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STAR FORMATION:
PROBLEMS AND PROSPECTS
Chris McKee
with thanks to Richard Klein, Mark
Krumholz, Eve Ostriker, and Jonathan Tan
THE BIG QUESTIONS IN STAR FORMATION:
Macrophysics: Properties determined by the natal gas cloud
What determines the rate at which stars form?
What determines the mass distribution of stars?
Microphysics: gravitational collapse and its aftermath
How do individual stars form in the face of angular
momentum, magnetic fields and radiation pressure?
How do clusters of stars form in the face of intense
feedback?
How does star formation lead to planet formation?
Length and Time Scales in Galactic Star Formation
Macrophysics: L ~ 0.01 pc -- 100 pc
(Cloud formation requires
larger scales)
t ~ 103 yr -- 107.5 yr
Microphysics: L ~ 1011 cm -- 1017 cm
t ~ 103.5 s -- 106 yr
(Planet formation requires
smaller scales)
(Not currently feasible)
ZENO’S PARADOX (ALMOST) IN COMPUTATIONS
OF STAR FORMATION
Time step Dt  1/(Gr)1/2
Truelove et al. (1998) calculations of star formation now:
Density increase of 109  Dt decrease of 104.5
ABN (2002) calculations of primordial star formation:
Density increase of 1017  Dt decrease of 108.5
In both cases, calculation stopped before formation of
protostar.
Currently impossible to numerically follow the hydrodynamics of
core collapse past the point of protostar formation
 need both analytic and numerical approaches
CHARACTERISTIC GRAVITATIONAL MASS
Kinetic energy/mass ~ gravitational energy/mass
M ~ r r3
(MJ = Jeans mass)
2 ~ GMJ/r
 MJ ~ 3/(G3 r)1/2 = 4/(G3 P)1/2
Maximum mass of isothermal sphere ( = cth) :
MBE = 1.18 cth3 /(G3 rs)1/2
(Bonnor-Ebert mass)
where rs is measured at the surface of the cloud
2D Jeans mass: In a self-gravitating cloud, P ~ G2,
where  is the mass/area of the cloud
 MJ, 3D~ 4/(G2 ) = MJ, 2D
I. MACROPHYSICS
FORMATION OF GIANT MOLECULAR CLOUDS (GMCs)
GMCs form by gravitational instability, not coagulation
“Top-down,” not “bottom-up” - (Elmegreen)
Characteristic mass is the 2D Jeans mass:
MGMC = 4 / (G2 )
= 7  105 ( / 6 km s-1)4 (100 Msun pc-2 / ) Msun
GMCs ARE GOVERNED BY SUPERSONIC TURBULENCE
Line-width size relation:
 ≈ 0.7 Rpc0.5 ± 0.05 km s-1
(Solomon et al. 1987)
Thermal velocity is only ~ 0.2 km s-1 at T ~ 10 K
 highly supersonic for R >~ 1 pc
Simulations Show Turbulence Damps Out in ~< 1 Crossing
Time, L / . How is It Maintained?
From formation--but then all clouds must be destroyed quickly
Injection by protostellar outflows, HII regions, or external
sources--but these are all highly intermittent
Significant issue: does turbulence damp out as quickly as
indicated by periodic box simulations?
CLOUD LIFETIMES
MAJOR ISSUE: ARE CLOUDS IN APPROXIMATE EQUILIBRIUM?
YES: 1. Star formation occurs in clusters over times long
compared to a crossing time (Palla & Stahler; Tan)
2. Cloud lifetimes are long compared to a crossing time:
GMCs are observed to be gravitationally bound:
Virial parameter vir = 52 R/GM ≈ Kinetic energy/Grav. energy
~1
GMCs must therefore be destroyed--they will not fall apart
Calculations show GMCs destroyed by photoionization:
tdestroy ~ 20 - 30 Myr >> crossing time L/ ~ 1.4Lpc1/2 Myr
CLOUD LIFETIMES
MAJOR ISSUE: ARE CLOUDS IN APPROXIMATE EQUILIBRIUM?
NO: 1. Star formation in a crossing time (Elmegreen)
Estimated time for star formation over a wide range of
length scales, reaching up to > 1 kpc: tsf  L
2. Critique of Palla & Stahler claim of long-term star formation
in Taurus (Hartmann)
3. OB associations can form in unbound clouds with
vir = 2 (Clark et al)
Possible partial resolution of debate:
Star formation in a crossing time valid for unbound structures,
including Taurus and the largest ones studied by Elmegreen.
But, is it possible to create the clumps with  ~ 1 g cm-2
characteristic of high-mass star forming regions in unbound
clouds?
PREDICTING THE PROPERTIES OF EQUILIBRIUM GMCs
(Chieze; Elmegreen; Holliman; McKee)
If cloud is in approximate equilibrium, virial theorem implies
<P> ≈ Psurface + 0.5 G2
( = surface density)
Stability requires <P> not much greater than Psurface.
Allowing for the weight of overlying HI and H2,
<P(CO)> ≈ 8 Psurface (Holliman) ,
where Psurface/k ≈ 2 104 K cm-3 (Boulares & Cox):
 GMC ≈ 100 Msun pc-2
Comparable to Solomon et al’s 170 Msun pc-2
PREDICTING THE CHARACTERISTIC STELLAR MASS
FROM THE WEIGHT OF THE ISM:
Gravitationally bound structures in equilibrium GMCs
(clumps and cores) have  ~ GMC ~ (8PISM/G)1/2
 m* ≈ Star formation efficiency  Bonnor-Ebert mass
≈ (1/2)  cs3 / (G3 r)1/2
SFE ~ 1/2 in core
(Matzner & McKee)
≈ (1/2)  cs4 / (G2 )
≈ 0.5 Msun for T = 10 K and  ~ GMC
Predicts that stellar masses are governed by the largescale properties of the ISM. Can be reduced by
subsequent fragmentation (cf Larson)
Possible problem: Works well for solar neighborhood, but
does it work elsewhere? (See later)
MAGNETIC FIELDS
“The strength of the magnetic field is directly proportional to
our ignorance” --- paraphrase of Lo Woltjer
Basic issue: Are magnetic fields of crucial importance in star
formation (Mouschovias), or are they negligible (Padoan & Nordlund) ?
Magnetic critical mass M : When magnetism balances gravity
B2 R3 ~ G M2 /R
 M = 0.12  / G1/2
Magnetically supercritical (M> M ): B cannot prevent collapse
Magnetically subcritical (M< M ): Collapse impossible without
flux loss or mass accumulation along field
MAGNETIC FIELDS: OBSERVATIONS
Crutcher finds M ≈ M and Alfven Mach number ~ 1
Caveats:
-Generally finds only upper limits at densities ~< 103 cm-3
(Recall that mean density of large GMC is ~ 100 cm-3, so
there are no data on large-scale fields.)
-If the clouds are flattened along B, then projection effects
imply that they are subcritical [M ≈ (1/2)M] (Shu et al.)
(But there is no evidence that clouds are sheet-like, and sheet-like
structure inconsistent with observed turbulent velocities.)
Determining the role of magnetic fields is one of the critical
problems in star formation.
THE IMF
Observations consistent with universal characteristic mass
~(1/3)Msun and high mass slope, dN/d ln m*  m*-1.35 (Salpeter)
Possible exceptions include paucity of O stars in the outer
parts of galaxies like M31
Slope of GMC mass distribution is flat (~  0.6), but the slope
of the core mass distribution is consistent with Salpeter:
Low-intermediate mass cores (Motte & Andre; Testi & Sargent)
High-mass cores (Beuther & Schilke)
THEORY: Universal slope requires universal physical
mechanism, turbulence (Elmegreen)
Derivation with many assumptions (Padoan & Nordlund)
Characteristic mass set by Jeans mass at average pressure
and possible subsequent fragmentation (described above)
CONCLUSION: IMF determined in molecular clouds
Computing the Star Formation Rate
From the Physics of Turbulence
• GMCs roughly virialized, turbulent KE ~ PE
• For sub-parts, linewidth-size relation  KE ~ r4
• PE ~ r5, so most GMC sub-parts are unbound. Only
overdense regions bound.
• Compute fraction f dense enough to be bound from PDF
of densities.
• SFR ~ f MGMC / tff
• Find f ~ 1% for any virialized object with high Mach no.
(Krumholz & McKee, 2005, ApJ, submitted)
SFR in the Galaxy
• Estimate cloud freefall times from direct
observation (Milky
Way) or ISM pressure
(other galaxies)
• SFR from molecular
mass, f, and tff
• Application to MW 
SFR = 2  5 Msun / yr.
• Observed MW SFR ~
3 Msun / yr
Result: SFR in Galactic Disks
The Kennicutt-Schmidt Law From First Principles
II. MICROPHYSICS: GRAVITATIONAL COLLAPSE
Paradigm: Inside-out collapse of centrally concentrated core
Accretion rate ~ Bonnor-Ebert mass per free-fall time
·
m* ~ mBE / tff ~ c3/(G3r)1/2  (Gr)1/2 ~ c3/G
Isothermal,  = p =1 (Shu)
Non-isothermal  = p  1 (McLaughlin & Pudritz)
Non-isentropic   p  1 (Fatuzzo, Adams & Myers)
If magnetic fields are important: Collapse of initially subcritical
clouds due to ambipolar diffusion (2D--Mouschovias)
Turbulent ambipolar diffusion can accelerate flux loss
(Zweibel; Fatuzzo & Adams; Heitsch)
THE CLASSICAL PROBLEMS OF STAR FORMATION
1. Angular momentum
Rotational velocity due to differential rotation of Galaxy is
~ 0.05 km s-1 in 2 pc cloud
 Specific angular momentum is j ~ rv ~ 3  1022 cm2 s-1
Angular momentum of solar system is dominated by Jupiter
and is much less: j ~ 1018 cm2 s-1
Protostars generally have accretion disks, but these have
angular momentum ~ solar system and << ISM value.
SOLUTION: Angular momentum removed by magnetic fields
2. Magnetic flux
Typical interstellar magnetic field ~ 5 mG
 Flux in 1 Msun sphere of ISM (r = 2 pc) is 6  1032 Mx
Net flux in Sun is ~ 1 G  p Rsun2 ~ 5  1021 Mx
How do protostars lose so much flux?
-Ambipolar diffusion: Flow of neutral gas through lowdensity, magnetized ions and electrons (ne/n < 10-6)
Most flux (in dex) must be lost in accretion disk; how does
ionization become low enough to allow this?
-Magnetic reconnection ?
-Issue not fully resolved yet.
PROTOSTELLAR JETS AND OUTFLOWS
Jet velocity v ~ 200 km s-1 ~ Keplerian
Mass loss rate in outflow ~ fraction of accretion rate onto star
PROTOSTELLAR JETS AND OUTFLOWS
Due to MHD winds driven by magnetic field threading the
accretion disk and/or the star. Detailed understanding lacking.
PROTOSTELLAR DISKS
ISSUE:
Generally believed that angular momentum transfer in
disks due to magnetorotational instability.
How can the coupling to the field be strong enough to enable
the MRI, yet weak enough to ensure observed flux loss?
ISSUE:
How do planets form out of protostellar accretion disks?
Enormous range of scales involved make this a very
formidible problem.
MASSIVE STAR FORMATION
HOW DO MASSIVE STARS FORM?
High-mass star-forming clumps (Plume et al. 1997)
Supersonically turbulent:  ~ 2.5 km s-1
Radius ~ 0.5 pc
 Virial mass ~ 4000 Msun

Surface density  ~ 1 g cm-2
Corresponding visual extinction: AV ~ 200  mag
Compare low-mass cores in Taurus (Onishi et al. 1996):
AV ~ 8 mag,
 ~ 0.03 g cm-2
EFFECT OF RADIATION PRESSURE
Wolfire &
Cassinelli
1987
Necessary condition: momentum in accretion flow at dust
destruction radius must exceed momentum in radiation field.
TURBULENT CORE MODEL FOR MASSIVE STAR FORMATION
McKee & Tan 2002, 2003
BASIC ASSUMPTION:
Star-forming clumps and cores within them are part of a selfsimilar, self-gravitating turbulent structure in approximate
hydrostatic equilibrium. Cores are supported in large part by
turbulent motions.
Consistent with observation:
* No characteristic length scales observed between the Jeans
length ~ ctff ~ c/(Gr)1/2 and the size of the GMC.
* All molecular gas in the Galaxy is observed to be in
approximate virial equilibrium.
TURBULENT CORE MODEL:
PROTOSTELLAR ACCRETION RATE
m* = f* mt *
[see Stahler, Shu & Taam 1980]
ff
·
m* = instantaneous protostellar mass
tff = (3p/32Gr)1/2 = free-fall time evaluated at r(m*)
f* = numerical parameter  (1)
In a turbulent medium, f*(t) could have large fluctuations.
On average:
f* >> 1 only in unlikely case of almost perfectly spherical inflow
f* << 1 only if supported by magnetic fields
Observations show fields do not dominate dynamics
(Crutcher 1999)
RESULTS FOR MASSIVE STAR FORMATION
Protostellar accretion rate for r  r -1.5:
·
m*  4.6 x 10-4 (m*f/ 30 Msun)3/4 3/4 (m*/m*f)1/2 Msun yr-1
Massive stars form in about 105 yr:
t*f = 1.3 x 105 (m*f/30 Msun)1/4 3/4
yr
Massive stars form in turbulent cores: velocity dispersion is
 = 1.3 (m*f/ 30 Msun)1/4 1/4
vs.
th = 0.3 (T/30 K)1/2
km s-1
km s-1
Accretion rate is large enough to overcome radiative
momentum:
·
m*  4.6 x 10-4 (m*f/ 30 Msun)3/4
3/4 (m*/m*f)1/2 Msun yr-1
Critique of Turbulent Core Model for Massive Star Formation
Dobbs, Bonnell, & Clark
Simulations of star formation in cores with r  r-1.5
Equation of state: isothermal or barotropic above 10^-14 g cm-3
Isothermal collapse results in many small fragments;
barotropic collapse in a few.
In no case did a massive star form (although simulation ran
only until ~ 10% of mass had gone into stars).
Require radiation-hydrodynamic simulations to address this
Massive Star Formation Simulations:
Required Physics
• Real radiative transfer and protostellar models are
required, even at early stages.
• Example: dM/dt = 10-3 Msun/yr, m* = 0.1 Msun, R*
= 10 Rsun  L = 30 Lsun!
• This L can heat 10 Msun of gas to 1000 K in ~ 300
yr. At nH = 108 cm-3, tff ~ 4000 yr  high
accretion rates suppress fragmentation.
• Most energy is released at sub-grid scales in the
final fall onto star. A barotropic approximation
cannot model this effect
NUMERICAL SIMULATIONS
2D: Yorke & Sonnhalter (2002)
Accurate grain opacities and multi-component grain model
120 Msun core

43 Msun star
(only 23 Msun with gray opacity)
3D: Krumholz, Klein, & McKee (2005)
AMR, flux-limited diffusion with gray opacity
Resolution ~ 10 AU, similar to Yorke & Sonnhalter
3D simulations with turbulent initial conditions, high accretion
rates, and radiative transfer (not barotropic approxmation) show
no fragmentation. Protostar has currently grown to > 20 Msun
ALTERNATE MODELS OF STAR FORMATION
COMPETITIVE ACCRETION (Bonnell et al.)
Protostellar “seeds” accrete gas that is initially unbound to
protostar
Does not work for m* > 10 Msun due to radiation pressure
(Edgar & Clarke)
Does not allow for reduction in accretion due to vorticity
(Krumholz, McKee & Klein)
STELLAR MERGERS (Bate, Bonnell, & Zinnecker)
Requires stellar densities ~ 108 pc-3, greater than ever observed
Not needed to form massive stars
Stellar mergers do occur in globular clusters (Fregeau et al.)
ISSUE: HOW DO STARS FORM IN CLUSTERS?
Most stars are born in clusters
All the problems of normal star formation are multiplied at
stellar densities that can be > 106 times local value
Solution unknown at present
NGC 3603
STAR FORMATION: PROBLEMS AND PROSPECTS
SUMMARY
MACROPHYSICS:
Key problem is FRAGMENTATION
Determines IMF and the rate of star formation
Theoretical progress: Major advance---star formation
occurs in supersonically turbulent medium
Importance of magnetic fields remains unclear
Equilibrium vs. non-equilibrium structure
Prospect for progress are good: AMR codes are
becoming widely available and are ideally suited for
multiscale problems
STAR FORMATION: PROBLEMS AND PROSPECTS
SUMMARY
MICROPHYSICS:
Problem: How do stars form--by gravitational collapse,
gravitational accretion, or stellar mergers?
Prospect: May require more computer power to resolve this,
since calculation of formation of even one star is a challenge.
Problem: How do massive stars form in the face of radiation
pressure?
Prospect: Good progress being made, but 3D calculations
with adequate radiative transfer and dust models are in
the future. Formation of clusters with massive stars is a
yet greater challenge.
Problem: Planet formation
Prospect: It will be some time before a single simulation can treat
the enormous range of scales needed for an accurate simulation.