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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
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Milky Way Igor Chilingarian, IMF workshop STScI 6/29/15 2 Photo credit: I. Chilingarian, 2015 Pipe nebula
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Interstellar medium Igor Chilingarian, IMF workshop STScI 6/29/15 3 Pipe nebula, dust extinction map Alves, Lombardi, Lada 2007 Dense core mass function
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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
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Universality of the power law exponent Igor Chilingarian, IMF workshop STScI 6/29/155 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? Igor Chilingarian, IMF workshop STScI 6/29/156 Lombardi et al. (2015)
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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
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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?
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Competitive accretion Igor Chilingarian, IMF workshop STScI 6/29/159 Accretion is competitive Cores grow by accretion
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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
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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
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Network science based approach Preferential attachment Fractality of ISM components Master equation Igor Chilingarian, IMF workshop STScI 6/29/1512
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Parcel attachment Igor Chilingarian, IMF workshop STScI 6/29/1513 Mean-field accretionParcel accretion Gravity Noise
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Distance factor Igor Chilingarian, IMF workshop STScI 6/29/1514 parcel j
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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
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Basins of attraction Igor Chilingarian, IMF workshop STScI 6/29/1516
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Two ISM phases Igor Chilingarian, IMF workshop STScI 6/29/1517 Turbulent bulk medium Dense cores “Sub-turbulent” medium Parcels VS
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Fractal interstellar medium Igor Chilingarian, IMF workshop STScI 6/29/1518 subdense
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Fractal ISM in projection Igor Chilingarian, IMF workshop STScI 6/29/1519 CO lines observations Vogelaar, Wakker 1994
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Supersonic turbulence Igor Chilingarian, IMF workshop STScI 6/29/1520 Kolmogorov 1941
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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” http://en.wikipedia.org/wiki/List_of_fractals_by_H ausdorff_dimension Kim, J. Kor. Phys. Soc., 46, 2 (2005)
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Fractal nature of parcels Igor Chilingarian, IMF workshop STScI 6/29/1522 Diffusion-limited aggregation
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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
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Dense core growth Igor Chilingarian, IMF workshop STScI 6/29/1524 Growth equation Linear growth Sublinear growth Choice of dense cores Choice of parcels
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Igor Chilingarian, IMF workshop STScI 6/29/1525 Accretion Source function Time stepping Master equation Mass balance in a bin:
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Continuous Master equation Igor Chilingarian, IMF workshop STScI 6/29/1526 Normalized source function
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Master equation as a filter Igor Chilingarian, IMF workshop STScI 6/29/1527 Lognormal Normal Dirac delta??? Nonlinear norm-preserving map Same tail!
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High-mass limit Igor Chilingarian, IMF workshop STScI 6/29/1528 Guaranteed power law Exponent handles
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Comparison with observations Igor Chilingarian, IMF workshop STScI 6/29/1529
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Bottom-heavy DCMF Igor Chilingarian, IMF workshop STScI 6/29/1530 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 Igor Chilingarian, IMF workshop STScI 6/29/1531
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