NGC 146–King 14: Unbound Young Cluster Pair

• NGC 146–King 14 cluster pair are young open clusters with a common molecular cloud origin yet remain dynamically unbound.
• Multiwavelength astrometric and photometric analyses using King model fits yield precise member counts and robust isochrone-derived ages.
• The system offers key insights into binary cluster formation, cluster dispersal mechanisms, and pulsational calibration from TESS photometry.

NGC 146 and King 14 form a spatially and kinematically associated pair of young open clusters in the Perseus spiral arm. Multiwavelength astrometric and photometric analyses demonstrate strong evidence for a co-moving origin in a common giant molecular cloud environment, yet current dynamical parameters indicate the system is unbound. These clusters serve as a critical astrophysical laboratory for understanding cluster pair formation, binary cluster dynamics, star formation in GMCs, and the early evolutionary processes driving cluster dispersal.

1. Probabilistic Member Identification

Precise cluster boundaries and member lists were established using a two-component, maximum-likelihood method based on Balaguer-Núñez et al. (1998), extending the Sanders (1971) and Vasilevskis et al. (1958) two-Gaussian proper-motion models. For each star $i$, cluster and field proper-motion distributions are described by:

$$\phi_c(\vec{\mu}_i) = \frac{1}{2\pi \sqrt{(\sigma_c^2+\epsilon_{x,i}^2)(\sigma_c^2+\epsilon_{y,i}^2)}}\exp\left\{-\frac{1}{2}\left[\frac{(\mu_{x_i} - \mu_{x_c})^2}{\sigma_c^2+\epsilon_{x,i}^2} + \frac{(\mu_{y_i} - \mu_{y_c})^2}{\sigma_c^2+\epsilon_{y,i}^2}\right]\right\}$$

$$\phi_f(\vec{\mu}_i) = \text{(two-dimensional correlated Gaussian; see full equation in source)}$$

with adopted parameters: $\sigma_c = 0.08$ mas yr$^{-1}$, RUWE ≤ 1.4, and PM errors ≤ 0.5 mas yr$^{-1}$. Membership probability is computed as $P_{\mu}(i) = \phi_c(\vec{\mu}_i)/\phi(\vec{\mu}_i)$. This yielded high-confidence member counts of 770 (NGC 146) and 690 (King 14), defined by $P_{\mu} \geq 80\%$ (Bisht et al., 3 Dec 2025).

2. Structural Properties: Radial Density Profiles

Empirical King model fits (King 1962) were applied to surface densities, using Poisson errors for bin uncertainties. The surface density profile is:

$\rho(r) = \rho_0[1+(r/r_c)^2]^{-1}$

with two-component King law providing best-fit parameters for both clusters:

Cluster	Core radius ($r_c$)	Tidal radius ($$\phi_c(\vec{\mu}_i) = \frac{1}{2\pi \sqrt{(\sigma_c^2+\epsilon_{x,i}^2)(\sigma_c^2+\epsilon_{y,i}^2)}}\exp\left\{-\frac{1}{2}\left[\frac{(\mu_{x_i} - \mu_{x_c})^2}{\sigma_c^2+\epsilon_{x,i}^2} + \frac{(\mu_{y_i} - \mu_{y_c})^2}{\sigma_c^2+\epsilon_{y,i}^2}\right]\right\}$$0)	Central Excess ($$\phi_c(\vec{\mu}_i) = \frac{1}{2\pi \sqrt{(\sigma_c^2+\epsilon_{x,i}^2)(\sigma_c^2+\epsilon_{y,i}^2)}}\exp\left\{-\frac{1}{2}\left[\frac{(\mu_{x_i} - \mu_{x_c})^2}{\sigma_c^2+\epsilon_{x,i}^2} + \frac{(\mu_{y_i} - \mu_{y_c})^2}{\sigma_c^2+\epsilon_{y,i}^2}\right]\right\}$$1)	Background ($$\phi_c(\vec{\mu}_i) = \frac{1}{2\pi \sqrt{(\sigma_c^2+\epsilon_{x,i}^2)(\sigma_c^2+\epsilon_{y,i}^2)}}\exp\left\{-\frac{1}{2}\left[\frac{(\mu_{x_i} - \mu_{x_c})^2}{\sigma_c^2+\epsilon_{x,i}^2} + \frac{(\mu_{y_i} - \mu_{y_c})^2}{\sigma_c^2+\epsilon_{y,i}^2}\right]\right\}$$2)
NGC 146	0.66 pc (0.93′)	11.0 pc	19.28 stars/arcmin$$\phi_c(\vec{\mu}_i) = \frac{1}{2\pi \sqrt{(\sigma_c^2+\epsilon_{x,i}^2)(\sigma_c^2+\epsilon_{y,i}^2)}}\exp\left\{-\frac{1}{2}\left[\frac{(\mu_{x_i} - \mu_{x_c})^2}{\sigma_c^2+\epsilon_{x,i}^2} + \frac{(\mu_{y_i} - \mu_{y_c})^2}{\sigma_c^2+\epsilon_{y,i}^2}\right]\right\}$$3	6.20 stars/arcmin$$\phi_c(\vec{\mu}_i) = \frac{1}{2\pi \sqrt{(\sigma_c^2+\epsilon_{x,i}^2)(\sigma_c^2+\epsilon_{y,i}^2)}}\exp\left\{-\frac{1}{2}\left[\frac{(\mu_{x_i} - \mu_{x_c})^2}{\sigma_c^2+\epsilon_{x,i}^2} + \frac{(\mu_{y_i} - \mu_{y_c})^2}{\sigma_c^2+\epsilon_{y,i}^2}\right]\right\}$$4
King 14	1.70 pc (2.20′)	14.0 pc	12.17 stars/arcmin$$\phi_c(\vec{\mu}_i) = \frac{1}{2\pi \sqrt{(\sigma_c^2+\epsilon_{x,i}^2)(\sigma_c^2+\epsilon_{y,i}^2)}}\exp\left\{-\frac{1}{2}\left[\frac{(\mu_{x_i} - \mu_{x_c})^2}{\sigma_c^2+\epsilon_{x,i}^2} + \frac{(\mu_{y_i} - \mu_{y_c})^2}{\sigma_c^2+\epsilon_{y,i}^2}\right]\right\}$$5	3.20 stars/arcmin$$\phi_c(\vec{\mu}_i) = \frac{1}{2\pi \sqrt{(\sigma_c^2+\epsilon_{x,i}^2)(\sigma_c^2+\epsilon_{y,i}^2)}}\exp\left\{-\frac{1}{2}\left[\frac{(\mu_{x_i} - \mu_{x_c})^2}{\sigma_c^2+\epsilon_{x,i}^2} + \frac{(\mu_{y_i} - \mu_{y_c})^2}{\sigma_c^2+\epsilon_{y,i}^2}\right]\right\}$$6

These spatial profiles confirm evolved dynamical states typical for young Galactic open clusters (Bisht et al., 3 Dec 2025).

3. Age, Distance, and Extinction Calibration

Ages and distances were determined using isochrone fitting with PARSEC v1.2S models at solar $$\phi_c(\vec{\mu}_i) = \frac{1}{2\pi \sqrt{(\sigma_c^2+\epsilon_{x,i}^2)(\sigma_c^2+\epsilon_{y,i}^2)}}\exp\left\{-\frac{1}{2}\left[\frac{(\mu_{x_i} - \mu_{x_c})^2}{\sigma_c^2+\epsilon_{x,i}^2} + \frac{(\mu_{y_i} - \mu_{y_c})^2}{\sigma_c^2+\epsilon_{y,i}^2}\right]\right\}$$7, applied to color-magnitude diagrams (Gaia, Pan-STARRS, 2MASS) and constrained by $$\phi_c(\vec{\mu}_i) = \frac{1}{2\pi \sqrt{(\sigma_c^2+\epsilon_{x,i}^2)(\sigma_c^2+\epsilon_{y,i}^2)}}\exp\left\{-\frac{1}{2}\left[\frac{(\mu_{x_i} - \mu_{x_c})^2}{\sigma_c^2+\epsilon_{x,i}^2} + \frac{(\mu_{y_i} - \mu_{y_c})^2}{\sigma_c^2+\epsilon_{y,i}^2}\right]\right\}$$8. Reduced $$\phi_c(\vec{\mu}_i) = \frac{1}{2\pi \sqrt{(\sigma_c^2+\epsilon_{x,i}^2)(\sigma_c^2+\epsilon_{y,i}^2)}}\exp\left\{-\frac{1}{2}\left[\frac{(\mu_{x_i} - \mu_{x_c})^2}{\sigma_c^2+\epsilon_{x,i}^2} + \frac{(\mu_{y_i} - \mu_{y_c})^2}{\sigma_c^2+\epsilon_{y,i}^2}\right]\right\}$$9 values ($$\phi_f(\vec{\mu}_i) = \text{(two-dimensional correlated Gaussian; see full equation in source)}$$0–$$\phi_f(\vec{\mu}_i) = \text{(two-dimensional correlated Gaussian; see full equation in source)}$$1) indicate robust fits.

• NGC 146: log $$\phi_f(\vec{\mu}_i) = \text{(two-dimensional correlated Gaussian; see full equation in source)}$$2 = 7.3 (age = $$\phi_f(\vec{\mu}_i) = \text{(two-dimensional correlated Gaussian; see full equation in source)}$$3 Myr); $$\phi_f(\vec{\mu}_i) = \text{(two-dimensional correlated Gaussian; see full equation in source)}$$4 mag ($$\phi_f(\vec{\mu}_i) = \text{(two-dimensional correlated Gaussian; see full equation in source)}$$5 kpc)
• King 14: log $$\phi_f(\vec{\mu}_i) = \text{(two-dimensional correlated Gaussian; see full equation in source)}$$6 = 7.7 (age = $$\phi_f(\vec{\mu}_i) = \text{(two-dimensional correlated Gaussian; see full equation in source)}$$7 Myr); $$\phi_f(\vec{\mu}_i) = \text{(two-dimensional correlated Gaussian; see full equation in source)}$$8 mag ($$\phi_f(\vec{\mu}_i) = \text{(two-dimensional correlated Gaussian; see full equation in source)}$$9 kpc)

Gaia DR3 parallax measurements, processed via Bailer-Jones Bayesian inference, yield independent distance estimates: $\sigma_c = 0.08$0 kpc (NGC 146) and $\sigma_c = 0.08$1 kpc (King 14). Reddening analyses from multi-band two-color diagrams provide $\sigma_c = 0.08$2 (NGC 146), $\sigma_c = 0.08$3 (King 14), consistent with Galactic average $\sigma_c = 0.08$4 (Bisht et al., 3 Dec 2025).

4. Kinematic Relations and Mass Functions

Mean Gaia DR3 proper motions:

• NGC 146: $\sigma_c = 0.08$5 mas yr$\sigma_c = 0.08$6
• King 14: $\sigma_c = 0.08$7 mas yr$\sigma_c = 0.08$8

Stellar mass functions calculated from PARSEC mass-luminosity relations yield present-day slopes in $\sigma_c = 0.08$9:

Cluster	MF Slope $^{-1}$0	Salpeter Reference ($^{-1}$1=1.35)
NGC 146	$^{-1}$2	Slightly steeper than Salpeter
King 14	$^{-1}$3	Slightly steeper than Salpeter

These profiles suggest normal IMF evolution within uncertainties. Both clusters show extinction following standard Galactic reddening (Bisht et al., 3 Dec 2025).

5. Spatial Configuration and Galactic Orbital Parameters

Three-dimensional mapping using the Subramaniam et al. (1995) method indicates a projected separation of $^{-1}$4 pc; full spatial separation is $^{-1}$5 pc.

Galactic orbit integration via the galpy MWPotential2014 model (Bovy 2015), using (α, δ), distance, proper motions, and average $^{-1}$6 yields:

Cluster	$^{-1}$7 (kpc)	$^{-1}$8 (kpc)	$^{-1}$9 (kpc)	$^{-1}$0 (kpc)	$^{-1}$1	$^{-1}$2 (Myr)
NGC 146	7.45	10.70	0.072	9.08	0.179	258
King 14	7.42	10.39	0.058	8.91	0.167	253

Orbits are nearly circular, disk-like, and confined to $^{-1}$3 kpc, supporting a shared dynamical origin (Bisht et al., 3 Dec 2025).

6. Dynamical Binding Status

With total mass $^{-1}$4 and a separation of $^{-1}$5 pc, the system’s escape velocity is:

$^{-1}$6

Observed relative velocity between NGC 146 and King 14 is several km s$^{-1}$7, exceeding the calculated $^{-1}$8, thereby establishing that the pair is presently unbound. This supports their status as a co-moving but dynamically unbound system (Bisht et al., 3 Dec 2025). This suggests ongoing dispersal, likely influenced by Galactic tidal forces and internal velocity dispersion.

7. Variable Star Census via TESS Photometry

Seven variable stars were detected in the combined TESS sectors, with detailed cluster-field separation:

TIC	Type	$^{-1}$9 (hours)	Likely Member	Comments
444457577	γ Doradus	0.78	No	Field
428480797	γ Doradus	1.64	No	Field
444457513	SPB	45.20	Yes	Cluster
419531798	Eclipsing Binary	197.1	No	Field
428479573	Miscellaneous	—	No	Field
428479652	Miscellaneous	—	No	Field
419524657	Miscellaneous	—	No	Field

The SPB member TIC 444457513 in King 14 exhibits $P_{\mu}(i) = \phi_c(\vec{\mu}_i)/\phi(\vec{\mu}_i)$0 K, $P_{\mu}(i) = \phi_c(\vec{\mu}_i)/\phi(\vec{\mu}_i)$1, $P_{\mu}(i) = \phi_c(\vec{\mu}_i)/\phi(\vec{\mu}_i)$2, $P_{\mu}(i) = \phi_c(\vec{\mu}_i)/\phi(\vec{\mu}_i)$3 mag, $P_{\mu}(i) = \phi_c(\vec{\mu}_i)/\phi(\vec{\mu}_i)$4 kpc, and falls within the theoretical SPB instability strip. All other variables are field contaminants (Bisht et al., 3 Dec 2025). A plausible implication is that bona fide cluster pulsators can anchor age and internal structure tests via asteroseismology.

8. Astrophysical Significance and Implications

NGC 146 and King 14 exemplify young open clusters originating in a common molecular cloud, sharing nearly identical distances, proper motions, and orbital characteristics. Their steep present-day MF slopes, standard interstellar reddening, and close spatial association ($P_{\mu}(i) = \phi_c(\vec{\mu}_i)/\phi(\vec{\mu}_i)$59 pc projected, $P_{\mu}(i) = \phi_c(\vec{\mu}_i)/\phi(\vec{\mu}_i)$632 pc full separation) strongly support a coeval formation event. The lack of dynamical binding, due to relative velocities exceeding the escape velocity, positions the system as a rare unbound, co-moving cluster pair. Such systems facilitate studies of tidal disruption, pair formation and breakup, and the transformation of clusters into unbound associations under Galactic field influences. Detection of an SPB pulsator within King 14 further enriches the value of this pair for cluster astrophysics, particularly for asteroseismic calibration. Future improvements in radial velocity precision, photometric depth, and forthcoming Gaia releases will enable more refined constraints on the fragmentation and dissolution mechanisms that seed the Galactic field with young stellar populations (Bisht et al., 3 Dec 2025).