Abstract
We present the most precise constraints to date for the mass and age distributions of single ultracool dwarfs in the solar neighborhood, based on an updated volume-limited sample of 504 L, T, and Y dwarfs within 25 pc. We develop a Monte Carlo approach using the statistic to correct for incompleteness and obtain a space density of pc−3 for spectral types L0–Y2. We calculate bolometric luminosities for our sample, using an updated "super-magnitude" method for the faintest objects. We use our resulting luminosity function and a likelihood-based population synthesis approach to simultaneously constrain the mass and age distributions. We employ the fraction of young L0–L7 dwarfs as a novel input for this analysis that is crucial for constraining the age distribution. For a power-law mass function , we find , indicating an increase in numbers toward lower masses, consistent with measurements in nearby star-forming regions. For an exponential age distribution b(t) ∝ e−βt we find β = −0.44 ± 0.14, i.e., a population with fewer old objects than often assumed, which may reflect dynamical heating of the Galactic plane as much as the historical brown dwarf birthrate. We compare our analysis to that of Kirkpatrick et al., who used a similar volume-limited sample. Although our mass function measurements are numerically consistent, their assumption of a flat age distribution is disfavored by our analysis, and we identify several important methodological differences between our two studies. Our calculation of the age distribution of solar neighborhood brown dwarfs is the first based on a volume-limited sample.
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1. Introduction
Brown dwarfs are substellar objects more massive than planets but less massive than stars (≈13–70 MJup; Dupuy & Liu 2017). Given this intermediate station, studies of brown dwarf formation have sought to determine whether the objects are extreme low-mass outcomes of standard star formation processes or high-mass products of planet formation (i.e., in disks around stars), or have some other origin. Observations of nearby (≈100–500 pc) star-forming regions indicate that brown dwarfs form like stars, via turbulent fragmentation and core collapse in molecular clouds (Luhman 2012), but it is unclear whether other mechanisms, such as massive disk instability (Boss 1997) or ejection from young multiple systems (Reipurth & Clarke 2001), may contribute significantly to the field population (Chabrier et al. 2014).
Closer to the Sun (i.e., within a few tens of parsecs), the lack of large and complete samples has hampered our understanding of the history of brown dwarf formation. Most brown dwarfs have been discovered by way of searches in wide-field sky surveys, such as the Sloan Digital Sky Survey (SDSS; York et al. 2000) and the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) that are severely magnitude-limited for the fainter brown dwarf spectral types. Selection of brown dwarfs from wide-field surveys has become efficient only with multiwavelength data (e.g., Mace et al. 2013a; Best et al. 2015), in particular including the red-optical bands from the Panoramic Survey Telescope And Rapid Response System 3π Survey (PS1; Chambers et al. 2024) and the mid-infrared (MIR) bands from the Wide-Field Infrared Survey Explorer (WISE; Wright et al. 2010). In addition, the parallaxes needed to confirm membership in volume-limited samples are expensive to obtain since most brown dwarfs are too faint to be observed by Gaia, and have only recently become available in sufficient numbers to build large, complete samples (e.g., Dupuy & Liu 2012; Faherty et al. 2012; Dahn et al. 2017; Smart et al. 2018; Best et al. 2020; Kirkpatrick et al. 2021).
Precise values for the local brown dwarf mass and age distributions would help discern the dominant formation mechanism(s), but these quantities have been difficult to ascertain (e.g., Marocco et al. 2015). Brown dwarfs cool throughout their lifetimes, with more massive brown dwarfs beginning as late-M dwarfs and progressing through L, T, and Y spectral types over billions of years, while less massive objects reach the cooler spectral types more quickly (e.g., Burrows et al. 1997; Kirkpatrick 2005). This continuous cooling means that a younger, less massive brown dwarf can have the same directly observable properties (e.g., Teff and luminosity) as an older, more massive brown dwarf. Further, there is no evolutionary phase analogous to the stellar main sequence where Teff can serve as a proxy for mass over any given age range. The masses and ages of field brown dwarfs are therefore not usually measurable from observations; exceptions include dynamical masses from orbital monitoring of binaries and spectroscopic features of unusually old and young brown dwarfs. Kinematic distributions have provided statistical measures of age (Wielen 1977; Faherty et al. 2009), but have not proven capable of distinguishing between a constant, evolving, or largely stochastic birth history (e.g., Burgasser 2004; Day-Jones et al. 2013; Kirkpatrick et al. 2021).
A statistical approach that can overcome the mass-age degeneracy is to synthesize model populations of brown dwarfs characterized by an initial mass function (IMF) and age distribution, evolve the populations to present-day luminosity and Teff using an evolutionary model, and compare the synthetic populations to observations of a well-characterized sample of brown dwarfs. If the IMF has not changed over time, and no external influences have altered the mass distribution of nearby ultracool dwarfs, then the present-day mass function will be the same as the IMF. Previous efforts in this vein established the first estimates of the local low-mass IMF and formation history (e.g., Burgasser 2004; Allen et al. 2005; Deacon & Hambly 2006; Metchev et al. 2008; Pinfield et al. 2008; Burningham et al. 2010b; Reylé et al. 2010; Kirkpatrick et al. 2012; Burningham et al. 2013; Day-Jones et al. 2013; Marocco et al. 2015). These studies used the best space density, luminosity function, binary fraction, and evolutionary models available at the time, but were unable to place significant constraints on either the mass or age distribution for ultracool dwarfs (spectral types M6 and later). Collectively, they only constrained the substellar mass function slope to −1 ≲ α ≲ 1 (for space density as a function of object mass given by Ψ(M) ∝ M−α ) and could only infer broad agreement with a constant formation rate over the history of the Galaxy. However, new well-characterized volume-limited samples of brown dwarfs in the solar neighborhood—those of Best et al. (2021, hereinafter Paper I) and Kirkpatrick et al. (2021, hereinafter K21)—have breathed new life into the population synthesis approach, enabling precise measurements of observable distributions such as bolometric luminosity (Lbol) or Teff needed to constrain the underlying mass and age functions. K21 presented a full-sky 20 pc volume-limited sample from which they derive a mass function estimate of α = 0.6 ± 0.1, assuming a uniform age distribution. In this paper, we present new simultaneous constraints on both the mass and age distributions of very low-mass stars and brown dwarfs in the solar neighborhood using bolometric luminosities from a 25 pc volume-limited sample of L, T, and Y dwarfs.
2. Volume-limited Sample
2.1. Sample Definition
In Paper I, we presented a volume-limited sample of 369 L0–T8 dwarfs out to 25 pc defined entirely by parallaxes. Our sample was the first to comprehensively map the L/T transition (spectral types ≈L8–T4), and its 22 young (≲200 Myr) members suggested a young-leaning age distribution. We have now updated our volume-limited sample to include recent nearby brown dwarf discoveries and parallax measurements from the literature (e.g., K21, Zhang et al. 2021; Gaia Collaboration et al. 2023). The boundaries of the sample remain the same—decl. −30° ≤ δ ≤ 60° (covering 68% of the sky) out to 25 pc from the Sun—but we have extended the coolest spectral type from T8 to Y2, i.e., to include the coldest known types. 6 We now also require parallax uncertainties to be less than one-seventh of the parallaxes, rather than one-fifth as in Paper I. This excised only one object from the Paper I sample: SDSS J152103.24+013142.7 (spectral type T3, parallax 43.3 ± 6.2 mas).
We present our updated volume-limited sample in Table 1. The sample now contains 504 objects, increasing from 369 in Paper I primarily due to the inclusion of 86 objects with spectral types T8.5 and cooler. Nearly all of these were discovered or discussed by K21. While we previously required parallaxes and spectroscopic confirmation for inclusion in our sample, we have allowed 73 objects (15% of our new sample) lacking one or both of these measurements that K21 concluded were bona fide brown dwarfs, 7 so that our volume-limited sample will more completely represent the coldest members of the solar neighborhood. All members of our updated volume-limited sample and associated photometry are tabulated in the UltracoolSheet 8 (which also identifies the Paper I volume-limited sample).
Table 1. Our Volume-limited 25 pc Sample of L0–Y2 Dwarfs
Object | Parallax a | Spectral Type a , b | Flag c | YMKO | JMKO | HMKO | KMKO | Mbol d | Source e | References |
---|---|---|---|---|---|---|---|---|---|---|
(mas) | (Optical/NIR) | (mag) | (mag) | (mag) | (mag) | (mag) | (Disc; ϖ; SpT; Flag; Phot) | |||
SDSS J000013.54+255418.6 | 70.8 ± 1.9 | T5/T4.5 | ⋯ | 15.80 ± 0.06 | 14.73 ± 0.03 | 14.74 ± 0.03 | 14.82 ± 0.03 | 16.75 ± 0.18 | D | 123; 73; 180,32; ...; 73,123 |
WISE J000517.48+373720.5 | 126.9 ± 2.1 | ⋯/T9 | ⋯ | 18.48 ± 0.02 | 17.59 ± 0.02 | 17.98 ± 0.02 | 17.99 ± 0.02 | 20.02 ± 0.05 | S | 158; 122; 158; ...; 134 |
2MASS J00132229-1143006 | 40.3 ± 3.1 | ⋯/T3pec | ⋯ | ⋯ | 16.05 ± 0.02 | 15.74 ± 0.22 | 15.76 ± 0.22 | 16.58 ± 0.24 | D | 110; 14; 110; ...; 14,15 |
2MASSW J0015447+351603 | 58.8 ± 0.3 | L2/L1.0 | ⋯ | 14.95 ± 0.05 | 13.74 ± 0.02 | 12.96 ± 0.02 | 12.25 ± 0.02 | 14.40 ± 0.10 | D | 114; 89; 114,6; ...; 15,14 |
PSO J004.6359+56.8370 | 46.5 ± 3.9 | ⋯/T4.5 | ⋯ | ⋯ | 16.22 ± 0.02 | ⋯ | ⋯ | 17.03 ± 0.23 | J | 14; 14; 14; ...; 14 |
2MASS J00282091+2249050 | 42.9 ± 1.6 | ⋯/L7: | ⋯ | 16.82 ± 0.05 | 15.49 ± 0.02 | 14.55 ± 0.06 | 13.77 ± 0.06 | 15.25 ± 0.14 | D | 39; 89; 39; ...; 15,14 |
WISE J003110.04+574936.3 | 71.0 ± 3.2 | ⋯/L9 | BY | 15.92 ± 0.05 | 14.770 ± 0.010 | 13.862 ± 0.014 | 13.21 ± 0.03 | 15.71 ± 0.13 | D | 217; 122; 12; 16,1; 15,78,12 |
PSO J007.9194+33.5961 | 45.4 ± 3.8 | ⋯/L9 | ⋯ | 17.48 ± 0.06 | 16.38 ± 0.02 | 15.46 ± 0.05 | 14.67 ± 0.05 | 16.06 ± 0.16 | D | 13; 14; 13; ...; 15,14 |
2MASS J00320509+0219017 | 41.3 ± 0.3 | L1.5/M9 | ⋯ | 15.443 ± 0.004 | 14.220 ± 0.002 | 13.446 ± 0.002 | 12.797 ± 0.002 | 14.15 ± 0.10 | D | 193; 89; 193,227; ...; 126 |
2MASS J00332386-1521309 | 43.5 ± 0.8 | L4 β/L1: fld-g | Y | 16.38 ± 0.08 | 15.22 ± 0.06 | 14.26 ± 0.04 | 13.39 ± 0.04 | 14.92 ± 0.12 | D | 96; 89; 54,3; 54,3; 15 |
Notes. This table updates the volume-limited sample of Paper I. This table lists all spectroscopically confirmed L0–Y2 dwarfs having declinations between −30° and +60° and parallax-determined distances less than 25 pc. Objects with photometric spectral types and/or distances from K21 that meet the other criteria are also included. YMKO, JMKO, HMKO, and KMKO photometry enclosed in single brackets indicates synthetic photometry (Paper I); double brackets indicate photometry converted from 2MASS into the MKO system using MK s ,2MASS and the polynomials of Dupuy & Liu (2017). The full table contains 504 rows.
References. (1) This work, (2) Albert et al. (2011), (3) Allers & Liu (2013), (4) Artigau et al. (2006), (5) Artigau et al. (2011), (6) Bardalez Gagliuffi et al. (2014), (7) Bardalez Gagliuffi et al. (2019), (8) Bardalez Gagliuffi et al. (2020), (9) Beamín et al. (2013), (10) Beichman et al. (2014), (11) Bernat et al. (2010), (12) Best et al. (2013), (13) Best et al. (2015), (14) Best et al. (2020), (15) Best et al. (2021), (16) W. Best et al. (2024, in preparation), (17) Bihain et al. (2013), (18) Bouy et al. (2003), (19) Bouy et al. (2005), (20) Bowler et al. (2010), (21) Burgasser et al. (1999), (22) Burgasser et al. (2000a), (23) Burgasser et al. (2000b), (24) Burgasser et al. (2002), (25) Burgasser et al. (2003a), (26) Burgasser et al. (2003c), (27) Burgasser et al. (2003b), (28) Burgasser et al. (2003d), (29) Burgasser et al. (2004), (30) Burgasser et al. (2005b), (31) Burgasser et al. (2005a), (32) Burgasser et al. (2006a), (33) Burgasser et al. (2006b), (34) Burgasser & McElwain (2006), (35) Burgasser et al. (2008a), (36) Burgasser et al. (2008b), (37) Burgasser et al. (2008c), (38) Burgasser et al. (2010b), (39) Burgasser et al. (2010a), (40) Burgasser et al. (2011), (41) Burgasser et al. (2016), (42) Burningham et al. (2008), (43) Burningham et al. (2009), (44) Burningham et al. (2010b), (45) Burningham et al. (2010a), (46) Burningham et al. (2011), (47) Burningham et al. (2013), (48) Castro & Gizis (2012), (49) Castro et al. (2013), (50) Chiu et al. (2006), (51) Cruz et al. (2003), (52) Cruz et al. (2004), (53) Cruz et al. (2007), (54) Cruz et al. (2009), (55) Cushing et al. (2011), (56) Cushing et al. (2014), (57) Cushing et al. (2016), (58) Cushing et al. (2018), (59) Cutri et al. (2003), (60) Dahn et al. (2002), (61) Dahn et al. (2017), (62) Deacon et al. (2005), (63) Deacon et al. (2011), (64) Deacon et al. (2012a), (65) Deacon et al. (2012b), (66) Deacon et al. (2014), (67) Deacon et al. (2017a), (68) Deacon et al. (2017b), (69) Delfosse et al. (1997), (70) Delfosse et al. (1999), (71) Delorme et al. (2008), (72) Dupuy et al. (2009), (73) Dupuy & Liu (2012), (74) Dupuy & Kraus (2013), (75) Dupuy et al. (2015a), (76) Dupuy & Liu (2017), (77) T. Dupuy (private communication), (78) Dye et al. (2018), (79) Faherty et al. (2012), (80) Faherty et al. (2014), (81) Faherty et al. (2016), (82) Faherty et al. (2018), (83) Fan et al. (2000), (84) Gagné et al. (2014), (85) Gagné et al. (2015b), (86) Gagné et al. (2015a), (87) Gagné et al. (2017), (88) Gagné & Faherty (2018), (89) Gaia Collaboration et al. (2023), (90) Gauza et al. (2015), (91) Geballe et al. (2002), (92) Gelino et al. (2014), (93) Gizis et al. (2000), (94) Gizis et al. (2001), (95) Gizis (2002), (96) Gizis et al. (2003), (97) Gizis et al. (2011b), (98) Gizis et al. (2011a), (99) Gizis et al. (2013), (100) Gizis et al. (2015), (101) Goldman et al. (2010), (102) Gomes et al. (2013), (103) Goto et al. (2002), (104) Greco et al. 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(2002), (138) Liebert et al. (2003), (139) Liu et al. (2002), (140) Liu & Leggett (2005), (141) Liu et al. (2010), (142) Liu et al. (2011), (143) Liu et al. (2012), (144) Liu et al. (2013), (145) Liu et al. (2016), (146) Lodieu et al. (2005), (147) Lodieu et al. (2007b), (148) Lodieu et al. (2012), (149) Looper et al. (2007), (150) Looper et al. (2008), (151) Lucas et al. (2010), (152) Lucas et al. (2012), (153) Luhman et al. (2007), (154) Luhman et al. (2012), (155) Luhman (2014b), (156) Luhman & Sheppard (2014), (157) Luhman (2014a), (158) Mace et al. (2013a), (159) Mace et al. (2013b), (160) Mamajek et al. (2018), (161) Manjavacas et al. (2013), (162) Marocco et al. (2010), (163) Marocco et al. (2013), (164) Marocco et al. (2015), (165) Martín et al. (1999), (166) Martín et al. (2010), (167) Martin et al. (2018), (168) McMahon et al. (2013), (169) McMahon et al. (2021), (170) Meisner et al. (2020b), (171) Meisner et al. (2020a), (172) Metchev et al. (2008), (173) Mugrauer et al. (2006), (174) Murray et al. (2011), (175) Muzic et al. (2012), (176) Nakajima et al. (1995), (177) Nilsson et al. (2017), (178) Peña Ramírez et al. (2015), (179) Phan-Bao et al. (2008), (180) Pineda et al. (2016), (181) Pinfield et al. (2008), (182) Pinfield et al. (2012), (183) Pinfield et al. (2014a), (184) Pinfield et al. (2014b), (185) Pope et al. (2013), (186) Potter et al. (2002), (187) Radigan et al. (2008), (188) Rebolo et al. (1998), (189) Reid et al. (2000), (190) Reid et al. (2001), (191) Reid et al. (2006b), (192) Reid et al. (2006a), (193) Reid et al. (2008), (194) Reylé (2018), (195) Ruiz et al. (1997), (196) Salim et al. (2003), (197) Schilbach et al. (2009), (198) Schmidt et al. (2010a), (199) Schmidt et al. (2010b), (200) Schneider et al. (2014), (201) Schneider et al. (2015), (202) Schneider et al. (2016), (203) Scholz & Meusinger (2002), (204) Scholz (2010a), (205) Scholz (2010b), (206) Scholz et al. (2011), (207) Scholz et al. (2012), (208) Scholz et al. (2014), (209) Scholz & Bell (2018), (210) Scholz (2020), (211) Skrutskie et al. (2006), (212) Smart et al. (2018), (213) Smith et al. (2014), (214) Smith et al. (2018), (215) Strauss et al. (1999), (216) Thalmann et al. (2009), (217) Thompson et al. (2013), (218) Thorstensen & Kirkpatrick (2003), (219) Tinney et al. (2003), (220) Tinney et al. (2005), (221) Tinney et al. (2018), (222) Torres et al. (2019), (223) Tsvetanov et al. (2000), (224) Vrba et al. (2004), (225) Warren et al. (2007), (226) Wilson et al. (2001), (227) Wilson et al. (2003), (228) Wright et al. (2013), (229) Zhang et al. (2017), (230) Zhang et al. (2019), (231) Zhang et al. (2021), (232) van Leeuwen (2007).
a Parallaxes and spectral types enclosed in curly braces were determined photometrically. b β, γ, and δ indicate classes of increasingly low gravity based on optical (Kirkpatrick 2005; Cruz et al. 2009) or NIR (Gagné et al. 2015b; Cruz et al. 2018) spectra. fld-g indicates NIR spectral signatures of field-age gravity, int-g indicates intermediate gravity, and vl-g indicates very low gravity (Allers & Liu 2013). c B = binary (presented here as a single unresolved source); C = resolved companion to a star or brown dwarf; Y = young (≲200 Myr); S = subdwarf. d Mbol for unresolved binaries (see Flag column) should be treated with caution. e Source of the Mbol value (Section 3.4). D = conversion from using the polynomial from Dupuy & Liu (2017); J = JMKO bolometric correction from Liu et al. (2010); K = KMKO bolometric correction from Liu et al. (2010); S = Super-magnitude (DK13; Appendix A).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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2.2. New Parallaxes
We present new J-band parallax measurements for four members of the volume-limited sample, shown in Table 2 and also included in Table 1. The objects were observed with CFHT/WIRCam as part of the Hawaii Infrared Parallax Program (Dupuy & Liu 2012). The observation strategies, data reduction, and astrometric solutions are all as described in Dupuy & Liu (2012) and Dupuy & Liu (2017).
Table 2. New Parallaxes and Proper Motions
Relative | Absolute | ||||||||||||
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Object | αJ2000 | δJ2000 | Epoch | ϖrel | μδ,rel | ϖabs | μδ,abs | Nep | Δt | χ2/dof | |||
(deg) | (deg) | (MJD) | (mas) | (mas yr−1) | (mas yr−1) | (mas) | (mas yr−1) | (mas yr−1) | (yr) | ||||
CFBDS J030135.11-161418.0 | 045.3965467 | −16.2382606 | 54436.31 | 48.0 ± 3.8 | 297.8 ± 3.0 | 126.1 ± 3.1 | 49.1 ± 3.8 | 301.9 ± 3.2 | 124.4 ± 3.4 | 8 | 8.06 | 10.3/11 | |
2MASSI J0512063-294954 | 078.0269272 | −29.8311129 | 57317.60 | 48.8 ± 2.8 | −1.0 ± 1.6 | 78.8 ± 2.0 | 49.6 ± 2.8 | 0.6 ± 1.8 | 79.9 ± 2.3 | 7 | 2.12 | 10.3/9 | |
WISE J064205.58+410155.5 | 100.5233880 | +41.0318223 | 56585.62 | 63.2 ± 1.2 | −0.5 ± 1.2 | −374.8 ± 1.2 | 64.1 ± 1.1 | −2.0 ± 1.4 | −377.4 ± 1.5 | 7 | 2.17 | 10.7/9 | |
2MASS J21543318+5942187 | 328.6376414 | +59.7040555 | 55050.45 | 63.3 ± 1.5 | −160.1 ± 0.5 | −463.8 ± 0.7 | 64.7 ± 1.5 | −166.0 ± 1.1 | −469.6 ± 0.9 | 6 | 5.16 | 9.3/7 |
Note. (αJ2000, δJ2000, epoch): coordinates and epoch for our first observation of that target.
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This is the first published parallax for CFBDS J030135.11-161418.0. K21 published parallaxes for the other three objects, which are generally consistent with ours. We opted to use our measurements because they are more precise. Theissen (2018) also published a parallax for 2MASS J21543318+5942187 using astrometry from WISE. This parallax is formally consistent with ours, but has a much larger uncertainty.
3. Demographics of L, T, and Y Dwarfs
3.1. Lutz–Kelker Bias
Parallax measurements are not exact, so each object in a parallax-defined sample may in fact lie closer or farther than the distance obtained by inverting the parallax. At the boundary of a volume-limited sample, the volume of space just outside the sample is larger than the volume just inside, so typically, there will be more objects with measured distances scattering inward than scattering outward, artificially inflating the number of objects in the sample. This is a manifestation of the bias that Eddington (1913) identified for data near the boundaries of sample bins. Lutz & Kelker (1973) realized that the same concept applies to parallax-defined samples: on average, objects are slightly farther away than measured, and their luminosities are slightly underestimated by the parallaxes. In the Lutz–Kelker formulation, the size of the bias depends on the parallax uncertainty as a fraction of the parallax. The mean fractional uncertainty of our sample is 5%, and the corresponding Lutz–Kelker correction implies that our parallaxes probe a volume of space ≈3% larger than their nominal distances indicate. Since we achieve an uncertainty of ≈10% for our space density calculation (Section 3.3), Lutz–Kelker bias is not a significant source of uncertainty, but we nevertheless account for this bias in our analysis.
Lutz & Kelker (1973) determined that the distribution of the true parallax ϖ about the measured parallax ϖ0 is
where σ is the standard deviation of ϖ0 (i.e., the measurement error). The expected value of ϖ from this distribution is
We corrected the parallaxes in our list to their expected values using Equation (2); the median correction was 0.4 mas. We stress that as the Lutz–Kelker correction is a statistical correction for samples of objects, we used these corrected parallaxes only for analysis of our volume-limited sample as a whole and do not quote them for individual objects. The effect of the correction was to reduce the membership of our volume-limited sample by ≈3%, confirming that this bias is not significant relative to our ≈10% uncertainty on the overall space density.
3.2. Completeness
We estimated the completeness of our volume-limited sample using the statistic (Schmidt 1968), employed by both K21 and our Paper I. Briefly, V is the volume of space with radius equal to the distance from the Sun of a given sample member, and is the volume of space with radius equal to the outer limit of the sample (for our full sample, this is 25 pc). Each object in the sample thus has a value between 0 and 1. If a sample has uniform spatial distribution—a valid assumption for our sample, which sits near the Galactic midplane—the expectation value will be . Our volume-limited sample is centered on the Sun, so we would expect any incompleteness to be in the more distant portions of our sample where objects appear fainter.
Figure 1 shows as a function of limiting distance for our volume-limited sample, for distances 8–25 pc. In addition, we show as a function of distance for five spectral type bins. Uncertainties for were determined using the method described in Paper I; briefly, we used Monte Carlo trials to incorporate the parallax uncertainties and statistical fluctuations (drawn from the binomial distribution) due to our limited sample size. Figure 1 makes clear that our volume-limited sample is close to completeness (≈92% at 25 pc) for spectral types L0–T4. Our sample is less complete for cooler spectral types, for which steadily decreases at larger distances and is well below 0.5 at 25 pc. The full sample has at 14 pc (143 objects). declines beyond this distance to at 25 pc, indicating ≈78% completeness for the full sample. This is lower than in Paper I because of the inclusion of T8.5 and later objects, which are prohibitively faint and difficult to observe beyond ≈15 pc. Overall, the completeness of our sample is very similar to that of Paper I, except for the impact of adding the late-T and Y dwarfs from K21.
Figure 1. as a function of distance for our entire volume-limited sample (spectral types L0–Y2, top panel) and for five spectral type bins (other panels). (This plot updates Figure 3 in Paper I.) The right-hand axes indicate approximate completeness values corresponding to values less than 0.5, estimated as twice the value. Our full sample has to within its uncertainty at 16 pc, consistent with completeness, with the declining trend of implying incompleteness beyond this distance. Most of the incompleteness in our full sample is for spectral types T5 and later, which is unsurprising given their faintness. Our sample is ≈92% complete for L0–T4.5 dwarfs.
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Standard image High-resolution image3.3. Space Density
To accurately calculate the space density of L, T, and Y dwarfs in the solar neighborhood, we needed to correct for incompleteness in our sample. We also needed to account for the uncertainties on the parallax measurements, which affect whether objects observed near the 25 pc boundary are actually inside the sample. To address these issues, we used the UltracoolSheet to compile a list of 1119 L0–Y2 dwarfs from the literature that met the same criteria as for our volume-limited sample (Section 2) but at any distance. We refer to this as the full parallax list. It is effectively our volume-limited sample augmented by L0–Y2 dwarfs with parallax measurements placing them beyond 25 pc. Using this list allowed us to incorporate the impact on our space density measurements of objects appearing to lie beyond 25 pc but whose uncertainties allowed a significant possibility of membership in our sample.
From this full parallax list, we drew new 25 pc volume-limited samples in a Monte Carlo fashion, perturbing the Lutz–Kelker-corrected parallaxes according to their uncertainties (assumed to be normally distributed) and rejecting objects with parallaxes <40 mas. These perturbations sometimes moved objects with parallaxes measured near 40 mas from outside to inside the 25 pc distance limit and vice versa, so the membership varied at the outer edges of the Monte Carlo samples.
We then needed to correct each Monte Carlo sample for incompleteness. In Paper I, we obtained a rough correction by dividing the number of objects in a sample by twice its value (Section 3.2). However, this approach provides only an estimate because varies with different spatial positions of objects within a sample even when the number of objects in the sample is fixed. We therefore developed a new method to correct for incompleteness that accounts for the distribution of objects within the sample and also gives a statistical estimate of uncertainty. For each Monte Carlo trial, we calculated at multiple distances from 2–25 pc (analogous to Figure 1), i.e., for a distance d, we identified the subsample of objects with distances less than d and calculated for that volume-limited subsample. We used steps of 0.1 pc, and identified the largest distance () at which a trial subsample had . We treated the Monte Carlo sample as complete at that distance, and calculated the space density for the sample using the number of objects enclosed in the volume defined by . Finally, we calculated , , and the space density for our volume-limited sample as the median and 68.3% confidence limits from all of the Monte Carlo trials.
Table 3 presents our final space density results and for the entire sample, as well as separately for L, T, and Y dwarfs, for five spectral type bins spanning the sample, for individual spectral types, and for single objects, binaries/triples, companions, and young objects. For the space density, we report two sets of confidence limits that include noise from the binomial and Poisson distributions, respectively. The calculation and purposes of these confidence limits are described in detail in Paper I. Briefly, the binomial uncertainties (σbinomial) reflect the uncertainty in our measurement of the space densities within 25 pc of the Sun, given that our sample only covers 68% of the associated volume. The Poisson uncertainties (σPoisson) reflect the uncertainty in our measurement of the space densities more broadly in our region of our Galaxy, of which our solar neighborhood is a small part. As in Paper I, we adopt the Poisson uncertainties in order to describe the space density of brown dwarfs in general and to enable direct comparison with previous estimates. We note that the uncertainties for our measurements in Table 3 have increased from Paper I in most categories, with our new values better reflecting the uncertainty in the completeness of our sample and its subsets.
Table 3. Space Density and for Our 25 pc Sample of L0–Y2 Dwarfs
Space Density | |||||||
---|---|---|---|---|---|---|---|
(10−3 objects pc−3) | |||||||
Objects | N25 pc | 25 pc a | Value | σbinomial | σPoisson | ||
L0 ≤ SpT < L1 | 16 | 0.49 ± 0.13 | 0.36 | ±0.05 | ±0.10 | ||
L1 ≤ SpT < L2 | 28 | 0.50 ± 0.10 | 0.63 | ||||
L2 ≤ SpT < L3 | 19 | 0.46 ± 0.11 | 0.48 | ±0.06 | |||
L3 ≤ SpT < L4 | 14 | 0.39 ± 0.13 | 0.60 | ||||
L4 ≤ SpT < L5 | 21 | 0.48 ± 0.11 | 0.50 | ±0.06 | |||
L5 ≤ SpT < L6 | 28 | 0.45 ± 0.10 | 0.72 | ||||
L6 ≤ SpT < L7 | 15 | 0.42 ± 0.13 | 0.44 | ±0.08 | |||
L7 ≤ SpT < L8 | 17 | 0.46 ± 0.13 | 0.41 | ||||
L8 ≤ SpT < L9 | 17 | 0.40 ± 0.12 | 22.5 ± 0.5 | 0.49 | ±0.07 | ||
L9 ≤ SpT < T0 | 17 | 0.40 ± 0.13 | 12 ± 2 | 0.46 | ±0.08 | ||
T0 ≤ SpT < T1 | 9 | 0.48 ± 0.18 | 8 ± 2 | 0.20 | ±0.04 | ||
T1 ≤ SpT < T2 | 9 | 0.50 ± 0.19 | 0.18 | ±0.04 | ±0.07 | ||
T2 ≤ SpT < T3 | 15 | 0.43 ± 0.14 | 0.41 | ||||
T3 ≤ SpT < T4 | 11 | 0.64 ± 0.16 | 25.0 ± 0.0 | 0.20 | ±0.07 | ||
T4 ≤ SpT < T5 | 14 | 0.44 ± 0.14 | 0.40 | ±0.08 | |||
T5 ≤ SpT < T6 | 40 | 0.41 ± 0.08 | 1.06 | ±0.12 | |||
T6 ≤ SpT < T7 | 34 | 0.35 ± 0.09 | 1.28 | ±0.17 | |||
T7 ≤ SpT < T8 | 49 | 0.35 ± 0.07 | 1.98 | ||||
T8 ≤ SpT < T9 | 67 | 0.32 ± 0.06 | 3.35 | ||||
T9 ≤ SpT < Y0 | 35 | 0.31 ± 0.08 | 1.81 | ||||
Y0 ≤ SpT < Y1 | 16 | 0.19 ± 0.11 | 8 ± 2 | 3.45 | |||
Y1 ≤ SpT < Y2 | 11 | 0.17 ± 0.12 | 6 ± 2 | 1.31 | |||
SpT = Y2 | 2 | 0.03 ± 0.13 | 1 ± 1 | 15.92 | |||
L0 ≤ SpT < L5 | 98 | 0.47 ± 0.05 | 20.1 ± 0.4 | 58 ± 5 | 2.51 | ||
L5 ≤ SpT < T0 | 94 | 0.43 ± 0.05 | 18.6 ± 2.7 | 2.63 | |||
T0 ≤ SpT < T5 | 58 | 0.49 ± 0.07 | 1.25 | ±0.11 | |||
T5 ≤ SpT ≤ T8 | 168 | 0.37 ± 0.05 | 77 ± 8 | 6.08 | ±0.71 | ||
T8.5 ≤ SpT ≤ Y2 | 86 | 0.28 ± 0.04 | 10.4 ± 0.2 | 8.23 | |||
L0 ≤ SpT < T0 | 192 | 0.45 ± 0.04 | 5.10 | ±0.29 | |||
T0 ≤ SpT ≤ T8 | 226 | 0.40 ± 0.04 | 16.4 ± 0.4 | 7.39 | |||
T0 ≤ SpT < Y0 | 283 | 0.37 ± 0.03 | 10.15 | ||||
Y0 ≤ SpT ≤ Y2 | 29 | 0.17 ± 0.07 | 5.57 | ||||
Single | 429 | 0.39 ± 0.02 | 15.77 | ||||
Binary/triple b | 48 | 0.48 ± 0.08 | 1.09 | ||||
Companion c | 30 | 0.35 ± 0.09 | 1.07 | ||||
Young | 20 | 0.41 ± 0.11 | 0.56 | ||||
All L0 ≤SpT ≤ T8 | 418 | 0.42 ± 0.03 | 12.46 | ||||
All | 504 | 0.39 ± 0.02 | 18.33 |
Notes. This table expands Table 2 from Paper I by adding spectral types T8.5–Y2, and adopts a new correction for incompleteness (Section 3.2). N25 pc: Number of objects in our full volume-limited sample. 25 pc: calculated for our full volume-limited sample. A sample with uniform spatial distribution will have . : median and 68% confidence limits for the largest distance at which (implying a complete sample) from our Monte Carlo trials. : median and 68% confidence limits for the number of objects in the sample out to . Space density: median and 68% confidence limits for divided by the volume of the sample at from our Monte Carlo trials. σbinomial describes how precisely our space density measurements represent the full 25 pc volume around the Sun. σPoisson describes how precisely our space density measurements represent brown dwarfs in our general neighborhood of the Galaxy. The calculation of σbinomial and σPoisson is described in Appendix B of Paper I.
a Mean and standard deviation from Monte Carlo trials that resample the parallaxes from their errors and incorporate binomial uncertainties to account for statistical fluctuations in our sample. b Close binaries and triples are counted as single objects with unresolved spectral types. c Three companions are themselves binaries (see Paper I for details) and are also included in the binary/triple bin.Download table as: ASCIITypeset image
For our full 25 pc sample of L0–Y2 dwarfs, we find pc, objects, and a space density of pc−3. However, proceeding through our sample toward colder spectral types, drops steadily beyond T3, while the number of objects per spectral type peaks at T8 and then drops rapidly. It is unclear to what extent this latter decline is due to an actually smaller number of T9 and Y dwarfs in the solar neighborhood, but the decreasing and imply that our sample is significantly incomplete and our space density estimates may be less accurate for these cold, faint spectral types. Table 3 therefore includes several subsamples with spectral types down to only T8 (not including T8.5) to facilitate comparison with our Paper I sample, and also to emphasize that our results for subsamples including T8.5 and later-type dwarfs should be used with caution.
For L0–T8 dwarfs, our space density estimate of (1.25 ± 0.10) × 10−2 pc−3 has increased by 27% from Paper I, more than the 13% increase in the number of objects with these spectral types. Inspecting the subsamples in Table 3, we note that for subsamples for which is close to 25 pc (implying completeness), we obtain space densities consistent with Paper I when accounting for the larger number of members of our updated sample. Where is significantly less than 25 pc, we systematically find larger space densities than in Paper I, which indicates that the method we used to correct for incompleteness in Paper I failed to account for the spatial distribution of objects within the subsamples and underestimated the space densities of incomplete subsamples.
Our updated pc−3 space density for L dwarfs, as well as our values for the L dwarf subclasses, are systematically ≈20%–80% larger than many previous estimates (Cruz et al. 2007; Reylé et al. 2010; Marocco et al. 2015), although they are formally different only at the ∼1σ–2σ level due to large uncertainties. Bardalez Gagliuffi et al. (2019) have also recently published space densities for M7–L5 dwarfs that are larger than previous estimates, but our early-L measurements are ≈70% smaller than theirs. We addressed this discrepancy in Paper I. Our L dwarf space density is consistent with the recent measurements of K21 based on their 20 pc all-sky volume-limited sample, which is somewhat surprising given that they treat the binary components as separate objects, whereas we do not (Section 3.5); however, assuming a ≈15% binary fraction for our sample, our space densities would differ by ≈10%, i.e., consistent within uncertainties. For T0–T8 dwarfs, a similar trend is clear: our updated pc−3 space density is larger than some previous estimates (Burningham et al. 2013; Marocco et al. 2015) but consistent with those of Metchev et al. (2008), Reylé et al. (2010), and K21.
3.4. Bolometric Magnitudes
Bolometric luminosities (Lbol) are fundamental, observable physical quantities of ultracool dwarfs that can be directly compared to evolutionary models, providing constraints on ages and masses. To determine these constraints for L, T, and Y dwarfs in the solar neighborhood, we calculated the bolometric luminosity (Lbol) of each member of our volume-limited sample. We used the log(Lbol/L⊙) versus polynomial of Dupuy & Liu (2017), derived from Lbol calculated by Filippazzo et al. (2015), for objects with mag, the magnitude range over which this polynomial is valid. This range included the brightest objects in our sample and provided Lbol for 370 objects, but did not include 133 objects with mag or which lacked K-band photometry.
For the remaining objects (mostly having spectral types ≥T8), we created an updated version of the "super-magnitude" method developed by Dupuy & Kraus (2013), applicable to objects with spectral types L4 and later. Our updated method is described in detail in Appendix A. Briefly, we used the Sonora Bobcat model atmospheres (Marley et al. 2021) to define polynomials that convert the combined absolute flux in several bands (usually JMKO, HMKO, and Spitzer/IRAC [3.6] and [4.5], or JMKO, HMKO, and AllWISE W1 and W2) into Lbol. The Sonora Bobcat models cover Teff down to 200 K, below the coldest objects in our sample, so using this super-magnitude method, we were able to calculate Lbol for 105 of the remaining 133 objects, many of which had no previous determination of Lbol.
We used the KMKO (when possible) or JMKO bolometric corrections of Liu et al. (2010) to determine Lbol for 26 of the remaining objects, leaving two for which we could not determine a value of Lbol because they lack the photometry needed for all of the above methods. These two objects, ULAS J074502.79+233240.3 and GJ 758B, were therefore not included in our volume-limited sample.
We converted Lbol to bolometric magnitude (Mbol) using Mbol,⊙ = 4.74 mag. For the super-magnitude-derived Lbol, we propagated the uncertainties from the objects' parallaxes and photometry through Monte Carlo trials to obtain uncertainties for the Mbol. We present our Mbol results in Table 1. We caution that for unresolved binaries our Mbol calculation should be treated as less reliable, particularly for pairs in which the secondary component contributes a significant amount of the total flux.
3.5. Luminosity Function for Single Objects
As discussed at length in Paper I, there is evidence suggesting that the components of binaries and companions to high-mass primaries may have different mass distributions and formation histories from single objects, e.g., the spectral type distribution of wide companions notably favors T dwarfs more than in the rest of our volume-limited sample. The number of binaries and companions in our volume-limited sample may also be impacted by different selection effects. These related but distinct populations therefore need to be treated as such. In addition, a proper analysis of binaries needs Lbol to be calculated separately for each component, requiring resolved absolute photometry, which is often not available, in particular in the MIR Spitzer/IRAC and WISE bands. We therefore focus our remaining analysis on the single objects in our volume-limited sample.
We used The UltracoolSheet and our own high-angular resolution imaging survey (W. Best et al. 2024, in preparation, described briefly in Paper I) to identify 48 binaries and multiples and 30 wide companions (separations typically >10'') to main-sequence primaries in our sample, which we removed to create a volume-limited sample of 429 single objects. Similarly, we removed binaries and wide companions from the full parallax list to create a list of single L0–Y2 dwarfs in our volume-limited sample and beyond. Using this list, we determined the bolometric luminosity function of the single objects in our volume-limited sample, correcting for incompleteness and incorporating Poisson uncertainties in the manner described in Section 3.3. We used bins of 1 mag in Mbol and Monte Carlo trials to calculate uncertainties. The mean effect of our Lutz–Kelker correction (Section 3.1) on our luminosity bins was −0.03 × 10−3 pc−3 per bin. This is less than the uncertainties on the space densities in those bins, confirming that our luminosity function was not significantly impacted by the correction.
We present our completeness-corrected luminosity function in Table 4, and plot it in Figure 2. This is the first bolometric luminosity function calculated for single brown dwarfs in the solar neighborhood. The function spans 13 ≤ Mbol ≤ 24 mag, and appears consistent with a flat distribution for Mbol ≥ 15 mag, although there is a suggestion of a gradual increase in space density toward fainter luminosities. The brightest bin (Mbol = 13–14 mag) corresponds to ≈M8–L1 dwarfs (Zhang et al. 2020) and is thus significantly incomplete due to our exclusion of M dwarfs from our sample, so we do not include this bin in our subsequent population synthesis analysis. On the faint end, the three bins with Mbol ≥ 21 mag have large uncertainties due to fewer objects and pc, so we likewise exclude these bins from our analysis.
Figure 2. The bolometric luminosity function for single objects in our volume-limited sample, plotted as the space density of bins of 1 bolometric magnitude. Along the bottom, we print the number of objects within the volume-complete distance () and the volume-complete distance () for each bin from Table 4; we use these numbers to calculate the space densities. Bins plotted in red have robustly measured space densities, which we use to infer population parameters in Section 4. Bins excluded from our analysis are plotted in gray: the brightest bin (13–14 mag) is incomplete due to our exclusion of M dwarfs from our sample, while the three faintest bins contain fewer than 10 objects within their . The luminosity function is consistent with a flat distribution for Mbol ≥ 15 mag, but the overall shape suggests an increase toward fainter magnitudes.
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Standard image High-resolution imageTable 4. Luminosity Function for Our 25 pc Volume-limited Sample of Single L0–Y2 Dwarfs
Space Density | |||||||
---|---|---|---|---|---|---|---|
(10−3 objects pc−3) | |||||||
Objects | N25 pc | 25 pc | Value | σbinomial | σPoisson | ||
13.0 ≤ Mbol < 14.0 a | 26 | 0.50 ± 0.10 | 0.60 | ||||
14.0 ≤ Mbol < 15.0 | 56 | 0.46 ± 0.07 | 22.6 ± 0.5 | 45 ± 5 | 1.37 | ||
15.0 ≤ Mbol < 16.0 | 55 | 0.44 ± 0.07 | 1.82 | ||||
16.0 ≤ Mbol < 17.0 | 58 | 0.42 ± 0.07 | 17.3 ± 0.7 | 30 ± 5 | 1.99 | ±0.27 | |
17.0 ≤ Mbol < 18.0 | 70 | 0.39 ± 0.07 | 1.96 | ±0.21 | |||
18.0 ≤ Mbol < 19.0 | 48 | 0.35 ± 0.08 | 15.0 ± 1.6 | 1.99 | |||
19.0 ≤ Mbol < 20.0 | 44 | 0.31 ± 0.08 | 1.62 | ||||
20.0 ≤ Mbol < 21.0 | 41 | 0.15 ± 0.08 | 12.9 ± 0.6 | 2.22 | |||
21.0 ≤ Mbol < 22.0 b | 22 | 0.07 ± 0.09 | 8.5 ± 0.1 | 5 ± 1 | 2.85 | ||
22.0 ≤ Mbol < 23.0 b | 7 | 0.02 ± 0.08 | 8.6 ± 0.1 | 3 ± 1 | 1.65 | ±0.53 | |
23.0 ≤ Mbol < 24.0 b | 1 | 0.00 ± 0.02 | 2.8 ± 0.0 | 15.92 |
Notes. The columns are the same as in Table 3, presented here for bins of 1 bolometric magnitude. , , and space densities were calculated and corrected for incompleteness using the same method as for the spectral type bins in Table 3. The space densities are plotted with Poisson uncertainties in Figure 2.
a This bin is incomplete due to our exclusion of M dwarfs from our sample. We do not use this Mbol bin in our comparison with synthetic populations (Section 4.5). b Space density is based on and has large uncertainties. We do not use this Mbol bin in our comparison with synthetic populations (Section 4.5).Download table as: ASCIITypeset image
4. Population Synthesis
4.1. Overview
To constrain the mass and age distributions underlying the observed luminosity function of ultracool dwarfs, we employed population synthesis, i.e., forward modeling of our volume-limited sample. Such modeling adopts parametrized distributions of masses and ages of objects, from which synthetic objects are drawn. Each synthetic object's mass and age can then be used to derive Lbol, Teff, and other physical properties using evolutionary models. We can then compare luminosity functions for our synthetic populations to that of our volume-limited sample to constrain the underlying age and mass parameters. This comparison enables us to simultaneously constrain the mass and age distributions of nearby brown dwarfs.
4.2. IMF
Studies of the stellar IMF typically adopt a power law with form , where M is stellar mass and N is the occurrence rate. This is the form used in the seminal work by Salpeter (1955), who determined that occurrence decreases toward larger masses, with Γ = 1.35 for stars with masses ≥ 1 M⊙ in nearby clusters. Previous work on ultracool dwarfs has commonly used the alternate power-law form (Kroupa 2001)
with Ψ representing space density, so we also used this form, which is related to the Salpeter form by α = Γ + 1 (so α = 2.35 for ≥1M⊙ stars). Figure 3 shows three examples of the brown dwarf mass distributions drawn from different choices of α, where α = 0 means a uniform distribution of masses. Chabrier (2003) has also shown that an IMF, which takes the form of a power law above 1 M⊙ and a log-normal distribution below 1 M⊙ (defined by a peak mass and a characteristic width) is consistent with data available at the time.
Figure 3. Left: examples of mass distributions drawn from Kroupa's power-law form of the IMF for α = {−0.5, 0, +0.5}. Right: examples of exponential age distributions with β = {−0.2, 0, +0.2}. Both vertical axis scales are arbitrary. Previous ultracool dwarf studies have found estimates spanning −1 ≲ α ≲ 1 with α = 0 corresponding to a uniform mass function, and have typically assumed a uniform age distribution (β = 0).
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Standard image High-resolution image4.3. Age Distribution
For age distributions, we used the exponential form
where b is the birthrate and t ∈ [0, 10] Gyr is the time since the formation of the Galaxy; hence, the present-day age of an object born at time t is (10 − t) Gyr. This is the most commonly used form in previous work that does not simply assume a flat age distribution (e.g., Allen et al. 2005; Deacon & Hambly 2006; Day-Jones et al. 2013). Figure 3 shows three examples of age distributions resulting from different choices of β. Note that β = 0 is equivalent to a flat age distribution, which is consistent with many estimates for the formation history of nearby stars (e.g., Soderblom et al. 1991; Gizis et al. 2002) but has been called into question by more recent work (e.g., Fantin et al. 2019; Mor et al. 2019).
4.4. Evolutionary Models
We used two sets of evolutionary models to generate Lbol and Teff for members of our synthetic ultracool dwarf populations (Section 4.5).
(1) The solar-metallicity "hybrid" evolutionary models of (Saumon & Marley 2008, hereinafter SM08) have to date provided the best matches to measured masses and luminosities for L through mid-T dwarfs (Dupuy et al. 2015b; Dupuy & Liu 2017; Chen et al. 2022). The SM08 models assume a gradual loss of cloud opacity as objects cool through the L/T transition, coupling cloudy models at 1400 K with cloudless models at 1200 K by linearly interpolating the surface boundary condition in Teff for each surface gravity in the model grid. The models are conjectural in that they do not derive from a physical explanation for the cloud clearing, but they are coupled to the cooling of the brown dwarf interiors. The SM08 models span a mass range of 0.002–0.085 M⊙ and an age range of 3 Myr–10 Gyr, so these are the boundaries of our synthetic populations.
(2) The AMES-COND evolutionary models (Baraffe et al. 2003, hereinafter COND) underlie the widely used BT-Settl model atmosphere grids (Allard et al. 2012). BT-Settl includes clouds for L dwarfs and a physical prescription for the transition to clear-photosphere T dwarfs, making these models well-suited for studying both L and T dwarf atmospheres as well as the L/T transition. However, the atmospheric component of the BT-Settl models is not coupled with the evolutionary component and does not impact the Lbol and Teff in the model grid. The evolutionary components of the BT-Settl models are thus identical to each other and to the original COND models, except that the BT-Settl versions have more densely sampled age grids and extend to higher masses (beyond the upper-mass limit of our analysis). We therefore refer to this evolutionary model grid as simply COND, although we use the solar-metallicity grid published with BT-Settl. The COND models have a wider mass range of 0.0005–0.1 M⊙ and age range of 1 Myr–10 Gyr, but for consistency we limit our synthetic populations to the boundaries required by the SM08 models.
We note that more recent evolutionary models are available, but they sample less of the brown dwarf parameter space or do not incorporate clouds. BHAC15 (Baraffe et al. 2015) has a lower-mass limit of 0.01 M⊙ and a lower Teff limit of 1200 K, and thus excludes most T dwarfs and all Y dwarfs. ATMO2020 (Phillips et al. 2020) has an upper-mass limit of 0.075 M⊙, and therefore does not include objects with Mbol ≤ 14 mag at ages ≳1 Gyr, thus excluding synthetic analogs of the low-mass stars in our volume-limited sample. In addition, ATMO2020 does not incorporate the impact of atmospheric clouds on evolution. The Sonora Bobcat models (Marley et al. 2021) have masses spanning 0.0005–0.1 M⊙ and an age range of 0.01 Myr–20 Gyr but do not incorporate clouds.
4.5. Construction and Comparison of Synthetic Luminosity Functions
We constructed synthetic populations with the same volume and space density as our corrected 25 pc sample (Table 4) of single objects, oversampled by a factor of 1000. We drew masses from the power-law distribution of Equation (3) and ages from the exponential distribution of Equation (4), and used the evolutionary models to interpolate Lbol and Teff for each object. We converted the synthetic Lbol to Mbol and binned synthetic objects into the same 1 mag Mbol bins we used for our volume-limited sample (Table 4). Figure 4 shows examples of these synthetic luminosity functions and illustrates the impact of varying the mass distribution (α) on synthetic luminosity functions with the age distribution (β) held constant. Larger values of α generate populations with more low-luminosity objects since larger α favors lower masses (Figure 3). Figure 5 shows the impact of varying β with α held constant: larger β similarly results in populations with more low-luminosity objects since those populations have larger numbers of older objects.
Figure 4. Synthetic bolometric luminosity functions based on power-law IMFs with three representative values of α (black) compared with our volume-limited sample's luminosity function (red/gray, from Figure 2). The synthetic Mbol were derived from the SM08 models and assume an exponential age distribution with β = −0.5. Increasing α (left to right) leads to populations with more faint objects because larger α favors the creation of more low-mass objects.
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Standard image High-resolution imageFigure 5. Synthetic bolometric luminosity functions based on exponential age distributions with three representative values of β (black) compared with the luminosity function for our volume-limited sample (red/gray, from Figure 2). The synthetic Mbol were derived from the SM08 models and assume a power-law IMF with α = 0.5. Increasing β (left to right) leads to populations with more faint objects because larger β favors older objects that have cooled to fainter luminosities.
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Standard image High-resolution imageWhen comparing the synthetic luminosity functions to our volume-limited sample, we did not use the 13 ≤ Mbol < 14 mag bin, which is significantly incomplete in our volume-limited sample due to our exclusion of M dwarfs. We also did not use any objects with Mbol ≥ 21 mag because our volume-limited sample contains fewer than 10 objects within the completeness distance () in each of those faint magnitude bins, so the completeness-corrected space densities in those bins have very large uncertainties. The total space density of the seven bins spanning 14 ≤ Mbol < 21 mag is pc−3. We normalized the luminosity functions of our synthetic populations to match this.
The SM08 models include masses of 0.002–0.085 M⊙ and ages of 3 Myr–10 Gyr, so we could only assign masses and ages to our synthetic objects within these limits. We explored whether this impacted the distribution of 14 ≤ Mbol < 21 mag objects in our synthetic populations. The SM08 model grid includes objects with Mbol < 14 mag at all ages, so the upper-mass limit of the models does not encroach on the Mbol ≥ 14 mag cutoff for our analysis. Objects with mass = 0.002 M⊙ have synthetic Mbol ≥ 21 mag for ages ≥ 100 Myr, so the lower-mass limit of the models only impacts the faint end of our analysis at young <100 Myr ages. The COND models have wider mass and age ranges (Section 4.4), so for consistency with our SM08 population synthesis, we only draw masses and ages from the same ranges. Limitations in model parameter space are not a significant issue in our analysis, given our wide Mbol range.
We constructed a grid of synthetic populations using −2 ≤ α ≤ 2 in steps of 0.02 and −5 ≤ β ≤ 1.5 in steps of 0.02. We initially calculated χ2 for the fit of our volume-limited sample's luminosity function to that of each synthetic population using our mean and rms for each Mbol bin, and we identified a single χ2 minimum at (α, β) = (−0.04, −0.06). However, we also noted significant asymmetry in some of the uncertainties on our volume-limited sample's Mbol bins (Table 4; Figure 2). We therefore sought a method to quantitatively compare our synthetic luminosity functions that did not assume a Gaussian distribution of luminosities in each bin of our volume-limited sample, as is inherent to the χ2 statistic.
For each bin of Mbol, we determined the space density histogram from our Monte Carlo trials (Section 3.5) in bins of 10−5 pc−3, and we smoothed these histograms using a boxcar width of 2.5 × 10−4 pc−3 (≈10% of the extent of the 95% confidence intervals for the Mbol bins). We normalized each smoothed histogram to a total of 1 and treated these as probability distribution functions (PDFs) for the space densities of the Mbol bins in our volume-limited sample. To compare each synthetic luminosity function to our volume-limited sample, we assigned the probability from the corresponding PDF to the synthetic space density in each Mbol bin. We took the product of these probabilities for each synthetic luminosity function to be its overall likelihood.
Figure 6 shows the distribution of likelihoods over the (α, β) parameter grid for the synthetic populations based on the SM08 models. We identified a maximum likelihood at (α, β) = (0.16, −0.12) (similar to the χ2 minimum), but also a large range of (α, β) having likelihoods near the maximum. For synthetic populations based on the COND models, we obtained a broadly similar distribution but with a maximum likelihood at (α, β) = (1.02, −1.52). The confidence limits for both sets of evolutionary models show that the mass function is constrained only to a broad −1 ≲ α ≲ 1, which encompasses all previous literature estimates. β is constrained to ≲0.5, but can extend to large negative values, implying that our volume-limited sample's luminosity function is consistent with a wide range of age distributions, excluding only very old populations. Plausible fits to the volume-limited sample within the 95% confidence limits extend to β < −5 where populations have unrealistically young age distributions, i.e., >40% of objects in the volume-limited sample would have ages less than 200 Myr, and there would be no objects older than ≈2 Gyr. Such populations are clearly in conflict with even the limited age constraints from lithium depletion, binaries with measured dynamical masses, and kinematics (e.g., Kirkpatrick et al. 2008; Dupuy & Liu 2017; Hsu et al. 2021).
Figure 6. Contour plot (smoothed) of likelihoods for the fits of our volume-limited sample's luminosity function to those from synthetic populations based on the α (IMF) and β (age distribution) parameters and the SM08 evolutionary models. The red cross at α = 0.16 and β = −0.12 indicates the maximum likelihood. Contours trace the 68.3%, 95.5%, and 99.7% confidence limits. The mass function is constrained only to a broad −1 ≲ α ≲ 1 range. β is only constrained toward negative values, and plausible (95.5% confidence) fits extend to β < −5 where populations are unrealistically young. Comparing our luminosity function to synthetic populations cannot by itself produce meaningful constraints for the mass and age distributions.
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Standard image High-resolution image4.5.1. A Constraint from the Young L0–L7 Fraction
The lack of strong constraints on β from this luminosity function analysis indicates the need for another source of constraint on the age distribution of our volume-limited sample. In Paper I, we identified young objects based on spectroscopic indicators of youth or kinematic association with a young moving group of stars. Here, we focus specifically on L0–L7 dwarfs in our sample, for which red-optical and near-infrared (NIR) spectroscopic indicators of low surface gravity are available for ages ≲200 Myr (Cruz et al. 2009; Allers & Liu 2013; Gagné et al. 2015b; Liu et al. 2016). Our volume-limited sample contains 12 single L0–L7 dwarfs having such indicators of youth, After correcting for incompleteness and Lutz–Kelker bias, 12 young out of 127 single L0–L7 dwarfs represents a fraction of in our volume-limited sample. For our analysis, we adopt an age of 0–200 Myr for these objects as suggested by Allers & Liu (2013) and Liu et al. (2016), with the caveat that no well-calibrated age scale has been established for the spectroscopic low-gravity indicators. (We discuss plausible age ranges for young L0–L7 dwarfs and their impact on our results in Section 5.2.) We note that our young L0–L7 dwarf fraction is consistent with the 7.6% ± 1.6% fraction of L dwarfs with ages <100 Myr found by Kirkpatrick et al. (2008) based on the presence of lithium absorption.
We converted the Teff values for our synthetic population members to spectral types by inverting the SpT–Teff relation of Stephens et al. (2009), identifying objects with synthetic spectral types L0–L7 (specifically, later than M9.75 and earlier than L7.25). We then calculated the percentage of L0–L7 dwarfs with ages less than 200 Myr in each of our synthetic populations. Figure 7 shows these percentages as a contour plot in the same β versus α space used in Figure 6, but for a narrower, more realistic range of α and β—in particular, setting −1.5 ≤ β ≤ 0.5 to exclude implausibly young age distributions. Within this more realistic range, synthetic populations with lower values of β have more young objects and thus naturally have a higher percentage of young L0–L7 dwarfs. Populations with higher values of α also have a higher percentage of young L0–L7 dwarfs because they have more low-mass objects that cool quickly to later spectral types, so the warmer L0–L7 dwarfs that are in the sample are more likely to be young. Figure 7 illustrates that the constraints in (α, β) from the young L0–L7 population are complementary to those from the luminosity function analysis (Figure 6).
Figure 7. Smoothed contours showing the percentage of single L0–L7 dwarfs in our synthetic populations that are young (<200 Myr), as a function of the α (mass function) and β (age distribution) parameters. Synthetic populations with a larger percentage of young L0–L7 dwarfs occur for higher α (more lower-mass objects) and lower β (more young objects). The brown-dashed contour and shaded region indicate where of L0–L7 dwarfs in the synthetic populations are young, corresponding to the percentage measured in our volume-limited sample.
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Standard image High-resolution imageFigure 8 combines the (α, β) constraints from the luminosity function analysis, recalculated for the grid used in Figure 7, with the constraints from the percentage of young L0–L7 dwarfs in our volume-limited sample. The two sets of contours are roughly perpendicular in the (α, β) plane. We multiplied the probability grids from each analysis to combine these two constraints, obtaining the final (α, β) constraints shown in Figure 8. This combined analysis establishes strong constraints on both α and β, which were not possible based on either analysis separately. We then marginalized the combined probability distribution over the (α,β) grid for each parameter to obtain their median values and confidence limits.
Figure 8. Left: constraints on the α (mass function) and β (age distribution) parameters from our luminosity function analysis using the SM08 evolutionary models, calculated for the same (α, β) grid as in Figure 7. The red cross indicates the likelihood maximum. The overlaid brown 68.3% (1σ), 95.5% (2σ), and 99.7% (3σ) contours represent the fraction of single young L0–L7 dwarfs in our volume-limited sample (shown in Figure 7). Right: product of the probability distributions from the luminosity function and young L0–L7 fraction analyses, with contours showing the 68.3%, 95.5%, and 99.7% confidence limits, and a maximum likelihood at and β = −0.44 ± 0.14. The intersection of the luminosity function and young L0–L7 fraction constraints provides a much stronger joint constraint on α and β than either analysis alone, thereby yielding the strongest statistical constraints to date on the mass and age distributions of nearby L0–Y2 dwarfs.
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Standard image High-resolution image5. Mass and Age Distribution Constraints
For the SM08 evolutionary models, we obtained and β = −0.44 ± 0.14 (median values with 68% confidence limits) with nearly symmetric posterior distributions (Figures 8 and 9). These are more precise statistical constraints on both parameters than from any previous analysis. The fit of the luminosity function for our volume-limited sample with that of the synthetic population generated using α = 0.58 and β = −0.44 yields χ2 = 10.8 for 4 degrees of freedom.
Figure 9. PDFs for the α (mass function, top) and β (age distribution, bottom) parameters marginalized from the combined probability distributions shown in Figures 8 and 10, using the SM08 (red) and COND (blue) evolutionary models, respectively. The median and 68% confidence limits are shown in the upper left in both plots. Both parameters are well constrained, and the significant overlap in the PDFs for each parameter confirms that the choice of the evolutionary model does not significantly impact our results.
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Standard image High-resolution imageRepeating the steps outlined in Section 4.5 but using the COND evolutionary models instead of SM08, we obtained α = 0.50 ± 0.16 and β = −0.58 ± 0.16, again from near-symmetric posteriors (Figures 9 and 10). These values are consistent with those found using the SM08 models, with the corresponding synthetic luminosity function providing a fit with χ2 = 7.2 (4 degrees of freedom). The similarity in the α and β posteriors indicates that the choice of the evolutionary model does not significantly impact our analysis, as the young L0–L7 fraction constraint exerts a strong influence.
Figure 10. Same as Figure 8, but using the COND evolutionary models, resulting in constraints of α = 0.50 ± 0.16 and β = −0.58 ± 0.16. The contours and median parameter values are consistent with those found using the SM08 models, indicating that the choice of the evolutionary model does not significantly impact our analysis.
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Standard image High-resolution imageOur sample includes 57 objects that have only photometric distances from K21 (no parallaxes; Section 2.1). We explored whether the bolometric luminosities we calculated for these objects could be systematically impacting our population synthesis results. We repeated our analysis using the SM08 models but excluding these objects, and obtained nearly identical contours, medians, and confidence limits for α and β. This perhaps surprising result is a consequence of our method of correcting for incompleteness (Section 3.3, which effectively discards objects beyond the largest distance at which a sample or subsample is complete () from our population analysis. These 57 objects without parallaxes are almost all farther away than the completeness distances for their respective bolometric magnitude bins, so removing them from our sample had little impact on our calculations.
5.1. Best-fit Synthetic Luminosity Functions
Figure 11 shows the synthetic luminosity function based on the SM08 and COND models for our respective best-fit α and β. The SM08 luminosity function is reasonably consistent with that of our volume-limited sample, but we note one apparent difference. The models predict a peak in the luminosity function at Mbol = 16–17 mag and a deficit at 17–18 mag. SM08 ascribe this feature to a release of entropy as brown dwarfs lose the clouds from their photospheres in the transition from L to T dwarfs, causing the objects to temporarily slow their rate of cooling at Mbol ≈ 16–17 mag followed by a more rapid decline. This scenario is supported by binaries with dynamically determined masses, whose mass–luminosity relation is better matched to the SM08 hybrid models than to cloudless models (Dupuy et al. 2015b; Dupuy & Liu 2017). Our volume-limited sample's luminosity function is essentially flat across Mbol = 15–21 mag and does not appear to reflect this predicted slowdown in cooling, although the difference is not strongly significant; our Mbol bins differ from the SM08 luminosity function by 1.5σ and 1.4σ in the Mbol = 16–17 mag and 17–18 mag bins, respectively. One possible explanation for the difference is that the transition from cloudy to cloudless objects occurs at lower temperatures in younger, lower-gravity brown dwarfs; this could wash out the luminosity peak seen in the SM08 models (e.g., Metchev & Hillenbrand 2006; Liu et al. 2016), in which the transition occurs over the same Teff range for all objects.
Figure 11. Top: our 25 pc luminosity function (red/gray) overlaid with the best-fit luminosity function (black) from a synthetic population based on the SM08 hybrid models with IMF and age distribution b(t) ∝ e−0.44t . The bins with Mbol < 14 mag and Mbol > 21 mag (gray) were not used in our analysis. Bottom: same as top, but showing the best-fit luminosity function from a synthetic population based on the COND models, with IMF and age distribution b(t) ∝ e−0.58t . The COND-based synthetic luminosity functions match that of our volume-limited sample better overall, in particular across the L/T transition (Mbol ≈ 15–17 mag).
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Standard image High-resolution imageThe COND-based synthetic luminosity function (Figure 11) appears more closely matched to our volume-limited sample, especially over the Mbol = 15–20 mag range, and does not show a peak/deficit or other feature that indicates an impact from the L/T transition cloud clearing on bolometric luminosity. This is expected from the cloudless COND models, but is more surprising in our volume-limited sample that contains many L/T transition objects, which are expected to be in various stages of cloud clearing. Future evolutionary models that provide a better accounting for the effects of clouds may shed more light on the impact of L/T transition cloud clearing on the luminosity function. Alternatives to a single power-law mass function and/or an exponential age distribution may also better represent the local ultracool dwarf population.
For the subsequent discussion, we adopt the values of α and β we obtained using the SM08 models because those models better represent the properties of our L0–Y2 sample due to the inclusion of clouds for L dwarfs.
5.2. Ages of Young L0–L7 Dwarfs
In our analysis, the choice of evolutionary models does not have a major impact on our results because the fraction of young L0–L7 dwarfs provides a powerful and complementary constraint for the mass and age distributions. However, the usefulness of the young L0–L7 dwarf fraction is determined by our ability to identify such objects in our volume-limited sample, estimate their ages, and understand the completeness of our young single L0–L7 dwarf sample. The objects in our sample have mostly been vetted for low-gravity spectroscopic signatures in the literature: of the 127 single L0–L7 dwarfs (optical or NIR spectral types) in our volume-limited sample, 109 (86%) have an optical or NIR spectroscopic surface gravity classification. None of the remaining 18 have other indications of youth, e.g., kinematic association with a young moving group, so our young single L0–L7 dwarf sample is likely to be complete. We assume these 18 objects are not young, so our young single L0–L7 dwarf fraction (Section 4.5.1) is technically a lower limit. The identification of more young single L0–L7 dwarfs in our sample would lead to our population analysis finding a more positive α and more negative β, further reinforcing the major trends of our results.
However, the ages corresponding to such low-gravity signatures are not well established. 9 The ≲200 Myr age limit for objects with low-gravity signatures quoted by Allers & Liu (2013) and Liu et al. (2016) derives from the ages of young moving groups, which have significant uncertainties at the older end where low-gravity signatures are weak and fewer such groups have been identified (e.g., Gagné et al. 2018). We explored the uncertainty in this 200 Myr limit by considering the ages of moving groups whose members have weaker low-gravity features. For example, the majority of members of the AB Doradus Moving Group (ABDMG; Zuckerman et al. 2004) with spectroscopic gravity classifications have intermediate int-g spectra (e.g., Liu et al. 2016), and the Myr age of ABDMG (Bell et al. 2015) has 3σ limits spanning ≈100–300 Myr. The Carina-Near Moving Group (Zuckerman et al. 2006), age 200 ± 50 Myr, has L0–L7 dwarfs with both int-g and fld-g (field-age gravity) but no vl-g (very low-gravity) classifications (UltracoolSheet, version 2.0.0, in preparation), and its 2σ age range spans 100–300 Myr.
We therefore reran our analysis using age limits spanning 100–300 Myr for the young L0–L7 dwarfs. This yielded parameters in the ranges of 0.36 ≤ α ≤ 0.88 for the mass function and −0.98 ≤ β ≤ −0.26 for the age distribution, with younger age limits corresponding to higher values of α and lower values of β, i.e., lower-mass and younger populations. These ranges are somewhat broader than the 68% confidence limits for the 200 Myr young L0–L7 age limit, but they extend the plausible intervals almost exclusively toward higher values of α and lower values of β. This firmly reinforces that α is positive and β is negative, pointing to a young and low-mass brown dwarf population in the solar neighborhood.
5.3. The Apparent Peak of the Mass Function
Many studies of the stellar IMF have found that it reaches a single maximum at ≈0.2–0.3 M⊙ (Bastian et al. 2010, and references therein). Our finding of a positive value for α, indicating an increase in numbers toward lower masses over the 0.002–0.085 M⊙ range of our analysis, may appear to be inconsistent with the oft-cited stellar IMF peak. We clarify that this peak is seen using log-normal forms of the mass function, or broken-power-law forms that parameterize the distribution of log(mass). Our analysis, and all analyses of the substellar mass function that use the form in Equation (3), parameterize the mass distribution in linear space, in which masses above the substellar regime are far more spread out. A mass function with α < 1 would appear in log(mass) space to decline toward lower masses, even though it rises in linear-mass space. Our result is therefore not inconsistent with the log-normal IMF peak at low stellar masses. In the linear-mass power-law parameterization of Equation (3), the peak of the mass function may in fact be at its extreme low-mass cutoff, which we discuss in Section 6.2.1.
6. Comparison with Previous Work
6.1. Nearby Ultracool Dwarfs
Previous modeling of the local ultracool population has produced relatively loose constraints on the IMF (Table 5). Our is consistent with the estimates from the pioneering works of Kroupa (2001, α = 0.3 ± 0.7) and Allen et al. (2005, α = 0.3 ± 0.6), although neither of these works constrains the age distribution. Our α is notably higher (indicating an IMF with more lower-mass objects) than the α ≈ 0 estimate of Metchev et al. (2008), based on T dwarf space density measurements that are consistent with ours but derived from a less complete sample, in particular for later-T dwarfs. Our α disagrees even more strongly with several studies that estimated −1 < α < 0 by visually comparing modeled space densities or luminosity functions to the space densities of late-M to late-T dwarfs or subsets thereof (Pinfield et al. 2008; Reylé et al. 2010; Kirkpatrick et al. 2012; Burningham et al. 2013; Day-Jones et al. 2013). These studies relied on smaller and usually magnitude-limited samples, which produced larger uncertainties on their space densities and luminosity functions than we have obtained with our 25 pc parallax-based sample, and did not fit for the age distribution of sample members. All of these previous results are consistent with the broad α distribution we found using only the luminosity function of our volume-limited sample (Figure 6); the inclusion of the young L0–L7 dwarf fraction in our analysis is key to constraining the mass distribution to positive values of α (Figures 8 and 10). We note that if we impose a flat age distribution (i.e., β = 0) on our analysis, as do two of the above studies, we find a maximum likelihood of , which is consistent with all of the above results for α. However, the flat age distribution is clearly disfavored by our analysis (Figure 9).
Table 5. Previous Estimates of Mass Function Exponent α and Age Distribution Exponent β
Source | Region | Range | Mass Function | Age Distribution |
---|---|---|---|---|
Alves de Oliveira et al. (2013) | ρ Ophiuchus | Mass: 5–80 MJup | α = 0.7 ± 0.3 | ≈1 Myr |
Da Rio et al. (2012) | Orion Nebula Cluster | Mass: 0.03–0.29 M⊙ | α = 0.6 ± 0.33 | ≈1–3 Myr |
Gennaro & Robberto (2020) | Orion Nebula Cluster | Mass: 0.005–0.16 M⊙ | α = 0.58 ± 0.06 | ≈1–3 Myr |
Alves de Oliveira et al. (2013) | IC 348 | Mass: 13–80 MJup | α = 1.0 ± 0.3 | ≈3 Myr |
Peña Ramírez et al. (2012) | σ Orionis | Mass: 0.006–0.35 M⊙ | α = 0.6 ± 0.2 | ≈3 Myr |
Damian et al. (2023) | σ Orionis | Mass: 0.004–0.19 M⊙ | α = 0.18 ± 0.19 | ≈2–4 Myr |
Lodieu et al. (2007a) | Upper Scorpius | Mass: 0.01–0.3 M⊙ | α = 0.6 ± 0.1 | ≈5 Myr |
Casewell et al. (2007) | Pleiades | Mass: ≈0.015–0.065 M⊙ | α = 0.62 ± 0.14 | ≈120 Myr |
Kroupa (2001) | Solar neighborhood | Mass: 0.01–0.08 M⊙ | α = 0.3 ± 0.7 | No constraint |
Allen et al. (2005) | Solar neighborhood | Mass: 0.04–0.1 M⊙ | α = 0.3 ± 0.6 | No constraint |
Deacon & Hambly (2006) | Solar neighborhood | Mass: 0.072–0.1 M⊙ | α = 0.95 ± 1.17 | β = −0.13 ± 0.17 |
Pinfield et al. (2008) | Solar neighborhood | SpT: T4–T8.5 | −1 <α <−0.5 | No constraint |
Metchev et al. (2008) | Solar neighborhood | SpT: T0–T8 | α ≈0 | No constraint |
Reylé et al. (2010) | Solar neighborhood | SpT: L5–T8 | α ≲0 | No constraint |
Kirkpatrick et al. (2012) | Solar neighborhood | SpT: T6–Y1 | −0.5 <α <0 | Assumed β = 0 |
Day-Jones et al. (2013) | Solar neighborhood | SpT: L4–T4.5 | −1 <α <0 | β ≲ 0.5 |
Burningham et al. (2013) | Solar neighborhood | SpT: T6–T8.5 | −1 <α <−0.5 | Assumed β = 0 |
K21 | Solar neighborhood | SpT: L0–Y2 | α = 0.6 ± 0.1 a | Assumed β = 0 |
This work (SM08 models) b | Solar neighborhood | SpT: L0–Y2 | α = | β = −0.44 ± 0.14 |
This work (COND models) | Solar neighborhood | SpT: L0–Y2 | α = 0.50 ± 0.16 | β = −0.58 ± 0.16 |
Notes. This table lists representative constraints on the exponent α of a power-law IMF (Ψ(M) ∝ M−α ) and the exponent β of an exponential age distribution (b(t) ∝ e−β t ) from the literature, along with our results at the bottom. The results from nearby clusters listed at the top show a selection with representative values of α from studies that used a power-law IMF. For solar neighborhood studies, only Deacon & Hambly (2006) and this work obtain a measurement for the age distribution; other work assumes a flat age distribution or obtains no significant constraint.
a Uncertainty is an estimate based on the similarity of χ2 for α = {0.5, 0.6, 0.7} from a comparison of their Teff function to synthetic populations. b We adopt these constraints on the mass and age distribution for our discussion in Section 6.Download table as: ASCIITypeset image
Previous efforts have produced very few constraints on the age distribution of ultracool dwarfs (Table 5), with some of those works simply assuming that brown dwarfs have been forming at a constant rate since the birth of the Galaxy ∼10 Gyr ago. Allen et al. (2005) demonstrated that evolutionary models predict systematic differences in the age distributions of ultracool spectral types, with late-L dwarfs being the youngest group overall, but found that their observed luminosity function was unable to sufficiently distinguish between increasing, constant, and decreasing birthrates for nearby ultracool dwarfs. Day-Jones et al. (2013) and Marocco et al. (2015) were able to exclude age distributions strongly favoring old objects using L0–T8 magnitude-limited samples, but did not directly fit for β or a similar parameter. The only previous study that calculated an age distribution, using the same exponential model that we use, found β = −0.13 ± 0.17 for masses between 0.072 and 0.1 M⊙ (early-L and late-M dwarfs; Deacon & Hambly 2006), based on a sample of 55 objects. Our value of β = −0.44 ± 0.14, based on our much larger volume-limited sample and—crucially—including constraints from the fraction of young L0–L7 dwarfs, implies a clearly younger population of L and T dwarfs. Figure 12 demonstrates the inconsistency of our result with a uniform age distribution.
Figure 12. Distribution of ages for our best-fit exponential age distribution β = −0.44 ± 0.14 (red), marginalized over the uncertainty on β. Our age distribution is clearly inconsistent with the uniform age distribution (black) assumed by many previous works, including the recent analysis by K21. Our result is more consistent with the age distribution found by Dupuy & Liu (2017) using dynamical masses of late-M to mid-T binaries.
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Standard image High-resolution image6.1.1. Galactic Dynamics
We note that our age distribution is qualitatively consistent with the age distribution found by Dupuy & Liu (2017) based on dynamical masses of late-M to mid-T binaries (Figure 12). That distribution also skews toward younger objects: They found a median age of 1.3 Gyr, with a 90% confidence interval of 0.4–4.2 Gyr. They highlight the fact that galactic dynamics excite objects out of the Galactic midplane over time (e.g., Robin et al. 2003; Mackereth et al. 2019) and thereby deplete the solar neighborhood of older objects. The skew toward young objects in our volume-limited ultracool sample may therefore be as much a reflection of this dynamical heating as it is the underlying birth history of the L and T dwarfs in the sample.
We also note that as dynamical heating removes objects from the Galactic midplane, the remaining mass distribution may be impacted as well, a scenario that has not been addressed to date. Further modeling of ultracool dwarfs in the Galactic thin disk, thick disk, and halo populations (e.g., Aganze et al. 2022) is needed to establish whether dynamical heating has altered the ultracool mass distribution of the solar neighborhood.
6.2. K21
K21 published the most recent effort to determine the mass function of ultracool dwarfs in the solar neighborhood. They based their analysis on a volume-limited sample of L, T, and Y dwarfs that they describe as complete for Teff ≥ 600 K (spectral types ≲T8), drawing upon the same set of parallaxes as our sample. K21 arrived at α = 0.6 ± 0.1, so in many ways, our works appear to be similar. However, the K21 sample construction and analysis differ from ours in several important ways. We describe these differences in detail.
- 1.The K21 sample covers the full sky but is volume-limited at 20 pc, making theirs a wider but shallower survey that encompasses a 25% smaller volume of space than ours. In contrast, our sample extends to 25 pc but only includes declinations −30° ≤ δ ≤ + 60° (68% of the sky). We chose these declination boundaries because this is the part of the sky that has been thoroughly searched for all spectral types ≤T8. In particular, the optical PS1 survey proved to be essential for constructing complete samples of L/T transition dwarfs (Best et al. 2015, 2020), and PS1 has a southern limit of δ = −30°. There has been no complete survey for L/T transition objects outside of the PS1 footprint because the JHK bands of NIR surveys, such as 2MASS, cannot distinguish L/T transition dwarfs from background M dwarfs (e.g., Reid et al. 2008), and the YJHK UKIDSS survey (Lawrence et al. 2007) and the optical SDSS only observed a subset of the PS1 footprint. If the spatial distribution of ultracool dwarfs within 25 pc of the Sun is significantly nonuniform (contrary to our assumption), our space density measurements could be impacted since our sample does not cover the whole sky, but no confirmed evidence of this has been found to date (Kirkpatrick et al. 2019; Paper I).
- 2.K21 included the resolved components of known binaries as distinct objects in their sample, whereas we excluded binaries in order to avoid mixing the potentially different mass distributions and formation histories of binaries with those of single ultracool dwarfs (Section 3.5). The K21 sample therefore contains more objects (525) even though it encompasses a 25% smaller volume of space than our 25 pc sample (504 objects).
- 3.K21 demonstrated that 150 K wide Teff bins of their sample are statistically consistent with completeness for Teff ≥ 600 K. However, the statistic indicates that their sample taken as a whole is ≈90% complete for such objects, and only ≈80% complete south of decl. −30°, where fewer searches and follow-up observations have occurred. K21 also apply 5%–13% corrections for incompleteness in the Galactic plane when calculating space densities. In contrast, we have calculated space densities only from volumes within our sample that are confirmed to be complete by the statistic (Section 3.3). Our space densities are therefore based on smaller but more robustly complete samples.
- 4.K21 constrain the mass function underlying their volume-limited sample using a population synthesis approach that is similar to ours, but with a few key differences. Notably, K21 draw ages for their synthetic populations from a uniform 0–10 Gyr distribution. This has been common practice in ultracool population studies, but we show it is inconsistent with the space density of young L dwarfs in the solar neighborhood (Section 4.5.1; Figure 7) and our subsequent finding that the overall age distribution clearly favors younger objects (Section 5; Figure 12).
- 5.Rather than using a luminosity function as we do, K21 conduct their analysis using the Teff function of their sample, with temperatures derived mostly from literature studies and H-band photometry, and compare this to synthetic populations to constrain the IMF. They argue that the determination of the bolometric luminosities for their sample should wait until more objects have broad spectral energy distribution (SED) measurements, in particular for the late-T and Y dwarfs. However, the Teff versus MH relation from Filippazzo et al. (2015, hereinafter F15) used by K21 for most objects with spectral types ≤T8 is itself based on Lbol values calculated by F15. The F15 Teff values were also derived from averaging Teff at ages 0.5 and 10 Gyr for all objects except those identified as members of young moving groups and clusters, reinforcing K21's assumption of a flat age distribution. 10 In the end, both K21's and our analyses use polynomial relations based on F15's bolometric luminosities for most of the objects in our samples to constrain the population properties. Our choice to use bolometric luminosities provides a less biased representation of the sample because they do not require an age assumption. In addition, our Mbol uncertainties are small (≈0.1–0.2 mag) relative to the 1 mag bin size of our luminosity function, whereas the K21 Teff uncertainties (mostly 79 K or 88 K) are usually >50% of their 150 K bin size. We also recognize that both studies use several alternative methods to determine Lbol or Teff for the small minority of sources that do not have the data necessary to use relations based on F15's Lbol, which could create inhomogeneities in the Lbol and Teff distributions used in population synthesis. This can best be addressed by more direct measurements of Lbol and Teff.
- 6.K21 allow the space density of their synthetic populations to be a free parameter when fitting their synthetic Teff functions to their volume-limited sample, whereas we fix our synthetic population space density to the value measured from our volume-limited sample. K21's fits therefore have an additional degree of freedom, but their best-match synthetic population may have a space density that is inconsistent with their volume-limited sample.
Despite these differences in our sample construction and analyses, K21 derive space densities for spectral subtype bins with similar precision to ours, and obtain a similar constraint on the exponent of a power-law IMF: α = 0.6 ± 0.1, with their quoted uncertainty representing that the χ2 for their Teff function fit differs little for α ∈ {0.5, 0.6, 0.7}. However, the apparent agreement of our with K21's result is in fact misleading, given the clear disagreement in our age distributions. Formally, the constant age distribution (i.e., β = 0) assumed by K21 is 3.1σ different from our result. If we were to also assume β = 0, we would find to be most likely, but this is 2.4σ different from the and β = −0.44 ± 0.14 we find in our full analysis (Figure 8), and 2.4σ different from the K21 value. It is not clear what difference(s) between our analyses could lead to such similar values for α despite the discrepant age distributions. No one difference is an obvious culprit, so it may be a combination of multiple factors.
K21 favor the SM08 hybrid models over the COND models, largely because they find an excellent match between their Teff distribution and that of SM08, in particular at the pileup in Teff in the L/T transition, which we do not see reflected in our luminosity function. This discrepancy would suggest that the SM08 models have correctly predicted a Teff pileup but incorrectly predicted an Lbol pileup in the L/T transition. However, such a situation is physically contradictory for objects in which cloud clearing is releasing trapped entropy. If these objects experience slower declines in Teff, causing the pileup, then we should also see slower declines in Lbol.
6.2.1. The Low-mass Cutoff
K21 explore one feature of their sample that we do not: the low-mass cutoff of the local brown dwarf population. K21 model this as a sharp cutoff at three possible masses: 10, 5, and 1 MJup, but are unable to obtain a constraint, finding that the best fits to their Teff function do not depend strongly on the cutoff value. They note that more accurate measurements of the space density in their lowest two Teff bins (spanning 150–450 K) will enable a constraint on the low-mass cutoff. The IMF constraint that we have found, like that found by K21, predicts an increase in the number of objects as mass decreases, so a cutoff is needed to avoid generating populations dominated by extremely low-mass objects. In our analysis, the minimum mass was implicitly set by the lowest mass in the SM08 model grid (0.002 M⊙). However, since objects at the faint end of the luminosity range we used for our population synthesis (Mbol = 21 mag) have higher masses at ages ≥ 100 Myr—as much as 0.026 M⊙ at 10 Gyr—our analysis is not sensitive to the low-mass cutoff. At present, the best available constraint is that the minimum brown dwarf mass cannot be greater than the ≈5 MJup of the known free-floating planetary-mass brown dwarfs in nearby star-forming regions (e.g., Luhman et al. 2009; Best et al. 2017; Zhang et al. 2021; Damian et al. 2023).
6.3. Star-forming Regions
Studies of star-forming regions and young clusters that estimate power-law IMFs have mostly found 0 < α < 1, with constraints tending toward the higher end of that range (Table 5; see also, e.g., Moraux et al. 2003; Scholz et al. 2013; Gennaro & Robberto 2020). Our therefore generally agrees with the IMF of nearby star-forming regions. This is consistent with a low-mass IMF that has not changed appreciably over the history of our Galaxy, and with galactic dynamics that have not significantly altered the ultracool mass function of the solar neighborhood.
7. Summary
We have updated the volume-limited sample of L and T dwarfs from (Paper I), expanding it to include L0–Y2 dwarfs and adding recent discoveries and parallax measurements. The sample now contains 504 members, covers 68.3% of the sky (δ = −30° to +60°), extends to 25 pc, and is defined by parallaxes for 85% of the sample, the exception being objects (mostly late-T and Y dwarfs) identified by K21 as brown dwarfs having photometric distances within 25 pc. Our sample is ≈78% complete overall but is ≈92% complete for spectral types L0–T4, indicating near-completeness through the L/T transition out to 25 pc. We corrected for incompleteness using the statistic to identify the maximum distance at which our sample is statistically complete. We included an additional correction for Lutz–Kelker bias, although we found it did not significantly impact our results. We calculated a space density of pc−3 for our volume-limited sample of L0–Y2 dwarfs, ≈20–80% larger than many previous estimates but consistent with that of K21.
We calculated bolometric luminosities and present a completeness-corrected luminosity function for single objects in our volume-limited sample. We used our luminosity function in combination with the fraction of young single L0–L7 dwarfs in our sample and synthetic populations based on the SM08 and COND evolutionary models to simultaneously constrain the mass and age distributions of single brown dwarfs in the solar neighborhood. The luminosity function and young L0–L7 dwarf fraction offered complementary constraints in our analysis, the latter being essential for obtaining a meaningful age distribution constraint. For a power-law mass function (Ψ(M) ∝ M−α ) and exponential age distribution (b(t) ∝ e−β t ), we find and β = −0.44 ± 0.14 using the SM08 models, and we find α = 0.50 ± 0.16 and β = −0.58 ± 0.16 using the COND models.
Our analysis used an age of 0–200 Myr for the young L0–L7 dwarfs, based on spectroscopic low-gravity features and membership in young moving groups, whose maximum ages are not precisely known. Alternate analyses using a wide range of 100–300 Myr for the maximum young L0–L7 dwarf age resulted in parameters spanning 0.36 ≤ α ≤ 0.88 for the mass function and −0.98 ≤ β ≤ −0.26 for the age distribution, somewhat broader than our 68% confidence intervals for a 0–200 Myr age range but clearly confirming that α is positive and β is negative.
Although the SM08 models have been shown to provide a better representation of individual brown dwarf luminosities due to their inclusion of clouds for L dwarfs and subsequent cloud depletion in the transition to T dwarfs, we find that synthetic populations from the COND models provide a better match to our volume-limited sample's luminosity function. However, the consistency of the constraints on the mass and age distributions that we find using the SM08 and COND models indicate that the choice of evolutionary model is not significant in our analysis, and we adopt the SM08-based constraints for α and β. These represent the most precise statistical constraints on both parameters to date, and the first calculation of the age distribution of brown dwarfs in the solar neighborhood based on a volume-limited sample.
Our mass distribution indicates an increase in space density toward lower brown dwarf masses, in tension with many previous estimates for the solar neighborhood (which favored fewer low-mass objects) but consistent with recent findings in nearby star-forming regions. Our age distribution clearly favors younger ultracool dwarfs rather than the commonly assumed uniform age distribution, which may be as much a result of galactic dynamics systematically removing objects from the midplane over time as it is a result of the historical birth rate.
We thank the anonymous referee for the careful review and helpful comments. W.B. acknowledges support from grant HST-GO-15238 provided by STScI and AURA. We acknowledge the grant provided under the John W. Cox Endowment for Advanced Studies in Astronomy by the Department of Astronomy at The University of Texas at Austin that supported A.S. for the duration of Summer 2020. This research was funded in part by the Gordon and Betty Moore Foundation through grant GBMF8550 to M.L. T.D. acknowledges support from UKRI STFC AGP grant ST/W001209/1. This work has benefited from The UltracoolSheet, maintained by W.B., T.D., M.L., A.S., R.S., and Z.Z., and developed from compilations by Dupuy & Liu (2012), Dupuy & Kraus (2013), Liu et al. (2016), Best et al. (2018), Paper I, Sanghi et al. (2023), and Schneider et al. (2023). This research has benefited from the SpeX Prism Library [and the SpeX Prism Library Analysis Toolkit], maintained by Adam Burgasser. 11 This work has made use of data from the European Space Agency (ESA) mission Gaia, 12 processed by the Gaia Data Processing and Analysis Consortium (DPAC). 13 Funding for the DPAC has been provided by national institutions, in particular, the institutions participating in the Gaia Multilateral Agreement. This research has made use of NASA's Astrophysical Data System and the SIMBAD and VizieR databases operated at CDS, Strasbourg, France. For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) license to any Author Accepted Manuscript version arising from this submission.
Appendix A: An Updated Super-magnitude Method for Calculating Bolometric Luminosities
The standard method for determining the bolometric luminosity of a source is to integrate its SED as a function of wavelength or frequency. Ideally, this means integrating a broad spectrum or multiband photometry spanning most of the SED (e.g., F15; Sanghi et al. 2023). For cold brown dwarfs, however, this method is currently not useful as most of the flux emerges in the MIR where spectra and broadband photometry are difficult to obtain from the ground, especially for fainter objects, and JWST has not yet observed a large sample of brown dwarfs in the MIR. Empirical relations in the literature mapping ultracool dwarf spectral types and broadband photometry to bolometric luminosities (e.g., Liu et al. 2010; Dupuy & Liu 2017, F15) are not valid for spectral types ≳T8 due to the lack of such cold objects with independently determined bolometric luminosities.
To overcome this barrier, (Dupuy & Kraus 2013, hereinafter DK13) added the absolute fluxes in several broadband NIR and MIR bandpasses to obtain a super-magnitude for each object. They used the JMKO, HMKO, and Spitzer/IRAC [3.6] and [4.5] bands to define the mJH12 super-magnitude whenever such photometry was available in all four bands for an object since these bands capture ≳50% of the total flux for late-T and Y dwarfs. When those four bands were not all available, they used super-magnitude from other combinations of bandpasses to match the available photometry. DK13 then calculated the super-magnitudes using the same bandpasses for model spectra (Morley et al. 2012) spanning appropriate ranges of temperature, surface gravity, and cloud thickness. They used the mean super-magnitudes and the corresponding model bolometric luminosities to derive bolometric corrections for the super-magnitudes, enabling bolometric magnitude determinations for T8 and later-type dwarfs.
We have updated this method in four ways:
- 1.We now use the flux tables for the Sonora Bobcat cloudless atmosphere models (Marley et al. 2021) to calculate model super-magnitudes using combinations of the JMKO, HMKO, [3.6], [4.5], and AllWISE W1 and W2 bands. The zero-points we used to convert J, H, [3.6], [4.5], W1, W2 fluxes into Vega magnitudes were, respectively, 9.31, 8.52, 2.13, 1.26, 4.94, and 1.90 × 10−11 W m−2 s−1. We calculated these by direct integration of the high-resolution model Vega spectrum from SpeXtool (Cushing et al. 2004) over each bandpass. The zero-points for our super-magnitudes are the sum of the relevant zero-point fluxes, e.g., for mJH12, the zero-point is 2.12 × 10−10 W m−2 s−1.
- 2.Rather than limiting our super-magnitude calculations to ranges of Teff and surface gravity appropriate only for late-T and Y dwarfs, we have calculated super-magnitudes using all of the Sonora Bobcat models (200 ≤ Teff ≤ 2400 K and ).
- 3.Using the super-magnitudes and luminosities from the Sonora Bobcat models, we fit fifth-order polynomials to convert the super-magnitudes directly into bolometric luminosities, eliminating the intermediate calculation of bolometric corrections. This change means that the polynomials require absolute super-magnitudes to accurately determine the bolometric luminosities.
- 4.As the Sonora Bobcat models are computed for three metallicities, [Fe/H] = {−0.5, 0.0, 0.5}, we derived separate polynomials for these three metallicities.
The polynomials converting super-magnitudes to bolometric luminosities are presented in Table 6, along with the rms of the residuals from the polynomial fits in super-magnitude bins. The residuals for the polynomial fits are shown in Figure 13. The residuals for each super-magnitude show little variation over the full range of metallicities. The polynomials give exceptionally tight relations (σ < 0.02 dex) between mJH12 and Lbol for mJH12 < 17, corresponding to Teff ≳ 400 K. At lower Teff, there is an expanding envelope of uncertainty about the polynomial relation. The residuals for the mJHW1W2 polynomials (replacing the Spitzer/IRAC [3.6] and [4.5] bands with the comparable WISE W1 and W2 bands) are very similar. For objects lacking photometry in the NIR J and H bands (which is the case for some known Y dwarfs), the m12 and mW1W2 super-magnitude polynomials have residuals with a spread of ≈0.10 mag, tightening down to ≈0.03 mag at m12 ≈ 15 mag and mW1W2 ≈ 16 mag (Teff ∼ 500 K).
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Standard image High-resolution imageFigure 13. Residuals for the fifth-order polynomial fits converting super-magnitudes to bolometric luminosities, using Sonora Bobcat cloudless atmosphere models.
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Standard image High-resolution imageTable 6. Polynomials Converting Super-magnitude to Mbol
Rms of Polynomial over Magnitude Bins | ||||||||||||||||||
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Super-magnitude | c0 | c1 | c2 | c3 | c4 | c5 | Valid M Range | Super-magnitude | 12–13 | 13–14 | 14–15 | 15–16 | 16–17 | 17–18 | 18–19 | 19–20 | 20–21 | 21–22 |
(mag) | (mag) | (mag) | (mag) | (mag) | (mag) | (mag) | (mag) | (mag) | (mag) | (mag) | ||||||||
[Fe/H] = −0.5 | [Fe/H] = − 0.5 | |||||||||||||||||
MJH12 | −6.4398e+00 | −1.4027e−01 | 2.3226e−01 | −3.0873e−02 | 1.4257e−03 | −2.2589e−05 | 12–22 | MJH12 | 0.011 | 0.009 | 0.012 | 0.016 | 0.012 | 0.025 | 0.038 | 0.049 | 0.065 | 0.026 |
MJHW1W2 | −2.7293e+01 | 6.0786e+00 | −4.9259e−01 | 1.0392e−02 | 2.7620e−04 | −1.0041e−05 | 12–22 | MJHW1W2 | 0.017 | 0.019 | 0.026 | 0.033 | 0.017 | 0.015 | 0.026 | 0.040 | 0.052 | 0.003 |
M12 | −1.5231e+02 | 5.0524e+01 | −6.5969e+00 | 4.1605e−01 | −1.2839e−02 | 1.5571e−04 | 12–20 | M12 | 0.080 | 0.069 | 0.030 | 0.020 | 0.036 | 0.049 | 0.065 | 0.028 | ⋯ | ⋯ |
MW1W2 | −1.5505e+02 | 4.9420e+01 | −6.2087e+00 | 3.7708e−01 | −1.1211e−02 | 1.3099e−04 | 12–20 | MW1W2 | 0.086 | 0.082 | 0.060 | 0.022 | 0.018 | 0.031 | 0.042 | 0.052 | ⋯ | ⋯ |
[Fe/H] = 0.0 | [Fe/H] = 0.0 | |||||||||||||||||
MJH12 | −2.7157e−01 | −2.6723e+00 | 6.1027e−01 | −5.7324e−02 | 2.3075e−03 | −3.3885e−05 | 12–22 | MJH12 | 0.010 | 0.012 | 0.012 | 0.010 | 0.010 | 0.027 | 0.046 | 0.062 | 0.073 | 0.086 |
MJHW1W2 | −1.0493e+01 | 5.6039e−02 | 3.3836e−01 | −4.5086e−02 | 2.0762e−03 | −3.2808e−05 | 12–22 | MJHW1W2 | 0.012 | 0.010 | 0.013 | 0.022 | 0.014 | 0.017 | 0.040 | 0.055 | 0.078 | 0.069 |
M12 | −2.4219e+02 | 7.7727e+01 | −9.8550e+00 | 6.0935e−01 | −1.8529e−02 | 2.2228e−04 | 12–20 | M12 | 0.068 | 0.063 | 0.033 | 0.019 | 0.043 | 0.062 | 0.073 | 0.089 | ⋯ | ⋯ |
MW1W2 | −2.8736e+02 | 8.9031e+01 | −1.0910e+01 | 6.5390e−01 | −1.9305e−02 | 2.2513e−04 | 12–20 | MW1W2 | 0.071 | 0.075 | 0.058 | 0.026 | 0.023 | 0.046 | 0.063 | 0.081 | ⋯ | ⋯ |
[Fe/H] = +0.5 | [Fe/H] = + 0.5 | |||||||||||||||||
MJH12 | −1.9805e+00 | −2.6266e+00 | 6.5022e−01 | −6.1584e−02 | 2.4614e−03 | −3.5652e−05 | 12–22 | MJH12 | 0.019 | 0.020 | 0.018 | 0.014 | 0.011 | 0.022 | 0.045 | 0.064 | 0.080 | 0.094 |
MJHW1W2 | −2.3089e+01 | 3.4414e+00 | −2.6539e−02 | −2.5080e−02 | 1.5099e−03 | −2.6089e−05 | 12–22 | MJHW1W2 | 0.015 | 0.007 | 0.010 | 0.018 | 0.014 | 0.017 | 0.042 | 0.063 | 0.075 | 0.095 |
M12 | −3.6928e+02 | 1.1681e+02 | −1.4623e+01 | 8.9825e−01 | −2.7232e−02 | 3.2676e−04 | 12–20 | M12 | 0.059 | 0.067 | 0.042 | 0.016 | 0.040 | 0.063 | 0.080 | 0.098 | ⋯ | ⋯ |
MW1W2 | −2.7307e+02 | 8.3427e+01 | −1.0076e+01 | 5.9445e−01 | −1.7261e−02 | 1.9793e−04 | 12–21 | MW1W2 | 0.068 | 0.074 | 0.066 | 0.030 | 0.023 | 0.050 | 0.068 | 0.085 | 0.097 | ⋯ |
Note. The polynomials are defined as , where M is the absolute super-magnitude of the object. The rightmost column in the top section gives the valid range for the absolute super-magnitude (not bolometric magnitude). The super-magnitudes used to derive these polynomials were calculated from Sonora Bobcat cloudless atmospheric models.
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Figure 14 compares our bolometric magnitudes derived from super-magnitudes to the Mbol from other sources and methods in the literature, presented as a function of and of spectral type for all L0–Y2 dwarfs with parallaxes and appropriate photometry. All underlying data (photometry and spectral types) used to compute the Mbol were taken from the UltracoolSheet. We show one set of Mbol derived from the mJH12 super-magnitude, using m12 in cases where JMKO or HMKO were unavailable; and a second set derived from mJHW1W2 and mW1W2. Literature sources include the Lbol computed from low-resolution spectra and broadband photometry by F15, the spectral-type-based JMKO and KMKO bolometric corrections of Liu et al. (2010), and the log(Lbol/L⊙) versus polynomial of Dupuy & Liu (2017).
Figure 14. Left: bolometric magnitudes calculated with our updated super-magnitude method for L0–Y2 dwarfs, using mJH12 (dark blue) or m12 (light blue, when JMKO or HMKO photometry were not available), shown as a function of (top) and of spectral type (bottom). Included for comparison are bolometric magnitudes determined (F15; black circles), or calculated using the JMKO and KMKO bolometric corrections of (Liu et al. 2010, brown squares and orange diamonds, respectively), or the log(Lbol/L⊙) vs. polynomial of Dupuy & Liu (2017, red circles). Right: same plots but showing Mbol calculated using the mJHW1W2 (dark blue) or mW1W2 super-magnitudes. All methods generally agree, but the super-magnitude-based Mbol is ≈0.3 mag fainter for mag (mid-T spectral types) and diverges sharply from other methods at spectral types earlier than ≈L4. The super-magnitude method extends the Mbol sequence to fainter and cooler objects than previous methods are able to reach and should be used to determine Mbol for spectral types ≈T8 and later.
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Standard image High-resolution imageIn general, all methods for computing Mbol agree well for most of the L, T, and Y dwarf regimes. The Mbol derived from mJHW1W2 and mW1W2 diverges sharply from other methods for and spectral types earlier than L4. (Insufficient [3.6] and [4.5] photometry exists for such objects to corroborate this trend using mJH12 and m12.) The super-magnitude-based Mbol are ≈0.3 mag fainter for mag (mid-T spectral types), suggesting that the atmosphere models are underpredicting the bolometric luminosities relative to the flux in passbands used in our super-magnitudes for these objects. The super-magnitudes clearly extend the Mbol sequence to fainter and cooler objects than previous methods are able to reach. The steady decline of bolometric luminosity as a function of appears to flatten significantly at mag, suggesting a more rapid decline in K-band flux as these cool objects become colder. Empirical calibration for these objects must await MIR spectra from the James Webb Space Telescope. Until then, we present our updated super-magnitude method as the best way to determine bolometric luminosities for objects with or spectral types T8 and later. For warmer objects, we note that the spectral-type-based KMKO bolometric corrections from Liu et al. (2010) and the polynomial of Dupuy & Liu (2017) give very similar results, and are both consistent with the calculations of F15. If direct calculation of Mbol from spectra and/or broadband photometry is not available or feasible, these methods or other similar conversions from the literature (e.g., F15; Faherty et al. 2016) should be sufficient.
Footnotes
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16 have no parallax measurement; 16 have no spectroscopic confirmation; 41 have neither.
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We also note that two objects we identify as young in our volume-limited sample, 2MASS J00332386-1521309 and 2MASS J10224821+5825453, have optical gravity class β, indicating moderately low gravity (Cruz et al. 2009) but kinematics indicating possible membership in the Galactic thick disk (Gonzales et al. 2019), which suggests a much older age.
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Formally, this is different from the 0–10 Gyr distribution used by K21 for their synthetic populations, so their analysis is not internally consistent, with the largest difference for objects younger than 0.5 Gyr.
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