A VOLUME-LIMITED SAMPLE OF 63 M7–M9.5 DWARFS. I. SPACE MOTION, KINEMATIC AGE, AND LITHIUM

We constructed a sample of known M dwarfs of spectral type M7–M9.5 that is almost volume-complete (d < 20 pc). The targets are taken from several catalogs and discoveries from the Deep Near Infrared Survey of the Southern Sky (DENIS) and the Two Micron All Sky Survey (2MASS) surveys.

Cruz et al. (2007) present a volume-limited sample from 2MASS that contains objects in the spectral range M7–L8. Distances are derived from spectrophotometry with M_J estimated from the spectral-type/M_J calibration from Cruz et al. (2003). This M_J is combined with photometry from the 2MASS Second Incremental Data Release Point Source Catalog (PSC) to yield $M_{K_s}$ and spectrophotometric distances. Uncertainties in distance are dominated by the uncertainties in spectral type (see Cruz et al. 2003). This sample covers about 40% of the sky. The sample taken from Cruz et al. (2007) was carefully constructed to yield a sample that is complete and volume-limited with d < 20 pc in the spectral range we consider. Nevertheless, Cruz et al. (2007) note that the sample may miss several objects at M7 due to the (J − K_s)>1.0 selection criterion.

Crifo et al. (2005) and Phan-Bao & Bessel (2006) present two parts of a sample from DENIS that covers most of the late-M dwarfs up to spectral type M8.5. Two later objects were discovered in Delfosse et al. (2001) and Phan-Bao et al. (2006), which make this sample almost complete up to M9.5 while covering ∼13% of the sky.

For 15 targets, we found parallax measurements in the literature, which we use instead of the spectrophotometric distances where available. One target, 2MASS J0041353−562112, was recently reported to show evidence of accretion, which implies that it is very young (∼10 Myr; Reiners 2009). At this age, the object is much brighter affecting the spectrophotometric distance, and we use the distance reported by Reiners (2009). Taking into account parallax and age information, three of the 63 targets are found to be located at a distance d>20 pc. We did not exclude these targets from the following investigation.

We constructed a joint sample of objects within 20 pc from the two surveys. Both surveys together cover more than 50% of the sky and can be considered as almost complete in the spectral range M7–M9.5. The full "20pc–2MASS–DENIS" survey contains 63 objects, four of which are found in both the 2MASS and the DENIS surveys. The constructed survey probably misses a few M7 and a few very late-M dwarfs, and at least three of the targets are likely to be located farther away than 20 pc. Nevertheless, the constructed sample probably provides a representative and robust picture of late-M dwarfs in the solar neighborhood. We show the distribution of spectral types in the sample in Figure 1. The sample targets are given with their apparent J-band magnitude and spectrophotometric distance in Table 1.

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Our sample contains four binaries that we were previously aware of; LHS 1070B (2MASS J0024442−270825B) has a companion (Leinert et al. 1994) that is about 02 away at the time of observation (Seifahrt et al. 2008). LP 349-25A (2MASS J0027559+221932A) was found to be a binary by Forveille et al. (2005), and companions to 2MASS J1124048+380805 and 2MASS J2206227−204706 were discovered by Close et al. (2003). In all four systems, the spectra are dominated by the brighter component so that we carried out our analysis in the same way as for single stars.

During our analysis, we discovered that one of our targets, LP 775-31 (2MASS J0435151−160657), shows a double-peaked cross-correlation profile indicative of a double-lined spectroscopic binary (SB2), i.e., it is a binary consisting of two nearly equally bright components.

Observations for our 63 sample targets were collected using the High-Resolution Echelle Spectrometer (HIRES) spectrograph at Keck observatory for targets in the northern hemisphere, and using the Ultraviolet and Visual Echelle Spectrograph (UVES) at the Very Large Telescope at Paranal observatory for targets in the southern hemisphere (PIDs 080.D-0140 and 081.D-0190). HIRES data were taken with a 115 slit providing a resolving power of R = 31, 000. The three HIRES CCDs cover the spectral range from 570 to 1000 nm in one exposure. UVES observations were carried out using a setup centered at 830 nm covering the spectral range 640–1020 nm. All data provide the Hα line as well as the molecular absorption bands of molecular FeH around 1 μm that are particularly useful for the determination of rotation and magnetic fields in M dwarfs. These will be investigated in a second paper.

Data reduction followed standard procedures including bias subtraction, two-dimensional flat-fielding, and wavelength calibration using ThAr frames. HIRES data were reduced using the MIDAS echelle environment. For the UVES frames, we used the ESO CPL pipeline, version 4.3.0. There are two important differences to the standard pipeline products. First, the version of the pipeline used in Period 81 produced wavelengths that are offset from the correct solution by a constant factor of 1 Å, this bug has been fixed in version 4.3.0. Second, the standard pipeline extracts the spectra before flat-fielding, which is not appropriate for spectra with strong fringing patterns. The new pipeline version allows to flat-field the original two-dimensional frames so that fringe patterns are effectively removed.

The apparent brightness of our targets are in the range J = 8.9–13.4 mag. Exposure times between 200 s and 3 hr yielded signal-to-noise ratios between 20 and 100 at 1 μm in all objects.

Our spectra cover the 6708 Å Li line that can be used for the "lithium-test" in low-mass objects (e.g., Magazzu et al. 1991; Basri 2000). In our spectra, the detection limit of the Li line is around 0.5 Å. While stars quickly deplete Li on a timescale of 100 Myr and less, brown dwarfs need longer for this process or do not entirely deplete Li at all. We show evolutionary tracks from D'Antona & Mazzitelli (1997) in Figure 2. The dashed blue line marks the region where Li is depleted by 99%, i.e., objects with detected Li lie on the left hand side of this line. For the temperatures of our sample stars, 2400 K ≲  T_eff ≲ 2800 K, this means that objects with Li are less massive than about 65 M_jup, and younger than ∼0.5 Gyr. This also means that they are less massive than the substellar limit at 75 M_jup, hence they are young brown dwarfs.

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We have detected Li in six of the 63 objects in our sample, i.e., about 10% of our sample are brown dwarfs younger than about 0.5 Gyr. The Li lines of all six objects are shown in Figure 3. Only one of the targets, LP 944-20 (2MASS J0339352−352544), was previously known as a young brown dwarf through the detection of its Li line (Tinney 1998). Youth and brown dwarf nature of the other five objects were unknown before. Cruz et al. (2007) classified 2MASS J0443376+000205 as a low-gravity, hence probably young, object. This classification is supported by our Li detection.

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Radial velocities were measured relative to a spectrum of a standard star. We used Gl 406 as radial velocity standard (v_rad = 19 ± 1 km s⁻¹; Martín et al. 1997) using the cross-correlation technique and correcting for barycentric motion. We corrected for wavelength calibration inconsistencies by cross-correlating telluric lines in the O₂ A band. Thus, uncertainties in our radial velocities are on the order of 1–3 km s⁻¹ depending on rotational line broadening. The uncertainties of the spectrophotometric distance are typically on the order of ∼10%. For our sample of nearby objects this introduces an uncertainty in the space motion components of typically ∼1 km s⁻¹. If an object is substantially younger than assumed for the spectrophotometric distance calculation, the error is much larger. This effect is illustrated for 2MASS J0041353−562112 in Reiners (2009), but is probably not important in the other objects.

Space motions U, V, and W were computed using the IDL procedure gal_uvw,⁴ but we use a right-handed coordinate system with U toward the Galactic center. For most of our targets, proper motions are reported in Faherty et al. (2009). Phan-Bao et al. (2001) and Phan-Bao et al. (2003) measured proper motions for some of our objects, and we use their values where available. One object, LHS 1070B (2MASS J0024442−270825B), is not included in either of the works, and we use the proper motion given in Salim & Gould (2003). All radial velocities and space motion vectors [U, V, W] together with the references for distance and proper motion are given in Table 1.

From our volume-limited sample, we can draw robust constraints on the fraction of young brown dwarfs among the ultracool dwarfs. Our results are summarized in Table 2. Reid et al. (2002b) found fractions of $F_{\rm {Li}} = (6 \,\pm\, 4)\%$ for M7–M9.5 dwarfs and $F_{\rm {Li}} = (10 \pm 7)\%$ for M8–M9.5 dwarfs. The Li fractions from our sample are somewhat higher than the ones reported in Reid et al. (2002b), but the discrepancy does not exceed the uncertainties due to the small sample size.

We conclude that the fraction of lithium brown dwarfs among M7–M9.5 dwarfs is at most only little higher than previously reported. This means that the mass function index α could perhaps be a little higher than reported in Reid et al. (2002b). Nevertheless, their conclusion that α < 2 for both the Arizona and Lyon models is still valid.

Kirkpatrick et al. (2008) investigated the fraction of objects with Li detections among L dwarfs together with a search for objects with spectroscopic signatures of youth, i.e., low gravity. They show that the fraction of Li detections is about 10% at spectral type L0 and rises to about 50%–60% at mid-L (see their Figure 17). After L6, the fraction turns over and declines toward the late-L types probably because Li disappears into molecules (Lodders 1999). Kirkpatrick et al. (2008) emphasize that Li is probably not detectable in very young L dwarfs, i.e., during the first few ten million years, because the Li line is not visible at very low gravity.

We plot the fraction of lithium brown dwarfs as a function of spectral types M7–L8 in Figure 4. M7–M9 dwarfs are from this work and from Reid et al. (2002b), L dwarfs are taken from Reiners & Basri (2008), and from the Li analysis of Kirkpatrick et al. (2008) including data from Kirkpatrick et al. (1999, 2000). While the Kirkpatrick et al. works employ low-resolution LRIS spectra, the other Li detections are based on spectra with higher spectral resolution. Nevertheless, the detection thresholds from the different samples are not very different because in L dwarfs it is usually not the spectral resolution that limits the detection of Li (note that the Li lines shown in Figure 3 are smoothed to a lower effective resolution). Comparing the subsamples taken from high- and low-resolution spectra, we see no difference in the fraction of Li detections, and the results for objects that are contained in both samples are consistent.

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The fraction of lithium brown dwarfs among ultracool dwarfs continually rises from M7 to M9, because late-M dwarfs in general are less massive, so that the fraction of objects with M ≲ 0.07 M_☉ is growing toward later spectral types. In Figure 4, there is also a general trend that the Li fraction rises from M7 to mid-L, which is consistent with a higher brown dwarf fraction at later spectral type. Note that the time required to deplete Li in high-mass brown dwarfs (≳60 M_jup) is longer among the L dwarfs (∼1 Gyr) than at late-M spectral type (∼300 Myr). However, the fraction of Li detections among M9 dwarfs is much higher than at M8, and, in particular, it is much higher than at spectral types L0 and L1. The number of objects in this sample is still fairly low, and the results are formally still consistent with a smooth rise in the range M7–L6 if the M9 bin is interpreted as an outlier. On the other hand, the discontinuity at spectral type M9 is fairly high while the overall scatter between the other bins is rather small. Three out of the 15 objects at spectral type M9 and M9.5 show Li, all three are most likely within 20 pc. The statistical significance of the M9 discontinuity is comparable to that of the apparent downturn in Li at late L types noted by Kirkpatrick et al. (2008). In the following, we discuss possible explanations.

The first possible explanation is that our combined sample is not strictly volume-limited, in particular the L dwarfs are not selected according to their distance. This implies a bias toward young objects and probably causes an excess of young dwarfs that still have Li. Most distances of our M dwarf sample are taken from spectrophotometry, which means that a few other young objects contained in our M dwarf sample may be situated outside 20 pc. However, this problem should apply even more to the early-L dwarfs, in which no effort was made to include faint old objects out to a certain distance. Thus, one would expect the early-L dwarfs to be biased toward higher fraction of Li detections, and this bias should be higher than the bias among the late-M dwarfs. Furthermore, the fraction of Li detections in our M9/M9.5 targets from the volume-limited sample is higher than the fraction in the full sample including the targets from Reid et al. (2002b). Thus, an observational effect that causes a bias toward young objects particularly at the M9 spectral bin is unlikely an explanation for the observed discontinuity.

A second potential reason for the discontinuity is that the L dwarf spectra are of lower quality compared to the M dwarf spectra taken for this work. This would lead to a higher Li detection threshold and hence a lower detection rate. While such an explanation might hold for the low-resolution spectra (a detection threshold of 3 Å is reported in Kirkpatrick et al. 1999, 2000, 2008), it does not apply to the high-resolution spectra of Reiners & Basri (2008). In the latter sample alone, the ratio of Li detections to total number of targets is 0/10 (0%) at spectral type L0 and 1/14 (7%) at spectral type L1, which is an even lower ratio than the one from low-resolution spectra in Kirkpatrick et al. (2008). Thus, differences in data quality do probably not lead to systematic differences in the Li fraction, and it seems unlikely that the detection threshold is the main reason for the low fraction of Li detections at early L dwarfs.

As a third potential reason, we speculate that the discontinuity is real, i.e., the fraction of objects with detectable Li peaks at spectral type M9. We see two possible ways to explain this. (1) Lithium brown dwarfs with temperatures of ∼2200 K may be classified as M9, but at older ages stars of the same temperature could be classified as L0. Classification of spectral types between M9 and L0 is based on molecular species that are temperature dependent and sensitive to the presence of dust, which is strongly influenced by gravity. In other words, the relation between temperature and spectral class may be a function of age. (2) The history of the star formation rate together with the detectability of Li as a function of age (see Kirkpatrick et al. 2008) may conspire so that today we observe an excess of Li detections at spectral type M9. This may imply a burst of star formation at a particular age. So far, however, we are not convinced that either of the two is a very likely scenario, and we will have to wait for a larger sample and better distance estimates to see whether the maximum disappears with better statistics.

Space motions of M dwarfs and ultracool dwarfs have raised a lot of interest during the last decades because the velocity dispersion of a sample can be tied to its age, which allows the study of the star formation rate and the time-dependence of physical processes like for example activity. Hawley et al. (1996) have shown in a volume-limited sample of M dwarfs that the velocity dispersion of active dMe stars is smaller than the dispersion of older non-active stars. Reid et al. (2002b) determined the velocity dispersion of ultracool objects finding that the velocity dispersion of their M7–M9.5 sample is smaller than the dispersion in the full dM sample of Hawley et al. (1996). Reid et al. (2002a) revisit the space velocities of ultracool M dwarfs suggesting that the early-M dwarf sample consists of members of both the thick disk and the thin disk. Late-M dwarfs, on the other hand, may have lost their thick disk component because the lower metallicity of these stars moves the hydrogen burning limit to earlier spectral types so that these stars have cooled toward later spectral types and don't show up as M7–M9.5 objects any longer.

We show the space velocities in U−V and U−W diagrams in Figure 5, and the distribution of U, V, and W in Figure 6. Lithium brown dwarfs are shown as red stars in Figure 5 and are overplotted as filled red histograms in Figure 6. The distribution of space velocities in ultracool dwarfs is clearly concentrated around the thin disk, and the six lithium brown dwarfs cluster in a very narrow range in the center of the thin disk in all three space velocities. This clustering implies a very small velocity dispersion, which is an indicator for a young sample (see Wielen 1977, and the following section). Thus, space velocities of lithium brown dwarfs are consistent with very low age. In the following, we will derive the age of our sample of M7–M9.5 dwarfs.

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Wielen (1977) has shown how a velocity dispersion of a sample of stars can be used to determine the age of the sample. More recently, Fuchs et al. (2001) refined this picture using more recent data from different works.

First, it is important to realize that the formulae provided by Wielen (1977) are valid for dispersions σ_U, σ_V, and σ_W that are weighted by their vertical velocity |W| (Equations (1)–(3) in Wielen 1977, see also Reid et al. 2002a). Although it might be arguable whether this weighting is useful or not when comparing dispersions from different samples, it has to be applied if the age of a sample should be determined from the formulae in Wielen (1977). The dispersion of space velocities in our sample is (σ_U, σ_V, σ_W) = (27.2, 22.6, 14.6) km s⁻¹, and the |W|-corrected velocity dispersions are (σ_U, σ_V, σ_W) = (30.7, 22.6, 16.0) km s⁻¹.

A second but much more crucial principle for the calculation of ages from velocity dispersions is the calculation of the total velocity dispersion,

$$\begin{equation} \sigma _{\rm tot} = (\sigma _U^2 + \sigma _V^2 + \sigma _W^2)^{1/2}. \end{equation} \tag{ 1 }$$

This means that σ_tot is not the dispersion of the total velocity scalar v_tot = (U² + V² + W²)^1/2, since the latter will always be smaller because motions into different directions partially cancel out each other (for example, a sample with velocities equally distributed into all directions but with the same total velocity will have zero dispersion around v_tot). The uncorrected total dispersion of our sample is σ_tot = 38.3 km s⁻¹, and the |W|-corrected total dispersion is σ_tot = 41.3 km s⁻¹.

It should be noted that, following the same argumentation as for σ_tot, the total dispersion in the tangential velocity scalar, v_tan, cannot be applied to calculate σ_tot using σ_tot = (3/2)^1/2σ_tan: the total dispersion in the two-dimensional velocity v_tan must be the squared sum of the two individual dispersion components. Thus, the total dispersion in v_tan cannot be used for the calculation of age from the Wielen relation.

Recently, kinematics of ultracool dwarfs and brown dwarfs have been presented by Schmidt et al. (2007), Zapatero-Osorio et al. (2007), and Faherty et al. (2009) using the velocity dispersion of a sample of stars to calculate the age of the sample following the work of Wielen (1977). Schmidt et al. (2007) derive an age of 3.1 Gyr for a sample of late-M and L dwarfs, Zapatero-Osorio et al. (2007) find ages of 4 Gyr for their late-M dwarfs and about 1 Gyr for L and T dwarfs, and Faherty et al. (2009) report ages between 2 and 3 Gyr for late-M, L, and T dwarfs. Unfortunately, all three of these papers contain the errors in interpretation of the total velocity dispersion pointed out in the previous two paragraphs.

Once the velocity dispersion σ_tot is determined for a sample, it can be translated into an age according to Wielen (1977). He presents three different equations to carry out this task, and it is often difficult to find out which prescription authors use when publishing an age for a sample of stars. The first prescription assumes a constant diffusion coefficient and is apparently not much used in recent work (Equation (8) in Wielen 1977). The two other formulae assume velocity-dependent diffusion coefficients. The two formulae, Equations (13) and (16) in Wielen (1977), differ in the way they treat the diffusion coefficient C_v, they are

$$\begin{equation} \sigma _v^3 = \sigma _{v,0}^3 + \frac{3}{2}\gamma _v\tau, \end{equation} \tag{ 2 }$$

$$\begin{equation} \sigma _v^3 = \sigma _{v,0}^3 + \frac{3}{2}\gamma _{v,p}T_\gamma (\exp {(\tau /T_\gamma)} - 1), \end{equation} \tag{ 3 }$$

with τ the age of the sample in yr, σ_v,0 = 10 km s⁻¹, γ_v = 1.410⁻⁵ (km s⁻¹)³ yr⁻¹, γ_v,p = 1.110⁻⁵ (km s⁻¹)³ yr⁻¹, and T_γ = 510⁹ yr. According to Wielen (1977), Equation (2) is inadequate for ages above 3 Gyr, which means that Equation (3) should be preferred.⁵

The determination of an age from a stellar sample assumes that the sample members (at least in a very qualitative sense) fulfill the requirement of homogeneity, i.e., their velocity distribution should not differ drastically from a Gaussian distribution. A robust way to investigate the velocity distribution in a sample is to use probability plots, or "probit" plots (Lutz & Upgren 1980). In a probability plot, the data points are sorted and assigned a probability bin, which is the distance to the mean in units of the standard deviation, that each of the sorted points in a strictly Gaussian distribution would have. If a distribution is Gaussian, the sample points will follow a straight line, and the value of the standard deviation, σ, is the steepness of this line. We show the probability plots for our sample in Figure 7. The largest part of the sample relatively well resembles a well-constrained line, which means that except for a only a handful of outliers, the distribution of space velocities is consistent with a Gaussian distribution. We overplot a fit to the inner ±1.75σ of the velocity distribution in Figure 7. Velocity dispersions and ages derived from the inner part of this distribution are statistically indistinguishable from results taken from the full sample.

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From the |W|-corrected space velocity dispersions, we have calculated the age of our sample as described above. Our results are summarized in Table 3, the age estimate for our volume-limited sample of ultracool dwarfs from this methodology is τ = 3.1 Gyr. We compare this result to the sample presented in Reid et al. (2002b). These authors also show that their sample matches a Gaussian distribution quite well. We calculated |W|-corrected space velocity dispersions from the 32 M7–M9.5 dwarfs (compare with their Equation (3)) given in their Table 5, and from the Wielen relation we derive an age for the sample of τ = 3.1 Gyr. Given the substantial uncertainties of this age determination, which are on the order of several hundred Myr, this is in remarkable agreement with the age derived from our sample. Thus, there is good evidence that the mean age of the local ultracool dwarf population is about 3 Gyr.