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.