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14 Dec 2011 the Stellar and Sub-Stellar IMF submitted to Stellar Systems and Galactic Structure, 2012 Vol. V, Planets, Stars & Stellar Systems, ed. by Gilmore, G. The stellar and sub-stellar IMF of simple and composite populations Pavel Kroupa1, Carsten Weidner2,3, Jan Pflamm-Altenburg1, Ingo Thies1, J¨org Dabringhausen1, Michael Marks1 & Thomas Maschberger4,5 1 Argelander-Institut f¨ur Astronomie (Sternwarte), Universit¨at Bonn, Auf dem H¨ugel 71, D-53121 Bonn, Germany 2 Scottish Universities Physics Alliance (SUPA), School of Physics and As- tronomy, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, UK 3 Instituto de Astrofsica de Canarias, C/ V´ıa L´actea, s/n, E38205 La Laguna (Tenerife), Spain 4 Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK 5 Institut de Plan´etologie et dAstrophysique de Grenoble, BP 53, F-38041 Grenoble C´edex 9, France arXiv:1112.3340v1 [astro-ph.CO] 14 Dec 2011 Abstract The current knowledge on the stellar IMF is documented. It is usually de- scribed as being invariant, but evidence to the contrary has emerged: it appears to become top-heavy when the star-formation rate density surpasses about 0.1 M /(yrpc3) on a pc scale and it may become increasingly bottom-heavy with increasing⊙ metallicity. It ends quite abruptly below about 0.1 M with brown dwarfs (BDs) and very low mass stars having their own IMF. The⊙ most massive star of mass mmax formed in an embedded cluster with stellar mass Mecl correlates strongly with Mecl being a result of gravitation-driven but resource- limited growth and fragmentation induced starvation. There is no convincing evidence whatsoever that massive stars do form in isolation. Massive stars form above a density threshold in embedded clusters which become saturated when mmax = mmax 150 M which appears to be the canonical physical upper mass limit of stars.∗ ≈ Super-canonical⊙ massive stars arise naturally due to stellar mergers induced by stellar-dynamical encounters in very young dense clusters. Various methods of discretising a stellar population are introduced: optimal sampling leads to a mass distribution that perfectly represents the exact form of the desired IMF and the m M relation, while random sampling results max − ecl in statistical variations of the shape of the IMF. The observed mmax Mecl correlation and the small spread of IMF power-law indices together suggest− that optimally sampling the IMF may be the more realistic description of star formation than random sampling. Composite populations on galaxy scales, which are formed from many pc scale star formatiom events, need to be described by the integrated galactic IMF. This IGIMF varies systematically in dependence of galaxy type and star formation rate, with dramatic implications for theories of galaxy formation and evolution. CONTENTS Contents 1 Introduction and Historical Overview 11 1.1 Solarneighbourhood ......................... 11 1.2 Starclusters.............................. 13 1.3 Intermediate-massandmassivestars . 14 1.4 The invariant IMF and its conflict with theory . 15 1.5 Aphilosophicalnote ......................... 17 1.6 Hypothesistesting .......................... 18 1.7 Aboutthistext............................ 19 1.8 OtherIMFreviews.......................... 20 2 Some Essentials 21 2.1 Unavoidable biases affecting IMF studies . 24 2.2 Discretising an IMF: optimal sampling and the m M relation 26 max − ecl 2.3 Discretising an IMF: random sampling and the mass-generating function ................................ 30 2.4 ApracticalnumericalformulationoftheIMF . 32 2.5 Statisticaltreatmentofthedata . 35 2.6 Binarysystems ............................ 37 3 The Maximum Stellar Mass 40 3.1 Ontheexistenceofamaximumstellarmass: . 40 3.2 The upper physical stellar mass limit . 42 3.3 The maximal stellar mass in a cluster, optimal sampling and saturatedpopulations . 43 3.3.1 Theory ............................ 44 3.3.2 Observationaldata. 47 3.3.3 Interpretation......................... 47 3.3.4 Stochasticorregulatedstarformation? . 48 3.3.5 Ahistoricalnote ....................... 49 3.4 Caveats ................................ 50 4 The Isolated Formation of Massive Stars? 50 5 The IMF of Massive Stars 55 6 The IMF of Intermediate-Mass Stars 57 7 The IMF of Low-Mass Stars (LMSs) 57 7.1 Galactic-field stars and the stellar luminosity function . 58 7.2 The stellar mass–luminosity relation . 59 7.3 Unresolved binary stars and the solar-neighbourhood IMF . 62 7.4 Starclusters.............................. 69 5 CONTENTS 8 The IMF of Very Low-Mass Stars (VLMSs) and of Brown Dwarfs (BDs) 78 8.1 BDandVLMSbinaries ....................... 79 8.2 ThenumberofBDsperstarandBDuniversality . 82 8.3 BDflavours.............................. 84 8.4 TheoriginofBDsandtheirIMF . 86 9 The Shape of the IMF from Resolved Stellar Populations 87 9.1 Thecanonical,standardoraverageIMF . 89 9.2 TheIMFofsystemsandofprimaries. 92 9.3 TheGalactic-fieldIMF........................ 92 9.4 Thealphaplot ............................ 94 9.5 The distribution of data points in the alpha-plot . 95 10ComparisonsandSomeNumbers 100 10.1 The solar-neighborhood mass density and some other numbers . 101 10.2 OtherIMFformsandcumulativefunctions . 103 11 The Origin of the IMF 107 11.1 Theoreticalnotions. 107 11.2 The IMFfromthe cloud-coremassfunction? . 112 12 Variation of the IMF 115 12.1 Trivial IMF variation through the mmax Mecl relation . 115 12.2 Variation with metallicity . .− . 116 12.3 Cosmological evidence for IMF variation . 117 12.4 Top-heavyIMFinstarburstinggas . 119 12.5 Top-heavyIMFintheGalacticcentre . 120 12.6 Top-heavyIMFin somestar-burstclusters. 121 12.7 Top-heavy IMF in some globular clusters (GCs) . 121 12.8 Top-heavyIMFinUCDs. 124 12.9 The current state of affairs concerning IMF variation with density andmetallicityandconcerningtheory . 129 13CompositeStellarPopulations-theIGIMF 134 13.1IGIMFbasics ............................. 134 13.2 IGIMF applications, predictions and observational verification . 136 14 The Universal Mass Function 143 15 Concluding Comments 143 6 LIST OF FIGURES List of Figures 1 Optimalvs.randomsampling:IMF . 28 2 Optimal vs. random sampling: mmax–Mecl ............. 29 3 Optimal vs. random sampling:m ¯ –Mecl .............. 30 4 SPPmethodtomeasureIMF .................... 36 5 The mmax Mecl relation ...................... 46 6 Aquadruplesystemejectedfromastarcluster− . 53 7 Evidence history for massive stars formed in isolation ...... 54 8 IMFslopesintheSMC,LMCandMW . 55 9 Luminosity function (LF) of solar-neighbourhood stars . 60 10 Stellar luminosity functions in star clusters . 61 11 Themass-luminosityrelationofstars. 63 12 DeviationsoftheMLRsfromempiricaldata. 64 13 LFpeakasafunctionofmetallicityI. 65 14 LFpeakasafunctionofmetallicityII.. 66 15 LF:modelvs.observationsI. 68 16 LF:modelvs.observationsII.. 71 17 DynamicalevolutionofPDMFs. 72 18 MassfunctioninthePleiades . 74 19 SystemmassfunctionintheONC . 76 20 MasssegregationintheONC . 77 21 BindingenergiesofBDandMdwarfbinaries . 81 22 MassfunctionsofBDs ........................ 85 23 BDdiscontinuity ........................... 90 24 Two-partpowerlawvs. log-normalIMF . 91 25 BDmassfunctionindependenceofbinarity . 93 26 The α plot............................... 96 27 Thedistributionofhigh-massstarIMFslopes . 98 28 ComparisonofIMFs ......................... 105 29 CumulativenumberplotsfordifferentIMFs . 108 30 CumulativemassplotsfordifferentIMFs . 109 31 IMF of massive stars (α3) as a function of cloud density . 126 32 α3 as a function of metallicity . 127 33 Variation of the IMF in dependence of metallicity . 131 34 α3 as a function of density and metallicity . 133 35 IGIMFasfunctionoftheSFR . 137 36 IGIMFandnumberofSNII . 141 37 The universal mass function of condensed structures . 144 7 LIST OF TABLES List of Tables 1 Stellarnumberandmassfractions . 102 2 Canonicalvs.SalpeterIMF . 104 3 SummaryofpublishedIMFs. 106 9 1 INTRODUCTION AND HISTORICAL OVERVIEW 1 Introduction and Historical Overview The distribution of stellar masses that form together, the initial mass function (IMF), is one of the most important astrophysical distribution functions. The determination of the IMF is a very difficult problem because stellar masses cannot be measured directly and because observations usually cannot assess all stars in a population requiring elaborate bias corrections. Indeed, the stellar IMF is not measurable (the IMF Unmeasurability Theorem on p. 23). Nevertheless, impressive advances have been achieved such that the shape of the IMF is reasonably well understood from low-mass brown dwarfs (BDs) to very massive stars. The IMF is of fundamental importance because it is a mathematical expres- sion for describing the mass-spectrum of stars born collectively in “one event”. Here, one event means a gravitationally-driven collective process of transfor- mation of the interstellar gaseous matter into stars on a spatial scale of about one pc and within about one Myr. Throughout this text such events are re- ferred to as embedded star clusters, but they need not lead to gravitationally bound long-lived open or globular clusters. Another astrophysical function of fundamental importance is the star-formation history (SFH) of a stellar system. The IMF and the SFH are connected through complex self-regulating physi- cal processes on galactic scales, whereby it can be summarised that for late-type galaxies the star-formation rate (SFR) increases with increasing galaxy mass and the deeper gravitational potential. For early-type galaxies the same is true except that the SFH was of short duration
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