An analytic explanation of the stellar initial mass function from the theory of spatial networks Andrei Klishin* (MIT Physics) & Igor Chilingarian (SAO)

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?

Igor Chilingarian, IMF workshop STScI 6/29/15

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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