Virial-based extraction of structures in numerical simulations: The vibes tool

Abstract: The processes that determine the stellar initial mass function (IMF) and its connection to the core mass function (CMF) are among the major open questions in star formation. The definition of a core remains unclear, yet the way they are extracted from simulations and observations critically shapes the CMF. Nowadays, cores are mostly detected through their density or intensity only. We aim to explore a new way to define cores in 3D numerical simulations based on a direct application of the virial theorem, and break free from some limitations induced by density-based methods. We intend to improve the accuracy and the physical meaning of the extracted cores. We developed vibes, an innovative method that makes full use of the virial theorem to extract overdensities in simulation snapshots. It works by building structures iteratively around density peaks, and applying the virial theorem to the structure at each iteration. Then, the structure boundary is set from the evolution of the its energy as it spatially grows. We used STARFORGE simulations to test the sensitivity of the extraction process to the main working parameters (constraints on the structure shape, iteration step, and peak selection criteria). This sensitivity is observed to be low. We compared our extraction with two density-based extraction algorithms, hop and dendrogram, that are observed to be very sensitive to their input density threshold parameter. Vibes returns structures that are coherent to each other and physically motivated, and it appears much more stable than existing 3D extraction tools. By defining the boundary of the cores on a physical criterion rather than on a user-defined set of density parameters, we expect such extracted cores to be closer to their forsaken definition: gas reservoirs that will form a single star or a close multiple system.

• The paper introduces a virial-energy-based method for defining core boundaries in 3D simulations, eliminating arbitrary density thresholds.
• It employs iterative structure growth with rigorous energy evaluation to ensure that extracted cores are physically stable and relevant to star formation theory.
• Parameter sensitivity studies reveal VIBES’ robustness and consistency compared to traditional density-based extraction techniques.

Virial-Based Core Extraction in Simulations: The VIBES Tool

Context and Motivation

The formation of stars within molecular clouds involves the evolution of dense, transient gas condensations known as "cores." The statistical mapping from the initial core mass function (CMF) to the stellar initial mass function (IMF) remains one of the most significant unsolved problems in star formation theory. A persistent challenge has been the ambiguous definition of a "core" itself, especially within 3D numerical simulation data: core identification typically relies on isodensity contour or thresholding methods, which are user-dependent and may not correspond to physically bound entities. These methods also often disregard the full multi-physics context (turbulence, magnetic fields, external pressure) and tend to generate samples whose properties are highly sensitive to parameter choices.

The authors propose a physically motivated alternative: utilize the virial theorem to define core boundaries. Their toolkit, vibes ("virial-based extraction of structures"), auto-detects spatial overdensities and assigns their boundaries based on virial energy criteria, decoupling the extraction process from arbitrary density thresholds. This approach is meant to enhance both the interpretability and robustness of extracted structures, mitigating the artifacts of conventional algorithms.

Algorithmic Framework

VIBES processes 3D simulation snapshots—both Eulerian and Lagrangian mesh codes—by iteratively constructing candidate structures around density maxima and rigorously evaluating their stability using the full (Eulerian) virial theorem as formulated by McKee & Zweibel (1992). The key features and steps of the methodology are as follows:

• Peak Detection and Pre-filtering: Local maxima in the density field are identified, using neighborhood-based relative contrast criteria to discard insignificant peaks. The algorithm employs a minimum density and a peak-to-saddle ratio, analogous to classical 'clumpfind' techniques but with more rigorous peak validation chainstopping on both absolute and relative basis.
• Iterative Structure Growth: Around each validated peak, 'vibes' incrementally adds the densest, shape-admissible neighboring cells to the nascent structure. Shape constraints are imposed through convexity and elongation ratios to avoid pathological or poorly defined regions. The iteration is performed layer-wise (enhancing computational scaling) and energy profiles are recorded at each step.
• Virial Energy Evaluation: The boundary selection for each candidate is based on local minima or inflections of the total virial energy function as the structure grows. Volume and surface terms—including gravitational, thermal, kinetic (with careful handling of inflow/outflow and rotation), magnetic, and a term approximating time derivatives from a snapshot—are computed using a combination of direct summation and divergence theorem methods, plus AMR grids for Lagrangian codes with ill-defined volumes.
• Boundary Assignment and Pruning: Final 'core' boundaries are chosen at the first energy minimum or significant inflection satisfying physical and numerical tolerances. Redundancy and overlap between extracted objects are resolved with a global mask, enforcing one-to-one mapping between volumes/cells and structures.

This approach allows core identification based on actual boundness and the underlying energetic balance, in contrast to fixed iso-density extraction, and includes external pressure, turbulence, bulk flows, and magnetic support. Additionally, the use of virial-tracked boundaries is particularly advantageous for characterizing dynamic and non-equilibrium features of the simulated interstellar medium (ISM).

Parameter Sensitivity and Algorithm Robustness

An extensive parameter study on the STARFORGE MHD simulations was carried out to probe the sensitivity of 'vibes' to user-controlled settings, including density/contrast thresholds, convexity/elongation limits, neighborhood width, and iteration step size (layer thickness). Key findings include:

• Low Sensitivity to Peak and Shape Parameters: For reasonable ranges (e.g., peak-to-saddle ratios 1.4–2.1, convexity from 0.55–0.95, elongation limits above ~4), the number, properties, and mass distributions of extracted structures remain highly stable—a significant improvement over classical density-based methods.
• Layering (Iteration Step) Effects: While extremely coarse or fine layering does modulate the computational load and can introduce some noise, the masses and sizes of recovered structures are not significantly altered for default or slightly coarser settings.

The resulting CMFs display robust slopes and turn-overs, generally unaffected by variations of the above parameters, especially in the mass regime above the method's effective resolution limit.

Comparative Analysis to Density-Based Methods

The study systematically benchmarks 'vibes' against HOP [Eisenstein & Hut 1998] and dendrogram [Rosolowsky et al. 2008]—the two most widely used 3D structure extractors—across a broad range of threshold parameters. Core findings:

• High Parameter Sensitivity of Density Algorithms: The number and mass distribution of structures identified by HOP/dendrogram vary wildly (up to orders of magnitude) with input density thresholds. Large-scale and high-mass structures are especially sensitive, and results generally lack convergence.
• Shape and Completeness of the CMF: The inferred CMFs using fixed density thresholds can spuriously echo or obscure theoretical expectations (e.g., Salpeter slope), depending on user parameterization.
• Physical Meaningfulness and Biases: Density algorithms tend to merge spatially-contiguous high-density regions into monolithic, extremely massive, and possibly unbound entities at low thresholds, or conversely, split contiguous structures at high thresholds, with little physical justification.
• VIBES Stability and Physical Relevance: The virial-extracted core set is more robust to algorithmic input and tracks genuine energy minima—structures are more likely to correspond to "gas reservoirs that may form a single star or close multiple system," the intended theoretical concept of a prestellar core.

Implications for Star Formation Theory and Computational Astrophysics

The development of energy-based core extraction tools such as 'vibes' represents a critical shift toward reconciling theory, simulation, and ultimately observations regarding the dense core population in the ISM. Key implications include:

• Physical Consistency of Statistical Inference: By removing arbitrary, nonphysical thresholds from the segmentation process, 'vibes' enables rigorous cross-simulation or cross-observation comparison of core populations and their CMFs, laying the ground for more secure mapping to the IMF.
• Tracking Core Evolution: Structures extracted with virial-tuned boundaries are more amenable to time-tracking and analysis of transient features, crucial for constraining lifetimes, accretion histories, and mass loss/gain.
• Extensibility to Observations: Although direct surface term evaluation is challenging in observations, the method opens pathways for inferring virial properties more consistently when applied to position-position-velocity cubes (or future high-quality 3D data).
• Machine Learning and Automated Analysis: By providing a physically justified core sample, 'vibes' could streamline AI/ML applications for ISM characterization, core tracking, and comparison across simulation suites.

Future Directions

Ongoing development should focus on a more comprehensive treatment of time-dependent terms, improved handling of feedback-driven 'sink' particles, and inclusion of additional physics (e.g., radiative transfer effects). Modular integration with major simulation ensembles and observational pipelines should accelerate adoption.

Conclusion

The "vibes" tool (2606.08494) constitutes a significant methodological advance for extracting physically meaningful structures in 3D astrophysical simulations. Its core innovation is the use of virial-energy-based criteria rather than user-defined density thresholds, resulting in object samples that are more physically interpretable and robust to parameter choices. Benchmark analyses demonstrate the tool's resilience to nuisance parameter variation, and comparative evaluation shows its superiority over density-centric extractors in assembling a core population suitable for rigorous study of star formation processes and the IMF-CMF mapping. The algorithm and its open-source implementation represent a valuable resource for computational and theoretical astrophysicists investigating multi-scale, multi-physics star-forming systems.