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

Robin Eappen, Pavel Kroupa

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 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.
Paper Structure (10 sections, 15 equations, 5 figures, 1 table)

This paper contains 10 sections, 15 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dynamical Framework and Definitions
  3. Data and Sample Selection
  4. Results: Dynamical Diagnostic Planes
  5. T1 vs. A
  6. tcross/tH vs. abar/a0
  7. DM vs. T
  8. Discussion
  9. Conclusions
  10. Supplementary Structural Trend: $\boldsymbol{D_M}$ versus Baryonic Mass

Figures (5)

  • Figure S1: The gas-consumption age $t_{\rm gas}$ as a function of baryonic mass for the SPARC galaxies, estimated from the gas fraction assuming an exponential depletion law with $\tau_{\rm dep}=3\,{\rm Gyr}$. LSB dwarfs populate the youngest part of the distribution, while HSB spirals show substantially older gas-consumption ages. This behaviour reflects the extended star-formation histories and high gas fractions characteristic of diffuse, low-acceleration systems.
  • Figure S2: Collisionality index, $\mathcal{T}_1$ as a function of acceleration index, $\mathcal{A}$. The diagnostic cleanly separates collisional stellar systems (MCOs) from collisionless galaxies. The colour bar depicts the age ($t_{\rm age}$ values and other as defined in Section \ref{['sec:data']}).
  • Figure S3: Dynamical maturity index, $\mathcal{T}$ as a function of $\mathcal{A}$. A continuous acceleration–timescale sequence links all galaxy types and MCOs. The vertical axis is shown on a linear scale to highlight the relative spread in $\mathcal{T}$ among the galaxy populations; MCOs, which occupy a very narrow range in $\mathcal{T}$, therefore appear cluttered near the lower edge of the diagnostic plane. The colour bar is as in Figure \ref{['fig:tcross_trelax']}.
  • Figure S4: MOND depth index, $D_M$ versus dynamical maturity index, $\mathcal{T}$, colour-coded by stellar population age. A continuous sequence emerges that links structural depth, dynamical maturity, and stellar age across all galaxy types. A linear scale is adopted for the $\mathcal{T}$-axis to enhance the visual spread of the galaxy populations along the horizontal direction; as a consequence, the MCOs, which occupy a very narrow range in $\mathcal{T}$, appear visually cluttered near the left edge of the diagnostic plane. The colour bar is as in Figure \ref{['fig:tcross_trelax']}.
  • Figure A1: MOND depth index $D_M$ as a function of total baryonic mass. The relation reflects the scaling of the MOND radius $r_M \propto M_{\rm bar}^{1/2}$, shown here for completeness as a supplementary structural trend within the sample. The colour bar is as in Figure \ref{['fig:tcross_trelax']}.