The Ultracool SpeXtroscopic Survey. I. Volume-limited Spectroscopic Sample and Luminosity Function of M7−L5 Ultracool Dwarfs

Targets for the sample were drawn from a number of literature sources, including surveys and previous compilations, each designed for its own scientific purposes and with a variety of follow-up. We attempt to average over the various biases from the original surveys by compiling as many sources as possible. Some of the known biases include a red J − K_S color bias (e.g., Cruz et al. 2003; Lépine et al. 2013, identified by Schmidt et al. 2015); incomplete compilations (e.g., Gagné et al. 2015c) or partial sky coverage, e.g., SDSS (Ahn et al. 2012; Alam et al. 2015), Deep Near-Infrared Southern Sky Survey (DENIS; Epchtein 1994), and UKIDSS (Lawrence et al. 2007); and targeted surveys (e.g., young objects, Shkolnik et al. 2009; wide binaries, Deacon et al. 2014; high proper motion surveys, i.e., SUPERBLINK, Lépine & Gaidos 2011). We believe biases due to proper-motion selection are negligible due to the completeness of the photometric selection surveys. While proper-motion surveys tend to be more incomplete, they also are less likely to scatter distant objects into the sample. Table 1 lists the literature sources used to consolidate a database of ∼16,000 candidate nearby UCDs. Table 2 summarizes the sequence of cuts leading to our final samples. Table 3 lists the sources compiled in the 25 pc and 1σ samples.

Table 2.  Cuts Leading to the Final Sample

Cut	Targets Remaining
Initial compilation	16,322
Deletion of duplicates	12,711
Optical, NIR, or "photometric" spectral type between M7 and L5	6226
Estimated distance ≤50 pc	1664
Compilation of photometry, recalculation of spectrophotometric distances
Deletion of nonstellar sources, giants, and compact and young stellar objects	1571
Estimated distance ≤30 pc	833
Compilation of Gaia astrometry, recalculation of trigonometric distances
Objects with literature optical, NIR, or SpeX spectral type within M7−L5 (including phototypes only)	595
Objects with trigonometric or NIR spectrophotometric distance ≤25 pc	${435}_{-20}^{+21}$ a
Objects with trigonometric or NIR spectrophotometric distance ≤25 pc+1σ	${470}_{-21}^{+22}$ a
Final samples
25 pc sample of M7−L5 dwarfs	${410}_{-20}^{+21}$ a
25 pc plus 1σ sample of M7−L5 dwarfs	${470}_{-21}^{+22}$ a

Note.

^aUncertainties based on Poisson statistics.

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Duplicate sources were removed with TOPCAT (Taylor 2005) through an internal match that organized sources in near-neighbor groups with a matching radius of 15'', large enough to catch binary components before deletion. This step reduced the number of entries to ∼12,700. We applied a spectral-type cut requiring optical or NIR spectral types or photometric spectral-type estimates (e.g., Skrzypek et al. 2015; Theissen et al. 2017) to be in the M7−L5 range, shrinking the database to ∼6200 sources. A rough distance cut eliminating objects farther than 50 pc trimmed this list to 1664 sources.

Galaxies, giants, T Tauri stars, and other non-UCD sources as reported in the literature were identified using SIMBAD and removed, reducing the sample to 1571 sources. After compiling photometric and astrometric data and recalculating spectrophotometric distances (see below), another distance cut at 30 pc was applied for those sources with astrometric parallaxes, yielding 833 sources.

Photometry from the 2MASS (Skrutskie et al. 2006), SDSS DR9 (Ahn et al. 2012), AllWISE (Wright et al. 2010; Mainzer et al. 2011), UKIDSS LAS (Lawrence et al. 2007), and Gaia DR2 (Gaia Collaboration et al. 2018) catalogs were collected for all sources, selecting the closest match up to 15'' through the VizieR interface to account for objects with large proper motions, using a custom routine¹⁶ built with the Astroquery Python package (Ginsburg et al. 2018). We obtained coordinates, epochs, identifiers, and GrizJHK_sKW1W2W3 magnitudes from Gaia, SDSS, 2MASS, UKIDSS, and AllWISE. Spectral types from SDSS spectroscopy were obtained when available. In addition to these surveys, we also obtained rizJHK_s magnitudes and uncertainties, spectral types, object types, and proper motions from SIMBAD with the same search radius.

Table 4 provides the photometry data for the sample. All sources in our final 25 pc sample (see Table 2) have NIR magnitudes,¹⁷ 88% have MIR magnitudes from AllWISE, and 39% have optical magnitudes from SDSS. Resolved NIR photometry on the Maunakea Observatory (MKO) filter system (Tokunaga et al. 2002) was obtained from the literature (e.g., Dupuy & Liu 2012; Best et al. 2018) and selected compilations,¹⁸ particularly for closely separated components of binary systems. We adopted 2MASS JHK_s magnitudes as the standard and used MKO JHK magnitudes if those were the only NIR ones available.

AllWISE includes a cross-match with the 2MASS catalog that we used to check for mismatches. We compared the JHK_s magnitudes from the 2MASS and AllWISE catalogs and kept the 2MASS magnitudes when the difference was within 0.05 mag (typical magnitude uncertainty for 2MASS JHK_s). Objects whose magnitude differences were >0.05 mag were flagged for visual examination in multiwavelength finder charts and comparison of SIMBAD and VizieR data sets. The mismatches between AllWISE and 2MASS JHK_s magnitudes were typically caused by the blending of a bright and faint source (Δm ∼ 3 mag) in the larger AllWISE pixels. In these cases, we assigned the 2MASS JHK_s magnitudes to the source and replaced the AllWISE W1W2W3 magnitudes with null entries. The same procedure was followed to consolidate JHK magnitudes from UKIDSS and literature sources. While UKIDSS uses MKO filters, we keep these measurements separate because the quantum efficiency of the various NIR detectors may differ.

Further inspection of mismatched photometry between SDSS, 2MASS, and AllWISE was done with color–color diagrams, as shown in Figure 1, and corrected by visual inspection using finder charts. Figure 1 illustrates the color loci of M7−L5 dwarfs from Schmidt et al. (2015). The most discriminating colors (e.g., z − J) use filters across surveys. Mismatches were corrected in a similar way as described above, using multiwavelength finder charts and comparing magnitudes.

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Astrometric data (positions, proper motions, and parallaxes) and radial velocities were drawn from SIMBAD when available. The sample was also cross-matched against the astrometric samples of Dupuy & Liu (2012) and Weinberger et al. (2016). Upon the release of Gaia DR2 (Gaia Collaboration et al. 2018), we cross-matched our preliminary sample against this data set to obtain five-parameter astrometric solutions. We used the following Astronomical Data Query Language query through the astroquery.Gaia package.

SELECT g.∗, t.∗

FROM gaiadr1.tmass_original_valid AS t

LEFT OUTER JOIN gaiadr2.tmass_neighbourhood AS xt ON xt.tmass_oid = t.tmass_oid

LEFT OUTER JOIN gaiadr2.gaia_source AS g ON xt.source_id = g.source_id

where 1 = CONTAINS(POINT(''ICRS'', t.ra, t.dec),CIRCLE(''ICRS'', {}, {}, 5./3600)).

The Gaia cross-match was done in two steps. First, we cross-matched the sample with the 2MASS–Gaia DR2 cross-match table (gaiadr2.tmass_neighbourhood) within a radius of 50 using 2MASS coordinates from our sample. Second, we joined this cross-match with the Gaia DR2 source table. We obtained 843 matches in 2MASS (10 objects with two matches each), 715 matched Gaia DR2 with a G magnitude, and 705 with parallaxes. To check the validity of our matches, we examined a color–magnitude diagram of G − RP versus absolute G magnitude. We considered sources as outliers if G − RP ≤ 1.25 and if M_G ≤ 5 to avoid giant stars. The 36 sources that failed our color–magnitude constraints were examined for cross-match accuracy, and we found 22 mismatches of true UCDs with erroneous Gaia data. The remaining 14 sources were dropped from the sample due to their small Gaia parallaxes (${\boldsymbol{\omega }}\ll 10\,\mathrm{mas}$), resulting in 825 sources. Figure 2 shows a Hertzprung–Russell diagram of our sample, including mismatches, against the full Gaia 25 pc sample.

2.2.1. Spectral Types

Most catalogs provide information on optical or NIR spectral classification or classification estimates from photometry (Skrzypek et al. 2015; Theissen et al. 2017). Given variations in classification schemes and intrinsic differences between optical and NIR classification (particularly for L dwarfs), we required at least one optical, NIR, or photometric type belonging to the M7−L5 range for sources to be included in the sample. Adopted literature spectral types were chosen by prioritizing optical, NIR, and phototypes, in that order. In the final 25 pc sample, the adopted spectral type is optical for 334 objects, NIR for 73, and photometric for four. The objects whose adopted literature type is photometric have SpeX observations (see Section 3) confirming their status as M7−L5 dwarfs. Figure 3 shows the distribution of adopted literature spectral types color-coded by the nature of their measurement. Table 5 lists the literature and measured SpeX spectral types, both by comparison to spectral standards and spectral indices (see Section 4.2).

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One hundred and eighty-nine objects have both optical and NIR measurements from the literature. With our SpeX observations (see Section 3), we have added 109 NIR classifications (see Section 4.2). Figure 4 shows a comparison between literature optical and NIR spectral types. The size of each circle is proportional to the number of overlapping sources. The scatter between spectral types is 0.95 subtypes; the 3σ boundaries are delineated by the dashed light gray lines.

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2.2.2. Distances

Trigonometric and spectrophotometric distances were calculated from parallaxes and spectrophotometric empirical relations in the NIR, respectively. Gaia DR2 provided most of the parallaxes in the sample, 80% of the total, or 327 in our 25 pc sample. Distances from Gaia were calculated simply as d = 1000/ω (mas), rather than using a likelihood function with Bayesian probabilities (e.g., Bailer-Jones et al. 2018), since we are concerned with sources with large parallaxes (ω ≥ 35 mas or d ≤ 28.5 pc to account for uncertainties beyond d = 25 pc) with small relative errors of the order of 0.04%–4%. Trigonometric distances from parallaxes predating Gaia DR2 were calculated in the same way. We also calculated trigonometric distances from WISE following the prescription of Theissen (2018) for 16 sources.

We calculated spectrophotometric distances using the adopted literature spectral type and the absolute magnitude empirical relations from Dupuy & Liu (2012). Distances were calculated for the NIR filters J, H, and K_s and averaged, weighted by their uncertainties. We adopt trigonometric distances, if available (for 93% of the sample), and use spectrophotometric distances for 29 sources that do not have a parallax measurement. Distances are reported in Table 6. Figure 5 summarizes the distance uncertainties for these measurements, and Figure 6 compares trigonometric to spectrophotometric distances for the 25 pc and 1σ samples. Trigonometric and spectrophotometric distances agree within 6.9% of each other, except for obviously overluminous sources.

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Table 6.  Trigonometric and Spectrophotometric Distances of Bona Fide and 1σ Samples of M7−L5 UCDs in the 25 pc Volume

	Spectral Type			Distance (pc)
Source Name	Adopted	Flag	Parallax (mas)	Trigonometric	Spectrophotometric (NIR)	References
25 pc Sample
LP 584−4	M9.0	NIR	48.05 ± 0.14	20.81 ± 0.06	15 ± 2	191
GJ 1001 C	L5.0	OPT	82.09 ± 0.38	12.18 ± 0.06	11 ± 1	191
GJ 1001 B	L5.0	OPT	82.09 ± 0.38	12.18 ± 0.06	⋯	191
2MASS J00044144−2058298	M7.0	NIR	66.33 ± 0.16	15.08 ± 0.04	23 ± 3	191
2MASS J00145575−4844171	L2.5	OPT	50.11 ± 0.39	19.96 ± 0.16	20 ± 2	191

References. (1) This paper, (2) Gaia Collaboration (2018), (3) Burgasser et al. (2015a), (4) Faherty et al. (2012), (5) Lépine et al. (2009), (6) Dieterich et al. (2014), (7) Marocco et al. (2013), (8) Dupuy & Liu (2012), (9) Gaia Collaboration et al. (2018), (10) Dittmann et al. (2014), (11) Weinberger et al. (2016), (12) Bartlett et al. (2017), (13) Lindegren et al. (2016), (14) Sahlmann et al. (2015), (15) van Leeuwen (2007), (16) Pravdo et al. (2005).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Using the best distance measure, a strict cut on 25 pc was applied to select our volume-limited sample with 410 sources whose measured literature optical or NIR spectral types lie within M7−L5 and whose distance was within 25 pc, i.e., excluding objects with only a photometric estimation of their spectral type. We assess Poisson uncertainties as described in Gagné et al. (2017) for our sample size in a subsequent analysis. Sources whose 1σ uncertainties placed them within 25 pc, amounting to 60 objects, were added to an expanded 25 pc + 1σ sample of 470 objects.

Figure 7 shows the spatial distribution of all of our targets. The 25 pc literature sample is evenly distributed across the sky, with the exception of the Galactic plane. Since 25 pc is a relatively small radius compared to the radius of the Milky Way (R_MW ∼ 25 kpc) and its vertical scale height (∼300 pc; Kent et al. 1991; Bochanski et al. 2010), we assume an isotropic distribution of sources within this volume. There are 217 sources at northern declinations and 193 at southern declinations. In Galactic coordinates, there are 228 sources above the plane of the galaxy and 182 below it. We convert the 381 sources with measured parallaxes in our 25 pc from equatorial to galactic X, Y, Z right-handed coordinates centered at the Sun. In the ${\boldsymbol{X}}$ direction, we find 161 objects between the Sun and the Galactic center and 220 between the Sun and the outer edge of the Galaxy. In the ${\boldsymbol{Y}}$ direction, we find 206 objects in the direction of the Sun's motion and 175 objects trailing behind it. In the ${\boldsymbol{Z}}$ direction, we find 207 objects above the plane of the Sun and 174 below it. All of these values are within 3σ of each other, considering the Poisson uncertainties, but not consistent at the 1σ level. Bihain & Scholz (2016) suggested an inhomogeneity in the spatial distribution of brown dwarfs compared to stars, most likely an effect of small number statistics and incomplete coverage of observations. The slight preference for northern sources is due to the larger number of panchromatic survey observations in the northern hemisphere (in particular SDSS). The Galactic plane looks sparse due to overcrowding and background source contamination, and this region is excluded from our space density analysis below (see Kendall et al. 2003, 2007).

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We compared our SpeX spectra to NIR spectral standards defined in Kirkpatrick et al. (2010) following the method described therein, which compares the 0.9–1.4 μm spectrum of an object to standards using a χ² minimization routine. The resulting distribution of spectral types is shown in Figure 8.

After classifying the spectra, we compared their literature and measured spectral types. For most objects, we measured a NIR spectral type within one subtype of the published literature type. Objects with only a photometric estimate from the literature and whose SpeX spectral type placed them outside of the M7−L5 range are in the 1σ sample.

Figure 9 compares the literature adopted optical or NIR classifications to the SpeX classification. The scatter for the literature optical–SpeX comparison is σ = 0.77 subtypes, the scatter for the NIR–SpeX comparison is σ = 1.06 subtypes, and the scatter in the adopted–SpeX comparison is σ = 0.82 subtypes. The larger scatter in the literature NIR–SpeX classifications may be due to poorly defined prior NIR types, sensitivity to surface gravity, metallicity, clouds, and variance in the spectral region used for NIR classification.

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We also classified our SpeX spectra using spectral indices from Allers et al. (2007), Burgasser (2007a), and Reid et al. (2001). These indices are applicable in the L0−T8, M5−L5, and M7−L8 spectral-type ranges, respectively. Figure 10 shows the comparisons from these index classification systems against optical and NIR spectral types reported in the literature. The points outside of the allowed classification ranges are plotted in light gray and are not included in the median offset and scatter calculations. The indices from Burgasser (2007a) have a systematic offset of +1.30 and +1.40 subtypes compared to optical and NIR types, respectively, and overestimate the spectral type of our sources. The Allers et al. (2007) indices are the most accurate at predicting optical spectral types with σ = 0.90 subtypes. The scatter is larger for NIR types (σ = 1.05 subtypes), with a slight tendency to predict spectral types earlier than measured in the literature (offset = −0.30 in both cases). For both optical and NIR types, the Reid et al. (2001) indices have the smallest offset (0.10 and 0.05 subtypes for optical and NIR spectral types, respectively) but slightly larger scatters than Allers et al. (2007), at σ = 1.21 and 1.42 subtypes, respectively. All spectral types for sample sources are summarized in Table 5.

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Young brown dwarfs (τ ≲ 200 Myr) are undergoing cooling and contraction and are both larger in radius and less massive than their older counterparts at a similar spectral type. These physical properties translate into lower surface gravities, affecting spectral features such as reduced collision-induced absorption and narrower alkali lines (Allers et al. 2007; Kirkpatrick et al. 2010). Due to their low surface gravity and typically dusty atmospheres, young brown dwarfs share physical properties with directly imaged exoplanets, making the former ideal analogs to the latter (Faherty et al. 2013a, 2016).

We obtained gravity classifications of our SpeX spectra, following the NIR scheme of Allers & Liu (2013), defined for the spectral-type range M5−L7, except that spectral types were determined from H₂O indices without a visual comparison of the J band with NIR standards.

Additionally, we obtain seven very low gravity (VL-G) and 64 intermediate-gravity (INT-G) candidate classifications from our spectra in the combined 25 pc and 1σ samples (Table 8). All low-gravity candidates were examined for visual signatures of low gravity, comparing the spectra band-by-band to low-gravity standards (see Gagné et al. 2015b and Cruz et al. 2018), leading to the rejection of 26 INT-G classifications. We labeled 11 sources with conflicting signatures as peculiar, such as blue J − K_S colors, indicating low-metallicity effects rather than low gravity (Aganze et al. 2016). Most VL-G sources are previously known, but we have identified 2MASS J1739+2454 as a new VL-G source. Thirteen of the 26 INT-G sources are first reported in this paper. The unresolved spectrum of the M8+M8 binary system 2MASS J0027+2219AB (Forveille et al. 2005) was also classified as an INT-G source. Since both components have the same spectral type and the system is coeval, we assume that both components would be independently classified as INT-G, leading to a final number of INT-G objects of 26 plus one more including the 1σ sample.

While 2MASS J1022+5825 (Reid et al. 2008), 2MASSW J2148+4003 (Looper et al. 2008), and 2MASS J0512−2949 (Cruz et al. 2003) were previously classified as having field gravity (FLD-G; Allers & Liu 2013; Faherty et al. 2016), our spectra yield INT-G classifications. Similarly, SDSS J0443+0002 was classified as a VL-G in Allers & Liu (2013), but our spectra yield an INT-G classification. These discrepancies may be due to instrumental or reduction differences.

We used BANYAN Σ (Gagné et al. 2018) on our low-gravity candidates to assess possible membership in 27 young moving groups using new kinematic data from Gaia DR2 (Gaia Collaboration et al. 2018), and we report the probabilities for young moving group membership in Table 8. The Allers & Liu (2013) gravity classification scheme is a spectroscopic test for youth, while BANYAN Σ uses kinematic information to determine membership in a young moving group. Many of our low-gravity sources are classified as 0% probability members of any young group by BANYAN Σ, which implies that these objects might be young and unassociated, field interlopers, or belonging to moving groups other than the 27 known associations included in BANYAN Σ, possibly as a result of ejection.

Figure 11 shows the distributions of gravity types from our SpeX spectra by spectral type, as classified by field standards. We find the VL-G and INT-G fractions for our 25 pc sample to be ${2.1}_{-0.8}^{+0.9} \%$ and ${7.8}_{-1.5}^{+1.7} \%$, respectively, with uncertainties based on Poisson statistics.

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The spectral types of our low-gravity objects were further refined using VL-G and INT-G spectral standards from Allers & Liu (2013). The comparison between classifications is shown in Figure 12. The seven VL-G sources in our sample have much earlier types (by one to three subtypes) when classified with a VL-G standard than with a field standard, although this is too small of a sample to precisely quantify the bias. Figure 13 shows the seven VL-G sources classified with a field standard and VL-G standard.

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For INT-G sources, there is a better correlation but larger scatter (σ = 1.67), particularly among L dwarfs, which are expected to show stronger gravity features even as INT-G. These differences highlight the strong role of gravity-sensitive features and reinforce the importance of comparing low-gravity sources to equivalent standards.

Red and blue J − K_S color outliers are empirically defined subpopulations. Their unusual color is likely a proxy for physical properties such as age, low or high surface gravity, atmospheric cloud content, opacity, and metallicity (Metchev & Hillenbrand 2006; Burgasser et al. 2008b; Looper et al. 2008; Faherty et al. 2009).

Clouds play a key role in J − K_S color evolution from late-M to L type, as increased opacity originating from condensates and possibly clouds reddens spectral energy distributions (e.g., Tsuji et al. 1996; Lodders & Fegley 2006). This is intrinsic reddening, as objects in the 25 pc sample should be minimally reddened by interstellar dust. The thickness of clouds may be an independent parameter (e.g., Ackerman & Marley 2001; Hiranaka et al. 2016) or may correlate with youth (e.g., Faherty et al. 2013b) and/or metallicity (e.g., Burgasser et al. 2003). Color outliers may also indicate the presence of an unresolved companion (e.g., Bardalez Gagliuffi et al. 2014). Unusually blue objects and subdwarfs have enhanced collision-induced H₂ opacity (Saumon et al. 1994; Burgasser et al. 2003) due to their metal-poor atmospheres.

To isolate the color outliers of our sample, we compared their J − K_S colors to the average colors and standard deviations as a function of spectral type from Faherty et al. (2016), defined over the M7−L8 range. We identified outliers as 2σ deviants, shown in Figure 14. From the 387 objects in the 25 pc whose adopted spectral type is within M7−L5,¹⁹ and with both J and K_S photometry,²⁰ 188 have J − K positive excesses, while 184 have negative color excesses, and 15 do not have a color excess. This even distribution of sources indicates that our sample does not have an NIR color bias, despite widely used 2MASS color selections (Schmidt et al. 2015), for which redder selection criteria were necessary to excise the background population.

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The individual outliers are listed in Table 9. In our 25 pc sample, 15 objects were found to have unusually blue J − K_S colors, and six have unusually red J − K_S colors. In the 1σ sample, we find two more unusually blue objects. Given the numbers of color outliers from the 25 pc sample, we infer fractions of ${1.4}_{-0.5}^{+0.6} \%$ for red and ${3.6}_{-0.9}^{+1.0} \%$ for blue M7−L5 dwarfs in the solar neighborhood (with Poisson uncertainties). Among the five red outliers, 2MASS J0355+1133, G 196−3B, and 2MASS J1741−4642 have been reported as young in the literature (Gagné et al. 2015b; Faherty et al. 2016), while LHS 2397aA and Kelu-1A are classified as having FLD-G but are also known binaries (Freed et al. 2003; Stumpf et al. 2008). From all sources with Gaia kinematics, we explored a reduced proper-motion diagram and found no potential subdwarfs, i.e., sources with high proper motion, high reduced proper motion, and blue G − G_RP colors.

Five blue sources were also classified as INT-G, cementing their stattus as metal-poor objects (see Section 4.3 and Aganze et al. 2016). Two unusually blue sources, G 203−50B and 2MASS J1721+3344, are also rejected spectral binary candidates, as blue sources tend to be contaminants in the identification of spectral binaries (Bardalez Gagliuffi et al. 2014).²¹

Additionally, we calibrated our SpeX spectra to 2MASS J and K_s magnitudes to find spectrophotometric J − K_S colors. These were compared against 2MASS J − K_S colors and found to have a scatter of 0.18 mag. The 2σ outliers or higher are highlighted in Figure 15 and could be due to intrinsic atmospheric variability (e.g., Radigan et al. 2012). These sources are LHS 5166B, 2MASS J1152+2438, 2MASS J1200+2048, Kelu-1A (unusually red), 2MASS J1416+1348A (unusually blue), and 2MASS J1438+6408. Kelu-1 has a variability detection in 410 Å with a peak-to-peak amplitude of 11.9 ± 0.8 mmag (Clarke et al. 2003), reported before the discovery of its nearby companion (Liu & Leggett 2005). Khandrika et al. (2013) reported marginal variability in the J band for 2MASS J1416+1348A. The remaining outliers have not been targeted in variability surveys.

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Spectral binaries of UCDs are systems composed of a late-M/L-type primary and a hidden T dwarf secondary, identifiable only by their peculiar blended-light spectrum in NIR wavelengths (Cruz et al. 2004; Burgasser et al. 2010; Bardalez Gagliuffi et al. 2014). Identifying these potentially closely separated binaries allows us to probe the VLM binary separation distribution at all scales and select potential systems for orbital measurement (see Bardalez Gagliuffi et al. 2015).

We applied the spectral binary technique of Bardalez Gagliuffi et al. (2014)²² to the SpeX spectral sample. The spectral binary technique consists of two parts, spectral index selection and binary template fitting, the second of which incurs a hypothesis test to determine whether binary template fits are statistically better fits to a candidate than single templates. The spectral binary candidates are listed in Table 10. Forty-two objects were selected by the index–index parameter spaces as candidates but rejected by the low confidence from hypothesis testing. Seven objects were rejected despite passing the spectral binary fitting due to their blue colors, as blue objects are known contaminants of the spectral binary technique (Bardalez Gagliuffi et al. 2014).

We found five previously identified and confirmed spectral binaries in our 25 pc sample: 2MASSW J0320284−044636 (Blake et al. 2008; Burgasser et al. 2008a), WISE J072003.20−084651.2 (Burgasser et al. 2015a), 2MASS J08053189+4812330 (Burgasser 2007b; Dupuy & Liu 2012; Burgasser et al. 2016), 2MASS J13153094−2649513 (Burgasser et al. 2011b), and 2MASS J22521073−1730134 (Reid et al. 2006). We recover the L4+T3 spectral binary 2MASS J0931+2802 (Bardalez Gagliuffi et al. 2014) outside our 25 pc sample. We identify two previously unreported spectral binary candidates in our spectral sample, both of which lie formally outside our 25 pc distance limit. Binary template fits to these spectral binary candidates are shown in Figure 16.

2MASS J14111847+2948515. Its spectrum shows a deep H-band dip at 1.62 μm and an angled J-band peak at 1.25 μm, both signs of a hidden T dwarf companion. The K_s band of the object is slightly fainter compared to the binary template, which could be an indication of slightly blue L dwarfs, known contaminants to the spectral binary technique. However, the best single fits to its SpeX spectrum fail to reproduce the dip in the H band and are fainter in the J and K_s bands in comparison to 2MASS J1411+2948. Its component spectral types are likely to be L4+T4. No parallax has been measured for this source, whose distance would be larger than the estimated spectrophotometric distance of 49 ± 6 pc if it is a binary.

2MASS J14211873−1618201. The spectrum of this source shows an angled J-band peak and a small dip in the H band. Its inferred component spectral types are M8+T5, similar to 2MASS J0320−0446 (Blake et al. 2008; Burgasser et al. 2008a), 2MASS J0006−0852 (Burgasser et al. 2012), and WISE J0720−0846 (Scholz 2014; Burgasser et al. 2015a). Our strict distance cut left this source outside of the 25 pc sample, yet it rests right at the 25 pc limit (d_t = 25.15 ± 0.14 pc; Gaia Collaboration et al. 2018).

To calculate the frequency of spectral binary systems, we used the definition of Reipurth & Zinnecker (1993; see Section 4.6), where the binary fraction is the number of binaries over the total number of systems. For this calculation, we only consider systems with a measured SpeX spectrum, since otherwise we would not be able to assess spectral binarity.²³ Since 2MASS J1421−1618 lies at our limit distance, we calculate two spectral binary fractions, assuming 24 pc (five spectral binaries/282 spectra) and 26 pc (six spectral binaries/312 spectra) volumes. The fractions are ${1.7}_{-0.7}^{+0.9}$ and ${1.9}_{-0.7}^{+0.8}$ for 24 and 26 pc, respectively, or an average of ${1.8}_{-0.5}^{+0.6} \%$ assuming Poisson errors. This fraction is significantly lower than the total fraction of resolved binaries in the sample (${7.5}_{-1.4}^{+1.6} \%$; see Section 4.6), but this is likely because spectral binary systems encompass a specific range of component spectral types to be selected. We analyze the spectral binaries in this sample and their implication for the brown dwarf binary fraction in a companion paper.

Binaries and multiple systems reported in the literature were identified in our sample through cross-matches with the Washington Double Star Catalog²⁴ (WDS; Mason et al. 2001), SIMBAD (Wenger et al. 2000), and vlmbinaries.org. Table 11 lists the UCD binaries with primary components between M7 and L5 found in our sample previously reported in the literature, as well as UCD companions to main-sequence stars. Our 25 pc sample contains 410 objects in 393 systems, 341 single systems, 42 binary systems, and 10 triple systems. Only 28 binaries and no triples have a primary with a spectral type M7 or later. Including the 1σ sample, we find four more binaries and one quintuple system, HD 114762, comprised of Aa, Ab, and Ac components F9+F8+F4 stars, an 11.0 ± 0.1 M_J (Kane et al. 2011) brown dwarf orbiting the F9 star (Latham et al. 1989), and an M6:: dwarf as the B component 130 au away from the F triple system (Patience et al. 2002).

We calculate several statistics to represent the multiplicity of the sample: the multiplicity fraction, which provides the probability that a given source is a multiple system; the companion star fraction, which is the probability for an object to be in a multiple system; the pairing factor, which is the mean number of companions per primary; and the companion frequency that indicates the mean number of companions per object. These equations are defined and explained in detail in Reipurth & Zinnecker (1993) and Goodwin et al. (2004). Since we have no triple systems with primaries M7 or later, our multiplicity fraction is effectively a binary fraction. We determine the binary fraction of the 25 pc sample to be ${7.5}_{-1.4}^{+1.6} \%$, including both spectral binaries and RV variable systems. The companion star fraction for this sample is ${14.1}_{-1.9}^{+2.1} \%$; the pairing factor is 1 ± 0.3, since there are no triple systems with primaries ≥M7; and the companion frequency is 0.14 ± 0.02 companions per object (following the definition of Goodwin et al. 2004).

Figure 17 shows the cumulative binary fraction as a function of distance. Out to a distance of 9 pc, the binary fraction oscillates around 13%–25%, and at larger distances, it begins to drop and settle around ∼7%. The resolved UCD binary fraction has been thoroughly studied (e.g., Bouy et al. 2003; Burgasser 2007a), leading to ∼10%–20% for separations >1 au, while sub-au systems comprise 1%–4% of the population (Allen 2007; Blake et al. 2010). However, this is the first time the UCD binary fraction has been calculated in a volume-limited sample,²⁵ and as seen in Figure 18, there may be a significant fraction of overluminous binaries that have not been confirmed by high-resolution imaging, astrometry, or RV monitoring yet. Additionally, in the previous section, we found that five out of the 25 binaries within 25 pc are spectral binaries. Since spectral binaries require specific combinations of spectral types to be identified as such, we do not expect them to dominate the binary detection yield. Yet in this study, ∼20% of our binaries are spectral binaries, supporting our hypothesis that the population of binaries in the 25 pc sample literature is incomplete. The incompleteness of the binaries is shown in Figure 19 as a cumulative histogram over distance that flattens beyond 20 pc compared to the general 25 pc sample. Fitting curves to the 5–10, 5–15, and 5–20 pc regions and extrapolating to 25 pc, we estimate a large binary incompleteness of 76%, 65%, or 56%, respectively.

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The area covered by our spectral survey is limited by the declinations accessible from IRTF, roughly −50° < δ < +67°. Additionally, our survey suffers from an inherent incompleteness of sources in the Galactic plane. We therefore restrict our analysis to the area of sky outside −15° < b < +15° and within −50° < δ < +67°, which corresponds to an area of 26,051.54 deg², or 63.2% of the sky.

Bright stars reduce the total available sky area by obscuring patches of sky where a UCD could otherwise be found. To account for this effect, we drew 1 million sources from our sample and reassigned them to random coordinates within our observable area. This list was cross-matched with the 2MASS catalog using TOPCAT with a 50 radius, returning 22,126 matches. Of these, 2345 stars were as bright or brighter than the simulated input targets within the search radius, thus effectively obscuring nearby UCDs. Accounting for this effect reduces the effective observable sky by 0.15%, to 25,990.45 deg².  While we note that 0.5% of the sky is obscured by bright stars and excluded from the 2MASS survey,²⁶ we do not take it into account in our calculations, since our sources also come from optical and MIR surveys.

A volume within 25 pc around the Sun is well embedded within the thin disk of the Galaxy (scale height ∼300 pc; Kent et al. 1991; Bochanski et al. 2010) and therefore should be relatively uniform in density. Assuming a uniform distribution of sources, the cumulative number of objects should increase with distance following an r³ relation. We estimate our volume completeness in trigonometric, spectrophotometric, and adopted distances by fitting power-law curves to the cumulative distribution of sources in the ranges 5–10, 5–15, and 5–20 pc, assuming completeness in those ranges, considering Poisson uncertainties (Figure 20), and extrapolating expected numbers to 25 pc. The ratio of the number of objects in our sample to the expected number is used to estimate our completeness. These values are summarized in Table 12.

Table 12.  Estimated Volume Completeness

	Predicted Numbers		Completeness
Fit Range (pc)	Trigonometric	Adopted Distance	Trigonometric	Adopted Distance
25 pc sample (N = 410)
5–10	592	592	${64}_{-7}^{+8} \%$	${69}_{-8}^{+9} \%$
5–15	552	583	${69}_{-8}^{+9} \%$	${70}_{-8}^{+9} \%$
5–20	484	511	${79}_{-8}^{+9} \%$	${80}_{-8}^{+9} \%$
25 pc M dwarfs (N = 223)
5–10	⋯	509	⋯	${44}_{-6}^{+7} \%$
5–15	⋯	357	⋯	${62}_{-7}^{+8} \%$
5–20	⋯	283	⋯	${78}_{-8}^{+9} \%$
25 pc L dwarfs (N = 187)
5–10	⋯	83	⋯	${226}_{-15}^{+16} \%$
5–15	⋯	226	⋯	${83}_{-9}^{+10} \%$
5–20	⋯	228	⋯	${82}_{-9}^{+10} \%$

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The completeness of late-M dwarfs is lower than that of L dwarfs. Using the 5–15 pc fit, which is a good trade-off between completeness and sample size, our sample contains ${62}_{-7}^{+8} \%$ of the late-M dwarfs within 25 pc and ${83}_{-9}^{+10} \%$ of the L dwarfs. Late-M dwarfs may have been missed in previous surveys due to color-selection biases designed to exclude more numerous and brighter mid-M dwarfs, as indicated by Schmidt et al. (2015). While most L dwarfs in the solar neighborhood have already been identified in previous searches, many may be hidden in crowded areas like the Galactic plane (e.g., the L8 dwarf recently identified at 11 pc; Faherty et al. 2018). From the trigonometric distances, we estimate our total sample completeness to be between 64% and 79%. Including spectrophotometric distances when parallaxes are not available, the sample completeness is between 69% and 80%, but we adopt the value for the 5–15 pc fit, ${70}_{-8}^{+9} \%$. This completeness value is used in Section 5.5 to scale the corrected number of sources in the 25 pc volume when measuring the luminosity function (see Equation (4)). We expect most of the incompleteness to come from missing sources beyond 20 pc, as seen in Figure 21, possibly including sources in the Galactic plane, the southern hemisphere, or UCD candidates recently identified in Reylé (2018) in need of spectroscopic validation.

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Additionally, we estimate $\langle V/{V}_{\max }\rangle$ averages suggested by Schmidt (1968) to evaluate the homogeneous spatial distribution of our sample. Here $\langle V/{V}_{\max }\rangle$ measures the number of sources in each half of a given volume, approaching 0.5 for a uniformly distributed sample with equal counts on each half-volume. Figure 22 shows the distribution of $\langle V/{V}_{\max }\rangle$ values. Uncertainties are calculated as $0.5-\tfrac{n/2-{a}_{\max }}{n}$, where a_max is the distance at which the value of $\langle V/{V}_{\max }\rangle$ last equals 0.68 (4 pc for the full sample and M dwarfs only and 8 pc for L dwarfs), corresponding to one Gaussian standard deviation. For M dwarfs, the largest distance at which $\langle V/{V}_{\max }\rangle$ approximates 0.5 is 13 pc, suggesting incompleteness of M7−M9.5 dwarfs at larger distances. Conversely, L dwarfs have $\langle V/{V}_{\max }\rangle$ consistent with 0.5 up until 25 pc, indicating a homogeneous distribution of L0−L5 dwarfs in our sample.

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Compiling a sample of objects starting from past literature compilations leads to a complicated selection function. Rather than determining the selection function of each selection process separately, we simulate a sample of UCDs in a volume larger than 25 pc, including unresolved binaries, and apply selections based on our spectral type and distance cuts from both parallaxes and spectrophotometric estimates. This procedure aims to measure systematic effects in the sample construction.

We simulate 10⁶ UCDs, assigning distances drawn from a uniform spatial distribution out to 50 pc. We calculate "true" parallaxes by inverting the distances. An underlying spectral-type distribution was derived by population simulations (see Burgasser 2004) using the Chabrier (2005) IMF, a uniform age distribution, the Burrows et al. (2001) evolutionary models, and the effective temperature to spectral-type empirical relations from Pecaut & Mamajek (2013), which cover the full stellar and substellar spectral-type range from O3 to Y2. From this distribution, 10⁶ "true" spectral types between M5 and L7 were randomly drawn and assigned to our simulated UCD sources.

We calculate absolute magnitudes empirically from the simulated spectral types using the following linear relations:

$$\begin{eqnarray}&&{M}_{J}=0.37\times \mathrm{SpT}+4.29,\mathrm{rms}=0.35,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{M}_{H}=0.32\times \mathrm{SpT}+4.61,\mathrm{rms}=0.29,\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{M}_{{K}_{S}}=0.29\times \mathrm{SpT}+4.67,\mathrm{rms}=0.29,\end{eqnarray} \tag{ 3 }$$

determined from a subset of 230 single M7−L5 dwarfs with parallax measurements and 2MASS magnitudes and not classified as VL-G, INT-G, unusually red, or unusually blue from our 25 pc sample. The scatter in these relations is slightly smaller than in other empirical relations covering broader spectral-type ranges (e.g., Dupuy & Liu 2012; σ = 0.4 mag). To simulate the intrinsic brightness distribution of the population, we add offsets to these empirical absolute magnitudes drawn from a Gaussian distribution centered at zero and scaled by the scatter in the empirical relations.

Parallax- and magnitude-limited samples are subject to different biases affecting the total number of included sample members. The Lutz–Kelker bias affects parallax-limited samples by allowing objects from outside a distance limit into the observed volume (Lutz & Kelker 1973). For an observed parallax π₀, there is a range of true parallaxes π₀ ± δπ for normally distributed measurement uncertainties. Assuming a uniform number density of stars, the number of objects per parallax bin will be proportional to N_* ∝ 1/π⁴, implying that the number of stars increases as the parallax decreases; i.e., there are more objects in the volume outside a given distance than within. Subsequently, this means that more stars will appear to have smaller true parallaxes than their observed parallaxes, and that the average distance for sample members will be farther than the distance limit (Lutz & Kelker 1973).

In magnitude-limited samples, intrinsically brighter sources (i.e., on the high end of the absolute magnitude distribution) and unresolved binaries will be selected in larger numbers than intrinsically fainter sources, again due to the larger volume sampled by the brighter sources, an effect known as the Malmquist bias (Malmquist 1922). Depending on the relative uncertainty in distance and magnitude measurements and intrinsic scatter in the population, the effect of the Malmquist bias can be significantly larger than that of the Lutz–Kelker bias. Since our sample is defined by both trigonometric and spectrophotometric distances, both effects are significant in our calculations, although the Lutz–Kelker bias plays a more significant role given the large number of parallaxes in our sample (93% of the sample).

We model the Lutz–Kelker bias in our simulation by adding an uncertainty offset to our parallax measurements drawn from the uncertainty distribution of our observed parallaxes (see Figure 23). We excluded 2246 simulated sources with observed negative parallaxes. We account for unresolved binarity by adding a magnitude offset to 20% of stars in our simulated sample, the fraction based on estimates of the underlying UCD binary fraction (Bouy et al. 2003; Gizis et al. 2003; Burgasser et al. 2007c). We randomly assigned mass ratios from a power-law distribution (∝q^1.8; Allen 2007) to compute secondary masses. Effective temperatures, spectral types, and absolute magnitudes for the secondaries were obtained in the same manner as the primaries, resulting in combined system absolute magnitudes. Magnitude offsets were in the range Δm = 0–0.75 mag.²⁷ For simplicity, we assumed that the addition of flux to the simulated binaries does not affect the spectral-type classification, which is likely true for late-M and early-L dwarf primaries but not necessarily for late-L+T dwarf systems (Cruz et al. 2004; Burgasser et al. 2010). The addition of magnitude offsets for simulated binaries and uncertainties to the true absolute magnitudes for all simulated sources models the effects from the Malmquist bias.

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To model observed spectral types, offsets were drawn from a Gaussian distribution with a standard deviation equal to 0.95 subtypes (see Section 2.2.1). Apparent magnitudes were assigned based on the distance modulus and absolute magnitudes, adding an observational uncertainty drawn from a Gaussian distribution with a standard deviation following the same photometric error distribution from our literature sample. Observed parallaxes were modeled by adding a Gaussian uncertainty to the true parallaxes.

We quantify four selection functions, one for trigonometric and one for spectrophotometric distance selections as functions of spectral type and absolute magnitude. First, we define our "intrinsic sample" as those simulated sources whose true distances are d ≤ 25 pc. We define "observed samples" by requiring observed trigonometric or spectrophotometric distances d ≤ 25 pc. In each sample, we select objects with an observed spectral type between M7 and L5 and organize them according to their true spectral type, given that we are concerned with modeling our observations but aware that the true subtype may be different from the observed one. For the selection function by absolute magnitudes, we organized this selected sample in bins of 0.5 mag observed absolute magnitudes. Our trigonometric and spectrophotometric selection functions are the ratio of objects selected by observations over the number of objects selected by their true parameters. These selection functions are summarized in Tables 13 and 14 and illustrated in Figure 24.

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Our trigonometric selection function is relatively high (92%–98%) for the central part of the M7−L5 spectral-type range, except at the edges, where the selection rate drops to 71% for M7 and 69% for L5. The spectrophotometric selection function runs parallel to the trigonometric one, following a similar shape at a lower rate, 77%–82% for M8−L4, and dropping to 65% for M7 and 53% for L5. The trigonometric selection function based on J-band absolute magnitudes steadily increases from 66% at 10.75 mag (roughly equivalent to M7) to 96% at 12.25 mag, then dropping to 92% and 76% in the subsequent fainter bins. The corresponding spectrophotometric selection function follows a similar shape at a lower rate as well, starting at 61% for 10.75 mag, reaching a peak of 80% at 11.25 mag, and decreasing toward fainter magnitudes down to 62% at 13.25 mag (roughly equivalent to L4). These results are presented in Tables 13 and 14. As expected, the edges of our sample suffer from higher contamination than the bulk of it. Contamination from bright sources that do not belong in the 25 pc M7−L5 sample is most noticeable in the low spectrophotometric selection rate of the brightest absolute magnitude bins.

We also calculated the proportion of true negatives and false positives per spectral subtype and absolute magnitude bin. True negatives are true M7−L5 dwarfs with true distances within 25 pc that are not selected by observed trigonometric or spectrophotometric cuts at 25 pc, i.e., true sources missed by our selections. The true negative fraction is 2% for any spectral subtype using a parallax cut, except for L0, where the missed fraction is 3%. However, for a spectrophotometric cut, the true negative fraction rises with spectral type from 8% to a maximum of 21% at L2, then decreasing again to 17% at L5. The true negative fraction by absolute magnitude bins is also 2% for trigonometric cuts and 7%–22% for spectrophotometric cuts, with the maximum at 12.25 mag. False positives are contaminants, either sources outside the M7−L5 spectral range within 25 pc or true M7−L5 dwarfs outside 25 pc selected by observations. The false-positive fraction for M7−L5 dwarfs varied between 6% and 9% for spectral-type bins selected by parallax and 10%–31% if selected by spectrophotometric distance. The false-positive rates by absolute magnitude bins are 2%–8% for trigonometric selections and 14%–30% for spectrophotometric selections. Thus, the true negative and false-positive rates for trigonometric and spectrophotometric selections are comparable across spectral type and absolute magnitude bins. Tables 15 and 16 show the fraction of simulated sources outside 25 pc with a given spectral type and their observed spectral type as selected by observed trigonometric and spectrophotometric distances. For example, in Table 15, 4% of the observed M8 dwarfs are actually M9 dwarfs outside of 25 pc. Overall, it appears that parallax selections are more resistant to scattering of earlier-type objects. Diagonal elements indicate objects of matching true and observed spectral subtype outside of 25 pc but falsely selected to be within the volume, possibly very close to the 25 pc limit (Lutz–Kelker bias) or brighter than most other objects of the same subtype (Malmquist bias).

5.5.1. Luminosity Function with Respect to Spectral Types

Luminosity functions are a result of the underlying mass function and stellar birth rates. Calculating a luminosity function of UCDs in the 25 pc volume around the Sun is the first step toward building a field IMF across the stellar/substellar boundary. To measure our luminosity function with respect to spectral types, we prioritize literature optical, SpeX, and literature NIR spectral types in that order, since optical classifications are more precise than NIR ones.²⁸ Since our study is concerned with the areas accessible by SpeX and outside of ±15° from the galactic plane, we excluded literature sources outside of these areas, reducing our sample to 331 sources. However, four sources do not have unresolved J-band magnitudes (see Section 2.2); hence, our effective sample includes 327 objects. From these, we find 306 sources in our 25 pc sample with prioritized spectral types within M7−L5 within decl. accessible by SpeX (−50° ≤ δ ≤ +67°) and outside galactic latitudes ±15° from the galactic plane.

To estimate the expected total number of objects in our 25 pc sample per spectral-type bin, we scale our counts by our selection functions and completeness. We proportionally apply the trigonometric and spectrophotometric selection functions (SF_plx and SF_phot, respectively) to each spectral-type bin by splitting our counts, N_bin = N_plx + N_phot, according to their type of adopted distance (trigonometric or spectrophotometric) and then scaled by the completeness percentage for the 5–15 pc fit from Section 5.2, i.e.,

$$\begin{eqnarray}&&{N}_{\mathrm{corrected}}=\left(\displaystyle \frac{{N}_{\mathrm{plx}}}{{{SF}}_{\mathrm{plx}}}+\displaystyle \frac{{N}_{\mathrm{phot}}}{{{SF}}_{\mathrm{phot}}}\right)\cdot \left(\displaystyle \frac{1}{\mathrm{completeness}}\right).\end{eqnarray} \tag{ 4 }$$

These corrected counts were divided over the volume estimated in Section 5.1 to obtain our luminosity function with respect to spectral types. Our number densities are listed in Table 17 and shown in Figure 25 with and without selection function and completeness corrections.

Table 17.  Number Densities by Spectral Subtype in Units of object pc⁻³

Spectral Type	Cruz et al. (2007)	Reid et al. (2008)	Kirkpatrick et al. (2012)	Na	Raw	SF-corrected
M5	⋯	⋯	(${6.99}_{-1.59}^{+2.05}$) × 10−3	⋯	⋯	⋯
M6	⋯	$({9.50}_{-4.70}^{+9.50}$) × 10−5	$({1.07}_{-0.20}^{+0.25})\times {10}^{-2}$	⋯	⋯	⋯
M7	$({1.08}_{-0.26}^{+0.34})\times {10}^{-3}$	$({9.50}_{-4.70}^{+9.50})\times {10}^{-5}$	$({1.40}_{-0.61}^{+1.07})\times {10}^{-3}$	64	$({1.54}_{-0.18}^{+0.21})\times {10}^{-3}$	$({3.20}_{-0.27}^{+0.29})\times {10}^{-3}$
M8	$({3.73}_{-0.52}^{+0.60})\times {10}^{-3}$	$({9.47}_{-1.89}^{+2.37})\times {10}^{-4}$	$({2.33}_{-0.84}^{+1.30})\times {10}^{-3}$	61	$({1.47}_{-0.18}^{+0.20})\times {10}^{-3}$	$({2.34}_{-0.23}^{+0.25})\times {10}^{-3}$
M9	$({9.95}_{-2.49}^{+3.32})\times {10}^{-4}$	$({8.53}_{-1.79}^{+2.26})\times {10}^{-4}$	$({9.33}_{-4.66}^{+9.33})\times {10}^{-4}$	44	$({1.06}_{-0.15}^{+0.17})\times {10}^{-3}$	$({1.58}_{-0.18}^{+0.21})\times {10}^{-3}$
L0	$({6.63}_{-1.97}^{+2.80})\times {10}^{-4}$	$({5.68}_{-1.42}^{+1.89})\times {10}^{-4}$	$({4.66}_{-2.88}^{+7.54})\times {10}^{-4}$	21	$({5.04}_{-0.99}^{+1.23})\times {10}^{-4}$	$({7.50}_{-1.23}^{+1.47})\times {10}^{-4}$
L1	$({4.97}_{-1.66}^{+2.49})\times {10}^{-4}$	$({2.37}_{-0.85}^{+1.32})\times {10}^{-4}$	⋯	28	$({6.72}_{-1.16}^{+1.40})\times {10}^{-4}$	$({1.02}_{-0.15}^{+0.17})\times {10}^{-3}$
L2	$({8.29}_{-2.24}^{+3.07})\times {10}^{-4}$	$({3.79}_{-1.12}^{+1.60})\times {10}^{-4}$	⋯	21	$({5.04}_{-0.99}^{+1.23})\times {10}^{-4}$	$({7.75}_{-1.26}^{+1.50})\times {10}^{-4}$
L3	$({4.14}_{-1.48}^{+2.31})\times {10}^{-4}$	$({2.37}_{-0.85}^{+1.32})\times {10}^{-4}$	⋯	16	$({3.84}_{-0.86}^{+1.10})\times {10}^{-4}$	$({5.80}_{-1.07}^{+1.31})\times {10}^{-4}$
L4	$({5.80}_{-1.82}^{+2.65})\times {10}^{-4}$	$({3.79}_{-1.12}^{+1.60})\times {10}^{-4}$	⋯	23	$({5.52}_{-1.05}^{+1.29})\times {10}^{-4}$	$({8.75}_{-1.34}^{+1.58})\times {10}^{-4}$
L5	$({4.97}_{-1.66}^{+2.49})\times {10}^{-4}$	$({3.32}_{-1.04}^{+1.51})\times {10}^{-4}$	$({4.66}_{-2.88}^{+7.54})\times {10}^{-4}$	28	$({6.72}_{-1.16}^{+1.40})\times {10}^{-4}$	$({1.44}_{-0.18}^{+0.20})\times {10}^{-3}$
L6	$({4.14}_{-1.48}^{+2.31})\times {10}^{-4}$	$({2.84}_{-0.95}^{+1.42})\times {10}^{-4}$	⋯	⋯	⋯	⋯
L7	$({7.46}_{-2.11}^{+2.94})\times {10}^{-4}$	$({4.70}_{-2.90}^{+7.70})\times {10}^{-5}$	⋯	⋯	⋯	⋯
L8	$({4.97}_{-1.66}^{+2.49})\times {10}^{-4}$	$({1.42}_{-0.62}^{+1.09})\times {10}^{-4}$	⋯	⋯	⋯	⋯
M7−M9.5	⋯	⋯	⋯	169	(4.1 ± 0.3) × 10−3	(7.1 ± 0.5) × 10−3
L0−L5	⋯	⋯	⋯	137	(3.3 ± 0.3) × 10−3	(5.4 ± 0.4) × 10−3
M7−L5	⋯	⋯	⋯	306	(7.3 ± 0.4) × 10−3	(12.6 ± 0.6) × 10−3

Notes. The units for all values are pc⁻³. These number densities take into account the sky coverage in each survey, but not the survey incompleteness. For the 20 pc sample of Cruz et al. (2007), the volume coverage is 36%; for the Reid et al. (2008) survey of the same volume, the coverage is 63%. For the Kirkpatrick et al. (2012) 8 pc sample, the volume coverage is 100%. For this study, the volume coverage is 63.6% ± 0.59%. Uncertainties are calculated from Poisson statistics.

^aNumber of sources within 25 pc, declinations accessible by SpeX (−50° ≤ δ ≤ +67°), and galactic latitudes outside of ±15° from the plane.

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Figure 26 compares our number densities to other UCD field studies, including the 20 pc samples of Cruz et al. (2007) and Reid et al. (2008) and the 8 pc sample of Kirkpatrick et al. (2012), extended into the substellar regime. Our number densities are consistently higher than those of Reid et al. (2008), particularly on the M dwarfs, although their study does not claim completeness on spectral types earlier than L0. Except for the M7 and L5 edges, our number densities are comparable within 2σ to those of Cruz et al. (2007) for all spectral types, albeit they claim only a lower limit on L dwarf densities. However, out densities are, on average, slightly higher than those of Cruz et al. (2007), except for the M8 bin. Cruz et al. (2007) found 99 objects between M7 and L8 in 20 pc with a sky coverage of 36%, which scales to 244 sources at 25 pc for our sky coverage of 63.5% and 69% completeness, yet we count 327 sources within a shorter spectral-type range. This ≥34% difference can be attributed to new discoveries, improvements in source color selection (i.e., Schmidt et al. 2015), and broader availability of parallaxes. The 8 pc sample of Kirkpatrick et al. (2012) is sparse on the L dwarf regime, with only one L5 within that volume, and while they include 11 M7−M9.5 dwarfs, they claim no completeness on the M dwarf range. We identify 19 M7−M9.5 sources in the literature within the 8 pc volume and therefore have larger number densities than Kirkpatrick et al. (2012), including a few new discoveries since then.

Table 17 also shows number densities for the M7−M9.5, L0−L5, and M7−L5 ranges. We find that the late-M dwarf raw number density agrees within 20% of Cruz et al. (2007), but our number density corrected by the selection function and incompleteness is ∼45% higher, largely driven by the latter. Our L dwarf densities cover a smaller spectral-type range than Cruz et al. (2007), and raw and corrected densities follow the same proportions as for the M dwarf regime. Taking the full range of M7−L5 spectral subtypes, we find 40% higher densities than Cruz et al. (2007), with a raw density of (7.3 ± 0.4) × 10⁻³ pc⁻³ and a corrected density of (12.6 ± 0.6) × 10⁻³ pc⁻³. Our volume density implies that the total number of M7−L5 dwarfs within the 25 pc volume could be as high as ∼820.

5.5.2. Luminosity Function with Respect to Absolute Magnitudes

We follow a similar procedure to calculate the luminosity function with respect to absolute magnitude in J. We use the subsample of 306 sources described in Section 5.5.1, but we organize it into absolute magnitude bins. Our luminosity function is described in Table 18. Figure 27 shows the resulting luminosity function, including Poisson error bars. Using the Filippazzo et al. (2015) empirical relations, we determine that the 10.3–14.2 mag range in the J band encompasses the M7−L5 dwarf range, including the 1σ (0.4 mag) relation uncertainties. Our luminosity function peaks at the 10.25–10.75 mag bin, which roughly corresponds to the peak at the M7−M8 spectral class, matching our spectral-type distribution from Figure 3. Our luminosity function then tapers off to a plateau after the 12.25 mag bin.

Table 18.  Luminosity Function

MJ	N	Ntrig	Nphot	SFplx	SFphot	Ncorrected	Density (mag pc−3)
9.75	4	4	0	0.15	0.16	38.65	$({9.28}_{-1.39}^{+1.62})\times {10}^{-4}$
10.25	51	41	10	0.27	0.27	273.75	$({6.57}_{-0.39}^{+4.14})\times {10}^{-3}$
10.75	68	60	8	0.66	0.61	150.76	$({3.62}_{-0.29}^{+0.31})\times {10}^{-3}$
11.25	50	49	1	0.92	0.80	79.00	$({1.90}_{-0.20}^{+0.23})\times {10}^{-3}$
11.75	41	39	2	0.96	0.79	62.55	$({1.50}_{-0.18}^{+0.20})\times {10}^{-3}$
12.25	30	28	2	0.96	0.78	45.99	$({1.10}_{-0.15}^{+0.18})\times {10}^{-3}$
12.75	25	24	1	0.92	0.76	39.71	$({9.54}_{-1.41}^{+1.65})\times {10}^{-4}$
13.25	26	24	2	0.76	0.62	50.44	$({1.21}_{-0.16}^{+0.18})\times {10}^{-3}$
13.75	16	14	2	0.53	0.38	45.91	$({1.10}_{-0.15}^{+0.18})\times {10}^{-3}$

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Our luminosity function follows from the faint end of the Reid et al. (2003b) luminosity function, as seen in Figure 28, matching it well within the uncertainties. Throughout the 10.75–13.75 mag range, our luminosity function resembles the downward slope of the Cruz et al. (2007) corresponding function.

The IMF is a direct outcome of the formation process. Measurements of the IMF across the hydrogen-burning limit have revealed that brown dwarfs are not a significant contributor to dark matter (Reid et al. 2003b), but brown dwarfs could be as abundant as stars (e.g., Mužić et al. 2017). The efficiency of the star formation process at low masses and the minimum mass allowed by the gravitational fragmentation of a molecular cloud can be determined by quantifying the IMF. Constraining the IMF at low masses is a necessary step toward determining the prevalence of different brown dwarf formation mechanisms (Reipurth & Clarke 2001; Padoan & Nordlund 2002; Whitworth & Zinnecker 2004; Stamatellos et al. 2007) and whether or not they depend on environmental conditions (e.g., Whitworth et al. 2007; Bate 2019).

Mass functions are typically derived from luminosity functions, a straightforward operation for main-sequence stars. For UCDs, however, the mapping is no longer one-to-one due to the long lifetimes of VLM stars and the mass–age–luminosity degeneracy of brown dwarfs. Substellar IMFs can be directly measured in clusters and young moving groups where the age is known for all members (e.g., Taurus, Luhman 2000; TW Hydrae, Looper 2011; Gagné et al. 2017). Measuring the substellar field IMF requires assumptions about the age distribution (Burgasser 2004). Nevertheless, the field luminosity function presented here is an important step toward measuring an accurate mass function across the hydrogen-burning limit in the field and the overall formation history and evolution of UCDs in the Milky Way.

This sample also has the potential to reveal UCD hosts to habitable-zone terrestrial planets like those orbiting TRAPPIST-1 (Gillon et al. 2016, 2017). Currently, this source is the only example of a planetary system around a UCD and the only planetary system known with three potentially habitable terrestrial worlds. With this volume-limited ultracool sample, planetary population studies around the lowest-mass stars and brown dwarfs can be approached in a systematic way (e.g., SPECULOOS; Delrez et al. 2018).