Radial Acceleration Relation in Galaxies

• The RAR is an empirical law that correlates observed galaxy accelerations with predictions from baryonic mass using a characteristic acceleration scale.
• It is derived from rotation curve measurements and photometric/21-cm gas data, resulting in a remarkably low intrinsic scatter across various galaxy types.
• Its universality challenges traditional dark matter models and provides key insights for refining galaxy formation simulations and exploring modified gravity.

The Radial Acceleration Relation (RAR) is an empirical law that tightly links the observed dynamical acceleration in galaxies to the radial acceleration predicted from their baryonic @@@@1@@@@ alone. This coupling persists across diverse galaxy morphologies, masses, surface brightnesses, and gas fractions, and it is most commonly parameterized by a single characteristic acceleration scale marking the transition between Newtonian and “dark-matter dominated” regimes. The existence of such a remarkably tight correlation calls into question conventional expectations from dark matter-dominated cosmologies and motivates both the refinement of galaxy formation models and consideration of alternative theories of gravity.

1. Definition, Empirical Formulation, and Key Characteristics

The RAR quantifies the correlation between the observed centripetal (radial) acceleration, $g_{\rm obs}$, at each radius in a galaxy and the acceleration, $g_{\rm bar}$, predicted by the distribution of baryonic mass (stars plus gas) at that radius. The primary operational definitions are

$g_{\rm obs}(R) = \frac{V^2(R)}{R}$

where $V(R)$ is the circular velocity measured from rotation curves, and

$$g_{\rm bar}(R) = \left|\,\frac{\partial\Phi_{\rm bar}}{\partial R}\,\right|$$

where $\Phi_{\rm bar}$ is the gravitational potential obtained by solving the Poisson equation for the observed baryon distribution (often from near-infrared photometry and 21 cm gas data) (McGaugh et al., 2016).

The relation between $g_{\rm obs}$ and $g_{\rm bar}$ is described by a single empirical function: $$g_{\rm obs} = \mathcal{A}(g_{\rm bar}) = \frac{g_{\rm bar}}{1 - \exp\left(-\sqrt{g_{\rm bar}/g_{\dagger}}\right)}$$ where $g_{\dagger}$ is the only free parameter, the characteristic acceleration scale. The best-fit scale is

$$g_{\dagger} = (1.20 \pm 0.02\ \text{(random)} \pm 0.24\ \text{(systematic)}) \times 10^{-10}\ \mathrm{m\,s}^{-2} \, .$$

The RAR is distinguished by its universality: 2693 points in 153 rotationally supported galaxies of differing morphologies, sizes, and gas fractions tightly follow this law with minimal scatter (width $\sim0.11$ dex), and the empirical relation is robust even in the dark-matter-dominated regime (McGaugh et al., 2016, Lelli et al., 2016).

2. Observational Basis and Methodology

The RAR is constructed by measuring rotation curves and light profiles for each galaxy:

• $g_{\rm obs}(R)$ is derived directly from measured $V(R)$ with radial increments.
• $g_{\rm bar}(R)$ is independently inferred. Stars are mapped using optical/infrared photometry and converted to a stellar mass profile adopting a mass-to-light ratio, while gas mass is traced using H I 21-cm emission. The sum yields the total baryonic mass as a radial function.

The gravitational potential from this baryonic mass distribution is computed via the Poisson equation, differentiated to yield $g_{\rm bar}$ at all radii, which is then compared on a point-by-point basis to $g_{\rm obs}$.

A rigorous error budget is constructed to account for uncertainties in rotation velocities, galaxy inclinations, distances, and mass-to-light ratios. The observed scatter in the RAR is found to be dominated by these uncertainties; little room remains for intrinsic scatter (McGaugh et al., 2016), reinforcing the view that the RAR is not the result of averaging over stochastic processes or sample selection.

3. Physical Interpretation and Theoretical Implications

The tightness and universality of the RAR imply a deep coupling between baryonic and total mass distributions in galaxies:

• In the high-acceleration regime ($g_{\rm bar} \gtrsim 10^{-10}\,\mathrm{m\,s}^{-2}$), $g_{\rm obs} \approx g_{\rm bar}$, consistent with Newtonian gravity and indicating baryonic dominance.
• In the low-acceleration regime, $g_{\rm obs}$ systematically exceeds $g_{\rm bar}$, leading to the apparent need for dark matter or modified gravitational laws.

The RAR enables the “prediction” of the dark matter contribution as fully specified by the observed baryon distribution: $g_{\rm DM} = g_{\rm obs} - g_{\rm bar}$ This natural law character is highlighted by the lack of any additional “second parameter” controlling $g_{\rm obs}$. Residuals show no significant correlations with radius, baryonic mass, gas fraction, galaxy size, or other global properties (Lelli et al., 2016).

The existence of a uniform acceleration scale $g_{\dagger}$ across galaxy types and environments challenges standard dark matter halo models, which generally predict more stochasticity in the baryon-dark matter connection. The RAR’s form is reminiscent of the MOND prescription. However, the relation also emerges within $\Lambda$CDM cosmologies as a result of baryons settling in dark matter halos with a correlated response (for example, adiabatic contraction or halo formation histories) (Lelli et al., 2016).

4. Extension Across Galaxy Populations and Comparison to Related Laws

Analysis spanning 153 galaxies (McGaugh et al., 2016) and extended to 240 including late-type, early-type, and dSph galaxies (Lelli et al., 2016), shows the RAR is independent of morphology or mass. At accelerations below $g_{\rm bar} \sim 10^{-12}\ \mathrm{m\,s}^{-2}$ (i.e., in ultrafaint dwarfs), there is tentative evidence for further flattening—possibly a mass “floor”.

The RAR naturally subsumes several classical galaxy scaling relations:

• Baryonic Tully-Fisher Relation (BTFR): In the outer, low-acceleration regime, the RAR ($g_{\rm obs} \propto \sqrt{g_{\rm bar} g_{\dagger}}$) reduces to $M_{\rm bar} \propto V_f^4$.
• Faber-Jackson relation: For pressure-supported (spheroidal) systems, a similar power law between velocity dispersion and luminosity/mass appears.
• Renzo’s Rule and Baryon-Halo Conspiracy: The one-to-one mapping between $g_{\rm bar}$ and $g_{\rm obs}$ encodes both the detailed correspondence of rotation curve features to photometric structure and the “conspiracy” of flat rotation curves.
• Central density relation: In the limit $R \to 0$, the RAR reduces to previously known central density scaling.

A tabular summary of representative scaling regimes:

Acceleration Regime	Behavior	Functional Form
High (baryons dominate)	$g_{\rm obs} \approx g_{\rm bar}$	$g_{\rm obs} = g_{\rm bar}$
Low (DM dominates)	$g_{\rm obs} \propto \sqrt{g_{\rm bar}}$	$g_{\rm obs} = \sqrt{g_{\rm bar}g_{\dagger}}$
Transition	Empirical fit	$$g_{\rm obs} = g_{\rm bar} / [ 1 - \exp( -\sqrt{ g_{\rm bar}/g_{\dagger} } ) ]$$

5. Departures, Limitations, and Ongoing Debates

While the RAR is exceptionally tight for rotationally supported galaxies, several areas have emerged prompting further investigation:

• Cluster Scales: Application of the RAR to galaxy clusters reveals a significantly larger intrinsic scatter and a higher effective acceleration scale, indicating the RAR is not strictly universal across all virialized structures (Chan et al., 2020, Tian et al., 2020).
• Intrinsic Scatter: Although most scatter is attributed to observational errors, precise measurement of the intrinsic (cosmic) scatter ($\sim$0.11 dex) allows discrimination between galaxy assembly models, baryonic feedback, and modifications to gravity (Stone et al., 2019).
• Physical Origin: Whether the RAR is a fundamental law of nature (as in MOND) or the outcome of complex galaxy formation processes in a $\Lambda$CDM universe remains under debate. The tight relation challenges models predicting stochastic baryon-dark matter couplings.
• Redshift Evolution: Evidence for the redshift dependence of the RAR, particularly at $z\gtrsim0.77$, may indicate evolution in the baryon-dark matter interplay or a breakdown of the empirical law in rapidly evolving environments (Navia, 2018).
• Low-Mass Dwarfs: Dwarf galaxies, especially at baryonic masses below $10^{7.5}\ M_{\odot}$, frequently exhibit increased scatter or loci systematically offset from the canonical RAR, suggesting either additional complexities in their dynamics or the breakdown of the law in extreme environments (Garaldi et al., 2017).

6. Theoretical Context, Interpretability, and Future Directions

The RAR’s empirical form is compatible with both $\Lambda$CDM-based and modified gravity paradigms:

• $\Lambda$CDM Context: The relation is interpreted as a consequence of baryonic physics shaping the dark matter halo structure; adiabatic contraction/expansion and baryon-driven feedback lead to correlated mass profiles. The RAR’s observed scatter and dependence on baryonic distribution must be matched by simulations.
• Modified Gravity (e.g., MOND): The RAR is a direct consequence of a modified force law, with $g_{obs} \propto \sqrt{g_{bar} a_0}$ in the low-acceleration limit. The characteristic scale $g_{\dagger}$ is identified with $a_0$.
• Diagnostic Tool: The RAR, and its extensions or breakdowns in different systems, provide sharp tests of the nature of dark matter, the validity of Newtonian gravity, and the universality of feedback-regulated galaxy formation.

Future research focuses on:

• Extending RAR measurements to more extreme environments (e.g., low-mass dwarfs, outer regions via weak lensing, galaxy clusters).
• High-precision modeling of the baryonic mass distribution and more sophisticated treatments of observational errors.
• Application of the RAR to redshift-independent distance measurements, given the small intrinsic scatter and “law-like” behavior (Lelli et al., 2016).

7. Summary

The RAR is a near-universal, empirical law that links the observed dynamical acceleration in galaxies to that predicted from the visible baryonic mass, parameterized by a single acceleration scale, and characterized by extraordinary tightness across galaxy populations. It presents a stringent benchmark for both galaxy-formation models and theoretical gravitation, encapsulates and generalizes multiple classical dynamical laws, and continues to drive both observational and theoretical research into the distribution of matter and the nature of gravity in galaxies (McGaugh et al., 2016, Lelli et al., 2016).