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An analytic explanation of the stellar initial mass function from the theory of spatial networks
Andrei Klishin* (MIT Physics) & Igor Chilingarian (SAO)* University of Michigan from Sep/2015
Milky Way
Igor Chilingarian, IMF workshop STScI 6/29/15
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Photo credit: I. Chilingarian, 2015
Pipe nebula
Interstellar medium
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Pipe nebula, dust extinction mapAlves, Lombardi, Lada 2007
Dense core mass function
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DCMF-IMF correspondence
Alves, Lombardi, Lada 2007
~ factor of 4
Dense core collapses…
…and leaves a star and debris
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Universality of the power law exponent
Same tail slope!
Bastian et al. 2010 ARA&A 48 339
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Open questions Why does the IMF have power-law tail? Why is the tail exponent universal while ISM
density distributions differ among star-forming regions? Lo
mbard
i et a
l. (20
15)
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Overview of previous approaches
• Numerical sampling from fractal clouds(Elmegreen 1997)
• Press-Schechter formalism (1974)• Hennebelle & Chabrier (2008)
Elmegreen 1997
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Scale-free physics
𝑑𝑁/𝑑
log𝑀
log𝑀
Q: Maximum is here, why?A: Threshold/Jeans mass?
log𝑀
𝑑𝑁/𝑑
log𝑀
Q: Break is here, why?A: Change of mechanism?
𝑑𝑁/𝑑
log𝑀
log𝑀
Q: No features here, why?A: Preferential attachment?
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Competitive accretion
Accretion is competitive
Cores grow by accretion
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Capital gain vs “labor” salaries
V. Yakovenko, J. Barkley Rosser Jr.Rev. Mod. Phys. 81, 1703 (2009)
Wage labor
Capital gains
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Networks
R. Albert, A-L Barabási, Rev. Mod. Phys. 74, 47 (2002)
a. Internet routersb. Movie actor collaborationc. HEP collaborationd. Neuroscience collaboration
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Network science based approach
Preferential attachment
Fractality of ISM components
Master equation
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Parcel attachment
Mean-field accretion Parcel accretion
Gravity
Noise
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Distance factor
parcel j
for Newtonian gravity
probability force
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Gravitational acceleration field
Strong gravityDominant attractor very clear
Weak gravityDominant attractor unclear
Stochastic competition of forces
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Basins of attraction
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Two ISM phases
Turbulent bulk mediumDense cores
“Sub-turbulent” mediumParcels
VS
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Fractal interstellar medium
𝑅
subdense
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Fractal ISM in projection
CO lines observationsVogelaar, Wakker 1994
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Supersonic turbulence
Kolmogorov 1941
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Is 2.33 high or what?
Image credit: David Wenman
“Every branch carries around 13 branches 3 times smaller”http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
Kim, J. Kor. Phys. Soc., 46, 2 (2005)
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Fractal nature of parcels
Diffusion-limited aggregation
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Two normalizations of probability
parcel jdense core i
I can attach to any core Any parcel can attach to meVS
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Dense core growth
Growth equation
Linear growth
Sublinear growth
Choice of dense cores
Choice of parcels
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𝑚
𝑝 (𝑚 , 𝑡) dense cores total
Accretion Source function
Time stepping
Master equation
Mass balance in a bin:
steady state
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Continuous Master equation
𝑚
𝑝 (𝑚 , 𝑡)
𝑚
𝑝 (𝑚 , 𝑡)
Normalizedsource functionGrowth exponent
regulates accretion speed
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Master equation as a filter
Lognormal Normal
Dirac delta ???
Nonlinear norm-preserving map
Same tail!
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High-mass limit
Guaranteed power law
Exponent handles
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Comparison with observations
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Bottom-heavy DCMF
Source function has be negative at some masses !!!
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Conclusions We obtained a fully analytical theory for the
DCMF shape Power law shape and exponent do not depend
on the source function (initial conditions or PDF)
Kroupa’s broken power law shape is acceptable as a fitting approximation of a smooth low-mass cutoff
Bottom-heavy IMF with the low-mass segment steeper than the high-mass one is ruled out
