs where v(ℓs) = cs (see below), ℓν = ℓ1/4s λ3/4mfp.
Using typical GMC parameters so that λmfp ∼ 1013 cm and ℓs ∼ 0.03 pc yields
ℓν ∼ 3× 10−5 pc. This is tiny compared to the sizes, ∼ 0.1 pc, of self-gravitating
cores in which individual stars form.
The length ℓs introduced above marks the scale at which the RMS turbulent
velocity is equal to the sound speed. At larger scales, velocities are supersonic and
compressions are strong; at smaller scales, velocities are subsonic and compres-
sions are weak. Taking v(ℓ) = v(ℓ0)(ℓ/ℓ0)q, the sonic scale is ℓs = ℓ0[cs/v(ℓ0)]1/q,

Theory of Star Formation 9
or ℓs ≈ ℓ0[cs/v(ℓ0)]2 when q ≈ 1/2. Density perturbations with characteristic
scales ∼ ℓs will have order-unity amplitude in an unmagnetized medium. In a
magnetized medium, the amplitude of the perturbation imposed by a flow of
speed v will depend on the direction of the flow relative to the magnetic field.
Flows along the magnetic field will be as for an unmagnetized medium, while
flows perpendicular to the magnetic field will create order-unity density pertur-
bations only if v > (c2s + v2A)1/2. Note that the thermal scale, at which the
line-of-sight turbulent velocity dispersion σv/
√
3 is equal to the one-dimensional
thermal speed σth, is larger than ℓs by a factor ≈ 3.
Another scale that is important for MHD turbulence in fully-ionized gas is the
resistive scale; below this scale Ohmic diffusion would smooth out strong bends
in the magnetic field, or would allow folded field lines to reconnect. The resistive
scale ℓη is estimated by equating the diffusion term ∼ ηB(ℓ)/(4πℓ2) to the flux-
dragging term ∼ v(ℓ)B(ℓ)/ℓ in the magnetic induction equation. Defining the
magnetic Reynolds number as Rm ≡ v(ℓ0)ℓ04π/η, and taking v(ℓ) ∼ cs(ℓ/ℓs)1/3
at small scales, this yields ℓη/ℓν = (Re/Rm)3/4. Since the “magnetic Prandtl
number” Rm/Re is very large (∼ 106), this means that the magnetic field could,
for a highly-ionized medium, remain structured at quite small scales (see Cho,
Lazarian, & Vishniac, 2002b for discussion of this in the diffuse ISM).
In fact, under the weakly-ionized conditions in star-forming regions, ambipolar
diffusion (ion-neutral drift) becomes important well before the resistive (or Ohmic
diffusion) scale is reached. Physically, the characteristic ambipolar diffusion scale
ℓAD is the smallest scale for which the magnetic field (which is frozen to the
ions) is well-coupled to the bulk of the gas, for a partially-ionized medium. An
estimate of ℓAD is obtained by equating the ion-neutral drift speed, ∼ B0δB(mi+
m)/(4πρiραinℓ), with the turbulent velocity, δv. Here, αin = 〈σin|vi − vn|〉 ≈
2 × 10−9 cm3s−1 is the ion-neutral collision rate coefficient (Draine, Roberge, &
Dalgarno, 1983), and mi,m and ρi, ρ are the ion and neutral mass and density.
The resulting ambipolar diffusion scale, assuming mi ≫ m (for either metal or
molecular cations) and δv ≈ δB/√4πρ, is
ℓAD =
vA
niαin
≈ 0.05 pc
( vA
3 km s−1
)( ni
10−3 cm−3
)−1
. (4)
Here, vA is the Alfve´n speed associated with the large-scale magnetic field B0.
The ambipolar diffusion scale (4) depends critically on the fractional ionization,
which varies greatly within star-forming regions. Regions with moderate AV <∼ 5
can have relatively large ionization fraction due to UV photoionization, whereas
regions with large AV are ionized primarily by cosmic rays (see §2.3.1). For
example, if the electrons are attached to PAHs in dense cores, the ion density
is ni ≈ 10−3(nH/104cm−3)1/2(ζCR/3 × 10−17s−1)1/2 (Tielens, 2005) where ζCR
is the cosmic ray ionization rate per H atom.. Since ni ∝ n1/2, we can express
equation (4) in terms of column density and magnetic field strength as ℓAD/ℓ =
0.09(B/10µG)/(NH/1021 cm−2).
For spatial wavelengths λ = 2π/k < πℓAD, MHD waves are unable to propagate
in the coupled neutral-ion fluid at all, because the collision frequency of neutrals
with ions, niαin, is less than (half) the wave frequency ω = kvA. For λ > πℓAD,
MHD waves are damped at a rate ωπℓAD/λ (Kulsrud & Pearce, 1969). Thus,
at scales ℓ <∼ ℓAD, the magnetic field will be essentially straight and uniform in
magnitude, and any further turbulent cascade will be as for an unmagnetized
medium. The scale ℓAD is also comparable to the thickness of the C-type shocks

Theory of Star Formation 11
for MF = 0.5 − 2.5.
Because the velocity field is spatially correlated, the density distribution will
also show spatial correlations over a range of scales. Density correlations can be
characterized in terms of the autocorrelation function, the power spectrum, and
structure functions of various orders (cf §2.1.1); usually, analyses are applied to
δρ ≡ ρ− ρ¯. Using delta-variance techniques, Mac Low & Ossenkopf (2000) show
that correlations in density decrease for wavelengths above the velocity driving
scale, and that there are relatively modest differences in the density correlations
between unmagnetized and magnetized models when all other properties are con-
trolled.
Kim & Ryu (2005) have analyzed the dependence of the spectral index on
Mach number for three-dimensional turbulence forced at large spatial scales, using
isothermal, unmagnetized simulations at resolution 5123. For M <∼ 1, the indices
nρ or n′ρ of the density power spectrum |δρ(k)|2 are similar to those of the velocity
field in incompressible turbulence – i.e. near n = 11/3 or n′ = 5/3; this is simply
because δρ(k)/ρ¯ ∼ −kˆ ·v(k)/cs for low-amplitude quasi-sonic compressions (note
that even when M = 1, the Mach number for the compressive component of the
velocity field is < 1). As the Mach number increases, the density power spectrum
flattens, reaching n′ρ ≈ 0.5 for M = 12. For comparison, a one-dimensional top
hat – corresponding to a large clump in 3D – would have n′ρ = 2, whereas a
one-dimensional delta function – corresponding to a thin sheet or filament in 3D
– would have n′ρ = 0. Note that for the density to take the form of multiple
delta functions, the velocity field must generally be a composite of step functions
– corresponding to shocks – and has n′ = 2 for the velocity power spectrum (as
discussed above). The low value of n′ρ at large Mach number implies the density
structure becomes dominated by curved sheets and filaments. Curved sheets
represent stagnation regions (of the compressive velocity field) where shocked
gas from colliding flows settles, and filaments mark the intersections of these
curved sheets.
Other statistical descriptions of density structure include fractal dimensions
(e.g., Elmegreen & Falgarone, 1996; Stutzki et al., 1998), multifractal spectra
(Chappell & Scalo, 2001), and hierarchical structure trees (Houlahan & Scalo,
1992); see Elmegreen & Scalo (2004) for a discussion. The spatial correlation
of density can also be characterized in terms of clump mass functions. Clump-
finding techniques have been applied to simulations of supersonic tubulent flows
by a number of groups; these results will be discussed and compared to observa-
tions in §3.3.
2.1.5 OBSERVATIONS OF TURBULENCE For observed astrophysical
systems, the intrinsic properties of turbulence cannot be directly obtained, due
to line-of-sight projection and the convolution of density and velocity in produc-
ing observed emission. A number of different techniques have been developed,
calibrated using simulations, and applied to observed data, in order to deduce
characteristics of the three-dimensional turbulent flow from the available obser-
vations, which include spectral line data cubes (from molecular transitions), con-
tinuum emission maps (from dust), maps of extinction (using background stars),
and maps of polarization (in extinction and emission from dust). Elmegreen &
Scalo (2004) review the extensive literature on observations of turbulence. Here,
we will mention just a few results.
The defining property of turbulent motion – in contrast to, for example, the

12 McKee & Ostriker
purely random motions of gas particles in a Maxwell-Boltzmann distribution or
the highly systematic motions of stars in a rotating system – is the stochastic yet
scale-dependent behavior of flow correlations. Larson (1981) was the first to draw
attention to the genuine “turbulent” nature of motions internal to star-forming
regions, as expressed by an empirical scaling law of the form (1) with q = 0.38.
Using more homogeneous data, Solomon et al. (1987) obtained a “linewidth-size”
scaling index q ≈ 0.5 for GMCs as a whole. Passot, Pouquet, & Woodward (1988)
pointed out that the linewidth-size scaling σv(ℓ) ∝ ℓ1/2 observed in star-forming
regions is indeed what would be predicted for “Burgers turbulence,” a more
appropriate model than Kolmogorov turbulence given the strongly supersonic
conditions.
Many subsequent studies have been made of observed scaling behavior of ve-
locities, both for subsystems of a given star-forming region, and for systems that
are spatially disjoint. A number of methods have been developed for these inves-
tigations, including autocorrelation analysis (Miesch & Bally, 1994) and delta-
variance analysis (Ossenkopf et al., 2006) applied to line centroids, the spectral
correlation function (Rosolowsky et al., 1999), velocity channel analysis (Lazar-
ian & Pogosyan, 2004; Padoan et al., 2006), and principal component analy-
sis (PCA) (Brunt & Heyer, 2002). Overall, analyses agree in finding power-law
linewidth-size relations, with similar coefficients and power-law expononents close
to q = 0.5. The lack of features in velocity correlations at intermediate scales,
and more generally the secular increase in velocity dispersion up to sizes compa-
rable to the whole of a GMC, indicates that turbulence is driven on large scales
within or external to GMCs (e.g. Ossenkopf & Mac Low, 2002; Brunt, 2003).
Interestingly, turbulence appears to have a “universal” character within most of
the molecular gas in the Milky Way, in the sense that the same scaling laws with
the same coeffcients fit both entire GMCs and moderate-density substructures
(observed via CO lines) within them. Using PCA, Heyer & Brunt (2004) find a
fit to the amplitudes of line-of-sight velocity components as a function of scale
following
δv = 0.9
(Lpca
1 pc
)0.56±0.02
km s−1 (6)
based on composite data of all PCA components from scales Lpca ∼ 0.03−30 pc in
a sample of 27 molecular clouds. Using data just within individual clouds, Heyer
& Brunt (2004) find a mean scaling exponent that is slightly lower, q = 0.49±0.15.
Note that the lengths Lpca entering the relation (6) are the characteristic scales
of PCA eigenmodes, and may differ from size scales defined in other ways. For
example, the effective GMC cloud diameters as measured by Solomon et al. (1987)
are on average about four times the maximum Lpca found in each cloud. Based
on the scaling law (6) the sonic length will be similar, ℓs ∼ 0.03 pc (allowing for
varying definitions of size), in all GMCs. We discuss this empirical result further
in §3.1; note that strongly self-gravitating clumps with high densities and surface
densities depart from the relation given in equation (6).
For evaluating the density distribution, the most unbiased measurements use
dust extinction maps (see Lada, Alves, & Lombardi, 2007 and references therein).
A promising new technique for observing the density distribution uses scattered
infrared light (Foster & Goodman, 2006), which can probe the structure of molec-
ular clouds for visual extinctions of 1 − 20 magnitudes at very high spatial res-
olution (Padoan, Juvela, & Pelkonen, 2006). Consistent with the prediction of

Theory of Star Formation 23
of the mass of a GMC, as much as ∼ 90%. FUV photoionization slows ambipolar
diffusion, and therefore star formation, when it dominates cosmic-ray ionization,
which occurs for visual extinctions AV . 4 mag from the surface or ∼ 8 mag along
a line of sight through the cloud (McKee, 1989). Suppression of star formation
in the outer layers of GMCs has been confirmed in the L1630 region of Orion
(Li, Evans, & Lada, 1997) and in Taurus (Onishi et al., 1998). To the extent
that ambipolar diffusion is essential for forming molecular cores, the absence or
near absence of sub-mm cores in the outer layers of Ophiuchus (Johnstone, Di
Francesco, & Kirk, 2004) and Perseus (Enoch et al., 2006; Hatchell et al., 2005) is
qualitatively consistent with this prediction. On the other hand, Strom, Strom,
& Merrill (1993) find that there is a substantial distributed population of young
stars in L1641, although this population is relatively old (5− 7) Myr.
2.3.1 Ionization The chemistry of molecular clouds is a full subject in its
own right. Here we summarize several developments that affect the ionization,
which governs the coupling between the gas and the magnetic field. (1) Photodis-
sociation regions (PDRs) are regions of the ISM that are predominantly neutral
and in which the chemistry and heating are predominantly due to far-UV radi-
ation (see the review by Hollenbach & Tielens, 1999). Most of the non-stellar
infrared radiation and most of the millimeter and submillimeter CO emission in
galaxies originates in PDRs. In the typical interstellar radiation field, photoion-
ization dominates ionization by cosmic rays for extinctions AV < 4 mag, which
includes most of the molecular gas in the Galaxy (McKee, 1989). (2) Polycyclic
aromatic hydrocarbons (PAHs), which contain a few percent of the carbon atoms,
often dominate the mid-IR spectrum of star-forming regions and galaxies. It is
frequently assumed that PAHs have a low abundance in molecular clouds due to
accretion onto dust grains; if this is not the case, they can dominate the ioniza-
tion balance, since electrons react with them very rapidly (Lepp et al., 1988). (3)
H+3 is a critical ion in initiating ion-molecule reactions in molecular clouds. For
many years, the rate of dissociative recombination, H+3 +e → H2 +H or H+H+H,
was uncertain, but careful laboratory experiments have shown that the rate co-
efficient for this reaction is large: a fit to the results of McCall et al. (2003) gives
αd(H+3 ) = 4.0 × 10−7(T/10 K)−0.52 cm3 s−1. In order to maintain the observed
abundance of H+3 in the face of this high recombination rate, these authors in-
ferred a very high cosmic-ray ionization rate, ζCR = 6×10−16 s−1 per H atom (in-
cluding secondary ionizations), in a diffuse molecular cloud along the line of sight
to ζ Per. Models involving two gas phases give somewhat lower values of ζCR (e.g.,
Dalgarno, 2006), but the correct value is now quite uncertain. In dense clouds,
Dalgarno (2006) concludes that the ionization rate is ζCR ≃ 2.5− 5× 10−17 s−1.
(4) Recent observations have established that carbon-bearing molecules freeze
out onto dust grains at high densities (nH ∼ 105 cm−3) in low-mass cores, with
nitrogen-bearing molecules freezing out at higher densities (di Francesco et al.,
2007). This affects the ionization, since it removes abundant ions such as HCO+
from the gas.
Although the chemistry determining the ionization in molecular clouds is com-
plex, simple analytic estimates are possible. In the outer layers of PDRs, carbon
is photoionized so that ne ≃ n(C). In regions ionized by cosmic rays, the degree
of ionization is given by
xe ≡
ne
nH
≃
( ζCR
αnH
)1/2
(24)

24 McKee & Ostriker
if the ionization is dominated by molecular ions (including PAHs), where α is
the relevant recombination rate in the chemistry that determines the ionization
fraction. If PAHs are depleted, then α ≃ 10−6 cm3 s−1 is the dissociative re-
combination rate for heavy molecules provided the density is high enough that
H+3 is destroyed primarily by reactions with such molecules; for lower densi-
ties, where the ionization is dominated by H+3 , one has α = αd(H+3 ). If PAHs
are sufficiently abundant that most of the electrons are attached to PAHs, then
α ≃ 3× 10−7 cm3 s−1 (Tielens, 2005) and ne in equation (24) includes the elec-
trons attached to PAHs. Metal ions can be readily included in the analytic theory
(McKee, 1989), but they do not appear to be important in dense cores (Maret,
Bergin, & Lada, 2006). Equation (24) is consistent with the results of Padoan
et al. (2004) at late times and at high densities for α ≃ αd(HCO+), which they
took to be 2.5× 10−6 cm3 s−1.
3 MACROPHYSICS OF STAR FORMATION
3.1 Physical State of GMCs, Clumps, and Cores
The molecular gas out of which stars form is found in molecular clouds, which
occupy a small fraction of the volume of the ISM but, inside the solar circle,
comprise a significant fraction of the mass. The terminology for the structure of
molecular clouds is not fixed; here we follow the discussion in Williams, Blitz, &
McKee (2000). Giant molecular clouds (GMCs) have masses in excess of 104 M⊙
and contain most of the molecular mass. Molecular clouds have a hierarchical
structure that extends from the scale of the cloud down to the thermal Jeans
mass in the case of gravitationally bound clouds, and down to much smaller
masses for unbound structures (Langer et al., 1995; Heithausen et al., 1998).
Overdense regions (at a range of scales) within GMCs are termed clumps. Star-
forming clumps are the massive clumps out of which stellar clusters form, and
they are generally gravitationally bound. Cores are the regions out of which
individual stars (or small multiple systems like binaries) form, and are necessarily
gravitationally bound. As remarked above, this terminology is not universal;
e.g., Ward-Thompson et al. (2007) use “pre-stellar core” to refer to a core, and
“cluster-forming core” to refer to a star-forming clump.
A molecular cloud is surrounded by a layer of atomic gas that shields the
molecules from the interstellar UV radiation field; in the solar vicinity, this layer
is observed to have a column density NH ≃ 2 × 1020 cm−2, corresponding to
a visual extinction AV = 0.1 mag (Bohlin, Savage, & Drake, 1978). A larger
column density, NH ≃ 1.4× 1021 cm−2, is required for CO to form (van Dishoeck
& Black, 1988). The layer of gas in which the hydrogen is molecular but the
carbon is atomic is difficult to observe, and has been termed “dark gas” (Grenier,
Casandjian, & Terrier, 2005).
The mass of a molecular cloud is generally inferred from its luminosity in the
J = 1 − 0 line of 12CO or 13CO. Because 12CO is optically thick, estimating
the column density of H2 molecules from the 12CO line intensity ICO (in units
of K km s−1) requires multiplication by an “X-factor,” an appropriate name
since it is not well understood theoretically; this is defined as X ≡ N(H2)/ICO.
Various methods have been used to infer the value of X in the Galaxy. In one
method, observations of γ-rays emitted by cosmic rays interacting with the ISM
give the total amount of interstellar matter; the mass of molecular gas follows by

Theory of Star Formation 27
As Larson pointed out, these three relations are not independent; any two
of them imply the third. Indeed, if we express the line width—size relation as
σ ≡ σpcR1/2pc , then
αvir =
(
5
π pc
) σ2pc
GΣ = 3.7
( σpc
1 km s−1
)2
(
100M⊙ pc−2
Σ
)
(27)
relates the three scaling laws. Observations supporting a “universal turbulence
law” (eq. 6) in the Galaxy, and thus small differences in σpc between inner- and
outer- Galaxy GMCs (Heyer & Brunt, 2004; Heyer, Williams, & Brunt, 2006),
then imply that the value of Σ is about the same for all GMCs with αvir ∼ 1,
regardless of Galactic location. Observational confirmation of this conclusion
would be valuable. Provided Larson’s Laws apply, the mean kinetic pressure
within GMCs is independent of mass and size, and is given by P¯kin = ρ¯σ2 =
3Σσ2pc/(4 pc). For inner-Galaxy GMCs, this is P¯kin/kB ≈ 3× 105 K cm−3.
These results can also be expressed in terms of the sonic length ℓs (see §2.1.3),
since σpc = cs(2/3ℓs, pc)1/2 if σ ≈ σnt ≫ cs, and ℓ = 2R. Gravitationally bound
objects (αvir ∼ 1) that obey a line width–size relation with an exponent ≃ 1/2
necessarily have surface densities Σ = (10/3παvir)c2s/(Gℓs) ∼ c2s/(Gℓs). The
small observed variation in Σ for the set of inner-Galaxy GMCs is then equiva-
lent to a small variation in ℓs in those clouds (since cs is observed to be about
constant). In terms of ℓs, the mean kinetic pressure in GMCs is P¯ = Σc2s/(2ℓs).
Do Larson’s laws apply in other galaxies? Blitz et al. (2007) summarize ob-
servations of GMCs in galaxies in the Local Group, in which the metallicity
varies over the range (0.1 − 1) solar. They find that the GMCs in most of these
galaxies would have luminous masses within a factor two of their virial masses
if X = 4 × 1020 cm−2 (K km s−1)−1; alternatively, if X has the same value
as in the Galaxy, then the GMCs are only marginally bound (αvir ≃ 2). They
conclude that metallicity does not have a significant effect on X since the ratio
of the virial mass to the CO luminosity is constant in M33, despite a factor 10
variation in metallicity. (Note, however, that Elmegreen 1989 argues that X de-
pends on the ratio of the metallicity to the intensity of the FUV radiation field,
which is not addressed by these results.) Although there is insufficient dynamic
range for clear evidence of a relationship between linewidth and size based on
current observations, the data are consistent with σ ∝ R1/2 but with values of
σpc (and therefore ℓs) that vary from galaxy to galaxy. Blitz et al. (2007) also
find that GMC surface densities have a relatively small range within any given
Local group galaxy, while varying from ∼ 50 M⊙ pc−2 for the SMC (L. Blitz,
personal communication) to > 100 M⊙ pc−2 for M33.
There are currently two main conceptual frameworks for interpreting the data
on GMC properties. One conception of GMCs is that they are dynamic, transient
entities in which the turbulence is driven by large-scale colliding gas flows that cre-
ate the cloud (e.g. Heitsch et al. 2005; Va´zquez-Semadeni et al. 2006; Ballesteros-
Paredes et al. 2007). This picture naturally explains why GMCs are turbulent
(at least in the initial stages), and why the line width–size relation within clouds
has an exponent of 1/2 – simply due to the scaling properties of supersonic tur-
bulence. However, it is less obvious why αvir ∼ 1 and why Σ has a particular
value, since small-scale dense structures may form (and collapse) at stagnation
points in a high-velocity compressive flow before sufficient material has collected
to create a large-scale GMC. Indeed, based on simulations with a converging flow

28 McKee & Ostriker
of ∼ 20 km s−1 with no stellar feedback, Va´zquez-Semadeni et al. (2007) find
that star formation occurs when the column density is NH ≈ 1021 cm−2, a factor
of ten below the mean observed value for GMCs. They also find that αvir remains
near unity after self-gravity becomes important, although the kinetic energy is
primarily due to the gravitational collapse of the cloud, not to internal turbulence.
Zuckerman & Palmer (1974) argued many years ago that GMCs cannot be in a
state of global collapse without leading to an unrealistically high star formation
rate. Proponents of the transient GMC picture counter by pointing out that most
of the gas in GMCs is unbound and never forms stars (e.g., Clark et al., 2005);
the global collapse is reversed by feedback from stars that form in the fraction of
the gas that is overdense and bound. Indeed, individual star formation proceeds
more rapidly than global collapse in essentially all turbulent simulations (see also
§3.2.2). However, dominance of global collapse and expansion over large-scale
random turbulent motions has not been confirmed from observations.
In the second conceptual framework, GMCs are formed by large-scale self-
gravitating instabilities (see §3.2.1), and the turbulence they contain is due
to a combination of inheritance from the diffuse ISM, conversion of gravita-
tional energy to turbulent energy during contraction, and energy injection from
newly formed stars (§3.2.2); the balance among these terms presumably shifts
in time. In the work of Chieze (1987); Elmegreen (1989); Maloney (1988); Mc-
Kee & Holliman (1999); McKee (1999), GMCs are treated as quasi-equilibrium,
self-gravitating objects, so that the virial parameter is near-unity by definition.
Whether or not equilibrium holds, the virial parameter is initially of order unity
in scenarios involving gravitational instability because GMCs separate from the
diffuse ISM as defined structures when they become gravitationally bound. For a
quasi-equilibrium, the mean surface density is set by the pressure of the ambient
ISM (see §2.2), which in turn is just the weight of the overlying ISM. Elmegreen
(1989) has given explicit expressions for how the coefficient in the line width–size
relation and the surface density depend on the external pressure, finding results
that are comparable with observed values. In particular, the cloud surface den-
sity scales with the mean surrounding surface density of the ISM. Even if GMCs
are not equilibria, if they are formed due to self-gravitating instabilities in spiral
arms (e.g. Kim & Ostriker 2002, 2006) they must initially have surface densities
a factor of a few above the mean arm gas density, consistent with observations.
Provided that stellar feedback destroys clouds within a few (large-scale) dynami-
cal times before gravitational collapse accelerates, the mean surface density would
never greatly exceed the value at the time of formation. Simple models of cloud
evolution with stellar feedback (e.g., Krumholz, Matzner, & McKee 2006) suggest
that the scenario of slow evolution with αvir = 1− 2 is self-consistent and yields
realistic star formation efficiencies, but more complete studies are needed.
The two approaches to interpreting GMC dynamics correspond to two alter-
nate views on GMC lifetimes. Elmegreen (2000) argued that, over a wide range
of scales, star formation occurs in about 1−2 dynamical crossing times of the sys-
tem, tcross ≡ 2R/(
√
3σ). Ballesteros-Paredes, Hartmann, & Va´zquez-Semadeni
(1999) and Hartmann, Ballesteros-Paredes, & Bergin (2001) focused on the par-
ticular case of star formation in Taurus, and argued that it occurred in about one
dynamical time. The alternate view is that GMCs are gravitationally bound and
live at least 2− 3, and possibly more, crossing times, tcross ≃ 10M1/46 Myr, where
M6 ≡ M/(106 M⊙) and a virial parameter αvir ∼ 1 − 2 is assumed. (Note that

Theory of Star Formation 31
with any plane-of-sky size may sample from the velocity field along the whole
line of sight, the linewidth varies only very weakly with size. Since a fraction of
the apparent clumps sample velocities from a range of line-of-sight distances no
larger than their transverse extent, however, Ostriker, Stone, & Gammie (2001)
argued that the lower envelope of the “clump” linewidth-size distribution should
follow the scaling defined by the true three-dimensional power spectrum; this is
generally consistent with observations (Stutzki & Guesten, 1990; Williams, de
Geus, & Blitz, 1994).
What about Larson’s third law? Since gravitationally bound clumps have
P ≃ GΣ2bd clump, and the mean pressure in a bound clump cannot be much
greater than the ambient pressure for a stable structure without an internal
energy source, it follows that typically Σbd clump is comparable to ΣGMC (Mc-
Kee, 1999). The high-mass, star-forming clumps studied by Plume et al. (1997)
violate this conclusion: they have Σ ∼ 4800M⊙ pc−2 ≃ 1 g cm−2 with consid-
erable dispersion, which is much greater than the typical GMC surface density
∼ 170M⊙ pc−2. There are several possible explanations for this, and it is impor-
tant to determine which is correct: Are these clumps just the innermost, densest
parts of much larger clumps? Do they have much higher pressures than their
surroundings but are avoiding gravitational collapse due to energy injection from
star formation? Or are they the result of a clump-clump collision that produced
unusually high pressures?
The density structure, velocity structure, and shape of cores offer potential
means for determining whether they are dynamic objects, with short lifetimes,
or quasi-equilibrium, gravitationally bound, objects. The observation of the Bok
globule B68 in near-infrared absorption revealed an angle-averaged density profile
consistent with that of a Bonnor-Ebert sphere to high accuracy (Alves, Lada, &
Lada, 2001). Since then, a number of other isolated globules and cores have been
studied with the same technique and fit to Bonnor-Ebert profiles, showing that
starless cases are usually close to the critical limit, while cases with stars often
match supercritical profiles (Teixeira, Lada, & Alves, 2005; Kandori et al., 2005).
Profiles of dense cores have also been obtained using sub-mm dust emission (see
di Francesco et al. 2007). A recent study by Kirk, Ward-Thompson, & Andre´
(2005) found that “bright” starless cores have density profiles consistent with
supercritical Bonnor-Ebert spheres.
Consistency of density profiles with the Bonnor-Ebert profile does not, however,
necessarily imply that a core is bound. Analysis of dense concentrations that
arise in turbulence simulations show that Bonnor-Ebert profiles often provide
a good fit to these structures (provided they are averaged over angles), even
when they are transients rather than true bound cores (Ballesteros-Paredes et
al., 2007). Even if a cloud with a Bonnor-Ebert profile is bound, however, it need
not be in equilibrium: Myers (2005) and Kandori et al. (2005) have shown that
density profiles of collapsing cores initiated from near-critical equilibria in fact
follow the shapes of static supercritical equilibria very closely. The reason for
this is that initially these cores collapse slowly, so that they are approximately
in equilibrium; at later times, they evolve via “outside-in” collapse to a state
that is marginally Jeans unstable everywhere (§4.1) so that ρ ∝ r−2 except in
a central core; highly supercritical Bonnor-Ebert spheres have profiles with the
same characteristic shapes. Thus, not only the density structure but also the level
of the internal velocity dispersion and detailed shape of the line profiles must be
used in order to distinguish between transient, truly equilibrium, and collapsing

32 McKee & Ostriker
objects (Keto & Field, 2005).
Core shapes also provide information on whether cores are transient or are
bound, quasi-equilibrium objects. In the absence of a magnetic field, a quasi-
equilibrium, bound cloud is approximately spherical. If the cloud is threaded by
a magnetic field that tends to support the cloud against gravity, it will be oblate;
if the field tends to compress the cloud (as is possible for some helical fields–Fiege
& Pudritz 2000), it will be prolate. If one assumes axisymmetry, the distribution
of observed axis ratios implies dense cores are primarily prolate (Ryden, 1996).
However, this conclusion appears to be an artifact of the assumption of axisym-
metry: using the method of analysis for triaxial clouds developed by Basu (2000),
Jones, Basu, & Dubinski (2001) and Jones & Basu (2002) concluded that cores
with sizes < 1 pc are in fact oblate. Basu (2000) showed that if the magnetic
field is aligned with the minor axis, as in most quasi-equilibrium models, the
projection of the field on the plane of the sky will not generally be aligned with
the projection of the minor axis, and he argued that the limited polarization
data available are consistent with the theoretical expectation that the field in the
cloud is aligned with the minor axis of the cloud. Kerton et al. (2003) showed
that larger structures, extending up to GMCs, are intermediate between oblate
and prolate, and are clearly distinct from the smaller objects. This is consistent
with the analyses of clump shapes in turbulence simulations by Gammie et al.
(2003); Li et al. (2004), who found that the majority of objects are triaxial. The
data on cores and small clumps are thus consistent with (but do not prove) that
they are bound, quasi-equilibrium objects. Large clumps and GMCs appear to
be farther from equilibrium.
A key feature of the cores that form individual low-mass stars is that they have
low nonthermal velocities, whether these cores are found in isolation or clustered
with other cores (di Francesco et al., 2007; Ward-Thompson et al., 2007). The
mean one-dimensional velocity dispersion in starless cores based on the sample of
Benson & Myers (1989) is 0.11 km s−1, such that the three-dimensional velocity
dispersion is approximately sonic. This places constraints on theoretical models,
and in particular may constrain the nature of turbulent driving. Klessen et al.
(2005) compared the results for cores identified in two (unmagnetized) simula-
tions with the same RMS Mach number ≈ 10, one driven on large scales and the
other driven on small scales; the time correlation of the driving force is short for
both cases. They found that only with large-scale driving is the mean turbulence
level within cores approximately sonic; in the small-scale driving case the prepon-
derance of cores are supersonic. Klessen et al. (2005) also found that the starless
cores in their large-scale-driving models are within a factor of a few of kinetic
and gravitational energy equipartition. In the 2D MHD simulations of Nakamura
& Li (2005) that implement driving by instantaneous injection of radial “wind”
momentum when (low-mass) stars are formed, the dense cores that are identified
also primarily have subsonic internal motions. Importantly, both types of models
show that dense, quiescent cores can form in a turbulent environment; the slow,
diffusive formation of quiescent cores central to the older picture of star formation
does not seem to be required.
What happens to dense cores once they form? Cores that have sufficient in-
ternal turbulence compared to their self-gravity will redisperse within a crossing
time. Cores that reach low enough turbulence and magnetization levels (allow-
ing for dissipation) within a few local free-fall times will collapse if M > Mcr.
Va´zquez-Semadeni et al. (2005) found that in globally supercritical 3D simula-

Theory of Star Formation 33
tions with driven turbulence, cores that collapse do so within 3–6 local free-fall
times of their formation. Nakamura & Li (2005) found via 2D simulations in-
cluding ambipolar diffusion that even when the mass in the simulation volume is
20% less than critical, supercritical cores can form; these then either collapse or
redisperse within several local free-fall times. In both cases (see also Krasnopol-
sky & Gammie 2005), only magnetically supercritical cores collapse, as expected.
Quiescent cores that are stable against gravitational collapse could in principle
survive for a long time (Lizano & Shu, 1989), undergoing oscillations in response
to fluctuations in the ambient medium (Keto & Field, 2005; Keto et al., 2006).
Because they are only lightly bound, however, such “failed cores” can also be
destroyed relatively easily by the larger-scale, more powerful turbulence in the
surrounding GMC. This process is clearly seen in numerical simulations; Va´zquez-
Semadeni et al. (2005) and Nakamura & Li (2005) found that the bound cores
that subsequently disperse do so in 1 – 6 ×tff . Quiescent, magnetically subcrit-
ical cores with thermal pressure ρcorec2s exceeding the mean turbulent pressure
ρ¯σ2nt (so that the core would collapse in the absence of magnetic support) can-
not easily be destroyed, however, and it is likely that they remain intact until
they merge with other cores to become supercritical. Simulations have not yet
afforded sufficient statistics to determine the mean time to collapse or dispersal
as a function of core properties and cloud turbulence level, or whether there is a
threshold density above which ultimate collapse is inevitable.
Observationally, core lifetimes can be estimated by using chemical clocks or
from statistical inference. The formation of complex molecules takes ∼ 105 yr at
typical core densities, but this “clock” can be reset by events that bring fresh C
and C+ into the core, such as turbulence or outflows (Langer et al., 2000). A
potentially more robust clock is provided by observations of cold H I in cores:
Goldsmith & Li (2005) infer ages of 106.5−7 yr for five dark clouds from the low
observed values of the H I /H2 ratio. These age estimates would be reduced if
clumping is significant and hence the time-averaged molecule formation rate is
accelerated, but, as in the case of complex molecules, they would be increased
if turbulent mixing were effective in bringing in fresh atomic hydrogen. In sim-
ulations of molecule formation in a turbulent (and therefore clumpy) medium,
Glover & Mac Low (2007) find that H2 formation is indeed accelerated when
compared with the non-turbulent case, although the atomic fractions they found
are substantially greater than those observed by Goldsmith & Li (2005). If con-
firmed, these ages, which are considerably greater than a free-fall time, would
suggest that these dark clouds are quasi-equilibrium structures.
Statistical studies of core lifetimes are based on comparing the number of star-
less cores with the number of cores with embedded YSOs and the number of
visible T Tauri stars. The ages of the cores (starless and with embedded YSOs)
can then be inferred from the ages of the T Tauri population, provided that
most of the observed starless cores will eventually become stars. The results
of several such studies have been summarized by Ward-Thompson et al. (2007),
who conclude that lifetimes are typically 3− 5 tff for starless cores with densities
nH2 = 103.5 − 105.5 cm−3. This is not consistent with dynamical collapse, nor
is it consistent with a long period (> 5tff) of ambipolar diffusion. It is consis-
tent with the ambipolar diffusion in observed magnetic fields (§2.3), which are
approximately magnetically critical. Of course, cores are created with a range of
properties, and observational statistics are subject to an evolutionary selection
effect: cores that are born or become supercritical evolve rapidly into collapse,

Theory of Star Formation 35
impact parameter Rcl). Expressed in terms of the cloud “gathering scale” Rgath ≡
[3Mcl/(4πρ¯)]1/3 = 190 pc(Mcl,6/n¯H)1/3 in the diffuse ISM, or in terms of the cloud
surface density Σcl ≡ Mcl/(πR2cl), the collision time is
tcollis =
√π
3
(ρcl
ρ¯
)2/3 Rgath
σ =
√π
4
Σcl
ρ¯σ . (31)
The mean intercloud separation is comparable to 2Rgath, which exceeds the
atomic disk scale height H ≈ 150 pc (Malhotra, 1995) for Mcl,6 ≡ Mcl/106 M⊙
>∼ 0.04. We can use equation (31) to estimate the collision time if all the dif-
fuse ISM gas were apportioned into equal-mass clouds with equal surface density.
Using σ ≈ 7 km s−1 for the nonthermal velocity dispersion in the diffuse ISM
at large ( >∼ H) scales (Heiles & Troland, 2003), Σcl ≈ 170 M⊙ pc−2 for the
mean GMC column (Solomon et al., 1987), and mean density n¯H = 0.6 cm−3
typical of the diffuse ISM at the Solar circle (Dickey & Lockman, 1990), this
yields a collision timescale > 5 × 108 yr. Gravitational focusing in principle de-
creases the cloud-cloud collision time, but in practice this does not help in form-
ing GMCs from atomic clouds since the reduction factor for the collision time,
[1+πGRclΣcl/σ2]−1, is near unity until the clouds are quite massive ( >∼ 105 M⊙).
Even if the background density were arbitrarily (and unrealistically) enhanced
by a factor 100 to approach ρcl, the total time of 40 Myr required to build clouds
from 104 M⊙ to 5× 106M⊙ (by successive stages of collisions) would still exceed
the estimated GMC lifetimes. These lifetimes are set by the time required to
destroy clouds by a combination of photodissociation and mechanical unbinding
by expanding HII regions (see §§3.2.2). Thus, if coagulation were the only way
to build GMCs, the process would be truncated by destructive star formation
before achieving the high GMC masses in which most molecular mass is actually
found.
Given the timescale problem and other difficulties of bottom-up GMC forma-
tion (e.g. Blitz & Shu, 1980), starting in the 1980’s the focus shifted to top-down
mechanisms involving large-scale instabilities in the diffuse ISM (e.g., Elmegreen,
1979, 1995). The two basic physical processes that could trigger growth of mas-
sive GMCs involve (1) differential vertical buoyancy of varying-density regions
along magnetic field lines parallel to the midplane, or (2) differential in-plane
self-gravity of regions with varying surface density. The first type of instability
is generically termed a Parker instability (Parker, 1966). The second type of
instability is generically a Jeans instability, although the simplest form of Jeans
instability involving just self-gravity and pressure cannot occur, due to galactic
(sheared) rotation (see §2.2). If the background rotational shear is strong, as
in the interarm regions of grand design spirals or in flocculent galaxies, there is
no true instability but instead a process known as swing amplification (Goldre-
ich & Lynden-Bell, 1965; Toomre, 1981); the dimensionless shear rate must be
d lnΩ/d lnR . −0.3 for “swing” to occur (Kim & Ostriker, 2001). If, on the
other hand, the mean background dimensionless shear rate is low (as in the inner
parts of galaxies where rotation is nearly solid-body, or as in spiral arms), another
type of gravitational instability can develop provided magnetic fields are present
to transfer angular momentum out of growing condensations (Elmegreen, 1987;
Kim & Ostriker, 2001); this is referred to as a magneto-Jeans instability (MJI).
The characteristic azimuthal spatial scale for Parker instabilities is λφ ≈ 4πH
(Shu, 1974). Growth rates are ∝ vA/H, which tends to increase in spiral arms;

36 McKee & Ostriker
thus these regions have traditionally been considered most favorable for growth
of Parker modes (Mouschovias, Shu, & Woodward, 1974). Numerical simulations
have shown, however, that Parker instability is not on its own able to create
structures resembling GMCs, because the instability is self-limiting and saturates
with only order-unity surface density enhancement (Kim et al., 1998; Santilla´n et
al., 2000; Kim, Ryu, & Jones, 2001; Kim, Ostriker, & Stone, 2002). Spiral arms
are also the most favorable regions for self-gravitating instabilities (Elmegreen,
1994), since the characteristic (thin-disk) growth rate ∝ GΣgal/cs is highest there.
(Here, Σgal is the mean gas surface density averaged over large [∼ kpc] scales in
the plane of the disk.) Since the spatial wavelengths of Parker and MJI modes
are similar, in principle growth of the former could help trigger the latter within
spiral arms (Elmegreen, 1982a,b). In fact, it appears that turbulence excited in
spiral shocks, together with vertical shear of the horizontal flow, may suppress
growth of large-scale Parker modes in arm regions (Kim & Ostriker, 2006). Thus,
while the Parker instability is important in removing excess magnetic flux from
the disk and in transporting cosmic rays (e.g. Hanasz & Lesch, 2000), it may be
of limited importance in the formation of GMCs.
Self-gravitating instabilities, unlike buoyancy instabilities, lead to ever-increasing
density contrast if other processes do not intervene. The same is true for the swing
amplifier if the (finite) growth is sufficient to precipitate gravitational runaway.
The notion that there should be a threshold for star formation depending on the
Toomre parameter,
Q ≡ κcsπGΣgal
= 1.4
( cs
7.0 km s−1
)
( κ
36 km s−1 kpc−1
)(
Σgal
13 M⊙ pc−2
)−1
, (32)
is based on the idea that star-forming clouds can form by large-scale self-gravitating
collective effects only if Q is sufficiently low. Here, κ2 ≡ R−3∂(Ω2R4)/∂R is the
squared epicyclic frequency, and cs is the mean sound speed of the gas. Numer-
ical simulations have been used to determine the nonlinear instability criterion,
finding that gravitationally bound clouds form provided Q < Qcrit ≈ 1.5 in model
disks that allow for realistic vertical thickness, turbulent magnetic fields, and a
“live” stellar component (Kim, Ostriker, & Stone 2003; Li, Mac Low, & Klessen
2005b; Kim & Ostriker 2007; these models do not include global spiral structure
in the gas imposed by variations in the stellar gravitational potential – see below.)
These results agree with empirical findings for the mean value of Q at the star
formation threshold radii in nearby galaxies (e.g. Quirk, 1972; Kennicutt, 1989;
Martin & Kennicutt, 2001). The masses of bound clouds formed via self-gravity
in galactic disk models where the background gas surface density is relatively
uniform are typically a few to ten times the two-dimensional Jeans mass,
MJ,2D ≡
c4s
G2Σgal
= 107 M⊙
( cs
7 km s−1
)4
(
Σgal
13 M⊙ pc−2
)−1
, (33)
depending on the specific ingredients of the model (Kim, Ostriker, & Stone, 2002,
2003; Kim & Ostriker, 2007).
Observations of external galaxies with prominent spiral structure show that
most of the molecular gas is concentrated in the spiral arms (e.g. Helfer et al.,
2003; Engargiola et al., 2003), and within the Milky Way the most massive clouds
that contain most of the mass and forming stars are strongly associated with
spiral arms (Solomon, Sanders, & Rivolo, 1985; Solomon & Rivolo, 1989; Heyer

38 McKee & Ostriker
into clumps. Since the mean density within GMCs is comparable to the typical
density of cold clouds in the atomic medium, the pre-existing cloudy structure
of the diffuse ISM would contribute to, but not dominate, the internal structure
within GMCs. In this “top-down” picture, the more massive, self-gravitating,
substructures within GMAs (or analogous atomic “superclouds”) would then be-
come gravitationally bound GMCs.
Although large-scale self-gravitating instabilities appear necessary for forming
massive GMCs, and many low-mass GMCs may form via fragmentation of mas-
sive GMCs or GMAs, it remains possible that a proportion of the low-mass GMCs
form through other mechanisms. Several recent studies have explored the pos-
sibility of GMC assembly via colliding supersonic flows (e.g Chernin, Efremov,
& Voinovich, 1995; Va´zquez-Semadeni, Passot, & Pouquet, 1995; Ballesteros-
Paredes, Hartmann, & Va´zquez-Semadeni, 1999; Heitsch et al., 2005; Va´zquez-
Semadeni et al., 2007); in this scenario the post-shock gas in the stagnation region
(which in fact becomes turbulent) represents the nascent GMC. For diffuse ISM
gas at mean density ρ¯ with relative (converging) velocity vrel, a total column of
shocked gas Σcl builds up over time
taccum =
Σcl
ρ¯vrel
= 1.6× 107yr
( NH
1021 cm−2
)
( nH
1 cm−3
)−1 ( vrel
20 km s−1
)−1
; (34)
note that, modulo order-unity coefficients, this time is the same as the result
in equation (31), with the velocity dispersion of the cloud distribution replaced
by the relative velocity of the converging flow. Correlated flows can only be
maintained up to the flow timescale over the largest spatial scale of the turbulence,
∼ 2H ∼ 300 pc. With vrel equal to the RMS relative velocity
√
6σ for a Gaussian
with 1D velocity dispersion σ ≈ 7 km s−1, this is ≈ 2× 107 yr, yielding a column
≈ 1021 cm−2 for n¯ ≈ 1 cm−3 (note that the shock velocity is about vrel/2).
If the interstellar magnetic field does not limit the compression of the shocked
gas (an artificial assumption, requiring flow along field lines only), the post-
shock gas would have high enough density for significant amounts of H2 to form
within the overall accumulation time, and the shielding from the diffuse UV is
sufficient for CO to begin to form (Hartmann, Ballesteros-Paredes, & Bergin,
2001; Bergin et al., 2004). However, it should be noted that for a shock velocity
of 10 km s−1, corresponding to a relative velocity of 20 km s−1, less than half
the C is in CO after 108 yr according to the 1D calculations of Bergin et al.
(2004). Turbulence-induced clumping can accelerate molecule formation rates
(Elmegreen, 2000; Glover & Mac Low, 2007), alleviating the timescale problem.
The molecule formation rate is proportional to the mass-weighted mean density
〈n〉M , which is larger than the volume-weighted mean density n¯ ≡ 〈n〉V in a
turbulent flow (see §2.1.4). Since 〈n〉M/n¯ ∼ 10 for typical turbulence levels
in GMCs, this reduces the typical molecule formation time (Tielens, 2005), ∼
2× 109 yr (T/10 K)−1/2/〈n〉M , to 1-2 Myr. Even so, the discussion above shows
that the maximum column density produced in the colliding-flow scenario is ∼
1021 cm−2, which is lower by an order of magnitude than the mean value of the
column of molecular gas in Milky Way GMCs; thus, this process can account for at
most a small fraction of the molecular gas mass in GMCs. Because gravitational
instabilities are suppressed by the high Q values in interarm regions, on the
other hand, the turbulent accumulation mechanism may be more important there.
Potentially, this may account for the observed difference (see above) between arm
and interarm GMC masses in the Milky Way, as well as for the very low surface

Theory of Star Formation 39
densities Σcl observed for many of the outer-Galaxy molecular clouds (Heyer,
Carpenter, & Snell, 2001).
Finally, we note that the dynamical considerations for gravitationally-bound
cloud formation apply whether the diffuse gas is primarily atomic, as is the case
in the Solar neighborhood and the outer portions of galaxies more generally, or
whether the diffuse gas is primarily molecular, as is true in the inner portions of
many galaxies. The time and length scales involved depend on the effective pres-
sure in the diffuse gas, which includes thermal as well as turbulent and magnetic
terms. If the diffuse gas is primarily molecular (or cold atomic) by mass, then the
mean turbulent and Alfve´n speeds will exceed the thermal speed in the dense gas.
The thermal sound speed cs in equations (32) and (33) must then be replaced
by an appropriately-defined ceff that incorporates the effects of turbulent kinetic
and magnetic pressures (the form of ceff would depend on the detailed multiphase
structure of the gas). Similarly, the characteristic timescale for self-gravitating
cloud formation becomes
tJ,2D =
ceff
GΣgal
= 3× 107yr
( ceff
10 km s−1
)
(
Σgal
100 M⊙ pc−2
)−1
. (35)
Turbulent velocity dispersions and magnetic field strengths are observed to be
similar in the cold and warm diffuse gas in the Solar neighborhood (Heiles &
Troland, 2003, 2005), and both observations and simulations (Piontek & Os-
triker, 2005) show that magnetic pressure is generally a factor two larger than
the thermal pressure.
The transition from having primarily atomic to primarily molecular gas typ-
ically occurs where the total gas surface density Σgal ≈ 12 M⊙ pc−2 (Wong &
Blitz, 2002; Blitz & Rosolowsky, 2004) and where the mean midplane pressure is
inferred to lie in the range P/k = 104 − 105 Kcm−3 (Blitz & Rosolowsky, 2006).
This transition occurs due to a combination of increased self-shielding (hence a
lower H2 dissociation rate) as Σgal increases, and increased density (hence in-
creased H2 formation rate) as both Σgal and the stellar surface density increase
toward the center of a galaxy (Elmegreen, 1993a; Blitz & Rosolowsky, 2004).
3.2.2 CLOUD EVOLUTION AND DESTRUCTION GMCs are born in
spiral arms downstream from the large-scale shock fronts, and begin life in a very
turbulent state. As they contract under the influence of gravity, this turbulence
decays, although the rate of decay is slowed by compression. Mestel & Spitzer
(1956) conjectured that turbulence would decay in about a crossing time, and
for that reason rejected turbulence as a mechanism for supporting clouds against
gravitational collapse (see also Mouschovias, Tassis, & Kunz, 2006, who argue for
magnetic support). Indeed, in simulations that do not include energy injection,
the contraction eventually evolves into free-fall collapse (cf. Va´zquez-Semadeni
et al., 2007, who simulated the formation of molecular clouds in colliding flows,
as discussed above), which is generally not observed. Furthermore, as discussed
in §3.1, turbulence in molecular clouds is observed to be ubiquitous, suggesting
that there is some mechanism acting to inject turbulent energy into clouds. How
important is energy injection to the structure and evolution of molecular clouds?
Broadly speaking, there are two modes of energy injection, external and in-
ternal. External mechanisms tap the turbulence in the diffuse ISM, and because
these modes are large scale, external driving would tend to yield a power spec-
trum that rises all the way to the largest scale in the GMC (see §2.1.1). In terms

44 McKee & Ostriker
1022 cm−2. Krumholz, Matzner, & McKee (2006) conjecture that such clouds can
occur in regions in which the mean density is not much less than the density in
the GMCs, so that external driving is more efficient; such conditions could occur
in starbursts. Testing and extension of these cloud evolution/destruction models
via full 3D numerical simulations has not yet been attempted, but development
and verification of the necessary computational codes is well underway (Mellema
et al., 2006; Mac Low et al., 2006; Krumholz, Stone, & Gardiner, 2006).
3.3 Core Mass Functions and the IMF
3.3.1 OBSERVATIONS OF THE STELLAR IMF AND THE CMF
How is the distribution of stellar masses, or initial mass function (IMF), estab-
lished? This is one of most basic questions a complete theory of star formation
must answer, but also one of the most difficult. Current evidence suggests that
the IMF is quite similar in many different locations throughout the Milky Way,
with the possible exception of star clusters formed very near the Galactic Cen-
ter (Scalo, 1998b presents evidence for significant variations in the IMF, but
Elmegreen, 1999 argues that much, if not all, of this is consistent with the ex-
pected statistical variations). The standard IMF of Kroupa (2001) is a three-part
power-law with breaks at 0.08 M⊙ and 0.5 M⊙; i.e. dN∗/d lnm∗ ∝ m−α∗ with
α = 1.3 for 0.5 < m∗/ M⊙ < 50, α = 0.3 for 0.08 < m∗/ M⊙ < 0.5, and
α = −0.7 for 0.01 < m∗/ M⊙ < 0.08. The slope of the IMF at m∗ >∼ M⊙ was
originally identified by Salpeter (1955), who found α = 1.35. Up to ∼ 1 M⊙,
a log-normal functional form dN∗/d ln(m∗) ∝ exp[−(lnm∗ − lnmc)2/(2σ2)] pro-
vides a smooth fit for the observed mass distribution (Miller & Scalo, 1979), with
Chabrier (2005) finding that mc ≈ 0.2 M⊙ and σ ≈ 0.55 applies both for indi-
vidual stars in the disk and in young clusters; the system IMF (i.e., counting
binaries as a single systems) has mc = 0.25M⊙. Thus, the main properties of the
IMF that any theory must explain are (i) the “Salpeter” power-law slope at high
mass, (ii) the break and turnover slightly below ∼ 1 M⊙, (iii) the upper limit on
stellar masses ∼ 150M⊙ (Elmegreen, 2000; Figer, 2005; Oey & Clarke, 2005), and
(iv) the universality of these features over a wide range of star-forming environ-
ments, apparently independent of the mean density, turbulence level, magnetic
field strength, and to large extent also metallicity. Theory predicts that there
should be a lower limit on (sub)stellar masses (Low & Lynden-Bell, 1976), but
this has not been confirmed observationally.
Important additional information has been provided by recent mm and submm
continuum surveys covering both cluster regions and larger areas in star-forming
systems (e.g. Motte, Andre, & Neri 1998; Testi & Sargent 1998; Johnstone et
al. 2000, 2001; Motte et al. 2001; Beuther & Schilke 2004; Reid & Wilson 2005,
2006a; Stanke et al. 2006; Enoch et al. 2006; Nutter & Ward-Thompson 2007).
Within continuum maps, high-density concentrations representing (starless) cores
have been identified in sufficient numbers (and with sufficent resolution) that
core mass functions analogous to the IMF can be defined. Similar core mass
functions (CMFs) may be derived using extinction data from well-sampled maps
(Lada, Alves, & Lombardi, 2007), and molecular line maps in high-density tracers
(Onishi et al., 2002). Studies of the CMF using extinction maps are only just
beginning, but they promise to be very important given the lower systematic
errors that are possible with this method. An excellent recent summary of the
statistical properties of observed cores is given by Ward-Thompson et al. (2007).

Theory of Star Formation 45
The CMFs derived from many independent studies and methods are in good
agreement with each other, and are remarkably similar in functional form to the
stellar IMF. In particular, regardless of the total mass and size of the star-forming
cloud, and regardless of whether cores are well separated or highly clustered, the
high-end CMF (above 1 M⊙) is consistent with a power law. Applying a uniform
analysis to data from 11 high- and low-mass star-forming regions, Reid & Wilson
(2006b) find α = 0.8 − 2.1, with the mean value α = 1.4. Observed CMFs
for relatively nearby clouds in the references cited above also show a peak and
turnover at low mass in the range ∼ 0.2−1 M⊙. For distant clouds, the peak core
mass is larger, but lack of resolution and hence low-mass incompleteness affects
these results. Observed well-resolved CMF distributions are thus very similar
to the stellar IMF, but shifted to higher mass by a factor of a few. For CMFs
derived from mm and submm observations, this factor involves some uncertainty
associated with conversion from dust emissivity to total mass. The CMF derived
from extinction in the Pipe nebula, which is not subject to this uncertainty, is
shifted to higher mass by a factor of 3 with respect to the standard stellar IMF
(Alves, Lombardi, & Lada, 2007).
The mirroring of the “universal” IMF by the (possibly also universal) CMF
suggests that the stellar mass distribution is imposed early in the star-forming
process. The final mass of a star appears to be controlled by the available reser-
voir of the core from which it forms, rather than, for example, being defined
by a termination of accretion due to internal stellar processes. The shift of the
observed CMF relative to the IMF nevertheless implies that stellar feedback and
other processes in the collapse or post-collapse stage affect stellar masses. Magne-
tized protostellar disk winds are believed to reduce the stellar mass compared to
core mass by a factor of a few (see discussion in §4.2.6). In particular, Matzner &
McKee (2000) predicted that the efficiency of a single star formation event in an
individual core is ǫcore ≃ 0.25− 0.7, depending on the degree of flattening, which
is comparable to the values implied by the observations cited above. Because
the efficiency is not sensitive to the parameters involved, this implies a similar
shift from CMF to IMF at all masses. Given the uncertainty in the CMF nor-
malization, the inefficiency of single star formation may account for essentially
the whole CMF → IMF shift. Some further fragmentation of presently-observed
massive cores during their collapse may also occur, but provided that the major-
ity of the mass goes into a single object, this will leave the high-mass end of the
CMF relatively unchanged. Since the CMF is already dominated by low-mass
cores (by mass as well as by number), the addition of low-mass stars formed as
fragments from collapsing high-mass cores would negligibly affect the low-mass
end of the IMF.
Molecular line observations of low-mass cores, whether found in isolation (as
in Taurus) or in close proximity to other cores in a dense, cluster-forming clump
(as in ρ Oph), show that these cores have very low nonthermal internal velocities
(Andre et al, 2006). Since weak internal turbulence implies that little density sub-
structure is present within these cores, they are unlikely to undergo subsequent
fragmentation during collapse, except to form binaries. The low-mass portion of
the CMF should therefore be conserved in mapping to the IMF, modulo mass re-
moval by outflows. Although cores in the high-mass end of the CMF are turbulent
and thus in principle subject to further fragmentation, the agreement between
the CMF and IMF suggests that this is not a dominant effect.
The environments of observed prestellar cores provide further clues to the pro-

46 McKee & Ostriker
cesses involved in their formation. Most stars form in clusters (Lada & Lada,
2003; see §4.3.5) and correspondingly, most (starless) molecular cores are part of
larger cluster-forming dense clumps. These cluster-forming clumps2, as observed
for example in Ophiuchus, Serpens, Perseus, and Orion, have supersonic internal
turbulent linewidths (even though the individual cores within them are subther-
mal). Compared to isolated cores, the cores in clusters tend to be more compact
in overall size and have higher densities and column densities; they are also lower
in mass (Ward-Thompson et al., 2007). The column densities of cluster-forming
clumps are generally quite large, and in particular they exceed the mean column
densities of the GMCs in which they are formed. In Perseus, where 80% of the
mm cores lie in groups and 50% are in clusters (Enoch et al., 2006), 50% of the
total cloud mass is at AV < 4 and 80% is at AV < 6, whereas 90% of the mass in
prestellar cores is in larger structures that have AV > 6, and 50% is at AV > 8
(Kirk, Johnstone, & Di Francesco, 2006). Similarly in Ophiuchus, the prestellar
cores are found in high-column density regions (AV > 15 for > 90% of the core
mass), while most of the cloud’s mass has much lower column densities (70% is
at AV < 7) (Johnstone, Di Francesco, & Kirk, 2004). The prestellar cores them-
selves represent only a tiny fraction of the total cloud mass: 5% in Perseus (Enoch
et al., 2006), and 3% in Ophichus (Johnstone, Di Francesco, & Kirk, 2004); this
is comparable to the net observed star formation efficiency over the lifetime of a
GMC (see §3.4). On the largest scales, GMCs generally consist of collections of
filaments, and both the clusters of cores and most of the individual isolated cores
are embedded in these filaments. The structure formation that produces cores,
and eventually stars, is therefore clearly a hierarchical process.
3.3.2 THEORETICAL PROPOSALS AND NUMERICAL RESULTS
Many theories have been proposed that aim to explain the IMF or some aspect of
it, and more recently to explain the CMF as well (see Elmegreen 2001 and Bonnell,
Larson, & Zinnecker 2007 for recent reviews). While none of the proposals to date
have won general acceptance, several have introduced elements that are likely to
be important in the eventual theory that is developed. Numerical simulations
have been valuable in demonstrating that the general characteristics of observed
CMFs arise naturally in turbulent, self-gravitating flows, and they have also been
useful in testing certain specific hypotheses. However, many features that are seen
in the simulations are not yet understood in a fundamental sense, and limited
numerical resolution may affect some existing results.
A variety of different numerical models have been used in computational stud-
ies of the mass distributions of bound and unbound condensations in turbulent
systems. Most models have adopted an isothermal equation of state: using SPH
techniques, Klessen & Burkert (2001), Bonnell, Bate, & Vine (2003), Bonnell,
Clarke, & Bate (2006), and Klessen (2001) analyzed decaying-turbulence mod-
els with a variety of power spectra, and Klessen (2001) and Ballesteros-Paredes
et al. (2006) analyzed driven turbulence models. Using grid-based codes in the
unmagnetized case, Ballesteros-Paredes et al. (2006) and Padoan et al. (2007) an-
alyzed driven-turbulence models. Using grid-based codes and including magnetic
fields, Gammie et al. (2003) analyzed decaying-turbulence models, and Vazquez-
Semadeni, Ballesteros-Paredes, & Rodriguez (1997), Ballesteros-Paredes & Mac
Low (2002), Li et al. (2004) and Padoan et al. (2007) analyzed driven-turbulence
models. Tilley & Pudritz (2004) analyzed decaying-turbulence unmagnetized
2Referred to as “cluster-forming cores” by Ward-Thompson et al. (2007)

48 McKee & Ostriker
star clusters like the Orion Nebula Cluster (Hillenbrand & Hartmann, 1998) and
globular clusters (Paresce & De Marchi, 2000), which are believed to have formed
at substantially greater pressures (McKee & Tan, 2003); the reason for this is not
clear.
A common numerical “shortcut” to studying cluster formation is to focus on
just a single cluster, rather than the whole hierarchical system; this allows the
collapse and fragmentation to be better resolved. Models of this kind initiate
a simulation at high density with comparable internal turbulent and gravita-
tional energy. However, this approach misses an aspect of the real situation
which may be quite important: self-gravitating massive condensations develop
out of non-self-gravitating gas in which pertubations have already been imposed
by turbulence. In simulations where the initial kinetic energy does not exceed
the gravitational energy, collapse occurs before the turbulence is able to imprint
a realistic density structure on the system, such that the subsequent fragmenta-
tion may also be unrealistic. In particular, this may lead to massive fragments
continuing to grow over time as they capture low-turbulence unstructured gas
from their surroundings (“competitive accretion” —see §4.1.2). To obtain a re-
liable measure of the high-end CMF from numerical models, it will be necessary
to perform simulations that include large scales as well as cluster scales, and
adequately resolve massive condensations both prior to and during collapse. In
addition, physical processes representing the feedback from star formation must
be properly included in order to impose realistic limits on fragmentation, coales-
cence, and accretion after collapse begins.
Another feature of numerical simulations that is at least qualitatively in accord
with observations is the presence of a resolved peak and turnover in the CMF.
Exactly how the location of this peak depends on model parameters, however, is
not well yet determined. In some simulations, the CMF peak is found to be at
masses comparable to the initial Jeans mass of the system (these are primarily
low-Mach-number simulations), while in other simulations the peak is at much
lower mass (these are primarily at high Mach number). The turbulent power
spectrum can also affect the position of the CMF peak, and in some simulations
the peak is seen to move to larger mass over time. In fact, the position of the
peak for an isothermal simulation with a fixed turbulence scaling law must be
a function of two dimensionless parameters, the ratio of the total mass in the
system to the initial Jeans mass, and the turbulent Mach number on the largest
scale. For magnetized simulations, an additional parameter is the ratio of mass to
the magnetic critical mass. Although limited dependence on parameters has been
explored, a comprehensive and controlled study has not yet been performed. Note
that the mass-weighted density in a turbulent system increases as the turbulent
Mach number increases (see §2.1.4), so that the Jeans mass at the “typical”
(mass-weighted) cloud density decreases as the turbulence level increases, for a
given mean (volume-weighted) density and Jeans mass. This probably accounts
for why the peak of the CMF was found to be far below the mean Jeans mass in
studies with high M, and close to the mean Jeans mass in studies with lower M.
A recurrent theme in star formation theory is that the characteristic mass –
defined by the peak of the IMF – is the Jeans mass at some preferred density.
An upper limit on the preferred density, and hence a lower limit on the fragment
mass, is the value at which which the optical depth is unity over a Jeans length;
this yields a minimum fragment mass ≈ 0.007 M⊙ (Low & Lynden-Bell, 1976).
More recently, Larson (2005) has argued that the thermal coupling of gas to dust

Theory of Star Formation 49
at densities above nH = nc ≈ 106 cm−3 results in a shift from weakly-decreasing
to weakly-increasing temperature as a function of density (T ∝ ρ−0.27 changes to
to T ∝ ρ0.07 at Tmin ∼ 5 K), and that the Jeans mass ∼ 0.3 M⊙ at this inflection
point sets the preferred mass scale in the IMF. Part of Larson’s argument is
that if structure is filamentary, then the filaments will contract radially while
γ < 1; fragmentation into protostellar cores would occur when the filament’s
central density reaches nc and γ exceeds unity. This argument does not take into
account, however, that the mass per unit length of a filament may be determined
primarily by the turbulence which originally creates it. In this case, the density
nc defines a Jeans length (see eq. 20 in §2.2), so that the mass scale that emerges
would be set by this (fixed) length scale ∼ λJ(nc) times the (variable) filament
mass per unit length. The simulations of Jappsen et al. (2005), which vary the
density nc at which the temperature minimum occurs, provide qualitative support
for Larson’s proposal in that the peak of the CMF moves to lower mass as nc
increases. The scaling of peak mass with nc in the simulations is not consistent
with the predicted mpeak ∝ n−0.95c scaling, however. In addition, these models
did not test dependence on other parameters that may be important, such as the
Mach number of the turbulence or the total mass of the system.
A recent comprehensive proposal to explain the CMF and IMF has been ad-
vanced by Padoan & Nordlund (2002, 2004) (hereafter “PN”). They argue that
because the strength of any given compression (in a shock) is related to its corre-
sponding (pre-shock) spatial scale ℓ, a power-law turbulence spectrum |v(ℓ)| ∝ ℓq
will result in a distribution of clump masses that itself follows a power law. In
particular, they propose that the clump mass function produced by turbulence
in a magnetized medium will obey dN (m)/d lnm ∝ m−3/(3−2q). They further
propose that at a given mass m, the fraction of clumps created by turbulence
that collapse is obtained by integrating the density PDF down to the density at
which that mass would be Jeans unstable, i.e. ρmin = π5σ6th/(36m2G3). With
this prescription, at high mass the limit of the integral ρmin → 0 and PN find
α = 3/(3 − 2q) ≈ 1.4. The position of the CMF peak would depend on the
properties of the density PDF; for a log-normal PDF (fM ; see eq. 5 in §2.1.4)
with µx = 0.5 − 2, the peak mass would be between 0.8 − 0.1 times the Jeans
mass at the mean (volume-weighted) density in the cloud.
The proposal of PN is attractive in its overall thrust, and analysis of numerical
simulations (Padoan et al., 2007) shows promising consistency with some of the
model predictions, such as a steepening of the CMF (larger α) with steeper
velocity power spectrum (larger q). The PN proposal, however, also suffers from
missing links in its theoretical underpinnings: (1) The effective value of vA is
defined by PN such that the typical compression ρ′/ρ in a shock moving at v is a
factor v/vA (in fact, compression factors depend on the magnetic field direction as
well as strength). This effective vA is assumed to be independent of scale, and for
numerical comparisons with data they adopt a value small compared to the typical
value in a GMC of ∼ 2 km s−1. (2) The argument used to obtain α = 3/(3− 2q)
for turbulent clumps assumes that each pre-shock volume ℓ3 maps to a number of
post-shock volumes of mass m that is independent of ℓ; i.e. ℓ3N (m)/L3 = const.
While this is plausible, other arguments can be made that draw on the scale-free
nature of turbulence, yet yield different results. Fleck (1996) and Elmegreen &
Falgarone (1996) have argued that in non-self gravitating turbulence one obtains
mN (m) = const. This scaling corresponds to converting a constant fraction of
the mass or volume behind every shock into clumps, ℓ′3ρ′N (m)/(L3ρ) = const.,

50 McKee & Ostriker
where ℓ′ = ℓvA/v = ℓρ/ρ′ is the post-shock scale. One might also propose
that the filling factor of post-shock clumps within the whole volume should be
constant, i.e. ℓ′3N (m)/L3 = const. This leads to N (m) ∝ m−(3−3q)/(3−2q),
or α = 0.75 for q = 1/2. While the assumption ℓ3N (m)/L3 = const. in the
PN formulation yields results that are in agreement with measured CMFs, a
physical argument is needed to explain why this is the correct choice among
several plausible alternatives. In particular, since PN’s argument for the slope
α = 3/(3 − 2q) involves only turbulence, why does this value of α disagree with
the distinctly-shallower empirical mass spectrum (α ∼ 0.5; see §3.1) of non-self-
gravitating clumps in GMCs? (3) The argument PN use to obtain a formula
for the mass function does not appear to take account of substructure within
clumps at a given mass scale although the presence of substructure is implicit
in their picture. In particular, they assume that any region that is unstable
by the thermal Jeans criterion will collapse. An implicit requirement for this
is that at each density, a contiguous volume containing a mass in excess of the
Jeans mass is present. More generally, since hierarchical density structures are
clearly important in nature (most cores and stars are clustered), any fundamental
theory should identify how this this comes about. Given these difficulties, it
appears premature to accept the PN proposal in its current form, although it is
promising as a basis for future development.
3.4 The Large-Scale Rate of Star Formation
Much of this review focuses on the detailed physical processes of star formation at
and below GMC scales. To understand the structure of a given galaxy, however,
or the evolution of a population of galaxies over cosmological timescales, often
only a very gross characterization of the star formation processes – such as the
overall star formation rate (SFR) and the resulting IMF – is adequate. Many
empirical studies of disk galaxies characterize the SFR in terms of the number
of stars formed per unit time per unit area Σ˙∗; this is usually reported using
either averages over the whole of a galaxy within some outer radius R, or using
azimuthal averages over an annulus of width dR to give Σ˙∗(R). Both of these
methods average over regions that may have widely varying SFRs, and the results
must be carefully interpreted as strong nonlinearities are involved. Fortunately,
with the data becoming available from large-scale galactic mapping surveys (e.g.
SONG and SINGS; Helfer et al., 2003; Kennicutt et al., 2003), it will soon be
possible to characterize SFRs on scales large compared to individual GMCs but
small enough to separately measure, e.g., SFRs for arm and interarm regions.
More fundamental than Σ˙∗ is the star formation or gas consumption timescale.
This is defined by tg∗ ≡ Σg/Σ˙∗ = Mg/M˙∗, where Σg is the gas surface density;
the second equality assumes that the same area average is used for the total gas
mass Mg and star formation rate M˙∗. The resulting timescale depends on the
gas tracer(s) chosen, which determines the range of gas densities included in Σg.
For a chemical species tracing gas in a class of structures denoted by S that have
mean internal gas density 〈ρ〉V = ρS , and total mass MS , a convenient fiducial
time for comparison to tS∗ ≡ MS/M˙∗ is the free-fall time obtained by using ρS
in equation (14). The star formation or gas consumption rate is then
M˙∗ ≡ ǫff,S
MS
tff,S
, (38)

52 McKee & Ostriker
order of magnitude or more to ∼ 0.1. Quiescent cores have individual lifetimes of
a few tff (see §3.1.2), and net efficiency of star formation in each core ∼ 1/3 (see
§3.3.1 and §4.2.6). These structures have evolved to have internal densities (and
hence self gravity) high enough that they can resist destruction by the ambient
turbulence. In regions such as forming clusters, where self-gravity causes strong
departures from the overall log-normal density distribution in GMCs and high
gravity is offset by locally-driven turbulence, the relation (39) would also not be
expected to apply.
Even within a given density regime, there may be significant cloud-to-cloud
variations in local conditions such that tS∗ need not be a universal quantity even
for structures observed in a given tracer. Indeed, Mooney & Solomon (1988)
showed that for Milky Way GMCs with virial masses (traced in CO)MCO = 104−
5 × 106 M⊙ and infrared luminosities LIR ∝ M˙∗, the ratio tGMC,∗ ∝ MCO/LIR
varies over two orders of magnitude and is not correlated with MCO. Williams &
McKee (1997) came to a similar conclusion from their analysis of OB associations
and GMCs in the Galaxy. With a total GMC mass ≃ 109M⊙ in the Galaxy and
a star formation rate of several M⊙ yr−1, the mean value of tGMC,∗ ≈ 3× 108 yr,
which translates to ǫff,GMC ∼ 0.01 if n¯H ∼ 100 cm−3 in GMCs. For dense gas
clumps in GMCs, however, it appears that conditions are more uniform, such
that there is less scatter in tS∗ for dense gas tracers. In particular, Wu et al.
(2005) show that the ratio LHCN/LIR ∝ Mdense clumps/M˙∗ measured in Milky
Way star-forming regions agrees with the same values measured in high-redshift
galaxies (Gao & Solomon, 2004), for which there is only one order of magnitude
scatter. Wu et al. (2005) estimate a corresponding star formation timescale of
tHCN,∗ = 8×107 yr. If the typical density of HCN-emitting gas is ∼ 105 cm−3, the
corresponding efficiency per free-fall time is ǫff,HCN ∼ 0.002. Krumholz & Tan
(2007) apply slightly different factors to convert total HCN and IR luminosities
to gas masses and star formation rates, and obtain ǫff,HCN ∼ 0.006. These values
of ǫff are small compared to those for individual cores (∼ 0.1), which in clustered
regions (where most stars form) have densities ∼ 107cm−3 (Ward-Thompson et
al., 2007) that are large compared to the densities traced by HCN.
In spite of the large scatter in tS∗ from one local region to another (in various
density tracers), empirical studies have shown that when averaged over large
scales, tg∗ is correlated with the global properties of gas in a galaxy. The early
studies of Schmidt (1959, 1963) sought to characterize the star formation rate as
a power law (with index > 1) in the mean gas density (both volume and surface
density); this would then translate to tg∗ (or tff/ǫff) that varies as a negative power
of gas density. More recently, following Kennicutt (1989), a number of empirical
studies of disk galaxies have identified and explored “Kennicutt-Schmidt” (or KS)
laws of the form Σ˙∗ ∝ Σp+1g , for which tg∗ ∝ Σ−pg . The original study of Kennicutt
investigated correlations of Σ˙∗(R) (based on Hα) with the total Σg(R) (including
both atomic and molecular gas); he found an index p = 0.3 for Σg(R) above a
threshold level. Kennicutt (1998) studied correlations of global averages of Σ˙∗
with Σg (again combining atomic and molecular gas). For the whole sample
including normal galaxies, the centers of normal galaxies, and starbursts, the
fitted index is p = 0.4; the index is slightly larger for just normal spirals. Recent
years have seen a number of other studies of the Σg – Σ˙∗ relationship based on
annular averages in galaxies, using Hα, radio continuum, or far-IR to measure
star formation, and using either the total gas surface density or just the molecular

Theory of Star Formation 53
gas contribution from CO observations (Wong & Blitz, 2002; Murgia et al., 2002;
Boissier et al., 2003; Heyer et al., 2004; Komugi et al., 2005; Schuster et al.,
2007). Most of these studies have found p in the range 0.3− 0.4, although larger
values of p have been obtained in some analyses that include both atomic and
molecular gas. For dense gas as traced by HCN, Gao & Solomon (2004) found a
linear relationship between the integrated star formation rate and the total mass
of dense gas, i.e. p = 0, based on a sample including both normal galaxies and
luminous/ultraluminous IR galaxies. For the same sample, the fitted SFR-gas
mass index is p = 0.7 for less-dense molecular gas observed in CO lines. All of
these fits involve (at least) an order of magnitude scatter about the mean relation.
Taken together, these results imply that the amount of dense gas available for
star formation increases nonlinearly with the global amount of lower-density gas,
but that the star formation rate in this dense gas is independent of global galactic
properties.
A second approach to characterizing global SFRs introduces the global timescale
associated with the galaxy, the orbital period torb = 2π/Ω. For grand design spi-
rals, the SFR is expected to be proportional to the rate at which gas passes
through spiral arms, since GMCs are expected (and observed) to form rapidly in
the high-surface-density gas behind the spiral shock (e.g. Roberts 1969; Kim &
Ostriker 2002, 2006; Shetty & Ostriker 2006). Shu (1973) appears to have been
the first to propose this idea, and showed that it is roughly consistent with obser-
vations of star formation in the Galaxy. Wyse (1986) proposed that GMCs, and
hence stars, are the result of atomic cloud-cloud collisions at a rate ∝ Σ2HI(Ω−Ωp),
where Ωp is the pattern speed. More generally, Wyse & Silk (1989) suggested
that the star formation rate should scale as Σ˙∗ ∝ ΣgΩ. This has been confirmed
by Kennicutt (1998); the resulting two forms for the KS law are
Σ˙∗ = 0.017ΣgΩ ≃ (2.5±0.7)×10−4
(
Σg
1 M⊙ pc−2
)1.4±0.15
M⊙ yr−1 kpc−2. (40)
The fact that there are two forms of the star formation law implies that there
is a correlation between Σg and Ω; Krumholz & McKee (2005) found Ω ∝ Σ0.49g
for a sample comprised of both normal and starburst galaxies (Kennicutt, 1998;
Downes & Solomon, 1998). The reason for this correlation is not known at
present, but may be related to an overall tendency for velocity dispersions to in-
crease at large surface densities (see below). The corresponding gas consumption
time is tg∗/torb ≈ 10 with tg∗ evaluated for the entire galaxy and torb evaluated
at the outer edge of the star formation. Subsequent observations have found
tmol,∗/torb ∼ 10 − 100 when considering the molecular gas alone (Wong & Blitz,
2002; Murgia et al., 2002).
Since most star formation is observed to take place within bound GMCs, it is
useful to introduce fGMC ≡ ΣGMC/Σg, i.e. the fraction of gas that is found in
GMCs. The surface densities must be averaged over a region containing a large
number of GMCs, since the specific star formation rate has very large fluctuations;
the average can be over a local patch of a galaxy, an azimuthal ring, or an entire
galaxy. Equation (38) implies
Σ˙∗ = ǫff,GMC
ΣgfGMC
tff,GMC
. (41)
This form of the star formation law is particularly useful if most of the gas is in
GMCs, fGMC ≃ 1. Since the gas density in the midplane ρg ∝ Ω2/GQ2 in terms

56 McKee & Ostriker
it does not give a sufficient condition: some of the galaxies in the Martin &
Kennicutt (2001) sample have Rth inside the radius at which the gas becomes
molecular, which in turn is inside the radius at which cold atomic gas first appears
(see also de Blok & Walter, 2006, who find evidence of a cold atomic component
even in non-star-forming regions). In these cases, it is possible that MRI-driven
turbulence in the outer disk maintains the effective Q greater than the critical
value even when some of the gas is cold (Piontek & Ostriker, 2007).
4 MICROPHYSICS OF STAR FORMATION
4.1 Low-Mass Star Formation
Star formation is traditionally divided into two parts: Low-mass stars form in
a time short compared to the Kelvin-Helmholz time, tKH = Gm2∗/RL, whereas
high-mass stars form in a time & tKH (Kahn, 1974). This distinction between
low-mass and high-mass protostars is not fully satisfactory, however, since for a
sufficiently high accretion rate any protostar would be classified as “low-mass.”
We somewhat arbitrarily divide low and high-mass stars at a mass of 8 M⊙.
Protostars that will form stars with masses significantly below this value have
luminosities dominated by accretion, and they form from cores that have masses
of order the thermal Jeans mass. Protostars above this mass have luminosities
that are dominated by nuclear burning unless the accretion rate is very high, and
if they form from molecular cores, those cores are significantly above the thermal
Jeans mass. Low-mass stars undergo extensive pre-main sequence evolution in
the Hertzsprung-Russell diagram, from the point on the “birthline,” where they
cease accreting and are revealed (Stahler, 1983; see also Larson, 1972), to the
main sequence. Here we briefly review the current understanding of how such
stars form.
4.1.1 Theory of core collapse and protostellar infall As dis-
cussed above, low-mass stars appear to form from gravitationally bound cores.
The time scale for the collapse of these cores determines both the time scale
for the formation of a star and the accretion luminosity. Note that the rate of
infall onto the star-disk system, m˙in, can differ from the rate of accretion onto
the protostar, m˙∗, since some of the infalling gas can be temporarily stored in
the disk. The collapse of such cores and the growth of the resulting protostars
has been reviewed by Larson (2003), and we draw on this work here. At the
outset of theoretical studies of star formation, it was realized that isothermal
cores undergoing gravitational collapse become very centrally concentrated, with
a density profile that becomes approximately ρ ∝ r−2 (Bodenheimer & Sweigart,
1968; Larson, 1969). Prior to the formation of the protostar, there is a central,
thermally supported region of size r ≃ λJ . Collapse of a marginally unstable core
begins near the outer radius. The r−2 density gradient is created as the wave of
collapse propagates inward, leaving every scale marginally unstable as the col-
lapse accelerates (cf. Larson, 2003). That is, since λJ ∼ cs/(Gρ)1/2 (eq. 20), a
sphere that is marginally unstable at each scale, r ∼ λJ , will have ρ ∼ c2s/(Gr2)
when the protostar is first formed; the corresponding infall rate is
m˙in ∼
MG
tG
=
c3s
G ⇒ m˙in = φin
c3s
G, (43)

Theory of Star Formation 57
where the gravitational mass and radius are defined in equation (13) and φin is a
numerical factor that is typically & 1. (When the effect of protostellar outflows
is included, the infall rate is reduced by a factor ǫcore < 1.) Although this
result was first derived for an isothermal sphere, Stahler, Shu, & Taam (1980)
emphasize that it should apply approximately to the collapse of any cloud that is
initially in approximate hydrostatic equilibrium, with c2s → c2eff = c2s + v2A + v2turb
including the effects of magnetic fields and turbulence as well as thermal pressure;
Shu, Adams, & Lizano (1987) suggest that ceff . 2cs, however. This infall rate
explicitly depends only on the sound speed, but it implicitly depends on the
density of the core: since the core was assumed to be initially in hydrostatic
equilibrium, equation (43) is equivalent to m˙in ∼ Mcore/tG ∝ Mcoreρ1/2.
There are two limiting cases for the gravitational collapse of an isothermal
sphere. In the first case, originally considered by Larson (1969) and Penston
(1969) and extended by Hunter (1977), one begins with a static cloud of constant
density and follows the formation of the r−2 density profile. At the time when
the protostar first forms (i.e., when the central density reaches infinity in this
idealized calculation), the collapse is highly dynamic, with an infall velocity of
3.3cs. The infall rate onto the star is large, rapidly increasing from m˙in = 29c3s/G
at the moment of protostar formation to m˙in = 47c3s/G. In the physically unreal-
istic case of an infinite, uniform medium, the accretion rate would remain at this
high value; in practice, the accretion rate rapidly declines after the formation of
a point mass (see below). In the opposite case, considered by Shu (1977), one
assumes that the evolution to the r−2 density profile is quasi-static (most likely
due to the effects of magnetic fields—see below), so that the infall velocities are
negligible at the moment of protostar formation. The resulting initial configu-
ration is the singular isothermal sphere (SIS), which is an unstable hydrostatic
equilibrium. The collapse is initiated at the center, and the point at which the
gas begins to fall inward propagates outward at the sound speed (the “expansion
wave”): Rew = cst. This solution is therefore termed an “inside-out” collapse.
For r ≥ Rew, the density is that of a SIS, ρ = c2s/(2πGr2); for r < Rew, the
gas accelerates until it reaches free fall, with v = −(2Gm∗/r)1/2 and ρ0 ∝ r−3/2.
The generalized post-core-formation solutions of Hunter (1977) share the same
density and velocity scalings at small radii. The infall rate for Shu’s expansion
wave solution is constant in time,
m˙in = 0.975c3s/G = 1.54× 10−6(T/10 K)3/2 M⊙ yr−1. (44)
The total mass inside the expansion wave at time t is 2m˙int, so that about half
this mass is in the protostar (i.e., few ≡ m∗/mew ≃ 1/2). Larson (2003) describes
the Larson-Penston-Hunter (LPH) and Shu solutions as “fast” and “slow” col-
lapse, respectively, and suggests that reality is somewhere in between. A general
discussion of the family of self-similar, isothermal collapse solutions has been
given by Whitworth & Summers (1985).
Observations suggest that the cores that form low-mass stars initially have
density profiles that approximate those of Bonnor-Ebert spheres (§3.1). Foster &
Chevalier (1993) used time-dependent simulations to follow the collapse of such
spheres under the assumption that support is by thermal pressure alone. They
found that the collapse of the innermost, nearly uniform, part of a critical Bonnor-
Ebert sphere (i.e., one with a center-to-edge density contrast of 14.1) approaches,
but does not reach, the Larson-Penston (LP) solution prior to and at the time of
core formation. Shortly thereafter, the infall rate begins to decline; there is no

62 McKee & Ostriker
which would thicken the disk and transport angular momentum, is unclear. The
infalling gas goes through two shocks, a C-shock (which has a structure dominated
by ambipolar diffusion—e.g., Draine & McKee, 1993) and a shock at the outer
edge of the centrifugally supported disk. When the protostar reaches 1M⊙, the C-
shock is at about 103 AU, and the centrifugal shock is at about 102 AU, consistent
with data on T Tauri systems (see §4.2.1 below). Within the self-similar frame-
work, they find that magnetic braking can be adequate to maintain accretion
onto the central protostar; in this case there would be no need for internal disk
stresses to drive accretion. The infall rate in their fiducial case is 4.7c3s/G; for a
gas at 10 K, this corresponds to a star formation time tsf = 1.3×105(m∗/M⊙) yr.
Their solution does not include an outflow, but they show how one might be in-
cluded and estimate that this could reduce the accretion rate by a factor . 3.
In sum, based on the theoretical work to date, it is clear that the infall rate is
proportional to c3eff/G, where ceff is an effective sound speed (Stahler, Shu, &
Taam, 1980), but the value of the coefficient and its time dependence have yet
to be determined in realistic cases.
The magnetic flux problem in star formation is that stars have very large
values of the mass-to-flux ratio (µΦ ∼ 104 − 105 in magnetic stars, ∼ 108 in the
Sun—Nakano, 1983), whereas they form from gas with µΦ ∼ 1. This problem
does not have an adequate solution yet, but it appears that it must be resolved
in part on scales . 1000 AU and in part on smaller (∼ AU) scales. Detailed
calculations of the ionization state of the infalling and accreting gas show that the
ionization becomes low enough that the field decouples from the gas at densities
of order 1010.5 − 1011.5 cm−3 (Nishi, Nakano, & Umebayashi, 1991; Desch &
Mouschovias, 2001; Nakano, Nishi, & Umebayashi, 2002); decoupling occurs at
a somewhat lower density after the formation of the central protostar, due to
the stronger gravitational force (Ciolek & Ko¨nigl, 1998). Li & McKee (1996)
showed that once the field decouples from the gas, magnetic flux accumulates in
the accretion disk as the gas flows through the field and onto the protostar. The
pressure associated with this field is strong enough to drive a C-shock (which has
a structure dominated by ambipolar diffusion) into the infalling gas. The radius
of the shock is predicted to be several thousand AU at the end of the infall phase
of a 1 M⊙ star; inside the shock, the field is approximately uniform (except close
to the star) and the gas settles into an infalling, dense disk that they identified
with the “outer disk” observed in HL Tau (Hayashi, Ohashi, & Miyama, 1993).
These results have been confirmed and improved upon by Contopoulos, Ciolek,
& Ko¨nigl (1998), Ciolek & Ko¨nigl (1998) and Krasnopolsky & Ko¨nigl (2002).
Tassis & Mouschovias (2005) have carried out 2D axisymmetric calculations with
careful attention to the evolution of the ionization and find that the location of
the shock oscillates, leading to fluctuations in the accretion rate; it is important
to determine if this effect persists in a full 3D simulation. Tassis & Mouschovias
(2007a,b,c) find that the magnetic field in the central region (r . 10 AU) is
about 0.1 G at the end of their calculation, when the central star has a mass
∼ 0.01M⊙; this is at the low end of the fields inferred in the early solar nebula
from meteorites, which are in the range 0.1 − 10 G (Morfill, Spruit, & Levy,
1993). They show that ohmic dissipation becomes as important as ambipolar
diffusion at densities & 1012.5 cm−3, but it does not affect the total magnetic
flux. However, even though these processes significantly reduce the field within
a few AU of the protostar, they are not sufficient to reduce the magnetic flux in
the protostar to the observed value (Nakano & Umebayashi, 1986; Li & McKee,

Theory of Star Formation 65
2005a; magnetic fields, which tend to suppress accretion, have not been consid-
ered yet). Bonnell and his collaborators (Bonnell & Bate, 2006 and references
therein) argue that the gas throughout star-forming clumps has a very low tur-
bulent velocity so that protostars in clusters can accrete efficiently. On the other
hand, Krumholz, McKee, & Klein (2005a) argue that stellar feedback and the
cascade of turbulence from larger scales ensure that the star-forming clumps are
sufficiently turbulent to be approximately virialized and to therefore have neg-
ligible competitive accretion. Analysis of data from several star-forming clumps
shows that stars in these clumps could grow by only (0.1 − 1)% in a dynamical
time, far too small to be significant. The timescale for the formation of star
clusters is an important discriminant between these models: Star clusters form
in about 2tff if turbulence is allowed to decay, whereas it can take significantly
longer if turbulence is maintained (Bonnell, Bate, & Vine, 2003). The observa-
tional evidence discussed by Tan, Krumholz, & McKee (2006) and Krumholz &
Tan (2007) favors the longer formation time. This controversy can be resolved
through more detailed observations of gas motions in star-forming clumps and
through more realistic simulations that allow for the evolution of the turbulent
density fluctuations as the clump forms and evolves to a star-forming state, and
that incorporate stellar feedback.
4.1.3 Observations of low-mass star formation The growth of proto-
stars can be inferred through observations of the mass distribution surrounding
the protostar, the velocity distribution of this circumstellar gas, and the non-
stellar radiative flux. The mass and/or temperature distribution on both small
and large spatial scales can be inferred by modeling the spectral energy distribu-
tion (SED) of the continuum. Protostellar SEDs are conventionally divided into
four classes, which are believed to represent an evolutionary progression ( Myers
et al., 1987 divided sources into two classes; Lada, 1987 introduced Classes I-III;
Adams, Lada, & Shu, 1987 discussed a similar classification; and Andre, Ward-
Thompson, & Barsony, 1993 introduced Class 0). Andre, Ward-Thompson, &
Barsony (2000) have summarized the classification scheme:
Class 0: sources with a central protostar that are extremely faint in the
optical and near IR (i.e., undetectable at λ < 10 µ with the technology of the
1990’s) and that have a significant submillimeter luminosity, Lsmm/Lbol >
0.5%. Sources with these properties have Menvelope & m∗. Protostars are
believed to acquire a significant fraction, if not most, of their mass in this
embedded phase.
Class I: sources with αIR > 0, where αIR ≡ d log λFλ/d log λ is the slope of
the SED over the wavelength range between 2.2 µ and 10–25 µ. Such sources
are believed to be relatively evolved protostars with both circumstellar disks
and envelopes.
Class II: sources with −1.5 < αIR < 0 are believed to be pre-main sequence
stars with significant circumstellar disks (classical T Tauri stars).
Class III: sources with αIR < −1.5 are pre-main sequence stars that are no
longer accreting significant amounts of matter (weak-lined T Tauri stars).
These classes can also be defined in terms of the “bolometric temperature,” which
is the temperature of a blackbody with the same mean frequency as the SED of
the YSO (Myers & Ladd, 1993).

68 McKee & Ostriker
file in which the blue wing is stronger than the red wing. Observations of samples
of starless cores (Lee, Myers, & Tafalla, 1999, 2001), Class 0 sources (Gregersen
et al., 1997), and Class I sources (Gregersen et al., 2000) show a “blue excess”
[(blue asymmetries - red asymmetries)/(number of sources)] of about 0.25−0.35,
indicating that many of these sources are undergoing collapse (Myers, Evans, &
Ohashi, 2000). Unfortunately, it has proved difficult to carry out unambiguous
observational tests of the theoretical models for protostellar accretion. Furuya,
Kitamura, & Shinnaga (2006) mapped the infall in a young Class 0 source and
found reasonably good agreement with the LPH solution (φin ≃ 20 in eq. 43);
this source appears to be very young, since for r > 100 AU there is no evidence for
the ρ ∝ r−3/2 density profile expected for accretion onto a protostar of significant
mass. Tafalla et al. (1998) and Lee, Myers, & Tafalla (2001) found that infall is
more extended than expected in inside-out collapse models, although this infall
may reflect the formation of small clusters rather than individual stars. They
also found that the infall velocity is faster than expected in standard ambipo-
lar diffusion models; however, the velocities are consistent with the collapse of
magnetically supercritical cores (Ciolek & Basu, 2000). A potentially important
result is that Ohashi et al. (1997) found that cores in Taurus are in solid body
rotation on scales & 0.03 pc but conserve angular momentum on smaller scales.
The physical significance of this length scale could be inferred by determining its
value in other molecular clouds.
Brown Dwarfs Since brown dwarfs represent the low-mass extreme of star
formation, they can shed light on the earliest stages of star formation. As a
result of a great deal of observational work over the past decade, it has been
established that most brown dwarfs form by the same mechanism as most stars
(Luhman et al., 2007; Whitworth et al., 2007): the initial mass function, velocity
and spatial distributions at birth, multiplicity, accretion rates, circumstellar disks,
and outflows are all continuous extensions of those for hydrogen-burning stars.
This is to be expected, since stars near the H-burning limit at 0.075M⊙ reach
their final mass long before hydrogen burning commences. Following Whitworth
et al. (2007) and Chabrier et al. (2007), we shall assume that brown dwarfs form
by gravitational instability on a dynamical timescale, and that their composition
reflects that of the ambient interstellar medium. By contrast, planets are believed
to form in circumstellar disks and to have an elemental composition with an
excess of heavy elements. With these definitions, the observational distinction
between giant planets with masses & MJ and small brown dwarfs is somewhat
indistinct, but should eventually be amenable to spectroscopic determination
(Chabrier et al., 2007). The lower limit to the mass of a brown dwarf is set
by the condition that the star become opaque to the radiation it emits while
undergoing gravitational collapse (Low & Lynden-Bell, 1976); including helium,
this is about 4 × 10−3M⊙ ≃ 4MJ (Whitworth et al., 2007). The smallest brown
dwarfs detected to date have masses ∼ (0.01 − 0.02)M⊙ (Luhman et al., 2007).
In order for a brown dwarf to form, its mass must exceed the Bonnor-Ebert
mass, even if it forms via shock compression (Elmegreen & Elmegreen, 1978); the
pressure at the surface of the core that forms the brown dwarf must therefore
be P/kB & 109(T/10 K)4(10−2 M⊙/mBD)2 K cm−3 at the surface of the brown
dwarf. Assuming that brown dwarfs form by turbulent fragmentation, Padoan
& Nordlund (2004) show that such pressures can be reached in a large enough

72 McKee & Ostriker
mospheres where the densities are low and stellar X rays strongly heat the gas
(Najita et al., 2006). Quite sophisticated radial-vertical radiative models (includ-
ing grain growth and settling) have been developed that agree well with observed
spectral energy distributions from µm to mm wavelengths (see e.g. Dullemond
& Dominik, 2004, D’Alessio et al., 2006 and references therein). The IR emission
signatures, including PAH features at 3− 13µ and edge-on silhouette images, as
well as scattered-light/polarization observations in optical and near-IR, indicate
that although some grains have grown to large sizes, small grains still remain in
disk atmospheres (see references and discussion in Dullemond et al., 2007 and
Natta et al., 2007).
Disk lifetimes are inferred based on stellar ages combined with IR and mm/sub-
mm emission signatures, which are sensitive to warm dust. Multiwavelength
Spitzer observations of the nearby star-forming cluster IC 348 (Lada et al., 2006)
show that for ∼ 70% of stars, disks have become optically-thin in the IR (implying
inner disks R <∼ 20AU have been removed) within the 2-3 Myr age of the system;
disk fractions are slightly higher (∼ 50%) for Solar-type stars than in those of
higher or lower mass. Observations of other clusters are consistent with these
results (Sicilia-Aguilar et al., 2006a). L-band observations of disk frequencies in
clusters spanning a range of ages (Haisch, Lada, & Lada, 2001) suggests that
overall disk lifetimes are ≈ 6 Myr. Even in the 10Myr old cluster NCG 7160,
however, a few percent of stars still show IR signatures of disks (Sicilia-Aguilar
et al., 2006a), and disk lifetimes appear to be inversely correlated with the mass
of the star (Hernandez et al., 2007). Signatures (or their absence) of dusty disk
emission are also well correlated with evidence (or lack) of accretion in gaseous
emission line profiles (see below) in systems at range of ages, indicating that gas
and dust disks have similar lifetimes (Jayawardhana et al., 2006; Sicilia-Aguilar
et al., 2006b). Andrews & Williams (2005) found, for a large sample of YSOs
in Taurus-Auriga, that in general those systems with near-IR signatures of inner
disks also have sub-mm signatures of outer disks, and vice versa; they conclude
that inner and outer disk lifetimes agree within 105 yr.
Accretion in YSO systems is studied using a variety of diagnostics (see e.g. Cal-
vet, Hartmann, & Strom (2000)), including continuum “veiling” of photospheric
absorption lines and optical emission lines, which are respectively believed to arise
from hot (shocked) gas on the stellar surface and from gas that is falling onto
the star along magnetic flux tubes. Gullbring et al. (1998) measured a median
accretion rate for million-year-old T Tauri stars of ∼ 10−8 M⊙ yr−1, and White
& Ghez (2001) found similar accretion rates for the primaries in T Tauri binary
systems. A recent compilation of observations (White & Basri, 2003; Muzerolle et
al., 2003; Calvet et al., 2004) shows an approximate dependence of the accretion
rate on stellar mass M˙disk ∝ m2∗, although with considerable scatter (Muzerolle
et al., 2005). This scaling of the accretion rate with stellar mass is potentially
explained by Bondi-Hoyle accretion from the ambient molecular cloud (Padoan
et al., 2005). However, such a model accounts only for the infall rate onto the
star-disk system, not the disk accretion rate; these need not agree. In addition,
it does not account for the accretion seen in T Tauri stars outside molecular
clouds (Hartmann et al., 2006). During their embedded stages (a few ×105 yr),
low-mass stars have typical disk accretion rates similar to or slightly larger than
those of TTSs (White et al., 2007). As discussed in §4.1.3, the infall rates from
protostellar envelopes typically exceed disk accretion rates by a factor 10-100, so
it is possible that mass is stored in the disk and released intermittently, in brief

Theory of Star Formation 73
but prodigious accretion events similar to FU Ori outbursts (Kenyon et al., 1990;
Hartmann & Kenyon, 1996).
For high-mass protostars, observations suggest that there are at least two
classes of disks (Cesaroni et al., 2007). In moderate-luminosity sources corre-
sponding to B stars (L . few ×104 L⊙), the disks appear to be Keplerian,
with masses significantly less than the stellar mass and time scales for mass
transfer ∼ 105 yr. In luminous sources (L & 105 L⊙), the disks are large
(4 − 30 × 103 AU) and massive (60 − 500 M⊙). Consistent with the discus-
sion in §§4.1.1 and 4.1.1, the disks are observed to be non-Keplerian on these
large scales. To distinguish these structures from the disks observed around B
stars, Cesaroni (2005) terms them “toroids.” The inferred infall rates in these
disks are of order 2 × 10−3 − 2× 10−2 M⊙ yr−1, corresponding to mass transfer
time scales of order 104 yr (Zhang, 2005). In view of their large size and mass,
they may be circumcluster structures rather than circumstellar ones. Indeed, one
of the best studied luminous sources, G10.8-0.4, is inferred to have an embedded
cluster of stars with a total mass ∼ 300 M⊙ (Sollins et al., 2005). Simulations of
the formation of an individual massive star in a turbulent medium give a disk size
∼ 103 AU, significantly smaller than the size of the toroids (Krumholz, Klein, &
McKee, 2005). To date, no disks have been observed in the luminous sources on
scales . 103 AU. Most likely, this is because of the observational difficulties in
observing such disks; it should be borne in mind, however, that there is no direct
evidence that these sources are in fact protostellar. Including disks around both B
stars and the toroids around luminous sources, Zhang (2005) finds that the mass
infall rate in the disks scales as M˙disk ∝ m2.2∗ , although there are substantial
uncertainties in the data for the luminous sources.
4.2.2 ACCRETION MECHANISMS The most fundamental theoretical
question about YSO disks is what makes them accrete; while many mechanisms
have been investigated, the problem is still open. In large part this is because the
accretion process depends on a complicated interplay of MHD, radiative transfer,
chemistry, and even solid state physics. The MHD is itself non-ideal, since the
medium is partially ionized, and in addition self-gravity is important in many
circumstances. Self-gravity effects and the level of electrical conductivity are
very sensitive to thermal and ionization properties, which in turn are determined
by chemistry and radiative transfer (including X-rays and cosmic rays), and the
latter are strongly affected by grain properties that evolve in time due to stick-
ing and fragmentation. Compounding the difficulty imposed by the interactions
among the physical processes involved is the lack of exact knowledge of initial and
boundary conditions: how does collapse of the rotating protostellar core shape
the distribution of mass in the disk, starting from the initial disk-building stage
and continuing (although at a reduced rate) with later infall? Finally, there is
the difficulty imposed by the huge dynamic range in space and time; disks them-
selves span a range of ∼ 104 in radius and 106 in orbital period, while the small
aspect ratio H/R ≪ 1 (where H is the scale height of the disk) implies a further
extension in dynamic range is required for numerical models that resolve the disk
interior.
Processes proposed to transport angular momentum in YSO disks generally
fall into one of three categories: purely hydrodynamic mechanisms, MHD mecha-
nisms, and self-gravitating mechanisms (e.g., see the reviews of Stone et al. 2000
and Gammie & Johnson 2005). Within the last decade, it has become possi-

74 McKee & Ostriker
ble to investigate mechanisms in each class using high-resolution time-dependent
numerical simulations in two and three dimensions, in which the stresses that
produce transport are explicitly obtained as spatial correlations of component
velocities, magnetic fields, and the density and pressure for a self-consistent flow.
Prior to the computational revolution that made these investigations possible,
and continuing into the present for modeling in which large radial domains and
long-term evolution is required, many studies have made use of the so-called “al-
pha prescription” for angular momentum transport. In this approach (Shakura
& Sunyaev, 1973; Lynden-Bell & Pringle, 1974; Pringle, 1981), a stress tensor is
defined that yields an effective viscous torque between adjacent rings in a dif-
ferentially rotating disk. On dimensional grounds, and using the fact that the
shear stress should be zero for solid-body rotation, this stress can be written as
TR,φ ≡ −αPd ln Ω/d lnR; i.e. the effective kinematic viscosity is taken to obey
ν ≡ ασ2th/Ω = ασthH. This effective viscosity Ansatz makes it possible to study
disk evolution with a purely hydrodynamic, one-dimensional model. While the
“α-model” approach has been essential to progress on modeling disk observables,
it is limited in its ability to capture realistic dynamics since the coefficient is ar-
bitrary (and usually taken as spatially constant) and the adopted functional form
for TR,φ, while dimensionally correct, may not reproduce the true behavior of non-
linear, time-dependent, three-dimensional flows (e.g., see Ogilvie 2003; Pessah,
Chan, & Psaltis 2006). For a Keplerian disk, −d lnΩ/d lnR = 3/2 and in steady
state the mass accretion rate is M˙disk = 3πΣν = 3πΣασ2th/Ω; i.e. the ratio of ra-
dial inflow speed to orbital speed is (vR/vφ) = (3/2)α(σth/vφ)2 = (3/2)α(H/R)2.
Observed accretion rates of TTSs require α ∼ 10−2 (Hartmann et al., 1998). Since
the effective viscosity is equal to a characteristic length scale for angular momen-
tum transport times a characteristic transport speed, the empirically-determined
viscosity corresponds to a few percent of the value that would obtain if transport
occurred at sonic speeds over distances comparable to the scale height of the disk.
Using the infall rate scaling of equation (43), the ratio of the disk accretion
rate to the infall rate is
M˙disk
m˙in
∼ αφin
(Mdisk
m∗
)(R
H
)(Tdisk
Tcore
)3/2
, (50)
where we have assumed that the gravitational potential is dominated by the star.
The outer-disk temperature is not much larger than the temperature in the core,
and R/H ∼ 10 for the outer disk, so the disk accretion rate is much lower than
the infall rate unless Mdisk/m∗ or α/φin exceeds ∼ 0.1. This is not the case
for TTSs, but during the embedded stages the disk masses may be larger, and
(possibly as a consequence of larger Mdisk and self-gravity; see below) the values
of α may be larger as well.
Hydrodynamic Mechanisms The simplest transport mechanisms would be
purely hydrodynamic. Turbulence generated either through convection (due to
vertical or radial entropy gradients), through shear-driven hydrodynamic insta-
bilities, or through external agents (such as time-dependent, clumpy infall) could
in principle develop velocity field correlations 〈ρδvRδvφ〉 of the correct sign (> 0)
to transport angular momentum outward. Ryu & Goodman (1992) showed, how-
ever, that convective modes tend to transport angular momentum inward, rather
than outward, and Stone & Balbus (1996) confirmed from three dimensional

Theory of Star Formation 75
numerical simulations with turbulence driven by convection that angular mo-
mentum transport is inward. Convection driven by radial entropy gradients also
transports angular momentum inward, and is generally stabilized by differential
rotation (Johnson & Gammie, 2006).
Several analytic studies have shown that purely hydrodynamic disturbances
in Keplerian-shear disks are able to experience large transient growth (Chagel-
ishvili et al., 2003; Klahr, 2004; Umurhan & Regev, 2004; Johnson & Gammie,
2005a; Afshordi, Mukhopadhyay, & Narayan, 2005), especially for the case of two-
dimensional (i.e. z-independent) columnar structures. Conceivably, transient
growth of sheared waves could lead to self-sustained turbulence with outward
transport of angular momentum, if new leading wavelets could be continually
reseeded in the flow via nonlinear interactions (Lithwick, 2007). While tran-
sient growth is indeed seen in two-dimensional (R − φ) numerical simulations,
it is subject to secondary Kelvin-Helmholtz instability that limits the growth
when |kRδvφ|/Ω >∼ 1 (Shen, Stone, & Gardiner, 2006). The turbulence that re-
sults also appears to decay without creating leading wavelets to complete the
feedback loop, but this may be due to limited numerical resolution. Other
numerical evidence, together with analytic arguments, suggest that nonlinear
shear-driven hydrodynamic instabilities are unable to maintain turbulence for
Rayleigh-stable rotational profiles (in which angular momentum increases out-
ward, i.e. κ2/Ω2 = 2d ln(ΩR2)/d lnR > 0) (Balbus, Hawley, & Stone, 1996;
Hawley, Balbus, & Winters, 1999). Since simulations using the same numerical
methods show that analogous Cartesian shear flows do exhibit nonlinear instabil-
ity, rotating systems are presumably stabilized by Coriolis forces and the epicyclic
motion that results. One potential concern is that the effective Reynolds numbers
of numerical experiments are too low to realize nonlinear shear-driven instabilities
and self-sustained turbulence. Very recently, however, Ji et al. (2006) reported
from laboratory experiments at Reynolds numbers up to millions that hydrody-
namic flows with Keplerian-like rotation profiles in fact show extremely low levels
of angular momentum transport, corresponding to α < 10−6.
Although it may be difficult to grow perturbations from instabilities in uni-
form Keplerian disks, it is still possible that disks are born with large internal
perturbations, and that ongoing infall at all radii can continually resupply them.
Simulations have shown that two-dimensional disks with non-uniform vorticity
tend to develop large-scale, persistent vortices that are able to transport angu-
lar momentum outwards (Umurhan & Regev, 2004; Johnson & Gammie, 2005b).
Three-dimensional simulations, however, show that vortex columns tend to be
destroyed (Barranco & Marcus, 2005; Shen, Stone, & Gardiner, 2006). While
off-midplane vortices can be long-lived (Barranco & Marcus, 2005), the angular
momentum transport in three-dimensional simulations is an order of magnitude
lower than for the two-dimensional case (Shen, Stone, & Gardiner, 2006), and
secularly decays. Further investigation of this process is needed, and it is par-
ticularly important to assess whether vorticity can be injected at a high enough
rate to maintain the effective levels of α ∼ 10−2 needed to explain observed TTS
accretion.
MHD Mechanisms The introduction of magnetic fields considerably alters
the dynamics of circumstellar disks. The realization by Balbus & Hawley (1991)
that weakly-or-moderately magnetized, differentially rotating disks are subject

78 McKee & Ostriker
that Q is near but not below the critical value ≈ 1.4, self-gravitational stresses
will be appreciable but not so large as to cause fragmentation. Analytic estimates
assuming steady state and accretion heating as well as irradiation (Matzner &
Levin, 2005; Rafikov, 2005) indicate that fragmentation is only possible in the
outer portions of disks, although more massive disks, around more massive stars,
are more subject to fragmentation (Kratter & Matzner, 2006). At temperatures
comparable to those in observed systems, disks with masses >∼ 0.1 M⊙ are candi-
dates for having significant mass transport due to self-gravitating torques (Mayer
et al., 2004). Thus, self-gravity is likely to be particularly important during the
embedded stage of disk evolution, when disk masses are the largest. Vorobyov
& Basu (2005b, 2006) propose, based on results of two-dimensional simulations,
that recurrent “bursts” of accretion due to self-gravity are likely to develop dur-
ing the early stages of protostellar evolution. A number of other results from
models of self-gravitating disk evolution (with an emphasis on criteria for planet
formation through fragmentation) are presented in the review of Durisen et al.
(2007).
4.2.3 DISK CLEARING While a large proportion of the mass in the disk
ultimately accretes onto the star, conservation of angular momentum requires
that some of the matter be left behind. MHD winds during the main lifetime
of the disk remove some of this material (see §4.2.5). What remains is either
incorporated into planets, or removed by photoevaporation. Although planet
formation is inextricably coupled to disk evolution, recent developments in this
exciting – and rapidly expanding – field are too extensive to summarize here. A
number of excellent recent reviews appear in Protostars and Planets V.
Disks can be irradiated by UV and X ray photons originating either in their
own central stars, or in other nearby, luminous stars (see e.g. reviews of Hol-
lenbach, Yorke, & Johnstone 2000 and Dullemond et al. 2007). EUV radiation
penetrates only the surface layer of the disk, where it heats the gas to ∼ 104 K
(the ionization and heating depth is determined by the Stro¨mgren condition);
FUV penetrates deeper into the disk (where densities are higher), but heats gas
to only a few 100 K (Hollenbach et al., 1994; Johnstone, Hollenbach, & Bally,
1998). The characteristic radial scale in the disk for a thermally-driven wind is the
gravitational radius rg = Gm∗µ/(kT ), where T is the temperature at the base of
the flow. Pressure gradients enable flows to emerge down to (0.1− 0.2)rg (Begel-
man, McKee, & Shields, 1983; Font et al., 2004; Adams et al., 2004). EUV-driven
winds are most important in the inner disk, since the gravitational potential there
is too deep for FUV-heated regions at modest temperatures to escape.
Observations discussed above (see also Simon & Prato 1995 and Wolk & Walter
1996) indicate that the inner and outer disks surrounding YSOs disperse nearly
simultaneously and on a very short (∼ 105 yr) timescale, based on the small
number of transition objects between classical and weak T Tauri systems and the
typical CTT lifetimes of a few to several Myr. Since the accretion time of the outer
disk itself determines the system lifetime, rapid removal of the outer disk must be
accomplished by other means; photoevaporation is the most natural candidate.
Models of photoevaporation that also include viscous disk evolution (which allow
spreading both inward and outward) have very recently shown that rapid and
near-simultaneous removal of the whole disk indeed occurs (Clarke, Gendrin, &
Sotomayor, 2001; Alexander, Clarke, & Pringle, 2006a,b). In this process, the
accretion rate declines slowly over time until the photoevaporative mass loss rate

Theory of Star Formation 79
at some location in the inner disk exceeds the rate at which mass is supplied
from larger radii. The inner disk, which is no longer resupplied from outside,
then drains rapidly into the star. At the same time, the radiative flux onto the
outer disk grows as it is no longer attenuated by the inner disk’s atmosphere; the
photoevaporation rate in the outer disk climbs dramatically, and it is removed as
well.
4.2.4 OBSERVATIONS OF YSO JETS AND OUTFLOWS Young
stellar systems drive very powerful winds. The clearest observable manifestations
of YSO winds are the central “Herbig-Haro” jets consisting of knots of ionized gas
(v >∼ 100 km s−1), and the larger-scale bipolar outflows consisting of expanding
lobes of molecular gas (v ∼ 10 km s−1). “Jet-like” outflows (i.e. high-v, narrow
molecular structures) are also observed in some circumstances (see below). The
high velocities of jets indicate that they represent (a part of) the primary wind
from the inner part of the star-disk system, while the low velocities and large
masses of (broad) molecular outflows indicate that they are made of gas from the
star’s environment that has been accelerated by an interaction with the wind.
In addition to these observed signatures, there may be significant gas in a large-
scale primary wind surrounding the jet, which remains undectected due to lower
excitation conditions (low density, temperature, and/or ionization fraction).
Outflows are ubiquitous in high-mass star formation as well as in low-mass star
formation (Shepherd & Churchwell, 1996). Outflows from high-mass protostellar
objects with L < 105 L⊙ (corresponding to m∗ < 25 M⊙—Arnett, 1996) are
collimated (Beuther et al., 2002b), but somewhat less so than those in low-mass
protostars (Wu et al., 2004). In some cases, jets are observed with the outflows,
and in these cases the momentum of the jet is generally large enough to drive the
observed outflow (Shepherd, 2005). No well-collimated flow has been observed in
a source with L > 105 L⊙; as remarked above, disks that are clearly circumstellar
have not been observed in such sources either. Beuther & Shepherd (2005) have
proposed an evolutionary sequence that is consistent with much of these data: A
protostar that eventually will become an O star first passes through the HMPO
stage with no H II region and with a well-collimated jet. When the star becomes
sufficiently massive and close to the main sequence that it produces an H II region,
the outflow becomes less collimated. The collimation systematically decreases
as the star grows in mass and the H II region evolves from hypercompact to
ultracompact (see §4.3.4). The remainder of this section focuses on winds and
outflows from low-mass stars, which have been observed in much greater detail
than their high-mass counterparts.
Recent reviews focusing on the observational properties of jets include those
of Eisloffel et al. (2000), Reipurth & Bally (2001), and Ray et al. (2007). Jets
are most commonly observed at high resolution in optical forbidden lines of O,
S, and N, as well as Hα, but recent observations have also included work in the
near-IR and near-UV. For CTTs, which are YSOs that are themselves optically
revealed, observed optical jets are strongly collimated (aspect ratio at least 10:1,
and sometimes 100:1), and in several cases extend up to distances more than a
parsec from the central source (Bally, Reipurth, & Davis, 2007). The jets con-
tain both individual bright knots with bow-shock morphology, and more diffuse
emission between these knots.
The emission diagnostics from bright knots are generally consistent with heat-
ing by shocks of a few tens of km s−1 (Hartigan, Raymond, & Hartmann, 1987;

82 McKee & Ostriker
Since j is dominated by the kinetic term at large distance (where the wind is
super-fast-magnetosonic), observations can be used to infer the ratio RA/Robs ≈√
2vφ,obs/vp,obs. For the low velocity component of DG Tau, Anderson et al.
(2003) find from calculating Ω0 as above that the wind launch point radii are
∼ 0.3−4AU, implying a disk wind. The high velocity component could originate
as either an x-wind or a disk wind from smaller radii. For DG Tau, the inferred
ratio RA/R0 ≈ 2 − 3 is also consistent with numerical solutions that have been
obtained for disk winds (see Pudritz et al. 2007 for a summary). This implies
that the angular momentum carried by the wind, M˙windΩ0R2A, which equals the
angular momentum lost by the disk, M˙diskΩ0R20, can drive accretion at a rate
M˙disk/M˙wind = (RA/R0)2 ∼ 4− 9.
The acceleration of MHD winds is provided by a combination of the centrifugal
“flinging” effect produced by rigid poloidal fields, and gradients in the toroidal
magnetic pressure in the poloidal direction (e.g. Spruit (1996)). Beyond the
Alfve´n surface, magnetic hoop stresses will tend to bend streamlines toward the
poles. Full cylindrical streamline collimation, in the sense of vp ‖ zˆ asymptoti-
cally, can only occur if BφR is finite for R → ∞ (Heyvaerts & Norman, 1989).
Using solutions in which all velocities scale as v, vA ∝ r−1/2 and the density and
magnetic field respectively scale as ρ ∝ r−q and B ∝ r−(1+q)/2, Ostriker (1997)
showed, however, that cylindrically-collimated disk winds are slow, in the sense
that the asymptotic value of vp/Ω0R0 is at most a few tenths. Since observed jets
are fast, they must either have their streamlines collimated by a slower external
wind, or else be collimated primarily in density rather than velocity. Time-
dependent simulations have also shown that the degree of collimation in the flow
depends on the distribution of magnetic flux in the disk; cases with steeper dis-
tributions of B with R tend to be less collimated in terms of streamline shapes
(Fendt, 2006; Pudritz, Rogers, & Ouyed, 2006).
The idea that nearly radially-flowing wide-angle MHD winds may produce a
“jetlike” core, with density stratified on cylinders, was first introduced by Shu et
al. (1995) in the context of x-winds. This effect holds more generally, however, as
can be seen both analytically (Matzner & McKee, 1999) and in simulations (see
below). Asymptotically, the density approaches ρ → |Bφ|Rk/(Ω0R2) where k is
the (conserved) mass flux-to-magnetic flux ratio (also termed the mass-loading
parameter). Since nearly radially-flowing winds must be nearly force-free, |Bφ|R
varies weakly with R, such that if the range of k/Ω0 over footpoints is smaller
than the range of R over which the solution applies (which is generally very
large), the wind density will vary as R−2. The R−2 dependence cannot continue
to the origin; Matzner & McKee (1999) suggested that precession, internal shocks
due to fluctuating wind velocity, or magnetic instabilities would result in a flat-
tening of the density close to the axis so that the momentum flux in the wind
ρvw ∝ (1 + θ20 − cos2 θ)−1, where θ is the angle of the flow relative to the axis
and θ0 ≪ 1 measures the size of the flattened region. This distribution gives
approximately equal amounts of momentum in each logarithmic interval of angle
for θ > θ0. Several time-dependent numerical MHD simulations have demon-
strated this density collimation effect for wide-angle winds (Gardiner, Frank, &
Hartmann, 2003; Krasnopolsky, Li, & Blandford, 2003; Anderson et al., 2005).
Magnetized winds are subject to a variety of instabilities (e.g. Kim & Ostriker
2000; Hardee 2004), which may contribute to enhancing the confinement of the
jet, structuring the jet column (yielding wanders, twists, and clumps), and mix-

Theory of Star Formation 85
greater densities and shorter collapse times than those supported by thermal
pressure alone. Caselli & Myers (1995) extended this to more massive stars and
found formation times > 106 yr for stars of 100M⊙, a significant fraction of the
main sequence lifetime. On the other hand, by modeling the spectral energy
distributions (SEDs) of high-mass protostars, Osorio, Lizano, & D’Alessio (1999)
inferred that high-mass stars form in somewhat less than 105 yr, and favored a
logatropic model for the density distribution of the core. Nakano et al. (2000)
inferred an accretion rate of 10−2 M⊙ yr−1 (corresponding to a formation time of
a few thousand years) for the source IRc2 in Orion based on the assumption that
the accretion rate is ∼ 10c3eff/G, with the effective sound speed ceff determined
from the observed line width.
The turbulent core model for high-mass star formation (McKee & Tan, 2002,
2003) follows from the assumption that such stars form in turbulent, gravitation-
ally bound cores (virial parameter αvir ∼ 1). The turbulence is self-similar on
all scales above the Bonnor-Ebert scale, where thermal pressure dominates. The
star-forming clump and the protostellar cores within it are assumed to be cen-
trally concentrated so that the pressure and density have a power-law dependence
on radius, P ∝ r−kP , ρ ∝ r−kρ . It follows that the cores are polytropes (§2.2),
and since the Bonnor-Ebert scale is small, the cores are approximately singular.
The protostellar infall rate is determined by the surface density of protostellar
core, which in turn is comparable to that of the clump in which it is embedded.
The regions of high-mass star formation studied by Plume et al. (1997) have sur-
face densities Σcl ∼ 1 g cm−2, corresponding to visual extinctions AV ∼ 200 mag;
these values are similar to those for observed star clusters in the Galaxy (e.g.,
∼ 0.2 g cm−2 in the Orion Nebula Cluster, 0.8 g cm−2 for the median globular
cluster and ∼ 4 g cm−2 in the Arches Cluster). By contrast, regions of low-mass
star formation have Σ ∼ 0.03 g cm−2, corresponding to AV ∼ 7 mag (Onishi et
al., 1996). The radius of a protostellar core is
Rcore =
( Mcore
πΣcore
)1/2
≃ 0.06
( m∗f
60ǫcore M⊙
)1/2 1
Σ1/2cl
pc, (51)
where m∗f is the final stellar mass. The second expression is based on the result
that the surface density of a typical core is comparable to that of the clump in
which it is embedded; cores near the center of a clump have higher surface densi-
ties, and the sizes are correspondingly smaller. Using the results of McLaughlin
& Pudritz (1997) for the inside-out collapse of a singular polytrope and adopting
kρ = 32 , a typical density power law from Plume et al. (1997), McKee & Tan
(2003) found that the typical infall rate and the corresponding time to form a
star of mass m∗f are
m˙∗ ≃ 0.5 × 10−3
( m∗f
60ǫcore M⊙
)3/4
Σ3/4cl
( m∗
m∗f
)0.5
M⊙ yr−1, (52)
t∗f ≃ 1.3 × 105
( m∗f
60ǫcore M⊙
)1/4
Σ−3/4cl yr, (53)
where Σcl is the surface density (in g cm−2) of the several thousand M⊙ clump
in which the star is forming. For typical values of Σcl ∼ 1 g cm−2, the star
formation time is of order 105 yr and the infall rate is of order 10−3 M⊙ yr−1.
This infall rate is large enough to overcome the effects of radiation pressure at the

90 McKee & Ostriker
bilities, which facilitate the escape of the radiation in low column regions
and the accretion of the gas in high column regions (Krumholz, Klein, &
McKee, 2005). There is no evidence that radiation pressure halts the ac-
cretion up to m∗ = 35 M⊙, a substantially higher mass than was found in
axisymmetric simulations with gray transfer (Yorke & Sonnhalter, 2002).
Turner, Quataert, & Yorke (2007) have shown that the dusty envelopes of
HMPOs are subject to the photon bubble instability, which further pro-
motes infall.
4.3.4 Photoionization feedback: H II regions The H II regions asso-
ciated with HMPOs provide strong feedback on infall and accretion, and may play
a role in defining the maximum stellar mass. They are classified into two types:
Ultra-compact H II (UCHII) regions have diameters (0.01 − 0.1) pc, densities
≥ 104 cm−3, and emission measures
∫
n2edl ≥ 107 pc cm−6 (Wood & Church-
well, 1989). Hypercompact H II (HCHII) regions have diameters < 0.01 pc with
emission measures ≥ 108 pc cm−6 (Beuther et al., 2007; for a slightly different
definition and a review of both types of H II region, see Hoare et al., 2007).
HCHII regions often appear in tight groups in high-mass star-forming regions,
and they often have broad radio recombination lines with widths that can exceed
100 km s−1.
The high accretion rates characteristic of HMPOs delay the point at which
the stars reach the main sequence (McKee & Tan, 2003; Krumholz & Thomp-
son, 2007), thereby delaying the time at which the photosphere is hot enough
to produce an H II region. High accretion rates also quench the emission of
ionizing photons once the star has reached the main sequence (Walmsley, 1995).
Close to the star—i.e., inside the gravitational radius rg = Gm∗/c2i = 3.2 ×
1015(m∗/30M⊙) cm, where ci ≃ 10 km s−1 is the isothermal sound speed of
the ionized gas—spherically accreting gas is in free fall, with ρ ∝ r−3/2. For an
ionizing photon luminosity S, the radius of the HCHII region is
RHCHII = R∗ exp(S/Scr), (55)
where
Scr =
α(2)m˙2∗
8πµ2HGm∗
= 5.6× 1050
( m˙∗
10−3 M⊙ yr−1
)2 (100M⊙
m∗
)
s−1, (56)
(Omukai & Inutsuka, 2002), where α(2) is the recombination rate to excited states
of hydrogen and where we have replaced the proton mass in their expression with
µH = 2.34 × 10−24 g, the mass per hydrogen nucleus. Provided the accretion
is spherical, the H II region is quenched for S . Scr. If S/Scr is not too large
(. 7), RHCHII is less than rg/2 and the infall velocity at the Stromgren radius
exceeds 2ci, the minimum velocity of an R-critical ionization front; as a result
there is no shock in the accretion flow and the H II region cannot undergo the
classical pressure-driven expansion (Keto, 2002). If the accretion is via a disk,
as is generally expected, then the ionizing photons can escape out of the plane
of the disk, and the H II region will not be trapped (Keto & Wood, 2006; Keto,
2007). Disk accretion is often associated with the production of winds, and
Tan & McKee (2003) have suggested that such winds confine HCHII regions:
the winds clear the gas along the axis, and the ionizing radiation then illumi-
nates the inner surfaces of the winds. If correct, this offers the possibility of

Theory of Star Formation 91
a powerful diagnostic for determining the nature of disk winds associated with
massive stars. van der Tak & Menten (2005) found very compact radio emission
aligned with the outflows in two high-mass protostellar sources, consistent with
this picture. When the ionizing luminosity becomes large enough, however, the
wind will become ionized and the H II region will evolve to a UCHII state. The
ionizing photons will photoevaporate the surface of the disk at a rate of order
m˙evap ∼ few ×10−5(S/1049 s−1)1/2 M⊙ yr−1; absorption of ionizing photons by
dust can significantly affect this (Hollenbach et al., 1994; Richling & Yorke, 1997).
This mass-loss rate is too small to be important in setting the maximum mass of
the star (although it can be important in primordial star formation—McKee &
Tan, in preparation). Absorption of ionizing photons by dust must also be taken
into account when inferring the ionizing luminosity of the central star from the
properties of the H II region (Dopita et al. 2006 and references therein).
4.3.5 Star Formation in Clusters Most stars are born in clusters (e.g.,
Lada & Lada, 2003; Allen et al., 2007), and this is particularly true of high-mass
stars. The mass distribution of clusters appears to obey a universal power law,
dNcluster/d lnM ∝ M−α, with α ≃ 1. With this distribution, MdNcluster/d lnM =
const: taken together, clusters in each decade of mass have the same total num-
ber of stars. Lada & Lada (2003) find that very young clusters within 2 kpc of
the Sun that are still embedded in their natal molecular clouds obey this power
law for M & 50 M⊙; the upper limit of the observed distribution is set by the
largest cluster expected in the area they surveyed. The mass distribution of OB
associations in the Galaxy also has a power-law distribution with α ≃ 1 (McKee
& Williams, 1997); they inferred that the distribution extended from ∼ 50 M⊙ to
2× 105 M⊙ and could account for all the stars formed in the Galaxy. Kennicutt,
Edgar, & Hodge (1989) found that the luminosity distribution of H II regions
in disk galaxies obeys dN/d lnL ∝ L−1±0.5, which is consistent with an M−1
distribution since the luminosity is proportional to mass for associations that are
large enough to fully sample the IMF. The distribution of OB associations in
the SMC has α = 1 from the largest associations down to associations with a
single OB star (Oey, King, & Parker, 2004). The star clusters in the “Antennae”
galaxies show α = 1 over the mass range 104 M⊙ < M < 106 M⊙ (Zhang &
Fall, 1999); this is one of the best determined cluster mass functions, and has an
error, including systematic errors, estimated as ±0.1. The mass distributions of
open clusters and globular clusters are also consistent with an M−1 distribution
at birth (Elmegreen & Efremov, 1997). Dowell, Buckalew, & Tan (2006) found
α ≃ 0.9 for clusters in irregular galaxies and α ≃ 0.75 in disk galaxies, but com-
ment that this result could be affected by the low spatial resolution of the data.
The M−1 mass distribution of clusters is intermediate between the high-mass
part of the IMF (M−1.35) on the one hand, and the observed mass distribution
of GMCs (M−0.6) and the clumps within them (M−0.3 to M−0.7; §3.1) on the
other. It is important to understand the origin of the differences among these
power laws, which appear to be real.
The structure of star clusters contains clues to their formation. High-mass
stars in Galactic clusters that are massive enough to contain a number of such
stars are observed or inferred to be segregated toward the center of the cluster.
The large fraction of O stars that are runaways can be naturally explained if they
originate in dense, mass-segregated clusters and undergo dynamical interactions
(Clarke & Pringle, 1992). Hillenbrand & Hartmann (1998) analyzed the spatial

Theory of Star Formation 95
arises nearest the central star, becomes collimated into a jet-like flow due
to magnetic hoop stresses.
• The impact of a wide-angle, stratified disk wind on the protostellar core
sweeps up much of the ambient gas into a massive molecular outflow. This
reduces the net efficiency of star formation to ∼ 1/3. The combined action
of many outflows also helps to energize dense, star-cluster-forming clumps.
• Massive stars form from cores that are considerably more massive than
a Bonnor-Ebert mass, and are most likely highly turbulent. Radiation
pressure strongly affects the dynamics of massive star formation, but can
be overcome by the combined action of disk formation, protostellar outflows
and radiation-hydrodynamic instabilities in the accreting gas. It is not clear
whether protostellar feedback determines the maximum mass of the stars
that form.
• Massive, luminous stars ionize their surroundings into HII regions. The
expansion of these regions into ambient gas at∼ 10 km s−1 energizes GMCs,
contributing to the large-scale turbulent power. However, this process is
difficult to regulate, and can unbind GMCs within a few dynamical crossing
times. By the time they are finally destroyed, GMCs may have lost much
of their original mass by photoevaporation.
• The destruction of GMCs returns almost all of the gas they contain to
the diffuse phase of the ISM, with a mean star formation efficiency over
the cloud lifetime of ∼ 5%. This low efficiency can be understood as a
consequence of the small fraction of mass that is compressed into clumps
dense enough that turbulence does not destroy them before they collapse.
• The return of GMC gas to the diffuse ISM completes the cycle of star
formation, which then begins anew.
The coming decade will test and revise this narrative of star formation, par-
ticularly with the advent of ALMA and JWST and the continued advances in
numerical simulation. Turning this narrative into a quantitative, predictive the-
ory will provide a foundation for addressing many of the outstanding questions
in astrophysics today, ranging from the formation of planets to the evolution of
galaxies and the origin of the elements.
Acknowledgements We are grateful to our expert readers, J. Bally, G. Basri,
S. Basu, E. Bergin, L. Blitz, R. Crutcher, B. Elmegreen, C. Gammie, L. Hart-
mann, M. Heyer, R. Kennicutt, S. Kenyon, M. Krumholz, C. Lada, Z.-Y. Li, C.
Matzner, S. Offner, P. Padoan, J. Tan, E. Vazquez-Semadeni, and E. Zweibel, for
their insightful comments on draft sections of the manuscript, and to our editor,
E. van Dishoeck, for her comments on the entire manuscript. We are also grateful
to C.-F. Lee for his help producing the figure of HH111 and to Nathan Smith
for his help with the figure of the Carina Nebula. The work of CFM and ECO
was supported by the National Science Foundation under grants AST 0606831
and AST 0507315, respectively. In preparing this review, we have relied upon
the search and archive facilities provided by NASA’s Astrophysics Data System
Bibliographic Services.

Theory of Star Formation 97
Balbus, S. A. 1988, ApJ, 324, 60
Balbus, S. A. 2003, ARAA, 41, 555
Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 214
Balbus, S. A., & Hawley, J. F. 1998, Reviews of Modern Physics, 70, 1
Balbus, S. A., Hawley, J. F., & Stone, J. M. 1996, ApJ, 467, 76
Ballesteros-Paredes, J. 2006, MNRAS, 372, 443
Ballesteros-Paredes, J., Gazol, A., Kim, J., Klessen, R. S., Jappsen, A.-K., &
Tejero, E. 2006, ApJ, 637, 384
Ballesteros-Paredes, J., Hartmann, L., & Va´zquez-Semadeni, E. 1999, ApJ, 527,
285
Ballesteros-Paredes, J., Klessen, R. S., Mac Low, M.-M., & Vazquez-Semadeni,
E. 2007, Protostars and Planets V, 63
Ballesteros-Paredes, J., & Mac Low, M.-M. 2002, ApJ, 570, 734
Ballesteros-Paredes, J., Va´zquez-Semadeni, E., & Scalo, J. 1999, ApJ, 515, 286
Bally, J., Reipurth, B., & Davis, C. J. 2007, Protostars and Planets V,
B. Reipurth, D. Jewitt, and K. Keil (eds.), University of Arizona Press, Tucson,
951 pp., 2007., p.215-230, 215
Bally, J., Stark, A. A., Wilson, R. W., & Langer, W. D. 1987, ApJL, 312, L45
Bally, J., & Zinnecker, H. 2005, AJ, 129, 2281
Banerjee, R., & Pudritz, R. E. 2006, ApJ, 641, 949
Barranco, J. A., & Goodman, A. A. 1998, ApJ, 504, 207
Barranco, J. A., & Marcus, P. S. 2005, ApJ, 623, 1157
Basu, S. 1997, ApJ, 485, 240
Basu, S. 2000, ApJL, 540, L103
Basu, S., & Mouschovias, T. C. 1994, ApJ, 432, 720
Basu, S., & Mouschovias, T. C. 1995a, ApJ, 452, 386
Basu, S., & Mouschovias, T. C. 1995b, ApJ, 453, 271
Bate, M. R., & Bonnell, I. A. 2005, MNRAS, 356, 1201
Bate, M. R., Bonnell, I. A., & Bromm, V. 2003, MNRAS, 339, 577
Beckwith, S. V. W., Henning, T., & Nakagawa, Y. 2000, Protostars and Planets
IV, 533
Beckwith, S. V. W., & Sargent, A. I. 1991, ApJ, 381, 250
Beckwith, S. V. W., Sargent, A. I., Chini, R. S., & Guesten, R. 1990, AJ, 99, 924
Begelman, M. C., McKee, C. F., & Shields, G. A. 1983, ApJ, 271, 70
Behrend, R., & Maeder, A. 2001, A&A, 373, 190
Bell, K. R., & Lin, D. N. C. 1994, ApJ, 427, 987
Bell, K. R., Lin, D. N. C., Hartmann, L. W., & Kenyon, S. J. 1995, ApJ, 444,
376
Bensch, F., Stutzki, J., & Ossenkopf, V. 2001, A&A, 366, 636
Benson, P. J., & Myers, P. C. 1989, ApJS, 71, 89

98 McKee & Ostriker
Bergin, E. A., Hartmann, L. W., Raymond, J. C., & Ballesteros-Paredes, J. 2004,
ApJ, 612, 921
Bertoldi, F. 1989, ApJ, 346, 735
Bertoldi, F., & McKee, C. F. 1990, ApJ, 354, 529
Bertoldi, F., & McKee, C. F. 1992, ApJ, 395, 140
Beuther, H., Churchwell, E. B., McKee, C. F., & Tan, J. C. 2007, Protostars and
Planets V, 165
Beuther, H., & Schilke, P. 2004, Science, 303, 1167
Beuther, H., Schilke, P., Gueth, F., McCaughrean, M., Andersen, M., Sridharan,
T. K., & Menten, K. M. 2002a, A&A, 387, 931
Beuther, H., Schilke, P., Sridharan, T. K., Menten, K. M., Walmsley, C. M., &
Wyrowski, F. 2002b, A&A, 383, 892
Beuther, H., & Shepherd, D. 2005, Cores to Clusters: Star Formation with Next
Generation Telescopes, 105
Beuther, H., Sridharan, T. K., & Saito, M. 2005, ApJL, 634, L185
Binney, J., & Tremaine, S. 1987, Galactic Dynamics, (Princeton, NJ, Princeton
University Press)
Biskamp, D. 2003, Magnetohydrodynamic Turbulence, by Dieter Biskamp,
pp. 310, Cambridge: Cambridge University Press
Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883
Blitz, L. 1993, Protostars and Planets III, 125
Blitz, L., Fukui, Y., Kawamura, A., Leroy, A., Mizuno, N., & Rosolowsky, E.
2007, Protostars and Planets V, 81
Blitz, L., & Rosolowsky, E. 2004, ApJL, 612, L29
Blitz, L., & Rosolowsky, E. 2006, ApJ, 650, 933
Blitz, L., & Shu, F. H. 1980, ApJ, 238, 148
Blitz, L., & Williams, J. P. 1997, ApJL, 488, L145
Bodenheimer, P., Burkert, A., Klein, R. I., & Boss, A. P. 2000, Protostars and
Planets IV, 675
Bodenheimer, P., & Sweigart, A. 1968, ApJ, 152, 515
Bohlin, R. C., Savage, B. D., & Drake, J. F. 1978, ApJ, 224, 132
Boissier, S., Prantzos, N., Boselli, A., & Gavazzi, G. 2003, MNRAS, 346, 1215
Boissier, S., et al. 2006, ApJS in press (astro-ph/0609071)
Boldyrev, S. 2002, ApJ, 569, 841
Boldyrev, S., Nordlund, A˚., & Padoan, P. 2002, ApJ, 573, 678
Boley, A. C., Mej´ıa, A. C., Durisen, R. H., Cai, K., Pickett, M. K., & D’Alessio,
P. 2006, ApJ, 651, 517
Bonazzola, S., Heyvaerts, J., Falgarone, E., Perault, M., & Puget, J. L. 1987,
A&A, 172, 293
Bonazzola, S., Perault, M., Puget, J. L., Heyvaerts, J., Falgarone, E., & Panis,
J. F. 1992, Journal of Fluid Mechanics, 245, 1

Theory of Star Formation 99
Bondi, H. 1952, MNRAS, 112, 195
Bonnell, I. A., & Bate, M. R. 2005, MNRAS, 362, 915
Bonnell, I. A., & Bate, M. R. 2006, MNRAS, 370, 488
Bonnell, I. A., Bate, M. R., Clarke, C. J., & Pringle, J. E. 1997, MNRAS, 285,
201
Bonnell, I. A., Bate, M. R., Clarke, C. J., & Pringle, J. E. 2001a, MNRAS, 323,
785
Bonnell, I. A., Bate, M. R., & Vine, S. G. 2003, MNRAS, 343, 413
Bonnell, I. A., Bate, M. R., & Zinnecker, H. 1998, MNRAS, 298, 93
Bonnell, I. A., Clarke, C. J., Bate, M. R., & Pringle, J. E. 2001b, MNRAS, 324,
573
Bonnell, I. A., Clarke, C. J., & Bate, M. R. 2006, MNRAS, 368, 1296
Bonnell, I. A., & Davies, M. B. 1998, MNRAS, 295, 691
Bonnell, I. A., Larson, R. B., & Zinnecker, H. 2007, Protostars and Planets V,
149
Bonnor, W. B. 1956, MNRAS, 116, 351
Bontemps, S., Andre, P., Terebey, S., & Cabrit, S. 1996, A&A, 311, 858
Boss, A. P., Fisher, R. T., Klein, R. I., & McKee, C. F. 2000, ApJ, 528, 325
Bourke, T. L., Myers, P. C., Robinson, G., & Hyland, A. R. 2001, ApJ, 554, 916
Brandenburg, A., Nordlund, A., Stein, R. F., & Torkelsson, U. 1995, ApJ, 446,
741
Bromm, V., & Larson, R. B. 2004, ARAA, 42, 79
Brunt, C. M. 2003, ApJ, 583, 280
Brunt, C. M., & Heyer, M. H. 2002, ApJ, 566, 276
Burgasser, A. J., Reid, I. N., Siegler, N., Close, L., Allen, P., Lowrance, P., &
Gizis, J. 2007, Protostars and Planets V, 427
Burkert, A., & Bodenheimer, P. 2000, ApJ, 543, 822
Calvet, N., Hartmann, L., & Strom, S. E. 2000, Protostars and Planets IV, 377
Calvet, N., Muzerolle, J., Bricen˜o, C., Herna´ndez, J., Hartmann, L., Saucedo,
J. L., & Gordon, K. D. 2004, AJ, 128, 1294
Calvet, N., Patino, A., Magris, G. C., & D’Alessio, P. 1991, ApJ, 380, 617
Carr, J. S. 1987, ApJ, 323, 170
Caselli, P., Benson, P. J., Myers, P. C., & Tafalla, M. 2002, ApJ, 572, 238
Caselli, P., & Myers, P. C. 1995, ApJ, 446, 665
Cassen, P., & Moosman, A. 1981, Icarus, 48, 353
Ceccarelli, C., Haas, M. R., Hollenbach, D. J., & Rudolph, A. L. 1997, ApJ, 476,
771
Cesaroni, R. 2005, Ap&SS, 295, 5
Cesaroni, R., Galli, D., Lodato, G., Walmsley, C. M., & Zhang, Q. 2007, Proto-
stars and Planets V, 197
Chabrier, G. 2005, ASSL Vol. 327: The Initial Mass Function 50 Years Later, 41

100 McKee & Ostriker
Chabrier, G., Baraffe, I., Selsis, F., Barman, T. S., Hennebelle, P., & Alibert, Y.
2007, Protostars and Planets V, 623
Chagelishvili, G. D., Zahn, J.-P., Tevzadze, A. G., & Lominadze, J. G. 2003,
A&A, 402, 401
Chakrabarti, S., & McKee, C. F. 2005, ApJ, 631, 792
Chandran, B. D. G. 2004, Ap&SS, 292, 17
Chandrasekhar, S. 1939, An Introduction to the Study of Stellar Structure,
(Chicago, Ill., The University of Chicago press) [1939],
Chandrasekhar, S. 1949, ApJ, 110, 329
Chandrasekhar, S. 1951a, Proc. Roy. Soc. A, 210, 18
Chandrasekhar, S. 1951b, Proc. Roy. Soc. A, 210, 26
Chandrasekhar, S., & Fermi, E. 1953a, ApJ, 118, 113
Chandrasekhar, S., & Fermi, E. 1953b, ApJ, 118, 116
Chappell, D., & Scalo, J. 2001, ApJ, 551, 712
Chernin, A. D., Efremov, Y. N., & Voinovich, P. A. 1995, MNRAS, 275, 313
Chiang, E. I., & Goldreich, P. 1997, ApJ, 490, 368
Chieze, J. P. 1987, A&A, 171, 225
Cho, J., & Lazarian, A. 2003, MNRAS, 345, 325
Cho, J., Lazarian, A., & Vishniac, E. T. 2002a, ApJ, 564, 291
Cho, J., Lazarian, A., & Vishniac, E. T. 2002b, ApJL, 566, L49
Cho, J., & Vishniac, E. T. 2000, ApJ, 539, 273
Ciolek, G. E., & Basu, S. 2000, ApJ, 529, 925
Ciolek, G. E., & Koenigl, A. 1998, ApJ, 504, 257
Ciolek, G. E., & Mouschovias, T. C. 1994, ApJ, 425, 142
Ciolek, G. E., & Mouschovias, T. C. 1995, ApJ, 454, 194
Ciolek, G. E., & Mouschovias, T. C. 1996, ApJ, 468, 749
Ciolek, G. E., & Mouschovias, T. C. 1998, ApJ, 504, 280
Clark, P. C., Bonnell, I. A., Zinnecker, H., & Bate, M. R. 2005, MNRAS, 359,
809
Clarke, C. J., Gendrin, A., & Sotomayor, M. 2001, MNRAS, 328, 485
Clarke, C. J., & Pringle, J. E. 1992, MNRAS, 255, 423
Contopoulos, I., Ciolek, G. E., & Ko¨nigl, A. 1998, ApJ, 504, 247
Crutcher, R. M. 1999, ApJ, 520, 706
Crutcher, R. M. 2005, AIP Conf. Proc. 784: Magnetic Fields in the Universe:
From Laboratory and Stars to Primordial Structures., 784, 129
Cunningham, A. J., Frank, A., Quillen, A. C., & Blackman, E. G. 2006, ApJ,
653, 416
Curry, C. L., & McKee, C. F. 2000, ApJ, 528, 734
Dale, J. E., Bonnell, I. A., Clarke, C. J., & Bate, M. R. 2005, MNRAS, 358, 291