The Stellar Initial Mass FunctionLarson, Richard B. ; Yale University, New Haven, USAEmail (RBL): [email protected] current status of both the obser vational evidence and thetheory of the stellar initial mass function (IMF) is reviewed,with particular attention to the two basic, apparently uni-versal features shown by all observations of nearby stellarsystems: (1) a characteristic stellar mass of the order of onesolar mass, and (2) a power-law decline of the IMF at largemasses similar to the original Salpeter law. Considerable evi-dence and theoretical work supports the hypothesis that thecharacteristic stellar mass derives from a characteristic scaleof fragmentation in star-forming clouds which is essentiallythe Jeans scale as calculated from the typical temperatureand pressure in molecular clouds. The power-law decline ofthe IMF at large masses suggests that the most massive starsare built up by scale-free accretion or accumulation processes,and the observed formation of these stars in dense clustersand close multiple systems suggests that interactions betweendense prestellar clumps or protostars in forming clusters willplay a role. A simple model postulating successive mergers ofsubsystems in a forming cluster accompanied by the accretionof a fract ion of the residual gas by the most massive protostarduring each merger predicts an upper IMF of power-law formand reproduces the Salpeter law with a plausible assumedaccretion efficiency.1 IntroductionThe stellar initial mass function (IMF), or distribution ofmasses with which stars are formed, is the most fundamentaloutput function of the star formation process, and it controlsnearly all aspects of the evolution of stellar systems. Theimportanc e of understanding the origin of the IMF and itspossible universality has therefore been a stimulus for muchresearch on star formation, both theoretical and observational,and interest in this subject is o f long standing, going back atleast to the pioneering study by Nakano (1966) of some ofthe processes that might be responsible for determining thestellar IMF. In recent years there has been much progress inobservational studies relating to the IMF, and somewhat moremodest progress in reaching a theoretical understanding of itsorigin; here I review briefly the current status of both theobservational evidence and the theoretical ideas concerningthe origin IMF. Other recent reviews of the observations andthe theory of the IMF have been given by Scalo (1998), Clarke(1998), Larson (1998, 1999), Elmegreen (1999), and Meyer etal. (2000).2 Basic Observed Features of the Stellar IMFNumerous observational studies have been carried out to mea-sure or constrain the IMF in systems with as w ide a rangein properties as possible in order to establish whether it isuniversal or whether it varies with place or time, dependingfor example on parameters such as metallicity. The regionsthat have been studied with direct star counts so far includethe local field star population in our Galaxy and many starclusters of all ages and metallicities in both our Galaxy andthe Magel lanic Clouds. As summarized below, this large bodyof direct evidence does not yet demonstrate convincingly anyvariability of the IMF, although the uncertainties are stilllarge. Some indirect evidence based on the photometric prop-erties of more distant and exotic systems suggests that theIMF may vary in extreme circumstances, possibly being moretop-heavy in star burst s and high-redshift galaxies (Larson1998), but this indirect evidence is less secure and will notbe discussed further here.As reviewed by Miller & Scalo (1979), Scalo (1986, 1998),Kroupa (1998), and Meyer et al. (2000), the IMF derived forthe field stars in the solar neighborhood exhibits an approxi-mate power-law decline with mass above one solar mass thatis consistent with, or somewhat steeper than, the originalSalpeter (1955) law; however, below one solar mass the IMF ofthe field stars clearly flattens, showing a possible broad peakat a few tenths of a solar mass in the number of stars per unitlogarithmic mass interval. If the logarithmic slope x of theIMF is defined by dN/d log m ∝ m−x, then the slope at largemasses is x ∼ 1.5, while the slope at small masses is x ∼ 0, therange of values or uncertainty in x being about ±0.5 in eachcase. The IMF inferred for the lo ca l field stars is subject to sig-nificant uncertainty, especially in the range around one solarmass, because it depends on the assumed evolutionary historyof the local Galactic disk and on assumed stellar lifetimes. Incontrast, the IMFs of individual star clusters can be derivedwith fewer assumptions and should be more reliable, since allof the stars in each cluster have the same age and since, atleast in the youngest clusters, all of the stars ever formed arestill present and can be directly counted as a function of masswithout the need for evolutionary corrections. Much effort hastherefore gone into determining IMFs for clusters with a widerange of properties in both our Galaxy and the MagellanicClouds. As reviewed by von Hippel et al. (1996), Hunter etal. (1997), Massey (1998), and Scalo (1998), the results ofthese studies are generally consistent with the IMF inferredfor the local field stars, and the values found for the slopex of the IMF above one solar mass gene rally scatter aroundthe Salpeter value x = 1.35 (see figure 5 of Scalo 1998). Inall cases in which it has been possible to observe low-massstars, the cluster IMFs also show a flattening below one solarmass. No clear evidence has been found for any systematicdependen ce of the IMF on any property of the systems studied,and this has led to the current widely held view that the IMFis universal, at least in the local universe.Recent studies have provided more information about veryfaint sta rs and brown dwarfs, and the IMF estimated for themremains approximately flat o r shows only a moderate declineinto the brown dwarf regime, consistent with an extrapolationof the IMF of lower main sequence stars and showing no evi-dence for any abrupt truncation at low masses (Basri & Marcy1997; Mart´ın et al. 1998; Bouvier et al. 1998; Reid 1998).Another area of recent progress has been the determinationof IMFs for a number of newly formed star clusters that stillcontain many pre-main-sequence stars ; as reviewed by Meyeret al. (2000), these results again show general consistency withthe field star IMF, including a similar flattening