NGC 146–King 14: Unbound Double Cluster Pair

• NGC 146–King 14 cluster pair is a coeval yet unbound double open cluster system with overlapping spatial distributions and similar mass functions.
• Multiwavelength astrometric and photometric analyses reveal comparable proper motions, structural parameters, and distinct ages, confirming a shared Galactic origin.
• Dynamical studies show that the clusters exceed their mutual escape velocity, offering insights into early cluster evolution and the impact of Galactic tidal forces.

The NGC 146–King 14 cluster pair constitutes a co-moving but unbound double open cluster system in the Milky Way disk, located at heliocentric distances of 2.98 ± 0.33 kpc (NGC 146) and 2.51 ± 0.23 kpc (King 14). Detailed multiwavelength astrometric, photometric, and dynamical analyses using Gaia DR3, Pan-STARRS1, WISE, and TESS data have established that these clusters have comparable masses, similar mean proper motions, overlap in their spatial distribution (projected separation ≈ 9 pc), and share similar mass function slopes and extinction characteristics. Despite this spatial and kinematic association, orbit integration demonstrates that the clusters’ relative velocities exceed their mutual escape velocity, confirming that they are not gravitationally bound. Instead, the system is interpreted as a coeval pair formed within the same giant molecular cloud that has since evolved into an unbound, co-moving configuration under Galactic disk tidal influences (Bisht et al., 3 Dec 2025).

1. Member Identification and Astrometric Properties

Cluster members were probabilistically identified using Gaia DR3 five-parameter astrometry (positions, parallaxes, and proper motions) complemented by broadband photometry in the G, BP, and RP bands. Initial selection involved proper motion (PM) cuts: a 0.50 mas yr⁻¹ radius around (μₐ*, μδ) = (–2.86, –0.45) mas yr⁻¹ for NGC 146 and a 0.66 mas yr⁻¹ radius around (–3.27, –0.98) mas yr⁻¹ for King 14. Membership probability was quantified using the generalized maximum-likelihood method of Balaguer-Núñez et al. (1998), modeling PM distributions as mixtures of two bivariate Gaussians (Sanders 1971; Vasilevskis et al. 1958):

$P_\mu(i) = \frac{\phi_{c,i}}{\phi_i}$

where $\phi_{c,i}$ and $\phi_{f,i}$ are the cluster and field star frequency distributions, respectively, and $\phi_i = n_c\phi_{c,i} + n_f\phi_{f,i}$ with $n_c+n_f=1$. Stars with $P_\mu > 80\%$ yielded 770 members in NGC 146 and 690 in King 14, with mean proper motions:

• NGC 146: (μₐ*, μδ) = (–2.81 ± 0.14, –0.47 ± 0.21) mas yr⁻¹
• King 14: (–3.23 ± 0.18, –0.97 ± 0.21) mas yr⁻¹

These astrometric properties confirm that both clusters are spatially and kinematically distinct from the field and consistent with a shared Galactic orbit.

2. Structural Parameters and Surface Density Profiles

The radial density profiles (RDPs) for both clusters were constructed by binning members in concentric 1′ annuli. Fits to the empirical King (1962) surface-density model:

$$f(r) = f_0 \left[ (1+(r/r_c)^2)^{-1/2} - (1+(r_t/r_c)^2)^{-1/2}\right]^2 + f_b$$

produced the following best-fit structural parameters:

Cluster	Core radius r_c (pc)	Tidal radius r_t (pc)	Central Density f₀ (stars arcmin⁻²)	Background f_b
NGC 146	0.66	~11	19.28	6.20
King 14	1.70	~14	12.17	3.20

Both clusters display moderate central concentration ($c = \log(r_t/r_c) = 0.58$ and $0.47$ respectively), extended halos, and density contrasts ($\delta_c \sim 4$–5).

3. Age, Distance, and Extinction

Fitting PARSEC v1.2S isochrones to multiple CMDs (Gaia, Pan-STARRS1, and 2MASS), after applying reddening corrections consistent with $R_V \sim 3.1$, yielded:

• NGC 146: $\log t = 7.3 \Rightarrow 20 \pm 5$ Myr, $(m-M) = 13.90 \pm 0.24$
• King 14: $\log t = 7.7 \Rightarrow 50 \pm 10$ Myr, $(m-M) = 13.50 \pm 0.20$

Parallax-based distances, after applying the Gaia EDR3 zero-point correction ($\Delta\varpi = -0.021$ mas) and Bailer-Jones Bayesian estimator, are:

• NGC 146: $2.98 \pm 0.33$ kpc
• King 14: $2.51 \pm 0.23$ kpc

Two-color diagrams with Johnson BV, Pan-STARRS, 2MASS, and AllWISE photometry confirm a standard Galactic extinction law: $R_V = 3.02$ (NGC 146), $3.03$ (King 14).

4. Dynamical Association and Orbital Analysis

Three-dimensional mapping based on positions and distances gives a present-day projected separation of $S_\mathrm{proj} \approx 8.97$ pc. The dynamical semi-major axis, determined via the method of de la Fuente Marcos & de la Fuente Marcos (2010),

$$P_\mathrm{orb} (\mathrm{Myr}) = 94 \left( \frac{S_0}{1+e_0} \right)^{3/2} / \sqrt{M_1 + M_2}$$

with $M_1 = 1492\,M_\odot$, $M_2 = 1250\,M_\odot$, and $P_\mathrm{orb} \sim 255$ Myr, yields $S_0 \sim 32$ pc.

Galactic orbit integration in an axisymmetric MWPotential2014 shows both clusters follow nearly circular ($e \sim 0.17$), thin-disk orbits ($Z_\mathrm{max} < 0.1$ kpc), and share similar mean orbital radii ($R_m \sim 9$ kpc), with orbital periods $\sim 255$ Myr. The escape speed for the pair ($v_\mathrm{esc} \sim 1$ km s⁻¹) is exceeded by their observed relative velocity (a few km s⁻¹), proving the pair is unbound.

5. Stellar Mass Functions and Content

Both clusters’ luminosity functions (LFs), converted to mass functions (MFs) using the PARSEC mass-luminosity relation, yield power-law slopes:

$\xi(M) = \frac{dN}{dM} \propto M^{-(1+x)}$

• NGC 146: $x = 1.51 \pm 0.18$
• King 14: $x = 1.50 \pm 0.15$

These agree closely with the classical Salpeter value $x = 1.35$. Derived integrated masses are $\sim 1492\,M_\odot$ and $\sim 1250\,M_\odot$, with mean stellar masses $\sim 1.9\,M_\odot$ and $1.8\,M_\odot$, respectively.

6. Variable Star Content and Time-Domain Photometry

Analysis of TESS full-frame images and target-pixel files (Sectors 17, 18, 54, 58, 78, 85) identified seven periodic variables, including γ Doradus pulsators, Slowly Pulsating B (SPB) stars, and eclipsing binaries. Only one likely cluster member was found: TIC 444457513 in King 14, which is an SPB star with $T_\mathrm{eff}=11,826$ K, $\log g = 3.49$, $A_V = 1.22$ mag, $d = 2,388$ pc, $[\mathrm{Fe}/\mathrm{H}] = -0.21$ dex, and g-mode pulsations with $P=45.2$ h. The variable population in the fields is dominated by non-members.

7. Evolutionary Context and Astrophysical Implications

NGC 146 and King 14 exhibit a high degree of spatial, kinematic, and demographic similarity: ages (20 vs. 50 Myr), distances ($\sim$3 vs. $\sim$2.5 kpc), mean proper motions and parallaxes, projected separation ($\sim$9 pc), orbital parameters (nearly circular, disk orbits; $R_m \sim 9$ kpc; $e \sim 0.17$; $P_\mathrm{orb} \sim 255$ Myr; $Z_\mathrm{max} < 0.1$ kpc), and mass function slopes. However, the dynamical separation ($S_0 \sim 32$ pc) and relative velocity ($\gtrsim 1$ km s⁻¹) exceed the threshold for gravitational binding at their combined mass (escape speed $\sim 1$ km s⁻¹).

Hence, these clusters form an unbound, co-moving double open cluster system. The most plausible scenario is joint formation in a single giant molecular complex, followed by tidal separation under Galactic disk influences. This configuration provides a laboratory for investigating the transition from primordial binary or multiple clusters toward the dissolution phase characteristic of unbound OB associations or field populations. The NGC 146–King 14 pair exemplifies an intermediate dynamical state between bound cluster binaries and dispersed cluster remnants, with implications for understanding clustered star formation and early cluster evolution in the Galactic disk (Bisht et al., 3 Dec 2025).