6. the substellar mass function

6.1 The Mass Function from Clusters

The long-term goal of searching for BDs in clusters is to discover whether there is a “universal” substellar mass function among clusters (or what variations there are, and why). In doing this, one must correct for unseen binaries (and stripping of wide binaries) and for mass segregation (and eventual evaporation) of low mass members caused by the cluster environment.

Several groups (Zapatero-Osorio 1997b,1999b; Stauffer et al 1998; Bouvier et al 1998; Hambly et al 1999) have now found a large number of Pleiades BD candidates (selected photometrically; Fig. 7), extending down to inferred masses as low as 35 jupiters (Martín et al 1998a). Having successfully applied the lithium test to this cluster to determine the luminosity of the substellar boundary, it is no longer necessary to test all these candidates for lithium. Establishment of cluster membership for objects faintward of the boundary is sufficient proof that they too are BDs. One must correct for contamination by non-members (which has been estimated from the percent of spectroscopic failures among photometric candidates). These estimates are still fairly uncertain because most of the candidates have not been fully tested. Testing can be done with proper motion, radial velocity, and perhaps Ha. Lack of lithium is excellent grounds for rejecting membership below the lithium boundary. The more of these tests used, the better. The entire cluster has not been surveyed (although this is being rectified with modern wide-field cameras). We do not expect mass segregation to have gone very far in the Pleiades, although BDs should be found preferentially nearer the periphery and will be the first objects to “evaporate” away. As always, one should correct the observed MF for the effects of binaries. Unfortunately we are still fairly ignorant of the binary fraction of these objects (see Section 5).

The substellar MF inferred from the Pleiades is gently rising. We can characterize it with the index a in the equation dN/dM=M^-a. It appears that this index has a value of about +0.5 (with uncertainty of a few tenths) for this cluster. The stellar population is well known in this cluster, and the age of all the objects is also known (this is a major advantage over field studies). The fit of the cluster sequence to models is also good (especially after using dust in models for the lowest mass objects). I therefore view this as the currently most reliable measurement of a substellar mass function. Work on several other clusters is rapidly approaching the point where substellar MFs can be checked in a variety of cluster environments (Section 3.4).

In order to reach all the way to the bottom of the MF one must study younger clusters, or star-forming regions. Of course, one never observes the MF directly, but rather the luminosity function. Theoretical models, tested against independently calibrated luminosity and mass observations, allow the conversion to the MF. See the article by Chabrier (this volume) for an assessment of the state-of-the-art. The recent work by Bejar et al (1999) on the s Ori cluster suggests that the substellar MF reaches down all the way to the deuterium-burning limit (and several other groups are coming to similar conclusions for other SFR).

6.2 The Mass Function for Binaries

The main source of BD candidates from PRV studies has been the work of Mayor et al (1997,1998). Basri & Marcy (1997) showed that the number of BD candidates was consistent with a flat or slowly rising mass function into the substellar domain. But recently Halbwachs et al (2000) used data from the Hipparcos project to lift the ambiguity of orbital inclination in many of those cases and found that half of them are definitely stellar. They show that this result is incompatible with the MF in clusters and the field: there are too few BDs. We cannot be sure of any of the PRV BD candidates at the present time; the remaining candidates must have their orbital inclinations determined. They conclude that current results are consistent with a very barren brown dwarf desert. Basri (2000) has argued that the new results are compatible with a lognormal distribution of mass ratios that peaks above 5 but clearly disfavors values above 10.

This means that binary companions (especially of solar-type stars) are not a good means of addressing the general substellar MF. They probably tell us more about the binary formation mechanism (itself a very interesting topic), than the general likelihood of forming substellar objects. A review of theories of binary formation (stellar and substellar) can be found in Bodenheimer et al (2000). The metaphorical “brown dwarf desert” should now be seen as merely a “desert island” which occurs for high mass-ratio systems. The binary formation mechanism probably cares more about the mass ratio than the absolute mass of the companion. As discussed below, when one searches for BDs in other contexts, there are verdant fields of them.

6.3 The Mass Function in the Field

Since 1997, the new NIR all-sky surveys (DENIS and 2MASS) have been uncovering nearby young BDs in the field at an increasing rate (and now the SDSS has begun to add to this tally). Close to 100 objects L stars are now known, though the surveys have not yet covered most of the sky. Not all of these are BDs, but some of them certainly are (those which show lithium or are cool enough). While this shows that BDs are not a rare class of object (the surveys reach out to less than 50 pc), the analysis of these results to yield a substellar MF is quite complicated.

The interpretation of field survey data requires two separate and difficult steps. The first is the correction of the survey for observational biases and effects. A survey with a given sensitivity will sample smaller total volumes for objects of cooler temperatures. There must also be a correction for completeness effects as a function of observed brightness in the various survey colors. One must convert observed intensities to luminosity or effective temperature. Finally binaries must be accounted for, as they both increase the numbers of objects and increase the survey volume (because they are brighter).

The second overriding problem lies in the nature of the BDs themselves. By definition, they never come onto the main sequence and so are continually fading with time. This should give rise to a deficit of objects just below the minimum main sequence (and greater numbers where typical BDs at average Galactic ages have reached). Most BDs should have cooled into methane dwarfs. Mercifully they all achieve similar radii as they age (slightly smaller than Jupiter), so the connection between effective temperature and luminosity is not too ambiguous. But there is a complete degeneracy in the relations between luminosity/temperature, mass, and age. Photometric observations, unfortunately, can only give us the first of these. Even that requires a spectral-type/temperature calibration, or the appropriate bolometric corrections and parallaxes. Spectroscopy cannot really resolve this problem (unless we get very precise at measuring gravity).

Most objects in the field will be older than 200 Myr (although there is a bias for finding younger objects which must be accounted for). This is the maximum time required for the depletion of lithium to run its course (and most objects will finish much earlier). So it will generally be true that if we see lithium in a field object, it must have a mass below 60 jupiters, and if we don't see lithium its mass must be higher. The ambiguity between stars and BDs is removed for objects cooler than the minimum main sequence temperature - they are all BDs. Thus, if one simply wants to know the ratio of VLMS to BDs (and does not demand the mass distribution), it can be found from the fraction of lithium-bearing objects cooler than spectral class M6, and the numbers of objects below the L subclass corresponding to the end of the main sequence (L3 +/-1?).

An excellent preliminary attack on the mass function has been accomplished by Reid et al (1999). They analyze the 2MASS and DENIS L star samples, carefully considering sources of observational bias. They find the mass function by modeling the luminosity function using current theory, and assume a constant star formation rate over the age of the galaxy. They do not attempt to correct for binaries. The bottom line is that the observations support a mass function with a below 2 (they suggest 1.3). This implies somewhat more BDs than the cluster result. Such a mass function means that the BDs are not a dynamically important mass constituent of the disk, and are unlikely to be major contributors to the baryonic dark matter (that would require a above 3).

The space density of BDs found by Delfosse et al (1998) and Reid et al (1999) is as high as 0.1 systems per cubic parsec. The total number of BDs could then easily exceed the total number of stars. This suggests that it is possible that our nearest neighbor may actually be a brown dwarf. If so, we have a pretty good chance of discovering it in the next decade (it would probably be an unusually bright methane dwarf). Such a discovery would certainly bring brown dwarfs to everyone’s attention! In any case, it is clear that many astronomers will be kept happy studying these fascinating objects for some time to come.