Deriving the Galactic Radial Acceleration Relation

From Measurement Quantization to Galactic Rotation Without a Galaxy-Specific Fit

One of the persistent problems in modern astrophysics appears in the rotation of galaxies. From the distribution of baryonic matter—stars, gas, and other ordinary matter—Newtonian gravity predicts that orbital velocities outside a sufficiently localized mass distribution should eventually decline with increasing distance from the galactic center. Observed rotation curves commonly remain elevated relative to the Newtonian prediction from the baryonic source alone.

The conventional explanation introduces an additional nonluminous gravitating component, dark matter. Measurement Quantization (MQ) approaches the same observations from a different direction. Rather than introducing an additional galactic mass component, MQ asks how the Newtonian response calculated from the baryonic source is realized when measure is resolved between the System Frame and Internal Frame through the Frames mapping.

This distinction leads to a specific result. MQ derives the functional form of the observed galactic radial acceleration relation (RAR) together with its acceleration scale before the relation is applied to galaxy rotation-curve data. The resulting gravitational-realization relation contains no galaxy-specific MQ parameter.

This is substantially different from the interpretation presented on earlier versions of this page. What was previously described as a single “quantization crossover” is more generally represented by a continuous transition between gravitational-realization regimes. The transition is not confined to one galactocentric radius, nor is a special crossover radius required as an input. Its local magnitude follows from the baryonic acceleration.

The Baryonic Starting Point

Consider circular motion at galactocentric radius R. Let v_New(R) denote the Newtonian circular velocity calculated from the baryonic mass model. The corresponding baryonic acceleration g_bar(R) is g_bar(R) = v_New²(R)/R. This quantity is not obtained from the observed rotation velocity. It is constructed from the independently modeled distribution of baryonic matter.

Let g_real(R) denote the realized centripetal acceleration. In the Newtonian limit, the realized response tends to the baryonic response. The difference between the two becomes significant as the baryonic acceleration decreases.

A Fixed MQ Acceleration Scale

The characteristic acceleration used in the MQ galactic calculation is not obtained by fitting individual galaxies. It follows from the frame-resolved cosmological relation below, where a_MQ is the MQ galactic acceleration scale, c is the speed of light, H_sys is the present-epoch System Frame expansion rate, and θ_si is the invariant MQ coefficient entering the Frames mapping.

At the present epoch, this gives a_MQ = 1.20040492178 × 10⁻¹⁰ m s⁻². This value is fixed upstream of the galaxy comparison. It is not determined from SPARC, the Milky Way rotation curve, or a galaxy-specific fit to the RAR.

The local dimensionless realization depth χ(R) is χ(R) = √[g_bar(R)/a_MQ]. MQ associates this depth with the radial realization fraction P(R), defined by P(R) = 1 − exp[−χ(R)]. This supplies the continuous transition that the earlier discussion of a “quantization crossover” was attempting to characterize.

The Galactic Radial Acceleration Relation

Applying the realization fraction to the baryonic gravitational response gives the central galactic realization relation.

Expressed in terms of orbital velocity, the same relation is

where v_real(R) is the realized circular velocity.

The functional form is the same exponential form reported empirically for the radial acceleration relation. The empirical relation is prior community knowledge and should be distinguished from its interpretation within MQ. In the empirical construction, the relation and its characteristic acceleration were inferred from galactic observations. In MQ, the acceleration normalization and realization mechanism are specified independently of galaxy-specific rotation-curve fitting, after which galaxy observations test the fixed relation rather than determine a separate MQ parameter for each galaxy.

Why Newtonian Gravity Is Recovered

The behavior of the realization relation becomes particularly clear in its limiting regimes. When g_bar is large compared with a_MQ, the realization depth χ becomes large. The exponential contribution tends toward zero, P(R) tends toward unity, and g_real tends toward g_bar. MQ therefore approaches the ordinary Newtonian baryonic result in the high-acceleration regime.

The Low-Acceleration Regime

When g_bar becomes small compared with a_MQ, define x = √(g_bar/a_MQ). As x tends to zero, (1 − e^−x)/x tends to unity. The exact realization law therefore produces the familiar square-root low-acceleration scaling while the realized acceleration still tends to zero more slowly than the Newtonian baryonic field.

Why Galactic Rotation Curves Become Flat

Outside a sufficiently localized baryonic mass distribution, the Newtonian acceleration has the inverse-square radial behavior determined by the enclosed baryonic mass M_bar. Combining that behavior with the low-acceleration limit changes the asymptotic radial dependence of the realized acceleration from inverse-square to inverse-radius. Circular orbital motion obeys v_real² = R g_real, so the radial factor cancels in the exterior weak-acceleration limit and the realized orbital velocity tends toward a radius-independent value.

This is the mathematical origin of asymptotically flat galactic rotation curves in the MQ construction. No additional halo mass is inserted into the MQ calculation to produce the flattening.

The Baryonic Tully-Fisher Relation

The same exterior weak-acceleration limit produces the fourth-power baryonic Tully-Fisher scaling. For finite enclosed baryonic mass, the exact asymptotic result is

Flat outer rotation curves, the radial acceleration relation, and the baryonic Tully-Fisher scaling are therefore not separate prescriptions within the MQ construction. They arise as different consequences of the same fixed gravitational-realization relation.

Testing the Relation in the Milky Way

The Milky Way provides a useful test because an independently constructed baryonic mass model can be compared with high-quality stellar kinematics. The principal MQ calculation uses a McMillan baryonic density model while excluding its dark-matter halo contribution. The Newtonian baryonic rotation curve used as the source is therefore constructed from ordinary matter rather than from a halo-adjusted gravitational source.

The statistical interpretation requires additional care. When the specified propagated covariance in the baryonic normalization is included, the generalized result is χ² = 39.57 for 38 degrees of freedom, with p = 0.400 under that covariance model. The appropriate conclusion is consistency with the Milky Way data under the stated model rather than a claim of covariance-complete precision validation.

Testing All 175 SPARC Galaxies

A much broader test is provided by the SPARC database, which contains 175 late-type galaxies with Spitzer photometry and high-quality rotation curves. The MQ analysis retains all 175 published SPARC galaxies and all 3,366 radial measurements before stratifying the sample by observational quality or galaxy properties. The same present-epoch acceleration scale, a_MQ = 1.20040492178 × 10⁻¹⁰ m s⁻², is used throughout. There is no galaxy-specific MQ adjustment.

The mean velocity ratio corresponds to an aggregate velocity-scale offset of about +0.019 percent. A near-zero pooled signed residual shows that the fixed relation introduces little catalog-wide signed normalization offset under the adopted source construction. It does not mean that every radial measurement or every galaxy is reproduced to that precision. Individual residuals remain, while uncertainties in stellar mass normalization, gas distributions, distances, inclinations, and other baryonic-source quantities remain material. The 3,366 radial measurements are also nested within 175 galaxies and must not be interpreted as 3,366 statistically independent galactic systems.

Observational Quality Matters

SPARC independently assigns observational-quality flags to its galaxies. These classifications provide a useful stratification because they are not determined from MQ residuals. Among the 99 galaxies in the highest-quality SPARC class used in the MQ analysis, the galaxy-level geometric MQ-to-observed velocity ratio is 1.0042, or about 0.42 percent above unity.

Lower-quality classes show larger normalization displacement in the MQ analysis. This dependence cautions against interpreting all residual structure as a failure of the gravitational relation itself. The quality of the observational inputs and baryonic reconstruction materially affects the comparison. Additional tests using alternative stellar-mass information likewise indicate that construction of the baryonic source remains an important part of the uncertainty budget.

An Out-of-SPARC Test

A further test uses galaxies outside the SPARC catalog. The LITTLE THINGS analysis applies the same fixed MQ relation to 15 prospectively screened dwarf galaxies comprising 288 radial comparisons. Using independently constructed source models, the galaxy-weighted geometric ratio of MQ realized to observed velocity is 0.9907. A galaxy-block bootstrap gives a 95 percent interval of 0.868–1.130, with p = 0.895 for the zero-mean logarithmic-offset test.

This analysis has lower precision than the principal SPARC comparison because figure digitization and reconstructed-source uncertainties contribute additional uncertainty. Its importance is methodological. It evaluates the frozen MQ relation in a population outside the catalog used for the principal large-sample analysis.

From a “Crossover” to a Continuous Realization Law

The earlier version of this page attempted to identify a particular galactic radius at which a quantization crossover occurred. That description captured part of the physical intuition but not the more general relation now used in MQ. There is no universal crossover radius.

The controlling quantity is the local baryonic acceleration relative to the fixed MQ acceleration scale, represented by the dimensionless realization depth χ(R) = √[g_bar(R)/a_MQ]. Different galaxies possess different baryonic distributions, so the radius at which the realized response begins to depart appreciably from the Newtonian baryonic prediction differs from galaxy to galaxy.

What is universal in the present construction is not a galactocentric radius but the realization relation. At large χ, gravitational realization tends toward the Newtonian baryonic result. As χ decreases, the realization fraction decreases continuously and the realized gravitational response increasingly exceeds the Newtonian baryonic acceleration. In the deep low-acceleration limit, the response acquires the square-root scaling described above.

What the Derivation Does—and Does Not—Establish

The radial acceleration relation was established empirically before the present MQ derivation, and its low-acceleration square-root scaling is closely related to behavior long associated with Modified Newtonian Dynamics (MOND). Agreement with an already known functional form cannot by itself establish MQ as the uniquely correct physical explanation.

The distinctive MQ claim is narrower and directly testable. The acceleration normalization and realization mechanism arise from quantities fixed independently of galaxy-specific rotation-curve fitting, after which the resulting relation is applied unchanged to the galaxies. If sufficiently precise baryonic reconstructions and kinematic measurements reveal systematic departures from the frozen relation that cannot be accounted for by observational or source-model uncertainties, the MQ galactic realization law would fail that test.

Its present significance within MQ instead lies in the range of galactic phenomenology generated by one relation. The same fixed construction connects the high-acceleration Newtonian regime, the observed radial acceleration relation, the emergence of asymptotically flat rotation curves, and the baryonic Tully-Fisher fourth-power scaling.

From the perspective of Measurement Quantization, the phenomenon conventionally attributed to additional galactic mass is therefore interpreted differently. The baryonic distribution remains the gravitational source used in the calculation. What changes with acceleration is the relation between that source and the gravitational response realized through the frame structure of measurement.

The original question posed by this page was where classical behavior gives way to a quantized galactic effect. The more complete result is that no single radius answers that question. The transition is encoded continuously in the ratio g_bar(R)/a_MQ, and its observable consequence is the galactic radial acceleration relation itself.