Star Formation, Fragmentation,
and Protoplanets in Discs
Matthew Bate
University of Exeter
Star Formation, Fragmentation,
and Protoplanets in Discs
Matthew Bate, University of Exeter
Kurosawa, Harries, Bate & Symington (2004)
Collaborators:
Ian Bonnell,
Volker Bromm,
Gennaro D’Angelo
Steve Lubow
Gordon Ogilvie
Daniel Price
St Andrews
Texas
Exeter
STScI
Cambridge
Exeter
Extra-solar Planets
• Discoveries (main sequence stars)
– 154 planets
– 136 planetary systems
– 14 multiple planet systems
–
http://www.obspm.fr/encycl/catalog.html
• Many giant planets in tight orbits
– Multiple-planet scattering
– Interaction between planet and disc
Planet-Disc Interaction
• Massive planet clears a gap in the gas disc
• Spirals in towards the star
– May explain extrasolar giant planets in close orbits
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
G. Bryden
Bate et al. 2003
Planet Migration
•
Difficult problem
– Gravity, hydrodynamics, magnetic fields, radiative
transfer
– High resolution for many dynamical times
•
Recent work
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
– `Runaway’ migration
•
Masset & Papaloizou 2003
– Protoplanets in turbulent MHD discs
•
•
Nelson & Papaloizou 2003
Winters, Balbus & Hawley 2003
– Multiple-planet systems
– `Radiative transfer’
•
•
D’Angelo, Henning & Kley 2003
Morohoshi & Tanaka 2003
– High resolution 2-D and 3-D a-accretion discs
•
•
D’Angelo, Kley & Henning 2003
Bate, Lubow, Ogilvie & Miller 2003
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
High-resolution 3-D Migration
• Low-mass planets
– D’Angelo et al. 2003; Bate et al. 2003
– Good agreement with linear theory
– Migration timescales of 104-106 years
• If neglect Hill sphere
– Including torques from within 1/2 Hill
sphere could give outward migration
– Resolution within Hill sphere
insufficient
• D’Angelo at Exeter
– Decided to try and get convergence
Bate et al. (2003)
`Runaway’ Migration
• Masset & Papaloizou 2003
– Increased disc mass
– Found super-linear increase in migration
Two Cases
1 MJ, low-mass disc
0.3 MJ, high-mass disc
Jupiter Mass, Low-mass Disc
Saturn Mass, High-mass Disc
Conclusions
• 3-D low-mass migration rates
– Good agreement with linear theory
• If neglect Hill sphere
– Migration timescales of 104-106 years
• High-mass migration rates
– Including Hill sphere slows migration by factor <2
• Achieving convergence can be difficult if include Hill sphere
– Resolution of ~100 zones/Hill diameter
• No numerical support for ‘runaway’ migration
– Converged results do not exhibit runaway
Star Formation
•
Physics
–
–
–
–
Self-gravity
Hydrodynamics
Radiative transfer
Magnetic fields
• Non-ideal: ambipolar diffusion, Hall conduction, resistivity
– Multi-fluid
• Chemistry, dust, ionisation
•
Scales
– Spatial scales: 10 orders of magnitude
• 10’s of parsecs to less than R
– Timescales: 12 orders of magnitude
• 107 yrs to <5 minutes
– Densities: >20 orders of magnitude
• <10-20 g cm-3 to >1 g cm-3
•
Sub-problems
– Turbulence, chaos, accretion discs, jets/outflows, chemical reactions
Star Formation
•
Physics
–
–
–
–
Self-gravity
Hydrodynamics
Radiative transfer (FLD in SPH: Whitehouse & Bate 2004)
Magnetic fields (Price & Monaghan 2003)
• Non-ideal: ambipolar diffusion, Hall conduction, resistivity
– Multi-fluid
• Chemistry, dust, ionisation
•
Scales
– Spatial scales: 10 orders of magnitude (5 orders of magnitude)
• 10’s of parsecs to less than R
– Timescales: 12 orders of magnitude
• 107 yrs to <5 minutes
– Densities: >20 orders of magnitude
•
•
<10-20
g
cm-3
to >1 g
cm-3
(~5 AU to 0.5 pc)
(8 orders of magnitude)
(~3 hours to 300,000 yrs)
(8 orders of magnitude)
(~10-19 to 10-11 g cm-3)
Sub-problems
– Turbulence, chaos, accretion discs, jets/outflows, chemical reactions
SPH
• Variable smoothing/softening lengths
– Lagrangian so resolution follows mass
• Individual timesteps
– Essential for following large range of timescales
– Can follow collapse to stellar densities
• Bate (1998): 1 solar mass cloud to form stellar core
– Need > 3x105 particles
• Sink particles
– Bate, Bonnell & Price (1995)
– Condensed objects (stars) replaced by point mass
• Accretes gas that falls within a certain radius
• Parallel
– OpenMP
Collapse to
Stellar Densities
•
Bate 1998
–
Collapse of molecular cloud core
R~5000 AU to stellar densities
Goal: Statistical Properties
• Understand the origin of and make predictions for:
– Initial mass function
– Star formation timescale
– Star formation efficiency
– Kinematics: velocity dispersion
– Frequency and properties of binaries and multiples
– Circumstellar disc properties
The Initial Mass Function
•
Many models for the origin of the IMF
– Usually analytic or semi-analytic
– Many try to reproduce the observed IMF (Salpeter, log-normal, etc)
•
Use numerical simulation to understand directly origin of the IMF
– Look for variations with initial conditions
– Understand using simple models
•
We find the IMF originates from accretion/ejection
– Present a simple accretion/ejection model that fits numerical simulations
•
Variation of the IMF?
– Results statistically independent of initial ‘turbulence’ velocity field
– Characteristic `stellar’ mass depends on mean thermal Jeans mass
– Minimum mass depends on opacity limit for fragmentation
Hydrodynamical Simulations
•
Initial supersonic (`turbulent’) velocity field
–
–
–
–
•
Resolve down to the opacity limit for fragmentation
–
–
–
–
•
E.g. Ostriker, Stone & Gammie 2001
Divergence-free random Gaussian velocity field
P(k) ~ k – 4 so that s(l) ~ l1/2 (Larson 1981)
Normalised so cloud contains one turbulent Jeans mass
Local Jeans mass always contains >75 particles (Bate & Burkert 1997; Bate et al. 2003)
Limited to 50 M of gas
Molecular cloud sizes 0.4-0.8 pc across
<100 stars and brown dwarfs
Four calculations
–
Standard: mean Jeans mass 1 M , Opacity limit at ~3 MJ
•
–
Bate, Bonnell & Bromm 2002a,b;2003
Denser cloud: mean Jeans mass 1/3 M , Opacity limit at ~3 MJ
•
Bate & Bonnell 2005
–
Lower metallicity: mean Jeans mass 1 M , Opacity limit at ~10 MJ
–
Different power spectrum: P(k) ~ k – 6
Opacity Limit for Fragmentation
•
•
Hoyle 1953; Low & Lynden-Bell 1976; Rees 1976
–
–
–
–
Low-density gas collapses isothermally ( T~10 K )
Compressional heating rate << cooling rate
Collapse accelerates
Heating rate > cooling rate
–
Occurs at ~ 10-13 g cm-3
• Masunaga & Inutsuka 2000
Pressure-supported core forms (Larson 1969)
–
–
•
Minimum mass
–
–
•
Size ~ 5 AU
Mass ~ 0.005 M (5 MJ)
0.007 M (7 MJ) Low & Lynden-Bell (1976)
0.01 M (10 MJ) Boss (1988)
Grows with time due to accretion from envelope
Brown Dwarf Formation
•
How do brown dwarfs form?
–
Bate, Bonnell & Bromm 2002
–
3/4 in massive circumstellar discs
via disc fragmentation
•
Bonnell 1994; Whitworth et al.
1995; Burkert et al. 1997
–
1/4 in dense collapsing filaments
–
Opacity limit for fragmentation sets initial
mass
–
Must avoid accreting to higher masses
–
Ejected from unstable multiple systems
(c.f. Reipurth & Clarke 2001)
•
Stops accretion before they attain stellar
masses
Standard Model
Denser Cloud
Low Metallicity
Turbulence P(k)~k-6
Resulting IMFs
Standard calculation
Low metallicity
Denser cloud
Turbulence P(k)~k-6
Variations with Initial Conditions?
• Denser cloud (1/3 of original thermal Jeans mass)
– Median mass decreased by factor of 3.04
– Higher proportion of brown dwarfs
– K-S test gives only 1.8% probability of same IMF
• Increasing minimum fragment mass by factor of 3
– Increases the minimum mass of a brown dwarf
– Median mass almost unchanged (20% smaller)
– 45% probability of being drawn from the same populations
• Dramatic change to initial velocity power spectrum
– No statistically-significant change in the IMF !
– 95% probability of being drawn from the same population!
Final Masses: Accretion vs Ejection
•
Plot time between formation and ejection versus final mass
–
Brown dwarfs have sub-stellar masses because they are ejected soon after they form
Simple Accretion/Ejection Model
• Assumptions (Bate & Bonnell 2005)
– Initial masses set by the opacity-limit (e.g. 3 MJ), then objects
accrete at a constant rate until ejected
– Individual accretion
rates drawn from log-normal distribution
_.
with mean M and variance s
– Ejections occur stochastically with characteristic timescale
teject (the same for all objects)
• Probability of not being ejected proportional to exp(-t/teject)
• See also Basu & Jones 2004
• Three
_.
_ parameters
– M = M teject
– s
– Minimum mass due to opacity limit (metallicity)
Reproduces Hydrodynamical Results
• Parameter values taken directly from simulations
Conclusions
•
Can perform simulations that form statistically significant numbers of
stars
– Much physics still missing or simplified
– Begin to understand the origin of stellar properties
•
IMF is quite robust against variations in initial conditions
–
•
In agreement with observations
Small variations may be detectable
–
Primarily in
•
•
•
Brown dwarfs
–
Denser clouds should produce more brown dwarfs
•
•
Location of the peak
Slope of the substellar IMF
c.f. Taurus vs Orion Trapezium (Briceno et al. 2002)
IMF originates from interplay between accretion/ejection
–
–
Stars and brown dwarfs form the same way
Brown dwarfs are those objects ejected sooner after they form
Numerical Issues
•
Chaos
– Star formation is a chaotic process
• At the very least, after it involves N>2 stars
– Probably before this
• e.g. Monaghan’s ABC gas flow
• Sensitive to small variations in physics
– Global statistical properties may still be derived
•
Disc fragmentation
– Fisher/Klein’s AMR and SPH results appear to differ on disc fragmentation
– Bonnell & Bate investigating dependence of fragmentation on resolution
•
Convergence
– SPH convergence can be slow
– Improvements can be made
• Viscosity switches that cut viscosity where its not required
• Including the W terms (due to variable smoothing lengths) dramatically improve
convergence
Chaos: Dependence on Opacity Limit
Disc Fragmentation
•
Ian Bonnell & I testing convergence
Bate & Burkert 1997 Resolution
9 times more particles
Disc Fragmentation
•
Ian Bonnell & I testing convergence
9 times more particles
81 times more particles (>8000 part/MJ)
AMR vs SPH Viscosity
• Effective viscosity of modelling disc in Cartesian grid
– Krumholz, McKee & Klein (2004) shows need Dx/r<1/100 for realistic
angular momentum transport
– Too larger viscosity may influence disc fragmentation
• SPH shear viscosity
h
 0.05
H
Dx
 0.05  ( 1) 
 0.5Dx /R
0.1R
AMR : a eff  1100(Dx /R) 2.37
a eff  0.05aSPH
• Shear equal at

– AMR: 275 zones/R (or more)
– SPH: h/H=1/27 (or less)
SPH Convergence
•
10K, 30K, 100K, 1M, 3M particles
–
–
–
Black: standard viscosity a=1, b=2
Red: no viscosity
Blue: 10K, W terms, standard viscosity
Conclusions on Resolution
• No substantial evidence for an SPH resolution problem
– Fragmentation does not go away with increased resolution
• Bonnell & Bate and Fisher’s results
• Shear viscosity in AMR is greater than that of SPH
– Extremely high resolution required before AMR beats SPH
– Likely to depend on implementation of AMR
• e.g. operator splitting?
– May influence disc fragmentation
• Convergence of standard SPH can be substantially improved
– Include variable smoothing length terms
– Viscosity switches
Comparison with Star-forming Regions
• Ophiuchus low-mass star-forming region
• 550 M within area of 2x1 pc (Wilking & Lada 1983)
• Contains 6 main dense cores
• Simulation similar to modelling Oph-F core (mass ~8 M)
Allen et al. 2001
Motte, Andre & Neri 1998
Bate & Burkert 1997
• Navarro & White (1993)
– Looked to see what number of SPH particles other people
required to resolve other problems
– Totally different problems
• Adiabatic collapse & virialisation of gas cloud
• Collision of two gas clouds
– Navarro & White recommend at least 100 particles
• Bate & Burkert used this as a STARTING POINT
– Our criterion was based on convergence testing of the
fragmentation of a cloud with an m=2 mode
– We found we needed roughly the same number of particles as
Navarro & White
Testing Bate & Burkert 1997
•
Happy that Bob Fisher finds fragmentation does not go away with
increasing SPH resolution
– Up to 32 times BB97’s resolution
•
Ian Bonnell & I have done similar testing
–
–
–
–
–
•
Collapse of a turbulent cloud to form protostar and fragmenting disc
Confirm Fisher’s results
SPH fragmentation is robust
Have increased resolution to 81 times BB97’s criterion
8000 particles per minimum Jeans mass during calculation
Disc fragmentation IS chaotic
– Spiral structure depends on growth of linear perturbations
• Different initial perturbations, different growth, different spirals
– When non-linear amplitude - true chaos
• Similar to several self-gravitating objects
• Chaotic evolution, c.f. a 3-body stellar system
Resolution Requirement for SPH
•
Bate & Burkert (1997)
– Jeans mass should always be resolved
– Empirically, roughly 2 Nneigh~100 particles required
– More particles, more accurate solution
•
Consequences of not resolving it depend on SPH formulation
– Gravity & hydrodynamics softened/smoothed on same length scale
• Collapse of Jeans unstable object slowed down
• Jeans-unstable collapses, Jeans-stable pressure supported
– Gravity softened on larger length scale than hydrodynamics
• Collapse of Jeans unstable objects inhibited
– Gravity softened on smaller length scale than hydrodynamics
• Artificial fragmentation possible
• Marginally Jeans unstable object may break into several objects
A Problem with SPH Resolution?
•
Klein, Fisher, McKee (unpublished)
– Performed AMR calculations of collapsing turbulence clouds
– Did not find as much fragmentation as we find with SPH
• In particular, fragmentation of massive discs
– Assumed their code was right, SPH must be wrong
•
More recently, Fisher has performed SPH simulations
– Preliminary results presented at Banff, July 2004
•
Calculations use the GADGET SPH code
–
–
–
Uses fixed gravitational softening length, variable hydrodynamic smoothing
Bate & Burkert (1997) warn against this
Can promote artificial fragmentation, may go away when
•
•
Resolution high enough
Softening/smoothing becomes equal at density of minimum Jeans mass
Plummer Law Summary
N Particles
tcollapse / tffc
tfrag/ tffc
Nfrag (t=1.80)
104
1.43
1.55
3
2 104
1.43
1.58
6
4 104
1.43
1.60
5
8 104
1.43
1.62
5
1.6 105
1.43
1.66
7
3.2 105
1.43
1.66
3
6.4 105
1.43
---
1 (t=1.66)
Binary Brown Dwarfs
•
Calculation 1
– 1 BBD from ~20 brown dwarfs: Frequency ~5%
– Close: 6 AU
– Still accreting, part of unstable multiple system
•
Calculation 2
– 3 BBD: 2 AU (acc), 21 AU (acc), 66 AU (stable)
– 2 BD+VLM (<0.09 M): 15 AU (acc), 136 AU (stable)
– Frequency: 5/60 ~8%
•
Observations
– Observed frequency ~15%
• Reid et al. 2001; Close et al. 2002,2003; Bouy et al. 2003; Burgasser et al.
2003; Gizis et al. 2003; Martin et al. 2003
– Almost all binary brown dwarfs close (<15 AU)
• Luhman (2004): Wide (200 AU) binary brown dwarf?
Potential Problems with Fisher’s
SPH
•
Banff presentation
– Fragmentation appears to be slightly delayed as resolution increases
– Highest resolution calculation not followed as long as the others
– His convergence test is biased against fragmentation
•
Calculations use the GADGET SPH code
– Uses fixed gravitational softening length, variable hydrodynamic smoothing
– Bate & Burkert (1997) warn against this
– Can promote artificial fragmentation, may go away when
• Resolution high enough
• Softening/smoothing becomes equal at density of minimum Jeans mass
Wengen Tests
•
Test 1
– 200,000 particles
• Standard viscosity, alpha=1, beta=2
•
Test 2
– 110,000 particles
• Standard viscosity, alpha=1, beta=2
•
Test 4
– 160,000 particles
• Standard viscosity
• Balsara
• Div v/curl v applied to standard
– 800,000 particles
• Standard viscosity
Test 4
Viscosity
Std
Bal
div/curl 1m
std
Planet Formation
• Protoplanets embedded in turbulent MHD discs
• Nelson & Papaloizou (Queen Mary)
Bate et al. (2003)
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
The Low-mass Cores
• Single object followed by
multiple disc fragmentation
• Forms stable triple
• Later 3rd disc fragmentation
• New filament fragmentation
•
Single object followed by
multiple disc fragmentation
•
Dynamically-unstable system
rearranges to more stable
configuration
Disc Fragmentation
•
Ian Bonnell & I testing resolution criterion of Bate & Burkert (1997)
BB97 resolution
81 times more particles
Opacity Limit for Fragmentation
• Pressure-supported core forms
– Size ~ 5 AU (Larson 1969)
• Minimum separation
– Close binaries (separations <10 AU)
– Wider binaries hardened?
• Minimum mass
– Initially 0.005 Mo (5 MJ) Larson (1969)
• Grows with time due to accretion from envelope
– 0.007 Mo (7 MJ) Low & Lynden-Bell (1976)
– 0.01 Mo (10 MJ) Boss (1988)
Opacity Limit for Fragmentation
•
Heating and cooling in collapsing molecular gas
– Masunaga & Inutsuka 2000
•
We use a barotropic equation of state
– Isothermal for r<10-13 g cm-3
– g=7/5 for r>10-13 g cm-3
•
When r>10-11 g cm-3 replace gas within r =5 AU with sink particle
•
Subfragmentation
During Second
Collapse: NO
Bate 1998, and Bate, in prep.
–
Collapse of molecular cloud core
R~5000 AU to stellar densities
–
Disc does not fragment due to
•
•
•
High thermal energy content
Gravitational torques
Conclusion:
–
Fragmentation can form all
binaries except separations < 10 AU
The Initial Mass Function
•
Many models for the origin of the IMF
– Usually analytic or semi-analytic
– Many try to reproduce the observed IMF (Salpeter, log-normal, etc)
•
Use numerical simulation to understand directly origin of the IMF
– Look for variations with initial conditions
– Understand using simple models
•
We find the IMF originates from accretion/ejection
–
–
–
–
Present a simple accretion/ejection model that fits numerical simulations
Results independent of initial ‘turbulence’
Predicts variation with initial conditions
Neither Salpeter or log-normal IMF
Performance of my SPH Code
• Performance depends on number of particles
– Efficiency with 105 (dashed) and 106 (solid) particles
Planet-Disc Interaction
G. Bryden
Velocity Dispersion
Stars & brown
dwarfs ejected
Vel. Disp.
Independent of
- mass
- binarity
3-D RMS
vel. disp.
2.1 km/s
4.3 km/s
4.6 km/s
4.1 km/s
Opacity Limit for Fragmentation
•
Hoyle 1953; Low & Lynden-Bell 1976; Rees 1976
–
–
–
–
Low-density gas collapses isothermally ( T~10 K )
Compressional heating rate << cooling rate
Collapse accelerates
Heating rate > cooling rate
–
–
Occurs at ~ 10-13 g cm-3 (Masunaga & Inutsuka 2000)
Not ~ when gas is optically-thick to IR radiation
•
The Dependence of the IMF on
Initial Conditions
Variation of peak mass
•
Accretion rate prop. to cs3/G
•
Ejection timescale prop. to (Gr)-1/2
• ~ crossing time for small groups
•
Varies with mean thermal Jeans mass
• Definition of Jeans mass: cs3/ (G3r)1/2
• Stellar groups more compact
• Ejection time decreases
•
Increase dispersion in accretion rates
•
•
•
Shallower slope of high-mass IMF
Near Salpeter for s ~ 0.7 dex
Dispersion higher for denser cloud
Ionization
from Massive Stars
Jim Dale & Cathie Clarke
Kurosawa, Harries, Bate & Symington (2004)
Three-colour Images (24, 70, 160
mm)
Spitzer Images: 24, 70, 160 mm
Spitzer Images: 24, 70, 160 mm
Spitzer Images: 24, 70, 160 mm
Spitzer Images: 24, 70, 160 mm
Individual SEDs
•
Spectral
Classification
Classify objects into Class 0, I, II, III
– Find a mixture of classes even though all objects < 105 yr old
– Later classes tend to be ejected objects
Conclusions
•
IMF is quite robust against variations in initial conditions
–
•
In agreement with observations
Small variations should be detectable
–
Primarily in
•
•
•
Brown dwarfs
–
Denser clouds should produce more brown dwarfs
•
–
–
c.f. Taurus vs Orion Trapezium (Briceno et al. 2002)
Binary frequency ~7%
Wide BBDs can form through simultaneous ejections
•
•
Location of the peak
Slope of the substellar IMF
Needs groups of ~30-40 objects to be efficient
IMF originates from interplay between accretion/ejection
–
–
–
Stars and brown dwarfs form the same way
Brown dwarfs are those objects ejected sooner after they form
Initial turbulence seems to be unimportant
The End
Numerical Simulations of the
Formation of Brown Dwarfs
Matthew Bate, University of Exeter
Collaborators:
Ian Bonnell,
Volker Bromm,
Univ. of St Andrews
Harvard-Smithsonian CfA
Variation of the Initial Mass
Function?
•
Many models for the origin of the IMF
– Usually analytic or semi-analytic
– Many try to reproduce the observed IMF (Salpeter, log-normal, etc)
•
Use numerical simulation to understand directly origin of the IMF
– Look for variations with initial conditions
– Understand using simple models
•
We find the IMF originates from accretion/ejection
–
–
–
–
Present a simple accretion/ejection model that fits numerical simulations
Turbulence is almost irrelevant
Predicts variation with initial conditions
Neither Salpeter or log-normal IMF
SPH with Flux-limited Diffusion
1-D tests published (Whitehouse &
Bate 2004)
3-D calculations underway
Low resolution
–
5000 particles
Opacity unrealistically high
–
No disc fragmentation
Currently redoing with
–
–
Correct opacities, eos
Higher resolution
SPH-TORUS: Monte-Carlo Radiative
•
Transfer
Cohl, Bate, Harries, Kurosawa, Symington
Binary
Formation
Low resolution
–
5000 particles
Opacity unrealistically high
–
No disc fragmentation
Currently redoing with
–
–
Correct opacities, eos
Higher resolution
Outline
• Star cluster formation
– Variation of star formation with initial conditions
• Radiation hydrodynamics
– Flux-limited diffusion within SPH
– Monte-Carlo radiative transfer plus SPH
• Fisher & Klein’s Crisis
Modelling Observations
•
Kurosawa, Harries, Bate & Symington 2004
•
Monte Carlo radiative transfer
– Use the TORUS code
• Written by Tim Harries (Exeter)
• Based on algorithm by Lucy (1999)
– Interpolate SPH density onto
AMR grid (Kurosawa, Symington)
– Input physics
• Stellar models
• Dust opacities
– Calculate
• Temperatures
• Emission at various wavelengths
• SEDs
Radiation Hydrodynamics
• Computationally expensive
– Kurosawa et al (2004):
• 2400 CPU hours, 1 dump
– Bate et al. (2003):
• 740 large timesteps, 95,000 CPU hours
– 128 CPU hours per large timestep
– Radiative transfer ~20 times slower
• ~740,000 timesteps (>100 particles evolved)
– Radiative transfer ~20,000 times slower
Radiation Hydrodynamics
• Dual approach
– Combining SPH and a Monte-Carlo RT code (TORUS)
• Cohl, Bate, Kurosawa & Harries, in prep.
– Implementing flux-limited diffusion in SPH
• Whitehouse & Bate (2004): 1-D tests
• 3-D calculations now being performed
• Complementary
– Monte-Carlo
• Frequency and direction dependent radiative transfer
• Slow in optically thick regime
– Flux limited diffusion (FLD)
• Frequency independent, diffusive
• SPH/FLD: slow in optically thin regime
Better Modelling of Gas
Temperatures
• Bate et al. (2003)
– Use Masunaga & Inutsuka (2000)
• Spherically symmetric model
• Frequency dependent RT
– Barotropic equation of state
• log T vs. log r relation
• In 3-D, radiative transfer won’t give single T vs r relation
– Fragmentation enhanced or inhibited?
• Discs and filaments
SPH with Flux-limited Diffusion
•
Whitehouse & Bate (2004, MNRAS)
•
Diffusion approximation for radiative transfer:
•
Flux limiter
– 1/3 replaced by function of grad E/E
– Limits propagation in optically thin regions to < c
•
Involves second order spatial derivative
– Second order derivatives
– Noisy in SPH if done directly (e.g. Brookshaw 1985)
Cleary & Monaghan (1999)
•
Wanted to study heat conduction
du 1
   (kT )
dt r
•
Reformulated equation as first-order derivative using Taylor series
expansion (see Jubelgas et al. 2004)
– Resulting SPH equation

•
W ij
ui N m j  4ki k j 

Ti - T j )


(


t j1 ri r j ki  k j 
rij
Has same form as diffusion approximation RT
– Apply to diffusion approximation RT?
– Jubelgas et al. (2004) apply it to thermal conduction in galaxy clusters

SPH with Flux-limited Diffusion
• Equations for specific radiation energy (x=E/r) and gas
energy (u=e/r) are
4

 u  
Dx
v : P   F
rx
- ac
-   


Dt
r
r
 a c v  
4

 u  
Du
p  v
rx
 ac 
-   


Dt
r
 a c v  
where a  4s /c.
• In SPH formalism

Explicit vs. Implicit Timesteps
•
Radiation in optically thin regions travels at c
•
Hence Courant condition for radiation is usually very small
– (characteristic length over characteristic velocity)
•
An implicit timestep is used
– Uses an iterative scheme (Monaghan 1997)
• Diffusion: pairwise exchange of radiation energy done in dt/N steps
•
Speed-up of implicit code depends on the problem
– Factors like the opacity and density
• Affect the timesteps
• And the number of iterations for the implicit scheme to converge
•
Implicit scheme can be as slow or orders of magnitude faster
– Generally faster for more optically thick problem
Transition between Optically
Thick and Thin Shocks
Matter & Radiation reach
Equilibrium
SPH
ZEUS (Turner & Stone 2001)
Optically Thin Radiation Pulse
SPH
ZEUS (Turner & Stone 2001)
Supercritical Shock
SPH
ZEUS
Subcritical Shock
SPH
ZEUS
Radiation
Shock
SPH Dominated
ZEUS
Exeter Tools: SPH & TORUS
•
SPH: Smoothed Particle Hydrodynamics
– Lagrangian, particle-based method for hydrodynamics
– Originally written by Willy Benz, modified and parallelized
by Matthew Bate
•
TORUS: 3-D Radiative Transfer
–
Monte Carlo scheme to model frequency dependent radiation
properties of gas/dust systems
– Based on algorithm developed by Lucy (1999)
– Implemented by Tim Harries, Ryuichi Kurosawa and Neil
Symington
•
Solution for Radiation Hydrodynamics
– Merge Together...
SPH-TORUS: Slow Together
•
Merged: SPH-TORUS
–
–
–
–
–
–
•
Evolve fluid using SPH through several timesteps
Calling TORUS from SPH
Calculate AMR grid densities from SPH particles
Use TORUS to calculate temperatures
Map temperatures back to SPH particles
Perform new fluid timesteps
But TORUS is slow compared to SPH in high optical
depth regions
–
–
Several orders of magnitude too slow to follow
hydrodynamical evolution
Random walk process...
Monte Carlo: High Optical Depth Regions
•
•
Photon packets are followed through absorptions and scatterings from
dust
– Optically thick cells have smaller mean free paths
– Photons experience many absorptions/scatterings
Cell temperatures calculated from
T4
L
k rl

4 k rsV
1
N photons
p
where the energy absorption rate
 k rl  
photons

k rl
absorptions, scatterings 
Monte Carlo Radiative Transfer:
A Faster Feasible Method
•
For an optically thick cell
– Follow a large number of photons through a cell of given T
and the sum of rL
– Record probabilities of exiting from each face
– Calculate mean value of energy absorption rate for
photons exiting each face
– Move photon packet manually (random number)
•
Then, calculation of energy absorption rate becomes
 k rl  

photons

k rl
absorptions, scatterings 
x
photons
where
x  absorptions,scatteringsk rl
Side 3 & 6 ...
Sample Photon Paths in 3-D
Side 3
Sample Photon Paths in 3-D
Side 6
Performance of Faster Method
• Currently we assume photons exit randomly from the cube
faces
– Likely future improvement is to give probability distributions
of exit location on each face
– Will depend on results of detailed testing
• The speed up depends on how optically thick the problem
is
– For the examples to follow, typically 50-100 times
SPH-TORUS: Validity Check
• Plot ratio of temperatures using
• Standard TORUS
• TORUS with optically-thick approximation applied to cells with t>10
SPH-TORUS: Important?
• Temperature versus density of each particle
– Compared with equation of state of Bate et al. (2003)
Summary of Radiation
Hydrodynamics
• Two working radiation hydrodynamics codes
– Have developed
• Flux-limited diffusion for SPH
• An approximation for optically-thick cells for SPH with Monte-Carlo
radiative transfer
• Next steps
– Already running star formation calculations with FLD code
• Studying effect of radiative transfer on fragmentation
• Binary & multiple star formation
– Test optically-thick approximation against standard tests
• e.g. Optically-thick dust shells, circumstellar discs
• Possibly improve optically-thick approximation
• Study effects of radiative feedback on star formation
Test 1
Column Density
Temperature
Test 2
Column Density
Temperature
Effect of the EoS
Brown Dwarfs and the IMF
•
Bate, Bonnell & Bromm (2003)
– Brown dwarfs formed by ejections of protostars
from multiple systems, terminating accretion
– Binary brown dwarfs
• Low ~8% (c.f. observations ~15%)
• Most should be close (<~10 AU) to avoid disruption
– Many originate from fragmentation of discs and filaments
• May be affected by incorrect modelling of thermodynamics
•
Full IMF itself originates from dynamical interactions
– Accretion onto stars and brown dwarfs terminated when ejected
– If overestimate fragmentation, more interactions
•
Correct treatment of radiation
– Crucial for understanding the origin of the initial mass function
Cluster Star Formation: Environment
•
Bate, Bonnell & Bromm (2003) - SPH
–
–
–
•
Performed largest simulation of star formation
to date
Produced 50 stars and brown dwarfs
Determined low-mass IMF
Four calculations now finished
–
Testing how star formation depends on
•
•
•
Thermal Jeans mass
Opacity limit for fragmentation
Initial turbulent power spectrum
Velocity Dispersion
•
Stars & brown dwarfs ejected due to
•
•
•
Independent of
•
•
•
Formation in unstable multiple systems
Falling into cluster that later dissolves
Mass
Binarity
RMS velocity dispersion
•
1st calculation
•
•
2.1 km/s (3-D), 1.2 km/s (1-D)
2nd calculation
•
Higher: ~3.5 km/s (3-D)
Effect of RT on Fragmentation
•
Using current codes, investigate effects of
– Heating of collapsing gas due to release of gravitational energy
• i.e., the opacity limit for fragmentation
• SPH/FLD code
– Radiative feedback from existing stars
• e.g., fragmentation of a disc surrounding a protostar
• SPH with Monte Carlo radiative transfer
•
Both effects together can only be modelled after the further code
development
– Aim: repeat Bate et al. (2003), but with radiative transfer
• Determine how IMF is affected by radiative transfer
• Compare with observational IMF studies (e.g. Naylor)
Radiation Magnetohydrodynamics
•
Collaborating with Daniel Price on 3-D MHD/SPH
– Have begun performing MHD calculations of binary formation
– Once understood behaviour of RT & MHD codes, combine
Applications of a RMHD Code
•
Star formation calculations
•
Magnetospheric accretion
– Better time dependent models for comparison with observations
•
Evolution of protoplanets embedded in discs
– MHD provides disc effective viscosity
– Radiation transport crucial for gas accretion onto solid cores
• Currently modelled in only one-dimension
•
Neutron star/neutron star and black hole collisions
– Collaborate with Stephan Rosswog (Germany)
– Neutrino transport is ‘optically’ thick (adapt FLD)
• Currently done as post-processing, no impact on hydrodynamics
– Magnetic fields may be crucial for producing low-baryon ‘jet’ region for GRB to
escape through
Initial Conditions
•
Limited to 50 M of gas
–
–
–
•
Initial supersonic (`turbulent’) velocity field
–
–
–
–
•
3.5 million SPH particles required to resolve to opacity limit
Molecular cloud sizes 0.4-0.8 pc across
Up to ~100 stars and brown dwarfs
E.g. Ostriker, Stone & Gammie 2001
Divergence-free random Gaussian velocity field
P(k) ~ k – 4 so that s(l) ~ l1/2 (Larson 1981)
Normalised so cloud contains one turbulent Jeans mass
Three calculations
–
Standard: mean Jeans mass 1 M , Opacity limit at ~5 MJ
•
Bate, Bonnell & Bromm 2002a,b;2003
–
Denser cloud: mean Jeans mass 1/3 M , Opacity limit at ~5 MJ
–
Lower metallicity: mean Jeans mass 1 M , Opacity limit at ~15 MJ
Results - IMF
• Lower Jeans mass, more brown dwarfs
• K-S test: 98.1% probability of being different IMF
• But, mean mass only 15% lower than in original calculation
– Mean thermal Jeans mass has only 2nd-order effect on IMF
Results - IMF
•
Minimum object mass increased by factor of 3
– To approximately 0.015 M
•
Results
– Formed 34 objects
– Mean mass 1.7 times greater than original calculation (0.20 M)
Brown Dwarfs Formed Due to
Ejection
• Plot final mass versus time between formation and ejection or end
of calculation
– Brown dwarfs have substellar masses because they are ejected soon
after their formation
•
2nd calculation: Greater fraction of objects ejected quickly
– Before they are able to accrete to stellar masses
The Goal of Star Formation
Studies
•
To understand the origin of the statistical properties of stars
–
–
–
–
–
•
Initial mass function
Star formation efficiency
Fractions of single, binary, multiple systems
Properties of multiple systems
Circumstellar disc properties (planets)
Implies that we study the formation of star clusters
– Most stars thought to form in stellar clusters
• Lada et al (1993); Clarke et al. (2000)
• N~100 (Adams & Myers 2001)
– Most relevant for determining stellar properties
•
Can now perform hydrodynamical calculations of clusters
– Chapman et al. 1992
– Klessen et al. 1998; Klessen & Burkert 2000, 2001
– Bate, Bonnell & Bromm 2002a,b;2003; Bonnell, Bate & Vine 2003
Bonnell, Bate & Vine
2003
Mass:
1000 M
Diameter: 1 pc
Under resolved
M>0.1 M
Fully-Resolved Calculations
•
Capture the formation of all stars and brown dwarfs
– Resolve fragmentation down to the opacity limit
•
•
Collapsing molecular gas
–
–
–
–
–
–
•
Low & Lynden-Bell 1976; Rees 1976; Boss 1988; Bate 1998
Low-density gas isothermal ( T~10 K )
Compressional heating << radiative cooling
Collapse accelerates
Heating rate > cooling rate at ~ 10-13 g cm-3
Pressure supported core forms, size ~ 5 AU
Fragmentation inhibited
• Boss (1988), Bate (1998)
Close binaries (sep < 10 AU)
– Cannot form directly
Collapse and
Fragmentation of
a Turbulent
Molecular Cloud
Bate, Bonnell,
& Bromm 2002a,b;2003
Download from:
http://www.astro.ex.ac.uk/
people/mbate
Orion Trapezium Cluster
• Stellar densities in this calculation ~103 pc-3
• Stellar densities in
Trapezium Cluster
– Centre ~2x104 pc-3
– Fall off as (R / 0.07 pc)-2
• Hillenbrand & Hartmann 1997
• Bate et al. 1998
• Simulation similar to
– Small region within cluster
– R ~ 0.3 pc from centre
– Especially if Orion was
originally sub-clustered
VLT, ESO PR 03a0/1
Variation with Initial
Conditions ?
•
Best guess at initial conditions in local SF regions
– IMF broadly consistent with observations
•
How does star formation depend on environment ?
– Mean Jeans mass of the cloud (density, temperature)
– Opacity limit for fragmentation
– Power spectrum of the turbulence, etc
•
What determines the `characteristic (mean) stellar’ mass?
– Mean Jeans mass, Opacity limit ?
•
2nd calculation
– Mean Jeans mass decreased by factor 3 (cloud 9 times denser)
•
3rd calculation
– Opacity limit: heating begins at 9 times lower density
• Minimum mass 3 times larger
Results – Star Formation
Timescale
Star formation occurs on dynamical timescale in localised bursts
-
-
-
Dense core forms
Stars form on core’s
dynamical timescale
Gas depleted
Potential well gathers
more gas
Star formation again
Main core undergoes
two bursts ~20000 yr
separated ~20000 yr
Results – Star Formation
Efficiency
•
For each core, local SF efficiency high
– Bursts
•
Overall efficiency for cloud low
Core
Initial Gas
Mo
Final
Mo
No. Stars
No. Brown
Dwarfs
Mass
Stars/BD
SF
Efficiency
1
3.0
3.7
17
21
5.0
58%
2
0.9
1.0
3
4
0.5
32%
3
1.1
1.1
3
2
0.4
30%
Cloud
50.0
44.1
23+
27-
5.9
12%
Observation and Past Theory
• Dynamical picture of star formation
– Pringle 1989; Elmegreen 2000; Hartmann et al. 2001
• Observations:
– Star formation efficiency generally low, ~10%
– Ophiuchus ( Wilking & Lada 1983 )
• 20-30% for cloud as a whole
• Local efficiency as high as 47%
– Problematic to measure
• Time dependent
• Stars disperse
Initial Mass Function
Originates from `competitive accretion’ and dynamical ejection
- All objects begin with ~ 5 MJ , then accrete to higher masses
Form equal numbers of stars & brown dwarfs, mean mass 0.12 M
(c.f. Reid et al. 1999; Chabrier 2003)
IMF consistent with slope
–1.35, M>0.5 M
0.0, 0.005<M<0.5 M
cut-off, M<0.005 M
where Salpeter –1.35
In agreement with
observations
(e.g. Kroupa 2001;
Luhman et al. 2000)
Results – Stellar Velocity
Dispersion
Stars and brown dwarfs ejected either due to
- Formation in unstable multiple systems
- Falling together into cluster which later dissolves
RMS velocity dispersion
2.1 km/s (3-D), 1.2 km/s (1-D)
Initial gas velocity dispersion
1.2 km/s (3-D)
Both gas velocity dispersion
and ejections contribute
Independent of
- mass
- binarity
Ophiuchus Main Cloud
Greyscale
550 Moof gas
(Wilking & Lada 1983)
Fields
HST young stars
(Allen et al 2001)
Perhaps the young stars
formed in the dense
cores and were
subsequently ejected?
- 2 km/s = 0.2 pc in 105 yrs
- Proper motions?
Binaries and Multiples
• Close
Close Binary Formation
•
Opacity limit for fragmentation sets
–
–
Minimum initial binary separation of ~10 AU
2 x Jeans length when gas becomes non-isothermal
•
7 close binaries (< 10 AU) out of 50 objects
•
Close binaries form through combination of
–
Dynamical encounters and exchanges
•
–
Gas accretion
•
–
e.g. Bate (2000)
Interaction with circumbinary and circumtriple discs
•
•
see Tokovinin (2000)
Pringle (1991); Artymowicz et al. (1991)
Results published in Bate, Bonnell & Bromm (2002)
Close Binary Formation
• Opacity limit for fragmentation sets
– Minimum initial binary separation of ~10 AU
– 2 x Jeans length when gas becomes non-isothermal
• 7 close binaries (< 10 AU) out of 50 objects
• Close binaries form through combination of
– Dynamical encounters and exchanges
• see Tokovinin 2000
– Gas accretion
• e.g. Bate 2000
– Interaction with circumbinary and circumtriple discs
• Pringle 1991; Artymowicz et al. 1991
Close Binary Properties
•
Close binary frequency: 7/43 = ~16%
– Duquennoy & Mayor (1991): ~20%
•
Frequency dependent on primary mass
– ~20 brown dwarfs, only one binary brown dwarf
– 11 stars with M>0.2 Mo, 5 in close binaries
– Due to exchange interactions which eject lowest-mass
•
Preference for equal mass ratios
– All have mass ratios q>0.3
– Due to accretion (Bate 2000) + exchange interactions
•
6 of 7 close binaries have wider companions
– Mayor & Mazeh (1987); Tokovinin (1997, 2000)
Results – Protoplanetary Discs
• Star formation highly dynamic and chaotic
– Many discs truncated by encounters
– Is this realistic?
• Stars
– 6/15 (40%) systems
with discs >20 AU
radius
– 1/10 (10%) finished
• Brown dwarfs
– 4/23 (17%) total
– 1/18 (5%) finished
Orion Nebula
and
Trapezium
Cluster
VLT ANTU & ISAAC
McCaughrean et al.
2001
HST (O’Dell and McCaughrean)
Courtesy of Mark McCaughrean, AIP
Discs in the Trapezium Cluster
• HST resolves silhouette discs to ~40 AU radius
– Resolves 41 discs from ~350 stars
– Only ~10% of stars have discs >40 AU radius
• Rodmann 2002; McCaughrean & Rodmann, in prep.
• ~80% stars have IR excess indicative of discs
– Lada et al. 2000
• Implies most discs have radii < 40 AU
– Serious implications for planet formation
Results - IMF
• Lower Jeans mass, more brown dwarfs
• K-S test: 98.1% probability of being different IMF
• But, mean mass only 15% lower than in original calculation
– Mean thermal Jeans mass has only 2nd-order effect on IMF
Comparison with Observations
•
Taurus star-forming region
– Low density, typical thermal Jeans mass: few M
•
Trapezium cluster
– High density, smaller inferred Jeans mass: ~ 1 M
•
Taurus has a factor of 2 fewer
brown dwarfs
– Briceno, Luhman, et al. (2002)
•
Brown dwarfs ejected?
– But velocity dispersion independent of mass
• Sterzik & Durisen (1998); Bate, Bonnell & Bromm (2003)
•
Higher thermal Jeans mass -> fewer brown dwarfs
Dependence on Properties of the
Turbulence
•
Above calculations use P(k)~k-4
– Consistent with Larson (1981) scaling relations
•
Delgado-Donate, Clarke & Bate (2003)
– 10 small calculations with power spectra -3 and -5
– Find slightly more brown dwarfs with power spectra -3
• More small-scale structure
• More disc fragmentation
Dependence on Properties of
Turbulence
• New large-scale calculation underway using P(k)~k-6
Results - IMF
•
Initial turbulent power spectrum slope = -6
– Less small-scale structure
•
Preliminary results (formed 33 objects)
– Mean mass 1.2 times greater than original calculation
– No obvious dependence on initial turbulent spectrum
The Characteristic Stellar Mass
•
Calculations suggest the `characteristic stellar’ mass is determined
both by
– The opacity limit for fragmentation
– The properties of the cloud
• Mean thermal Jeans mass
• Perhaps turbulence
•
Dependence on opacity limit
– Expect mass function to move to higher masses at lower metallicity
• Low & Lynden-Bell (1976): Mmin  Z-1/7
– Implies clusters with [Fe/H] = -2 should have mean mass ~1.5 times higher
than local SF regions
• Consistent with globular cluster mass functions
• Paresce & De Marchi 2000; Chabrier 2003
Brown Dwarf Properties
•
Frequency of binary brown dwarfs
–
–
1st calculation: 1:18 at end of calculation (~5%)
2nd calculation: 4:45 at end of calculation (~10%)
•
•
–
•
Generally close (< ~20 AU) to avoid disruption during ejection
Discs
–
Most discs truncated to ~10-20 AU due to dynamical interactions
•
•
Includes a 1300 AU binary brown dwarf!
Does not include two VLM star (0.085 & 0.088 M) + BD systems
One 60 AU disc in 1st calculation, none >20 AU in 2nd calculation
Observations
–
Binary brown dwarf frequency: 15±7%
•
•
–
All binary brown dwarfs are close (<15 AU)
•
–
Reid et al. 2001; Close et al. 2003; Martin et al. 2003
Magnitude-limited samples
Consistent with ejection
Discs detected around brown dwarfs (e.g. Natta & Testi 2001)
•
Size distribution unknown
Winds From Massive Stars
• Ian Bonnell modelling the effect of winds from massive
stars
Winds From Massive Stars
• Winds confined by dense surrounding gas
Ionization from Massive
Stars
Dale, Clarke, Bonnell,
& Bate 2004
Required: Observations to Test
Models
•
Initial mass function
–
–
–
Low-mass and substellar IMFs in different star-forming environments
Mean cloud density/temperature
Different metallicities
•
•
–
–
•
Degree of turbulence (e.g. near galactic centre)
Detection of minimum mass due to opacity limit for fragmentation
Brown dwarfs
–
Multiplicity
•
•
–
–
•
Binary fraction, triples?
Distribution of separations (all < 15 AU separation; only 2 spectroscopic BBDs known)
Frequency as wide companions to stars
Disc sizes - most should be truncated to ~10-20 AU
Velocity dispersion
–
–
•
Extragalactic star-forming regions?
Globular clusters? (e.g. transits of tidal-capture BDs ; Bonnell et al. 2003)
Do NOT expect brown dwarfs to have higher velocity dispersion than stars
Proper motions: Are stars/brown dwarfs formed in dense cores and then ejected?
Protoplanetary discs
–
Size distribution as a function of stellar environment and stellar mass
The End
•
Conclusions
Star formation highly dynamical and complex
–
Fragmentation, competitive accretion, disc and dynamical interactions
•
Large-scale hydrodynamical calculations required to understand the origin of
statistical properties of stars
•
Star formation in turbulent clouds:
–
IMF consistent with observations
•
•
Brown dwarf frequency varies with molecular cloud density (mean Jeans mass)
Characteristic stellar mass primarily determined by opacity limit for fragmentation
–
In dense regions, most large discs truncated to < 20 AU radius : planets?
–
Lots of brown dwarfs, binary brown dwarfs rare
–
Close binaries (<10 AU)
•
•
Cannot be formed in situ (opacity limit for fragmentation)
Formed via hardening through
–
Accretion, circumbinary discs, dynamical interactions
Disc Detection from IR Colours
• Longer wavelength observations better at detecting discs
Results – Binary Formation
• Almost exclusively formed via fragmentation
• Only two star-disc captures to form binaries or multiple
systems
– Most star-disc encounters simply truncate discs
– c.f. Clarke & Pringle (1991)
Fully-Resolved Calculations
•
Capture the formation of all stars and brown dwarfs
–
Resolve fragmentation down to the opacity limit
•
•
Limited to 50 M of gas
–
–
•
Molecular cloud sizes 0.4-0.8 pc across
Up to ~100 stars and brown dwarfs
Three calculations
–
Standard: mean Jeans mass 1 M , Opacity limit at ~5 MJ
•
•
Low & Lynden-Bell 1976; Rees 1976; Boss 1988; Bate 1998
Bate, Bonnell & Bromm 2002a,b; 2003
–
Denser cloud: mean Jeans mass 1/3 M , Opacity limit at ~5 MJ
–
Lower metallicity: mean Jeans mass 1 M , Opacity limit at ~15 MJ
What determines the `characteristic (mean) stellar’ mass?
–
–
–
Mean Jeans mass
Opacity limit for fragmentation
Properties of turbulence
Opacity Limit for Fragmentation
• Collapsing molecular gas
–
–
–
–
–
–
Low-density gas behaves isothermally ( T~10 K )
Compressional heating rate << radiative cooling rate
Collapse accelerates
Heating rate > cooling rate
Occurs at ~ 10-13 g cm-3
Occurs approximately when gas is
optically-thick to IR radiation
– Pressure supported core forms,
size ~ 5 AU
– Fragmentation inhibited
• Boss (1988), Bate (1998)
• Close binaries (sep < 10 AU)
– Cannot form directly
Sizes of Protoplanetary Discs
•
Star formation highly dynamic
–
–
•
Many discs truncated by encounters
Is this realistic?
1st calculation
–
Stars
•
•
–
Brown dwarfs
•
•
•
~10% final systems with discs
>20 AU radius (1/9 systems)
Upper limit ~47%
~5% ejected BDs with >20 AU discs
Upper limit ~15%
2nd calculation
–
Disc frequency even lower due to
denser cluster
Formation of the Arches Cluster
• Isothermal calculations scale-free
• Scale to 104 M
– Mean thermal Jeans mass 10 M
• How to get a mean thermal Jeans mass of 10 M
– Cloud temperature 70 K
• Jeans mass increases by T3/2 to 18.5 M
– Increase cloud density by factor 4
• Radius decreased by factor 41/3 = 1.5 to 0.33 pc
• Publish a new paper
Bonnell, Bate & Vine
20XX
Mass:
10000 M
Diameter: 0.33 pc
Under resolved
M>1.0 M
1
10
100
Comparison with Observations
•
Taurus star-forming region
– Low density, typical thermal Jeans mass: few M
•
Trapezium cluster
– High density, smaller inferred Jeans mass: ~ 1 M
•
Taurus has a factor of 2 fewer
brown dwarfs
– Briceno, Luhman, et al. (2002)
•
Higher thermal Jeans mass -> fewer brown dwarfs
Klessen, Burkert & Bate 1998
• Ch
•
Clusters
Observed IMF
– Luhman et al. 2000: similar conclusions for IMF
– Flat or slowly increasing up to 0.6 Mo
– Salpeter above 1 Mo
•
Solar-neighbourhood
– Reid et al. 1999: roughly equal numbers of stars and BDs
•
‘Universal IMF’
– Kroupa 2001
• Salpeter slope above 0.5 Mo
• slope – 0.3 for 0.5 – 0.08 Mo
• substellar slope +0.7
– Recent cluster surveys
• Substellar slope converging to ~ +0.5
– i.e. Roughly flat in range 0.5 – 0.01 Mo
Required: Observations to Test
Models
•
Able to produce realistic clusters
–
–
•
Generate synthetic observations of clusters
Compare with SIRTF and other infrared facilities
Initial mass function
–
–
Low-mass and substellar IMFs in different star-forming environments
Detection of minimum mass due to opacity limit for fragmentation
–
Mean cloud density/temperature
•
–
Degree of turbulence
•
–
e.g. near galactic centre
e.g. near galactic centre
Different metallicities
•
•
Extragalactic star-forming regions?
Globular clusters? (e.g. transits of tidal-capture BDs ; Bonnell et al. 2003)
Results – Binaries and Multiples
•
4 multiple systems
–
–
–
–
•
1 pure binary
Quadruple
System of 7 objects
System of 11 objects
Upper limit to C.S.F.
– Companions/systems
– 20/30=67%
•
Low-mass companions
– All wide members of multiples
– c.f. Delgado-Donate et al. 2002
Results – Star Formation
Timescale
• Distribution of intervals between formation events not
Poisson
• Excess of short
intervals
– K-S test 0.7% chance
• Due to:
– Bursts
– Binary formation from
filaments
– Multiple fragmentation
of discs
Results - IMF
• Lower Jeans mass produces greater frequency of brown
dwarfs
• K-S test: only 1.9% probability of being same IMF
• Objects form with higher space-density
– Greater fraction of objects ejected from cloud before they
accrete to stellar masses
Objects in the Trapezium Cluster
HST (O’Dell and McCaughrean)
Three-colour Images (24, 70, 160
mm)
•
Kurosawa, Harries, Bate & Symington (2004)
–
–
–
Take hydrodynamical simulations
Run Monte Carlo radiative transfer code (TORUS)
Generate synthetic images, cluster SEDs
•
Edge-on discs clearly visible
•
Deeply embedded objects still visible at long wavelengths
Three-colour Synthetic
Images (1,10,100 mm)
Kurosawa & Harries: Using simulations as input to Monte Carlo radiative
transfer code (TORUS), generating synthetic images
Column Density
Synthetic Image
Three-colour Synthetic
Images (1,10,100 mm)
Column Density
Synthetic Image
Individual Wavebands
Temperature
10 mm
1 mm
100 mm
Three-colour Synthetic
Images (1,10,100 mm)
Without <10-AU discs
With <10-AU discs
Three-colour Synthetic
Images (1,10,100 mm)
Without <10-AU discs
With <10-AU discs
Individual Wavebands - No
Discs
Temperature
10 mm
1 mm
100 mm
The Goal of Star Formation
Studies
•
To understand the statistical properties of stars
–
–
–
–
–
•
Initial mass function
Star formation efficiency
Fractions of single, binary, multiple systems
Properties of multiple systems
Circumstellar disc properties (planets)
Implies formation of stellar groups or clusters
– Most stars thought to form in stellar clusters
• Lada et al (1993); Clarke et al. (2000)
• N~100 (Adams & Myers 2001)
– Most relevant for determining stellar properties
•
Few hydrodynamical calculations of clusters
– Chapman et al. 1992
– Klessen et al. 1998; Klessen & Burkert 2000, 2001
• Limited to IMF
The Next Generation of
Hydrodynamical Calculations
•
Form statistically-significant numbers of stars
– Mass: 50 Mo
– Size: Molecular cloud 0.2-0.4 pc across
– ~50-100 stars and brown dwarfs
•
Fully-resolved
– Resolves fragmentation down to the opacity limit
• Low & Lynden-Bell 1976; Rees 1976; Boss 1988; Bate 1998
– Captures the formation of all stars and brown dwarfs
– Binary systems followed down to ~1 AU separation
– Circumstellar discs resolved down to ~10 AU radius
•
Evolve over dynamical timescale of the cluster
– Final properties
Initial Conditions
•
50 Mo, uniform density, spherical cloud
– 3.5 million SPH particles required to resolve to opacity limit
•
•
Diameter 0.38 pc, 77000 AU
Temperature 10K
– Contains 50 thermal Jeans masses ( 1 Mo )
•
Free-fall time 190,000 years
•
Initial supersonic (`turbulent’) velocity field
–
–
–
–
–
–
E.g. Ostriker, Stone & Gammie 2001
Divergence-free random Gaussian velocity field
P(k) ~ k – 4
s(l) ~ l1/2 (Larson 1981)
Kolmogorov spectrum, P(k) ~ k – 11/3
Matches amplitude scaling of Burgers supersonic turbulence
associated with an ensemble of shocks
– Normalised so cloud contains one turbulent Jeans mass
– RMS Mach number = 6.4 (i.e. 1.2 km/s)
Large-scale Calculations
Bonnell, Bate & Vine
2003
Mass:
1000 Mo
Diameter: 1 pc
Under resolved
M>0.1 Mo
•
Conclusions
Mulitple system formation highly complex
– Fragmentation, accretion, disc and dynamical interactions
•
Large-scale hydrodynamical calculations required to understand the
origin of statistical properties of stars
•
Preliminary results promising
– IMF consistent with observations
• Brown dwarf frequency varies with molecular cloud density
– Close binaries (<10 AU)
• Cannot be formed in situ (opacity limit for fragmentation)
• Formed via hardening through
– Accretion, circumbinary discs, dynamical interactions
– Binary formation
• Fragmentation more important than star-disc encounters
– Twins may be explained by high-angular momentum accretion
• Mass ratios evolve towards unity
• Frequency should increase with smaller separations
• Dynamical interactions may also play an important role
Conclusions
• New era in star formation
– Predicting wide range of stellar properties for comparison with
observation
• Surprises
– Highly dynamic, chaotic process
– Most large discs truncated to < 20 AU radius : planets?
– Lots of brown dwarfs, binary brown dwarfs rare
• Hydrodynamical fragmentation of turbulent molecular cloud
indistinguishable from observations
– Challenge to observers – surveys of BDs and discs
– Do magnetic fields have an important role in star formation?
– How does feedback alter the picture?
The Next Generation of
Hydrodynamical Calculations
•
Form statistically-significant numbers of stars
– Size: Molecular cloud ~ 0.2-0.4 pc across
– Mass: 50 Mo
– 50-79 stars and brown dwarfs
•
Fully-resolved
– Resolves fragmentation down to the opacity limit
• Low & Lynden-Bell 1976; Rees 1976; Boss 1988; Bate 1998
– Captures the formation of all stars and brown dwarfs
– Binary systems followed down to ~1 AU separation
– Circumstellar discs resolved down to ~10 AU radius
•
Evolve over dynamical timescale of the cluster
– Final properties
Binary Formation in Stellar
Clusters
• Involves all of the above processes
–
–
–
–
Fragmentation
Accretion
Disc interactions
Dynamical interactions
• Need to perform large-scale high-resolution calculations
– Take into account all processes
– Learn which are the most important
– Determine the origin of the statistical properties of stars and
brown dwarfs
• IMF, properties of circumstellar discs,
frequencies of single, binary,multiple stars
The Future – Initial Conditions
• Identical calculation with lower initial Jeans mass
– Denser cloud, to finish within 6 weeks
•
•
Future – Larger Scale
Fully-resolved Orion Nebula Cluster in 8 years
Begin now with low resolution: 500 - 1000 Mo
– Resolution ~ 0.1 Mo ~200 AU
•
Include feedback processes : Ionisation, Outflows
Processes in Binary Formation
• Fragmentation
– Favoured mechanism for binary formation
– The close binary problem
• Accretion
– Effects on binary/triple properties
• Discs
• Dynamical Interactions
• Multiple Systems in Clusters
– Accretion, disc and dynamical interactions
• New and Future Calculations
Fragmentation
• Numerical calculations since late 1970s
• e.g. Boss & Bodenheimer (1979)
• Two main types of fragmentation
– Prompt fragmentation
• e.g. Boss (1986)
– Disc fragmentation
• e.g. Bonnell (1994); Whitworth et al. (1995); Burkert, Bate &
Bodenheimer (1997)
• Together can produce most observed binary systems
– Except close binaries (<10 AU)
Collapse and
Fragmentation of
a Turbulent
Molecular Cloud
Bate, Bonnell,
& Bromm 2002
Bate,
Bonnell &
Price (1995)
Burkert,
Bate,
Bodenheimer
(1997)
Accretion
• Fragmentation only 1st step in the formation of a multiple system
– May increase their masses by ~104
• Accretion
– Alters masses, mass ratios
– Alters orbital separations,
eccentricities
– Responsible for disc formation
• Artymowicz (1983); Bate (1997);
Bate & Bonnell (1997); Bate (2000)
• Mass to accrete before reaching
final masses depends on
• Initial separation
• Final stellar masses
Bate & Bonnell (1997)
Evolution of Accreting
Binaries
Bate (2000)
Uniform, Solid-body rotation
Effects of Accretion (Bate 2000)
• Mass ratios
– Long-term evolution is towards equal-mass binaries
– Close binaries (<10 AU) should be biased towards equal
masses relative to wider binaries (e.g. Mazeh et al. 1992)
– Massive binaries should have more equal masses than lowermass binaries
• Brown dwarfs
– Difficult to form brown dwarf companions to solar-type stars in
close orbits (<10 AU)
– Predict greater frequency of brown dwarfs companions
• At greater separations
• Around low-mass primaries
Accretion in Clusters
• Specific angular momentum of accreted gas likely to be low
(Bate 2001)
– Binaries motions through gas uncorrelated with gas motions
• Lead to massive binaries having unequal masses
– Close binaries still biased towards equal masses
• Specific angular momentum of gas higher relative to close binaries than
wide binaries
– Fewer circumbinary discs
Effect of Accretion on Triple
Systems
• Alters the stability of hierarchical triple systems
– Smith, Bonnell & Bate (1997)
• Depends on mass ratios of components and angular
momentum of material being accreted
– Separation of single-binary
• Decrease, destablise
• Increase, stablise
– Separation of binary
• Increase, destablise
• Decrease, stablise
The Effect of a Circumbinary Disc
• Gravitational torques transfer orbital angular momentum to
circumbinary disc
– Can remove arbitrary amounts of orbital angular momentum in
the long-term
– Pringle (1990); Artymowicz et al. (1991)
– Binary’s separation decreases
– Binary’s eccentricity grows
• Accretion from circumbinary disc equalises mass ratio (Bate
2000)
– e.g. DQ Tau, q=0.97, e=0.55, accreting circumbinary disc
• Probable formation mechanism for close (< 10 AU) ‘twins’
Binary Formation via Star-Disc
Encounters
• Fragmentation not the only way to form binaries
• Star-disc encounter (Larson 1990)
– Excess energy from unbound orbit dissipated
– Two unbound stars become bound
Hall, Clarke &
Pringle (1995)
• May be efficient in small-N groups
• Large-N groups discs simply truncated
– Clarke & Pringle (1991)
The Effects of Dynamical
Interactions on Binaries
• Break up wide binaries
• Kroupa (1995a,b,c; 1998)
• Closer systems hardened & undergo exchange interactions
–
–
–
–
Binary + single object
Usually results in binary with lowest-mass object ejected
Results in preference for equal-mass ratio binaries
Results in higher binary frequency for more massive stars
• Can dynamical interactions in a cluster broaden a separation
distribution over a few Myr ?
– Produce close binaries?
– Not significantly: Kroupa & Burkert (2001)
Binary Evolution in Clusters
Kroupa & Burkert (2001)
Example of Circumbinary Disc
Formation
Results – Brown Dwarf
Properties
•
Frequency of binary brown dwarfs < ~5%
– 1 at end of calculation, but still accreting
– Must be close (< ~10 AU) to avoid disruption
•
Frequency of brown dwarfs with large discs ~5%
– 1 of 18 ejected brown dwarfs has disc radius > 20 AU
– Most brown dwarfs have encounters < 20 AU
•
Observations
– Binary brown dwarfs exist (e.g. Reid et al. 2001: 4/20)
• Frequency unclear: magnitude-limited samples
• All binary brown dwarfs are close
– ~65% brown dwarfs have discs in Orion
• Muench et al. 2001
• But radius distribution?
Sub-fragmentation During
Second Collapse?
•
Dissociation of molecular hydrogen, T>2000 K
– Second phase of dynamic collapse
– Possibility of sub-fragmentation within ~5 AU fragment
– Minimum mass ~1 MJ
•
Boss 1989
– Fragmentation: non-axisymmetric structure grows when b>0.27
– Fragments merged due to gravitational torques
Sub-fragmentation During
Second Collapse?
• Bonnell & Bate 1994a,b
– Single object formed by second collapse
– But fragmentation of massive disc when b>0.27
Close Binary Animations
QuickTime™ and a YUV420 codec decompressor are needed to see this picture.
QuickTime™ and a YUV420 codec decompressor are needed to see this picture.
Evolution of Accreting
Binaries
Bate (2000)
Uniform, Solid-body rotation
Density~1/r, Solid-body rotation
Uniform, W~1/r
High angular momentum accretion
Effect of Accretion on Mass
Ratios
Effect of Accretion on Separation
Disc Formation
Hierarchical Fragmentation
• Hoyle (1953)
– Jeans mass
MJ T
3/ 2
r
-1/ 2
– Barotropic equation of state
–
pr
h
Tr
h -1
3
(h -1)
4
MJ  r
Jeans mass decreases during collapse if h < 4/3
– Collapse of low-density molecular gas
– Gas compressed
– Gravitational energy released, freely radiated
– Gas remains isothermal: h~ 1
– Jeans mass decreases during collapse
– Possibility of hierarchical fragmentation
Minimum Mass
• Masunaga & Inutsuka (1999, 2000)
– Does not depend on t=1
– Depends on opacity, temperature of pre-collapse gas
– Minimum around 0.006 Mo (6 MJ)
Accretion, Magnetic Fields
• Minimum mass is just that
– Fragment will be embedded in collapsing envelope
– Must avoid accreting in order to retain a low mass
• Fragments ejected by dynamical interactions
– Reipurth & Clarke (2001)
– see also Watkins et al (1998)
• Boss (2001)
–
–
–
–
Magnetic fields turn collapse into bounce
Gas cools due to adiabatic expansion, lower Jeans mass
Masses down to ~1 MJ
Field may also slow collapse, slows rate of heating
– Boss’ calculations not include magnetic tension
– See Hosking (2002)
Conclusions for Minimum Mass
• Opacity limit for fragmentation appears to be real
– No sub-fragmentation during second collapse phase
• Minimum mass in range 0.001 - 0.010 Mo (1-10 MJ)
• Best guess, perhaps 6-7 MJ
• Must avoid subsequent accretion
– Dynamic ejection from multiple systems?
Observation and Past Theory
• Observation
– Taurus-Auriga low-mass SF region
• < 2km/s (Hartmann et al. 1991)
– Orion Trapezium Cluster
• 2.2 km/s (1-D), (Jones & Walker 1988; Tian et al. 1996)
• Theory of break-up of small-N clusters
– Pure N-body (Sterzik & Durisen 1998)
– Gas+N-body (Delgado-Donate et al. 2002)
– Velocity dispersion independent of mass
– Smaller velocity dispersion for binaries
• Smaller-N systems and/or lack of dissipation
Discs Around Stars & Brown
Dwarfs
•
•
Lack of large discs around brown dwarfs
Brown dwarfs
• Fragments ejected before can accrete to stellar masses
• Ejected before can accrete gas with high specific angular momentum
• Ejection truncates disc
•
Stars
• Can remain in cloud accreting indefinitely
• More time for close encounters
• But also can replenish discs if not ejected from cloud
Minimum Encounter
Distance
Number of Objects
Stars Brown Dwarfs
Resolved Discs
Stars Brown Dwarfs
<1 AU
19
4
11
0
1-10 AU
3
6
1
0
10-100 AU
1
6
1
1
100-1000 AU
0
2
0
0
Li, Norman,
Heitsch, Mac Low
2001
Results – Brown Dwarf
Formation
•
¾ by fragmentation
of massive
circumstellar discs
•
¼ in collapsing
dense filaments
•
First Fully-resolved Largescale Star Formation
Calculation
Large-scale
–
–
–
•
Size: Molecular cloud ~ 0.4 pc across
Mass: 50 Mo
To form 50 stars and brown dwarfs
Fully-resolved
–
Resolves fragmentation down to the opacity limit
•
–
–
–
Low & Lynden-Bell 1976; Rees 1976; Boss 1988; Bate 1998
Captures the formation of all stars and brown dwarfs
Binary systems followed down to ~1 AU separation
Circumstellar discs resolved down to ~10 AU radius
•
Evolve over dynamical timescale of the cluster
•
Why do they end up with substellar masses?
–
–
–
–
–
Form in unstable multiple systems
Ejected from dense gas
Accretion stopped before stellar mass attained
Watkins et al. 1998; Reipurth & Clarke 2001
First calculation to determine frequency and properties
The Goal of Star Formation
Studies
• To understand the statistical properties of stars
–
–
–
–
–
Initial mass function
Star formation efficiency
Fractions of single, binary, multiple systems
Properties of multiple systems
Circumstellar disc properties (planets)
• Most past hydrodynamical calculations
– Limited to a few stars: binary / multiple systems
– No statistical information
• Few hydrodynamical calculations of clusters
– Chapman et al. 1992
– Klessen et al. 1998; Klessen & Burkert 2000, 2001
• Limited to IMF
The Next Generation of
Hydrodynamical Calculations
• To form statistically-significant numbers of stars
–
–
–
–
Resolve all fragmentation
Resolve binaries and multiple systems
Resolve circumstellar discs
Follow evolution until stars finish forming
• Final properties
• Implies formation of stellar groups or clusters
– Most stars thought to form in stellar clusters
• Lada et al (1993); Clarke et al. (2000)
• N~100 (Adams & Myers 2001)
– Most relevant for determining stellar properties
•
Clusters
–
–
•
Reid et al. 1999: roughly equal numbers of stars and BDs
‘Universal IMF’
–
–
•
Luhman et al. 2000: similar conclusions for IMF
Flat or slowly increasing up to 0.6 Mo, Salpeter or steeper above 1 Mo
Solar-neighbourhood
–
•
Observed IMF
Kroupa 2001
Substellar slope uncertain
Ophiuchus
–
–
30% objects could be BDs
Allen et al. (2001)
Break at ~0.5 Mo
Upper limit of +0.5 for
IMF below 0.5 Mo
(Luhman & Rieke 1999)
Influence on Stellar Properties
• Disruption of wide binaries
– e.g. Kroupa 1995
• Hardening of close binaries
• Binary formation via star-disc capture
– Larson 1990; Clarke & Pringle 1991
• Competive accretion of gas
– Initial mass function
– Zinnecker 1982; Bonnell et al. 1997, 2001a,b
• Evaporation and truncation of circumstellar discs
Previous Calculations of
Cluster Formation
• Chapman et al. 1992
– Shock-compressed layer between colliding clouds
– Found high frequency of binary and multiple systems
– Unable to follow dynamical evolution of systems
• Klessen, Burkert & Bate 1998; Klessen & Burkert 2000,
2001; Klessen 2001
– Clumpy and turbulent molecular clouds
– Form ~ 60 protostellar cores
• Li, Norman, Heitsch, Mac Low 2001
– Turbulent cloud, including magnetic fields
– Forms ~35 dense cores with disc-like structures
•
Cluster
Formation
Star clusters form in molecular clouds
• Molecular clouds
– Thermal support unimportant (Jeans unstable)
– Supersonic ‘turbulent’ motions
• Problem (Zuckerman & Evans 1974)
– Molecular gas in Galaxy / free-fall time
– Two orders of magnitude > observed star-formation rate
• Until recently, thought to be held up by
magnetohydrodynamic (MHD) turbulence
– Arons & Max (1975)
– But HD or MHD turbulence decays on crossing time
• (e.g. MacLow et al. 1998; Stone et al. 1998)
Decay of HD and MHD
Turbulence
Stone, Ostriker,
& Gammie 1998
Cluster Formation
• Alternative, star formation on crossing-time
– Observational support, e.g. Elmegreen (2000)
• High star-formation rate avoided
– Low star formation efficiency
•
•
•
•
Molecular cloud forms
A few percent of the mass converted to stars
Cloud destroyed by feedback
Time taken before gas recycled into new cloud
• Orion Trapezium Cluster
– Contains 4500 Mo in radius ~2 pc
• (Hillenbrand & Hartmann 1998)
– Free-fall time (uniform density) ~700,000 years
– Mean age of stars ~106 years
Orion Nebula
and
Trapezium
Cluster
Photo taken
with
VLT ANTU
and ISAAC
ESO PR
03a0/1
(15 Jan 2001)
Klessen, Burkert & Bate 1998
• Calculation scale-free (isothermal)
– Peak of IMF ~ mean initial Jeans mass, could be scaled
– Width of log-normal IMF independent of scaling
•
But
– Sub-fragmentation?
Li, Norman, Heitsch, Mac Low
2001
• 5123 ZEUS simulation of turbulent molecular cloud
– Includes magnetic fields
– 115000 CPU hours, on latest Pittsburgh Supercomputer
– Forms 35 dense cores with disc-like structures
Summary of Past Calculations
• Initial mass function perhaps from combination of
– cloud fragmentation
– competitive accretion
– dynamical interactions
• Formation of dense cores in magnetised clouds on
dynamical timescale
• Problems
– Klessen et al.
• No information about protostellar discs and binary stars
• Neglects sub-fragmentation
– Li et al.
• Lack of resolution (discs at ~100 AU scale)
• Cannot follow dynamical evolution of cluster
Star Clusters on UKAFF
• UK Astrophysical Fluids Facility
– 128-processor SGI Origin 3800
– Based in Leicester
• National facility
– Telescope-like application process
– Apply for number of CPU hours
• Bate, Bonnell & Bromm
– Allocated 95000 CPU hours during 2001
– 9% of UKAFF during 2001
– To perform one large star cluster formation calculation
Comparison with Star-forming
Regions
• Ophiuchus low-mass star-forming region
– 550 Mo within area of 2x1 pc (Wilking & Lada 1983)
– Contains 6 main
cores
– Simulation similar to
core
dense
modelling one
Allen et al. 2001
Past Hydrodynamic Star
Formation Calculations
• Collapse and fragmentation to form single, binary and
multiple stellar systems
–
–
–
–
Larson 1969; Boss & Bodenheimer 1979; Boss 1986
Bonnell et al. 1991; Nelson & Papaloizou 1993;
Burkert & Bodenheimer 1993; Bate, Bonnell & Price 1995
Truelove et al. 1998
• Difficult to determine statistical properties
– Do not understand initial conditions well enough to select
representative sample of initial conditions
– Neglect interactions between stellar systems
• Large-scale calculations of molecular clouds
– Extremely computationally intensive
Discs Around Other Young Stars
Particle-based Fluid Dynamics
• Smoothed Particle Hydrodynamics (SPH)
– Several variations
– Originally written by Willy Benz or Joe Monaghan
• Physical processes
– Compressible fluid dynamics
– Self-gravity
• Additional features
– Modifications by Matthew Bate
• Individual time steps for particles
• Sink particles (point masses) for condensed objects (e.g.
stars)
SPH
• Does not need computational grid
• Lagrangian method – mass associated with particles that
move under forces
• SPH particles interact with neighbours via
– Gravitational forces
– Repulsive forces
• Pressure forces (the gradient of which gives acceleration due to
pressure gradients)
• Viscous forces (required to enable shock capturing)
– Typically, a particle interacts with ~50 neighbours
• Thus, resolution is mass-limited
SPH on UKAFF
• Two SPH codes currently used on UKAFF
– Both parallelised using OpenMP
• With no previous parallel programming experience, I
TimeinStep
parallelised my SPH code
9 days
e Gravity
earest
bours
Calculate Densities
and Pressures
Calculate Forces
• OpenMP has some choice for loop parallelisation
– `Guided’ strategy works best for most SPH subroutines
Walk– Tree
`Static’ or `Dynamic’ best for ZEUS codes
Up
a
Ophiuchus Main Cloud
6 Dense cores
- Masses 8-62 Mo
- n(H2)~105-106 cm-3
- Sizes 0.05 - 0.2 pc
Motte, Andre & Neri 1998
Scientific Studies on UKAFF
Project Category
Thousands of CPU hours
allocated during 2001
Accretion Discs
115
Planet Formation
150
Stellar Collisions
260
Star Formation
180
Other
293
Total
998
The End
The Goal of Star Formation
Studies
•
To understand the statistical properties of stars
–
–
–
–
–
•
Initial mass function
Star formation efficiency
Fractions of single, binary, multiple systems
Properties of multiple systems
Circumstellar disc properties (planets)
Implies that we study formation of stellar groups or clusters
– Most stars thought to form in stellar clusters
• Lada et al (1993); Clarke et al. (2000)
• N~100 (Adams & Myers 2001)
– Most relevant for determining stellar properties
•
Can now perform hydrodynamical calculations of clusters
– Chapman et al. 1992
– Klessen et al. 1998; Klessen & Burkert 2000, 2001
– Bonnell, Bate & Vine 2003
Star Formation
• World’s largest simulations of star cluster formation
• Bate, Bonnell & Bromm, (Exeter, St Andrews, Cambridge)
The End