Characterization of Stellar and Substellar Members in the Coma Berenices Star Cluster

Distance determination is based on parallax measurements, whenever available, by Gaia/DR2 (Gaia Collaboration et al. 2018). Gaia is a space mission designed for astrometry by the European Space Agency, launched on 2013 December 19. The latest data release (DR2), including the first 22 months of the nominal mission lifetime, contains celestial positions and apparent brightness for ∼1.7 billion sources, among which 1.3 billion also have parallaxes and proper motions available (Lindegren et al. 2018). For our study the empirical limit for the Gaia/DR2 is J ∼ 15 mag, and only measurements with $\varpi /{\rm{\Delta }}\varpi \gt 10$ are considered in the analysis, where ϖ is the parallax and Δϖ is the error (Lindegren et al. 2018).

For an object with no parallax data, we estimate its spectral type via multiband photometry, from which the distance is derived. Photometric data in optical wavelengths include those of SDSS/DR12 (Alam et al. 2015) and PS1 (Panoramic Survey Telescope and Rapid Response System, Chambers et al. 2016). In a few cases we utilize the SDSS flags to distinguish a star from a galaxy. For photometry extending to mid-infrared wavelengths, "ALLWISE" (Cutri et al. 2013) has been used, which combines the data of the Wide-field Infrared Survey Explorer (WISE) (Wright et al. 2010) in the cryogenic phase of the mission, and NEOWISE (Mainzer et al. 2011) in the first post-cryogenic phase.

Proper motions are taken from Gaia/DR2 when available, as long as the measurements are reliable, again with ϖ/Δϖ > 10. Alternatively, proper motions are extracted from URAT1 (Zacharias et al. 2015), which is an astrometric catalog as a follow-up project of UCAC. In addition to proper motions, with typical errors 5–8 mas yr⁻¹, URAT1 provides photometry in one single "f" band (between R and I). URAT1 covers almost the entire northern sky and extends down to decl. −15° in some areas, cataloging over 228 million objects at a mean epoch around 2013 May. A large fraction (83%) of the URAT1 entries (3'' matching radius) list 2MASS J, H, and K_s magnitudes. Some 16% of URAT1 sources are supplemented with five-band photometry (BVgri) from the AAVSO Photometric All-Sky Survey (APASS).

In our analysis, photometry is taken from 2MASS whenever available. The 2MASS Point Source Catalog (Skrutskie et al. 2006) has 10σ detection limits of J ∼ 15.8 mag, H ∼ 15.1 mag, and K_s ∼ 14.3 mag, and  saturates around J ∼ 9 mag, H ∼8.5 mag, and K_s ∼ 8 mag.

The UKIDSS/GCS aimed to measure the very-low-mass end of the stellar mass functions in 10 star clusters. As for other UKIDSS surveys, the Large Area Survey (LAS) covered only the edge of Coma Ber, whereas the Galactic Plane Survey (GPS) did not include Coma Ber at all. Proper motions are available for UKIDSS/GCS starting with DR9 (Collins & Hambly 2012; Smith et al. 2014). For the work reported here, we use the latest data release DR10, but its spatial coverage is incomplete within the surveyed sky of 78.5 deg² toward the cluster (see Figure 1), missing a sky area of about 7 deg² in the Z- and Y-bands, 2.5 deg² in the J-band, and 3 deg² in the H-band, due to poor data quality (Boudreault et al. 2012). The K-band observations were taken at 2 epochs to enable proper motion estimates. The typical proper motion error of the GCS in our data is about 5 milli-arcseconds (mas) per year. We have made use of the ZYJHK data, with the detection limits at an error of 0.15 mag Z = 20.5, Y = 20.3, J = 19.5, H = 18.8, K1 = 18.0, and K2 = 18.1 mag, respectively, and with the saturation limits Z = 11.3, Y = 11.5, J = 11.0, H = 11.3, and K1 = 9.9 mag (Lodieu et al. 2012). The photometric sensitivity of each band is depicted in Figure 2. Our investigation is limited to UKIDSS sources with a probability greater than 70% of being a star, using the UKIDSS database flag to distinguish a star from a galaxy, with a photometric error less than 0.15 mag and being fainter than J = 12 mag. The sky area of our study, limited by the UKIDSS/GCS coverage of 78.5 deg², or about a 5° radius toward Coma Ber, is chosen as the "cluster region." In addition, a patch of sky of a 3° radius roughly 11° to the east from the cluster center is used in experimental design as the "control field."

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We analyze evolved members to constrain the age. In Coma Ber, any post-main-sequence members are too bright to render reliable 2MASS photometry, so we characterize them with optical photometry. Table 1 lists the parameters of the five brightest stars in the region. For 18 Com, a subdwarf F5 IV with a Gaia distance 59.8 ± 0.4 pc and proper motions $({\mu }_{\alpha }\cos \delta ,{\mu }_{\delta })=(-17.57\pm 0.17,0.68\pm 0.12)$ mas yr⁻¹,⁷ its deviation from theoretical isochrones (shown in Figure 3) suggests that it is not a part of the cluster. The other four stars have distance, photometry, and kinematics consistent with membership. The star 12 Com, a known member, is a double-lined spectroscopic binary (Griffin & Griffin 2011) consisting of an A2/A3 dwarf and a mid-typed giant (Abt (2008, F6 III), Griffin & Griffin (1986, G7 III)). The coeval age of the binary system 670 Myr (Griffin & Griffin 1986) and a Gaia distance 84.5 ± 1.7 pc both indicate membership.

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The star 31 Com, a G0 IIIp giant, known to have varying $v\sin i$ (Massarotti et al. 2008), suggestive of binarity, is located at 6.8 deg from the cluster center, i.e., outside our analysis range, but it has photometry, astrometry, and distance consistent with membership. It has been considered a member by Casewell et al. (2006) and by Mermilliod et al. (2008), and is also included in our member list.

Using the Padova isochrone (Bressan et al. 2012), assuming null reddening ($E(B-V)=0.006$, Nicolet 1981) and solar metallicity (Friel & Boesgaard 1992; Netopil et al. 2016), an age of 800 Myr gives an overall better fit than younger ages, as evidenced in Figure 3, where for each star the absolute magnitude is computed using the Gaia/DR2 parallax and the apparent magnitude taken from the Bright Star Catalog, without correction for extinction or reddening. The fit is considered satisfactory, given the known binarity of 12 Com and 31 Com, and non-membership of 18 Com. We therefore conclude Coma Ber to be about 800 Myr old.

A bright candidate is selected as having proper motions, from Gaia/DR2 or from URAT1, within 17 mas yr⁻¹ from (μ_α cos δ, μ_δ) = (−11.21, −9.16) mas yr⁻¹, a range judiciously chosen to include all known proper motion members in the literature. Figure 4 illustrates how this range encompasses the literature candidates. The concentration is more obvious for the samples of Casewell et al. (2006) and Kraus & Hillenbrand (2007), which included proper motions in their membership criteria, than for those of Mermilliod et al. (2008) or Melnikov & Eislöffel (2012), which applied no proper motion criteria.

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Furthermore, a bright candidate is selected as being brighter than J = 14 mag with photometric errors <0.15 mag, and along the PARSEC isochrone (Bressan et al. 2012; Chen et al. 2014, 2015; Tang et al. 2014) bracketed with a color range (−0.07, +0.3) in the J versus J − K_s color–magnitude diagram (CMD), (−0.25, +0.15) in J versus J − H, and (−0.07, +0.11) in H versus H − K_s. With these criteria, binary systems would still be selected. After excluding 23 candidates, all fainter than about J ∼ 12 mag, considered as galaxies by SDSS (class = 3), a total of 450 sources satisfy the initial proper motion and CMD scrutiny. Of these, 393 have Gaia/DR2 counterparts.

Figure 5 plots the distance distributions of (a) all Gaia stars in the cluster region (within 5° radius), and all Gaia stars in the control field (3° radius), with a sky area 9/25 of the cluster region, and (b) the 393 preliminary candidates with Gaia measurements available. The clustering around 85 pc stands out clearly, particularly in (b). The fact that in (b) away from the peak the number does not increase much with distance, hence the space volume, in contrast to the case in (a), indicates an effective winnowing by proper motions and CMD.

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Gaia Collaboration et al. (2017) analyzed a radius 104 around Coma Ber, and reported 50 members based on Gaia/DR1 data. All their members have been confirmed by our selection (40 within and 9 outside the 5° cluster-centric radius), except BD+27 2139, which should have been in their list but is not, perhaps because of an editing glitch (Gaia Collaboration et al. 2017, their Table D.2 containing only 49 entries, though there should have been 50).

To bootstrap the three proper motion data sets used in this study, we compare the Gaia/DR2, URAT1, and UKIDSS/GCS measurements in the cluster region, shown in Figure 6. The Gaia/DR2 and URAT1 measurements are consistent with each other, and are used to supplement each other for bright candidates. There is, however, a systematic offset of UKIDSS/GCS measurements relative to those of URAT1, computed for all stars with J = 12–15 mag, i.e., common in both data sets, (Δμ_αcosδ, Δμ_δ) = (−3.57, −0.61). After the offset is applied, the UKIDSS/GCS proper motion vector center (−7.64, −8.55) mas yr⁻¹ is used to select faint members.

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Distance is a critical parameter in membership identification. Although the apparent magnitude of a star in isochrone fitting and the proper motion both implicitly incorporate the distance criterion, ambiguity exists. In addition to direct parallax measurements, we have developed a distance estimator using the photometry data taken from the PS1, SDSS, and YJHK from UKIDSS, plus W1 and W2 from WISE. Our algorithm first estimates the spectral type of a star. This part is adapted from the photo-type method developed by Skrzypek et al. (2015), in which a combination of photometric colors of a target is compared against a database of templates of different spectral types, from which a match is chosen, in a least-squares sense, as the most probable spectral type. The work by Skrzypek et al. (2015) was devised for late-M, L, and T type dwarfs only, and we expand the templates to include earlier spectral types (see the Appendix). Once the spectral type is determined, the distance is then derived by comparison of the apparent magnitude and absolute magnitude in each band, rendering a median distance when all bands are considered. Our experiments using different stellar data sets with spectral types known to be K or M types indicate an accuracy within 1–2 subtypes in most cases, with the majority of interlopers being extragalactic or post-main-sequence objects, both expectedly rare in our field. Earlier than K type, the estimator still works reasonably fine, albeit with larger scattering; see Figure 7. Currently, no interstellar reddening is taken into account, and the algorithm is validated only for main-sequence stars. This distance estimator, which we call phot-d, offers an effective filtering by distance of stars that would have contaminated our member sample by chance inclusion in proper motion and CMD selection. More details on phot-d are given in the Appendix.

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Figure 8 illustrates the J versus J − K_s CMD toward Coma Ber. Member candidates chosen on the basis of proper motions, distances, and isochrone are marked, together with those satisfying proper motion and isochrone conditions but having inconsistent distances, which would have been disguised as contaminants if no distance information were available. Also shown is the false-positive sample of the control field processed following the same selection procedure used for the cluster region. In this analysis, the distance range has been taken as 50–120 pc to account for the possible uncertainty of the phot-d distance. It is encouraging that the false-positive rate is low, particularly for the bright candidates (≳1 M_☉) Within this distance range, there are 131 bright Gaia members, averaging 87.0 pc with a standard deviation (dispersion) 8.1 pc.

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For faint stars, we have adopted the proper motion vector center (μ_α cos δ, μ_δ) = (−7.64, −8.55) mas yr⁻¹ for UKIDSS/GCS, also within a radius of 17 mas yr⁻¹ for membership selection. In addition, a candidate is selected to be fainter than J = 12 mag, with photometric errors <0.15 mag, and along the DUSTY isochrone bracketed with a color range (−0.35, +1.2) in the Z versus Z − K CMD, and (−0.1, +0.5) in J versus J − K, as shown in Figure 9. The ranges of colors are chosen to be deliberately wide to allow for uncertainties in photometry/color and also in isochrones. The distance range, as chosen for the bright sample, is chosen to be 50–120 pc for either the Gaia or the phot-d distances.

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Note that in J versus J − K, which is often used to identify substellar objects, the isochrone passes through populated regions, so such a CMD is not as discriminating as those involving shorter wavelengths such as Z versus Z − K for low-mass objects. Our investigation hence relies primarily on Z versus Z − K, though for very cool objects a marked flux suppression sometimes renders detection at neither Y nor Z in UKIDSS/GCS. For these, analysis with J versus J − K would be applied. The lesson is that there are no preferred colors to identify cool objects, and a combination of CMDs must be iterated. As a comparison, Figure 10 presents how literature candidates behave in our diagnostic 2MASS J versus J − K_s and UKIDSS/GCS Z versus Z − K CMDs, overlaid with several theoretical 800 Myr isochrones adopting the average distance 85 pc. We note that for a nearby cluster such as Coma Ber, assuming a single distance in a CMD would introduce intrinsic scattering unless absolute magnitudes are plotted (see for example Figure 3). While the bright literature candidates by and large follow the model isochrones, the faint ones are discrepant.

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Combining the bright (Section 3.2) and faint (Section 3.3) candidates together results in 194 candidates within the cluster region. This sample is spatially complete, notwithstanding the UKIDSS voids, and forms the basis of our characterization of the cluster. The candidates are listed in Table 2. The first column gives the running number. Columns 2 and 3 are the coordinates, followed by columns 4 to 9 representing J mag, its error, H mag, its error, and K mag and its error. Columns 10 to 12 give the proper motions and the error. Column 13 lists the distance, followed by column 14, which contains the references if the star is a known literature candidate. The last column indicates the data source, where "1" stands for 2MASS photometry (JHK_s) plus Gaia/DR2 proper motions, "2" stands for 2MASS photometry (JHK_s) plus URAT-1 proper motions, "3" stands for UKIDSS/GCS (JHK) plus Gaia/DR2 proper motions, and "4" means that JHK photometry and proper motion are both from UKIDSS/GCS, with no transformation between 2MASS and UKIDSS photometric measurements.

Candidates in Table 2 are categorized as (1) those with parallax distances (Nos. 1–154; this is the most reliable member list to our knowledge to date of the Coma Ber cluster); and (2) the other 38 with distances estimated by phot-d (Nos. 155–192). For sources with parallax measurements but ϖ/Δϖ < 10, their photo-d distances are adopted. In each category the entries are in ascending R.A. order.

The criterion of the ratio ϖ/Δϖ > 10 for Gaia/DR2 data is biased against a distant source for which ϖ would be small (so is the ratio) even though Δϖ is already relatively small. These sources tend to be distributed in the vertical segment of the J versus J − Ks CMD. They are hence likely background giants; as such, phot-d, which is valid for dwarfs only, would underestimate the distances. At the moment we do not have an effective method to remove these individual contaminants from the member list, except by statistical subtraction by the control sample.

There are individual objects considered as member candidates in the literature but outside the 5° radius. A total of 15 have been reaffirmed by our analysis, including, for example, 31 Com, presented above as a post-main-sequence member (Section 3.1), and two reported by Kraus & Hillenbrand (2007): HD 111878 with a Gaia distance 85.1 pc, and HD 109390 at 120.1 pc (close to the limit of our distance range). This selected sample is spatially incomplete and thus is not included in further analysis, but is listed in Table 3 for reference, with the same table format as in Table 2, except with no last column because all data are from 2MASS and Gaia/DR2.

For potential usefulness, we also summarize in Table 5 the literature candidates rejected by our selection. The table is in a two-column format arranged in ascending R.A. order. For each star, the coordinates, references for candidacy, and an offending code in our analysis are given: 1 = rejection by proper motions, 2 = rejection by CMD, 4 = rejection by distance. The code is additive, so, for example, a literature candidate that has consistent proper motions but is inconsistent with being a member in CMD position and in distance has a code = 6.

Any member fainter than Z = 17.3 mag has a mass less than 0.08 M_☉, and is therefore a brown dwarf. There are several lines of evidence to further substantiate its brown dwarf nature. First, late-M, L, and T dwarfs are known to have UKIDSS colors different from those of field stars (Hewett et al. 2006), and all our brown dwarf candidates indeed have colors, shown in Figure 11, consistent with being M- or L-type objects. Second, all these candidates have spectral types estimated by phot-d as being brown dwarfs.

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Moreover, while very-low-mass objects have distinctly red $W1-W2$ colors (Kirkpatrick et al. 2011) owing to the lack of methane absorption at W2 (4.6 μm) relative to the flux at W1 (3.4 μm), their W3 (12 μm) and W4 (22 μm) fluxes often fall below the sensitivity limits of WISE unless they are located in the solar vicinity (see, for example, Scholz et al. 2011).

Among the efforts to identify brown dwarfs in Coma Ber, the spectroscopic study by Casewell et al. (2014) led to confirmation of an M9 (their cbd34, R.A. = 12:23:57.37, decl. = +24:53:29.0, J2000, J = 15.94 mag), an L1 (their cbd67, R.A. = 12:18:32.71, decl. = +27:37:31.3, J2000, J = 17.68 mag), and an L2 (their cbd40, R.A. = 12:16:59.89, decl. = +27:20:05.5, J2000, J = 16.30 mag), among which cbd40 was stripped of its membership on the basis of its brightness and colors. Our candidate list includes cbd34 but not cbd67, which satisfies neither the proper motion nor the photometric criterion, and is therefore a field brown dwarf. West et al. (2011) compiled a catalog of spectroscopic M dwarfs on the basis of the Sloan Digital Sky Survey DR7. Among the M dwarfs in our cluster region, 10 satisfy our selection criteria of proper motions, CMD, and distance, and indeed have been included in our list.

Table 4 summarizes the properties of the substellar objects, a subset of the member candidates in Table 2. Following the same identification numbers as in Table 2 in the first column, the next columns list, respectively, the coordinates, UKIDSS Z, Y, J, H, and K1 magnitudes, and then the UKIDSS proper motions. The last two columns compare the spectral type determined with spectroscopy, as reported in the literature or observed by us, and the spectral type estimated with phot-d. In general, the spectral typing with phot-d is in agreement within 1–2 subtypes with observations. This gives us confidence in our phot-d method. The first six sources in Table 4 are all of late-M types known in the literature (West et al. 2011; Casewell et al. 2014), and we have confirmed their membership. The M9 objects, namely Nos. 176, 55, and 130, were the coolest known members in Coma Ber before our work.

In Table 4 there are a few miscellaneous objects , A, B, and C, that are not classified as member candidates but are worth clarification. Stars A and B have similar infrared colors (from Z to WISE W2) as those of known brown dwarfs, as seen in the two-color diagrams shown in Figure 11. Yet, at shorter wavelengths object A has PS1 measurements ${g}_{P1}=19.97$ mag, ${r}_{P1}=18.73$ mag, and ${i}_{P1}=17.61$ mag, and object B has ${g}_{P1}=20.55$ mag, ${r}_{P1}=19.30$ mag, and ${i}_{P1}=18.03$ mag. Compared with the mean values and standard deviation of the M-type brown dwarfs in Table 4, g_P1 = 21.51 ± 0.56 mag, r_P1 = 21.94 ± 0.19 mag, and i_P1 = 19.54 ± 0.32 mag, the two stars stand out as significantly brighter. The phot-d analysis suggests both to be of early-M types. While the binarity of a hot plus a cold component may explain the brightness inconsistency, we do not have evidence at the moment as to the nature of either star.

Object C has JHK magnitudes and proper motions consistent with being substellar, but has Z and Y magnitudes below the UKIDSS/GCS limits. It is the only source in the table that has been clearly detected not only in W1 and W2, but also in W3 and W4 (see Figure 12.) It is likely a galaxy.

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3.4.1. Follow-up Spectroscopy

Confirmation spectra of the brown dwarf candidates were acquired with Palomar/TripleSpec or with Gemini/GNIRS. Three targets, No. 159, A, and B, were observed using TripleSpec (Herter et al. 2008) on 2016 December 14, with a slit width 1''. The sky was clear and the observations were executed at an airmass of about 1.20. The standard A-B-B-A nodding sequence was followed along the slit to record target and sky spectra. The exposure time per pointing was 300 s, with a total integration of 1200 s for each target. Flat-fields and argon lamp spectra were taken after every set of target observations. A nearby A0 V star was observed for each target for telluric correction, as well as for flux calibration.

SPEXTOOL package version 4.1 (Vacca et al. 2003; Cushing et al. 2004) has been used for processing of the data taken with TripleSpec, which includes the pre-processing, aperture extraction, and wavelength calibrations. The individual extracted and wavelength-calibrated spectra from a given sequence of observations, each with their own A0 standard star, were then scaled to a common median flux and combined using XCOMBSPEC in SPEXTOOL. The combined spectra were corrected for telluric absorption and flux-calibrated using the respective telluric standards with XTELLCOR. All calibrated sets of observations of a given target were then median-combined to produce the final spectrum.

The spectra of candidates No. 191 and 160 were acquired by Gemini Fast Turnaround GN-2017B-FT-18, on 7 and on 11 December 2017, respectively, both under good sky conditions, using the cross-dispersed mode of GNIRS (Elias et al. 2006a, 2006b), covering 0.9–2.5 μm simultaneously with a resolving power R ∼ 1200. The short blue camera with 32 lines/mm grating was selected with a slit width 045 for No. 191 and 0675 for No. 160. The individual exposure time under the A-B-B-A nodding sequence was 150 s for No. 191, and 60 s for No. 160. Three sets of nodding sequence were observed for No. 191, and one set was taken for No. 160.

The GNIRS raw data were reduced by PyRAF using the Gemini and GRNIRS packages. We first cleaned the pattern noise, radiation events, flat-field, and sky-subtraction, then did the wavelength calibration by spectra of arc lamps. Each spectrum was extracted from the combined ABBA exposure files. Telluric absorption lines were removed with two A2 V standard stars (HIP 58297 and HIP 63006) observed before or after the observing run.

Each reduced Palomar or Gemini 1D spectrum was then compared with the low-dispersion template spectra of brown dwarfs from SpeX (Rayner et al. 2003), from which the "best" match, judged by eye examination, was determined. Three candidates turned out to be bona fide substellar objects, with one late-M (No. 160), one early-L (No. 159), and one mid-L type (No. 191), shown in Figure 13. The classification was warranted by the characteristic absorption features due to methane/water near 1.4 μm.

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The Palomar/TripleSpec spectra for objects A and B are exhibited in Figure 14, together with template spectra from SpeX. Both spectra show the CN band feature near 1.1 μm characteristic of giants and supergiants (Loidl et al. 2001; Lançon et al. 2007). The Gaia measurements, however, yield ϖ = 2.5 ± 0.2 mas for star A, and ϖ = 2.6 ± 0.3 mas for star B, respectively, placing each at ∼400 pc. Their optical brightness (g_P1 ∼ 20 mag) is hence consistent more with M dwarfs (M_V = 9–10 mag) than with giants (M_V ∼ −0.3) (Cox 2000). Further spectroscopic observations are required to provide information on the nature of these two stars, such as, for example, whether they belong to the dwarf carbon population. In any case, neither of them is associated with the cluster.

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The age we derive for Coma Ber, ∼800 Myr, is based on the consistency of evolved members with model isochrones (see Section 3.1), for which no rotation is taken into account. Rotation affects the equations of stellar structure, and with elevated centrifugal force, acts to reduce the stellar mass, thereby prolonging the main-sequence lifetime and core helium burning phase (Kippenhahn et al. 1970). This age is older than the 400–600 Myr often quoted in the literature.

Lithium abundances have been measured for solar-type members in Coma Ber (Jeffries 1999; Ford et al. 2001), particularly in the context of convective mixing in stellar interior in terms of metallicity versus age, e.g., in comparison with stars of similar spectral types in the Hyades and Praesepe, or in the much younger Pleiades (see for example Ford et al. 2001). With an older age, it is not clear if the lithium depletion boundary is still applicable as a chronometer (Martín et al. 2018). In any case, our candidate list contains quite a number of solar-type or cooler members to shed light on the subject.

Adopting a distance 85 pc, the angular radius 5° of the cluster corresponds to a linear radius ∼7 pc, to be compared with the tidal radius of the cluster 6–7 pc (Casewell et al. 2006; Kraus & Hillenbrand 2007). The Galactic distribution of our candidates, including those in Tables 2 and 3, is illustrated in Figure 15. Given the high Galactic latitude (ℓ ≈ + 84°) position of the cluster, the distribution in the X–Y plane, i.e., the top view on the Galactic plane, resembles what appears in the celestial sphere, e.g., in R.A. and decl. coordinates, for which members are concentrated within a linear extent about 15 pc n roughly circular shape. The effect of mass segregation is clearly manifest, namely, with more massive members (with larger-sized and lighter-shadowed symbols) concentrating more toward the center. The number of members is markedly reduced beyond a radius of ∼10 pc, indicative that the cluster is not more extended as projected in the sky than the sky coverage in our analysis.

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However, the situation in the X–Z plane, namely the side view of the plane, is different. The apparent cone shape from the bottom upward is the consequence of our pencil-beam view, and the longer extent in the Z direction results from our analysis volume, as we started out with a larger distance range (50–120 pc) to search for candidates than in the angular extent in the sky (within 5° radius of the UKIDSS/GCS coverage). That there are more distant sources beyond ∼100 pc than nearby ones closer than ∼60 pc is the consequence of the space volume effect, hence many must be false positives. The grouping between Z ∼ 65–95 pc represents the cluster, with a linear size twice as extended as in the X–Y plane, stretching toward the Galactic plane. Odenkirchen et al. (1998) reported a heliocentric space motion for the cluster (U, V, W) = (−2.3, −5.5, −0.7) km s⁻¹ with an error of 0.2 km s⁻¹ in each component. As such, the motion of the cluster is primarily along V, i.e., in the Galactic rotation, and the cluster is almost at its highest location above the plane. The prolate spheroidal shape is likely the consequence of the tidal pull by the disk.

Analysis of the control field (see Section 3.2) with the same selection criteria led to 14 false positives. Considering the sky area of the control field relative to the cluster region, and assuming the same field distribution toward the cluster region as in the control field, this means out of the 192 candidates, there are roughly 44 field stars that coincidentally share the same ranges of proper motions, distances, and color–magnitude relation as true members, but are not physically associated with the cluster. Additional scrutiny of members against field stars in the same volume would have to rely on metallicity or chemical abundances, e.g., (Ford et al. 2001, by lithium abundances). For the faint sample, no control field is available because of the limit of the UKIDSS/GCS spatial coverage. After field subtraction for the bright sample, plus the entire faint candidate sample, there are 148 members. We emphasize that this is a statistical sample in the sense that out of the 196 candidates, there are 148 true members, but we do not know for sure individually which ones are true members. This is the sample we use to derive member statistics such as the luminosity function, mass function, total stellar mass, etc.

In studies of cluster membership, some researchers would assign a membership probability for each star, considering its location relative to the cluster center, proper motions, radial velocity, etc., and decide on a threshold probability for membership. As such, the probabilistic nature has to be taken into account when the total mass and the luminosity function are derived, e.g., a candidate with an 80% probability should have its mass weighted by 0.8 and be counted as 0.8 stars. The uncertainty arises, then, if a 0.8 star is really worth twice as much as a 0.4 star in derivation of cluster parameters. In contrast, our analysis exploits a control field to remove sample contaminations. The likelihood of a star being a member of the cluster region depends on the number of false positives in the control field. As seen in Figure 8, our member list is highly reliable for almost the entire bright sample, i.e., for candidates more massive than ∼0.1–0.2 M_☉ (J ≲ 13 mag), with only a few false positives.

4.3.1. Luminosity Function and Mass Function

The cluster's J-band luminosity function is depicted in Figure 16. For the bright members, the luminosity function has been derived by subtraction of the J-magnitude distribution toward the cluster region by that toward the control field. The luminosity function at the faint end, lacking a control field, is given as is. This does not affect our result significantly because each of our substellar candidates has been examined spectroscopically, so contamination is expected to be low. Moreover, the unique colors at very low masses, e.g., in Z − K in Figure 9, would result in little field confusion, hence creditable candidacy. The luminosity function derived in this work, therefore, is reliable except for J = 14–16 mag, which has not been corrected for field subtraction.

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The J-band luminosity function of the cluster increases with magnitude up to J ∼ 12 mag, and falls off rapidly toward fainter magnitudes. In the K-band luminosity of Coma Ber derived by Casewell et al. (2006), they found a paucity around K ∼ 8–12 mag, but otherwise no difference fainter than K ∼ 12 mag between the cluster and a controlled sample chosen by proper motions. Our results show no such shortfall.

The present-day mass function of Coma Ber is exhibited in Figure 17. We convert from the J magnitude to mass according to PARSEC, or, toward the fainter magnitudes, the AMES-Dusty model, based on the cluster luminosity function shown in Figure 16(b). The number of members increases in general from high toward low masses until about 0.3 M_☉, a phenomenon commonly seen in star clusters or associations (Bastian et al. 2010). A linear least-squares fit gives a slope, in the sense of ${dN}/{dm}={m}^{-\alpha }$, α ≈ 0.49 ± 0.03, if the two most massive bins for post-main-sequence objects with uncertain masses are excluded. This is close to the value reported by Kraus & Hillenbrand (2007) α = 0.6 ± 0.3 for 0.1–1 M_☉, but much shallower than either the nominal Salpeter α = 2.35 initial mass function in the solar neighborhood for 1–10 M_☉, the present-day mass function of field M dwarfs α ≈ 1.3 for 0.1–0.7 M_☉ (Reid et al. 2002), or, in nearby young star clusters or associations, α ≈ 1.3 M_☉ (Hillenbrand & White 2004).

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Currently the statistics of the substellar population in star clusters are poorly constrained; less than a handful of stellar systems have been surveyed comprehensively, and even these may be subject to contamination. Furthermore, the age dependence (hence model dependence) of the spectral type with mass hampers derivation of a reliable substellar mass function. Melnikov & Eislöffel (2012) derived a mass function with α ≈ 0.6 from 0.2 M_☉ to 0.14 M_☉, and α ≈ 0 toward lower-masses from 0.14 M_☉ to 0.06 M_☉. The sky area of their analysis, ∼27 in radius, however, covered only part of the core of the cluster. Moreover, lacking a control sample, their member list could be considerably polluted. Among the five photometric brown dwarf candidates these authors proposed (their Table 2), F9−1134 is not detected by UKIDSS/GCS; C3−250 satisfies the proper motion criterion, but has an inconsistent Z − K color; both D3−1251, with (μ_α cosδ ≈ μ_δ) ≈ −130 mas yr⁻¹, and G1−3083, with (μ_α cosδ, μ_δ) ≈ (−100, −79) mas yr⁻¹, have too large proper motions as members. Only E3−5219 turns out to be an M9 member (Casewell et al. 2005, 2014), and also was recovered by us (No. 176).

For our sample, the mass function below 0.3 M_☉ declines monotonically with decreasing mass, and continues into the substellar regime, with a slope α = −1.69 ± 0.14, if the anomaly near 0.1 M_☉ corresponding to the unreliable bin in the luminosity function, is excluded. This contrasts with the flat slope that Kraus & Hillenbrand (2007) found in Coma Ber, or Goldman et al. (2013) derived in Hyades, which is also an intermediate-age cluster (650 Myr Perryman et al. 1998). The slopes at the low masses have been found even to be positive in some very young star clusters (a few Mys), e.g., α = 0.3–0.8 for λ Orionis (Bayo et al. 2011), or α ∼ 0.5 for ρ Ophiuchi (Luhman & Rieke 1999; Bastian et al. 2010). However, as we learn in our study, recognition of true members is susceptible to field confusion. The situation must be escalated in star-forming regions where dust extinction, often variable, becomes excessive.

4.3.2. Dynamical Status