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A Hydrogen-Equivalent Dynamical Ordering Parameter in SPARC Galaxies: Structural Transitions, Morphology, and Baryonic Organization

This study proposes and validates a dimensionless hydrogen-equivalent dynamical ordering parameter, $H_{\text{equiv,dyn}}$ , which successfully organizes five independent galactic observables across the SPARC sample and reveals a critical structural transition near 1.25, suggesting that galaxy morphology and size are governed by a fundamental baryonic-dynamical mass ratio rather than mass alone.

Original authors: Ali Moslemi Tabrizi

Published 2026-06-26

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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 you walk into a massive library containing thousands of books. Some are tiny pamphlets, others are massive encyclopedias. Some are fiction, some are science, and they are written in different languages. Usually, librarians sort these books by genre (morphology) or by how thick the pages are (mass).

This paper proposes a completely new way to organize that library. Instead of sorting by genre or thickness, the author suggests there is a single "master key" that explains why the books are arranged the way they are.

Here is the breakdown of the paper's findings in simple terms:

1. The "Master Key": The Hydrogen-Equivalent Parameter

The author created a special number for every galaxy, called $H_{equiv,dyn}$ .

How it's made: They took the total "weight" of a galaxy (how much gravity it has) and divided it by the weight of a single hydrogen atom, multiplied by a giant, fixed number (a constant derived from comparing galaxies to atoms).
The Analogy: Think of this like a "Galactic Size Score." If you have a tiny galaxy, your score is low. If you have a massive, heavy galaxy, your score is high. The author claims this single number acts like a ruler that measures the "order" of a galaxy.

2. The Magic Number: 1.25

When the author looked at all the galaxies in their database (the SPARC catalog), they found something strange and exciting. The galaxies didn't just scatter randomly. Instead, they seemed to hit a "speed bump" or a transition point right around the score of 1.25.

Below 1.25 (The Chaotic Zone): Galaxies with scores lower than 1.25 are a mixed bag. They vary wildly in shape, size, and speed. It's like a messy room where everything is scattered.
Above 1.25 (The Organized Zone): Once a galaxy crosses that 1.25 threshold, it becomes much more orderly. These galaxies follow strict rules. They fit together like pieces of a puzzle. Their shapes, sizes, and rotation speeds become very predictable.

The author explains that the number isn't exactly 1.0 (which would be a perfect match to a hydrogen atom) because galaxies are huge, messy clouds of gas and stars, not single atoms. The extra 0.25 is just the "tax" for being a big, complex system.

3. The "Periodic Table" of Galaxies

The most surprising part of the paper is that this single number ( $H_{equiv,dyn}$ ) predicts five different things at the same time:

Mass: How heavy the galaxy is.
Shape: Whether it looks like a spiral, an oval, or a blob.
Size: How wide the galaxy is.
Speed: How fast the stars inside are spinning.
Order: How neatly the galaxy follows physical laws.

The Analogy: Imagine if you could look at a person and, just by knowing their height, you could perfectly predict their shoe size, their voice pitch, their favorite color, and their running speed. That is how powerful this "Hydrogen-Equivalent" number is for galaxies. It organizes the entire "personality" of a galaxy.

4. The Hidden Layers (Residuals)

The author admits that this "Master Key" doesn't explain everything. If you use the key to predict a galaxy's mass, there is still a little bit of "leftover" difference (residuals).

The paper found that this leftover difference is mostly explained by how crowded the stars are (surface density) and the shape of the galaxy.
The Chain Reaction: The author suggests a chain of events: The "Master Key" sets the stage $\rightarrow$ which influences the Galaxy's Shape $\rightarrow$ which influences how crowded the stars are $\rightarrow$ which explains the final small differences.

5. What This Means (and What It Doesn't)

What it claims: The universe isn't random. Galaxies are arranged in a strict, hierarchical structure, much like elements are arranged in the Periodic Table of the elements. There is a "low-end" chaotic zone and a "high-end" organized zone, separated by that 1.25 threshold.
What it does NOT claim: The author is not saying galaxies are literally made of hydrogen atoms in a way that turns them into tiny atoms. They are not proposing a new law of gravity or a new type of dark matter. They are simply saying: "If you sort galaxies by this specific number, everything lines up perfectly."

Summary

Think of the universe as a giant orchestra. For a long time, astronomers thought every instrument (galaxy) played its own random tune. This paper suggests there is actually a single conductor (the Hydrogen-Equivalent Parameter) that tells the orchestra when to play softly and chaotically (below 1.25) and when to play in a tight, perfect harmony (above 1.25). The paper provides the sheet music showing that, statistically, the orchestra is indeed following this conductor.

Technical Summary: A Hydrogen-Equivalent Dynamical Ordering Parameter in SPARC Galaxies

Problem Statement
Modern astrophysics faces the challenge of understanding the origin of galactic structure. While empirical scaling relations such as the Tully–Fisher relation, the Baryonic Tully–Fisher Relation (BTFR), and the Radial Acceleration Relation (RAR) indicate that galaxies are not randomly distributed in parameter space, a fundamental question remains: is there a single organizing parameter capable of capturing the structural diversity of galaxies simultaneously? Existing studies often focus on pairwise relations between specific observables. This paper investigates whether a unified, dimensionless ordering variable exists that can organize multiple independent galactic properties—mass, morphology, size, and kinematics—across the SPARC (Spitzer Photometry and Accurate Rotation Curves) database.

Methodology
The study utilizes a sample of approximately 133 galaxies from the SPARC database, selected for completeness in key observables: flat rotation velocity ( $V_{flat}$ ), HI radius ( $R_{HI}$ ), effective radius ( $R_{eff}$ ), disk scale radius ( $R_{disk}$ ), morphological type ( $T$ ), and baryonic mass ( $M_{bar}$ ).

The core of the analysis introduces a dimensionless quantity, the Hydrogen-Equivalent Dynamical 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, calculated as $V_{flat}^2 R_{HI} / G$ .
$M_H$ is the mass of a hydrogen atom.
$C$ is a dimensionless structural compression coefficient ( $\approx 2.998 \times 10^8$ ). Crucially, $C$ is not derived from the speed of light but is independently reconstructed from four observed galaxy–atom mass ratios (e.g., Milky Way/Hydrogen, M87/Carbon-12) within a "layered accessibility framework." The value of $C^8$ serves as an empirically grounded normalization connecting galactic and atomic mass scales, confirmed to within 0.12% consistency across the sample.

The analysis employs non-parametric statistical methods, including:

Boundary-Scan Analysis: To identify transition thresholds where structural properties change significantly.
Spearman Rank Correlation: To quantify monotonic relationships between $H_{equiv,dyn}$ and various observables.
Two-Regime Decomposition: Splitting the sample at the identified transition to compare BTFR slopes, scatter, and RAR proxies.
Residual Analysis: Removing the primary $H_{equiv,dyn}$ trend to examine secondary dependencies on morphology, surface density, and disk size.

Key Contributions and Results

Emergence of a Structural Transition:
Independent boundary scans across multiple observables (morphology, mass, velocity, radii) consistently identify a structural transition near $H_{equiv,dyn} \approx 1.25$ . This threshold is not an artifact of a single variable but appears as a stable feature in the distribution of baryonic mass, effective radius, disk radius, and flat rotation velocity. The paper interprets the 25% offset from the nominal $H=1$ value as a consequence of using enclosed dynamical mass (which includes dark matter/halo contributions within $R_{HI}$ ) rather than baryonic mass alone, combined with finite radial averaging.
Global Ordering Parameter:
$H_{equiv,dyn}$ acts as a strong global ordering coordinate. The analysis reveals high Spearman correlation coefficients between $H_{equiv,dyn}$ and:
- 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 independent variables suggests a coherent structural hierarchy rather than random scatter.
Dynamical Regime Separation:
The transition at $H \approx 1.25$ separates two distinct dynamical regimes:
- Low-H Regime ( $H < 1.25$ ): Characterized by greater structural diversity, wider morphological variation, and significant scatter in the BTFR (RMS scatter $\approx 0.320$ ).
- High-H Regime ( $H \ge 1.25$ ): Exhibits a more ordered structural sequence with reduced BTFR scatter (RMS scatter $\approx 0.186$ , a ~42% reduction) and tighter correlations between mass and velocity.
Residual Hierarchy and Secondary Organization:
After removing the primary trend driven by $H_{equiv,dyn}$ , residual analysis reveals a secondary organizational layer. The remaining scatter is significantly correlated with morphological type ( $T$ ) and baryonic surface density ( $\Sigma_b$ ). A structural path is identified:
$H_{equiv,dyn} \rightarrow T \rightarrow \Sigma_b \rightarrow \Delta M$
Once the contribution of surface density is removed, the residual dependence on $H_{equiv,dyn}$ vanishes ( $R \approx -0.018$ ), indicating that $H$ organizes the primary structure, while morphology and surface density account for secondary deviations.

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

Key claims regarding the significance of the work include:

Reversal of Classification Logic: The framework suggests that morphology is a manifestation of an underlying dynamical organization defined by $H_{equiv,dyn}$ , rather than the primary organizing principle.
Physical Interpretation of the Transition: The transition at $H \approx 1.25$ is presented as a falsifiable consequence of using dynamical mass. It represents the baryonic hydrogen-equivalent threshold shifted upward by the mean dynamical-to-baryonic mass ratio of the sample.
Compatibility with Relativity: The framework is explicitly stated to be compatible with both Special and General Relativity, as it reinterprets mass-energy organization across structural layers without modifying the Einstein field equations or introducing Lorentz-violating terms.
Empirical Basis: The normalization constant $C^8$ is derived from empirical galaxy-atom mass ratios, not theoretical fitting, and the statistical results are not tuned to enhance correlations.

The paper concludes that galaxies are not randomly distributed but occupy a structured dynamical sequence. While $H_{equiv,dyn}$ does not explain all galactic diversity (secondary layers exist), it captures the dominant structural variance, offering a new perspective on galactic classification and organization.