Formation of Isotopically Heterogeneous Molecular Cloud Cores in Filamentary Molecular Clouds

Technical Summary: Formation of Isotopically Heterogeneous Molecular Cloud Cores in Filamentary Molecular Clouds

Problem Statement
Meteorite analysis reveals that early solar system solids, such as calcium–aluminum-rich inclusions (CAIs), exhibit isotopic inhomogeneity (e.g., variations in $^{54}\text{Cr}/^{52}\text{Cr}$ and $^{50}\text{Ti}/^{47}\text{Ti}$ ratios). While thermal processing in protoplanetary disks and inheritance from inhomogeneous parent molecular cloud cores are proposed explanations, the extent to which isotopic heterogeneity survives the collapse of filamentary molecular clouds into dense cores remains unclear. Specifically, it is necessary to quantify how much of the initial isotopic variation present in a filamentary cloud is preserved or smoothed out during gravitational fragmentation and core formation, and how this heterogeneity translates to the resulting circumstellar disk.

Methodology
The authors utilized Smoothed Particle Hydrodynamics (SPH) simulations, specifically employing the Godunov smoothed particle magnetohydrodynamic (GSPM) method, to model the gravitational fragmentation of filamentary molecular clouds.

• Simulation Setup: The simulations modeled filaments with a critical line mass ($M_{\text{line,crit}} \approx 18 M_\odot \text{pc}^{-1}$) and a width of $\sim 0.1$ pc, consistent with observations. The initial conditions included a hydrostatic equilibrium density profile, Kolmogorov turbulence ($\sigma = 2c_s$), and an initial magnetic field of $10 \mu\text{G}$ perpendicular to the filament axis. Hydrodynamic-only cases were also run for comparison.
• Isotopic Tracing: Isotopic species were treated as passive tracers. The authors constructed six distinct models (Table 1) to impose initial isotopic ratio gradients ($C_0$) along different axes: the longitudinal axis ($z$), the minor axes ($x, y$), and combinations thereof. The isotopic ratio was normalized between 0 and 1 over the computational domain.
• Analysis: The evolution of the isotopic ratio was tracked by following SPH particles. The authors defined spherical shells around the density peak of forming cores to calculate the average isotopic ratio ($\bar{C}$) and the difference from the core center ($|\Delta C|$). They also employed a semi-analytical approach to estimate the initial center-of-mass displacement of shells ($\delta z_{\text{ini}}$) based on turbulent velocity fields and density power spectra.

Key Results

1. Survival of Heterogeneity: The simulations demonstrate that isotopic inhomogeneities are not completely smoothed out during filament fragmentation. Cores retain approximately 0.1% to 1% of the initial isotopic inhomogeneity present over the initial filament length of 1 pc.
2. Geometric Dependence: The magnitude of the retained heterogeneity depends on the orientation of the initial isotopic gradient relative to the filament geometry.
   • Gradients along the longitudinal axis ($z$-direction, Model $M_z$) result in the largest isotopic differences between shells ($|\Delta C| \sim 0.01$).
   • Gradients along the minor axes ($x$ or $y$) result in significantly smaller differences. This is attributed to the filament's geometry, which restricts the displacement of fluid elements along the minor axes, thereby limiting the mixing of material with different isotopic ratios.
3. Mechanism of Inhomogeneity: The primary driver of the residual isotopic difference between shells is the initial difference in the center-of-mass of the shells, caused by the turbulent velocity field. The semi-analytical estimates, which account for the initial displacement of fluid elements, agree with simulation results within a factor of 2, confirming that the initial turbulent configuration dictates the final isotopic profile.
4. Magnetic Field Influence: The presence of a magnetic field ($10 \mu\text{G}$) does not significantly alter the internal profile of the isotopic ratio compared to the purely hydrodynamic case.
5. Circumstellar Disk Implications: By estimating the centrifugal radius of accreting material, the authors found that isotopic heterogeneity persists in the circumstellar disk. While the disk inherits the core's heterogeneity, the anisotropic distribution of angular momentum means that gas accretes from various spherical shells. The results suggest that roughly 1% of the initial filamentary heterogeneity remains in the disk.

Significance and Claims
The paper claims to provide a quantitative link between the isotopic anomalies observed in meteorites and the astrophysical conditions of the parent molecular cloud. By combining their findings with the reduction factor reported by Jacquet et al. (2019) (which suggests a factor of $\sim 3$ reduction from core to disk), the authors infer that the residual heterogeneity in the final circumstellar disk is approximately 0.03% to 0.3% of the initial filamentary variation.

The authors assert that this level of residual heterogeneity is sufficient to explain the observed isotopic anomalies in Solar System materials (e.g., $^{54}\text{Cr}/^{52}\text{Cr}$ and $^{50}\text{Ti}/^{47}\text{Ti}$ variations of 1–10 parts in 10,000) without requiring extreme thermal processing or exotic nucleosynthetic injection scenarios post-collapse. The study concludes that the isotopic inhomogeneity of the solar system's building blocks could naturally survive the entire process from filament fragmentation to disk formation, provided the parent cloud possessed an appropriate level of initial inhomogeneity. The authors note that high-resolution simulations are required for future work to fully resolve the realistic properties of isotopic ratios in the circumstellar disk, particularly regarding anisotropic infall and outflows.