Serna-Gerbal Dendrogram Method

• The Serna-Gerbal dendrogram method is a hierarchical, energy-based clustering algorithm that computes gravitational binding energies to group galaxies.
• It leverages galaxy positions, redshifts, and luminosity-derived masses to determine pairwise binding strengths and reveal detailed substructure in clusters.
• Its applications extend to dynamical analysis and evolutionary studies, offering insights into merger histories and the interplay between cluster dynamics and the intracluster medium.

The Serna-Gerbal dendrogram method is a hierarchical, energy-based clustering algorithm for identifying gravitationally bound substructures in galaxy systems. Initially developed for the dynamical analysis of galaxy clusters and groups, its applications have extended to the study of group evolutionary stages, galaxy luminosity function interpretation, and the connection between cluster dynamics and the intracluster medium (ICM). Conceptually, the method leverages spectroscopic and photometric data to compute pairwise gravitational binding energies, hierarchically grouping galaxies using a dendrogram structure that encodes merger energetics.

1. Fundamental Principles and Algorithmic Framework

The Serna-Gerbal (SG) method uses galaxy positions (sky coordinates, redshifts) and luminosity-derived mass estimates to assess the gravitational binding energy for every galaxy pair. The basic energy formula adopted is

$E_{ij} = -\frac{G\,M_i\,M_j}{r_{ij}}$

where $G$ is the gravitational constant, $M_i$ and $M_j$ are masses (converted from magnitudes or luminosities using a constant mass-to-light ratio), and $r_{ij}$ is the separation. This approach presumes that each galaxy’s mass is proportional to its measured luminosity, with the proportionality fixed by the adopted mass-to-light ratio (e.g., M/L = 400 for Abell 222/223 (Durret et al., 2010), M/L = 100 for Abell 3376 (Durret et al., 2013)).

The algorithm executes as follows:

• For all galaxy pairs, compute $E_{ij}$.
• Hierarchically group galaxies starting with the most negative (i.e., strongest) binding energies, merging lower-level groups when their joint binding energy is sufficiently negative.
• The process constructs a dendrogram—a tree-like diagram—where the abscissa represents the binding energy level at which groups combine. The structure illustrates the hierarchy of substructures from small, tightly bound groups to the larger system.

Minimum membership criteria (e.g., n > 3 or n > 5) are commonly imposed to avoid statistical insignificance from chance alignments, and velocity information is leveraged to exclude dynamically unrelated lines-of-sight components.

2. Dynamical Substructure Identification and Interpretation

The SG dendrogram method is especially effective in delineating distinct gravitationally bound systems within complex environments, even for clusters or groups with overlapping sky positions and velocity distributions. For example, in Abell 222/223 (Durret et al., 2010), the dendrogram exhibits two well-separated branches, corresponding to members of each cluster (55 for Abell 222, 64 for Abell 223). Similarly, in Abell 3376 (Durret et al., 2013), the method reveals a main substructure with 82 galaxies and further subdivisions (25 and 6 galaxies), as well as independent minor substructures.

This capability extends to group-scale systems, as demonstrated in the analysis of fossil groups and their candidates (Ramos et al., 23 Sep 2025). The SG dendrogram dissects the group membership according to binding energies, revealing inner cores and peripheral subgroups, tracking ongoing accretion and interaction processes. For instance, in the NGC 4104 near-FG, only minor subgroups were identified besides the dominant central galaxy—consistent with an advanced evolutionary stage.

Hierarchical dendrogram labeling (e.g., G1, G11, G111, G2, etc.) encodes substructure nesting, with deeper branches marking tightly bound cores and back branches marking peripheral or less bound aggregates.

3. Quantitative Estimation of Dynamical Properties

The SG method facilitates the derivation of key dynamical parameters:

• Velocity dispersions for the substructures, traceable directly from the redshift data (1014 km/s for Abell 222, 1170 km/s for Abell 223 (Durret et al., 2010)).
• Virial masses for each detected substructure, albeit with the caveat that masses are generally underestimated due to incomplete redshift sampling and the assumption that galaxies are the sole mass carriers.
• Crossing times, calculated for subgroups, provide insight into merger efficiency (short crossing times indicate rapid merging and advanced dynamical evolution).
• Relative masses and member counts, tabulated for substructures (e.g., S1 mass ~0.3× cluster mass (Durret et al., 2013)).

These metrics allow a detailed quantitative characterization of the cluster/group's dynamical state and facilitate the connection to observable properties such as luminosity functions and star formation rates.

4. Role in Evolutionary Studies: Fossil Groups and Assembly History

Application to galaxy groups in various evolutionary stages (fossil, non-fossil, near-fossil) positions the SG dendrogram method as a diagnostic of merger history and environmental influence (Ramos et al., 23 Sep 2025). Results show:

• Groups with fewer, highly bound substructures (short crossing times, dominant core) are characteristic of advanced or fossil stages, displaying older stellar populations and higher metallicities.
• More dynamically active systems (e.g., the XCLASS 1330 group) exhibit multiple substructures with varying levels of binding, indicating ongoing assembly and accretion.
• A correlation is observed: lower-mass substructures tend to have more recent starbursts, as confirmed by FIREFLY spectral analyses; massive cores are associated with older stars.
• The fossil status appears tied to the large-scale environment: groups in low-density environments with limited accretion can evolve into fossil systems, while those in dense cosmic web regions maintain complex substructure and younger populations.

This suggests that the SG method helps reconstruct group assembly history, demonstrates the physical link between dynamical structure and stellar evolution, and enables identification of merger-driven transitions in galaxy populations.

5. Integration with Multiwavelength and Population Studies

A strength of the SG dendrogram method is its synergy with multiwavelength analyses, notably linking optical (galaxy properties and luminosity functions) and X-ray (ICM temperature, metallicity) diagnostics (Durret et al., 2010). Factual examples include:

• Disentangling cluster membership (Abell 222 vs. Abell 223) enables separate GLF derivation; Abell 223's need for an extra bright-magnitude component reflects recent merger influence and ICM disturbance.
• Perturbed GLFs and X-ray inhomogeneities are co-located with complex substructure in the SG dendrogram, supporting ram-pressure stripping and induced star formation scenarios.

In evolutionary studies (Ramos et al., 23 Sep 2025), SG substructure identification correlates with spectral analysis results (FIREFLY, PIPE_VIS), connecting recent star formation episodes with dynamical mergers in lower-mass subgroups, and demonstrating that dynamical properties (binding energy, crossing time) influence stellar metallicity and age distributions.

6. Broader Methodological and Theoretical Context

The SG energy-based dendrogram is one concrete realization within a generalized family of hierarchical clustering algorithms, related but not restricted to single linkage or minimax methods. In representation learning, single-linkage dendrograms produce minimax distances, but generalized frameworks allow arbitrary dendrograms with flexible node-level functions (Chehreghani et al., 2018). The SG approach can be recovered by selecting linkage as the level function in a single-linkage configuration; extensions using other level functions or linkage criteria permit adaptation to nonuniform densities or complex geometries.

In categorical and representation-theoretic contexts, hierarchical decomposition via dendrograms finds analogies with the clustering of module categories in gerbal (projective 2-) representation theory—where successive induction from subgroup data yields a hierarchical (dendrogram-like) structure, enriched by twisted phase information (Ganter et al., 2014). The connection is formalized using induction functors and twisted Drinfeld doubles (i.e., $D^\alpha(G) \simeq {}^{\tau(\alpha)}[\Lambda G]$).

7. Limitations and Interpretive Caveats

The SG dendrogram method is subject to several limitations:

• Virial mass estimates are sensitive to the completeness of spectroscopic sampling and the adopted M/L ratio; galaxies are assumed to be the sole mass contributor (neglecting dark matter and gas).
• Smaller substructures may not be statistically robust if group membership is low.
• Velocity outliers and projection effects can lead to ambiguous substructure assignments, particularly in cases with line-of-sight contamination.
• In non-fossil groups embedded in dense environments, the SG method may detect several low-mass substructures with recent mergers, complicating interpretation without auxiliary spectral or environmental data.

A plausible implication is that integration with comprehensive photometric and kinematic catalogues, machine learning-based aggregation algorithms, and multiwavelength diagnostics offers a more reliable probe of dynamical and evolutionary processes.

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The Serna-Gerbal dendrogram method is empirically validated for identifying gravitationally bound galaxy substructures, quantifying dynamical states, and interpreting evolutionary pathways in cluster and group environments. Its versatility lies in robust hierarchical decomposition, compatibility with multiwavelength population analysis, and conceptual ties to categorical group representation frameworks. Continued refinement of input catalogues, expansion to broader linkage and level function choices, and integration with ensemble clustering or spectral modeling approaches suggest ongoing methodological relevance.