🔭 astrophysics

Surface Density of Disk Galaxies in MOND

This paper extends Milgrom's (2009) findings by applying a double exponential disk model to spiral galaxies, confirming the existence of a quasi-universal surface density for galaxies with high mean inner acceleration while demonstrating a distinct behavior for those with low surface density.

Original authors: Antonino Del Popolo, Morgan Le Delliou

Published 2026-02-04

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Antonino Del Popolo, Morgan Le Delliou

Original paper licensed under CC BY 4.0 (http://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, bustling city. For a long time, astronomers have tried to understand how this city is built using a blueprint called the Lambda-CDM model. This blueprint says that most of the city's weight comes from invisible "dark matter" bricks that we can't see but feel through gravity.

However, this blueprint has some cracks when we look at the smaller neighborhoods (like individual galaxies). The math predicts that these neighborhoods should be dense and "spiky" in the center, but when we look, they often look "fluffy" and spread out. This is known as the "Cusp/Core" problem.

To fix these cracks, some scientists propose a different idea called MOND (Modified Newtonian Dynamics). Instead of adding invisible bricks, MOND suggests that the rules of gravity itself change when things get very slow or very far apart. It's like saying the traffic laws change in a quiet suburb compared to a busy highway.

The Big Question: Is There a "Universal Density"?

In 2009, two groups of scientists (Donato et al. and G09) looked at many different galaxies and found something fascinating. They noticed that if you measure the "surface density" (how much mass is packed into a specific area) of the dark matter in these galaxies, it seems to be almost the same number for almost every galaxy, from tiny dwarfs to massive giants.

They called this a "quasi-universal surface density." It's as if every house in the city, regardless of size, was built with exactly the same amount of bricks per square foot.

The New Study: Testing the Rules

The authors of this paper, Del Popolo and Le Delliou, wanted to check if this "universal rule" holds up when you apply the MOND theory, specifically to spiral galaxies (which look like flat, spinning disks, like a pizza).

They took a previous calculation by a scientist named Milgrom, which worked well for simple, point-like objects (like a single star), and expanded it to fit the complex shape of a spiral galaxy.

The Findings: Two Different Worlds

The paper reveals that the "universal rule" isn't actually universal. It depends entirely on how strong the gravity is in that specific galaxy.

The "Highway" Galaxies (High Acceleration):
For galaxies where the gravity is strong (like in the inner parts of big, bright spiral galaxies), the rules of normal physics (Newtonian regime) apply. Here, the paper confirms that there is a quasi-universal surface density. It's like the busy highway where the traffic laws are strict and predictable; the density stays constant.
The "Suburb" Galaxies (Low Acceleration):
For galaxies where the gravity is weak (like in the outer edges of galaxies or in tiny, faint dwarf galaxies), the MOND rules kick in. Here, the "universal" constant disappears. The surface density drops significantly. It's as if in the quiet suburbs, the traffic laws change, and the density of cars (or mass) becomes much lower and varies from place to place.

The Analogy: The "Magic Paint"

Imagine you are painting a wall.

The Old View (D09/G09): You think that no matter what kind of wall you have, you always use exactly 10 cans of paint per square meter. It's a universal rule.
The New View (This Paper): The authors say, "Wait a minute." If you are painting a wall in bright sunlight (strong gravity), you might indeed use 10 cans. But if you are painting a wall in the deep shade (weak gravity/MOND regime), the paint behaves differently. You might only need 5 cans, or the amount varies wildly depending on the wall's texture.

Why Did the Old Studies Miss This?

The paper suggests that previous studies found a "universal" number because they mostly looked at the "highway" galaxies (the bright, massive ones) and didn't have enough data on the "suburb" galaxies (the faint, low-density ones). When they did look at the faint ones, the data was messy and full of errors, so they might have missed the fact that the density was actually dropping.

The Bottom Line

This paper argues that the idea of a single, universal surface density for all galaxies is incorrect.

Strong gravity? Yes, there is a consistent, universal density.
Weak gravity? No, the density drops and changes.

This supports the MOND theory, which predicts that gravity behaves differently in low-acceleration environments, and it aligns with other recent studies that found surface density is not the same for every galaxy. The "universal constant" is actually a conditional rule that only applies in specific, high-gravity neighborhoods of the universe.

Technical Summary: Surface Density of Disk Galaxies in MOND

Problem Statement
The paper addresses the debate regarding the existence of a "quasi-universal" central surface density ( $\mu_{0D} = \rho_0 r_0$ ) for dark matter (DM) halos across galaxies of varying mass and morphology. Previous studies, specifically Donato et al. (2009, D09) and Gentile et al. (2009, G09), claimed that this surface density is constant for all galaxy types, from dwarfs to ellipticals, when modeled using a Burkert profile. However, this claim conflicts with Modified Newtonian Dynamics (MOND), which predicts that such universality should only hold for systems in the Newtonian regime (where mean acceleration $g \gg a_0$ ). Furthermore, observational studies (e.g., Napolitano et al., Cardone and Tortora) have argued that surface density correlates with halo mass or luminosity, contradicting the universality hypothesis. A specific limitation in the D09 and G09 analyses is their assumption that all galaxies possess a flat inner spherical density profile (Burkert profile), which is known to be a poor fit for giant galaxies and ellipticals.

Methodology
The authors extend the theoretical framework established by Milgrom (2009), who demonstrated the quasi-universal surface density for point masses in the Newtonian regime and its breakdown in the deep-MOND regime. The primary methodological contribution of this paper is the extension of Milgrom's point-mass calculation to spiral galaxies modeled as double exponential disks with cylindrical symmetry.

The analysis proceeds as follows:

Theoretical Formulation: The authors utilize the MOND field equation, defining the "phantom mass density" ( $\rho_P$ ) which arises from the modification of gravity. They decompose the phantom density into two components: $\rho_{P1}$ (dependent on the gradient of the potential) and $\rho_{P2}$ (dependent on the baryonic density and the interpolating function $\mu$ ).
Disk Modeling: The baryonic mass distribution is modeled as $\rho = \rho_0 e^{-R/h_r - |z|/h_z}$ . The authors calculate the column density (surface density) by integrating the phantom density along the line of sight.
Regime Analysis: The study distinguishes between two regimes based on the dimensionless acceleration $x = g/a_0$ $x = g / a_{0}$ :
- Newtonian Regime ( $x \gg 1$ ): Where the interpolating function $\mu \approx 1$ .
- Deep-MOND Regime ( $x \ll 1$ ): Where $\mu \approx x$ .
  The authors derive analytical expressions for the total surface density ( $F$ ) in both regimes, utilizing an interpolating function $\mu(x) = x/\sqrt{1+x^2}$ (corresponding to $n=1$ ) to simplify the algebra while noting that results hold for $n=2$ as well.
Approximation: The gravitational potential is approximated using a spherical expression to evaluate the MOND acceleration, a method the authors note introduces errors of only a few percent in the inner disk and up to 10% in the outer disk, which is acceptable for the qualitative behavior being investigated.

Key Results
The paper derives the behavior of the MOND surface density ( $\Sigma^*_c$ ) as a function of the ratio of baryonic surface density to the characteristic MOND surface density ( $\Sigma_b / \Sigma_M$ ) and the dimensionless acceleration.

High Surface Density (Newtonian Regime): For galaxies with large surface density ( $\Sigma_b \gg \Sigma_M$ ) or high acceleration ( $x \gg 1$ ), the calculated surface density approaches a constant value, $\Sigma_M = a_0 / 2\pi G$ . This confirms the existence of a quasi-universal surface density for massive, high-acceleration systems, consistent with Milgrom's point-mass results and the D09/G09 findings for these specific systems.
Low Surface Density (Deep-MOND Regime): For galaxies with low surface density ( $\Sigma_b \ll \Sigma_M$ ) or low acceleration ( $x \ll 1$ ), the surface density is not universal. Instead, it decreases significantly below the value $\Sigma_M$ , scaling approximately as $\sqrt{\Sigma_b/\Sigma_M} \Sigma_M$ (or linearly with $x$ in the deep-MOND limit).
Discrepancy with D09/G09: The results demonstrate that the claim of a universal surface density for all galaxies (including low surface brightness dwarfs) is inconsistent with MOND. The paper notes that specific low-surface-brightness galaxies (e.g., NGC 3741, KK98 250, KK98 251) exhibit surface densities ( $\sim 56\text{--}66 M_\odot/\text{pc}^2$ ) far below the proposed universal value ( $\sim 100\text{--}138 M_\odot/\text{pc}^2$ ).

Significance and Claims
The authors claim that their work resolves the tension between the "universal surface density" hypothesis and MOND theory by clarifying the conditions under which universality holds.

Validation of MOND: The paper supports the MOND prediction that a quasi-universal surface density is a feature of the Newtonian regime only. It predicts that systems with mean inner accelerations smaller than $a_0$ will not exhibit this universal behavior.
Critique of Previous Universality Claims: The authors argue that the D09 and G09 conclusions of universality across all galaxy types are likely artifacts of sample selection bias (few low-surface-density galaxies in their samples) and model assumptions (forcing a Burkert profile on systems that do not fit it).
Agreement with Observational Counter-evidence: The theoretical results align with observational studies (Napolitano et al., Cardone and Tortora, etc.) that found correlations between surface density and mass/luminosity, thereby supporting the view that surface density is not a universal constant for all galaxy types.

The paper concludes that while a quasi-universal surface density exists for high-acceleration (Newtonian) systems, the extension of this universality to low-acceleration (deep-MOND) systems is theoretically unfounded within the MOND framework and contradicted by specific observational data.