19 | UCL |  $\rho$ Lup | * rho Lup | 246 |  315.12${}^{+6.44}_{-3.47}$ |  9.83${}^{+3.78}_{-3.09}$ |  8.16${}^{+0.55}_{-0.57}$ |  123${}^{+9}_{-8}$ |  87${}^{+7}_{-8}$ |  -82${}^{+9}_{-13}$ |  21${}^{+7}_{-6}$
20 | UCL |  $\nu$ Cen | * nu Cen | 1737 |  318.50${}^{+9.60}_{-8.02}$ |  17.45${}^{+4.13}_{-4.08}$ |  7.21${}^{+0.67}_{-0.69}$ |  139${}^{+15}_{-12}$ |  99${}^{+22}_{-19}$ |  -87${}^{+18}_{-12}$ |  41${}^{+12}_{-9}$
21 | LCC | Centaurus-Far | HD 121808 | 99 |  310.69${}^{+2.02}_{-3.30}$ |  -1.18${}^{+2.14}_{-2.74}$ |  5.25${}^{+0.35}_{-0.31}$ |  190${}^{+12}_{-12}$ |  122${}^{+15}_{-12}$ |  -142${}^{+6}_{-15}$ |  -4${}^{+7}_{-9}$
22 | LCC |  $\sigma$ Cen | * sig Cen | 1805 |  301.56${}^{+5.56}_{-5.32}$ |  8.35${}^{+4.99}_{-4.74}$ |  8.71${}^{+0.87}_{-1.05}$ |  115${}^{+16}_{-10}$ |  60${}^{+12}_{-11}$ |  -96${}^{+10}_{-13}$ |  17${}^{+11}_{-10}$
23 | LCC | Acrux | * zet Crux | 394 |  300.27${}^{+3.20}_{-1.71}$ |  -1.98${}^{+2.05}_{-3.53}$ |  9.41${}^{+0.47}_{-0.37}$ |  106${}^{+4}_{-5}$ |  54${}^{+5}_{-3}$ |  -91${}^{+7}_{-4}$ |  -4${}^{+4}_{-6}$
24 | LCC | Musca-foreground | HD 107947 | 95 |  300.64${}^{+1.69}_{-2.18}$ |  -10.27${}^{+1.81}_{-1.92}$ |  9.79${}^{+0.30}_{-0.40}$ |  102${}^{+4}_{-3}$ |  52${}^{+3}_{-4}$ |  -86${}^{+3}_{-4}$ |  -18${}^{+3}_{-4}$
25 | LCC |  $\epsilon$ Cham | * DX Cha | 39 |  300.34${}^{+0.71}_{-0.27}$ |  -15.97${}^{+0.68}_{-0.76}$ |  9.81${}^{+0.18}_{-0.26}$ |  102${}^{+3}_{-2}$ |  50${}^{+1}_{-1}$ |  -85${}^{+3}_{-3}$ |  -28${}^{+1}_{-2}$
26 | LCC |  $\eta$ Cham | * eta Cha | 30 |  292.49${}^{+1.56}_{-0.48}$ |  -21.60${}^{+2.30}_{-0.32}$ |  10.14${}^{+0.40}_{-0.13}$ |  99${}^{+1}_{-4}$ |  35${}^{+2}_{-1}$ |  -85${}^{+2}_{-1}$ |  -36${}^{+5}_{-1}$
27 | Pipe | B59 | Em* AS 218 | 32 |  357.10${}^{+0.38}_{-0.30}$ |  7.11${}^{+0.59}_{-0.63}$ |  6.23${}^{+0.12}_{-0.16}$ |  160${}^{+4}_{-3}$ |  159${}^{+4}_{-3}$ |  -8${}^{+1}_{-1}$ |  20${}^{+2}_{-2}$
28 | Pipe | Pipe-North | HD 155427 | 42 |  4.92${}^{+1.29}_{-1.15}$ |  12.85${}^{+2.53}_{-1.98}$ |  7.69${}^{+0.69}_{-0.30}$ |  130${}^{+5}_{-11}$ |  126${}^{+6}_{-12}$ |  11${}^{+3}_{-3}$ |  29${}^{+6}_{-5}$
29 | Pipe |  $\theta$ Oph | HD 158704 | 98 |  359.71${}^{+1.09}_{-2.75}$ |  7.05${}^{+2.66}_{-1.95}$ |  6.79${}^{+0.29}_{-0.20}$ |  147${}^{+5}_{-6}$ |  146${}^{+4}_{-6}$ |  -1${}^{+3}_{-7}$ |  19${}^{+5}_{-6}$
30 | CrA | CrA-Main | HD 177076 | 96 |  359.87${}^{+0.38}_{-0.66}$ |  -17.65${}^{+0.70}_{-0.32}$ |  6.46${}^{+0.13}_{-0.13}$ |  155${}^{+3}_{-3}$ |  147${}^{+3}_{-3}$ |  0${}^{+1}_{-2}$ |  -47${}^{+2}_{-1}$
31 | CrA | CrA-North | HD 172910 | 351 |  359.02${}^{+1.17}_{-1.77}$ |  -13.97${}^{+2.74}_{-1.91}$ |  6.70${}^{+0.25}_{-0.29}$ |  149${}^{+7}_{-5}$ |  145${}^{+7}_{-5}$ |  -2${}^{+3}_{-5}$ |  -36${}^{+8}_{-5}$
32 | CrA | Scorpio-Sting | HD 157864 | 132 |  350.59${}^{+5.27}_{-3.11}$ |  -3.04${}^{+4.72}_{-3.77}$ |  7.49${}^{+0.66}_{-0.52}$ |  134${}^{+10}_{-11}$ |  131${}^{+9}_{-11}$ |  -22${}^{+12}_{-6}$ |  -7${}^{+11}_{-9}$
33 | Cham | Chamaeleon-1 | V* CR Cha | 192 |  297.22${}^{+0.21}_{-0.52}$ |  -15.52${}^{+0.90}_{-0.37}$ |  5.25${}^{+0.18}_{-0.10}$ |  191${}^{+4}_{-6}$ |  84${}^{+2}_{-3}$ |  -164${}^{+6}_{-4}$ |  -50${}^{+2}_{-2}$
34 | Cham | Chamaeleon-2 | V* BF Cha | 54 |  303.69${}^{+0.23}_{-0.37}$ |  -14.72${}^{+0.74}_{-0.38}$ |  5.08${}^{+0.13}_{-0.16}$ |  197${}^{+6}_{-5}$ |  105${}^{+4}_{-3}$ |  -159${}^{+4}_{-5}$ |  -50${}^{+3}_{-4}$
35 | NE | L134/L183 | HD 141569 | 24 |  358.40${}^{+5.78}_{-3.05}$ |  36.84${}^{+0.79}_{-2.22}$ |  8.76${}^{+0.24}_{-0.32}$ |  114${}^{+4}_{-3}$ |  93${}^{+4}_{-4}$ |  -3${}^{+9}_{-5}$ |  67${}^{+4}_{-1}$
36 | NE | Oph-Southeast | HD 154922 | 61 |  4.54${}^{+1.49}_{-1.01}$ |  13.06${}^{+0.68}_{-0.63}$ |  4.80${}^{+0.19}_{-0.24}$ |  208${}^{+11}_{-8}$ |  202${}^{+11}_{-8}$ |  16${}^{+6}_{-4}$ |  48${}^{+3}_{-4}$
37 | NE | Oph-NorthFar | BD-06 4472 | 28 |  9.60${}^{+2.56}_{-1.57}$ |  24.98${}^{+1.32}_{-1.33}$ |  5.06${}^{+0.24}_{-0.37}$ |  198${}^{+15}_{-9}$ |  176${}^{+16}_{-8}$ |  29${}^{+11}_{-5}$ |  85${}^{+2}_{-6}$

191919In Cols. 6–12 we list the medians of the positional parameters for each
cluster including all cluster members (without considering any stability cut).
The given lower and upper uncertainties represent the 1$\sigma$ scatter around
the median. In this scatter the original measurement uncertainties of single
stellar sources are not considered.

Table 4: Median velocity parameters for the 37 SigMA clusters in Sco-Cen.

SigMA | Group Name | Nr. | $\mu_{\alpha}^{*}$ | $\mu_{\delta}$ | $v_{\alpha}$ | $v_{\delta}$ | $v_{\alpha,\mathrm{LSR}}$ | $v_{\delta,\mathrm{LSR}}$
---|---|---|---|---|---|---|---|---
| (short version) |  | (mas/yr) | (km/s) | (km/s)
1 (US) |  $\rho$ Oph | 535 |  -6.81${}^{+1.36}_{-1.56}$ |  -25.91${}^{+1.78}_{-1.58}$ |  -4.50${}^{+0.89}_{-1.01}$ |  -16.95${}^{+0.91}_{-1.28}$ |  1.36${}^{+0.84}_{-0.94}$ |  -4.20${}^{+1.01}_{-1.15}$
2 |  $\nu$ Sco | 150 |  -8.47${}^{+0.74}_{-0.81}$ |  -24.50${}^{+0.76}_{-0.98}$ |  -5.62${}^{+0.55}_{-0.56}$ |  -16.22${}^{+0.61}_{-0.73}$ |  1.21${}^{+0.37}_{-0.43}$ |  -4.52${}^{+0.58}_{-0.60}$
3 |  $\delta$ Sco | 691 |  -12.24${}^{+2.04}_{-2.54}$ |  -23.92${}^{+1.27}_{-1.35}$ |  -8.22${}^{+1.40}_{-1.99}$ |  -16.27${}^{+0.88}_{-0.75}$ |  -0.77${}^{+0.88}_{-1.09}$ |  -3.88${}^{+0.57}_{-0.66}$
4 |  $\beta$ Sco | 285 |  -9.39${}^{+1.37}_{-1.07}$ |  -21.69${}^{+1.00}_{-1.73}$ |  -6.87${}^{+0.86}_{-0.77}$ |  -15.86${}^{+0.77}_{-1.50}$ |  0.26${}^{+0.56}_{-0.59}$ |  -4.13${}^{+0.58}_{-0.96}$
5 |  $\sigma$ Sco | 544 |  -10.68${}^{+1.47}_{-1.37}$ |  -21.79${}^{+1.34}_{-1.87}$ |  -8.12${}^{+1.09}_{-0.87}$ |  -16.49${}^{+0.92}_{-1.23}$ |  -1.68${}^{+0.78}_{-0.61}$ |  -3.48${}^{+0.91}_{-1.31}$
6 | Antares | 502 |  -11.00${}^{+1.59}_{-1.25}$ |  -23.31${}^{+2.15}_{-2.00}$ |  -7.24${}^{+0.87}_{-0.69}$ |  -15.52${}^{+1.29}_{-0.72}$ |  -1.29${}^{+0.84}_{-0.74}$ |  -2.43${}^{+0.86}_{-1.17}$
7 |  $\rho$ Sco | 240 |  -15.98${}^{+1.96}_{-2.08}$ |  -24.14${}^{+1.65}_{-1.72}$ |  -10.44${}^{+1.22}_{-1.18}$ |  -15.82${}^{+0.74}_{-0.85}$ |  -3.64${}^{+0.52}_{-0.59}$ |  -2.78${}^{+0.62}_{-0.78}$
8 | Sco-Body | 373 |  -8.04${}^{+2.92}_{-1.91}$ |  -26.82${}^{+3.67}_{-3.55}$ |  -5.30${}^{+2.16}_{-1.70}$ |  -17.64${}^{+1.06}_{-1.08}$ |  -1.38${}^{+0.69}_{-0.39}$ |  -2.83${}^{+0.52}_{-0.67}$
9 | US-fg | 276 |  -19.97${}^{+2.74}_{-2.86}$ |  -31.25${}^{+4.09}_{-3.02}$ |  -10.63${}^{+1.33}_{-0.96}$ |  -16.28${}^{+1.20}_{-1.07}$ |  -3.33${}^{+1.16}_{-0.73}$ |  -3.86${}^{+0.88}_{-0.77}$
10 (UCL) | V1062-Sco | 1029 |  -12.14${}^{+0.96}_{-2.02}$ |  -21.15${}^{+0.93}_{-0.82}$ |  -10.18${}^{+0.75}_{-1.43}$ |  -17.73${}^{+0.75}_{-0.61}$ |  -5.12${}^{+0.41}_{-0.52}$ |  -2.15${}^{+0.66}_{-0.55}$
11 |  $\mu$ Sco | 54 |  -11.75${}^{+0.42}_{-0.83}$ |  -22.61${}^{+0.55}_{-0.37}$ |  -9.22${}^{+0.54}_{-0.53}$ |  -17.61${}^{+0.49}_{-0.44}$ |  -5.28${}^{+0.30}_{-0.25}$ |  -2.11${}^{+0.41}_{-0.31}$
12 | Libra-S | 71 |  -14.87${}^{+1.55}_{-1.76}$ |  -20.92${}^{+1.22}_{-1.03}$ |  -11.15${}^{+1.11}_{-0.86}$ |  -15.44${}^{+0.45}_{-0.71}$ |  -1.00${}^{+0.53}_{-0.62}$ |  -3.71${}^{+0.43}_{-0.42}$
13 | Lup 1-4 | 226 |  -10.54${}^{+1.51}_{-1.67}$ |  -23.39${}^{+1.01}_{-1.02}$ |  -8.07${}^{+1.18}_{-1.41}$ |  -17.77${}^{+0.95}_{-0.74}$ |  -0.76${}^{+0.79}_{-1.07}$ |  -2.85${}^{+0.75}_{-0.92}$
14 |  $\eta$ Lup | 769 |  -17.63${}^{+2.28}_{-2.01}$ |  -27.84${}^{+2.12}_{-3.93}$ |  -11.21${}^{+1.70}_{-1.34}$ |  -17.83${}^{+0.95}_{-0.83}$ |  -4.15${}^{+0.79}_{-0.85}$ |  -3.18${}^{+0.83}_{-0.65}$
15 |  $\phi$ Lup | 1114 |  -20.99${}^{+2.67}_{-4.71}$ |  -25.60${}^{+3.00}_{-4.16}$ |  -13.31${}^{+1.48}_{-1.24}$ |  -15.82${}^{+0.90}_{-1.38}$ |  -3.37${}^{+1.05}_{-1.00}$ |  -2.45${}^{+0.85}_{-1.03}$
16 | Norma-N | 42 |  -27.93${}^{+2.68}_{-2.41}$ |  -42.80${}^{+2.50}_{-4.59}$ |  -14.07${}^{+1.61}_{-1.03}$ |  -21.39${}^{+1.22}_{-1.69}$ |  -5.73${}^{+0.82}_{-0.83}$ |  -6.05${}^{+0.73}_{-1.40}$
17 |  $e$ Lup | 516 |  -20.80${}^{+2.73}_{-4.62}$ |  -21.67${}^{+1.78}_{-1.60}$ |  -14.44${}^{+1.17}_{-1.46}$ |  -15.07${}^{+1.42}_{-0.67}$ |  -3.81${}^{+0.54}_{-0.50}$ |  -1.05${}^{+0.79}_{-0.60}$
18 | UPK606 | 131 |  -20.07${}^{+1.27}_{-0.73}$ |  -17.01${}^{+0.71}_{-1.17}$ |  -15.96${}^{+0.87}_{-0.49}$ |  -13.69${}^{+0.44}_{-0.60}$ |  -3.20${}^{+0.44}_{-0.33}$ |  -1.14${}^{+0.47}_{-0.31}$
19 |  $\rho$ Lup | 246 |  -26.22${}^{+2.43}_{-2.95}$ |  -23.13${}^{+3.50}_{-2.37}$ |  -15.51${}^{+1.52}_{-1.06}$ |  -13.14${}^{+1.35}_{-1.36}$ |  -1.95${}^{+0.45}_{-0.73}$ |  -1.18${}^{+0.41}_{-0.40}$
20 |  $\nu$ Cen | 1737 |  -23.33${}^{+5.86}_{-4.69}$ |  -20.27${}^{+2.26}_{-2.61}$ |  -15.28${}^{+2.99}_{-1.84}$ |  -13.53${}^{+1.99}_{-1.64}$ |  -1.78${}^{+0.79}_{-0.73}$ |  -1.71${}^{+0.61}_{-0.82}$
21 (LCC) | Cen-Far | 99 |  -18.40${}^{+1.87}_{-2.36}$ |  -11.75${}^{+2.75}_{-2.67}$ |  -16.64${}^{+1.24}_{-1.26}$ |  -10.51${}^{+2.22}_{-2.15}$ |  -2.79${}^{+1.28}_{-0.83}$ |  0.31${}^{+1.72}_{-1.50}$
22 |  $\sigma$ Cen | 1805 |  -33.23${}^{+4.14}_{-3.71}$ |  -13.67${}^{+3.91}_{-5.10}$ |  -18.27${}^{+1.02}_{-0.69}$ |  -7.70${}^{+2.33}_{-2.30}$ |  -2.20${}^{+0.50}_{-0.67}$ |  -0.41${}^{+0.77}_{-0.63}$
23 | Acrux | 394 |  -37.73${}^{+1.83}_{-1.67}$ |  -11.36${}^{+2.84}_{-4.43}$ |  -19.10${}^{+0.68}_{-0.40}$ |  -5.69${}^{+1.32}_{-2.00}$ |  -2.75${}^{+0.32}_{-0.37}$ |  -0.17${}^{+0.45}_{-0.35}$
24 | Musca-fg | 95 |  -39.37${}^{+1.82}_{-1.57}$ |  -9.33${}^{+4.36}_{-3.95}$ |  -19.20${}^{+0.61}_{-0.33}$ |  -4.49${}^{+2.05}_{-2.02}$ |  -2.85${}^{+0.31}_{-0.22}$ |  -0.23${}^{+0.44}_{-0.40}$
25 |  $\epsilon$ Cham | 39 |  -41.23${}^{+2.27}_{-0.87}$ |  -6.05${}^{+2.04}_{-2.99}$ |  -19.85${}^{+0.73}_{-0.42}$ |  -2.92${}^{+1.01}_{-1.44}$ |  -3.31${}^{+0.50}_{-0.51}$ |  -0.54${}^{+0.49}_{-0.46}$
26 |  $\eta$ Cham | 30 |  -30.16${}^{+1.93}_{-4.40}$ |  26.86${}^{+1.24}_{-5.90}$ |  -14.06${}^{+0.69}_{-2.04}$ |  12.55${}^{+0.67}_{-3.18}$ |  -2.58${}^{+0.47}_{-0.25}$ |  2.09${}^{+0.59}_{-0.51}$
27 (Pipe) | B59 | 32 |  -0.49${}^{+0.70}_{-1.16}$ |  -18.84${}^{+0.41}_{-0.60}$ |  -0.37${}^{+0.53}_{-0.89}$ |  -14.48${}^{+0.71}_{-0.76}$ |  2.13${}^{+0.70}_{-0.63}$ |  -0.61${}^{+0.68}_{-0.83}$
28 | Pipe-N | 42 |  -4.78${}^{+1.51}_{-2.56}$ |  -23.36${}^{+0.62}_{-1.65}$ |  -3.08${}^{+1.04}_{-1.30}$ |  -14.55${}^{+0.79}_{-0.36}$ |  -0.27${}^{+0.44}_{-0.65}$ |  -2.79${}^{+0.38}_{-0.28}$
29 |  $\theta$ Oph | 98 |  -4.71${}^{+0.45}_{-0.94}$ |  -21.85${}^{+0.67}_{-2.08}$ |  -3.29${}^{+0.39}_{-0.73}$ |  -15.41${}^{+0.35}_{-0.93}$ |  -1.16${}^{+0.38}_{-0.40}$ |  -2.03${}^{+0.45}_{-0.70}$
30 (CrA) | CrA-Main | 96 |  4.57${}^{+1.19}_{-0.70}$ |  -27.11${}^{+0.88}_{-1.33}$ |  3.33${}^{+0.98}_{-0.47}$ |  -19.83${}^{+0.63}_{-1.21}$ |  -1.76${}^{+0.83}_{-0.49}$ |  -4.77${}^{+0.67}_{-1.04}$
31 | CrA-North | 351 |  0.92${}^{+1.84}_{-2.34}$ |  -27.60${}^{+1.17}_{-1.00}$ |  0.65${}^{+1.32}_{-1.59}$ |  -19.51${}^{+0.68}_{-0.63}$ |  -3.16${}^{+0.59}_{-0.80}$ |  -4.25${}^{+0.54}_{-0.62}$
32 | Sco-Sting | 132 |  -10.03${}^{+2.98}_{-2.27}$ |  -29.75${}^{+1.78}_{-4.47}$ |  -6.12${}^{+1.33}_{-1.49}$ |  -19.51${}^{+1.49}_{-0.82}$ |  -5.15${}^{+0.52}_{-0.43}$ |  -3.70${}^{+0.65}_{-0.44}$
33 (Cham) | Cham-1 | 192 |  -22.55${}^{+0.72}_{-0.91}$ |  0.38${}^{+1.21}_{-1.11}$ |  -20.31${}^{+0.64}_{-0.71}$ |  0.35${}^{+1.06}_{-1.01}$ |  -3.96${}^{+0.74}_{-0.70}$ |  -0.79${}^{+0.86}_{-0.79}$
34 | Cham-2 | 54 |  -20.16${}^{+0.76}_{-0.94}$ |  -7.55${}^{+0.77}_{-0.64}$ |  -18.95${}^{+0.65}_{-0.73}$ |  -7.07${}^{+0.83}_{-0.53}$ |  -3.24${}^{+0.55}_{-0.57}$ |  -0.22${}^{+0.51}_{-0.64}$
35 (NE) | L134/L183 | 24 |  -17.63${}^{+1.02}_{-0.97}$ |  -20.28${}^{+1.00}_{-1.52}$ |  -9.75${}^{+0.92}_{-0.62}$ |  -11.13${}^{+0.94}_{-0.92}$ |  -0.83${}^{+0.17}_{-0.36}$ |  -2.44${}^{+0.34}_{-0.17}$
36 | Oph-SE | 61 |  -5.65${}^{+0.61}_{-0.54}$ |  -11.44${}^{+0.66}_{-0.75}$ |  -5.63${}^{+0.54}_{-0.60}$ |  -11.58${}^{+1.26}_{-0.63}$ |  -2.77${}^{+0.51}_{-0.46}$ |  0.30${}^{+1.04}_{-0.69}$
37 | Oph-NF | 28 |  -8.57${}^{+1.88}_{-1.46}$ |  -16.75${}^{+0.99}_{-1.59}$ |  -7.78${}^{+0.96}_{-1.11}$ |  -15.89${}^{+0.41}_{-0.96}$ |  -2.83${}^{+0.44}_{-0.85}$ |  -7.11${}^{+0.52}_{-0.79}$

202020In Cols. 4–9 we list the medians of the velocity parameters for each
cluster including all cluster members (without considering any stability cut).
The given lower and upper uncertainties represent the 1$\sigma$ scatter
(velocity dispersion) around the median. In this scatter the original
measurement uncertainties are not considered. See Table 3 for the positional
parameters.

Table 5: Hipparcos astrometry from van Leeuwen (2007) of bright stellar
members in Sco-Cen. The Gaia DR3 astrometry is given if available.

HIP | Name | SigMAaa$a$Col. 3 gives the index of the SigMA cluster that is hosting the given star. | SpT | Hipparcos | Gaia DR3
---|---|---|---|---|---
|  |  |  | $l$ | $b$ | $\varpi$ | $\mu_{\alpha}^{*}$ | $\mu_{\delta}$ | $\varpi$ | $\mu_{\alpha}^{*}$ | $\mu_{\delta}$
|  |  |  | (deg) | (mas) |  (mas yr-1) | (mas) |  (mas yr-1)
80473 | * rho Oph | 1 | B2V | 353.69 | 17.69 |  9.03$\pm$0.90 |  -5.53$\pm$0.89 |  -21.74$\pm$0.93 |  7.26$\pm$0.13 |  -4.38$\pm$0.19 |  -23.30$\pm$0.16
79374 | * nu Sco (Jabbah) | 2 | B2IV | 354.61 | 22.70 |  6.88$\pm$0.76 |  -7.65$\pm$0.71 |  -23.71$\pm$0.47 |  7.06$\pm$0.22 |  -7.44$\pm$0.25 |  -28.20$\pm$0.16
78401 | * del Sco (Dschubba) | 3 | B0.2IV | 350.10 | 22.49 |  6.64$\pm$0.89 |  -10.21$\pm$1.01 |  -35.41$\pm$0.71 |  |  |
78820 | * bet Sco (Acrab) | 4 | B0.5V | 353.19 | 23.60 |  8.07$\pm$0.78 |  -5.20$\pm$0.92 |  -24.04$\pm$0.64 |  |  |
80112 | * sig Sco (Alniyat) | 5 | B1III | 351.31 | 17.00 |  4.68$\pm$0.60 |  -10.60$\pm$0.78 |  -16.28$\pm$0.43 |  |  |
79404 | * c02 Sco | 5 | B2V | 348.12 | 16.84 |  6.81$\pm$0.16 |  -10.38$\pm$0.18 |  -23.94$\pm$0.14 |  6.78$\pm$0.14 |  -12.26$\pm$0.18 |  -24.25$\pm$0.14
81266 | * tau Sco | 6 | B0V | 351.53 | 12.81 |  6.88$\pm$0.53 |  -9.89$\pm$0.61 |  -22.83$\pm$0.55 |  |  |
80763 | * alf Sco (Antares) | 6 | M1Ib+B2.5V | 351.95 | 15.06 |  5.89$\pm$1.00 |  -12.11$\pm$1.22 |  -23.30$\pm$0.76 |  |  |
78104 | * rho Sco (Iklil) | 7 | B2IV/V | 344.63 | 18.27 |  6.91$\pm$0.19 |  -15.68$\pm$0.21 |  -24.88$\pm$0.19 |  7.32$\pm$0.29 |  -17.03$\pm$0.33 |  -23.66$\pm$0.27
78265 | * pi Sco | 9 | B1V+B2V | 347.21 | 20.23 |  5.57$\pm$0.64 |  -11.42$\pm$0.78 |  -26.83$\pm$0.74 |  9.67$\pm$0.70 |  -12.40$\pm$0.52 |  -26.21$\pm$0.57
81477 | V* V1062 Sco | 10 | Ap Si | 343.57 | 5.18 |  7.54$\pm$0.61 |  -10.25$\pm$0.69 |  -21.59$\pm$0.45 |  5.72$\pm$0.03 |  -12.45$\pm$0.04 |  -21.17$\pm$0.04
82545 | * mu02 Sco (Pipirima) | 10 | B2IV | 346.20 | 3.86 |  6.88$\pm$0.12 |  -11.09$\pm$0.13 |  -23.32$\pm$0.11 |  5.66$\pm$0.28 |  -12.11$\pm$0.30 |  -22.57$\pm$0.27
82514 | * mu01 Sco (Xamidimura) | 11 | B1.5IV+B | 346.12 | 3.91 |  6.51$\pm$0.91 |  -10.58$\pm$0.87 |  -22.06$\pm$0.74 |  |  |
78533 | V* LL Lup | 13 | Ap Si | 339.64 | 11.41 |  8.61$\pm$0.69 |  -12.65$\pm$0.75 |  -24.14$\pm$0.68 |  6.18$\pm$0.03 |  -12.18$\pm$0.03 |  -24.02$\pm$0.02
78384 | * eta Lup | 14 | B2.5IV | 338.77 | 11.01 |  7.38$\pm$0.18 |  -16.96$\pm$0.19 |  -27.83$\pm$0.19 |  7.59$\pm$0.43 |  -17.59$\pm$0.49 |  -27.26$\pm$0.37
71865 | * b Cen | 15 | B2.5V | 325.90 | 20.10 |  9.62$\pm$0.18 |  -29.92$\pm$0.14 |  -30.68$\pm$0.13 |  10.03$\pm$0.31 |  -29.83$\pm$0.37 |  -31.91$\pm$0.52
76945 | * psi02 Lup | 15 | B5V | 338.48 | 16.08 |  8.97$\pm$0.27 |  -21.37$\pm$0.30 |  -29.98$\pm$0.25 |  |  |
75304 | * phi02 Lup | 15 | B4V | 333.84 | 16.75 |  6.28$\pm$0.20 |  -18.24$\pm$0.22 |  -20.72$\pm$0.16 |  6.49$\pm$0.22 |  -18.47$\pm$0.30 |  -20.41$\pm$0.29
71860 | * alf Lup | 17 | B1.5III | 321.61 | 11.44 |  7.02$\pm$0.17 |  -20.94$\pm$0.14 |  -23.67$\pm$0.14 |  |  |
74449 | * e Lup | 17 | B3IV | 327.83 | 11.43 |  6.47$\pm$0.21 |  -22.01$\pm$0.18 |  -21.75$\pm$0.19 |  7.24$\pm$0.20 |  -21.67$\pm$0.20 |  -21.90$\pm$0.18
76371 | * d Lup | 17 | B3IVp | 331.02 | 8.76 |  7.62$\pm$0.43 |  -20.53$\pm$0.33 |  -21.23$\pm$0.31 |  6.85$\pm$0.13 |  -20.79$\pm$0.15 |  -20.79$\pm$0.11
71536 | * rho Lup | 19 | B5V | 320.13 | 9.86 |  10.32$\pm$0.16 |  -28.26$\pm$0.12 |  -28.82$\pm$0.13 |  9.35$\pm$0.20 |  -28.92$\pm$0.22 |  -29.21$\pm$0.32
72800 | V* V1019 Cen | 20 | B7II/III | 327.93 | 19.11 |  6.63$\pm$0.22 |  -20.48$\pm$0.23 |  -19.20$\pm$0.22 |  6.96$\pm$0.17 |  -20.60$\pm$0.16 |  -20.35$\pm$0.20
67464 | * nu Cen | 20 | B2IV | 314.41 | 19.89 |  7.47$\pm$0.17 |  -26.77$\pm$0.12 |  -20.18$\pm$0.08 |  8.05$\pm$0.35 |  -26.14$\pm$0.42 |  -21.23$\pm$0.94
63945 | * f Cen | 22 | B5V | 305.47 | 14.34 |  8.36$\pm$0.25 |  -29.85$\pm$0.18 |  -15.17$\pm$0.15 |  8.16$\pm$0.16 |  -29.56$\pm$0.12 |  -15.69$\pm$0.10
60823 | * sig Cen | 22 | B3V | 299.10 | 12.47 |  7.92$\pm$0.18 |  -32.36$\pm$0.15 |  -12.51$\pm$0.13 |  7.92$\pm$0.23 |  -32.17$\pm$0.18 |  -13.17$\pm$0.16
60009 | * zet Cru | 23 | B2.5V | 299.32 | -1.36 |  9.12$\pm$0.45 |  -33.80$\pm$0.48 |  -10.15$\pm$0.43 |  |  |
60718 | * alf Cru (Acrux) | 23 | B0.5IV | 300.13 | -0.36 |  10.13$\pm$0.50 |  -35.83$\pm$0.47 |  -14.86$\pm$0.43 |  |  |
58484 | * eps Cha | 25 | B9Vn | 300.21 | -15.62 |  9.02$\pm$0.36 |  -40.34$\pm$0.38 |  -8.30$\pm$0.40 |  |  |
42637 | * eta Cha | 26 | B9IV | 292.40 | -21.65 |  10.53$\pm$0.16 |  -28.89$\pm$0.16 |  27.21$\pm$0.16 |  10.17$\pm$0.07 |  -29.43$\pm$0.18 |  26.83$\pm$0.12
84970 | * tet Oph | 29 | B2IV | 0.46 | 6.55 |  7.48$\pm$0.17 |  -7.37$\pm$0.18 |  -23.94$\pm$0.10 |  |  |
86670 | * kap Sco | 32 | B1.5III | 351.04 | -4.72 |  6.75$\pm$0.17 |  -6.05$\pm$0.21 |  -25.54$\pm$0.13 |  |  |

212121Shown are mostly B-type stars that are either part of the SigMA selected
clusters, or which are the name-givers of some clusters. This is not a
complete list of B-stars in Sco-Cen. The Hipparcos astrometry can be compared
to average parameters of the SigMA clusters as derived from Gaia astrometry in
Tables 3–4, to evaluate possible correlations. The Hipparcos parallaxes should
be treated with caution, since significant deviations to Gaia parallaxes are
possible (on the order of about 60 pc when converted to distances), while
proper motions show deviations on the order of about $\pm 2$ mas yr-1, when
comparing sources which are both in Hipparcos and Gaia DR3 within about 500 pc
from the Sun.

###  5.1 Overview of the seven subregions in Sco-Cen

In the following, and to help compare SigMA results with the literature, we
give a brief overview for each sub-region within Sco-Cen (US, UCL, LCC, Pipe,
CrA, Cham, and NE). We then give a more detailed comparison to recent works in
Sect. 5.2. The listed seven subregions include four regions that are not a
traditional part of the Sco-Cen OB association, namely CrA, Pipe, Cham, and NE
clusters, while we find them to be be part of Sco-Cen because they are co-
moving with the complex itself, and have ages below of about the 20 Myr age
cut we used for the association. Even if we assign each stellar group to one
of the seven subregions, we stress that this classification should not be seen
as physically distinct regions inside Sco-Cen, but simply to help compare our
results with the literature.

####  5.1.1 Upper Scorpius (US)

Toward US we identify nine clusters containing in total 3596 stellar sources,
which are partially extending beyond the traditional borders (Fig. 10). Of
these nine clusters, seven appear higher surface density and tend to be
associated with prominent B-stars, as already pointed out above, namely $\rho$
Oph/L1688, $\beta$ Sco, $\delta$ Sco, $\nu$ Sco, $\sigma$ Sco, Antares, and
$\rho$ Sco (see Tables 3–5). The remaining two clusters appear more extended,
which we name US-foreground and Scorpio-Body.

The clusters $\rho$ Oph/L1688, Antares, and $\rho$ Sco show significant
overlap in the same volume in space, while separating in velocity space. In a
recent paper (Grasser et al., 2021) we studied the $\rho$ Oph/L1688 cluster
with Gaia EDR3 data and identified two kinematically distinct populations
within the same volume (Pop 1 and Pop 2). These two populations coincide with
the $\rho$ Oph/L1688 and Antares clusters, respectively. In detail, the cross-
matched Pop 1 sample contains $\sim$93% of the $\rho$ Oph/L1688 group and few
matches with other clusters (Antares, $\sigma$ Sco, $\beta$ Sco, $\delta$
Sco). The cross-matched Pop 2 sample contains $\sim$75% of the Antares group
and few matches with other clusters ($\rho$ Sco, $\sigma$ Sco, US-foreground).
Luhman (2022a) point out that “new” $\rho$ Oph/L1688 members in Grasser et al.
(2021) have already been identified previously by other literature as being
part of US. We clarify here that the “new” sources in Grasser et al. (2021)
refer to sources that have not been assigned previously as members of the
young $\rho$ Oph/L1688 star-forming event. The two intertwining distinct
populations within the same volume have been mentioned for the first time in
Grasser et al. (2021). In this work, we add another stellar population to the
region, $\rho$ Sco, which also seems to occupy a similar volume in space,
partially overlapping with the two populations while having distinct
velocities from these.

The group US-foreground is located in front of the more compact clusters,
visible in 3D space (Fig. 12), hence the chosen name. Finally, the Scorpio-
Body group extends from US toward the Galactic South, beyond the traditional
borders of US, with a significant fraction located in UCL and in the direction
of CrA (Sect. 5.1.5). It spans the Scorpius constellation’s central body,
hence the name. The nine clusters toward US reveal a complex star formation
history, which will be further discussed in a follow-up paper.

####  5.1.2 Upper Centaurus Lupus (UCL)

We identify rich substructure within UCL separated into 11 SigMA clusters
(5935 stellar sources), as listed in Table 3. The most prominent cluster in
the region is V1062 Sco (Röser et al., 2018), lying towards the far side of
Sco-Cen. This cluster was picked up easily by visual selection methods (e.g.,
by Damiani et al., 2019 or Luhman, 2022a; see Sects. 5.2.1, 5.2.4). We
identify a second cluster close to V1062 Sco, which we call $\mu$ Sco, since
its members are scattered around the bright B-star * mu01 Sco. We find that
the positions and velocities of the two SigMA clusters are very similar, and
members of both clusters are part of V1062-Sco-selections in previous work
(Sect. 5.2), also named UPK 640 in Cantat-Gaudin & Anders (2020). The star *
mu01 Sco, which is the name giver of $\mu$ Sco lies in the center of the
cluster, while the star * mu02 Sco is part of the SigMA selected members for
V1062 Sco, located at the periphery of this cluster. This suggests a possible
connection between the two clusters, but this statement is tentative.

Lupus 1–4 appears correlated with regions of high dust column-density,
matching with previous selections of Lupus–3 and 4 stellar members (e.g.,
Damiani et al., 2019; Kerr et al., 2021), which are merged in the SigMA
selection. The average distance to Lupus 1–4 matches well with cloud distance
estimates from Zucker et al. (2021, derived from , ), who report a distance
between 155–198 pc, or an average of about 165 pc for the Lupus 1–4 clouds
(see also Teixeira et al., 2020).

At the heart of UCL lie the clusters $\eta$ Lup, $\phi$ Lup, and $e$ Lup,
which likely belong to the oldest parts of Sco-Cen, probably the clusters
where the first supernovae in Sco-Cen originated from (Zucker et al., 2022).
To the North of the traditional UCL borders we find a clustering, which has
not been isolated in previous works, named Libra-South, based on its location
within that constellation.

There is one cluster slightly in front and to the south of the main UCL body,
called Norma-North, named after its location in that constellation. This is a
new clustering, which does not have a clear counterpart in the literature.
Another SigMA cluster lies to the far side of UCL and the Galactic west of the
Lupus constellation. This cluster correlates with UPK 606 when compared to
Cantat-Gaudin & Anders (2020) (see also Kerr et al., 2021 and Table 10).
Finally, to the Galactic west, UCL connects with LCC via the clusters $\nu$
Cen and $\rho$ Lup.

####  5.1.3 Lower Centaurus Crux (LCC)

We find six SigMA clusters (2462 stellar sources) toward the LCC region (see
Table 3), which is now reaching farther below the Galactic plane compared to
most of the work in the literature. For the SigMA extraction, the young local
associations $\epsilon$ Cha and $\eta$ Cha are part of LCC, located at the
Southern most tip, confirming the results of Mamajek et al. (1999, 2000) or
Fernández et al. (2008). Toward LCC, SigMA extracts a cluster that seems
unrelated to the main body of LCC, which we name Centaurus-Far, since it lies
about 60 pc to the back of it, at a distance similar to that of the Chamaeleon
clouds (Sect. 5.1.6). This cluster was already identified in Kerr et al.
(2021), as part of the TLC21 group (Cham-group) as EOM3, and named Cen-South
(see Sect. 5.2.2 and Table 10). Consequently, the main body of LCC is composed
of five subgroups, which seem to constitute an age gradient (e.g., Kerr et
al., 2021) from north to south ($\sigma$ Cen, Acrux, Musca-foreground,
$\epsilon$ Cha, and $\eta$ Cha), which will be analyzed in future work.

####  5.1.4 Pipe Nebula

Although not traditionally considered part of the Sco-Cen association, we find
three SigMA clusters toward the Pipe nebula (172 stellar sources), including
B59, Pipe-North, and $\theta$ Oph. The group B59 seems to be closely related
to the star forming B59 cloud (e.g., Lombardi et al., 2006; Brooke et al.,
2007; Román-Zúñiga et al., 2007, 2010). This is supported not only by
projection in the sky towards cluster and cloud but also by the cloud distance
between 147–163 pc (Zucker et al., 2021), compatible with the cluster distance
of about 160 pc. The $\theta$ Oph cluster, surrounding the B2 star * tet Oph,
is located at about the same distance to B59 and is close to the stem of the
Pipe Nebula cloud, giving ground to studies of a possible interaction between
the B2 star and the cloud (Gritschneder & Lin, 2012). Pipe-North lies slightly
in front of the other two clusters (at about 130 pc) and to the Galactic north
of the Pipe Nebula, as the name suggests.

####  5.1.5 Corona Australis (CrA)

The possible physical connection between CrA and the Sco-Cen association was
already pointed out in previous studies (e.g., Mamajek & Feigelson, 2001;
Preibisch & Mamajek, 2008; Kerr et al., 2021) and confirmed by our work. We
count three SigMA clusters to the CrA region, containing in total 579 sources.
We identify a distinct cluster projected on top of the CrA molecular cloud and
the embedded Coronet clusters, which we call the CrA-Main group. The cluster
distance fits to the cloud distance of about 136–179 pc (Zucker et al., 2021).
To the Galactic north we identify a second more extended group, called CrA-
North, which was already discussed in Galli et al. (2020) or Esplin & Luhman
(2022). Additionally, we identify a third group to the Galactic northwest of
the two other clusters, apparently building a bridge to the main body of Sco-
Cen. This group we name Scorpio-Sting since its projected location matches the
sting of the Scorpio constellation. Sco-Sting has only one clear counterpart
in the literature, namely the TLC22/EOM7 group in Kerr et al. (2021) (see
Sect. 5.2.2 and Table 10), while they identify a smaller sub-sample of this
group (12 members in Kerr et al., 2021 versus 132 members in this work).

####  5.1.6 Chamaeleon (Cham)

The well-known star-forming molecular clouds of Chamaeleon are seen through
the same line-of-sight as the southern tip of LCC, but lie clearly towards the
back of LCC when seen in 3D (Fig. 12). We identify two clusters with a total
of 246 stellar sources in Chamaeleon 1 & 2, which are likely directly related
to the two molecular clouds of the same name and are already characterized
with Gaia (e.g. Roccatagliata et al., 2018; Galli et al., 2020; Kerr et al.,
2021, see also Sect. 5.2.2). Due to their youth, position, and tangential
velocities we assume that the Cham clusters and clouds are part of the Sco-Cen
star formation event, but this must be confirmed by tracebacks of the young
population (see, e.g., Großschedl et al., 2021). Similar suggestions appear in
Lépine & Sartori (2003) or Sartori et al. (2003).

####  5.1.7 Northeast clusters (NE)

We identify three extra clusters to the Galactic north and east of Sco-Cen,
which we like to discuss separately in this section, which are L134/L183, Oph-
southeast, and Oph-north-far. We assign these clusters to a separate region,
which we call the Northeast clusters (NE), based on their location relative to
Sco-Cen in Galactic coordinates, since they do not fit to any other of the
Sco-Cen subregions.

The cluster L134/L183 is a small, newly identified stellar group to the
Galactic north of US (with 24 stellar members). This stellar group is likely
associated with the small molecular clouds L134 and L183 (or MBM 36 and 37,
Magnani, Blitz, & Mundy, 1985), that are currently non star-forming (Pagani et
al., 2003, 2004, 2005). The distances to the clouds in Zucker et al. (2019)
are about 105–120 pc, which match the cluster distance of about 114 pc. The
presence of the young stellar group close by the clouds suggests that (1) the
clouds are remnants of a larger cloud that formed the newly identified SigMA
cluster and (2) that the newly identified sources might be playing a role in
the observed “cloudshine” phenomenon towards this cloud (Steinacker et al.,
2010, 2015).

The cluster Ophiuchus Southeast (Oph-SE, 61 members) lies at a similar
projected position as Pipe-North, while being at a farther distance, about 50
pc in the back (hence, we do not count it to the Pipe region). This stellar
group was already selected by Kerr et al. (2021) as TLC 4 with 31 members
(Sect. 5.2.2). Finally, the group Ophiuchus Northfar (Oph-NF, 28 members)
appears to be a newly identified stellar group, located at a similar distance
as Oph-SE. This new group needs more investigations in the future, since the
stability of the selected members, as determined by the SigMA algorithm, is
generally very low (stability ¡ 11%).

Table 6: Overview of the recent Literature to which we compare the SigMA Sco-
Cen clustering results in more detail.

Reference | Sect. | Data | Studied Area | Number statistics
---|---|---|---|---
|  |  |  |  Nr. in Ref.aa$a$Number of stellar members from the given reference. If there was a distinction in the literature between members in a more clustered or diffuse mode (which are generally differently defined in each reference), then the numbers are given below. | SigMAbb$b$Number of stellar cluster members from SigMA in the given studied area (volume), out of the total 13,103 SigMA stellar cluster members. |  Matchescc$c$Number of matches between the given reference and the SigMA clusters. If a distinct comparison with clustered or diffuse sources was given in the literature, then the numbers and matches with these are given below. |  Recovereddd$d$The fraction of recovered and rejected (or not-recovered) sources when comparing the sample sizes from the literature (Col. $a$) with the number of SigMA matches (Col. $c$). |  Rejecteddd$d$The fraction of recovered and rejected (or not-recovered) sources when comparing the sample sizes from the literature (Col. $a$) with the number of SigMA matches (Col. $c$).
Damiani et al. (2019)ee$e$For DPP19 we only give the number of sources within their clustered or diffuse populations within $1000/\varpi_{\mathrm{EDR3}}<200$ pc after a cross-match with Gaia EDR3, and without IC 2602. | 5.2.1 | DR2 |  $(l=360^{\circ}$ to $280^{\circ}$, $b=0^{\circ}$ to $30^{\circ})$ OR | 10,421 | 11,796 | 9328 | 89.5% | 10.5%
|  |  |  $(l=315^{\circ}$ to $280^{\circ}$, $b=-10^{\circ}$ to $0^{\circ})$ | 1732 clustered (17%) |  | 1719 | 99.2% | 0.8%
|  |  |  FOV = 2750 deg2, $d<$200\text{\,}\mathrm{p}\mathrm{c}$$ | 8689 diffuse (83%) |  | 7609 | 87.6% | 12.4%
Kerr et al. (2021)ff$f$For KRK21 we do not give the surveyed area, since they extracted the clusters from all-sky data within 333 pc from the Sun. We show a comparison to their whole TLC22 group (their main Sco-Cen group), then the TLC22 without the older EOM groups, and finally the TLC22 members without older groups and which are in on of the younger EOM clusters. | 5.2.2 | DR2 | The whole TLC22 stellar group | 7394 | 12,796 | 6270 | 84.8% | 15.2%
|  |  | TLC22 without EOM 1–5 | 7138 | 12,796 | 6270 | 87.8% | 12.2%
|  |  | 22 EOMs of TLC22 (without EOM 1–5) | 3453 | 12,796 | 3447 | 99.8% | 0.2%
Schmitt et al. (2022)gg$g$The X-ray selected sources from SCF22 included velocity-clustered and velocity-diffuse sources. The separation of these was applied by us by hand, guided by Fig. 7 in SCF22. Hence, the fractions are only given roughly. The fraction of potential IC 2602 members is also given. The SigMA clusters have only matches with their velocity-clustered population. | 5.2.3 | EDR3 & |  de Zeeuw et al. (1999) borders: | 6150 | 11,348 | 3385 | 55.0% | 45.0%
|  | eROSITA |  US $(l=343^{\circ}$ to $360^{\circ}$, $b=10^{\circ}$ to $30^{\circ})$ OR |  $\sim$69% vel-clustered |  | 3385 |  $\sim$80% |  $\sim$20%
|  |  |  UCL $(l=312^{\circ}$ to $350^{\circ}$, $b=0^{\circ}$ to $25^{\circ})$ OR |  $\sim$26% vel-diffuse |  | 0 |  |
|  |  |  LCC $(l=285^{\circ}$ to $312^{\circ}$, $b=-10^{\circ}$ to $22^{\circ})$, |  $\sim$5% IC 2602 |  | 0 |  |
|  |  |  FOV = 2050 deg2, $d\sim 60$–$200\text{\,}\mathrm{p}\mathrm{c}$ |  |  |  |  |
Luhman (2022a) | 5.2.4 | EDR3 |  $l=2^{\circ}$ to $285^{\circ}$, $b=-12^{\circ}$ to $35^{\circ}$, | 10,509 | 12,215 | 9838 | 93.6% | 6.4%
|  |  |  FOV = 3252 deg2, $d\sim 90$–$250\text{\,}\mathrm{p}\mathrm{c}$ |  |  |  |  |
Žerjal et al. (2021) | 5.2.5 | DR2 |  $l=40^{\circ}$ to $240^{\circ}$, $b=-60^{\circ}$ to $70^{\circ}$, | 8185 | 12,943 | 7671 | 93.7% | 6.3%
|  |  |  FOV = 36,400 deg2, $d\sim 83$–$200\text{\,}\mathrm{p}\mathrm{c}$ |  |  |  |  |
Squicciarini et al. (2021)hh$h$SGB21 only studied the US region, finding sources in a more clustered mode and sources in a more diffuse mode, while the latter are simply the residuals of their clustering procedure. They study a subsample of sources with $v_{r}$ information in the 6D phase space ($\sim$28%), which is not further discussed in this work. | 5.2.6 | EDR3 |  $\alpha=236^{\circ}$ to $251^{\circ}$, $\delta=-29^{\circ}$ to $-16^{\circ}$ | 2745 | 2717 | 2575 | 93.8% | 6.2%
(only US, |  |  |  FOV = 195 deg2, $d\sim 125$–$175\text{\,}\mathrm{p}\mathrm{c}$ | 1442 clustered (53%) |  | 1435 | 99.5% | 0.5%
subsample with RVs) |  |  |  | 1303 diffuse (47%) |  | 1140 | 87.5% | 12.5%
Miret-Roig et al. (2022b)ii$i$Similarly as h, MR22 only studied the US region. They study a subsample of sources with $v_{r}$ information in the 6D phase space ($\sim$30%). The numbers of sources in the MR22 bona-fide-5D and bona-fide-6D samples are listed separately, with “Rest” being sources in neither of the two. Note that the 6D sample is contained withing the 5D sample. | 5.2.7 | DR3 |  $\alpha=235^{\circ}$ to $252^{\circ}$, $\delta=-30^{\circ}$ to $-17^{\circ}$ | 2810 | 3089 | 2683 | 95.5% | 4.5%
(only US, |  |  |  FOV = 221 deg2, $d\sim 80$–$200\text{\,}\mathrm{p}\mathrm{c}$ | 2190 5D (78%) |  | 2145 | 97.9% | 2.1%
subsample with RVs) |  |  |  | 670 6D (24%) |  | 667 | 99.6% | 0.4%
|  |  |  | 620 Rest (22%) |  | 538 | 86.8% | 13.2%

222222

###  5.2 Comparison with previous work

In the following we compare the SigMA selected stellar clusters with recent
results from the literature (Table 6), including eight publications. The
studies by Damiani et al. (2019), Schmitt et al. (2022), Luhman (2022a), and
Žerjal et al. (2021) discuss the whole Sco-Cen region, slightly extending
beyond the traditional Sco-Cen borders, while excluding the regions to the
Galactic South (CrA and Cham). The first three of these studies select members
within broad selection borders decided by hand, which we call in this paper
visual selection methods. Squicciarini et al. (2021), Miret-Roig et al.
(2022b), and Briceño-Morales & Chanamé (2022) focus only on the US region and
extract clusters using a combination of Gaia astrometry and radial velocities.
Kerr et al. (2021) present an all-sky study of young stars within 333 pc,
hence covering the new extended view of the Sco-Cen association, using an
unsupervised machine learning approach, which is more similar to our work then
the aforementioned studies.

The literature samples are cross-matched with the SigMA clusters using the
Gaia DR3 source_id, as specified in Appendix A. We provide an overview of the
discussed literature samples in Table 6, giving the total number of sources of
each literature sample, the total number of sources of SigMA Sco-Cen cluster
members within the respective studied areas, and the number of total matches.
Finally, we list the fraction of sources that we recover or reject (or do not
recover) when compared to the individual literature samples.

####  5.2.1 Comparison with Damiani et al. (2019)

Damiani et al. (2019, hereafter DPP19) analyze Sco-Cen using Gaia DR2 data and
a traditional approach, selecting by hand over-densities in velocity and
position space, followed by selecting pre–main-sequence (PMS) stars from an
HRD. Their field of view (FOV) goes slightly beyond the traditional borders of
the association (see Table 6). They discuss eight compact clusters, which are
prominently peaked in projection and in velocity space (hence, easier to
identify with visual selection methods); these are UCL-1, UCL-2, UCL-3, Lupus
3, LCC-1, US-far, US-near, and the well studied IC 2602. Although SigMA easily
detects IC 2602, we do not discuss this cluster since its age ($\sim$ 30 Myr)
excludes it as a part of the recent Sco-Cen star formation event, as mentioned
above. DPP19 also discuss four diffuse populations (D1, D2a, D2b, US-D2),
which are generally distributed across large parts of the traditional Blaauw
Sco-Cen OB association. Moreover, their catalog includes sources, which have
not been assigned to any group (labeled with “N” in Table 9).

The DPP19 catalog contains in total $14,437$ sources, of which 1734 are in
their seven clustered Sco-Cen populations (350 in IC 2602), 8727 are in their
four diffuse populations, and the rest 3626 have not been assigned to any
population (labeled with “N”). When cross-matching the DPP19 Gaia DR2 sample
with DR3 astrometry, we find that 201 stars (1.4%) are rejected when applying
the distance criteria from DPP19 ($d<200$ pc), due to updated parallaxes in
DR3. The majority of these sources have not been assigned to any group or
belong to one of the diffuse populations. When now considering only the
sources in the clustered and diffuse populations within 200 pc (and without
IC2602), then there are 10,425 potential Sco-Cen members in DPP19, or 10,421
when additionally applying the box and quality criteria form Sect. 2.

There are in total 9635 cross-matches between the SigMA clusters and DPP19,
while 9328 of these (89.5% out of 10,421) belong to either the clustered or
diffuse populations (307 are not assigned, “N”). Of the 9328 cross-matches,
7609 belong to one of the four diffuse populations. Comparing this number to
their total diffuse population (8689 within 200 pc), we find that about 88%
are a match with the SigMA clusters, calling into question the existence of
any physically meaningful diffuse stellar population. In most cases, more than
one DPP19 group (both clustered or diffuse) fits to one of our clusters, and
vice versa (see Table 9). In particular, their diffuse groups each contain
sub-parts of about 10–20 of the SigMA clusters.

Focusing on the 1732 DPP19 sources in compact clusters, there are 1719 matches
with SigMA clusters (99%) within 200 pc. The better consensus considering
their compact samples highlights the higher robustness of these samples (see
also Table 6). Focusing on individual samples, we find that their US-near and
US-far can not be assigned clearly to only one of the SigMA clusters (see
Table 9). US-near correlates best with $\rho$ Oph/L1688 (containing fractions
of $\delta$ Sco, $\nu$ Sco, Antares, and $\beta$ Sco), and US-far with
$\sigma$ Sco (containing fractions of Antares, $\beta$ Sco, $\delta$ Sco, and
$\rho$ Sco). In particular, Antares is distributed almost equally among these
two clusters. The Antares group is partially occupying the same volume as
$\rho$ Sco and in particular $\rho$ Oph/L1688 (see Sect. 5.1.1 and Grasser et
al., 2021). This highlights the capability of SigMA to untangle young
populations that share the same volume but have different space motions. The
rest of the DPP19 compact clusters correlate best with SigMA clusters as
follows: UCL-1 with V1062 Sco and $\mu$ Sco, UCL-2 with UPK 606, UCL-3 with
$\phi$ Lup, LCC-1 with Acrux, and Lup III with Lupus 1–4. Finally, about 9% of
the SigMA cluster members correlate with unassigned sources in DPP19 (N),
within their FOV and our box criteria.

Concerning the different approaches, comparing the DPP19 visual section method
and the SigMA unsupervised clustering method, we first note that the method
used by DPP19 starts with a selection of stars by hand in velocity space,
followed by a selection by hand of PMS stars on the HRD. Such an approach will
generally find more candidates than an unsupervised method and it will deliver
the most prominent clusters. However, somewhat less dense clusters can not be
identified easily, when compared to unsupervised machine learning tools, like
SigMA, and their method is less sensitive to possible spatial and kinematical
structure in the Sco-Cen population. For example, a look at Figs. 2, 3, and 4
in DPP19 will make clear that the total number of member candidates using this
approach is a strong function of the size of the selection shapes used in
tangential velocity space and the HRD. These selection borders will
necessarily select a larger number of true positives than an unsupervised
method, while also the total number of false positives is likely higher.

When focusing on a comparison of the total number of Sco-Cen members in DPP19
stellar clusters (compact and diffuse within 200 pc, 10,421 sources) to the
number of matched SigMA cluster members (9328 within 200 pc) (see Table 6), we
find there are 1093 sources only in DPP19, implying that we could be missing
about 10% of possible members if all 10,421 sources were good members. We did
not perform a detailed comparison but find that the 1093 sources also contain
sources that seem to be older than the SigMA clusters when investigated in an
HRD (similar as in Fig. 14), hence the difference based on this comparison is
likely lower than 10% (see also comparison with Luhman, 2022a). As mentioned
above, we expect SigMA to be missing possible candidates when compared with a
method that selects broad regions in various 2D planes of the phase space, but
also expect the SigMA sample to be less contaminated. Nevertheless, the SigMA
sample contains in total 11,796 sources inside the DPP19 FOV, implying that,
in the end, we find about 10% more Sco-Cen members. This could partially be
caused by the different data sets, DR2 versus DR3, while the different
methodologies likely cause more severe disagreements. A deeper analysis is
needed, although not warranted in this paper.

####  5.2.2 Comparison with Kerr et al. (2021)

Recently, Kerr et al. (2021, hereafter KRK21) presented a study of nearby
young stellar populations within 333 pc from the Sun. They use the HDBSCAN
clustering algorithm (see Sect. B) on Gaia DR2 parallaxes and proper motions
on a pre-selected sample of PMS stars with ages $\lesssim$ 50 Myr. They
identify 27 top-level clusters (TLC), including Chameleon as TLC 21 and the
Sco-Cen association as TLC 22. The latter was further broken down into another
27 sub-groups based on the excess of mass (EOM) method, selecting the most
persistent clusters in the clustering tree. Three of these EOM sub-groups (EOM
12 Lupus; EOM 17 US; and EOM 27 LCC) were further broken down into leafs,
which are nodes of the clustering tree.

The TLC 22 covers the main Sco-Cen association and TLC 21 the Chamaeleon
region. Additionally, there were cross-matches with members of the group TLC
4, which is called Ophiuchus Southeast in KRK21. These three TLC groups
combined show a similar extent to our Sco-Cen extraction. SigMA finds in total
a slightly lower number of groups toward Sco-Cen (37 in this work versus 45 in
KRK21), while the TLC 22 sub-groups in KRK21 also include older or unrelated
populations (e.g., $\beta$ Pic, IC 2602, Platais 8, and EOM-2 & 5), which are
not included in our final Sco-Cen sample, as outlined above. Consequently,
only 39 of the KRK21 groups toward Sco-Cen fall within the 37 selected SigMA
clusters from this work.

In Table 10, we show an overview of the matches of SigMA groups with
corresponding KRK21 groups. Overall, the SigMA Sco-Cen groups are more richly
populated compared to the KRK21 groups. In most cases, there is at least some
overlap between our groups and the TLC 22 main Sco-Cen group (and with TLC 21,
Cham; or TLC 4, Oph-SE), while some of our groups also distinctly correspond
to EOM subgroups (or leafs). For about 40% of the SigMA groups, a clear
accordance with a single EOM group (or leaf group) is not possible due to
overlaps with more than one SigMA group or due to no or only insignificant
overlap (see also Table 10).

Some differences between the SigMA and KRK21 clustering results might arise
from the different data input since we use Gaia DR3 and KRK21 use DR2, while
this would only create minor deviations. Although both HDBSCAN and SigMA
approximate the hierarchical cluster tree, we expect discrepancies in
clustering results. The primary reason for this difference is the cluster tree
pruning strategy discussed in Appendix B. The EOM heuristic prioritizes large
clusters over their children when they maintain a long lifetime in the density
hierarchy. The resulting children fail to exceed the parent’s EOM. Conversely,
our pruning strategy does not depend on cluster lifetimes but only cares about
substantial density valleys between neighboring density peaks.

The additional leaf separations in KRK21 were applied to the Lupus, US, and
LCC regions (in TLC 22, EOM-12,17,27) since they found that there are
substructures that have not been identified by the EOM method. Some leaf
clusters match quite well with SigMA clusters; in contrast, the SigMA clusters
are significantly richer and mostly more extended, and in some cases, they are
differently separated (see details in Table 10). Compared to the EOM heuristic
or SigMA’s multi-modality considerations, leaf clusters do not come with
statistical guarantees. The clustering result is highly susceptible to random
density fluctuations since leaf nodes are extracted only considering the
minimum cluster size criterion (Stuetzle & Nugent, 2010); see Sect. B for more
details. Without any additional pruning strategy, which deals with spurious
clusters, leaf clustering results need to be taken with a grain of salt.
Nevertheless, some of the leafs in US ($\rho$ Oph/L1688, $\nu$ Sco, $\delta$
Sco, $\beta$ Sco) show good agreement with the SigMA US cluster separations,
indicating the robustness of these clusters (see also Sect. 5.2.7).

When comparing all members in the TLC22 group (7394), which encompasses the
main Sco-Cen association, with our Sco-Cen SigMA extraction (12,796 members
without Cham or Oph-SE) we find 6270 cross-matches in total (Table 6). Hence,
1124 ($\sim$15% of TLC22) sources are only in TLC22, and 6526 are only in
SigMA ($\sim$50% of SigMA). We find that the KRK21 TLC22 sample contains at
least 256 sources from older stellar groups, which gets apparent from their
Table 6 (EOM 1–5, including $\beta$ Pic, IC 2602, and Platais 8), and 456
sources of TLC22 match with sources that are in older SigMA clusters.
Combined, this leaves 6895 potential younger TLC22 Sco-Cen members, hence 625
possible extra sources ($\sim$8% of TLC22). For these extra sources, a clear
separation of the younger Sco-Cen stellar groups as discussed in this work,
and the somewhat older groups is not straightforward, since about 50% of the
sources in the TLC22 group have not been assigned to a separate sub-cluster
(EOM or leaf). The somewhat older sources can also be estimated when
investigating an HRD or the velocity space. In the HRD no clear separation of
older or younger sources can be identified. In tangential velocity space,
there are sources that have slightly deviating motions from expected Sco-Cen
motions or which coincide with velocity spaces of the KRK21 older EOM groups
or older SigMA groups. Taking all this into account, the fraction of TLC
22-only sources is likely below 8%.

The reason for these extra potential Sco-Cen members in the KRK21 TLC22 group
is similar to the mentioned reasons above (e.g., in Sect. 5.2.1). The TLC22
group represents a cluster root, enveloping the whole Sco-Cen region and
somewhat beyond, and no additional substructure was extracted (yet). In the
following step KRK21 use the EOM and leaf methods to identify individual
clusters, while in this step they lose almost 50% of the original TLC22 group,
as mentioned above. Focusing only on the TLC22 members that are in one of the
22 younger EOM sub-groups, we find that we recover 99.8% of these sources as
Sco-Cen members. Finally, the TLC22 group seems to be overall more incomplete
compared to the SigMA Sco-Cen extraction, since we find in total more members
($\sim$40%, 12,796 versus 7394), and also somewhat different substructure. At
the same time, the sub-clusters themselves are significantly richer compared
to KRK21.

In conclusion, the comparison with KRK21 highlights the differences that can
arise with different unsupervised machine learning tools. Compared to
applications of HDBSCAN, we find that SigMA is able to extract similar sub-
structure, however, with only one clustering step (no sub-steps like EOM or
leafs are needed), while at the same time extracting significantly higher
numbers of members per cluster.

####  5.2.3 Comparison with Schmitt et al. (2022)

Recently, Schmitt et al. (2022, hereafter SCF22) used eROSITA232323Extended
ROentgen Survey with an Imaging Telescope Array. A wide-field X-ray telescope
on-board the Russian-German “Spectrum-Roentgen-Gamma” (SRG) observatory.
(Merloni et al., 2020) to search for low-mass Sco-Cen members by cross-
correlating the eRASS1 source catalog with the Gaia EDR3 catalog. They discuss
6190 X-ray observed sources within the traditional borders (Blaauw, 1964a; de
Zeeuw et al., 1999) which could be Sco-Cen members. They include sources
within a distance range of 60 to 200 pc (Table 6), restricted to low-mass
stars ($G_{\mathrm{BP}}-G_{\mathrm{RP}}>1$, following Pecaut & Mamajek, 2013).
The 6190 sources include 40 double Gaia sources that match the same eROSITA
source. We will only discuss the 6150 single Gaia sources.

Since X-ray emitting sources are expected to be young (e.g., Schmitt, 1997;
Neuhäuser, 1997; Feigelson & Montmerle, 1999; Bouvier et al., 2014), the
sources detected by eROSITA in the direction of Sco-Cen, as discussed in
SCF22, are potential members of Sco-Cen. They found X-ray sources down to
about 0.1 M⊙, and, unexpectedly, they also found the existence of a population
of young X-ray emitting stars that appear to be more diffuse in velocity
space242424We applied the separation of kinematically clustered and diffuse
populations by hand in $v_{\alpha}$/$v_{\delta}$ space, as indicated in Fig. 7
of SCF22., calling into question search approaches relying on kinematic
selections.

We cross-matched the 6150 SCF22 X-ray selected sources with the SigMA
selection in the same FOV (containing 11,348 SigMA sources, see Table 6). We
find in total 3385 cross-matches, while none of these belong to their
velocity-diffuse population. The latter is expected since SigMA only selects
clusters confined in position-velocity space, which naturally excludes any
such velocity-diffuse sources. SCF22 claim that the diffuse population is
largely composed of young stars, only somewhat older compared to the
kinematically confined Sco-Cen members. We confirm the general youth of the
sources by inspecting the two populations in an HRD. However, we see a
relatively clear age separation between the velocity-clustered and velocity-
diffuse populations. X-ray sources that occupy similar velocity spaces as the
Sco-Cen members have ages between 0.1–20 Myr, while X-ray sources that are
velocity-diffuse have ages between 10–1000 Myr, with the majority at about
30–100 Myr. While these are technically young stars, they seem too old to be
related to the Sco-Cen association.

The origin of this co-spatial but velocity diffuse population remains
mysterious. Since these sources are older than Sco-Cen, they are unlikely to
result from stellar interactions in Sco-Cen (an a priori unlikely process
given the low stellar density of Sco-Cen). The diffuse population, or the co-
eval part, could be related to a relatively older star-formation episode,
sharing today the volume space of Sco-Cen, a plausible scenario in the Milky
Way (Fürnkranz et al., 2019). We posit here that the SCF22 velocity-diffuse
young sources are unlikely to be part of Sco-Cen, but represent a mystery that
needs to be solved. As SCF22 points out, the sensitivity of eROSITA will allow
in the near future to detect virtually all young Sco-Cen low-mass members. A
combination of eROSITA future releases and Gaia data in Sco-Cen will be
crucial to increase statistics and better understand the relation between
observed X-ray luminosity with distance, age, stellar masses, and the origin
of the velocity-diffuse population.

When concentrating on the velocity-coherent sample in SCF22 (without IC 2602),
we find that there are about 20% in the whole SCF22 sample that could be
additional Sco-Cen candidate members. These are only in SCF22 and have similar
velocities as SigMA Sco-Cen members. When investigating possible older-star
contaminants in an HRD (similar to Fig. 14), this fraction would reduce to
about 10%. These extra potential members might result from the broad selection
conditions in SCF22, based on all X-ray detected sources within the Blaauw
borders in a distance range of 60 to 200 pc (Table 6). These broad conditions,
which do not attempt to identify any underlying clustered structure, will
naturally pick up more candidate members, although more false positives too,
as discussed in Sect. 5.2.1 and 5.2.9.

####  5.2.4 Comparison with Luhman (2022)

Luhman (2022a, hereafter L22A) recently investigated the Sco-Cen region using
Gaia EDR3 data to identify 10,509 kinematic candidate members of Sco-Cen (see
Table 6). L22A includes selections for US, UCL/LCC, V1062 Sco, Ophiuchus, and
Lupus (the Southern parts of Sco-Cen are not discussed in L22A), and
concentrates on established stellar groups in Sco-Cen to guide the selection.
The visual selection approach of L22A is not suitable to separate the
underlying kinematical substructure of the Sco-Cen population. For example, it
is clear from Fig. 4 in L22A (bottom panel) that the UCL/LCC group contains
several over-densities in $l/b$ space, but these are not extracted or
identified. The L22A selection is based on global kinematic criteria,
extracting candidates exhibiting proper motions similar to expected proper
motions of known members.

Cross-matching the 10,509 L22A Sco-Cen candidate members with the SigMA
clusters gives a total of 9838 matches (93.6%), 671 L22A only sources (6.4%),
and 2377 SigMA only sources within the L22A studied area (Table 6), where the
SigMA sample contains 12,215 sources in total. A more detailed comparison of
the SigMA clusters with the L22A subgroups252525Subgroups are given in Table 1
of L22A in columns “kin” and “pos”., which are generally larger scale groups,
shows no clear correlation between single groups. Virtually each SigMA cluster
has several matches with various L22A subgroups (and vice versa). When
investigating the 671 L22A only sources, we find that about 50% of these
sources do not show significant signs of being older than 20 Myr, and the
majority of the sources do not show significant deviating motions from SigMA
Sco-Cen cluster velocities. These extra L22A sources, or part of them, could
be Sco-Cen members, meaning we might be missing up to about 6% of the
candidates in L22A. This is not surprising because methods based on visual
selection, using broad selection borders, will naturally find more candidates,
but also more false positives, as also discussed in Sect. 5.2.1. Nevertheless,
the SigMA samples contain in total more Sco-Cen member candidates within the
same field of view.

####  5.2.5 Comparison with Žerjal et al. (2021)

Žerjal et al. (2021, hereafter ZIC21) present another clustering result for
the Sco-Cen association using Chronostar, a clustering tool developed by
Crundall et al. (2019). This is a Bayesian tool to kinematically decompose
stellar groups using the full 6D kinematic data, also performing a kinematic
age determination. They identify eight distinct kinematic components
containing in total 9556 sources26262618 sources are outside of our box and
quality criteria, leaving 9538 ZIC21 sources. Moreover, there are 25 sources
that are outside of the ZIC21 parallax range due to updates from DR2 to DR3..
The 9556 stellar members are both within dense and also diffuse stellar
groups. They also include two known clusters that we excluded from the final
Sco-Cen sample, which are IC 2602 and Platais 8 (H and I in ZIC21). Without
these clusters their sample contains 8185 stars, of which 7671 (94%) match
with the SigMA groups.

The groups are C–US, E–US-multi272727ZIC21 define group E as a complex, multi-
population component., D–UCL-V1062-Sco, F–UCL-V1062-Sco, G–UCL-East, T–UCL-
West, A–LCC-North, and U–LCC-South. Hence, with their method they are
splitting US, UCL, LCC, and also V1062 Sco, each into two parts. We list all
matches of SigMA clusters with ZIC21 in Table 9. Generally, it can be seen
that there is significant mixing of various groups in both the ZIC21 and the
SigMA groups. In particular, the ZIC21 groups encompass larger areas, often
containing several or up to 20 of the SigMA groups. Concerning the groups D
and F, we find that both match with V1062-Sco and $\mu$ Sco, while D has a
slightly higher correlation with V1062-Sco and F with $\mu$ Sco.

Within their FOV also other groups exist (see Fig. 8 in ZIC21), like CrA or
the Cham cloud regions, while they were not selected as kinematic members of
Sco-Cen. We speculate that their initial sample by Gagné et al. (2018c), which
includes sources from US, UCL, and LCC, limits their ability to select these
additional groups. Their low signal-to-noise ratio, positional distance, and
slightly deviating motions from bulk Sco-Cen sources282828Possibly caused by
internal feedback mechanisms in the history of Sco-Cen (e.g., Zucker et al.,
2022). likely prevented their classification as Sco-Cen groups.

Their method fits a mixture of Gaussians to data. Instead of allowing
arbitrary covariance matrices, Chronostar constrains 6D Gaussian distributions
in XYZUVW to the following form. Present-day observations are assumed to
follow a ballistically evolved Gaussian in Galactic potential. The free-
fitting parameters are a cluster’s mean birth position in phase space, its
birth positional and kinematic variance (the covariance matrix is assumed
uncorrelated), and its age. The fitting is done via a modified EM algorithm
where the number of components is determined via the BIC. ZIC21 state that
they see evidence for substructure in several groups. Thus, it has to be
investigated if the groups found in ZIC21 will eventually break in subgroups.
When compared to SigMA we can already see that the individual ZIC21 groups
generally contain more than one SigMA cluster (see Table 9).

####  5.2.6 Comparison with Squicciarini et al. (2021)

Squicciarini et al. (2021, hereafter SGB21) studied 2745 potential US members
(see Table 6) by selecting subgroups solely based on kinematics. They divided
the region into eight groups which they call the clustered population (1442
stars), and into an older diffuse population (1303), which is, however,
differently defined than the velocity-diffuse population in DPP19 or SCF22.

When comparing the SGB21 selection to the SigMA clusters, we find that there
are 2575 cross-matches ($\sim$94%) in total out of the 2745 sources in SGB21.
Hence we miss 170 SGB21 US candidate members, while 13 sources are lost due to
our box and quality criteria (Sect. 2). Focusing on the SGB21 candidate
members in the clustered populations (1442), there are only seven sources that
are only in SGB21 and not in SigMA, while we miss 163 of the SGB21 diffuse
members, which are overall more uncertain members. In total we find almost a
factor two more clustered sources with SigMA in the same FOV as SGB21, 2717
versus 1442 (see Table 6). The 2575 sources match with nine of the SigMA
clusters, while only seven SigMA clusters have a significant number of
matches.

We list the cross-matches of SigMA with SGB21 in Table 11. We highlight more
significant matches here. Groups 1, 2, 3, and 4 match best with $\rho$
Oph/L1688, $\nu$ Sco, $\delta$ Sco, and $\beta$ Sco, respectively, while Group
6 also has significant matches with $\beta$ Sco. Group 5 matches best with
$\sigma$ Sco, while the majority of $\sigma$ Sco is in the SGB21 diffuse
population. Group 7 and Group 8 match best with Antares, while the majority of
Antares is also in their diffuse population. Generally, the Antares group
seems to split up into more than one cluster, also in other previous work. The
two groups US-foreground and $\phi$ Lup have only few matches with the diffuse
population. The SGB21 diffuse population is largely contained within the SigMA
groups $\sigma$ Sco, Antares, $\rho$ Sco, and $\delta$ Sco, with some diffuse
members distributed among each mentioned group (see Table 11). This indicates
that the diffuse population is not a separate older group but it contains
stars that were not clustered by the methodology in SGB21, while they are
clustered in SigMA.

The differences in the final cluster definition in US likely arise from the
different clustering methodologies. To better understand the SGB21 approach we
outline the basics here. SGB21 use a semi-automated approach based on
iterative k-means clustering on a 4D sample, using 2D sky positions and 2D
tangential velocities. The authors propagate the sky positions $15$ Myr into
the past and future, producing a new 4D data set at each step; tangential
velocities are constant throughout individual data sets. By studying the sky
distribution of each slice, SGB21 visually identify over-densities. These
over-densities are extracted via k-means clustering in 4D space at a given
time step. Subsequently, the clustered data points are removed from the data
set, and the process of looking for over-densities starts anew. The clustering
process terminates when the authors cannot find any apparent density peaks in
the sky distribution.

Besides the feature space difference, SigMA has significant differences
compared to SGB21’s iterative clustering approach. First, the k-means
algorithm cannot deal with the observed non-convex cluster shapes in projected
coordinates. The extracted clusters are 4D Voronoi cells292929As far as we
know, scaling between sky coordinates and tangential velocities was not
considered. which can have very elongated shapes. Second, SGB21 analyze 2D
projections of the high-dimensional data to identify clusters visually. Thus,
cluster selection is influenced by projection effects and human judgment.
Conversely, SigMA employs a modality test directly in the high-dimensional
phase space, taking into account multi-dimensional relationships between data
axes. These rather different approaches to extract clusters in US make it
clear that the results can not be compared at face value, while fractions of
the most robust clusters ($\rho$ Oph/L1688, $\nu$ Sco, $\delta$ Sco, $\beta$
Sco, $\sigma$ Sco, and Antares) have been identified by either method.

####  5.2.7 Comparison with Miret-Roig et al. (2022)

Miret-Roig et al. (2022b, hereafter MR22) recently applied a Gaussian mixture
model to Gaia DR3 data of the US region. They include radial velocities, when
available, to select the stellar groups. They identify seven stellar groups
within a FOV of about 220 deg2 that is centered on US, containing 2810 sources
(see also Table 6). A cross-match with the SigMA members gives 2683 matches,
hence we miss 127 sources, of which 23 are outside of our box and quality
criteria (Sect. 2). This leaves 104 potential members that the SigMA algorithm
did not select. Within the same FOV SigMA contains 3089 Sco-Cen members.

When comparing the SigMA groups to the individual seven MR22 groups in more
detail, we find largely good agreement, especially for $\rho$ Oph, $\nu$ Sco,
$\delta$ Sco, and $\beta$ Sco (see also Table A.1 in MR22 and Table 11). The
SigMA Antares and $\sigma$ Sco groups seem to be mixed in MR22, with
significant fractions in both MR22’s $\alpha$ Sco and $\sigma$ Sco. Finally,
the MR22 $\pi$ Sco group, which lies largely in the foreground of the other US
groups, coincides largely with two of the SigMA groups: US-foreground (which
we identified as a foreground population to the traditional US) and $\rho$
Sco. The latter is overlapping in space with Antares and $\rho$ Oph. The
larger volume investigated with SigMA allows selecting a more complete sample
of US-foreground, since the MR22 FOV cuts off the outer edges of what they
call $\pi$ Sco.

Since MR22 also use radial velocity information for a subsample ($\sim$30%),
the classification of these 6D selected sources might be more robust compared
to a 5D selection. Investigating the bona-fide 6D selected members in MR22, we
still find some mixing of SigMA clusters within MR22 clusters (and vice
versa), especially concerning Antares and $\sigma$ Sco. These differences need
more investigations in the future, and dedicated cluster studies are called
for (e.g., with Uncover, Ratzenböck et al., 2020).

####  5.2.8 Comparison with Briceño-Morales & Chanamé (2022)

Briceño-Morales & Chanamé (2022, hereafter BMC22) present another recent study
on US, where they seem to find a similar substructure. We do not include a
detailed comparison with BMC22, since we do not have access to their final
catalog303030The paper only appeared on the arXiv after the first submission
of our paper draft, and we did not (yet) get access to their data tables..
Therefore, we only give a qualitative comparison here.

BMC22 use Gaia EDR3 data to obtain a clustering solution for US by first
combining the convergent point method (Perryman et al., 1998) with a Gaussian
Mixture fit (Pedregosa et al., 2011) to identify kinematic groups with a
Bayesian approach, and second, they use OPTICS (Ankerst et al., 1999) to
identify spatial substructure in XYZ. This is complemented with age estimates
based on Gaia photometry. They select in total 4028 US members, while their
astrometrically clean sample contains 3114 sources, but most of them seem to
be contained within a diffuse population ($\sim$66%). In the same
FOV313131BMC22 FOV: $343^{\circ}<l<360^{\circ}$, $10^{\circ}<b<30^{\circ}$,
$\varpi>5$ mas. we find 3418 clustered Sco-Cen members.

They find that US contains three main kinematic groups, and about 34% are
contained in seven spatial substructures. They find a clear correlation
between cluster density and cluster age, suggesting that these substructures
expand at a measurable rate. Finally, they argue that four potential
supernovae happened in the US region based on literature information (e.g.,
Breitschwerdt et al., 2016; Neuhäuser et al., 2020; Forbes et al., 2021),
cluster mass estimates, and on certain “voids” that are identified in the 3D
distribution of their cluster extraction. Notably, we do not see such voids in
the distribution of the SigMA clusters.

Based on Table 2 in BMC22 we identify the following qualitative matches of
SigMA clusters with BMC22 clusters, when comparing average positions and
velocities. The clusters $\rho$ Oph/L1688, $\beta$ Sco, $\delta$ Sco and $\nu$
Sco seem to match well with the same named clusters in BMC22. Their $\alpha$
Sco seems to match our $\sigma$ Sco cluster (not with Antares), while they
mention that both stars, * alf Sco and * sig Sco, are likely members of this
cluster. This connection is of interest, since also other works seem to mix
the SigMA Antares and $\sigma$ Sco clusters, which needs further
investigations in the future. Their $\pi$ Sco seems to match best with US-
foreground. Their $\omega$ Sco seems to match approximately with Antares. A
detailed cross-match with their data is needed to make a more quantitative
statement, which is not possible at this stage.

####  5.2.9 Concluding remarks on the comparison with the literature

Although SigMA finds the larger number of Sco-Cen member candidates of all the
methods presented in the literature, visual selection methods (e.g., Damiani
et al., 2019; Luhman, 2022a) contain about 6–10% candidate members not
detected by SigMA (Table 6). This is mainly due to these methods “select-by-
eye” approach, in projected sub-spaces of the multi-dimensional phase space,
to identify Sco-Cen candidates. Unsupervised machine learning methods, on the
other hand, find more spatial and kinematical substructures in the Sco-Cen
population and produce samples with lower contamination levels compared to
visual selection methods. More importantly, the SigMA method reveals a more
complex velocity structure across the entire Sco-Cen, critical for a physical
description the formation process of OB associations such as Sco-Cen. When
comparing SigMA to other unsupervised methods, which studied the whole Sco-Cen
area (in particular Kerr et al., 2021), the SigMA clusters are generally
richer in members.

Focusing on the US region, we find generally good agreement between cluster
selections from SGB21, KRK21, MR22, BMC22, and SigMA. Not surprisingly, the
denser clusters in US ($\rho$ Oph, $\nu$ Sco, $\delta$ Sco, $\beta$ Sco) have
been recovered well by the different approaches. The Antares, $\sigma$ Sco,
$\rho$ Sco, and US-foreground clusters are slightly more dispersed and they
have less clear matches across the methods (e.g., merged or distributed
differently in different samples). The newly identified velocity substructure
in the US region, as revealed by Gaia data, is relevant to understand the star
formation processes at play in Sco-Cen and will continue to be an obvious
target for future studies and surveys. Adding radial velocity information will
be critical to further characterize these clusters, as already presented, for
example, by Miret-Roig et al. (2022b).

The somewhat different clustering results for the US region, considering the
five mentioned studies, call for a re-analysis of this important young region,
to better understand its star formation history. In conclusion, SigMA finds
significant numbers of sources not present in other samples (up to 15% more),
while missing candidates when compared with visual selection methods (of the
order of 10%). More work is needed to understand the sources SigMA misses. A
possible way forward toward a most complete sample of Sco-Cen members is to
use SigMA cluster members and 3D velocities (by including radial velocities)
as training sets to the Uncover method, a validated bagging classifier of one-
class support vector machines (see the application in Ratzenböck et al., 2020
to the Meingast-1 stellar stream, Meingast et al., 2019).

In the near future, improved membership lists will allow a more precise
analysis of the star formation history of Sco-Cen, the initial mass function
of each cluster, and the dynamical state of the Sco-Cen complex.

##  6 Summary

In this paper, we present SigMA, a method that explores the topological
properties of a density field to define significant structure. To test and
validate SigMA, we apply it to Gaia DR3 data of the nearest OB association to
Earth, Sco-Cen. The main results of this work can be summarized as follows:

  1. 1.

SigMA is a novel clustering method that interprets density peaks, which are
separated by dips, as significant clusters. Using a graph-based approach, the
technique detects peaks and dips directly in the multi-dimensional phase
space.

  2. 2.

SigMA is fine-tuned to large-scale surveys in astrophysics. In this context,
this new method is able to identify co-spatial and co-moving groups with non-
convex shapes and variable densities, with a measure of significance. SigMA is
able to properly incorporate 5D astrometric uncertainties, does not require
any photometric pre-filtering of stellar populations, and scales to millions
of points.

  3. 3.

SigMA is capable of finding clusters in Gaia DR3 data, reaching stellar volume
densities as low as 0.01 sources/pc3 and tangential velocity differences of
about 0.5 km s-1 between clusters.

  4. 4.

We validate SigMA on two simulated data sets and highlight its merits in
relation to established clustering techniques. Our comparison shows that SigMA
can significantly outperform competing methods, especially in environments
where clusters are densely packed, such as the Sco-Cen OB association.

  5. 5.

SigMA identifies more than $13\,000\text{\,}$ Sco-Cen members located in 37
individual clusters of co-spatial and co-moving young stars. The HRD for each
cluster shows a well-defined sequence. Because SigMA is not aware of a star’s
brightness nor color, the well-defined sequences constitute a validation test
to the ability of SigMA to extract coeval populations. A large fraction of
clusters is seen toward well-known Sco-Cen massive stars, too bright to be in
Gaia DR3, and we (tentatively) associated with them. Because SigMA is not
aware of these massive stars, their association with SigMA clusters also
constitutes a validation test to SigMA.

  6. 6.

When comparing the 37 SigMA stellar populations in Sco-Cen to previous results
from the literature we find mostly agreement, however, several discrepancies
exist. When compared to visual selection methods used recently on Gaia data of
Sco-Cen we find that we might be missing roughly 10% of candidates, while at
the same time finding a higher total number of stellar members. Unsupervised
methods like SigMA find more spatial and kinematical substructure for the same
data set and produce samples with lower contamination levels.

In the future, in particular, when combined with auxiliary radial velocity
surveys, a detailed comparative study of the different clustering methods is
fully warranted. The application of SigMA to upcoming Gaia data releases
promises the unveiling of detailed cluster distributions like the one
presented here but for all the near star-forming regions. Reconstructing an
accurate and high-spatial-resolution Star Formation History of the last 50 Myr
in the Local Milky Way with Gaia data is within reach.

###### Acknowledgements.

We thank the anonymous referee for their detailed and very helpful comments.
We thank Núria Miret-Roig, Maruša Žerjal, Vito Squicciarini, and J. H. M. M.
Schmitt for providing their data ahead of final publication. S. Ratzenböck
acknowledges funding by the Austrian Research Promotion Agency (FFG,
https://www.ffg.at/) under project number FO999892674. Additionally, S.
Ratzenböck extends his gratitude to the research network Data Science @ Uni
Vienna and Petra Schönfelder for excellent administrative support. J.
Großschedl acknowledges funding by the Austrian Research Promotion Agency
(FFG) under project number 873708. This work has used 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). 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 Python, https://www.python.org, of Astropy, a community-developed core
Python package for Astronomy (Astropy Collaboration et al., 2013, 2018), NumPy
(van der Walt et al., 2011), Matplotlib (Hunter, 2007), Galpy (Bovy, 2015),
and Plotly (Plotly Technologies Inc., 2015). This research has used the SIMBAD
database operated at CDS, Strasbourg, France (Wenger et al., 2000), of the
VizieR catalog access tool, CDS, Strasbourg, France (Ochsenbein et al., 2000),
and of “Aladin sky atlas” developed at CDS, Strasbourg Observatory, France
(Bonnarel et al., 2000; Boch & Fernique, 2014). This research has made use of
TOPCAT, an interactive graphical viewer and editor for tabular data (Taylor,
2005).

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##  Appendix A Gaia DR3 data retrieval and details on quality criteria

The Gaia DR3 data was downloaded from the Gaia
Archive323232https://gea.esac.esa.int/archive/ using the following ADQL query:

    SELECT * FROM gaiadr3.gaia_source
    WHERE (1000./parallax*COS(l*PI()/180)*COS(b*PI()/180))>-50.
    AND (1000./parallax*COS(l*PI()/180)*COS(b*PI()/180))<250.
    AND (1000./parallax*SIN(l*PI()/180)*COS(b*PI()/180))>-200.
    AND (1000./parallax*SIN(l*PI()/180)*COS(b*PI()/180))<50.
    AND (1000./parallax*SIN(b*PI()/180))>-95.
    AND (1000./parallax*SIN(b*PI()/180))<100.
    AND parallax_over_error>4.5

The first six expressions give the XYZ box conditions, while X is positive
toward the Galactic center, Y is positive in direction of Galactic rotation,
and Z points toward the Galactic north-pole. XYZ can also be calculated using
astropy.coordinates.SkyCoord from Astropy v4.0. The parameter fidelity_v2 from
Rybizki et al. (2022) was retrieved with the following ADQL query, using the
Topcat TAP Query and the GAVO service333333https://dc.zah.uni-heidelberg.de/,
German Astrophysical Virtual Observatory:

    SELECT gaia.*
    FROM gedr3spur.main AS gaia
    JOIN tap_upload.t1 AS mine
    USING (source_id)

We support our quality criteria choices from Sect. 2 as follows. Using
fidelity_v2 $>$ 0.5 is suggested as separator into “good” and “bad” sources by
Rybizki et al. (2022) (see also Zari et al., 2021). Using a threshold of 0.9
would give slightly cleaner data, however only few sources lie in the range
between 0.5 and 0.9 ($\sim$2% in the box), and we decide for the less
conservative value. The additional cut using parallax_over_error, which is
similar to the signal-to-noise ratio ($\mathrm{S/N}_{\varpi}$), is used to
reduce further parallax uncertainties. The choice of the threshold
$\mathrm{S/N}_{\varpi}>4.5$ is further supported by Rybizki et al. (2022),
where they apply different classifiers (high- and low-S/N classifiers) for
sources above and below this threshold. To avoid inhomogeneous data we decide
to include this S/N threshold. Moreover, we want to avoid too high parallax
errors. As mentioned, we use the inverse of the parallax to estimate the
distance to a source, which gets unreliable if the uncertainties are too
large. For more distant sources, or intrinsically faint sources with high
parallax errors, the distance estimate becomes a non-trivial inference problem
(e.g., Luri et al., 2018; Bailer-Jones et al., 2021). In Table 7 we list the
typical uncertainties of the various parameters for sources inside the box
after the applied quality criteria, and also separately for sources that are
selected as members of the 37 Sco-Cen clusters (given in brackets in Table 7).
It can be seen that the majority of the sources in the box (2$\sigma$) have
parallax uncertainties below 0.6 mas.

For SigMA we use the 5D phase space, as mentioned in Sect. 2, for which we use
tangential velocities in km s-1. The proper motions
($\mu_{\alpha}^{*}=\mu_{\alpha}\cos(\delta)$, $\mu_{\delta}$) are transformed
from mas yr-1 to tangential velocities in km s-1 as follows, where the
conversion constant 4.74047 is in units of km yr s-1.

$\displaystyle v_{\alpha}=4.74047\cdot\mu_{\alpha}^{*}/\varpi$ (21)
$\displaystyle v_{\delta}=4.74047\cdot\mu_{\delta}/\varpi$

Moreover, we correct the tangential motions for the Sun’s reflex motion,
resulting in tangential motions relative to the LSR, using the values by
Schönrich et al. (2010). This conversion is accomplished with the help of
Astropy, by defining the below sky coordinates. For a conversion of
heliocentric proper motions to proper motions relative to the LSR, the radial
velocity can be set to an arbitrary value in the sky-coordinate definition of
Astropy (here set to $0$), since different RV values do not change the outcome
of this conversion.

    from astropy.coordinates import ICRS, LSR
    from astropy import units as u

    skyicrs = ICRS(ra = ra * u.deg,
                   dec = dec * u.degree,
                   distance = 1000./parallax * u.pc,
                   pm_ra_cosdec = pmra * u.mas/u.yr,
                   pm_dec = pmdec * u.mas/u.yr,
                   radial_velocity = 0. * u.km/u.s)

    pma_lsr = skyicrs.transform_to(LSR()).pm_ra_cosdec.value
    pmd_lsr = skyicrs.transform_to(LSR()).pm_dec.value

    v_a_lsr = 4.74047 * pma_lsr / parallax
    v_d_lsr = 4.74047 * pmd_lsr / parallax

Table 7: Typical parameter uncertainties. The listed uncertainties give the
threshold below which the given percentage of sources can be found in the
whole box (or in the SigMA Sco-Cen selection, given in brackets).

|  1, 2, 3 $\sigma$ percentiles
---|---
Uncertaintiesaa$a$Given are the uncertainties for the following parameters: parallax, distance, proper motions in direction of $\alpha$ and $\delta$, tangential velocities in direction of $\alpha$ and $\delta$, radial velocities from Gaia DR3, and XYZ. The RVs are available for a subsample of 367,127 sources (37%) inside the box (or 4967, 40%, in the SigMA Sco-Cen selection). For the derived parameters we give lower and upper uncertainty thresholds, since the errors, gained via a Monte Carlo approach, get asymmetric for large parallax errors. | 68.3% | 95.5% | 99.7%
$\sigma_{\varpi}$ (mas) $<$ | 0.13 (0.08) | 0.56 (0.34) | 1.17 (0.94)
$\mathrm{S/N}_{\varpi}$ $>$ | 47.6 (83.2) | 9.3 (20.8) | 4.8 (7.5)
$\sigma_{d}$ (pc) $<$ | ${}^{+3.8\,(+1.8)}_{-3.6\,(-1.7)}$ | ${}^{+26.0\,(+7.4)}_{-21.0\,(-6.7)}$ | ${}^{+65.2\,(+23.8)}_{-43.7\,(-18.0)}$
$\sigma_{\mu_{\alpha}}$ (mas yr-1) $<$ | 0.13 (0.09) | 0.60 (0.39) | 1.47 (1.06)
$\sigma_{\mu_{\delta}}$ (mas yr-1) $<$ | 0.12 (0.08) | 0.54 (0.32) | 1.34 (0.89)
$\sigma_{v_{\alpha}}$ (km s-1) $<$ | ${}^{+0.4\,(+0.2)}_{-0.4\,(-0.2)}$ | ${}^{+3.2\,(+0.6)}_{-2.7\,(-0.6)}$ | ${}^{+12.1\,(+2.1)}_{-8.9\,(-1.7)}$
$\sigma_{v_{\delta}}$ (km s-1) $<$ | ${}^{+0.4\,(+0.2)}_{-0.4\,(-0.2)}$ | ${}^{+3.3\,(+0.8)}_{-2.8\,(-0.7)}$ | ${}^{+12.1\,(+2.4)}_{-8.9\,(-1.9)}$
$\sigma_{v_{r}}$ (km s-1) $<$ | 3.3 (5.8) | 8.3 (13.0) | 26.2 (37.3)
$\sigma_{X}$ (pc) $<$ | ${}^{+2.3\,(+1.4)}_{-2.2\,(-1.4)}$ | ${}^{+20.1\,(+6.3)}_{-16.5\,(-5.7)}$ | ${}^{+54.6\,(+20.3)}_{-36.8\,(-16.0)}$
$\sigma_{Y}$ (pc) $<$ | ${}^{+1.6\,(+0.6)}_{-1.5\,(-0.6)}$ | ${}^{+13.4\,(+2.9)}_{-11.2\,(-2.7)}$ | ${}^{+41.5\,(+12.1)}_{-28.1\,(-9.8)}$
$\sigma_{Z}$ (pc) $<$ | ${}^{+0.9\,(+0.4)}_{-0.9\,(-0.4)}$ | ${}^{+6.5\,(+1.8)}_{-5.4\,(-1.7)}$ | ${}^{+19.6\,(+6.4)}_{-13.4\,(-5.1)}$

343434

In Sect. 5.2 we compare the SigMA clusters with recent literature samples. To
this end we cross-match the samples using the Gaia DR3/EDR3 source_id. This
cross-match is straight forward for the samples of Schmitt et al. (2022),
Squicciarini et al. (2021), Luhman (2022a), and Miret-Roig et al. (2022b) who
used Gaia DR3/EDR3 data, which allows a direct match with the source_id. In
the case of Damiani et al. (2019), Kerr et al. (2021), and Žerjal et al.
(2021), who used Gaia DR2 data, we first retrieve the Gaia DR3 source_id using
the gaiadr3.dr2_neighbourhood catalog from the Gaia Archive, since the DR2 and
DR3 source_ids are not generally the same. Such a cross-match delivers few
sources that have several possible matches of DR3 with DR2 sources (see Torra
et al., 2021; Gaia Collaboration et al., 2021a). In such cases we choose the
closer match, using the provided angular_distance parameter.

##  Appendix B Related work: cluster analysis

In the following we present a cross section of related work upon which SigMA
rests. From the vast corpus of data mining and statistics literature we focus
specifically on identifying stable groups using density-based clustering
methods.

### Hierarchical, density-based clustering

The strength of level-set formulation, see Eq. (3) and further discussions in
Sect. 3.1.3, lies in the natural emergence of a cluster tree, a clustering
hierarchy which arises from sweeping the density threshold $\lambda$ from
$\infty\to-\infty$. Under a continuous change of $\lambda$, the number of
connected components changes when the threshold passes through a critical
point in $f$, thus $\nabla f=\@vec{0}$. A new cluster is born when $\lambda$
reaches the height of a mode in $f$. On the other hand, a cluster dies when
$\lambda$ traverses a saddle point or a local minimum, in which case the two
connected components merge into a single one. The cluster creation and merging
process is schematically shown in Fig. 1.

However, estimating the connected components of level-sets, while easy in one
dimension, gets nontrivial in higher dimensions. Consequently, algorithmic
realizations of the Hartigan (1975) level-set idea rely on graph heuristics
and graph theory in which connected components arise naturally. Early
implementations by Azzalini & Torelli (2007) and Stuetzle & Nugent (2010) and
subsequent theoretical analyses (Chaudhuri & Dasgupta, 2010; Kpotufe & von
Luxburg, 2011; Chaudhuri et al., 2014) adopt a graph $G(\lambda)$ over the
data samples where vertices and/or edges are filtered according to $\lambda$,
thus $\\{\@vec{x}\in X\leavevmode\nobreak\ :\leavevmode\nobreak\
\hat{f}(\@vec{x})\geq\lambda\\}$353535Edges are commonly assigned the minimum
density sampled along the path connecting two vertices..

However, the use of graphs to represent the connectivity comes with its own
limitations. This scheme guarantees that two samples from one connected
component of $G(\lambda)$ are to be found in a connected component in
$L(\lambda)$. However, as Stuetzle & Nugent (2010) point out, the reverse
implication is not necessarily given. This means, samples from the same
connected component in $L(\lambda)$ may end up in different connected
components of $G(\lambda)$. Since density estimates are inherently noisy,
usually too many clusters arise from this iterative filtration procedure. To
counteract this over-clustering, the resulting graph cluster tree is usually
pruned in a post-processing step during which spurious clusters are identified
and merged back into the “mother cluster” (Stuetzle & Nugent, 2010; Kpotufe &
von Luxburg, 2011; Chaudhuri et al., 2014).

### The HDBSCAN algorithm

A well-known algorithm belonging to the family of hierarchical level-set
methods is the HDBSCAN algorithm (Campello et al., 2013) (Hierarchical
Density-Based Spatial Clustering of Applications with Noise), which recently
has been gaining attention in the astronomical community (e.g., Kounkel &
Covey, 2019; Kounkel et al., 2020; Hunt & Reffert, 2021; Kerr et al., 2021).
In order to prevent over-clustering, the authors introduce the _minimum
cluster size_ parameter which provides an interpretable pruning strategy.

At each cluster split decision, the smaller cluster created is merged back
into the “mother cluster” if it has less than _minimum cluster size_ points,
otherwise, a new cluster is created. To obtain a flat clustering result from
the cluster tree, HDBSCAN estimates the stability of a cluster in the
hierarchy via the concept of _relative excess of mass_ (EOM). Similar to the
concept of excess mass (Muller & Sawitzki, 1991), it measures the lifetime and
size of a cluster. The heuristic favors more prominent and stable clusters
that live longer in the cluster tree. For example, a group that persists for a
long time as a single connected component should be preferred over the two
small clusters it breaks into and which quickly vanish.

However, the EOM criterion tends to produce too large clusters in practice. If
a large group persists in the hierarchy for a long enough time, its children
are unlikely to exceed the parent’s EOM. Alternatively, the HDBSCAN
implementation by McInnes et al. (2017) offers the opportunity to extract the
leaf nodes from the cluster tree. Since the leaf nodes are extracted only
considering the _minimum cluster size_ criterion, the resulting clusters lack
any stability guarantee; thus, the clustering result is highly susceptible to
random density fluctuations. In general, these methods suffer from complex and
hard-to-interpret pruning procedures and parameters, which affect the
confidence and interpretability of the clustering result.

### Topological methods

Extracting a flat clustering from the cluster tree requires a notion of
cluster stability. As discussed, the concept of relative excess of mass, which
inherently depends on the pruning process, can lead to too coarse clusters. A
related pruning heuristic comes from considering the topological persistence
of each mode in $\hat{f}$, introduced by Chazal et al. (2013). Persistence is
defined as the lifespan of each connected component. The notion of persistence
is shown to be stable under small perturbations to the initial density $f$
(Edelsbrunner et al., 2000; Zomorodian & Carlsson, 2005; Ghrist, 2008).

A variation on the persistence formulation is proposed by Ding et al. (2016),
who instead of thresholding the cluster lifetime, use cluster _saliency_
$\nu$, defined by the ratio of birth and death density, as a cluster stability
criterion. By varying $\nu$ between 0 and 1 the cluster tree is revealed and
the most stable and long-lived configuration is chosen as an appropriate
clustering result.

While easy to interpret, these stability parameters can get quite tedious to
select in practice. In the large data and cluster regime, the separation
between stable and unstable clusters becomes less apparent. In these limiting
cases, selecting the input parameters again warrants a proper parameter
search.

### Extracting stable and significant clusters

Compared to the notion of persistence, there is also growing research to apply
statistical methods that test the modality structure of the data. These
methods offer the advantage of an interpretable and meaningful parameter
$\alpha$, defining the significance level of a corresponding hypothesis test.
The null hypothesis H0 commonly assumes that the data, or subsets of it, are
sampled from an uni-modal density, whereas the alternative hypothesis H1
suggests multi-modality. The null hypothesis is rejected at a significance
level $\alpha$ if the $p$-value from the corresponding test procedure exceeds
this significance level.

We identify first applications of hypothesis test procedures in the clustering
literature in the context wrapper methods around the _k_ -means and EM
frameworks. G-means (Hamerly & Elkan, 2004) employs the Anderson-Darling
statistic to test the hypothesis that each cluster is generated from a
Gaussian distribution. Instead of testing on a per-cluster basis, Pg-means
(Feng & Hamerly, 2007) tests the whole Gaussian mixture model (GMM) at once.
Dip-means (Kalogeratos & Likas, 2012) proposes an incremental clustering
scheme for selecting $k$ in $k$-means which employs Hartigan’s dip statistic
(Hartigan & Hartigan, 1985). In case the distance distribution of one or more
points to their co-cluster members exhibit a significant multimodal structure,
the cluster is split.

Skinny-dip (Maurus & Plant, 2016) also implements Hartigan’s dip test and
applies it to one-dimensional linear projections of the data set. Distinct
density peaks are to be identified based on the gradient of the projected cdf.
By projecting the data iteratively into multiple axes, the samples are
partitioned into clusters. Skinny-dip is specifically able to handle
background noise very well, however, it considers noise samples to be
uniformly distributed and clusters to be axis-parallel.

These algorithms, however, are intrinsically tied to convex or Gaussian
cluster assumptions. The recently proposed M-dip (Chronis et al., 2019) is
able to deal with arbitrary oriented and shaped clusters, which applies a
simulation strategy to approximate values for smallest density dips of uni-
modal data sets of the same size and density. However, we do not want to
depend on simulations but instead directly obtain a measure of significance
from given data.

##  Appendix C Testing for uni-modality

Here we highlight the work of Burman & Polonik (2009) more closely, who’s
modality test procedure we adopt in this work. The modality procedure is tied
to the notion of a density dip along a path between two points in the data
set. In the following, we aim to define the concept of such a path formally.

### A formal description of the test procedure

We consider directed, continuous paths from $\@vec{x}_{1}$ to $\@vec{x}_{2}$
through input space $\mathcal{X}$. By assuming there exists a parametrization
$\@vec{r}(t)$, with $t\in[0,1]$, the path becomes the image of $\@vec{r}(t)$.
With this map, we can uniquely express every point on the path via the
parameter $t$. For example, its start and endpoints are given by
$\@vec{x}_{1}=\@vec{r}(0)$ and $\@vec{x}_{2}=\@vec{r}(1)$, respectively.

Let $f$ be the underlying density function and $\@vec{x}_{1}$ and
$\@vec{x}_{2}$ two candidate modes of $f$. We assume, without loss of
generality, that $f(\@vec{x}_{1})<f(\@vec{x}_{2})$. If all possible paths
undergo a density dip when moving from $\@vec{x}_{1}$ to $\@vec{x}_{2}$, both
points are found in two distinct modal regions:

$\exists\,t\in(0,1):f(\@vec{r}(t))<f(\@vec{x}_{1})$ (22)

Conversely, if we can find a path between $\@vec{x}_{1}$ and $\@vec{x}_{2}$
where all points have a higher density than $\@vec{x}_{1}$, both points are
part of the same modal region:

$f(\@vec{r}(t))\geq f(\@vec{x}_{1})\quad\forall\,t\in(0,1]$ (23)

Eq. (23) describes the case of single-modality, which constitutes the null
hypothesis we aim to reject. For general pairs of modal candidates it becomes:

$f(\@vec{r}(t))\geq\text{min}(f(\@vec{x}_{1}),f(\@vec{x}_{2}))\quad\forall\,t\in(0,1]$
(24)

An equivalent and useful formulation is obtained by taking the logarithm on
both sides; after that the left side is subtracted from the inequality.

$\text{SB}(t):=-\text{log}f(\@vec{r}(t))+\text{min}(\text{log}f(\@vec{x}_{1}),\leavevmode\nobreak\
\text{log}f(\@vec{x}_{2}))\\\ $ (25)

Using the variable $\text{SB}(t)$, we can formulate the null hypothesis as
follows:

$\text{H}_{0}:\text{SB}(t)\leq 0\qquad\forall\leavevmode\nobreak\ t\in(0,1)$
(26)

Rather than testing H0 across the full path a pointwise test
H${}_{0,t}:\text{SB}(t)\leq 0$ for some values of $t$ is employed.

Since we do not have access to the underlying density $f$, we cannot test the
hypothesis in Eq. (26) directly. Instead, we have a data set of
$d$-dimensional random variables drawn from $f$. Given proper normalization of
the coordinate axes (see Sect. 3.3.3), Burman & Polonik (2009) show that the
following expression is asymptotically standard normal distributed and
converges – up to a constant factor – to SB$(t)$ as the number data samples
approaches infinity:

$\widehat{\text{SB}}(t)=d\sqrt{k/2}\left[\text{log}\leavevmode\nobreak\
d_{k}(\@vec{r}(t))-\text{max}(\text{log}\leavevmode\nobreak\
d_{k}(\@vec{x}_{1}),\leavevmode\nobreak\ \text{log}\leavevmode\nobreak\
d_{k}(\@vec{x}_{2})\right]$ (27)

Here $d_{k}(\@vec{x})$ denotes the distance to the $k$’th nearest neighbor of
the point $\@vec{x}$. The distance is an approximation to the density $f$. Due
to their inverse proportionality the sign is flipped between Eq. (25) and Eq.
(27); and the minimum is replaced with the maximum function.

Since the corresponding test statistic $\widehat{\text{SB}}(t)$ is
approximately standard normally distributed, the null hypothesis is rejected
at significance level $\alpha$ if

$\widehat{\text{SB}}(t)\geq\Phi^{-1}(1-\alpha)$ (28)

where $\Phi$ is the standard normal cdf. Therefore, if any $t\in(0,1)$
fulfills condition (28), H0 is rejected.

Due to the employment of $k$ nearest neighbor technique, this test procedure
applies naturally to multivariate data without the need of projecting the data
onto a one-dimensional line, as is the case for most modality tests.
Furthermore, nearest neighbor queries have access to very efficient algorithms
such as the kd-tree (Bentley, 1975) which reduces neighbor searches to only
$\mathcal{O}(\text{log}N)$ distance computation. Thus, these considerations
allow us to study the modality structure of the data set at Gaia data scales
without careful projection loss considerations.

Figure 15: Test statistic distribution $\widehat{\text{SB}}$ of our reduced
test procedure on a uni-modal data set. The distribution is stable under
variation of different parameters and respective values (as shown in
corresponding rows and columns).

### Empirical results

Burman & Polonik (2009) describe the iterative application of the test
procedure to modal candidates to cluster the data into significant modal
regions. However, the test is employed along the straight line path connecting
two modes, which limits the procedure to convex cluster shapes only. Moreover,
to detect significant dips reliably, enough samples need to be tested along
the path.

We provide a natural extension to the presented procedure, which applies to
arbitrary cluster shapes while reducing the number of test evaluations to a
single one. In Sect. 3.2 we describe the modification in more detailed and
argue that the reduction of the original method to a single pointwise
evaluation at the saddle point leaves the test unchanged.

To substantiate this statement, we empirically validate our method on
simulated data. In particular, we aim to show that these changes do not affect
the distribution of the test statistic under the null hypothesis. According to
Burman & Polonik (2009) the test statistic $\widehat{\text{SB}}$ is standard
normally distributed under $H_{0}$, which assumes uni-modality between two
points in the input space.

Figure 15 shows the test statistic distribution $\widehat{\text{SB}}$ of our
reduced test procedure on a uni-modal data set. To faithfully test the
distribution under H0 we require a uni-modal data set with Gaia and Sco-Cen-
like positions and kinematic properties, realistic errors, and suitable size.
Since the Gaia eDR3 mock catalog (Rybizki et al., 2020) reproduces kinematic
features found in the Milky Way, its uni-modality in guaranteed.

Instead, we directly use data within the box defined around Sco-Cen in Eq.(1).
To remove any local over-densities we shuffle the observations randomly in
each respective feature individually. This procedure leaves marginal
distributions unchanged while fully decorrelating the data. Since all marginal
distributions are uni-modal, the joint distribution (which is implicitly
constructed as a factorization of marginals) is equally uni-modal.

To gauge parameter effects on the test statistic distribution we vary the
sample size and three SigMA parameters across different values363636While
varying one parameter, the remaining ones are set to their default values as
described in Sect. 3.3. The parameters are, the overall sample size, the $k$
parameter of $k$-nearest neighbor density estimation method (see Sect. 3.3.2),
the $\beta$ parameter of the underlying $\beta$-skeleton graph (see Sect.
3.3.1), and the velocity scaling factor (see Sect. 3.3.3).

As shown in Fig. 15, varying the given parameters within a sensible range does
not modify the test statistic distribution. Due to its stability, a single
test statistic distribution under H0 can be universally assumed over different
parametrizations of SigMA.

The distribution of $\widehat{\text{SB}}$ on uni-modal data follows closely a
zero-mean Gaussian. However, in our tests, the standard deviation differs
slightly from unity as stated by Burman & Polonik (2009). We update its value
in our tests to the average standard deviation across our test of
$\sigma=0.78$.

##  Appendix D Scaling factor distribution

Here we discuss the derivation of the scaling factor distribution, which we
use to weigh the velocity sub-space in the clustering process. For a more
detailed motivation see Sect. 3.3.3.

We replace the scaling factor variable $c_{v}$ with $y$ to simplify and
shorten the reading flow. Additionally, compared to the main text, we denote
the distance to a cluster with $r$ instead of $d$. This notation makes the
integration alongside the differential $dr$ easier to read (otherwise the
differential would be $dd$).

Our goal is to obtain the distribution $f(y\mid r_{0}\leq r\leq r_{1})$, which
describes the behavior of the scaling factor $y$ for a given range of
distances to groups of interest. A simple way to find this distribution is to
interpret the empirical linear model $g(r)$ and associated Gaussian
uncertainties as an improper probability function $f(r,y)$373737Since the
marginal distribution $f(r)\propto 1$ is a uniform distribution over
$\mathbb{R}^{+}$, the joint distribution $f(r,y)$ is improper as it does not
integrate to unity..

As we are dealing with an improper pdf, we consider the following
proportionality condition and handle the normalization of the left hand side
later.

$\displaystyle\begin{aligned} f(y\mid r_{0}\leq r\leq
r_{1})&\propto\int_{r_{0}}^{r_{1}}f(r,y)\,dr\\\
&\propto\int_{r_{0}}^{r_{1}}f(y\mid r)f(r)\,dr\end{aligned}$ (29)

Since $f(r)\propto 1$ is independent on the distance $r$ we can add it to the
yet unknown constant normalization factor and move it out of the integral.
Hence we can write the target distribution as:

$f(y\mid r_{0}\leq r\leq r_{1})\propto\int_{r_{0}}^{r_{1}}f(y\mid r)\,dr$ (30)

Thus, to obtain an analytic solution to Eq. (30) we need an expression for the