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A Hydrogen Equivalent Dynamical Parameter Reveals Structural Transitions and Baryonic Organization in SPARC Galaxies

This paper proposes a dimensionless hydrogen-equivalent dynamical parameter, $H_{\text{equiv,dyn}}$ , which serves as a global ordering variable for SPARC galaxies by organizing five independent structural observables and revealing a significant structural transition near 1.25 that reduces Baryonic Tully–Fisher scatter beyond simple mass normalization.

Original authors: Ali Moslemi Tabrizi

Published 2026-08-13

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CC BY 4.0

Original authors: Ali Moslemi Tabrizi

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic library. For a long time, astronomers have been trying to organize the books on the shelves. These "books" are galaxies—huge, swirling collections of stars, gas, and dust. Some look like flat pancakes, others like fuzzy blobs, and they come in all sizes and weights. Scientists have noticed that these galaxies aren't just thrown together randomly; they follow strict rules, like how heavier galaxies spin faster. This is the field of galactic astronomy, where researchers try to find the "periodic table" for galaxies: a single rule or number that explains why a galaxy looks the way it does, how big it is, and how it moves. It's a bit like trying to figure out if there's a secret code that connects the size of a house to the number of people living inside it. If we can find that code, we might finally understand the deep structure of the universe, rather than just listing facts about individual galaxies.

Now, enter a new idea from a researcher named Ali Moslemi Tabrizi. This paper asks a wild question: What if we could measure a galaxy using the same "ruler" we use for a single atom of hydrogen? The author proposes a special number called the "Hydrogen Equivalent Dynamical Parameter." Think of it like a cosmic conversion chart. Usually, we measure a galaxy's weight by how fast its stars are spinning (its dynamical mass). But this new method takes that weight and divides it by a giant, pre-calculated number (based on the mass of a hydrogen atom) to see if the galaxy fits a specific "slot" in a universal hierarchy. It's as if the researcher is asking, "If a galaxy were a giant atom, what element would it be?"

The paper doesn't claim that galaxies are actually made of atoms or that they are giant versions of hydrogen. Instead, it suggests that galaxies might organize themselves in a pattern similar to how elements are organized in the Periodic Table. The study looked at a massive database of 133 real galaxies (from the SPARC collection) and ran them through this new "hydrogen ruler." The result? The author found a distinct "tipping point" or transition zone. When the galaxies' scores hit a specific number—around 1.25—something interesting happens. Below this number, galaxies are a bit of a mess: they vary wildly in shape, size, and speed. But once they cross that 1.25 threshold, they suddenly become much more orderly, fitting into a tight, predictable pattern.

The paper suggests that this single number acts like a master switch for galactic structure. It organizes everything at once: how heavy the galaxy is, how big it is, what shape it has, and how fast it spins. The author argues that this isn't just a coincidence. By splitting the galaxies into "low score" and "high score" groups, the messy, scattered data on the "low" side becomes a neat, straight line on the "high" side. It's like sorting a pile of mixed-up LEGO bricks; once you find the right way to group them, the chaotic pile suddenly reveals a clear, structured design. The paper concludes that while we don't know exactly why this happens yet, the hydrogen-equivalent number seems to be the key to unlocking the hidden order of the galaxy universe, suggesting that the universe might have a deeper, more organized structure than we previously realized.

Technical Summary: A Hydrogen Equivalent Dynamical Parameter Reveals Structural Transitions and Baryonic Organization in SPARC Galaxies

Problem Statement
Modern astrophysics faces the challenge of understanding the origin of galactic structure. While empirical scaling relations such as the Baryonic Tully–Fisher Relation (BTFR) and the Radial Acceleration Relation (RAR) demonstrate that galaxies are not randomly distributed in parameter space, it remains an open question whether a single, global ordering variable exists that simultaneously organizes the diverse structural properties of galaxies (mass, size, morphology, and kinematics). This study investigates whether a dimensionless "hydrogen-equivalent dynamical parameter" can serve as such a global ordering variable for the SPARC (Spitzer Photometry and Accurate Rotation Curves) galaxy sample.

Methodology
The analysis utilizes a working sample of approximately 133 SPARC galaxies selected for data completeness. The core of the methodology involves the definition and application of a dimensionless parameter, $H_{equiv,dyn}$ , defined as:
$H_{equiv,dyn} = \frac{M_{dyn}}{C^8 M_H}$
where $M_{dyn}$ is the dynamically inferred galactic mass ( $V_{flat}^2 R_{HI} / G$ ), $M_H$ is the mass of a hydrogen atom, and $C \approx 2.998 \times 10^8$ is a dimensionless structural compression coefficient.

The value of $C$ is not derived from the speed of light but is empirically reconstructed from four independent galaxy–atom mass ratios (Milky Way/Hydrogen, M31/Deuterium, M87/Carbon-12, IC1101/Iron-56), yielding a consistent value within 0.12%. The author provides five distinct consistency routes (mass hierarchy, density scaling, Friedmann-assisted, length hierarchy, and Hamiltonian photon) that converge on this value, though they acknowledge that a first-principles field-theoretic derivation remains an open theoretical question.

The study employs a multi-faceted statistical approach:

Boundary-Scan Analysis: A non-parametric search to identify transition thresholds ( $H_c$ ) that maximize the statistical separation between galaxy populations across multiple observables.
Correlation Analysis: Spearman rank correlations between $H_{equiv,dyn}$ and independent observables (baryonic mass, morphology, effective radius, disk radius, flat rotation velocity).
Two-Regime Decomposition: Splitting the sample at the identified transition to compare BTFR slopes, scatter, and RAR proxies.
Residual and Mediation Analysis: Examining the remaining scatter after removing the primary $H$ -dependent trend to determine the roles of morphology and baryonic surface density ( $\Sigma_b$ ).

Key Contributions and Results

Emergence of a Structural Transition: A systematic boundary scan reveals a statistically stable structural transition near $H_{equiv,dyn} \approx 1.25$ . This threshold appears independently in morphology, baryonic mass, effective radius, disk radius, and velocity distributions. The author interprets the $\approx 25\%$ offset from the nominal value of $H=1$ as a consequence of using enclosed dynamical mass (which includes dark matter/halo contributions within the HI radius) rather than baryonic mass alone, combined with finite radial averaging effects.
Global Ordering Parameter: $H_{equiv,dyn}$ $H_{e q u i v, d y n}$ exhibits strong Spearman correlations with multiple independent observables:
- Baryonic Mass ( $R \approx 0.97$ )
- Morphology ( $R \approx -0.82$ )
- Disk Radius ( $R \approx 0.81$ )
- Effective Radius ( $R \approx 0.71$ )
  The simultaneous organization of these distinct properties suggests $H_{equiv,dyn}$ acts as a large-scale ordering coordinate rather than a simple scaling variable.
Dynamical Regime Separation: The transition separates two distinct dynamical regimes. Galaxies in the high- $H$ regime ( $H \ge 1.25$ ) exhibit significantly reduced scatter in the BTFR (RMS scatter drops from 0.320 to 0.186, a ~42% reduction) and follow a tighter mass-velocity sequence compared to the structurally heterogeneous low- $H$ regime.
Residual Hierarchy: Residual analysis indicates that while $H$ organizes the primary structural sequence, secondary variations are mediated by morphology and baryonic surface density. A hierarchical pathway is identified: $H_{equiv,dyn} \rightarrow T \text{ (morphology)} \rightarrow \Sigma_b \text{ (surface density)} \rightarrow \Delta M \text{ (residual mass)}$ . Crucially, once the surface-density contribution is removed, the direct dependence of residuals on $H$ vanishes ( $R \approx -0.018$ ), suggesting $H$ influences structure primarily through the organization of these secondary layers.
Robustness Checks: The transition remains stable under partial correlation analysis controlling for flat rotation velocity ( $V_{flat}$ ), bootstrap resampling, and train/test procedures where the threshold is located on a subset of data and applied to a held-out set without re-optimization.

Significance and Claims
The paper claims to provide the first evidence that a single dimensionless parameter can simultaneously organize five independent galactic observables across the full SPARC sample with high statistical significance. The author posits that $H_{equiv,dyn}$ functions as a "large-scale ordering parameter" analogous to an order parameter in statistical physics, defining a macroscopic structural state within which specific galactic properties (mass, size, morphology) emerge.

The work explicitly frames this as an empirical classification framework rather than a new dark matter model or a modification of General Relativity. The author states that the framework is compatible with both special and general relativity, as it reinterprets mass-energy organization across structural layers without altering the Einstein field equations. The significance of the $H \approx 1.25$ transition is presented not as a prediction of a specific atomic isotope, but as an effective dynamical threshold derived from the interplay between galactic dynamics and the empirically calibrated normalization constant $C^8$ .

The study concludes that galaxies are not randomly distributed in parameter space but occupy a structured dynamical sequence governed by this hydrogen-equivalent scale, with morphology and surface density providing secondary refinements to this primary ordering. The author emphasizes that while the results are statistically robust within the SPARC sample, independent validation on other galaxy catalogs and a complete first-principles derivation of the compression coefficient $C$ remain necessary future steps.

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