The Astrophysical Journal, 726:27 (20pp), 2011 January 1 doi:10.1088/0004-637X/726/1/27 C 2011. The American Astronomical Society. All rights reserved. Printed in the U.S.A. AN INITIAL MASS FUNCTION FOR INDIVIDUAL STARS IN GALACTIC DISKS. I. CONSTRAINING THE SHAPE OF THE INITIAL MASS FUNCTION Antonio Parravano1, Christopher F. McKee2, and David J. Hollenbach3,4 1 Universidad de Los Andes, Centro de F´ısica Fundamental, Merida´ 5101a, Venezuela 2 Physics Department and Astronomy Department, University of California at Berkeley, Berkeley, CA 94720, USA 3 NASA Ames Research Center, MS 245-3, Moffett Field, CA 94035, USA 4 SETI Institute, 515 N. Whisman Road, Mountain View, CA 94043, USA Received 2010 September 22; accepted 2010 October 28; published 2010 December 10 ABSTRACT We derive a semi-empirical galactic initial mass function (IMF) from observational constraints. We assume that the IMF, ψ(m), is a smooth function of the stellar mass m. The mass dependence of the proposed IMF is determined by five parameters: the low-mass slope γ , the high-mass slope −Γ (taken to be −1.35), the characteristic mass mch (∼ the peak mass of the IMF), and the lower and upper limits on the mass, m and mu (taken to be 0.004 and −Γ γ +Γ 120 M, respectively): ψ(m)d ln m ∝ m {1 − exp[−(m/mch) ]}d ln m. The values of γ and mch are derived from two integral constraints: (1) the ratio of the number density of stars in the range m = 0.1–0.6 M to that in the range m = 0.6–0.8 M as inferred from the mass distribution of field stars in the local neighborhood and (2) the ratio of the number of stars in the range m = 0.08–1 M to the number of brown dwarfs in the range m = 0.03–0.08 M in young clusters. The IMF satisfying the above constraints is characterized by the parameters γ = 0.51 and mch = 0.35 M (which corresponds to a peak mass of 0.27 M ). This IMF agrees quite well with the Chabrier IMF for the entire mass range over which we have compared with data, but predicts significantly more stars with masses <0.03 M; we also compare with other IMFs in current use and give a number of important parameters implied by the IMFs. Key words: evolution – stars: formation – stars: luminosity function, mass function sys 1. INTRODUCTION 2. the system IMF, ψ (m), in which each multiple stellar system and each single star is counted as one object with The initial mass function (IMF) is a fundamental ingredient its total mass; and for modeling any system containing stars—from clusters to 3. the effective system IMF (or effective IMF for short), galaxies to the luminous universe. Since the pioneering work ψeff(m), in which unresolved binaries are counted as single by Salpeter (1955), the IMF has been derived for a variety of stars with an effective mass meff. Note that Chabrier (2003b) systems, such as clusters of different ages, field stars, the galactic adopted a different terminology: he used the term “system bulge, globular clusters, and nearby galaxies. Comprehensive IMF” for ψeff and did not consider what we define as ψsys. studies by Miller & Scalo (1979), Scalo (1986), Kroupa (2001), The relation between the effective mass and the system mass and Chabrier (2003b, 2005) have provided the community depends on how the mass is determined. For stars with known with standard IMFs that have allowed the construction of distances (such as those in clusters), the absolute magnitude innumerable models with a common basis. A comprehensive of the system is observed and the effective mass is inferred review of the current state of IMF studies has been given by from a mass–magnitude relation. For field stars that do not Bastian et al. (2010). Earlier reviews of the theoretical and have independent distance determinations, a color–magnitude observational studies of the IMF are in Scalo (1998a) and relation is used to infer the absolute magnitude of the system (1998b), respectively. It is impressive that after more than and then a mass–magnitude relation is used to infer the effective half a century, the Salpeter work is still the accepted IMF for mass. Interestingly, Gould et al. (1996) showed that the stellar intermediate and high-mass stars. At low masses, the situation mass per unit area of Galactic disk inferred from the effective has evolved much more. The ever increasing sensitivity and IMF agrees reasonably well with that from the true IMF. It resolution of observations, together with the theoretical means of should be noted that theories of the IMF, such as those of inferring the stellar masses from the observable quantities, have Padoan & Nordlund (2002; see also Padoan et al. 2007) and ∼ resulted in the discovery of a turnover in the IMF at 0.5 M of Hennebelle & Chabrier (2008, 2009) predict the system and in a progressive reduction of the errors in the low-mass IMF. IMF, whereas observations that cannot resolve close binaries Only a few years ago, the existence of a flattening or a turnover determine an effective IMF. in the IMF at subsolar masses was controversial, but now its The goal of this paper is to infer the individual-star IMF for existence is well established. However, uncertainties in the IMF stars in the disk of the Galaxy. In this paper, we shall use the grow as mass decreases below the hydrogen-burning mass limit term “star” to refer to star-like objects, whether they burn nuclear ∼ ( 0.08 M ) into the realm of brown dwarfs (BDs). fuel or not, as well as stellar remnants. Thus, BDs, white dwarfs, One of the problems that plagues attempts to determine the neutron stars, and stellar-mass black holes (which are sometimes IMF is that of unresolved binary (or multiple) star systems. One referred to as “collapsed stars”) are included in the term “star.” can distinguish three different IMFs: The terminology for objects below the hydrogen-burning limit 1. the individual-star IMF, or “true” IMF, ψ(m), in which has not been fixed yet (e.g., see Basri 2003, Chabrier et al. each star that is a member of a multiple system is counted 2005); we use the term “brown dwarfs” for all stars below the separately; hydrogen-burning limit. Evolved objects such as white dwarfs 1 The Astrophysical Journal, 726:27 (20pp), 2011 January 1 Parravano, Mckee, & Hollenbach do not enter the IMF; the IMF thus covers BDs and main- which it forms, the rest being ejected by protostellar outflows sequence (i.e., hydrogen-burning) stars. We distinguish stars and, for massive stars, by photoevaporation; the formation of from planets, which form in circumstellar disks and have an binaries, etc., is due to fragmentation of the core; and finally, the excess of heavy elements (Chabrier et al. 2007; Whitworth multiplicity of stellar systems evolves with time. In each case, et al. 2007). In principle, free-floating planets could be confused the associated physical processes could in principle leave their observationally with BDs (e.g., Liu et al. 2002); based on mark on the IMF. We assume that these effects are not large and observations of extrasolar planets, such free-floating planets are in any case they tend to be averaged out by considering the total expected to have masses less than 10 Jupiter masses. population of stars from many different star-forming regions. In addressing this problem, we assume that the IMF for stars A significant departure from simplicity would occur if there in galactic disks is both universal and simple. is more than one distinct star formation mechanism. Indeed, Universality. Although significant variations in the IMF are several mechanisms have been suggested for the formation of observed in the disk of the Galaxy (e.g., Scalo 1998a), for the very low mass stars (VLMSs) and BDs (see the review by most part, these are consistent with random sampling from a Whitworth et al. 2007). Thies & Kroupa (2007) have argued that universal IMF (Elmegreen 1997, 1999; Kroupa 2002; Bastian observations of the distribution of the distances of separation et al. 2010). Fluctuations about the mean IMF can arise in IMF of binaries support the existence of a separate mechanism for models based on deterministic chaos (Sanchez´ & Parravano the formation of BDs and VLMSs. As they point out, there 1999). Evidence for evolution of the IMF with redshift is is no reason for the star formation mechanism to depend on summarized by Elmegreen (2009), but none of this evidence the hydrogen-burning limit, so there should be a significant deals specifically with disk galaxies. Treu et al. (2010) find that mass range in which the two mechanisms are both operative. In for a sample of early-type galaxies a Salpeter IMF provides that case, the total IMF would presumably still be smooth and light-to-mass ratios more consistent with observation than does relatively simple. a Chabrier (2003b) IMF, and conclude that massive early-type Contrary to our assumption of simplicity, some observations galaxies cannot have both a universal IMF and a universal do show features in the inferred IMF; for example, the mass dark matter halo. Van Dokkum & Conroy (2010) have directly function of nearby stars measured by Reid et al. (2002) has observed a large population of low-mass stars in elliptical upticks both at m 0.1 and at m 1.0 (throughout this galaxies and conclude that the IMF in massive star-forming paper, m is the stellar mass in units of solar masses).