The MOND Depth Index and Dynamical Maturity Clock: Toward a Universal Classification of Galaxies and Star Clusters

Abstract: Mass discrepancies in galaxies are empirically known to appear only below a characteristic acceleration scale a0. Here we show that this behaviour is not limited to galaxies: it extends continuously across the full hierarchy of self-gravitating stellar systems, from gas-rich dwarfs and spirals to massive early-type galaxies, and further down to compact stellar clusters. We introduce the Milgromian dynamics (MOND) depth index DM, together with dynamical maturity index T = tcross/tH, dynamical collisionality index T1 = tcross/trelax, with tcross being the crossing time, tH the Hubble time and trelax the median two-body relaxation time, and the MOND acceleration index A = abar/a0. We uncover a well-defined two-dimensional dividing surface in dynamical space. The "dark matter phenomenon" is found only in systems that are both in the deep-MOND regime (abar < a0) and collisionless (trelax > tH), while high-acceleration, collisional systems (abar > a0, trelax << tH), including globular clusters and UCDs, show no evidence for a mass discrepancy. This clean dynamical separation defines a new, physically motivated classification scheme for stellar systems, unifying galaxies and clusters under one framework. The observed division emerges naturally within the MOND framework and provides a useful diagnostic for examining how different gravitational paradigms account for the origin of the mass discrepancy.

• The paper introduces the MOND Depth Index and Dynamical Maturity Clock to classify self-gravitating systems based solely on baryonic dynamics.
• It employs dimensionless metrics—collisionality and acceleration indices—to clearly distinguish collisionless galaxies from collisional clusters.
• The method reveals continuous dynamical sequences linking system age, stellar evolution, and dark-matter-like effects, offering a challenge to standard ΛCDM models.

The MOND Depth Index and Dynamical Maturity Clock: A Universal Dynamical Classification for Galaxies and Star Clusters

Introduction and Motivation

The longstanding ambiguity in distinguishing galaxies from star clusters arises from intrinsic mass discrepancies in low-mass stellar systems, where Newtonian gravitational dynamics fail to account for observed kinematics based solely on baryonic content. Traditional morphology-based classifications and operational definitions relying on relaxation timescales inadequately capture the continuous spectrums and evolutionary pathways evident from dwarf galaxies to ultra-compact dwarfs (UCDs), globular clusters, and massive early-type galaxies (ETGs).

This work proposes a comprehensive dynamical framework rooted in Milgromian Dynamics (MOND), introducing the MOND Depth Index ($D_M$) and dynamical maturity index ($\mathcal{T}$), along with two complementary dimensionless metrics: the collisionality index ($\mathcal{T}_1$) and acceleration index ($\mathcal{A}$). Collectively, these baryon-based measures unify the classification of self-gravitating stellar systems irrespective of mass, morphology, or kinematic support.

Dynamical Diagnostics: Definitions and Physical Basis

The framework employs the following indices:

• MOND Depth Index ($D_M$): Quantifies the fraction of baryonic mass in the deep-MOND regime ($r > r_M$), where $r_M \propto M_\text{bar}^{1/2}$ is the MOND radius defined by internal accelerations matching the universal acceleration scale $a_0$.
• Dynamical Maturity Index ($\mathcal{T}$): Ratio of crossing time $t_\text{cross}$ to Hubble time $t_H$, representing the number of dynamical cycles experienced over cosmic time.
• Collisionality Index ($\mathcal{T}_1$): Ratio of crossing time to median two-body relaxation time, effectively distinguishing collisionless galaxies ($\mathcal{T}_1 \ll 1$) from collisional clusters ($\mathcal{T}_1 \gg 1$).
• Acceleration Index ($\mathcal{A}$): Ratio of mean internal baryonic acceleration $\bar{a}$ to $a_0$.

This approach establishes a firm physical distinction between collisional and collisionless regimes not merely through the standard relaxation criteria but explicitly via the MOND field equation, which encodes the transition between Newtonian and deep-MOND dynamics using observable baryonic distributions.

Data and Sample Characterization

A comprehensive sample is assembled including:

• Early-type galaxies (ETGs) from ATLAS$^{3D}$
• Spirals, LSB/HSB disks, and dwarfs from SPARC
• Massive Compact Objects (MCOs): Globular clusters and UCDs from the Dabringhausen et al. 2008 catalog

Gas-rich dwarfs and LSB disks are assigned gas-consumption ages as proxies for relative evolutionary state, while ETGs employ spectroscopically determined luminosity-weighted ages. All indices are computed solely from baryonic mass, radius, and velocity, ensuring a universal and scalable application.

Figure 1: The gas-consumption age $t_{\rm gas}$ as a function of baryonic mass for the SPARC sample, illustrating younger ages and slower evolution in LSB dwarfs compared to HSB spirals.

Results: Continuous Dynamical Sequences

Analysis in several diagnostic planes reveals robust, well-ordered sequences:

Collisionality vs. Acceleration

Plotting $\mathcal{T}_1$ versus $\mathcal{A}$ distinctly separates collisional (MCOs) and collisionless (galaxies) stellar systems. All galaxies are deeply in the collisionless regime, regardless of morphology or mass.

Figure 2: Collisionality index $\mathcal{T}_1$ as a function of acceleration index $\mathcal{A}$, cleanly separating star clusters from galaxies.

Dynamical Maturity vs. Acceleration

The ($\mathcal{T}$, $\mathcal{A}$) plane displays a monotonically organized structural hierarchy: ETGs and MCOs cluster at high $\mathcal{A}$ and low $\mathcal{T}$, while LSB galaxies and dwarfs populate the diffuse, dynamically young, low-acceleration regime.

Figure 3: Dynamical maturity index $\mathcal{T}$ versus acceleration index $\mathcal{A}$, showing a continuous acceleration–timescale sequence connecting all system types.

MOND Depth vs. Dynamical Maturity

The ($D_M$, $\mathcal{T}$) plane presents the strongest empirical sequence, where ETGs and MCOs (low $D_M$, low $\mathcal{T}$) form the dynamically old extreme, HSB spirals lie intermediate, and diffuse LSB disks and dwarfs inhabit the dynamically shallow, young, high-$D_M$ regime.

Figure 4: MOND depth index $D_M$ versus dynamical maturity index $\mathcal{T}$, displaying a sequence that correlates with stellar population age and evolutionary state.

Implications and Theoretical Consequences

The coherence of these diagnostic planes demonstrates that structural depth and dynamical maturity—measured from baryonic content—systematically link to stellar age and evolutionary history. Within the MOND paradigm, systems with most baryons inside $r_M$ collapse rapidly to dynamical maturity, emerging as compact and old populations (ETGs, MCOs), while more diffuse systems remain dynamically and chemically unevolved due to extended evolution in the deep-MOND regime.

Critically, the detection of a two-dimensional dividing surface in dynamical space indicates that the "dark matter phenomenon" only manifests in deep-MOND, collisionless systems (high $D_M$, $\mathcal{T}_1 \ll 1$), while high-acceleration, collisional systems lack such mass discrepancies. This is a direct, quantitative test that distinguishes MOND from standard $\Lambda$CDM models, where baryonic structure and dynamical depth are decoupled owing to stochastic halo formation and baryon-dark-matter feedback interactions.

Additionally, these results argue against any strict dichotomy between galaxies and clusters, supporting the view that MCOs, UCDs, dwarfs, and disks comprise a continuous family in dynamical and structural terms governed by baryonic depth in the MOND framework.

Figure 5: The empirical correlation of the MOND depth index $D_M$ with dynamical indicators sharply divides dark-matter-like and Newtonian baryonic systems.

Systematics, Limitations, and Future Directions

Potential sources of systematic uncertainty include heterogeneity in characteristic radii, uncertainty in star formation histories (especially for dwarfs), and the assumptions employed in the analytic baryonic profiles. Despite these, the observed continuity and order across >8 orders of magnitude in mass and surface density are robust, indicating the utility of the indices $D_M$, $\mathcal{T}$, $\mathcal{T}_1$, and $\mathcal{A}$ as physically meaningful dynamical descriptors.

Extension to galaxy groups and clusters, especially at higher redshift or environments with substantial non-isolated systems, will provide further opportunities to test the limits of the MONDian dynamical sequence. These diagnostic planes also offer a framework for analyzing stellar system evolution in future deep and high-resolution surveys (e.g., JWST).

Conclusion

This work introduces and validates the MOND Depth Index as a unifying, baryon-based dynamical descriptor that, together with $\mathcal{T}$, $\mathcal{T}_1$, and $\mathcal{A}$, enables a continuous, physically motivated classification of all self-gravitating stellar systems. These indices capture the principal axes of structural maturity, collapse history, and acceleration regime, reproducing empirical trends (such as downsizing) and providing discrimination between "dark matter" and collisionless/collisional behaviors in a manner consistent with MONDian phenomenology. This formalism constitutes a substantial advancement in the dynamical taxonomy of galaxies and star clusters, with significant implications for theoretical galaxy evolution and gravitational paradigm testing.

Reference: "The MOND Depth Index and Dynamical Maturity Clock: Toward a Universal Classification of Galaxies and Star Clusters" (2603.18135)