Transcript Mass Limits to Primordial Star Formation from Protostellar

The Accretion Physics
of Primordial Protostars
Jonathan C. Tan
Christopher F. McKee
What are the initial conditions for
primordial star formation?
Chemical Composition
—  H— + 
TraceCore
H2 formation:
H
+
e
Mass H + H—  H + e—
2
Core Size
Density Structure
Bulk Velocities: Rotation
Radial infall
Internal Velocities: Turbulence
4 cm-3
Tmin ~= 200 K, ncrit ~= 10
Sound
Speed
MBE = 380Msun cs=1.2 km/s
Chemical Composition
What are the initial conditions for
primordial star formation?
Tmin ~= 200 K, ncrit ~= 104 cm-3
MBE = 380Msun cs=1.2 km/s
-2.2
Density
structure:
~self-similar,
r
Centrally concentrated cloud
quasi-hydrostatic, subsonic contraction
More chemistry: at high density >108cm-3
H+H+H -> H2 + H
completely moleculer core ~Msun
efficient continuum cooling ->
dynamical collapse
Rotation: core forms from mergers and
n>n
(T=200K)=indep
 T
collapse
along
filaments: expect
J>0
crit , tcool
rises
fKep  vEcirc
vKep  ~ 0.5
grav/=-GM/r
r -k, k≈2.2
Abel,
Abel,Bryan,
Bryan,Norman
Norman(2002)
(2002)
The Accretion Rate
and Formation Timescale
r
-k
Density structure: self-similar, r -k, k≈2.2
~singular polytropic sphere in virial
and hydrostatic equilibrium P = K , =1.1
.
Accretion rate: m*= * m / tff(m)  f(m,K)
Abel et al. 2002
“Isentropic Accretion”
K=1.9x1012(T/300K)(nH/104cm-3)-0.1cgs
K’=K/ 1.9x1012cgs
*=1.4 (Hunter 1977)
.
m*=0.026K’15/7(m*/Msun)-3/7Msun/yr
t* =41 K’-15/7(m*/Msun)10/7yr
 m*(t=2Myr) ≈ 2000Msun
Collapse to a Disk
Geometry of Streamlines
rd = f2Kepr0  3.4 (M/Msun)9/7 AU
Anticipate accretion driven
by large scale grav. instabilities
and gravitational viscosity
(Gammie 2001)
Conserve J during free-fall
inside sonic point (Ulrich 1976)
viscosity
= cs h,
<0.3
fragmentation
tcool< 3-1
Disk Models
=0.3 : look for fragmentation condition, Q<1 in >1 region
Surface density
Thickness
Ionization
Temperature
Tc , Teff
Toomre Q
.
m*
17x10-3Msun/yr 6.4x10-3Msun/yr 2.4x10-3Msun/yr
Evolution of the Protostar
Assume polytropic structure
Energy: E = -agGm*2/(2r*)
dE/dt = - L
-
fD Dm*
.
- (1-fk)Gm*m*/r*
Luminosity: L = f ( Lint + Lacc ) ; Lacc = fk Gm. *m*/r*
Advection: f = exp( - 3 vff / c)
Deuterium burning for Tc>106K
Structural rearrangement after tKelvin
Eddington model for 
Solve for r*(m*), until reach main sequence
Evolution of the Protostar
Initial condition
m* = 0.04 Msun
r* = 14 Rsun
(Ripamonti et al. 02)
:Radius
Photosphere
Accretion Shock
Main Sequence
(Schaerer 2002)
Protostar is large (~100 Rsun) until it is older than tKelvin
Contraction to Main Sequence
Accretion along Main Sequence
Evolution of the Protostar:Luminosity
Total
Boundary Layer
Accretion Disk
Internal
:Ionizing
Luminosity
Evolution of the Protostar
Total
fKep=0.5
Internal
Boundary Layer
fKep=0.05
Accretion Disk
Spherical, fKep=0
Main Sequence
(Schaerer 2002)
Spectrum depends on initial rotation
Feedback Processes
When does accretion end?
See Poster: JT, McKee, Blackman
Ly- Radiation Pressure
m* >~ 20-30 Msun (polar)
Ionization
m* >~ 100 Msun
Disk Evaporation
m* ~ 100-200 Msun
Hydromagnetic Outflows
m* >~ 100-500 Msun
JT & Blackman (2003)
Conclusions
Understand initial conditions for star formation: set by H2 physics
Analytic model for collapse with angular momentum:
accretion rate is large and declining
most material collapses to disk
Analytic model for disk accretion:
ionization energy important
.
no fragmentation
Analytic model for protostellar evolution:
large protostars, contract to main sequence m*>20Msun
predict feedback is quite strong compared to spherical case
Feedback processes are complicated:
m* probably >30Msun, perhaps several 100Msun
Implications of massive star formation in each mini-halo?
How effective is external feedback?
Are low-mass zero metallicity stars possible?
Growth of the HII Region
Balance ionizing flux vs
recombinations and infall
Infall likely to be suppressed
for rHII>rg , where vesc=ci
Find stellar mass at breakout
rHII = rg
polar; equatorial
Breakout mass vs rotation
Ly- and FUV Radiation Pressure
1 in HI around HII region : P = u/3 = 4J/3c
Photons diffuse in freq. and space
Normalize J to appropriate F 1/r2
Velocity field: Voigt profile D ; x /D
Line profile: damping wings x = <> x ; x = a/x2
L
Escape after n scatterings, or 2 photon decay
freq. shift xe = n1/2 ; mean free path at escape le = 1/<>e
diffusion scale n1/2 le = n1/2/<>e must equal size of region, L=<>L/<>
n1/2 = <>L e and xe = <>L e ≈ (a <>L)1/3
total path length of photons is n1/2L so mean intensity boosted by factor
n1/2 = <>L a/xe2 ≈ (a <>L )1/3
(Neufeld 90) :
P / (F/c) = 36.7 NH,201/3 / vD,62/3
Evaluate NH from harmonic mean of sightlines from star
Disk
Photoevaporation
Weak wind case:
.
mevap= 6.1x10-5 S491/2
(m*/100Msun)1/2Msun/yr
=1.7x10 -4(m*/100Msun)5/4Msun/yr
for zero age main sequence
Equate with mass accretion rate
m*max = 480 K’60/47 40/47 Msun
Hollenbach et al. 94
Mass Limits vs. Core Rotation
Disk Photoevaporation
Overview of Structure Formation
1. Recombination z ≈1200, start of “dark ages”
2. Thermal equilibrium matter-CBR until z ≈160
MJeans ≈ 105Msun(T/1/3)3/2 : independent of z
e.g. globular clusters
3. Thermal decoupling, T (1+z)2 ; MJeans (1+z)3/2
4. “First Light”
5. Reionization, T ≈104K, MJeans ≈ 109-10 ((1+zion)/10)3/2Msun
e.g. galaxies
Madau (2002)
Numerical Simulations: Results
Abel, Bryan, Norman (2002):
1. Form pre-galactic halo ~105-6Msun
at intersection of filaments
2. Form quasi-hydrostatic gas core
inside halo: M≈4000Msun, r ≈10pc,
nH ≈10cm-3, fH2 ≈10-3, T>=200K
H2 formation: H+e—H— + 
H+H— H2+e-—
Gradual contraction driven by cooling
in dense central region. Rapid 3-body
H2 formation for nH>1010cm-3:
fully molecular region; strong cooling
supersonic inflow. Line cooling is
optically thick for n>1013cm-3:end of sim.
3. 1D simulations (Omukai & Nishi 1998): Form quasi-hydrostatic protostar
nH ≈1016-17cm-3, T ≈2000K: optically thick, adiabatic contraction
hydrostatic core with m* ≈0.005Msun,r* ≈14Rsun (also Ripamonti et al.02)
Abel, Bryan, Norman (2002)
Initial Conditions for Star Formation from
Abel, Bryan, Norman 02
Mass Limits vs. Core Rotation
core rotation (ABN)
Stellar Evolution to Supernovae
Primordial high-mass main sequence is relatively stable with
little mass loss (Baraffe, Heger, Woosley 01)
Calculations of stellar evolution and supernovae
(Fryer, Woosley, Heger 01)
8<M<40
NS
“normal”
enrichment
40<M<130
BH
inefficient
enrichment
130<M<260
no remnant
efficient
enrichment
260<M
BH
inefficient
enrichment
Nucleosynthetic yields may reveal themselves in metal-poor
stars (Aoki et al. 02; Christlieb et al. 02)
Hydrogen Ionizing Luminosities
along the Primordial Main Sequence
Tumlinson & Shull 00; Bromm et al. 01;
Ciardi et al. 01; Schaerer 02
Rotating Infall
fKep  vcirc / vKep 0.5 (ABN)
rd = f2Kepr0  3.4 (M/Msun)9/7 AU
Geometry of Streamlines
Density along radii, =0,/3, 9/20
Conserve J during free-fall
(Ulrich 76)
Optical Depth