2. Estimates of the local mass density of brown dwarfs
2.1. Calan-ESO proper motion survey
The Calan-ESO proper motion survey has at present led to the discovery of one brown dwarf, Kelu-1 (Ruiz et al. 1997). The authors estimate that their effective search volume had a size of 23 pc3, which implies a number density of 0.04 pc-3 or a mass density of 0.0028 pc-3, if a typical mass of a brown dwarf of 0.065 is assumed. However, these numbers may underestimate the actual density. First, the survey detects only stars with proper motions exceeding 0.25 arcsec/yr, so that some stars with low space velocities might not be found. Furthermore, the plate limits imply that even among the very nearby brown dwarfs such as Kelu-1 only the bright brown dwarfs will be detected. Ruiz et al. (1997) and Tinney (1998) argue that these objects have ages less than one Gyr. In order to correct for both selection effects we assume that the distribution function of brown dwarfs in phase space is the same as that of the luminous stars. The velocity distribution is then described by Schwarzschild distributions, i.e. three-dimensional Gaussians with anisotropic axial ratios. As is well known from nearby stars (Wielen 1977), the velocity dispersions of the Schwarzschild distributions increase with the ages of the stars. We assume the same behaviour for the brown dwarfs, because the dynamical evolution of a population of brown dwarfs is expected to be the same as for luminous stars. The velocity distribution of young brown dwarfs with ages less than 1 Gyr is then modelled as
In Eq. (1) denote the space velocity components of the brown dwarfs in the direction of the galactic center, the direction of galactic rotation and towards the galactic north pole, respectively. are the corresponding velocity dispersions. They are assumed to increase with age as
like the velocity dispersions of the luminous stars (Wielen 1977). The axial ratios of the velocity ellipsoid have been chosen according to the most recent discussion of the kinematics of nearby stars by Jahreiß & Wielen (1997) as . The diffusion coefficients follow the same ratios and have an absolute value of = 6 10-7 (km/s)2/yr. Initial velocity dispersions of = 100 (km/s)2 have been assumed (Wielen 1977, Jahreiß & Wielen 1997). For the solar motion with respect to the local standard of rest standard values of () = (9, 12, 7) km/s have been adopted. The velocity distribution f is calculated by a superposition of Schwarzschild distributions for each generation of stars weighted by the star formation rate , defined per surface density, which we have assumed to be constant with respect to time. Stars with higher vertical velocity dispersions settle with larger vertical scale heights in the galactic gravitational potential than stars with low vertical velocities , so that there will be a thinning-out of the density of such stars at the galactic midplane. This is modelled in Eq. (1) by an extra term. takes care of all the normalization constants.
The expected number density of brown dwarfs in a cone towards the direction of Kelu-1 (l=, b=) is given by
where is the radial length of the probing volume investigated by Ruiz et al. (1997). The observed number density of brown dwarfs in the cone, , can be expressed similarly to Eq. (3), if the proper motion limit of 0.25 arcsec/yr of the Calan-ESO poper motion survey is taken into account in the integration over the velocity components. To implement this proper motion limit one has to transform the cartesian velocity components , and W in Eq. (3) to radial and tangential velocities and from that to proper motions. The integrand in Eq. (3) depends then explicitly on the radial distance r. We set the upper boundary of the r-integration at = 10 pc. This is based on the observation that Kelu-1 is very similar to DENIS-P J1228.2-1547 (Tinney et al. 1997), implying an absolute magnitude of = 19.5 mag, which is about the plate limit of the Calan-ESO survey (Ruiz et al. 1997). 1 Integrating Eq. (3) and its modification numerically we find a ratio of
which changes to 0.28 or 0.44, if is increased or decreased by a factor of 1.6 according to an estimated uncertainty of the distance modulus of 1 mag, respectively. Up to now we have considered only bright and young brown dwarfs. Very likely there are also fainter and older brown dwarfs, which will contribute to the mass budget as well, although they have not yet been found in the surveys. Assuming again a constant star formation rate and an age of the galactic disk of 10 Gyrs we estimate from Eq. (1)
Combining this with the estimate (4) we conclude that the contribution of brown dwarfs to the local mass density is 10 times the density of brown dwarfs deduced straightforward from the Calan-ESO survey. The same correction factor is given by Ruiz et al. (1997), although the authors do not explain in detail how they arrive at that estimate.
2.2. DENIS mini survey
The DENIS mini survey with a field size of 230 deg2 has led to the discovery of three brown dwarf candidates (Delfosse et al. 1977). High resolution spectroscopy by Tinney et al. (1997) has clearly confirmed the brown dwarf nature of DENIS-P J1228.2-1547 by detecting a Li absorption line. However, Tinney et al. (1997) conclude that the other two objects must be also very cool low-mass objects. If we adopt the absolute K magnitude of DENIS-P J1228.2-1547 and the plate limit in the K band given by Delfosse et al. (1997), we find that the search volume of the DENIS mini survey for brown dwarfs is 162 pc3. This implies a local number density of brown dwarfs of 0.019 pc-3. In order to account for the fainter and older brown dwarfs, which have not yet been detected by the DENIS survey, the correction factor given in Eq. (5) has to be applied.
2.3. BRI survey
High resolution spectroscopy of very late type stars of the BRI survey (Irwin et al. 1991) by Tinney (1998) has led to the identification of the brown dwarf LP 944-20 = BRI 0337-3535. Tinney (1996, 1998) has also measured the parallax, proper motion and radial velocity of this brown dwarf. Its age is estimated as about 500 Myrs (Tinney 1998). The low space velocity of 7 km/s is consistent with such a young age (Wielen 1977). The BRI survey covers 1000 deg2 and its plate limit is at 19.0 mag in the R band. In the NLTT catalogue (Luyten 1979) the apparent R magnitude of LP 944-20 is given as R = 17.5 mag. This implies that the search volume of the BRI survey for brown dwarfs is 99 pc3. The local number density of brown dwarfs is then according to this determination 0.01 pc-3. Since LP 944-20 is younger than the brown dwarfs found in the other surveys (Tinney 1998), the correction factor for taking into account the fainter and older brown dwarfs is 6.8 (cf. Eq. (5)).
2.4. UK Schmidt survey
Spectroscopy of very red stars of the UK Schmidt survey of ESO/SERC field 287 (Hawkins et al. 1998) has led to the identification of three brown dwarf candidates at distances between 37 and 48 pc. The plate limit is at 23.1 mag in the R band, and the field area is 25 deg2, although crowding reduces the effective area of the survey by as much as a factor of 4. Using the apparent R magnitudes and the parallaxes given by Hawkins et al. (1998) we estimate that the survey is complete up to a distance of 50 pc. This implies an effective search volume of 80 pc3 and the local number density of brown dwarfs is then 0.038 pc-3. Hawkins et al. (1998) assume that the ages of the brown dwarfs, which they have found, are about one Gyr. Thus the correction factor for the older brown dwarfs given in Eq. (5) has to be applied.
© European Southern Observatory (ESO) 1998
Online publication: October 21, 1998