The abundance of brown dwarfs

Abstract

The amount of mass contained in low-mass objects is investigated anew. Instead of using a mass±luminosity relation to convert a luminosity function to a mass function, I predict the mass±luminosity relation from assumed mass functions and the luminosity functions of Jahreiss & Wielen and Gould, Bahcall & Flynn. Comparison of the resulting mass± luminosity relations with data for binary stars constrains the permissible mass functions. If the mass function is assumed to be a power law, the best-fitting slope lies either side of the critical slope, a ˆ 22, below which the mass in low-mass objects is divergent, depending on the luminosity function adopted. If these power-law mass functions are truncated at 0.001 M(, the contribution to the local density from stars lies between 0.013 and 0.10 M( pc 23 depending on the mass at which the mass function is normalized and the adopted value of a . Recent dynamical estimates of the local mass density rule out stellar mass densities above , 0:05 M( pc: Hence, power laws steeper than a ˆ 22 that extend down to 0.001 M( are allowed only if one adopts an implausible normalization of the mass function. If the mass function is generalized from a power law to a low-order polynomial in log(M), the mass in stars with M , 0:1 M( is either negligible or strongly divergent, depending on the order of the polynomial adopted. Key words: stars: luminosity function, mass function. 1 I N T R O D U C T I O N The abundance in galaxies of low-mass objects is of fundamental importance because we know that near the Sun at least half of the mass density of the Galaxy is made up of stars fainter than MV ˆ 10, and uncertainty in the expected mass-to-light ratio of the Galactic disc is dominated by uncertainty in the number of stars with MV * 16, which are extremely hard to detect despite being intrinsically numerous. The traditional way to determine the density of the lowest-mass stars is the use of a wide-field proper-motion survey to pick up faint but nearby stars, for which photometry and perhaps parallaxes can be obtained. More recently, two alternative strategies have become available: (i) searches for gravitational microlensing events, and (ii) narrow-field surveys to identify extremely red stars. Microlensing surveys detect stars through their gravitational fields rather than their radiation, so microlensing surveys should provide a powerful probe of the mass contained in low-mass stars. Unfortunately, there are two large problems. First, to obtain even the projected density of deflectors along lines of sight towards the survey stars, one must know how the deflectors are distributed along the line of sight. Secondly, to determine the mass function of the deflectors one requires a model of their kinematics. In consequence of these difficulties, there is no consensus regarding the nature of the deflectors that have caused the observed microlensing events towards either the Galactic Centre or the Magellanic Clouds. The work with the Hubble Space Telescope (HST) of Gould, Bahcall & Flynn (1997, hereafter GBF) and Gould, Flynn & Bahcall (1998) pushes the strategy of counting extremely red stars to its ultimate form, in which the limiting magnitude of the survey becomes extremely faint and the field becomes very narrow. Consequently, most objects detected are of low luminosity and rather distant , 2 kpc† (as are objects detected by microlensing surveys), and it is a non-trivial task, that involves the adoption of a large-scale model of the Galaxy, to infer the local luminosity function from the data. Despite these difficulties, the luminosity function of the Galactic disc is now well determined at MV & 13 and is usefully constrained down to MV , 19 ± see Fig. 1. This paper is concerned with the problem of converting a known luminosity function into a mass function. The conventional procedure involves the adoption of some mass±luminosity relation. The mass±luminosity relation for cool stars is complex and hard to determine either theoretically or observationally, while we expect, a priori, that the mass function is simple. Therefore, following Kroupa, Tout & Gilmore (1990) I assume plausible mass functions and use measured luminosity functions to infer mass±luminosity relations. Comparison of these inferred relations with the data for binary stars clarifies the amount of mass that may