The Baryonic Faber-Jackson Relation and Fundamental Plane of Galaxy Groups, Elliptical Galaxies, and Dwarf Galaxies

This study of 1,400 pressure-supported systems reveals that the baryonic Faber-Jackson relation and fundamental plane systematically transition at an acceleration scale of $a_0 \simeq 1.2\times10^{-10}$ m s$^{-2}$, where high-acceleration systems follow the Newtonian fundamental plane while low-acceleration systems adhere to a baryonic Faber-Jackson relation consistent with MOND predictions.

The Gist

Imagine the universe as a giant, bustling city. In this city, there are different types of neighborhoods: massive, crowded skyscraper districts (giant elliptical galaxies), quiet suburban cul-de-sacs (dwarf galaxies), and entire towns made of smaller houses (galaxy groups).

For decades, astronomers have been trying to understand the "traffic laws" of this cosmic city. Specifically, they wanted to know: How much "stuff" (stars and gas) is in a neighborhood, and how fast are the cars (stars) zooming around inside it?

This paper is like a massive traffic study that looked at 1,400 different neighborhoods, ranging from tiny hamlets to huge metropolises, spanning a massive range of sizes. The researchers discovered that the rules of the road change depending on how "busy" or "accelerated" the neighborhood is.

Here is the breakdown of their findings in simple terms:

1. The Two Different Rulebooks

The researchers found that the universe doesn't use just one set of rules. It uses two, and the switch happens based on acceleration (how hard gravity is pulling on the stars).

• The "Busy City" Rule (High Acceleration):
  In massive, dense elliptical galaxies, the stars are packed tight and gravity is strong. Here, the traffic follows the standard "Newtonian" laws we learned in school. If you know how fast the cars are moving, you can predict the size of the city and the amount of stuff in it. It's a bit like a crowded highway where the traffic flow depends heavily on the road width and the number of cars. In this zone, the relationship involves three variables: mass, speed, and size. This is called the Fundamental Plane.

• The "Quiet Country" Rule (Low Acceleration):
  In dwarf galaxies and galaxy groups, the stars are spread out, and gravity is very weak. Here, the "Busy City" rules break down. The researchers found that in these low-gravity zones, the size of the neighborhood doesn't matter anymore.
  Instead, there is a simple, direct link: The total amount of "stuff" (mass) is directly tied to the speed of the stars. If you double the speed, the mass goes up by a huge factor (specifically, the speed to the fourth power). It's as if the cars in a quiet country town are driving at a speed that is perfectly locked to the total weight of the town, regardless of how spread out the houses are. This is the Baryonic Faber-Jackson Relation (BFJR).

2. The "Magic Switch"

The paper identifies a specific "magic switch" point (an acceleration scale called $a_0$).

• Above the switch: The universe behaves like standard physics (Newton).
• Below the switch: The universe behaves differently, following a simpler, tighter rule where size disappears from the equation.

The data showed that when the researchers looked only at the "quiet" neighborhoods (low acceleration), the relationship between mass and speed was incredibly tight and precise. It was so precise that it suggested a fundamental law of nature that standard physics struggles to explain without inventing invisible "dark matter" that magically arranges itself perfectly in every single galaxy.

3. The MOND Connection

The paper suggests that these findings align perfectly with a theory called MOND (Modified Newtonian Dynamics).

• The Analogy: Imagine a car engine. In a standard car (Newton), the engine needs a specific amount of fuel to go a certain speed, and the weight of the car matters a lot. In a MOND car, once you get to a certain low speed, the engine changes its behavior. It becomes super-efficient, and the relationship between fuel and speed becomes a simple, unbreakable rule that ignores the car's size.
• The authors argue that the universe seems to have this "engine switch." When gravity is weak (low acceleration), the laws of motion change slightly, creating the tight relationship they observed.

4. Why This Matters

The researchers are essentially saying: "We looked at 1,400 different cosmic neighborhoods. We found that the 'quiet' ones all follow one simple, perfect rule, while the 'busy' ones follow the old, complex rules."

This is a big deal because:

1. It challenges the "Dark Matter" story: In the standard view, dark matter is a mysterious substance that clumps around galaxies. But for dark matter to make 1,400 different types of galaxies (from tiny dwarfs to huge groups) all follow this exact same simple rule, it would have to be "fine-tuned" with incredible precision. It's like if every single car in the world, regardless of make or model, automatically adjusted its suspension to be perfect just because of the road it was on.
2. It supports a simpler gravity theory: The findings fit the prediction that gravity itself changes behavior when things get very slow and spread out, removing the need for invisible dark matter to explain the speed of stars.

Summary

The paper is a massive data check that found a universal "speed limit" rule.

• High Gravity: Complex rules involving size and mass (Standard Physics).
• Low Gravity: Simple, size-independent rules where mass and speed are locked together (Modified Gravity/MOND).

The authors conclude that the universe seems to naturally switch between these two modes of operation, and this switch explains the behavior of galaxies better than the current standard model of invisible dark matter.

Technical Summary

Technical Summary: The Baryonic Faber-Jackson Relation and Fundamental Plane of Galaxy Groups, Elliptical Galaxies, and Dwarf Galaxies

Problem Statement
Scaling relations between the visible mass of galaxies and their kinematics are critical for understanding the interplay between baryons, dark matter (DM), and gravity. The classic Faber-Jackson Relation (FJR) links luminosity to stellar velocity dispersion in elliptical galaxies, while the Fundamental Plane (FP) adds the effective radius as a third variable. In the standard $\Lambda$CDM paradigm, these relations are expected to emerge from complex stochastic processes involving dark matter halo growth and baryonic feedback. However, this framework faces a "fine-tuning" problem: it requires precise coupling between baryons and dark matter to place DM-dominated dwarf galaxies and baryon-dominated giant galaxies on the same scaling relations. Conversely, Modified Newtonian Dynamics (MOND) predicts a universal scaling relation for isolated, self-gravitating systems in the deep-MOND regime (low internal acceleration), where the baryonic mass scales as the fourth power of velocity dispersion ($M_{\text{bar}} \propto \sigma^4$) without dependence on system size. The authors investigate whether the properties of the Baryonic Faber-Jackson Relation (BFJR) and the FP systematically depend on the internal acceleration of the systems, testing the transition between Newtonian and MONDian regimes.

Methodology
The authors compiled a homogeneous dataset of 1,400 pressure-supported systems spanning eight orders of magnitude in baryonic mass ($M_{\text{bar}} \simeq 10^5 - 10^{13} M_\odot$). The sample includes:

1. 63 galaxy groups in the Local Supercluster.
2. 1,218 elliptical galaxies from the MaNGA survey.
3. 26 elliptical galaxies from the ATLAS3D survey.
4. 34 dwarf ellipticals in the Virgo cluster.
5. 31 dwarf ellipticals in the Fornax cluster.
6. 28 dwarf spheroidals in the Local Group.

Key quantities were homogenized across datasets:

• Baryonic Mass ($M_{\text{bar}}$): Defined as stellar mass plus hot gas mass. For dwarfs with negligible gas, $M_{\text{bar}} \approx M_{\text{star}}$. For ellipticals and groups, hot gas mass was estimated using scaling relations or X-ray observations.
• Velocity Dispersion ($\sigma_{\text{los}}$): Measured as the stellar velocity dispersion within one effective radius ($\sigma_e$) for individual galaxies, and as the velocity dispersion of member galaxies for groups.
• Internal Acceleration ($\langle g_{\text{bar}} \rangle$): The median Newtonian gravitational acceleration due to observed baryons within one effective radius (or mean projected radius for groups), assuming spherical symmetry.

The BFJR was modeled as a linear relation in logarithmic space ($\log M_{\text{bar}}$ vs. $\log \sigma_e$) using Bayesian orthogonal and vertical MCMC fitting (via BayesLineFit), accounting for errors in both variables and intrinsic scatter. The Fundamental Plane was tested by comparing $M_{\text{bar}}$ against the Newtonian virial estimator ($5R_e\sigma_e^2/G$).

Key Results

1. Acceleration-Dependent Scaling: The properties of the BFJR and FP are not universal but depend systematically on the mean internal acceleration $\langle g_{\text{bar}} \rangle$.

   • Low-Acceleration Regime ($\langle g_{\text{bar}} \rangle < 0.6 a_0$): This regime includes dwarf galaxies and galaxy groups. The BFJR converges to a tight power law:
     $$\log_{10}(M_{\text{bar}}/M_\odot) = (4.19 \pm 0.10) \log_{10}(\sigma_e/\text{km s}^{-1}) + (2.55^{+0.16}_{-0.16})$$
     The orthogonal intrinsic scatter is remarkably small at $0.11 \pm 0.01$ dex. Crucially, the residuals of this relation show no correlation with the effective radius, indicating that a third structural variable cannot reduce the scatter.
   • High-Acceleration Regime ($\langle g_{\text{bar}} \rangle \gtrsim 6 a_0$): This regime is dominated by massive elliptical galaxies. These systems follow the Newtonian Fundamental Plane ($M \propto R \sigma^2$) with a slope consistent with unity when plotting $M_{\text{bar}}$ against the virial estimator. The residuals of the BFJR for these systems correlate with the effective radius, confirming the necessity of the third variable (the FP) for high-acceleration systems.

2. Transition Scale: A transition occurs around the characteristic acceleration scale $a_0 \simeq 1.2 \times 10^{-10} \text{ m s}^{-2}$. Systems with $\langle g_{\text{bar}} \rangle < 0.6 a_0$ deviate from the Newtonian FP but follow a tight BFJR, while systems with $\langle g_{\text{bar}} \rangle > 6 a_0$ follow the Newtonian FP.

3. External Field Effect (EFE): The low-acceleration BFJR agrees with the MOND prediction for isolated systems. The authors note that while some sample dwarfs are satellites or in clusters, the EFE appears to play a secondary role, as the data still aligns with the isolated MOND prediction. A weak anti-correlation between BFJR residuals and $\langle g_{\text{bar}} \rangle$ is observed, which is qualitatively consistent with EFE expectations.

Significance and Claims
The paper claims that its results provide a stringent test for both $\Lambda$CDM galaxy formation models and MOND.

• Support for MOND: The findings empirically corroborate three core MOND predictions: (1) the existence of an acceleration scale $a_0$ marking a transition in dynamical behavior; (2) a BFJR slope of exactly 4 in the low-acceleration regime; and (3) a normalization consistent with specific non-relativistic MOND theories (AQUAL/QUMOND). The disappearance of the radial dependence in the low-acceleration regime (BFJR) versus its presence in the high-acceleration regime (FP) aligns with the MOND virial theorem versus the Newtonian virial theorem.
• Challenge to $\Lambda$CDM: The authors argue that the tightness of the BFJR across eight orders of magnitude in mass, particularly the fact that dwarf galaxies and galaxy groups (systems shaped by vastly different physical processes) lie on the same relation, poses a significant "fine-tuning" challenge for $\Lambda$CDM. It requires complex feedback mechanisms to conspire to produce a relation that mimics the a-priori prediction of MOND.

The authors conclude that the BFJR and FP emerge as fundamental scaling relations whose acceleration dependence offers a powerful empirical testbed for distinguishing between competing theories of gravity and galaxy formation.