# Kinematic Age of the $\beta$-Pictoris Moving Group

Jinhee Lee KASI-Arizona Fellow  Korea Astronomy and Space Science Institute
(KASI), 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea
Steward Observatory, The University of Arizona, 933 N. Cherry Ave, Tucson, AZ
85721, USA<EMAIL_ADDRESS>Inseok Song Department of Physics and Astronomy,
The University of Georgia, Athens, GA 30602, USA

###### Abstract

Accurate age estimation of nearby young moving groups (NYMGs) is important as
they serve as crucial testbeds in various fields of astrophysics, including
formation and evolution of stars, planets, as well as loose stellar
associations. The $\beta$-Pictoris moving group (BPMG), being one of the
closest and youngest NYMGs, has been extensively investigated, and its
estimated ages have a wide range from $\sim$10 to 25 Myr, depending on the age
estimation methods and data used. Unlike other age dating methods, kinematic
traceback analysis offers a model-independent age assessment hence the merit
in comparing many seemingly discordant age estimates. In this study, we
determine the kinematic ages of the BPMG using three methods: probabilistic
volume calculation, mean pairwise distance calculation, and covariance matrix
analysis. These methods yield consistent results, with estimated ages in the
range of 14 to 20 Myr. Implementing corrections to radial velocities due to
gravitational redshift and convectional blueshift increases the ages by $\sim
2-4$ Myr. Conversely, considering data uncertainties decreases the estimated
ages by 1 to 2 Myr. Taken together, our analysis determined the kinematic age
of BPMG to be 16.3${}^{+3.4}_{-2.1}$ Myr. This age is significantly younger
than the commonly accepted age of the BPMG ($\sim$24 Myr) determined primarily
from the lithium depletion boundary analysis. This younger kinematic age may
point to the discrepancy between the luminosity evolution and lithium
depletion models or the presence of unaccounted systematic error in the
method. This result underscores the necessity for systematic reevaluations of
age-dating methods for nearby, young moving groups.

Stellar associations (1582) — Stellar kinematics (1608) — Stellar ages (1581)

††software:  NumPy (Harris et al., 2020), SciPy (Virtanen et al., 2020),
Matplotlib (Hunter, 2007), KDEpy (Odland, 2018), scikit-image (Van der Walt et
al., 2014)

##  1 Introduction

The $\beta$-Pictoris moving group (BPMG) stands out as one of the most
extensively studied stellar groups due to its proximity and youth, as
evidenced by various works (e.g., Song et al. 2003; Alonso-Floriano et al.
2015; Elliott & Bayo 2016; Shkolnik et al. 2017; Cronin-Coltsmann et al.
2023). When the mass function of the currently known BPMG members was compared
to those of other stellar groups (e.g., $\alpha$Per, Pleiades, and Praesepe in
Lodieu et al. 2019), it appears that the membership survey of BPMG is mostly
complete down to $\sim$0.035 solar mass ($\sim$late T type, e.g., Lee et al.
2023). Consequently, BPMG serves as an ideal benchmark for studying the
evolution of stellar groups, as well as the lowest mass stars and brown
dwarfs. In addition, similar to other nearby young moving groups (NYMGs), BPMG
members are prime targets for exoplanet direct imaging; well-known exoplanets
discovered by direct imaging ($\beta$-Pictoris b and 51 Eri b; Lagrange et al.
2010; Macintosh et al. 2015) are members of the BPMG and they have become
testbeds for studying the early evolutionary stages of wide-orbit massive
planets (Marois et al., 2008; Nielsen et al., 2013; Bowler, 2016; Baron et
al., 2019).

Most stars are born in clusters, with over 90% expected to dissipate into the
field within 1 Gyr (Krumholz et al., 2019; Lamers & Gieles, 2006). Young stars
formed in a diffuse environment, such as NYMGs, are gravitationally unbound
and can be spread on the sky plane over $100^{2}$ degrees or more. This is
similar to stars in the Taurus star-forming region, which likely represents an
earlier stage of the NYMG kinematic evolution. The fraction of stars in
gravitationally bound groups (open and globular clusters) in the Milky Way is
small ($\sim 10\%$; Krumholz et al. 2019), which implies that most stars are
formed in unbound groups like NYMGs. Gaia data, combined with ground-based
high-resolution spectroscopic surveys that can establish radial velocities
(RVs) with a precision of $<1$ km s-1, provide the means to obtain detailed
kinematic evolutionary histories for NYMGs. These kinematic fingerprints can
reveal the dynamical mechanisms through which, and timescales over which,
members of NYMGs disperse into the young field star population. Numerous
structures of stellar populations, known as tails or streams, in the Milky Way
were discovered recently (Meingast et al., 2021; Tarricq et al., 2022;
Bhattacharya et al., 2022) with extensions over hundreds of parsecs in most
cases. Only young stellar groups are detectable in (or as) such structures
because they are dissipated in $\sim 100$ Myr (Krumholz et al., 2019). Given
their age range ($10-100$ Myr), NYMGs offer unique opportunities to study
various evolutionary stages of sparse stellar groups so as to address several
fundamental questions.

Studying the kinematic evolution of NYMGs provides another piece of crucial
information, i.e., age. The age of an ensemble of young stars such as an NYMG,
can be determined using the theoretical isochronal fitting, lithium depletion
boundary, IR-excess, and stellar activity indicators such as Ca H & K lines,
X-ray luminosity, NUV magnitudes, and H-$\alpha$ lines (Soderblom, 2010).
Based on these methods, the age of BPMG has been estimated to be $\sim$12 to
25 Myr (e.g., Zuckerman et al. 2001; Mamajek & Bell 2014; Binks & Jeffries
2014). A dispersing stellar group can provide a model-independent age by
determining the time when its members occupied the smallest volume in the
past. This age-dating method, known as kinematic expansion or traceback age-
dating, has been successfully applied to some young NYMGs (Ducourant et al.,
2014; Miret-Roig et al., 2018; Galli et al., 2023). Generally, age estimates
from the kinematic expansion method for BPMG show systematically younger ages
($12-20$ Myr; Ortega et al. 2002; Song et al. 2003; Crundall et al. 2019;
Miret-Roig et al. 2020; Couture et al. 2023) compared to the widely accepted
lithium depletion boundary age of $21-26$ Myr (Binks & Jeffries, 2014; Malo et
al., 2014), prompting more systematic studies on the ages of NYMGs.

The wide range of different kinematic ages obtained for BPMG illustrates the
sensitive dependence on input data and methods in calculating kinematic ages.
Despite the precision of astrometric data (R.A., Dec., parallax, and proper
motion) from Gaia and sub-km/sec RV precision, the propagation of uncertainty
in position plays a major factor. For typical NYMG members, their positions in
3D space (e.g. XYZ in Cartesian coordinate) can be precisely calculated thanks
to the superb precision of Gaia data. Therefore, uncertainties of traceback
positions are generally limited by the precision of the RV measurements. For
example, with a velocity uncertainty as small as 1 km s-1, it contributes to a
1 parsec over 1 Myr of traceback calculation. For the upper age bound of BPMG
at $\sim 25$ Myr, the velocity uncertainty of km s-1 will result in a 25 pc
uncertainty in traced back positions. In addition to the effect of uncertainty
propagation, kinematic age dating results are sensitive to the input list of
members and employed methods.

In this study, we derive the kinematic age of BPMG using three different
methods to investigate the dependency of the resultant age on the method
choice. In Sections 2 and 3, we will explain data and methods, respectively.
Then, we will present the results and discussion in Sections 4 and 5,
respectively. The conclusion is presented in Section 6.

##  2 Membership List and Input Data

###  2.1 Member List

Figure 1: A color-magnitude diagram (CMD) of BPMG members is displayed, and
input data are obtained from the Gaia DR3 catalog. Stars that meet the CMD
criteria are indicated by blue circles, while those inconsistent with other
BPMG members in the CMD are denoted as orange stars, with their corresponding
star ID numbers and names referenced in Table 1. Two theoretical isochrone
models with an age of 30 Myr are displayed to show the youth of the members.
The PARSEC model (Bressan et al., 2012) is displayed for the higher mass
regime ($\geq 0.8M_{\odot}$; AFG types), while the BHAC15 model (Baraffe et
al., 2015) is presented for the lower mass regime ($<0.8M_{\odot}$; KM types).
Members of other NYMGs and field stars are presented for comparison (TWA:
$\sim$8 Myr, Tuc-Hor: $\sim$30 Myr, and AB Dor: $\sim$100 Myr). Figure 2:
Stellar orbits were computed for all input members, tracing back from the
present to 30 Myr ago using galpy (Bovy, 2015). The resulting backtraced
trajectories are depicted in the Galactocentric Cartesian coordinate system.
Gray ellipsoids illustrate the 1 to 3-$\sigma$ of backtraced positions of the
members 30 Myr ago. Circles annotate two kinematic outliers exceeding the
3-$\sigma$ threshold.

In the analysis of the kinematic age of the BPMG, using a reliable list of
members is critical. We adopted the list of BPMG members from Couture et al.
(2023), comprising $N=76$ members divided into “core samples” ($N=25$) and
“extended samples” ($N=51$). We carefully examined and selected members based
on six criteria for outlier evaluation: (1) known spectroscopic binaries from
literature, (2) RV variability (RV variations greater than 2 km s-1), (3) RV
uncertainties larger than 1 km s-1, (4) RUWE values from Gaia DR3 larger than
2, (5) inconsistent positions on the color-magnitude diagram (CMD) compared to
other BPMG members (referred to as CMD outliers), and (6) kinematic outliers.
Astrometric and photometric data are primarily sourced from Gaia DR3 (Gaia
Collaboration et al., 2016, 2023; Babusiaux et al., 2023), while RV data were
obtained from Couture et al. (2023), which compiled data from the literature
(references therein). The assessment results for criteria (1), (2), and (4)
are from Couture et al. (2023).

Figure 1 presents a color-magnitude diagram (CMD) of the entire input samples,
with a theoretical isochrone model at 30 Myr displayed to assess the youth of
the members. We used two isochrone models, PARSEC (Bressan et al., 2012) and
BHAC15 (Baraffe et al., 2015), for spectral types of AFG (M$\geq$0.8M⊙) and KM
(M$<$0.8M⊙), respectively. Faint colored circles represent other well-known
NYMG members with different ages (TW Hydrae Association:$\sim$8 Myr, Tucana-
Horologium Association:$\sim$30 Myr, and AB Doradus moving group:$\sim$100
Myr), as well as 500 randomly selected Gaia DR3 field stars within 100 pc.
Most members align well between the sequences of TWA and AB Dor, indicating
the intermediate age of BPMG and the well-prepared membership list. However,
some members are fainter than expected for BPMG members of similar colors,
suggesting that these stars may not be genuine BPMG members or that there are
some peculiarities in their input data. 2MASS J04433761+0002051, StHA 182, and
HD 207043 (star IDs: 34, 69, and 74 in Table 1) appear to be older than the
other BPMG members.

Kinematic outliers were assessed using backtraced stellar positions at 30 Myr
in the past, approximately the upper age limit of the BPMG. Stellar positions
were calculated by adopting the Milky Way’s gravitational potential model
(MWPotential2014) using galpy111http://github.com/jobovy/galpy  (Bovy, 2015).
Figure 2 displays the orbits of the entire input members on a Galactocentric
Cartesian coordinate frame, tracing back from the present to 30 Myr ago. The
Sun’s present position is at (X, Y) = ($+8,0$) kpc, and nearly all NYMG
members are concentrated around this position as expected. The Sun’s position
30 Myr ago is at around (X, Y) = ($+5,-6$) kpc. Stars located outside the
3-$\sigma$ range at 30 Myr ago from the averages in the X, Y, or Z coordinates
are defined as kinematic outliers. In Figure 2, one star notably shows a
deviating orbit from the other members. This star, UCAC3 89-246860 (star ID:
45), is classified as a kinematic outlier. After removing these kinematic
outliers, we additionally excluded 2MASS J18430597-4058047 (star ID: 61) due
to its discrepant velocity toward the Galactic anti-center ($U=-0.75$ km s-1)
compared to the other adopted members ($U=-9.3\pm 2.5$ km s-1).

While not classified as outliers, one object was additionally excluded due to
its lack of RV measurements. PSO J318.5338-22.8603 (star ID: 71) has only one
RV measurement, and its RV variability cannot be evaluated.

Couture et al. (2023) used only “core samples” ($N=25$) in their kinematic
analysis of the BPMG, and 21 of these stars pass our outlier evaluation
criteria. These stars are referred to as core members in this study. One
reason for using our own evaluation of membership is to allow for the
inclusion of more members from the “extended samples”. Including 14 members
from the “extended samples” of Couture et al. (2023) that pass our criteria,
we generated a larger list called inclusive members. Table 1 provides the
entire input samples along with the results of our outlier evaluation. Their
kinematic data are provided in Appendix A. In this study, we primarily present
the results obtained from inclusive members. The number of stars is 35 with
mean and maximum RV uncertainties of 0.5 and 1.0 km s-1, respectively. Results
based only on core members ($N=$21, with mean and maximum RV uncertainties of
0.5 and 1.0 km s-1, respectively.), for easy comparison of the effects solely
based on the chosen methods, are available in Table 5 in Appendix C.

Table 1: Member Selection Results

Star IDaaStar IDs are arranged in ascending order based on R.A., with core members listed first. | Designation | R.A.bbPositions are sourced from Gaia DR3, except for PSO J318.5338-22.8603, which is obtained from the 2MASS survey (Cutri et al., 2003). All positions are at the 2016.0 epoch. | Dec.bbPositions are sourced from Gaia DR3, except for PSO J318.5338-22.8603, which is obtained from the 2MASS survey (Cutri et al., 2003). All positions are at the 2016.0 epoch. | SpT | Membership statusccThe membership statuses are coded as follows: (1) core member, (2) member but not core member, and (N) non-member or outlier. | Outlier categoryddThe outlier categories are defined as follows: (1) known spectroscopic binary in literature, (2) RV variable, (3) RV uncertainties $>$1 km s-1, (4) RUWE $>2$, (5) inconsistent CMD locations compared to other BPMG members, and (6) kinematic outlier.
---|---|---|---|---|---|---
|  | (hh:mm:ss.ss) | (hh:mm:ss.s) | Type | (1, 2, N) | (1$-$6)
1 | RBS 38 | 00:17:23.82 | -66:45:12.6 | M2.5 | 1 | $-$
2 | GJ 2006 A | 00:27:50.37 | -32:33:07.1 | M3.5 | 1 | $-$
3 | HD 14082 B | 02:17:24.84 | +28:44:29.3 | G2 | N | 3
4 | AG Tri A | 02:27:29.35 | +30:58:23.4 | K8 | 1 | $-$
5 | CD-57 1054 | 05:00:47.20 | -57:15:24.3 | K8 | 1 | $-$
6 | V1841 Ori | 05:00:49.30 | +15:26:59.8 | K2 | 1 | $-$
7 | BD-21 1074 A | 05:06:49.97 | -21:35:09.4 | M1 | N | 3
8 | AO Men | 06:18:28.18 | -72:02:40.3 | K3.5 | N | 3
9 | UCAC2 12510535 | 17:02:40.13 | -45:22:00.9 | M2 | 1 | $-$
10 | HD 164249 A | 18:03:03.41 | -51:38:57.8 | F4.5 | 1 | $-$
11 | UCAC4 299-160704 | 18:04:16.18 | -30:18:29.0 | M4 | 1 | $-$
12 | 2MASS J18092970-5430532 | 18:09:29.72 | -54:30:55.0 | M4 | 1 | $-$
13 | 1RXS J184206.5-555426 | 18:42:06.98 | -55:54:26.8 | M3.5 | 1 | $-$
14 | Smethells 20 | 18:46:52.58 | -62:10:37.9 | M1 | 1 | $-$
15 | CD-31 16041 | 18:50:44.50 | -31:47:48.5 | K8 | N | 3
16 | HD 181327 | 19:22:58.99 | -54:32:18.3 | F6 | 1 | $-$
17 | 1RXS J192338.2-460631 | 19:23:38.24 | -46:06:32.6 | M0 | 1 | $-$
18 | UCAC4 314-239934 | 19:48:16.55 | -27:20:32.8 | M2 | 1 | $-$
19 | TYC 7443-1102-1 | 19:56:04.41 | -32:07:38.8 | K9 | 1 | $-$
20 | UCAC4 284-205440 | 20:01:37.21 | -33:13:15.0 | M1 | 1 | $-$
21 | HD 191089 | 20:09:05.26 | -26:13:27.6 | F5 | 1 | $-$
22 | AU Mic | 20:45:09.88 | -31:20:33.0 | M0 | 1 | $-$
23 | GSC 06354-00357 | 21:10:05.46 | -19:19:59.1 | M2 | 1 | $-$
24 | CPD-72 2713 | 22:42:49.26 | -71:42:22.0 | K7 | 1 | $-$
25 | WW PsA | 22:44:58.19 | -33:15:03.7 | M4 | 1 | $-$
26 | HD 203 | 00:06:50.20 | -23:06:27.9 | F3 | N | 2
27 | LP 525-39 | 00:32:34.91 | +07:29:25.8 | M3 | N | 1
28 | GJ 3076 | 01:11:25.63 | +15:26:19.9 | M5 | N | 1
29 | Barta 161 12 | 01:35:14.03 | -07:12:52.2 | M4.3 | N | 1, 3
30 | PM J01538-1459A | 01:53:50.97 | -14:59:51.5 | M3 | N | 1, 3
31 | EPIC 211046195 | 03:35:02.15 | +23:42:34.4 | M8.5 | N | 3
32 | StKM 1-433 | 03:57:33.97 | +24:45:09.9 | M2 | 2 | $-$
33 | 51 Eri | 04:37:36.18 | -02:28:25.8 | F0 | N | 3
34 | 2MASS J04433761+0002051 | 04:43:37.67 | +00:02:03.4 | M9 | N | 5
35 | V1005 Ori | 04:59:34.88 | +01:46:59.2 | K8 | 2 | $-$
36 | LP 476-207 | 05:01:58.83 | +09:58:57.1 | M4 | N | 1, 2, 3
37 | AF Lep | 05:27:04.78 | -11:54:04.3 | F7 | N | 2
38 | UCAC4 287-007048 | 05:29:44.71 | -32:39:14.1 | M4.5 | 2 | $-$
39 | V1311 Ori | 05:32:04.51 | -03:05:30.0 | M2 | N | 1, 4
40 |  $\beta$ Pic | 05:47:17.10 | -51:03:58.1 | A6 | N | 1, 3, 4
41 | GSC 06513-00291 | 06:13:13.30 | -27:42:05.6 | M3.5 | N | 1, 3, 4
42 | TWA 22 A | 10:17:26.58 | -53:54:26.5 | M5 | N | 1, 4
43 |  $\alpha$ Cir | 14:42:29.93 | -64:58:34.2 | A7 | N | 4
44 | V343 Nor A | 15:38:57.45 | -57:42:28.8 | G8 | N | 1, 3, 4
45 | UCAC3 89-246860 | 16:12:05.11 | -45:56:26.0 | M4 | N | 2, 3, 6
46 | TYC 8726-1327-1 | 16:57:20.24 | -53:43:33.0 | M3 | N | 1, 4
47 | 2MASS J17020937-6734447 | 17:02:09.32 | -67:34:46.3 | M4 | 2 | $-$
48 | 2MASS J17092947-5235197 | 17:09:29.45 | -52:35:20.9 | M3 | N | 3, 4
49 | HD 155555 A | 17:17:25.45 | -66:57:05.9 | G5 | N | 2, 3
50 | CD-54 7336 | 17:29:55.07 | -54:15:49.7 | K1 | N | 1
51 | HD 160305 | 17:41:49.03 | -50:43:29.1 | F9 | 2 | $-$
52 | UCAC4 184-193200 | 17:44:42.57 | -53:15:48.5 | M3 | 2 | $-$
53 | UCAC3 74-428746 | 17:48:33.74 | -53:06:12.8 | M2 | 2 | $-$
54 | UCAC3 66-407600 | 18:05:54.92 | -57:04:31.9 | M2 | 2 | $-$
55 | HD 165189 | 18:06:49.91 | -43:25:32.5 | A6 | N | 1, 3
56 | V4046 Sgr | 18:14:10.49 | -32:47:35.4 | K4 | N | 1, 2, 3
57 | UCAC3 63-445434 | 18:16:12.41 | -58:44:07.9 | M3.5 | 2 | $-$
58 | HD 168210 | 18:19:52.22 | -29:16:33.6 | G3 | N | 3
59 | UCAC4 229-170667 | 18:28:16.52 | -44:21:48.6 | M2 | 2 | $-$
60 | UCAC4 226-179372 | 18:28:35.25 | -44:57:28.9 | K7 | 2 | $-$
61 | 2MASS J18430597-4058047 | 18:43:05.99 | -40:58:06.0 | M2 | N | 6
62 | HD 172555 | 18:45:26.98 | -64:52:18.9 | A7 | N | 2, 3
63 | PZ Tel | 18:53:05.90 | -50:10:51.3 | G9 | N | 1, 2, 3
64 | TYC 6872-1011-1 | 18:58:04.17 | -29:53:05.4 | K8 | N | 1, 3, 4
65 | 2MASS J19243494-3442392 | 19:24:34.99 | -34:42:40.6 | M4 | 2 | $-$
66 | UCAC3 116-474938 | 19:56:02.98 | -32:07:19.8 | M4 | N | 1, 2, 3, 4
67 | SCR J2010-2801 | 20:10:00.11 | -28:01:42.2 | M2.5 | N | 1, 3, 4
68 | SCR J2033-2556 | 20:33:37.66 | -25:56:53.3 | M4.5 | 2 | $-$
69 | StHA 182 | 20:43:41.17 | -24:33:55.0 | M3.7 | N | 1, 3, 5
70 | HD 199143 | 20:55:47.74 | -17:06:52.0 | F7 | N | 1, 2, 3, 4
71 | PSO J318.5338-22.8603 | 21:14:08.18 | -22:51:38.2 | L7 | N |  $-$eeThis object has only a single RV measurement, making it impossible to evaluate its variability.
72 | TYC 9114-1267-1 | 21:21:29.00 | -66:55:07.9 | K7 | N | 1, 2, 3, 4
73 | 2E 4498 | 21:37:40.28 | +01:37:12.7 | M5 | N | 2, 3, 4
74 | HD 207043 | 21:47:55.47 | -52:55:51.7 | G5 | N | 5
75 | HD 213429 | 22:31:18.49 | -06:33:20.3 | F8 | N | 1, 2, 3
76 | BD-13 6424 | 23:32:31.01 | -12:15:52.8 | M0 | 2 | $-$

Note. — Members are sourced from Couture et al. (2023).

###  2.2 Input Data

Although the concept of kinematic age-dating analysis, finding the time when a
group of stars occupies the smallest volume, is straightforward, implementing
the method and selecting input data to demonstrate the concept is challenging.
As illustrated before, a 1 km s-1 uncertainty in the input data causes about a
1 pc positional uncertainty over 1 Myr of traceback time. For typical NYMG
members, Gaia data provide very precise astrometric information ($\lesssim 20$
milli-arcsecond), and the resultant uncertainty in UVW caused by the typical
Gaia uncertainty is smaller than 0.02 km s-1. However, typical uncertainties
in RV are an order of magnitude larger at least. Therefore, it is crucial to
use the most precise input data and consider even a small systematic effect in
the traceback age-dating. Furthermore, employing a method that systematically
takes the uncertainty in input data into account is critical. Couture et al.
(2023) demonstrated that stellar RV measurements can be systematically
affected by two effects, gravitational redshift and convectional blueshift.
These effects can alter RVs by $\sim 0.6$ km s-1. By applying the RV
corrections, they concluded that the RV correction could contribute to
$\sim+3$ Myr adjustment to the obtained kinematic age. However, the RV
correction was applied only to the “core samples” in their analysis. When the
correction is applied to a larger population of candidate BPMG members, some
additionally accepted BPMG members due to the RV correction can act in the
opposite direction (i.e., lowering the kinematic age). We included the RV
correction effect in our analysis as well, and our results with and without
the RV correction are denoted as “$\rm RV_{corr}$” and “$\rm RV_{obs}$”,
respectively.

Large positional uncertainties in backtraced positions can result in a failed
convergence at the birth time (e.g., Mamajek & Bell 2014). Most of the
previous kinematic age analyses on NYMGs did not incorporate the position
uncertainties in the method. In our methods, we formally include the effect of
input data uncertainties. Specifically, one of our three methods takes the
positional uncertainties into account naturally by treating each star’s
position as a density function instead of treating it as a point source. For
the other two methods we adopted, to assess the effects of uncertainties in
backtraced positions, we employed a bootstrap resampling method of input data
from the Gaussian distributions defined by uncertainties. We denote the
results with formally assessed positional uncertainties as “+Uncer”, and cases
without considering the effect of positional uncertainties are presented with
“$-$Uncer” denotation.

##  3 Methods

In this study, we estimated the age of the BPMG through the application of
three distinct methods: “probabilistic volume calculation”, “mean pairwise
distance calculation”, and “covariance matrix calculation”. As mentioned in
Section 2.1, we computed the trajectories of stars using galpy (Bovy, 2015) by
adopting the Milky Way’s gravitational potential model, MWPotential2014. These
stellar orbits were traced from their current positions back in time to 30 Myr
ago.

###  3.1 Method 1: Probabilistic Volume Calculation

This method involves analyzing the spatial distribution of stars in an NYMG,
considering measurement uncertainties, and calculating the volume they occupy.
In the context of studying an NYMG of stars spread over several tens of
parsecs, individual stellar sizes are negligible, allowing us to treat each
star as a point in 3D space. However, due to the presence of measurement
uncertainties and their propagation in backtraced positions, we represent the
true position of a star using a probability density function, which we call a
position probability density function (PPDF). A PPDF takes the form of a 3D
ellipsoid with axes determined by uncertainties in X, Y, and Z, where those
uncertainties arise from uncertainties in the input parameters (R.A., Dec.,
and parallax). When tracing back positions in time, the PPDF expands with time
due to additional parameters involved (e.g., proper motion and RV) and their
uncertainties. To calculate the volume occupied by $N$ stars while accounting
for these positional uncertainties, we treat an NYMG as a collection of $N$
PPDFs. At each grid point in our coordinate system, we sum the values of these
$N$ PPDFs to create a summed PPDF (sPPDF). The total star density of the NYMG
can be obtained by summing sPPDF values across the entire grid domain,
starting from group’s center and extending infinitely. We define a volume to
represent the space that contains 70% of the total star density of the group.
To calculate this enclosing volume, we employ the isosurface approach, using
scikit-image.marching_cubes (Van der Walt et al., 2014) to create an
isosurface that encompasses 70% of the total star density. The volume within
this isosurface can be calculated using the vertices and faces of the 2D
surface mesh. We refer to this method as “Probabilistic Volume Calculation
(PVC)”.

As we have the relatively small number of members ($N=21$ for core members and
$N=35$ for inclusive members), the sPPDF requires smoothing to reduce the
influence of outliers. To achieve this, we applied data smoothing techniques
to sPPDF, utilizing Kernel Density Estimation (KDE) in KDEpy (Odland, 2018)
with a Gaussian kernel. The bandwidth ($bw$) of the Gaussian kernel can be
determined using Scott’s rule-of-thumb (Scott, 1992), which suggests:
$bw=1.06\sigma n^{-1/5}$ where $\sigma$ represents the standard deviation of
data and $N$ is the number of data. This rule is for data with a normal
distribution. Because positions of NYMG members are not necessarily following
a normal distribution and BPMG members show a larger dispersion along the
X-axis, we decided to experiment with a range of $bw$ (7, 10, 15, 20 pc)
instead of using a single value.

To establish age upper and lower limits, we employed a leave-4-out jackknife
resampling method, which includes all core members while randomly selecting
80% of the remaining members ($N=14$). This results in 364 possible cases
(${}_{14}C_{11}$). The PVC method inherently calculates the case of $+$Uncer,
and we only provide $+$Uncer calculations. We also applied RV corrections for
these two sets of calculations ($\rm RV_{corr}$ and $\rm RV_{obs}$).

###  3.2 Method 2: Mean Pairwise Distance Calculation

Figure 3: Distances between all pairs of inclusive members are depicted as
gray lines over time. The mean distances, calculated by averaging after the
removal of outliers2, are shown as a red solid line. The time corresponding to
the minimum distance is $-15.2$ Myr, written in the upper left corner.

In a dispersing NYMG, its members gradually move farther apart from each other
over time. This means that the average distance in all pairwise distance
combinations (${}_{35}C_{2}$) decreases with traceback times until the group’s
birth time. Beyond the birth time, the average pairwise distance increases
because of the imaginary expansion of the group before its existence. We
attempted to find the time when the average pairwise distance is at its
minimum. The approach employed in our calculation shares similarities with the
one utilized in Kerr et al. (2022), although there are differences in details.
In Kerr et al. (2022), they computed a median distance for each star to all
other stars and determined the time at which the median distance was minimal.
The minimal distance time (i.e., age) is obtained for all stars, and the final
age is obtained from the distribution of ages. In our analysis, we obtained a
time at which the mean separation of all pairs is minimum. From a single data
set, we obtain an age instead of distribution of ages. The number of available
pairs for $N$ stars is $N(N-1)/2$, resulting in 595 cases. Figure 3
illustrates an example calculation with inclusive members. However, there are
notably large separated pairs that can introduce a bias into the mean distance
calculation. To mitigate this bias, we removed such outliers222Outliers are
defined if the data is either $<$Q1-1.5*(Q3-Q1) or $>$Q1+1.5*(Q3-Q1), where Q1
and Q3 are 25th and 75th percentiles of the data, respectively. before
computing the mean distance. The yellow shaded area illustrates a region of
data after excluding outliers, and the mean distance at each time is presented
as a red solid line. The minimum mean pairwise distance occurs at $-15.2$ Myr
in this example.

To estimate the age uncertainty, we used bootstrap resampling, performing the
calculation 10,000 times. This calculation is performed considering data
uncertainties (+Uncer and $-$Uncer) and RV corrections ($\rm RV_{corr}$ and
$\rm RV_{obs}$).

###  3.3 Method 3: Covariance Matrix Calculation

The positions of stars in 3D space can be fitted as an ellipsoid, whose volume
can be represented as a covariance matrix. We computed the geometric mean of
the standard deviation along the principal axes (i.e., determinant of the
covariance matrix) at each time. Subsequently, we found the time at which this
mean size reaches its minimum (a similar calculation was performed in the
study by Miret-Roig et al. 2020).

In addition to the above, akin to our approach for pairwise distance
calculations, we evaluated ages in cases of considering RV correction and data
uncertainties and employed the bootstrap resampling technique.

##  4 Results

###  4.1 Method 1: Probabilistic Volume Calculation (PVC)

Figure 4: An example calculation of group size ($\sim$volume1/3) with a
bandwidth ($bw$) of 10. The group size is estimated in two different
scenarios, considering data uncertainties and RV correction. The volume is
defined by the isosurface enclosing 70% of the total star density. The times
corresponding to the minimum volume for each scenario are denoted in the upper
left corner of the figure. Figure 5: Distributions of age (left) and mean size
(right) from PVC with $bw=10$. These results are obtained through jackknife
resampling, which includes all core members and 80% randomly selected
remaining inclusive members in each resampling. The medians with 68%
confidence intervals are denoted in the upper of the figure.

The volume ($V$) obtained from the PVC method is translated into the mean
group size ($R$) with a simplifying assumption ($V=4/3\pi R^{3}$). Figure 4
illustrates the mean size of the group as a function of time. At the present
time ($t=0$), the mean size of the group is $\sim 34$ pc. As we trace back in
time, the mean group size ($R$) decreases to reach a minimum and subsequently
increases for further back in time. We conducted these calculations for the
mean group size as a function of time in two different scenarios, taking into
account the effects of RV correction. The age from each scenario was
determined by finding the time of the minimum mean group size. In Figure 4,
the ages are 14.7 and 16.7 Myr for case of $\rm RV_{obs}$+Uncer and $\rm
RV_{corr}+$Uncer, respectively. These calculations were carried out using
jackknife resampling. The 364 calculations are performed for each scenario.
The left panel of Figure 5 displays distribution of the ages, while the right
panel shows the mean group size $R$. The median and 68% confidence intervals
are indicated as the age (left panel) and the mean group size (right panel)
with uncertainties. Table 2 summarizes the results under various scenarios,
including different bandwidths for KDE.

When we analyzed data without considering RV correction [$\rm
RV_{obs},+$Uncer], the estimated ages are 14$-$15 Myr with all $bw$ values.
When RV correction is applied [$\rm RV_{corr},+$Uncer], the estimated ages
increase by $1-2$ Myr, resulting in an age of approximately $16-17$ Myr. These
trends are consistent with all $bw$ values, but larger $bw$ values tend to
yield slightly younger age estimates. Nonetheless, the differences in age
estimates between different $bw$ values are generally less than 1 Myr.

The mean group size at the time of minimum volume (i.e., its birth time) is
approximately 30 pc as seen in Figures 4 and 5 (right). The mean sizes
resulting from full calculations with resampling and different $bw$ values are
provided in Appendix B. These sizes vary from 25 to 45 pc, with an increase in
the $bw$ values. More smoothing of the density cube with a larger $bw$ results
in larger group size estimates, as expected.

Table 2: Ages Derived Through Three Distinct Methods

|  $-$Uncer |  $+$Uncer
---|---|---
| (Myr) | (Myr)
Method 1: Probabilistic Volume Calculation (PVC)
$bw$=7
$\rm RV_{obs}$ | $-$ | 14.8$(^{+0.5}_{-0.5})$
$\rm RV_{corr}$ | $-$ | 16.7$(^{+0.9}_{-0.9})$
$bw$=10
$\rm RV_{obs}$ | $-$ | 14.6$(^{+0.6}_{-0.6})$
$\rm RV_{corr}$ | $-$ | 16.2$(^{+0.8}_{-0.8})$
$bw$=15
$\rm RV_{obs}$ | $-$ | 14.4$(^{+0.6}_{-0.6})$
$\rm RV_{corr}$ | $-$ | 15.9$(^{+0.7}_{-0.7})$
$bw$=20
$\rm RV_{obs}$ | $-$ | 14.2$(^{+0.6}_{-0.6})$
$\rm RV_{corr}$ | $-$ | 15.6$(^{+0.8}_{-0.8})$
Method 2: Mean Pairwise Distance Calculation
$\rm RV_{obs}$ | 15.6$(^{+0.9}_{-1.0})$ | 14.5$(^{+1.3}_{-1.4})$
$\rm RV_{corr}$ | 17.8$(^{+1.2}_{-1.5})$ | 15.9$(^{+1.9}_{-2.2})$
Method 3: Covariance Matrix Calculation
$\rm RV_{obs}$ | 16.7$(^{+0.9}_{-0.8})$ | 15.6$(^{+1.4}_{-1.3})$
$\rm RV_{corr}$ | 20.5$(^{+3.1}_{-1.5})$ | 18.3$(^{+2.9}_{-2.1})$
SummaryaaThe summary ages were determined through a weighted mean of three
methods. We utilized an age with $bw$=10 for the PVC, and we adopted larger
uncertainties.
$\rm RV_{obs}$ | 16.2$\pm$0.7 | 14.7$\pm$0.5
$\rm RV_{corr}$ | 18.3$\pm$1.4 | 16.3$\pm$0.7

Note. — The results are obtained using the inclusive members. The reported
uncertainties correspond to the 68% confidence interval.

###  4.2 Method 2: Mean Pairwise Distance Calculation

Figure 6: Distribution of age (left) and mean pairwise distance (right) from
Method 2. These results are obtained through 10,000 rounds of bootstrap
resampling. The medians with 68% confidence intervals are denoted in the upper
of the figure.

Mean pairwise distances between all pairs of stars were calculated as a
function of time. For a given dataset, the time of the minimum mean distance
was found and considered as the birth time of the group. The calculations were
performed through 10,000 rounds of bootstrap resampling. The results of
calculations are presented in Figure 6. The application of RV correction ($\rm
RV_{corr}$) tends to increase the estimated age, while considering data
uncertainties (+Uncer) has the opposite effect, resulting in a decrease in the
calculated age.

The calculated ages for the four scenarios are listed in Table 2. These
findings align closely with the case using a $bw$ of 15 in the PVC analysis.
Furthermore, the estimated mean pairwise distances are approximately $22-28$
pc (presented in Appendix B).

###  4.3 Method 3: Covariance Matrix Calculation

Figure 7: Distributions of age (left) and determinant of covariance matrix
(right). These results are obtained through 10,000 rounds of bootstrap
resampling. The medians with 68% confidence intervals are denoted in the upper
of the figure.

Similar to the previous analyses conducted using the two methods described
earlier, we computed the covariance matrix considering data uncertainties and
RV correction. The time of the minimum size of the position ellipsoid was
considered as the birth time of the group hence the age of the group, and age
upper and lower limits were determined through bootstrap resampling. The
results are depicted in Figure 7 and summarized in Table 2.

Consistent with the outcomes of the two preceding methods, we observe that the
application of RV correction tends to increase the estimated ages, whereas
considering data uncertainties leads to a reduction in the resultant ages.

The lengths of the semi-major axes of the ellipsoid correspond to the standard
deviation (1-$\sigma$), which would encompass approximately $\sim 30\%$ of the
group ($0.68^{3}$) assuming a normal distribution of members. The estimated
mean size obtained from the covariance matrix is approximately 10 pc (provided
in Appendix B), and it notably appears to be smaller than the sizes obtained
through the PVC method, primarily due to the size estimation differences.
However, when mean sizes are calculated using 2-$\sigma$ ($0.95^{3}\sim
0.86$), the range becomes 20 pc, aligning well with the results obtained from
the PVC method with $bw=7$.

##  5 Discussion

Kinematic analysis serves as a semi-fundamental method for age dating a
stellar group, offering the advantage of being insensitive to chosen models
(Soderblom, 2010). However, this method is sensitive to the composition of the
member list, and the presence of outliers can significantly bias age
estimations (Miret-Roig et al., 2020). The member lists utilized in this study
were meticulously generated, with discrepant members excluded as explained in
Section 2.1. The calculations presented here were performed using inclusive
members ($N$=35), and the calculations derived from core members ($N$=21)
provide consistent results (provided in Appendix C). The differences in age
estimations obtained with inclusive members and core members are typically
less than 1 Myr.

The choice of the size estimation metric plays a role in kinematic age
estimation. In this study, three methods were employed, and they consistently
yield similar results, as summarized in Table 2. Ages obtained from the PVC
and mean pairwise distance calculation exhibit similarities, whereas ages
derived from the covariance matrix calculation tend to be approximately $1-2$
Myr older on average.

The major factors that influence age estimations are data uncertainties and RV
correction. When data uncertainties and RV corrections are not considered
[$\rm RV_{obs},-$Uncer], the age estimates range from 16 to 17 Myr. The
inclusion of RV correction to the data [$\rm RV_{corr},-$Uncer] resulted in an
increase in the age estimation to $18-21$ Myr as reported in Couture et al.
(2023). Conversely, when we consider data uncertainties [$\rm
RV_{obs}$,+Uncer], the age estimates decreased to $\rm 14-16$ Myr. Finally,
when both data uncertainties and RV correction are taken into account [$\rm
RV_{corr}$,+Uncer], the age estimation falls within the range of $\rm 16-18$
Myr. Miret-Roig et al. (2018) and Couture et al. (2023) have demonstrated that
taking input data uncertainties into the calculation always decreases the
kinematic age estimates. The primary reason for such a behavior could be due
to the fact that the kinematic age dating is a calculation with a half-open
boundary whose anchor point is the values defined by the present-day positions
and uncertainties. The volume of a group members at present is well restricted
with small uncertainties while the volume is more uncertain in the past
because of the effect of the positional uncertainty propagation. When one adds
errors in simulated input data sets as in the case of Miret-Roig et al.
(2018)’s simulation, the minimum volume obtained by the calculation becomes
larger than the true value. Since the volume is anchored at the present, the
only way the curve of the volume as a function of traceback time can move with
added uncertainties is to the direction of the upper-right corner in Figure 3
of Miret-Roig et al. (2018). As a result, a simulation with added
uncertainties in the input data ends up with a younger minimum volume time and
a larger minimum volume always. The magnitude of this effect should be
dependent on the relative magnitudes of the intrinsic velocity dispersion at
birth and observational uncertainties. As the intrinsic dispersion is unknown,
predicting the contribution of observational uncertainties to the decrease in
kinematic age is challenging. Therefore, the younger ages observed in the
$+$Uncer cases ($\sim$2 Myr younger) in this study cannot solely be attributed
to the effect of the half-open boundary problem.

The final reported age of BPMG was determined from the weighted mean of all
age estimates considered in this paper. We obtained a weighted mean age from
the three methods in Table 2 for the [$\rm RV_{corr}$+Uncer] case. The upper
and lower age limits were manually selected to encompass the age ranges of the
four ages listed in the summary panel of Table 2, as the resultant age
uncertainty from the weighted mean is too small ($<$1 Myr) and we cannot tell
if any one method out of three produces more reliable result than other
methods. Our final BPMG kinematic age reported in this paper is
16.3${}^{+3.4}_{-2.1}$ Myr.

###  5.1 Comparison with Other Studies on the Kinematic Age of BPMG

In addition to the three methods applied in this study, several different
approaches have been employed in previous studies to determine the kinematic
ages of the BPMG. Ortega et al. (2002) determined the expansion age by
calculating the maximum distance between members. Song et al. (2003) observed
the time when the BPMG members were most compact in the XY space. The standard
deviation of members along each axis ($\sigma_{X}$, $\sigma_{Y}$, and
$\sigma_{Z}$) was used in Mamajek & Bell (2014). They concluded that the
traceback analysis is not applicable to the BPMG with a large range of the
kinematic age ($13-58$ Myr). Crundall et al. (2019) assumed a group model at
the group’s birth time in 6D space and evaluated the evolution of the model
along the Galactic potential, then determined the kinematic age by comparing
it to the observation data. The covariance matrix, median absolute deviation,
and minimum spanning tree were used in studies of Miret-Roig et al. (2018,
2020) and Couture et al. (2023).

The differences in age estimations between these earlier studies and our
present study can be attributed to variations in the group size estimation
metric, the composition of member lists, and input data. Our age estimate
($16.3^{+3.4}_{-2.1}$ Myr) falls within a range between the estimates from
earlier studies (Song et al. 2003; 12 Myr, Ortega et al. 2002, 2004; $11-12$
Myr), and more recent studies (Couture et al. 2023; 20.4 Myr, Miret-Roig et
al. 2020; 18.5 Myr, Crundall et al. 2019; 17.8 Myr).

###  5.2 Comparison with Ages Determined by Other Methods

The age of BPMG has been estimated using various methods, including isochrone
fitting, lithium depletion boundary (LDB), and empirical approaches based on
signs of activities.

Zuckerman et al. (2001) reported seventeen members and estimated the age as 12
Myr based on the H-R diagram, H-$\alpha$ emission, X-ray counts from ROSAT,
and equivalent width of lithium. During the same period, Barrado y Navascués
et al. (1999) determined the age of group as 20$\pm$10 Myr through isochrone
fitting.

In the analysis of age of the BPMG using kinematic ages, Mamajek & Bell (2014)
also incorporated isochronal analyses. They claimed that the A-type members
are all on the zero-age main sequence, implying an age of greater than 20 Myr.
They also noted that late-F and G pre-main sequence stars align with the
isochronal age of 22 Myr.

Bell et al. (2015) determined the isochronal age of the BPMG using a CMD
($V-J$ vs $M_{V}$) and four isochronal models. They concluded that the best-
matching age for the BPMG with 97 members is $24\pm 3$ Myr. Recently, Ujjwal
et al. (2020) assessed the age of the BPMG using photometric data from Gaia
DR2, resulting in an estimated age of 19.38 Myr. In a study of Lee et al.
(2022), the age of the BPMG was evaluated using combined isochrones that
incorporated the PARSEC model (Bressan et al., 2012) for massive stars ($\geq
0.7M_{\odot}$) and the BHAC15 model (Baraffe et al., 2015) for low-mass stars
($<0.7M_{\odot}$). The age estimation was approximately 10 Myr, primarily
influenced by the low-mass members for which age estimations were from the
BHAC15 model. Figure 8 presents color-magnitude diagrams of the inclusive
members in comparison to the theoretical isochrone models (Bressan et al.,
2012; Baraffe et al., 2015). Most of the members are in good agreement with
the models within the range of 15 to 20 Myr, while a few members align with
the 10 Myr or 30 Myr models.

As a stellar group ages, the fraction of stars with gaseous or dusty disks
decreases. The BPMG is recognized as a disk-rich group with a large debris
disk fraction ($>37\%$; Rebull et al. 2008, $57-75\%$; Najita et al. 2022).
Najita et al. (2022) shows that the dust luminosity of the BPMG is similar to
those of stars with ages of $10-40$ Myr.

The lithium depletion boundary (LDB) method has been actively employed to
estimate the age of stellar groups including BPMG, as the method is believed
to be based on better constrained stellar physics than CMD isochrone fittings
where stellar luminosities can oftentimes be more affected by various effects
such as episodic accretion history, stellar spots, etc. Several studies
assessed the LDB age of the BPMG, ranging from approximately 20 to 25 Myr
(Binks & Jeffries, 2014; Malo et al., 2014; Messina et al., 2016). Notably,
Messina et al. (2016) considered the impact of activity by correcting Li and
employing models with magnetic fields, resulting in an age estimate of
25$\pm$3 Myr. However, it is essential to note that the LDB age determination
is typically applicable within a range of 20 to 200 Myr, despite its
relatively high precision of $\pm 10\%$ (Burke et al., 2004). The LDB age of
the BPMG falls close to the lower limit of LDB method’s applicability,
suggesting that the LDB age may be subject to greater uncertainty.

Figure 8: Color-magnitude diagrams of the BPMG members compared to the
theoretical isochrones (PARSEC (Bressan et al., 2012) and BHAC15 (Baraffe et
al., 2015) for $M\geq 0.8M_{\odot}$ and $M<0.8M_{\odot}$, respectively).

###  5.3 Origin of the BPMG

The origin of the BPMG has been hypothesized to result from interaction with
the nearby OB association, the Scorpius Centaurus complex (Sco-Cen). Ortega et
al. (2002, 2004) analyzed the relationship between the BPMG and two of these
subgroups, Lower Centaurus Crux (LCC) and Upper Centaurus Lupus (UCL). Figure
9 illustrates the traceback orbits of these groups from the present to 15 Myr
ago. The members of Sco-Cen are taken from de Zeeuw et al. (1999), Mamajek et
al. (2002), and Song et al. (2012). Astrometric data, including RV, are taken
from Gaia DR3 (Gaia Collaboration et al., 2023). This improved quality of data
provides a more accurate relationship between these three groups. In Figure 9,
the group sizes are represented as circles with radii determined by the
geometric mean of the standard deviation (1-$\sigma$) of the member
distribution. The groups are well separated at present, while they largely
overlapped 15 Myr ago, at the time of around these groups’ formation. This
supports the idea that the formation of these three groups might be closely
related, as argued in Ortega et al. (2002, 2004).

Zucker et al. (2022) provided a comprehensive picture of the formation and
evolution history of young nearby stars. They explained that LCC and UCL were
formed at $\sim 15-16$ Myr ago at the center of the local bubble due to the
explosion of $N\sim 15$ supernovae. The age of the BPMG, around $16$ Myr,
naturally supports the idea that the origin of BPMG is closely related to the
formation of the Sco-Cen complex. In the outskirt region, with relatively
lower-mass members and lower numbers, the BPMG might have formed at around the
time of the formation of the local bubble. This scenario is consistent with
the findings of Song et al. (2012), which suggest that the BPMG is slightly
older than LCC and UCL according to the LDB analysis.

Figure 9: Traceback positions of BPMG members and two subgroups (LCC and UCL)
of the Sco-Cen complex. The members of Sco-Cen are taken from de Zeeuw et al.
(1999), Mamajek et al. (2002), and Song et al. (2012). The mean traceback
positions for each group are presented as thick solid lines. The time
integration is taken from the present to $-15$ Myr. For better presentation,
the positions of members are presented only at the time of the present
(triangles) and the $-15$ Myr (dots).

##  6 Conclusions

As a benchmark NYMG, the age of the BPMG holds significant importance in
shaping our understanding of young stars and the astrophysical properties of
planets, as well as the evolution of loose stellar associations.

In this study, we estimated the kinematic age of the BPMG using three distinct
methods: probabilistic volume calculation, mean pairwise distance calculation,
and covariance matrix calculation. Notably, we carefully accounted for the
effects of data uncertainties. We also considered RV corrections due to
convectional blueshift and gravitational redshift, as demonstrated in Couture
et al. (2023).

All three methods employed in our study yielded consistent age estimations,
falling within the range of $14-20$ Myr. RV correction was observed to
increase the age estimates by approximately $2-4$ Myr, while data
uncertainties had the opposite effect, reducing age estimations by $1-2$ Myr.
When accounting for all these influences [$\rm RV_{corr}$,+Uncer], we arrived
at an age estimation of $16.3^{+3.4}_{-2.1}$ Myr.

While this age estimate falls within the known BPMG age range, which spans
from 10 to 25 Myr, it is notably younger than the recently accepted age of
$\sim 20-25$ Myr. This relatively young age estimation underscores the
necessity for a revisitation of systematic analyses of the BPMG’s age through
other age estimation methods and recalibration to achieve consistent age
estimations.

##  7 Acknowledgments

We acknowledge Jae-Joon Lee and the anonymous referee for discussion and
comments that greatly improved the this paper. J. L. acknowledge the support
by the Korea Astronomy and Space Science Institute (KASI) grant funded by the
Korean government (MSIT; No. 2024-1-860-02, International Optical Observatory
Project).

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium).

The authors acknowledge that the computational work reported on in this paper
was (partly) performed on the KASI Science Cloud platform supported by Korea
Astronomy and Space Science Institute.

##  Appendix A Input data

Table 3: Kinematic Data of the Full Input Samples

Star ID | Designation | SpT | R.A. | Dec. | Plx | eplx | pmRA | epmRA | pmDE | epmDE | RV | eRV | RUWE
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  | (deg) | (deg) | (mas) | (mas) | (mas yr-1) | (mas yr-1) | (mas yr-1) | (mas yr-1) | (km s-1) | (km s-1) |
1 | RBS 38 | M2.5V | 4.349247 | -66.753500 | 27.16 | 0.02 | 102.89 | 0.02 | -16.83 | 0.02 | 10.30 | 0.40 | 1.0
2 | GJ 2006 A | M3.5Ve | 6.959886 | -32.551962 | 28.55 | 0.04 | 110.20 | 0.04 | -47.19 | 0.04 | 8.80 | 0.40 | 1.8
3 | HD 14082 B | G2V | 34.353506 | 28.741465 | 25.23 | 0.02 | 85.88 | 0.02 | -71.14 | 0.03 | 4.00 | 2.00 | 0.9
4 | AG Tri A | K8V | 36.872305 | 30.973178 | 24.42 | 0.02 | 79.54 | 0.02 | -72.13 | 0.01 | 6.00 | 0.30 | 0.9
5 | CD-57 1054 | K8Ve | 75.196664 | -57.256741 | 37.21 | 0.01 | 35.39 | 0.01 | 74.11 | 0.02 | 19.30 | 0.80 | 1.0
6 | V1841 Ori | K2IV | 75.205434 | 15.449933 | 18.83 | 0.02 | 18.31 | 0.02 | -58.72 | 0.01 | 18.00 | 1.00 | 1.0
7 | BD-21 1074 A | M1V | 76.708210 | -21.585937 | 50.43 | 0.02 | 47.19 | 0.02 | -15.38 | 0.02 | 21.00 | 2.00 | 1.3
8 | AO Men | K3.5V | 94.617421 | -72.044516 | 25.57 | 0.01 | -7.71 | 0.02 | 74.41 | 0.01 | 16.00 | 2.00 | 0.8
9 | UCAC2 12510535 | (M2) | 255.667191 | -45.366924 | 31.30 | 0.02 | -20.10 | 0.02 | -137.85 | 0.02 | -3.00 | 0.50 | 1.0
10 | HD 164249 A | F4.5V | 270.764224 | -51.649392 | 20.29 | 0.02 | 2.33 | 0.02 | -86.23 | 0.02 | 0.20 | 0.40 | 1.1
11 | UCAC4 299-160704 | (M4) | 271.067437 | -30.308057 | 18.15 | 0.02 | 3.42 | 0.02 | -65.22 | 0.02 | -7.40 | 0.10 | 0.7
12 | 2MASS J18092970-5430532 | (M4) | 272.373822 | -54.515278 | 25.67 | 0.03 | 4.71 | 0.03 | -108.17 | 0.02 | -1.90 | 0.30 | 1.4
13 | 1RXS J184206.5-555426 | M3.5V | 280.529082 | -55.907457 | 19.44 | 0.02 | 12.01 | 0.02 | -79.07 | 0.01 | 0.60 | 0.20 | 1.3
14 | Smethells 20 | M1Ve | 281.719094 | -62.177194 | 19.72 | 0.02 | 13.24 | 0.02 | -80.28 | 0.02 | 1.80 | 0.50 | 1.0
15 | CD-31 16041 | K8IVe | 282.685436 | -31.796817 | 20.22 | 0.01 | 17.27 | 0.02 | -72.34 | 0.01 | -8.00 | 3.00 | 1.0
16 | HD 181327 | F6V | 290.745786 | -54.538414 | 20.93 | 0.03 | 24.40 | 0.02 | -82.19 | 0.02 | -0.10 | 0.20 | 1.0
17 | 1RXS J192338.2-460631 | M0V | 290.909327 | -46.109045 | 14.03 | 0.02 | 18.07 | 0.02 | -57.25 | 0.01 | -0.20 | 0.20 | 1.3
18 | UCAC4 314-239934 | (M2) | 297.068959 | -27.342442 | 15.47 | 0.02 | 25.15 | 0.02 | -53.38 | 0.01 | -6.26 | 0.05 | 1.3
19 | TYC 7443-1102-1 | K9IVe | 299.018392 | -32.127436 | 19.49 | 0.02 | 33.60 | 0.02 | -68.53 | 0.01 | -6.10 | 0.40 | 1.1
20 | UCAC4 284-205440 | M1V | 300.405042 | -33.220831 | 16.68 | 0.02 | 29.23 | 0.02 | -61.39 | 0.01 | -4.00 | 0.60 | 1.1
21 | HD 191089 | F5V | 302.271932 | -26.224333 | 19.96 | 0.02 | 40.34 | 0.02 | -67.42 | 0.01 | -7.00 | 1.00 | 0.9
22 | AU Mic | M0Ve | 311.291183 | -31.342500 | 102.94 | 0.02 | 281.32 | 0.02 | -360.15 | 0.02 | -4.50 | 0.60 | 0.9
23 | GSC 06354-00357 | M2V | 317.522749 | -19.333074 | 30.90 | 0.03 | 90.61 | 0.03 | -90.99 | 0.02 | -6.20 | 0.80 | 1.2
24 | CPD-72 2713 | K7IVe | 340.705235 | -71.706122 | 27.23 | 0.01 | 94.85 | 0.01 | -52.38 | 0.01 | 7.80 | 0.60 | 0.9
25 | WW PsA | M4IVe | 341.242468 | -33.251032 | 47.92 | 0.03 | 179.95 | 0.02 | -123.19 | 0.02 | 2.70 | 0.40 | 1.1
26 | HD 203 | F3V | 1.709162 | -23.107750 | 25.16 | 0.03 | 96.97 | 0.04 | -47.25 | 0.02 | 7.00 | 1.00 | 0.9
27 | LP 525-39 | M3Ve | 8.145477 | 7.490485 | 28.43 | 0.06 | 104.36 | 0.08 | -62.91 | 0.06 | -3.70 | 0.60 | 1.2
28 | GJ 3076 | M5 | 17.856810 | 15.438867 | 58.00 | 7.30 | 192.00 | 8.00 | -130.00 | 8.00 | 2.50 | 0.30 | –
29 | Barta 161 12 | M4.3 | 23.808453 | -7.214511 | 26.82 | 0.05 | 96.76 | 0.07 | -48.94 | 0.03 | 10.00 | 4.00 | 1.5
30 | PM J01538-1459A | M3V | 28.462387 | -14.997647 | 29.64 | 0.04 | 106.72 | 0.04 | -40.60 | 0.03 | 9.00 | 4.00 | 1.4
…
…
…

Note. — The full machine-readable table is provided online.

##  Appendix B Mean Group Sizes Using inclusive members

Table 4: Mean Group Sizes Derived Through Three Distinct Methods

|  $-$Uncer |  $+$Uncer
---|---|---
| (pc) | (pc)
Method 1: Probabilistic Volume Calculation (PVC)
$bw$=7
$\rm RV_{obs}$ | $-$ | 24.1$(^{+0.3}_{-0.3})$
$\rm RV_{corr}$ | $-$ | 25.7$(^{+0.3}_{-0.3})$
$bw$=10
$\rm RV_{obs}$ | $-$ | 28.3$(^{+0.3}_{-0.3})$
$\rm RV_{corr}$ | $-$ | 29.9$(^{+0.3}_{-0.3})$
$bw$=15
$\rm RV_{obs}$ | $-$ | 35.9$(^{+0.2}_{-0.2})$
$\rm RV_{corr}$ | $-$ | 37.2$(^{+0.2}_{-0.2})$
$bw$=20
$\rm RV_{obs}$ | $-$ | 44.0$(^{+0.2}_{-0.2})$
$\rm RV_{corr}$ | $-$ | 45.3$(^{+0.2}_{-0.2})$
Method 2: Mean Pairwise Distance Calculation
$\rm RV_{obs}$ | 22.3$(^{+1.7}_{-1.9})$ | 24.8$(^{+2.0}_{-1.9})$
$\rm RV_{corr}$ | 24.9$(^{+1.8}_{-2.0})$ | 27.6$(^{+2.1}_{-2.1})$
Method 3: Covariance Matrix Calculation
$\rm RV_{obs}$ | 8.4$(^{+0.6}_{-0.6})$ | 9.8$(^{+0.9}_{-0.8})$
$\rm RV_{corr}$ | 9.4$(^{+0.6}_{-0.7})$ | 10.9$(^{+1.0}_{-0.9})$

Note. — The results are obtained using the inclusive members. The reported
uncertainties correspond to the 68% confidence interval.

##  Appendix C Results Using core members

Table 5: Ages for core members

|  $-$Uncer |  $+$Uncer
---|---|---
| (pc) | (pc)
Method 2: Mean Pairwise Distance Calculation
$\rm RV_{obs}$ | 16.0$(^{+0.9}_{-1.3})$ | 14.8$(^{+1.6}_{-2.1})$
$\rm RV_{corr}$ | 18.9$(^{+1.6}_{-2.6})$ | 16.5$(^{+2.8}_{-4.2})$
Method 3: Covariance Matrix Calculation
$\rm RV_{obs}$ | 16.4$(^{+1.1}_{-0.8})$ | 15.7$(^{+1.9}_{-1.7})$
$\rm RV_{corr}$ | 20.6$(^{+4.8}_{-1.7})$ | 18.9$(^{+5.9}_{-2.9})$
SummaryaaThe summary ages were determined through a weighted mean of two
methods. We utilized an age with $bw$=10 for the PVC, and we adopted larger
uncertainties.
$\rm RV_{obs}$ | 16.2$\pm$0.8 | 15.3$\pm$1.4
$\rm RV_{corr}$ | 19.3$\pm$2.3 | 17.3$\pm$3.4

Note. — The reported uncertainties correspond to the 68% confidence interval.

Table 6: Mean Group Sizes for core members

|  $-$Uncer |  $+$Uncer
---|---|---
| (pc) | (pc)
Method 2: Mean Pairwise Distance Calculation
$\rm RV_{obs}$ | 21.0$(^{+1.8}_{-2.0})$ | 23.1$(^{+2.2}_{-2.2})$
$\rm RV_{corr}$ | 23.3$(^{+1.9}_{-2.2})$ | 25.7$(^{+2.3}_{-2.4})$
Method 3: Covariance Matrix Calculation
$\rm RV_{obs}$ | 7.4$(^{+0.6}_{-0.7})$ | 8.7$(^{+0.9}_{-0.9})$
$\rm RV_{corr}$ | 8.2$(^{+0.7}_{-0.8})$ | 9.7$(^{+1.1}_{-1.0})$

Note. — The reported uncertainties correspond to the 68% confidence interval.

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