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 2 Photo credit: I. Chilingarian, 2015 Pipe nebula

Interstellar medium Igor Chilingarian, IMF workshop STScI 6/29/15 3 Pipe nebula, dust extinction map Alves, Lombardi, Lada 2007 Dense core mass function

DCMF-IMF correspondence Igor Chilingarian, IMF workshop STScI 6/29/154 Alves, Lombardi, Lada 2007 ~ factor of 4 Dense core collapses… …and leaves a star and debris

Universality of the power law exponent Igor Chilingarian, IMF workshop STScI 6/29/155 Same tail slope! Bastian et al ARA&A

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? Igor Chilingarian, IMF workshop STScI 6/29/156 Lombardi et al. (2015)

Overview of previous approaches Igor Chilingarian, IMF workshop STScI 6/29/157 Numerical sampling from fractal clouds (Elmegreen 1997) Press-Schechter formalism (1974) Hennebelle & Chabrier (2008) Elmegreen 1997

Scale-free physics Igor Chilingarian, IMF workshop STScI 6/29/158 Q: Maximum is here, why? A: Threshold/Jeans mass? Q: Break is here, why? A: Change of mechanism? Q: No features here, why? A: Preferential attachment?

Competitive accretion Igor Chilingarian, IMF workshop STScI 6/29/159 Accretion is competitive Cores grow by accretion

Capital gain vs “labor” salaries Igor Chilingarian, IMF workshop STScI 6/29/1510 V. Yakovenko, J. Barkley Rosser Jr. Rev. Mod. Phys. 81, 1703 (2009) Wage labor Capital gains

Networks Igor Chilingarian, IMF workshop STScI 6/29/1511 R. Albert, A-L Barabási, Rev. Mod. Phys. 74, 47 (2002) a. Internet routers b. Movie actor collaboration c. HEP collaboration d. Neuroscience collaboration

Network science based approach  Preferential attachment  Fractality of ISM components  Master equation Igor Chilingarian, IMF workshop STScI 6/29/1512

Parcel attachment Igor Chilingarian, IMF workshop STScI 6/29/1513 Mean-field accretionParcel accretion Gravity Noise

Distance factor Igor Chilingarian, IMF workshop STScI 6/29/1514 parcel j

Gravitational acceleration field Igor Chilingarian, IMF workshop STScI 6/29/1515 Strong gravity Dominant attractor very clear Weak gravity Dominant attractor unclear Stochastic competition of forces

Basins of attraction Igor Chilingarian, IMF workshop STScI 6/29/1516

Two ISM phases Igor Chilingarian, IMF workshop STScI 6/29/1517 Turbulent bulk medium Dense cores “Sub-turbulent” medium Parcels VS

Fractal interstellar medium Igor Chilingarian, IMF workshop STScI 6/29/1518 subdense

Fractal ISM in projection Igor Chilingarian, IMF workshop STScI 6/29/1519 CO lines observations Vogelaar, Wakker 1994

Supersonic turbulence Igor Chilingarian, IMF workshop STScI 6/29/1520 Kolmogorov 1941

Is 2.33 high or what? Igor Chilingarian, IMF workshop STScI 6/29/1521 Image credit: David Wenman “Every branch carries around 13 branches 3 times smaller” ausdorff_dimension Kim, J. Kor. Phys. Soc., 46, 2 (2005)

Fractal nature of parcels Igor Chilingarian, IMF workshop STScI 6/29/1522 Diffusion-limited aggregation

Two normalizations of probability Igor Chilingarian, IMF workshop STScI 6/29/1523 parcel j dense core i I can attach to any coreAny parcel can attach to me VS

Dense core growth Igor Chilingarian, IMF workshop STScI 6/29/1524 Growth equation Linear growth Sublinear growth Choice of dense cores Choice of parcels

Igor Chilingarian, IMF workshop STScI 6/29/1525 Accretion Source function Time stepping Master equation Mass balance in a bin:

Continuous Master equation Igor Chilingarian, IMF workshop STScI 6/29/1526 Normalized source function

Master equation as a filter Igor Chilingarian, IMF workshop STScI 6/29/1527 Lognormal Normal Dirac delta??? Nonlinear norm-preserving map Same tail!

High-mass limit Igor Chilingarian, IMF workshop STScI 6/29/1528 Guaranteed power law Exponent handles

Comparison with observations Igor Chilingarian, IMF workshop STScI 6/29/1529

Bottom-heavy DCMF Igor Chilingarian, IMF workshop STScI 6/29/1530 Source function has be negative at some masses !!!

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 Igor Chilingarian, IMF workshop STScI 6/29/1531