Numerical Study of the IMF - Paolo Padoan Troels Haugbølle and Åke Nordlund
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Numerical Study of the IMF
Paolo Padoan
(ICREA - Institute of Cosmos Sciences - University of Barcelona)
Troels Haugbølle and Åke Nordlund
(STARPLAN and Niels Bohr Institute - University of Copenhagen)
Friday, November 30, 12
The Effect of Turbulence
If u >> cS and L >> λJ --> Turbulent Fragmentation:
• Velocity fields fragment the medium “independently” of gravity
• The strongest density peaks are then taken over by gravity
• We can use the universal statistical properties of the turbulence for a
statistical model of the SFR and the stellar IMF
Friday, November 30, 12
density
P enhancements,
= MP BE =
+ 1.182
P but
= Pfar from(1 +! clear
M 2MBE,t
) that ⌘ turbulent
1.182 pressure can be used (11)
The universality
0 th,0 of
dyn,0
as a source of internal support G PIMF
the
3/2 1/2
th,0
to peak
define a from
critical turbulent
mass G 3/2
for P fragmentation
1/2
collapse).
0
2 th,0
P0 = Pth,0 + Pdyn,0 = Pth,0 (1 + M )
The characteristic
Substituting
If turbulence mass
intocan
is present, we be
thehave derived
previous as the
expression
to include “turbulent” , weBonnor-Ebert
of MBEpressure
its dynamic get in a modified
the totalmass:
turbu-
ex-
ternallent
Substituting BE
pressure,mass:
into and
the if the turbulence
previous expression4 is M
of highly
BE , wesupersonic,
get a its
modified dynamic
4 turbu- pressure
th th
is dominant,
nt BE mass: M BE = 1.182 1/2
! M BE,t ⌘ 1.182 1/2
(11)
3/2
G MBE,0 Pth,0(1 + M ) 2 1/2 3/2
G MP0 1
MBE,t = ⇡ MBE,0 (12)
2 1/2 1
P =This M
If0turbulence + is
Pth,0modified
BE,t = M
present,
Pdyn,0 =
BE,0 P(1
we+ M
have
(1 + )to
M 2
)⇡
include
⇡ M P its
BE,0 MM 2
dynamic ) pressure
M (12)
in
⇡ the
M total
M 1
ex-
turbulent
th,0 BE mass is the characteristic mass
th,0 BE,t of turbulent
BE,0 frag-
ternal pressure,
mentation, and if
and predicts the turbulence is highly supersonic, its dynamic pressure
This modified turbulent BE massquite is theaccurately
characteristic themassIMFofpeak.turbulent frag-
So, even if the functional
is dominant,
Substituting into form
the of the IMFexpression
previous were universal of (for
M BEexample
, we universal
get a Salpeter’
modified s slope),
turbu-
mentation, and predicts quite accurately the IMF peak.
the
lentIMF
BEpeakmass:MUST be sensitive to the environment. But under average Larson relations, the IMF
peak is roughly
P0 = Pconstant,
th,0 + Pso no
dyn,0 obvious
= P th,0IMF
(1 +variations
M 2
) in our neighborhood:
2 1/2 1/2 1
MBE,0 ⇠ ⇢ BE,t M1/2 = M
1/2
⇠ L BE,0 (1 +
and M ⇠ v ⇠ L M ) ⇡ M ) M
BE,0MBE,t = const ⇥ T0(12) (13)
1/2
Substituting1/2 into the previous 1/2 expression of M= , we get a modified turbu-
MMore ⇠ ⇠
BE,0 generally, let’s justand
⇢ L M ⇠ ⇠ L ) M BEconst ⇥ T
assume that stars are formed in clusters whose parent gas clouds have rms
v BE,t 0 (13)
This
lent modified
Describe
BE mass: turbulent
density BE mass is the characteristic mass of turbulent frag-
dependence
velocity of order virial (so we do not hide the normalization of the Larson relations):
mentation, and predicts quite 4accurately the IMF peak.
1
) M BE,t = th const ⇥ T 0 N (Padoan
2 et al. 2007) (14) 1
1/2
MBE,t ⇡ 1.182 1/2
= MBE,0 (1 + M )
col ⇡ MBE,0 M (12)
G3/2 P0
Perhaps SF in the main halo of an early-type galaxy 2 can be viewed as a single turbulent
1/2 1/2 2 1/2
M
systemBE,0 This
⇠ modified
⇢ ⇠ L turbulent
and M BE ⇠ mass
(huge turbulence outer scale), hence vlarge column ⇠ isL the characteristic
) MBE,t
density, hence=mass
lowerof
const ⇥ turbulent
IMFT0peak?
(13)
fragmentation, and predicts quite accurately the IMF peak.
Friday, November 30, 12
General conditions to study the IMF numerically
1. Good AMR code (unless you can do MHD turbulence some other ways, e.g. with SPH)
2. Sink particles with large creation density, ρsink > 14 max(ρturb), to avoid artificial sink
creation. At Mach=40 (pre-gravity MHD) we can reach 3e5 , so ρsink~ 1e7.
3. Well resolved turbulence, from large enough root grid and/or shear refinement criterion
(to be quantified soon).
4. Huge dynamic range:
a. We must resolve both the IMF peak (possibly BDs) and measure the Salpeter slope,
while keeping the SFE realistically low.
b. We must create clusters with realistic initial and boundary conditions, so they must
form out of a much larger region.
Friday, November 30, 12
Most massive region simulated to date has M=1e4 M⊙: Full IMF with SFE=0.15
Bonnell et al. 2010 (mass resolution 0.017 M⊙):
Problem: IMF peak 0.08 M⊙. Including winds and outflows should be 0.04 M⊙
Next most massive regions simulated to date have M=1000 M⊙:
They barely reach 10 M⊙ stars with SFE~20%, and barely resolve the IMF peak.
Bonnell et al. 2006 (mass resolution 0.15 M⊙):
SFE=0.33 SFE=0.32 SFE=0.20
Krumholz et al. 2012 (mass resolution 0.05 M⊙):
Friday, November 30, 12
Bate 2012, 500 M⊙, mass resolution 0.001 M⊙, SFE=0.18:
• Too small total mass to constrain Salpeter’s slope
• Too short evolution (0.1 Myr of star formation) to get the IMF peak settled (only 0.05 M⊙ for fully formed stars!)
• Unphysical initial conditions (random initial velocity not correlated with density, because there is no initial
turbulence-developing phase prior to gravity)
• Unphysical boundary conditions (isolated system, no accretion).
• No magnetic fields
Krumholz et al. 2012: they have a pre-gravity turbulence-developing phase, but SFR still huge (as it should be
without B field).
They actually run for only 0.02 Myr, so they must have an even larger ambiguity in the IMF peak, between
accreting stars and fully formed ones.
Friday, November 30, 12
Krumholz et al. 2012: much too large SFR in HD even if you
• develop the initial turbulence
• account for radiation
• include outflows
Padoan et al. 2011,2012: you need B field (and turbulence driving) for realistic SFR.
Including radiation or B fields has the same effect? Not even close!
The SFR is quite wrong without B fields, so why should the IMF be right?
OUR APPROACH:
• BIG volume, 20-60 pc, with 6.e4-2.e5 M⊙ (depending on Larson relations), for realistic boundary and initial
conditions for clusters
• Run for a long time, approaching 1 Myr, so no IMF ambiguity between accreting and fully formed stars
• Realistic B field, so realistic SFR
• Develop the turbulence for many dynamical times pre-gravity, for self-consistent ICs
• Continue to drive after including self-gravity (even GMCs are not isolated)
• BIG dynamic range 32-128 times larger than Krumholz et al. 2012 in linear size (or 5-7 extra AMR levels),
so comparable spatial resolution, 30-90 AU
• We form over 2,000 stars with realistic SFE~0.04, from BDs to 100 M⊙, so huge sample for the IMF.
Friday, November 30, 12
AMR Simulations of Star Formation
(Vast convergence study, several Mhr on NASA/Ames Pleiades supercomputer)
• Periodic boundary conditions
• Random, large-scale force
• Uniform initial magnetic and density fields
• Random initial velocity field
• Isothermal equation of state
• Root grid: 2563-1,0243
• AMR levels: 9-11
• Sink particles, ρsink~ 1e7
AMR code:
Ramses, with a hybrid layout, using
OpenMP within individual nodes,
and MPI between nodes.
Friday, November 30, 12
Ramses AMR simulation of star formation - 2563 + 9 AMR levels
0.2 pc
Maximum resolution
equivalent to 131,0723:
Range of length scales:
107 AU -- 80 AU
4pc
Range of time scales:
106 yr -- 1 yr
We simulate a full GMC for
~1 Myr, forming over 2,000
stars and brown dwarfs,
with SFE~0.04 20pc
Friday, November 30, 12The birth of 2,000 stars, from 0.01 to 100 M⊙
(Full box)
Friday, November 30, 12The birth of 2,000 stars, from 0.01 to 100 M⊙
(Main cluster)
Friday, November 30, 12Numerical convergence versus root-grid size:
Numerical convergence versus AMR levels:
Friday, November 30, 12The Next Challenge Resolve the formation of individual stars within a galactic fountain volume, for realistic initial and boundary conditions for whole GMCs Friday, November 30, 12
What I don’t know (yet)
Effect of realistic initial and boundary conditions of whole GMCs, not
just of clusters inside GMCs
Convergence with respect to various numerical parameters still
underway
Convergence with respect to magnetic field strength
Convergence with both B and radiation
Some more work is needed, but we are close to constrain (and
understand) the IMF numerically.
Friday, November 30, 12Friday, November 30, 12
Friday, November 30, 12
The Cosmic History of the IMF and its M/L
z~4
z~2➞0
z~10
Friday, November 30, 12You can also read
