Analysis of Karin and Koronis2 asteroid families: new findings and challenges

TL;DR

The study refines the Karin asteroid family membership using an updated proper-element catalog and a novel convergence metric tied to the initial velocity field, revealing a robust age of $T=5.72^{+0.16}_{-0.19}$ Myr and a kilometer-scale size-frequency distribution slope of $\alpha=3.20\pm0.01$ for $D\in(0.8,3)$ km. It dissects the overlapping Koronis2 population, finding that large members often fail to converge within the previously quoted ages, suggesting the Koronis2 age may exceed $7.6$ Myr and potentially require $>10$ Myr or direct numerical modeling. The analysis shows sub-kilometer Karin members exhibit bimodal $\dot{a}$ distributions due to YORP-induced pole tilts, reinforcing a link between Yarkovsky drift and spin-state evolution, and demonstrates the need for broken-power-law extrapolations in the size-frequency distributions to connect kilometer-scale fragments with meter- to micron-scale debris linked to meteorites and dust bands. Collectively, the work provides a more accurate Karin census, introduces a forward-connected convergence approach, and outlines the challenges and future steps for constraining Koronis2’s age and the meter-sized fragment population with upcoming surveys and numerical simulations.

Abstract

We use our home catalog of the asteroid proper elements to study the Karin family. The hierarchical clustering method provides formal identification with 3,863 members, but this set also includes objects from the neighboring Koronis2 and Kuitaisi families, as well as interlopers originating from the much older Koronis family. By tracking the trajectories of cluster objects backward in time, we identified 2,161 asteroids whose orbits converged with that of their parent body (832) Karin at $5.72\pm 0.09$ My ago ($95$\% C.L.). This method of calculating the family's age is based on a novel convergence metric that is directly related to the velocities at which fragments were ejected from (832) Karin. We analyze the extent to which members $\leq 1.5$ km in diameter had drifted in semimajor axis due to Yarkovsky thermal forces and find it reflects the tilt of their rotation poles away from the ecliptic, recording the influence of the YORP torque. Karin's size frequency distribution in the $\simeq(0.8-3)$ km range follows a power-law with a cumulative slope index $-3.20\pm 0.01$. Removing members of the Karin family from the original group, we examine the Koronis2 family, whose members are associated with (158) Koronis. We find it difficult for large members of the Koronis2 family to converge with the orbit of (158) Koronis within its previously estimated age of $7.6$ My. Achieving such convergence would require the Koronis2 family to be older than $10$ My, but our result must be verified with a direct numerical approach in the future.

Figures (15)

• Figure 1: Convergence of the proper longitude of node (middle panel) and perihelion (bottom panel) of $370$ Karin family members with $H\leq 16.7$ at $T=5.72$ My. Darker lines for larger members. The top panel shows behavior of $\mathcal{C}^\prime_\Omega$ (red) and $\mathcal{C}^\prime_\varpi$ (blue) defined in (\ref{['conv1']}). Both show sharp minima at nearly the same epoch, as required by a satisfactory global solution of $\mathcal{C}$, attaining the best-fit values $1.29^\circ$ for nodes and $1.05^\circ$ for perihelia. The gray dashed level at the top panel corresponds to $\mathcal{C}^\prime$ value of a uniform distribution for comparison.
• Figure 2: Behavior of the target function $\mathcal{C}^\prime$ near its minimum at $T=5.72$ My (vertical gray line) combining convergence of proper nodes and perihelia of $370$ Karin family members with $H\leq 16.7$. The dashed gray horizontal line indicates 95% confidence level of the age solution assuming $\mathcal{C}^\prime$ represents a merit function of minimization procedure with $\simeq 1.5^\circ$ uncertainty of the $\Delta \Omega_j$ and $\Delta\varpi_j$ values nr2007.
• Figure 3: Adjusted values of mean semimajor axis drift-rate values $\dot{a}$ for $367$ Karin family members with $H\leq 16.7$ as a function of their size $D$. Red symbols correspond to the solution with minimum dispersion of the secular angles expressed by $\mathcal{C}$ function in (\ref{['tf']}). The black symbols, and vertical bars, are the mean and standard deviation of their distribution from solutions with $T$ sampling the 95% C.L. interval of Karin family ages, namely between $5.53$ My and $5.88$ My (Fig. \ref{['fig2']}). The enveloping thick gray lines are the estimated maximum values $\dot{a}_{\rm max}=\pm 3.1\times 10^{-4}/D$ au My$^{-1}$ from the linearized heat conduction model on spherical bodies and bulk density of $2.5$ g cm$^{-3}$vok1998. The positive and negative maximum values correspond to $0^\circ$ and $180^\circ$ obliquity. The thin gray lines are for intermediate obliquity values incremented by $30^\circ$. For sake of completeness, we also displayed the $\dot{a}$ values corresponding to 7719, 33143 and 40510 (blue symbols), whose membership in Karin or Koronis2 families is ambiguous (see Sec. \ref{['karin']}).
• Figure 4: Differential distribution of $\mathcal{C}^\prime_j=\sqrt{(\Delta\Omega_j^2+\Delta\varpi_j^2)/2}$ values by which each of the $3217$ small ($H>16.7$) Karin family candidates contributes to the convergence target function (\ref{['tf']}) at fixed epoch $T=5.72$ My. The distribution is clearly bimodal with $1866$ inputs having $\mathcal{C}^\prime_j\leq 7.5^\circ$ which we adopted a limit for satisfactory convergence (red segment); these asteroids define the population of small Karin members. The remaining 1217 objects do not exhibit satisfactory convergence to (832) Karin at epoch of the Karin family formation (blue segment). Given the logarithmic binning at the abscissa, the dashed line characterizes a uniform distribution of $\mathcal{C}^\prime_j$.
• Figure 5: Left panels: Karin family members (black symbols) projected onto 2D planes of proper semimajor axis $a$ at the abscissa, and proper sine of inclination $\sin I$ (top) and eccentricity $e$ (bottom) at the ordinate. The largest body (832) Karin shown by red circle, and members with size $\geq 3$ km by blue symbols. The gray dots show unrelated background asteroids in the Koronis family, including the Koronis2 cluster (at $\sin I\simeq 0.0376$). Right panel: Cumulative size distribution for $2161$ members in our conservative identification of Karin family (red curve). Sizes of $21$ largest objects from WISE observations, while those for the remaining members determined from their absolute magnitude $H$ and assumed albedo $p_V=0.23$, a mean value of the Koronis family from WISE data as well mas2011mas2013. In between $0.85$ and $3$ km the distribution is fairly well matched by a power-law $N(>D)\propto D^{-\alpha}$ with a slope index $\alpha=3.2$ (dashed line). For sake of comparison we also show distribution that would include also asteroids 7719, 33143 and 40510 with an ambiguous membership in either Karin or Koronis2 families (see Sec. \ref{['karin']}; blue curve).