Galaxy Rotation Curves and Acceleration Relations

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A galaxy rotation curve compares orbital speeds with the gravitational field inferred from stars, gas, and any unseen component. The same data can be described by general relativity in its weak-field limit plus a dark halo, or by modifying the relation between baryonic sources and acceleration. The sharp scientific question is therefore not whether outer rotation curves are non-Keplerian; they are. It is whether versioned kinematic and baryonic measurements predict held-out radii and galaxies better under competing, fully specified source-to-force maps once distance, inclination, selection, and intrinsic diversity are propagated. The assessment here is current through 2026-08-19.

Required background. Classical Field Theory supplies source fields and potential equations, while Probability and Statistical Inference supplies hierarchical likelihoods, priors, covariance, and predictive comparison.

Helpful background. Weak-Field Gravity supplies the metric-to-acceleration limit, and Black-Hole Astrophysics supplies the broader galaxy and baryonic modeling context without turning it into a gravity test.

Rotation speeds and baryonic acceleration

For a nearly circular orbit in an axisymmetric potential $\Phi(R)$ ,

g_{\rm obs}(R)=\frac{V_{\rm rot}^2(R)}{R}\simeq \frac{\partial\Phi}{\partial R}.

The measured line-of-sight speed is approximately $V_{\rm los}=V_{\rm sys}+V_{\rm rot}\sin i\cos\theta$ before noncircular motion, pressure support, beam convolution, and warp geometry are added. Thus inclination $i$ enters nonlinearly and becomes dangerous for nearly face-on disks. Distance $D$ changes physical radii and luminosities together, correlating points within one galaxy.

An often-used baryonic decomposition is

V_{\rm bar}^2(R)=V_{\rm gas}^2(R)+\Upsilon_d V_d^2(R)+\Upsilon_b V_b^2(R), \qquad g_{\rm bar}=\frac{V_{\rm bar}^2}{R},

where the stellar curves computed at unit mass-to-light ratio are scaled by disk and bulge ratios $\Upsilon_d$ and $\Upsilon_b$ . Each contribution means $R\,\partial\Phi_j/\partial R$ , not a measured speed of a separate component. The displayed sum assumes each contributes an inward radial force. A disk with a central hole can exert an outward force locally; a tabulated signed velocity proxy then contributes $V_j\lvert V_j\rvert$ , retaining the force sign instead of blindly squaring it Lelli, McGaugh, and Schombert 2016, §3.3, Eq. (2). In a halo model, $V_{\rm model}^2=V_{\rm bar}^2+V_{\rm halo}^2$ ; in a modified-acceleration model, one specifies a relation between $g_{\rm obs}$ and $g_{\rm bar}$ , together with an external-field or geometry prescription. Halo population priors, stellar populations, feedback-informed correlations, interpolation functions, and external fields carry different predictive content.

SPARC provides 3.6-micron photometry and quality-controlled rotation curves for 175 disk galaxies across broad morphology, luminosity, surface-brightness, and gas-fraction ranges Lelli, McGaugh, and Schombert 2016, §§2–3 and Tables 1–2. Its downloadable files at the official SPARC data site are versionless. A numerical workflow must therefore record retrieval date, individual filenames, byte hashes, and any local corrections rather than citing “SPARC” as if it were a version identifier.

What an acceleration relation predicts

A commonly used empirical summary is

g_{\rm obs}=\frac{g_{\rm bar}} {1-\exp[-\sqrt{g_{\rm bar}/g_\dagger}]},

where $g_\dagger$ is an acceleration scale to be inferred with the nuisance model McGaugh, Lelli, and Schombert 2016, Eq. (4). This fit is not by itself a field equation. Its limits explain its attraction: for $g_{\rm bar}\gg g_\dagger$ the denominator tends to one, while for $g_{\rm bar}\ll g_\dagger$ it gives $g_{\rm obs}\simeq\sqrt{g_{\rm bar}g_\dagger}$ . Far outside a finite baryonic source, $g_{\rm bar}\simeq GM_b/R^2$ then implies $V^4\simeq GM_b g_\dagger$ , approximately independent of radius. A general nonspherical modified-gravity theory need not obey this algebraic fit point by point; its field equation and environment must still be solved.

Worked calibration check. At fixed angular radius, major-axis line-of-sight speed, and surface brightness, $R\propto D$ and $V_{\rm rot}\propto(\sin i)^{-1}$ . Hence

\delta\ln g_{\rm obs} =-\delta\ln D-2\cot i\,\delta i,

with $\delta i$ in radians. Under fixed mass-to-light ratio and a fixed angular mass geometry, the inferred baryonic mass scales as $D^2$ and its radius squared as $D^2$ , so $g_{\rm bar}$ is approximately independent of distance. A $10\%$ distance increase lowers $g_{\rm obs}$ by about $9\%$ without the same shift in $g_{\rm bar}$ . Likewise, a $1^\circ$ inclination shift at $i=20^\circ$ changes $g_{\rm obs}$ by about $9.6\%$ to first order. These shared shifts explain why a relation built from many radii cannot treat each point’s calibration error independently.

Likelihood, calibration, and sample definition

Let $d_{gi}$ be the measured velocity at radius $R_{gi}$ in galaxy $g$ . A conditional Gaussian model illustrates the covariance structure,

p(\mathbf d_g\mid\Theta_g,\Lambda,\mathcal M) =\frac{\exp[-\tfrac12(\mathbf d_g-\mathbf m_g)^{\mathsf T}C_g^{-1}(\mathbf d_g-\mathbf m_g)]} {(2\pi)^{N_g/2}\det(C_g)^{1/2}},

where $\Theta_g$ includes distance, inclination, systemic velocity, gas conversion, mass-to-light ratios, warp or pressure corrections, and halo or modified-gravity parameters; $\Lambda$ describes population distributions. The calibration layer records velocity zero points, beam and channel response, photometric zero point, point-spread function, gas map and conversion, and adopted solar luminosity. Shared distance and inclination uncertainty produces correlated radial residuals, so diagonal error bars are not enough.

The selection function includes the parent surveys, availability and quality of resolved H I or H $\alpha$ kinematics, inclination and asymmetry cuts, distance limits, surface-brightness sensitivity, and analyst quality flags. SPARC was assembled to span galaxy properties, not as a volume-complete census. Population claims must therefore condition on that construction or model its completeness. A predictive analysis should state priors, interval convention, intrinsic-scatter distribution, sampler diagnostics, posterior-predictive checks, and whether radii, entire galaxies, or both are withheld.

Results and limitations through 2026-08-19

Position or evidence	Result or claim	Limitation or discriminating boundary
Baryon-plus-halo models fit diverse rotation curves in the weak-field GR limit	Dark matter provides flexible galaxy-by-galaxy mass distributions and a cosmological formation context	Halo response, feedback, concentration and mass priors, and baryonic covariance affect out-of-sample predictions
The radial acceleration relation is tight across SPARC’s sampled dynamic range	Kinematics correlate strongly with the observed baryonic acceleration McGaugh, Lelli, and Schombert 2016, Figs. 2–4	Shared distance, inclination, mass-to-light assumptions, quality selection, and intrinsic scatter complicate causal interpretation
Modified-dynamics maps can predict rotation curves from baryons with few local parameters	Low-acceleration regularity is directly encoded and can be tested on held-out radii	External fields, interpolation choice, relativistic completion, clusters, and galaxy-formation selection remain model-dependent

The evidence establishes a strong empirical baryon–acceleration connection, not a unique cause. The conservative position attributes the remaining force to dark matter whose distribution is correlated with baryons through formation physics. The competing position treats the relation as evidence that the force law changes at low acceleration. Neither a flexible per-galaxy halo fit nor a tuned modified relation alone is decisive. It remains unresolved which complete model best predicts new galaxies under common population priors and realistic selection.

An adversarial predictive test

An independent check generates mock galaxies from known halo and modified-acceleration populations, passes them through realistic distance, inclination, beam, and quality selection, and verifies recovery and coverage. The real-data fit then refits $D$ , $i$ , $\Upsilon_d$ , and $\Upsilon_b$ rather than freezing catalog central values. An adversarial comparison withholds outer radii and entire low-surface-brightness galaxies, uses the same effective number of population freedoms, and scores predictive density rather than best-fit residual alone. Fits should be repeated under alternative stellar-population priors and noncircular-motion models.

A software pipeline must pin the downloaded files and hashes, unit conversions, quality cuts, covariance construction, model code, priors, sampler settings, and random seeds. Reproducing a published scatter from a rounded table is not enough; the recovered distribution should also match residuals versus radius, inclination, surface brightness, gas fraction, and galaxy quality.

What could change the conclusion. The comparison could change after a versioned SPARC replacement, a calibration or distance revision, a substantially larger resolved-kinematics sample with a documented selection function, or a blinded mock challenge in which one complete framework predicts withheld galaxies robustly better under shared nuisance treatment. A change in the empirical relation alone is insufficient unless the source and selection layers are traced.

Exercise

For a nearly face-on galaxy, why can a small inclination error mimic an acceleration anomaly across many radii rather than producing independent random residuals?

Solution

The deprojected rotation speed is approximately $V_{\rm rot}=(V_{\rm los}-V_{\rm sys})/\sin i$ . When $i$ is small, the derivative of $1/\sin i$ is large, so a small inclination shift rescales velocities strongly. Because the same inclination applies across much of the disk, the shift is coherent over radii; $g_{\rm obs}\propto V_{\rm rot}^2$ then changes coherently as well. Treating each point as an independent velocity error would therefore exaggerate evidence for a force-law mismatch.

References

Lelli, F., McGaugh, S. S., and Schombert, J. M. (2016), “SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves,” Astronomical Journal 152, 157. DOI.
McGaugh, S. S., Lelli, F., and Schombert, J. M. (2016), “Radial Acceleration Relation in Rotationally Supported Galaxies,” Physical Review Letters 117, 201101. DOI.
SPARC Team (n.d.), “SPARC,” data portal (accessed 2026-08-19). Official data site.