3 Discussion

3.1 A large sample of very low mass stars and brown dwarfs

Our infrared spectroscopy, then, indicates that the source list contains 17 objects with $M_K \geq 11$, and 3 objects with $M_K \geq 11.5$.Alternatively, we can consider the $1.51-1.57\,\mu$m H₂O index as an estimator of spectral type, in which case Table 3 indicates that objects M8V and later will have a H₂O index $\raisebox{-0.6ex}{$\,\stackrel{\raisebox{-.2ex}{$\textstyle\gt$}}{\sim}\,$}2.5$. There are 19 such objects in our sample.

Thus the analysis of only a small fraction of the existing DENIS data has significantly increased (i.e. by $\approx\,60\%$ - Kirkpatrick 1998) the known inventory of very late dwarfs. Seven of those have spectral type of M9-M9.5V or later (based on their $\rm 1.51-1.57\,\mu m\; H_2O \gt 4$), where eight were previously known (Kirkpatrick 1998). These very late objects are DENIS-PJ0205-3357, J1228-2415, J0021-4244, J2146-2153, J1058-1548, J1228-1547 and J0205-1159.

We have obtained excellent infrared spectra for all except DENIS-PJ2146-2153 (Fig. 7) and they are very similar to those of the M9V templates. The spectrum of DENIS-PJ2146-2153 (not shown in Fig. 7) is noisier, but also consistent with dwarfs cooler than M9V. Exact typing on the current M system will require optical spectra, but all are certainly new M9-M9.5 dwarfs. At least half (DENIS-PJ0205-1159, DENIS-PJ1228-1547, J1058-1548 and J0021-4244) have photometric distances placing them inside 25pc, and are candidates for inclusion in the Catalogue of Nearby Stars.

Optical spectra have been obtained for a number of the Mini-survey sources, and are discussed in more detail by Tinney et al. (1998, hereafter TDFA), who find that optical spectral typing of those objects produce a $T_{\rm eff}$ ordering consistent with that produced by the H₂O indices described above.

The reddest three objects detected in the DENIS Mini-survey are DENIS-P J0205-1159, J1058-1548 and J1228-1547. Both their DENIS colours, and their infrared spectra indicate they are as late in type as GD165B, or later. They are discussed in more detail by Delfosse et al. (1997, hereafter D97), TDFA, Martín et al. (1997, hereafter MBDF) and Tinney et al. (1997, hereafter TDF). Briefly, the optical (TDFA,TDF) and infrared (D97) spectra of DENIS-P J1058-1548 are very similar to those of GD165B. It must have the same effective temperature, for which the current best estimate is $1900 \pm 100$K, based on dusty atmospheric models (Kirkpatrick et al. 1998). The visible and infrared spectra of DENIS-P J0205-1159 and J1228-1547 are both cooler than GD 165B. DENIS-P J0205-1159 is the coolest isolated dwarf identified to date. DENIS-P J1228-1547 has an effective temperature intermediate between those of GD 165B and J0205-1159. It has been demonstrated to be a brown dwarf by both MBDF and TDF. They have shown that lithium is abundant in the photosphere of this fully convective object, and therefore that its mass has to be lower than the lithium burning threshold of $\sim$0.06 solar masses. Taken together, the data and the available models therefore imply that DENIS-P J1228-1547 is a brown dwarf. DENIS-P J0205-1159 has a high probability of being a brown dwarf, because of its very low effective temperature. The effective temperature scale below 2000 K is however sufficiently uncertain that it could also be a very low mass star. DENIS-P J1058-1548 may also be a brown dwarf, but we consider its status - like GD 165B - more uncertain.

The TiO and VO bands which constitute the defining characteristic of the M spectral class are absent, or very weak, in the optical spectrum of these very cool dwarfs (TDF, MBDF). This is almost certainly due to depletion of heavy elements in their cool photospheres by the formation of dust grains - in particular solid VO and perovskite (CaTiO₃) (Sharp & Huebner 1990; Allard 1998). Kirkpatrick (1998) has advocated that a new spectral class should be defined to describe these spectra. Both Kirkpatrick et al. (1998) and MBDF have proposed the designation "L'' for this new spectral class. To date, five published objects can be considered L-type dwarfs: GD 165B, Kelu-1 (Ruiz et al. 1997), DENIS-P J0205-1159, J1058-1548 and J1228-1547. The optical spectrum of Gl229B shows no TiO or VO absorption (Oppenheimer et al. 1998). However, it also shows none of the molecular features seen in the remaining L-type dwarfs, so whether or not it should be considered an L-dwarf remains problematic. Though the existing sample is too small to define a detailed typing scheme, work on this is progressing (Kirkpatrick & Reid 1998).

The on-going processing of the DENIS survey should eventually provide a sample of about 300 L-dwarfs. Already the present Mini-survey has produced the largest sample of very low mass dwarfs, and its follow-up programs (astrometry and visible spectroscopy) will provide much needed observational constraints for low mass dwarf interior and atmospheric models.

3.2 The space density of GD165B-like L-dwarfs  

Any estimate of the space density of L-dwarfs will, of necessity, be very uncertain. This is due to both the large uncertainties in the colour-luminosity relations required to evaluate the distances to objects, and to the small number of objects (3) detected in an homogenous survey. On the observational side only one L-dwarf, GD 165B, has a known parallax and absolute magnitudes. This is of course insufficient to properly calibrate the necessary color-luminosity relations. On the theoretical side, the most recent tracks which consistently combine internal structure and atmospheric models (e.g. Chabrier & Baraffe 1997) still rely on non-dusty model atmospheres. The influence of dust formation on the emerging spectra of extremely cool dwarf is now clearly established (Jones & Tsuji 1997; Allard 1998) and dusty model atmospheres are needed to provide an acceptable description of the spectra of L-dwarfs.

We therefore adopt the absolute magnitude of GD 165B ($M_K=11.7\pm 0.2$,$M_J=13.3\pm 0.2$, $M_I=16.7\pm 0.2$, C. Dahn, private communication), as representive of all three DENIS L-dwarfs, in spite of their slightly different colours and spectra. Under these assumptions, the generalised 1/V$_{\rm max}$ estimator (Schmidt 1968; Felten 1976; Stobie et al. 1989) of the space density is $\Phi_1=0.02\pm 0.01\,{\rm pc}^{-3}$. As two of these object are cooler than GD165B and certainly fainter, this value is a firm lower limit on the space density.

If on the other hand we use the M_K derived from the slope of the infrared spectra in Sect. 2.4.2 ($M_K=11.5\pm 0.5$ for DENIS-P J1058-1548; $M_K=12.0\pm 0.5$for DENIS-P J1228-1547 and $M_K=12.1\pm 0.5$ DENIS-P J0205-1159) we obtain a second estimate of the space density, $\Phi_2=0.04\pm 0.02\,{\rm pc}^{-3}$. This estimate is also uncertain, as it relies on a spectral shape to luminosity calibration that extrapolates on the absolute K magnitude of GD 165B, which itself is only moderately well determined. We consider that it is more likely to overestimate the true luminosity function than to underestimate it.

In both of the above cases, however, the raw luminosity function will be systematically overestimated. This is because the considerable uncertainty in our absolute magnitudes will bias us to sample larger volumes than we expect - i.e. we have a classical Malmquist bias (Stobie et al. 1989). In this case the classical bias dominates over all other biases because, while our infrared spectra give systematically uncertain M_K values, we can be certain that no more luminous dwarfs have been scattered into our GD165B-like sample. We can derive an estimate of the magnitude of this effect using Eq. (12) of Stobie et al., assuming a locally flat luminosity function, and a scatter of 0.5 magnitude (uncertainty on our absolute magnitudes). We find that our raw luminosity function will be systematically overestimated by about a third, giving corrected luminosity function estimates of $\Phi^\prime_1=0.013\pm 0.007\,{\rm pc}^{-3}$and $\Phi^\prime_2=0.026\pm 0.014\,{\rm pc}^{-3}$.

The field star bolometric luminosity function has a maximum of $\approx 0.02\,{\rm pc}^{-3}$mag^-1 at $M_{\rm bol} \approx 10$, and then decreases to a plateau of $\sim 0.005\,{\rm pc}^{-3}$mag^-1 at $M_{\rm bol} \gt 11$(e.g. Kroupa 1995; Tinney 1992). If we assume our GD165B-like luminosity function bin covers approximately M_K=11.5-12.5 or $M_{\rm bol}=14.7-15.9$ (Tinney et al. 1993), we derive a lower bound to the bolometric luminosity function at $M_{\rm bol}=15.3$ of $0.011\pm 0.006$pc^-3mag^-1, allowing the possibility that the luminosity function may increase through the begining of the brown dwarf domain.

On-going analysis of a larger fraction of the existing DENIS data will soon provide a much larger sample of GD 165B-like brown dwarfs, and considerably reduce the present large statistical uncertainties. It will also efficiently select the handful of cool Gl229B-like brown dwarfs which will be present in the survey. The other dominant uncertainty for the luminosity function in this range is our poor knowledge of the colour-luminosity relation. We are presently addressing this problem by measuring the parallaxes of the objects discussed here, and plan a parallax follow-up of all brown dwarfs found in the full DENIS survey. This will provide both a well defined set of color-luminosity relations, and a luminosity function for L-dwarfs significantly free of the Malmquist-type biases which have plagued the study of more massive stars via photographic plates.

Analysis and follow-up of the DENIS survey data will also have important implications for VLM research, since it will be able to identify a complete distance-limited sample of nearby stars for which parallaxes can be easily measured. This will enable us to settle the nearby-star vs photometric luminosity function question once and for all by "superseding'' the photometric function with a well sampled and almost "bias-free'' nearby star luminosity function.

Acknowledgements

We warmly thank the members of the DENIS consortium whose work made these results possible. We are grateful to Isabelle Baraffe, Gilles Chabrier, France Allard, Davy Kirkpatrick, Neill Reid and Eduardo Martín for useful discussions, and to Hugh Jones and Tom Geballe for providing digital copies of their GD 165B and Gl229B spectra (respectively). We would also like to thank the technical and astronomical support staff at the AAT for their most professional assistance throughout this observing program. The DENIS project is partly funded by the European Commission through SCIENCE and Human Capital and Mobility grants. It is also supported in France by INSU, the Education Ministry and CNRS, in Germany by the Land of Baden-Würtenberg, in Spain by DGICYT, in Italy by CNR, in Austria by the Fonds zur Förderung der Wissenshaftlichen Forschung and Bundesministerium für Wissenschaft und Forschung, in Brazil by FAPESP, and in Hungary by an OTKA grant and an ESO C&EE grant. X.D. acknowledges a "Lavoisier'' grant of the French ministère des affaires étrangères.