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 (Planet formation requires smaller scales)

t ~ 103.5 s -- 106 yr (Not currently feasible)

ZENO’S PARADOX (ALMOST) IN COMPUTATIONS OF STAR FORMATION

Time step t 1/(G)1/2

Truelove et al. (1998) calculations of star formation now:

Density increase of 109 t decrease of 104.5

Density increase of 1017 t decrease of 108.5

ABN (2002) calculations of primordial star formation:

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

2 ~ GMJ/r

M ~ r3 MJ ~ 3/(G3 )1/2 = 4/(G3 P)1/2

(MJ = Jeans mass)

Maximum mass of isothermal sphere ( = cth) :

MBE = 1.18 cth3 /(G3 s)1/2 (Bonnor-Ebert mass)

where s is measured at the surface of the cloud

2D Jeans mass: In a self-gravitating cloud, P ~ G2, where is the mass/area of the cloud

MJ, 3D~ 4/(G2 ) = MJ, 2D

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

I. MACROPHYSICS

GMCs ARE GOVERNED BY SUPERSONIC TURBULENCE

≈ 0.7 Rpc0.5 ± 0.05 km s-1 (Solomon et al. 1987)

Line-width size relation:

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?

GMCs are observed to be gravitationally bound:

Virial parameter vir = 52 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

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:

YES:

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?

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)

PREDICTING THE PROPERTIES OF EQUILIBRIUM GMCs

If cloud is in approximate equilibrium, virial theorem implies

<P> ≈ Psurface + 0.5 G2

(Chieze; Elmegreen; Holliman; McKee)

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

surface density)

Gravitationally bound structures in equilibrium GMCs (clumps and cores) have ~ GMC ~ (8PISM/G)1/2

PREDICTING THE CHARACTERISTIC STELLAR MASS FROM THE WEIGHT OF THE ISM:

m* ≈ Star formation efficiency Bonnor-Ebert mass

≈ (1/2) cs4 / (G2 )

≈ 0.5 Msun for T = 10 K and ~ GMC

≈ (1/2) cs3 / (G3 )1/2 SFE ~ 1/2 in core

(Matzner & McKee)

Predicts that stellar masses are governed by the large-scale 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 M2 /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 free-fall 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 DisksThe Kennicutt-Schmidt Law From First Principles

II. MICROPHYSICS: GRAVITATIONAL COLLAPSE

Paradigm: Inside-out collapse of centrally concentrated core

m*· ~ mBE / tff ~ c3/(G3)1/2 (G)1/2 ~ c3/G

Accretion rate ~ Bonnor-Ebert mass per free-fall time

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

SOLUTION: Angular momentum removed by magnetic fields

Protostars generally have accretion disks, but these have

angular momentum ~ solar system and << ISM value.

2. Magnetic flux

Typical interstellar magnetic field ~ 5 G

Flux in 1 Msun sphere of ISM (r = 2 pc) is 6 1032 Mx

Net flux in Sun is ~ 1 G Rsun2 ~ 5 1021 Mx

-Issue not fully resolved yet.

How do protostars lose so much flux?

-Ambipolar diffusion: Flow of neutral gas through low-density, magnetized ions and electrons (ne/n < 10-6)

-Magnetic reconnection ?

Most flux (in dex) must be lost in accretion disk; how does ionization become low enough to allow this?

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?

Compare low-mass cores in Taurus (Onishi et al. 1996):

AV ~ 8 mag, ~ 0.03 g cm-2

Supersonically turbulent: ~ 2.5 km s-1

Surface density ~ 1 g cm-2

Corresponding visual extinction: AV ~ 200 mag

(Plume et al. 1997)

Virial mass ~ 4000 Msun

Radius ~ 0.5 pc

High-mass star-forming clumps

Wolfire & Cassinelli 1987

Necessary condition: momentum in accretion flow at dust destruction radius must exceed momentum in radiation field.

EFFECT OF RADIATION PRESSURE

TURBULENT CORE MODEL FOR MASSIVE STAR FORMATIONMcKee & Tan 2002, 2003

BASIC ASSUMPTION:

Star-forming clumps and cores within them are part of a self-similar, 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/(G)1/2 and the size of the GMC.

* All molecular gas in the Galaxy is observed to be in approximate virial equilibrium.

PROTOSTELLAR ACCRETION RATE

m*· = *

m*tff

m* = instantaneous protostellar mass

tff = (332Gfree-fall time evaluated at (m*)

* = numerical parameter (1)

In a turbulent medium, *(t) could have large fluctuations.On average:

* >> 1 only in unlikely case of almost perfectly spherical inflow* << 1 only if supported by magnetic fields

Observations show fields do not dominate dynamics (Crutcher 1999)

[see Stahler, Shu & Taam 1980]

TURBULENT CORE MODEL:

Protostellar accretion rate for r -1.5:

m*· 4.6 x 10-4 (m*f/ 30 Msun)3/4 (m*/m*f)1/2 Msun yr-1

RESULTS FOR MASSIVE STAR FORMATION

Massive stars form in about 105 yr:

t*f = 1.3 x 105 (m*f/30 Msun)1/4 yr

Massive stars form in turbulent cores: velocity dispersion is

= 1.3 (m*f/ 30 Msun)1/4 km s-1

vs.

th = 0.3 (T/30 K)1/2 km s-1

Accretion rate is large enough to overcome radiative momentum:

m*· 4.6 x 10-4 (m*f/ 30 Msun)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-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.

Require radiation-hydrodynamic simulations to address this

In no case did a massive star form (although simulation ran only until ~ 10% of mass had gone into stars).

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.)

NGC 3603

ISSUE: HOW DO STARS FORM IN CLUSTERS?

All the problems of normal star formation are multiplied at stellar densities that can be > 106 times local value

Most stars are born in clusters

Solution unknown at present

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.