The Intrinsic Scatter of the Radial Acceleration Relation

Connor Stone; Stephane Courteau

REVIEW 5 major objections 5 minor 58 references

The Intrinsic Scatter of the Radial Acceleration Relation

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The radial acceleration relation carries intrinsic scatter of 0.11 dex.

desk verdict A careful Monte Carlo error model on a 2,500-galaxy catalog yields a nonzero RAR intrinsic scatter around 0.11 dex that favors LCDM over L17's null, but the value leans heavily on the adopted M/L uncertainty and is less secure than the abstract's ±0.02 suggests. read the letter →

arxiv 1908.06105 v2 pith:UKHZ6GUN submitted 2019-08-16 astro-ph.GA

Connor Stone , Stephane Courteau This is my paper · ORCID

classification astro-ph.GA

keywords radialaccelerationrelationintrinsicscatterMonteCarloerrormodelgalaxyscalingrelationsformationLCDMpredictionsMONDspiralrotationcurves

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the radial acceleration relation (RAR)—the tight empirical link between a galaxy's observed rotation acceleration and the acceleration expected from its stars alone—has a real intrinsic scatter of $0.11\pm0.02$ dex once measurement errors are removed. This matters because a previous analysis of the baryonic RAR reported near-zero intrinsic scatter, a result that favored modified Newtonian dynamics over the standard cosmological model. The authors assembled more than 2500 spiral galaxies from six surveys and modeled every major source of observational uncertainty with a Monte Carlo simulation. Subtracting that simulated error scatter in quadrature from the observed $0.17$ dex scatter leaves $0.11\pm0.02$ dex, consistent with ΛCDM predictions of $0.06$–$0.08$ dex. If correct, the RAR's tightness is real but not perfect, and it no longer singles out modified gravity.

What carries the argument

The load-bearing machinery is a Monte Carlo error model that simulates a universe with zero intrinsic scatter and then asks how much scatter observational errors alone would produce. Each observed point is first projected onto the fitted RAR by maximizing a likelihood over all measured parameters—distance, inclination, intrinsic disk flattening, velocity, luminosity, and stellar mass-to-light ratio—while requiring the point to lie exactly on the relation. Those zero-scatter values are then resampled from their uncertainty distributions, and the mock data are run through the same quality cuts and fitting pipeline as the real data. Comparing the mock scatter to the observed scatter in quadrature isolates the intrinsic component, a step that a simple first-order error propagation cannot do because many uncertainties are shared across points within a galaxy.

What would settle it

Measure the stellar mass-to-light ratio independently per galaxy with spatially resolved stellar-population modeling on a PROBES subsample and include noncircular motions modeled from high-resolution simulations; if the simulated scatter from these errors alone rises to the observed $0.17$ dex, the intrinsic scatter is zero, and the $0.11$ dex claim collapses.

Watch

Extended reading notes

Core claim

The central discovery is that the apparent tightness of the RAR hides a non-zero cosmic scatter whose size matches the expectations of cosmological galaxy formation. Using the forward residuals of the stellar RAR, the median observed scatter across six surveys is $0.17$ dex, while a Monte Carlo model that resamples all known observational errors produces a scatter of $0.12$ dex. The difference in quadrature gives an intrinsic scatter of $0.11\pm0.02$ dex, close to, though slightly larger than, the $0.06$–$0.08$ dex predicted by ΛCDM simulations and incompatible with the null value previously inferred. The intrinsic scatter also decreases with galaxy mass, from about $0.14$ dex at low mass to about $0.10$ dex at high mass, mirroring simulation trends.

Load-bearing premise

The Monte Carlo error model has to capture every significant observational uncertainty—especially the $0.13$ dex stellar mass-to-light ratio scatter, inclination recovery from axis ratios, magnitude errors, and the neglect of noncircular motions—because any underestimate inflates the claimed intrinsic scatter.

Editorial extensions

If this is right

A non-zero intrinsic scatter of about $0.11$ dex replaces the null value reported for the baryonic RAR, so the relation no longer discriminates against ΛCDM galaxy formation.
To reduce the intrinsic scatter to zero, the measurement uncertainties would have to be inflated by factors of about 2 for the stellar mass-to-light ratio up to factors greater than 10 for other parameters.
The intrinsic scatter decreases with galaxy mass (median $0.144$, $0.103$, and $0.095$ dex from low to high mass), matching the mass dependence seen in ΛCDM simulations and the diversity of dwarf rotation curves.
In the outer regions beyond one effective radius, where noncircular motions are weaker, the intrinsic scatter is $0.10\pm0.01$ dex, confirming the result.
The forward versus inverse choice of residuals changes the measured scatter substantially (median forward $0.17$ dex versus inverse $0.24$ dex), a consideration that affects all galaxy scaling relations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the intrinsic scatter is genuinely near $0.1$ dex, any modified-gravity theory that demands a perfectly tight relation must be either revised or supplemented by an astrophysical source of scatter.
The same Monte Carlo projection-and-resample recipe could be applied to the baryonic RAR once gas maps are available for all PROBES galaxies; a similar estimate would directly test the assumption that stellar and baryonic RARs share their scatter.
The method's logic implies a falsifiable hierarchy: future higher-resolution, lower-error surveys should show the observed scatter shrinking toward the simulated error scatter, keeping the quadrature residual near $0.1$ dex rather than falling to zero.
The mass-dependent trend of the intrinsic scatter suggests that a single universal acceleration scale may be an oversimplification; the low-mass tail is where the relation's scatter is largest and where tests of universality should focus.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Read the Stone & Courteau RAR scatter paper. The headline: they build a Monte Carlo error model that finally treats the RAR's correlated, shared galaxy-level errors properly, run 2,600+ galaxies from six surveys through it, and infer an intrinsic stellar RAR scatter of 0.11 ± 0.02 dex — nonzero, in line with ΛCDM's 0.06–0.08 dex, and in tension with L17's claimed null. That's a new result, and the method is the real contribution. Anyone separating intrinsic from observational scatter in a scaling relation should borrow the approach.

Credit where due: the mock data pass through the same quality cuts and fitting code as the real data; the forward/inverse axis issue that partly manufactured L17's tight relation is exposed clearly (Appendix A); and Table 5 is an honest stress test — the paper reports the factors by which each uncertainty would have to inflate to erase σ_int, and the smallest is 1.7 on the optical M/L error. They also check beyond one Re (0.10 dex) and flag noncircular motions and possible underestimated errors themselves. The circularity burden is genuinely low: the simulated scatter is forward-modeled, not tuned to match.

Now the soft spot, and it is proportional: everything hinges on the completeness of that error model. A 1.7× inflation of the adopted 0.13 dex optical M/L uncertainty — the upper end of the Roediger & Courteau range, not independently validated for this heterogeneous catalog — zeros out the signal in the paper's own Table 5. The test holds other errors fixed while scaling one; if distance, velocity, and M/L are each modestly underestimated together, no individual factor needs to reach 1.7. The quoted ±0.02 dex propagates only survey-to-survey scatter, not this systematic sensitivity. Also telling: the best-fit g† values in Table 1 differ by factors up to ten across surveys, which says the M/L modeling carries survey-level systematics much larger than the adopted random term. That does not directly bias a scatter measurement, but it should temper confidence in the error model's completeness. The PROBES catalog and code are not yet released — a real obstacle for a result that leans this heavily on an error model.

My take: the direction of the claim — nonzero intrinsic scatter around a tenth of a dex — is probably right and the paper deserves a serious referee; it was in fact published in ApJ. But the specific 0.11 ± 0.02 is softer than the abstract implies. If I were refereeing, I'd ask for the catalog release and a joint multi-parameter sensitivity analysis. I'd bring it to reading group and I'd cite the method.

Referee Report

5 major / 5 minor

Summary. This paper develops a Monte Carlo model of observational errors to infer the intrinsic scatter of the stellar radial acceleration relation (RAR). Using a new PROBES compilation of roughly 2500 spiral galaxies from six surveys, the authors fit the MOND-inspired RAR (Eq. 1), project each galaxy onto the zero-scatter relation with a likelihood that accounts for shared parameters (Eq. 10), resample mock observations from the uncertainty model of Sec. 4, and process the mocks through the same fits and quality cuts as the data. Comparing the observed scatter (Table 1, median 0.17 dex) with the simulated scatter (Table 3, median 0.12 dex) in quadrature yields a median intrinsic scatter of 0.11 +/- 0.02 dex (Sec. 6.2), which the authors claim agrees with LCDM predictions of 0.06-0.08 dex and contradicts the null scatter reported by L17. Sensitivity tests in Table 5 and a beyond-one-Re check are presented in support of the result.

Significance. If the inference is robust, the paper would resolve an important controversy: it would show that the apparent tightness of the RAR is not exactly zero intrinsic scatter, alleviating the tension with LCDM galaxy formation simulations and weakening a simple MOND-based requirement. The study's strengths are its large, heterogeneous sample; the explicit modeling of shared per-galaxy errors; and the use of identical processing code for real and mock data, which avoids several common circularity pitfalls. The sensitivity table (Table 5) and the beyond-one-Re analysis are commendable. However, the central number is only as good as the error model in Sec. 4, and several components of that model are either unvalidated for the heterogeneous samples or defined inconsistently. These issues currently prevent me from endorsing the quantitative claim as established.

major comments (5)

[§4.1, Eq. (4)] The distance uncertainty model is dimensionally inconsistent. The text states sigma_D = max(0.15D, 300), with 300 km/s being a peculiar velocity dispersion, while Eq. (4) uses sigma_D/D. As written, the first term treats sigma_D as a distance and the second as a velocity; the intended expression must involve sigma_pec/H_0 or an explicit unit conversion. Since Table 5 lists a distance-error scaling factor of 2.1, this ambiguity directly affects the inferred sigma_sim and sigma_int and must be corrected before the error model is usable.
[§5.2 and §2.2] The mock galaxies are treated as spherically symmetric, while the observed g* values are computed from a flattened disk density (Eq. 7). The paper justifies this by noting that g* depends linearly on luminosity and mass-to-light ratio, but the disk geometry changes the radial profile of g* and hence the covariance structure of the forward residuals entering sigma_sim. The internal consistency argument only holds if the spherical simplification is demonstrated not to bias the scatter; I recommend running the Monte Carlo for at least one survey with a disky potential, or otherwise quantifying the effect.
[§4.5 and Table 5] The derived intrinsic scatter is highly sensitive to the adopted random stellar mass-to-light ratio uncertainty. Table 5 shows that scaling that uncertainty by a factor of 1.7 (from 0.13 to roughly 0.22 dex) is sufficient to make the median sigma_int zero. The adopted 0.13 dex is a representative upper value from Roediger & Courteau (2015) for optical bands, not a validated value for each heterogeneous PROBES sample, and random M/L scatter estimates in the literature can reach about 0.2 dex. The single-parameter stress test also leaves open the possibility that modest combined underestimates of velocity, distance, and M/L errors remove the signal; a multi-parameter stress test or a prior-weighted marginalization over M/L uncertainty is needed to support the claim that the nonzero scatter is robust.
[§6.2 and §4.3] Noncircular motions are not included in the error model. The beyond-one-Re check gives 0.10 +/- 0.01 dex, but this only removes the inner disk, where noncircular motions are strongest; the outer disk can still host warps, bars, or spiral perturbations that contribute to the observed scatter. A concrete test using the SPARC Q=1 versus Q=2 flags, or a comparison of H-alpha and H I rotation curves, would bound the magnitude of this neglected term. Without such a test, part of sigma_RAR may be misattributed to intrinsic scatter.
[§6.2, Tables 4 and 7] The reported sample-median value conceals strong survey-to-survey variation. SPARC alone gives sigma_int = 0.040 +/- 0.013 dex, and several mass-binned entries in Table 7 are negative (sigma_int approximately -0.02 to -0.05 dex). The surveys with adopted rather than measured uncertainties (M92, M96, and to some extent C97) drive the median upward. The authors should either restrict the headline claim to samples with per-point errors, or justify why the heterogeneous and adopted-error samples can be combined in a single quadrature subtraction.

minor comments (5)

[§3.1.5] The exclusion of the 12 Q=3 SPARC galaxies is mentioned in Sec. 3.1.5, but the general quality-cut description in Sec. 3.2 does not state whether Q=3 objects are removed from the RAR analysis; please make this explicit.
[§4.2] The text says axis ratio uncertainties are computed from isophotal ellipticity variations beyond Re, but the exact estimator (e.g., standard deviation of the mean, median absolute deviation) is not specified; Fig. 2 shows the resulting distributions but not the recipe.
[§5.1] The maximum-likelihood projection in Eq. (10) is a methodological novelty, but the manuscript does not describe how the optimization is initialized or how convergence is verified for galaxies with many points; a brief technical note would improve reproducibility.
[Table 5] The entries reported as '>10' for total luminosity and intrinsic disk flattening have no upper bound; please specify the search procedure or the maximum scaling factor that was tested.
[§1] The claim that the stellar and baryonic RAR scatters are comparable is supported in part by a private communication (A. Dutton 2019); please replace this with a published or otherwise publicly documented analysis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Monte Carlo error model is independent of the observed scatter, and the derived intrinsic scatter is not an artifact of the adopted inputs.

full rationale

The paper's central number is obtained by subtracting the Monte Carlo error-scatter (median sigma_sim ≈ 0.12 dex, Table 3) from the observed scatter (median sigma_RAR ≈ 0.17 dex, Table 1) in quadrature (Sec. 6.2, Table 4). This is not circular: sigma_sim is produced by projecting the data onto the fitted RAR under an explicit zero-intrinsic-scatter null and resampling the pre-specified uncertainty distributions of Sec. 4; no element of that error model is tuned to match the observed 0.17 dex. The paper's own stress test (Table 5) shows that individual uncertainties would have to be inflated by factors of 1.7 (stellar mass-to-light ratio), 2.1 (distance), 3.0 (velocity), 5 (zero-point), 7 (intrinsic flattening), and >10 (luminosity, axis ratio) to erase the inferred scatter, which is evidence that the simulated scatter was not constructed to reproduce the observation. The adopted optical M/L uncertainty of 0.13 dex is a literature input from Roediger & Courteau (2015) (Sec. 4.5); although an author overlaps, it is an external stellar-population result, not an output of this paper's fit, so it does not make the subtraction definitionally circular. The RAR functional form (Eq. 1, McGaugh et al. 2016) is used for fitting and projection, but the scatter metric is a running median (Sec. 3.3), so the scatter is not forced by the fitted function. Sec. 6.2 explicitly acknowledges unmodeled noncircular motions and the possibility of underestimated uncertainties; the stellar/baryonic scatter equivalence is supported partly by L17 and partly by a private communication (A. Dutton 2019). These are limitations and supportability concerns, not reductions of the conclusion to its inputs. The derived 0.11±0.02 dex is therefore not equivalent by construction to the adopted error model.

Assumptions & free parameters 8 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities and no new fitted constant for the target quantity. The free parameters are measurement-error inputs and a per-survey normalization; the central intrinsic scatter is a derived quantity obtained by quadrature subtraction, not a fit to the observed scatter.

free parameters (8)

Per-survey g-dagger normalisation (Eq. 1 fit) = 0.813, 0.839, 0.558, 5.224, 1.790, 1.345 in 10^-10 m s^-2 for SV, SP, SF, C97, M92, M96
Fitted separately per survey for the RAR template used in the zero-scatter projection and mock generation. Bootstrap errors only; systematics could shift values.
Optical stellar mass-to-light ratio random uncertainty = 0.13 dex
Adopted conservative value from Roediger and Courteau (2015); controls the size of simulated scatter and thus sigma_int.
3.6 micron stellar mass-to-light ratio uncertainty = 0.11 dex
Adopted for SPARC from Meidt et al. (2014).
M92/M96 surface brightness uncertainty function parameters = (a,b,c) = (0.00075, 0.52, 31.97)
Fit to the Shellflow distribution and applied as a proxy because M92/M96 report no per-point magnitude errors.
Intrinsic disk flattening q0 by T-type = 0.20 +/- 0.03 (T 1-3), 0.17 +/- 0.03 (T 4), 0.12 +/- 0.02 (T 5-10)
Adopted from previous studies and used in inclination uncertainties and disk scale height.
Hubble flow distance uncertainty = sigma_D = max(0.15 D, 300 km s^-1 converted with H0)
Assumed peculiar velocity dispersion of 300 km s^-1 with a 15 percent distance floor.
Uniform velocity uncertainty for M92/M96 = 6 km s^-1
Assumed as the median uncertainty of the other surveys because M92/M96 do not report per-point velocity errors.
Luminosity uncertainty = 0.04 dex
Adopted from L17 and retained in the first-order and Monte Carlo models.

assumptions (4)

domain assumption The stellar and baryonic RAR have comparable scatter.
Used to compare the stellar RAR result to LCDM predictions made for the baryonic RAR. Justified with L17 and a private communication from Dutton, not independently checkable here.
domain assumption The adopted uncertainty distributions in Section 4 describe the true measurement errors, including Gaussian and truncated models for q, q0, V, L, D and M/L.
The entire quadrature subtraction depends on these distributions; underestimates inflate sigma_int and overestimates can produce negative sigma_int as seen in Table 7.
domain assumption Noncircular motions can be ignored in the scatter model.
Not included in the Monte Carlo error model; the authors test points beyond one Re and find similar sigma_int, but inner-disk noncircular motions remain a potential source of underestimated observational scatter.
domain assumption Mock galaxies can be treated as spherically symmetric for error propagation.
Poisson inversion is only applied to observed galaxies; the mock model uses spherical symmetry, relying on the linear dependence of g* on luminosity and mass-to-light ratio for the uncertainty behavior to be equivalent.

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Cite this review

Pith. "Pith review of The Intrinsic Scatter of the Radial Acceleration Relation." pith.science (2026). https://pith.science/paper/UKHZ6GUN

@misc{pith2026190806105,
  author       = {Pith},
  title        = {Pith review of: The Intrinsic Scatter of the Radial Acceleration Relation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UKHZ6GUN}},
  note         = {Machine review of arXiv:1908.06105}
}

abstract

We present a detailed Monte Carlo model of observational errors in observed galaxy scaling relations to recover the intrinsic (cosmic) scatter driven by galaxy formation and evolution processes. We apply our method to the stellar radial acceleration relation (RAR) which compares the local observed radial acceleration to the local Newtonian radial acceleration computed from the stellar mass distribution. The stellar and baryonic RAR are known to exhibit similar scatter. Lelli+2017 (L17) studied the baryonic RAR using a sample of 153 spiral galaxies and inferred a negligible intrinsic scatter. If true, a small scatter might challenge the LCDM galaxy formation paradigm, possibly favoring a modified Newtonian dynamics interpretation. The intrinsic scatter of the baryonic RAR is predicted by modern LCDM simulations to be ~0.06-0.08 dex, contrasting with the null value reported by L17. We have assembled a catalog of structural properties with over 2500 spiral galaxies from six deep imaging and spectroscopic surveys (called PROBES for the "Photometry and Rotation curve OBservations from Extragalactic Surveys") to quantify the intrinsic scatter of the stellar RAR and other scaling relations. The stellar RAR for our full sample has a median observed scatter of 0.17 dex. We use our Monte Carlo method, which accounts for all major sources of measurement uncertainty, to infer a contribution of 0.12 dex from the observational errors. The intrinsic scatter of the stellar RAR is thus estimated to be 0.11$\pm$0.02 dex, in agreement with, though slightly greater than, current LCDM predictions.

Figures

Figures reproduced from arXiv: 1908.06105 by the authors.

**Figure 3.** Inclination uncertainty model. The q and q0 distributions shown on the x-axis are folded through Eq. 3 (red dashed line) in a Monte Carlo simulation to determine the resulting inclination and its uncertainty distribution. The i distribution is taken from the q distribution shown on the x-axis. All i uncertainty distributions can be determined with vertical slices in the contour plot, which has levels chosen evenl… view at source ↗

**Figure 4.** Rotational velocity uncertainties for the 4 surveys (as in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Magnitude uncertainties for the 3 surveys (as in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: Representation of the variations introduced by each observed variable in the RAR calculation. For each parameter, ±1σ variations are shown as colored bars radiating from a central point laying exactly on the RAR. For reference, the grey distribution in the background …

**Figure 7.** Figure 7: presents the simulated data with the same analysis as in Sec. 3.3. Many qualitative elements from [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

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Works this paper leans on

58 extracted references · 45 canonical work pages

[1]

& Hooper, D

Bertone, G. & Hooper, D. 2018, Rev. Mod. Phys., 90, 90

work page 2018
[2]

2005, PhR, 405, 279

Bertone, G., Hooper, D., & Silk, J. 2005, PhR, 405, 279

work page 2005
[3]

P., Jord´ an, A., Mei, S., et al

Blakeslee, J. P., Jord´ an, A., Mei, S., et al. 2009, ApJ, 694, 556

work page 2009
[4]

D., Aaronson, M., Schommer, B., et al

Bothun, G. D., Aaronson, M., Schommer, B., et al. 1985, ApJS, 57, 423

work page 1985
[5]

B., Santos-Santos, I., & Stinson, G

Brook, C. B., Santos-Santos, I., & Stinson, G. 2016, MNRAS, 459, 459

work page 2016
[6]

& Heiles, C

Burstein, D. & Heiles, C. 1982, AJ, 87, 1165

work page 1982
[7]

& Heiles, C

Burstein, D. & Heiles, C. 1984, ApJS, 54, 33

work page 1984
[8]

2013, ARA&A, 51, 393

Conroy, C. 2013, ARA&A, 51, 393

2013

Show all 58 references

[9]

1996, ApJS, 103, 363

Courteau, S. 1996, ApJS, 103, 363

1996
[10]

1997, AJ, 114, 2402

Courteau, S. 1997, AJ, 114, 2402

1997
[11]

S., et al

Courteau, S., Cappellari, M., de Jong, R. S., et al. 2014, Rev. Mod. Phys., 86, 86

2014
[12]

A., van den Bosch, F

Courteau, S., Dutton, A. A., van den Bosch, F. C., et al. 2007, ApJ, 671, 203

2007
[13]

M., Dressler, A., & Willick, J

Courteau, S., Faber, S. M., Dressler, A., & Willick, J. A. 1993, ApJ, 412, L51

1993
[14]

2000, ApJ, 544, 636 de Blok, W

Postman, M. 2000, ApJ, 544, 636 de Blok, W. J. G., Walter, F., Brinks, E., et al. 2008, AJ, 136, 2648 de Grijs, R. 1998, MNRAS, 299, 595

2000
[15]

A., Conroy, C., van den Bosch, F

Dutton, A. A., Conroy, C., van den Bosch, F. C., et al. 2011, MNRAS, 416, 322

2011
[16]

A., Courteau, S., de Jong, R., & Carignan, C

Dutton, A. A., Courteau, S., de Jong, R., & Carignan, C. 2005, ApJ, 619, 218

2005
[17]

A., Macci` o, A

Dutton, A. A., Macci` o, A. V., Obreja, A., & Buck, T. 2019, MNRAS, 485, 1886

2019
[18]

2007, ApJ, 654, 27

Courteau, S. 2007, ApJ, 654, 27

2007
[19]

2012, MNRAS, 425, 2741

Zhu, Y. 2012, MNRAS, 425, 2741

2012
[20]

Haynes, M. P. & Giovanelli, R. 1984, AJ, 89, 758

1984
[21]

& Ostriker, J

Hernquist, L. & Ostriker, J. P. 1992, ApJ, 386, 375

1992
[22]

Jerjen, H., Binggeli, B., & Barazza, F. D. 2004, AJ, 127, 771

2004
[23]

2001, SciPy: Open source scientiﬁc tools for Python, [Online; accessed July 2018]

Jones, E., Oliphant, T., Peterson, P., et al. 2001, SciPy: Open source scientiﬁc tools for Python, [Online; accessed July 2018]

2001
[24]

Keller, B. W. & Wadsley, J. W. 2017, ApJL, 835, L17

2017
[25]

C., & de Grijs, R

Kregel, M., Van Der Kruit, P. C., & de Grijs, R. 2002, MNRAS, 334, 334

2002
[26]

C., & Freeman, K

Kregel, M., van der Kruit, P. C., & Freeman, K. C. 2005, MNRAS, 358, 503

2005
[27]

1982, ESO/Uppsala survey of the ESO(B) atlas

Lauberts, A. 1982, ESO/Uppsala survey of the ESO(B) atlas

1982
[28]

1998, VizieR Online Data Catalog, VII/34C

Lauberts, A. 1998, VizieR Online Data Catalog, VII/34C

1998
[29]

S., & Schombert, J

Lelli, F., McGaugh, S. S., & Schombert, J. M. 2016, AJ, 152, 157

2016
[30]

S., Schombert, J

Lelli, F., McGaugh, S. S., Schombert, J. M., Desmond, H., & Katz, H. 2019, MNRAS, 484, 3267

2019
[31]

S., Schombert, J

Lelli, F., McGaugh, S. S., Schombert, J. M., & Pawlowski, M. S. 2017, ApJ, 836, 152

2017
[32]

D., Ben´ ıtez-Llambay, A., Schaller, M., et al

Ludlow, A. D., Ben´ ıtez-Llambay, A., Schaller, M., et al. 2017, PhRvL, 118, 118

2017
[33]

Mathewson, D. S. & Ford, V. L. 1996, ApJS, 107, 97

1996
[34]

S., Ford, V

Mathewson, D. S., Ford, V. L., & Buchhorn, M. 1992, ApJS, 81, 413

1992
[35]

B., & Roediger, J

McDonald, M., Courteau, S., Tully, R. B., & Roediger, J. 2011, MNRAS, 414, 414

2011
[36]

McGaugh, S. S. 2004, ApJ, 609, 652

2004
[37]

McGaugh, S. S. 2012, AJ, 143, 40

2012
[38]

S., Lelli, F., & Schombert, J

McGaugh, S. S., Lelli, F., & Schombert, J. M. 2016, PhRvL, 117, 201101 Scatter of the Radial Acceleration Relation 15

2016
[39]

Blok, W. J. G. 2000, ApJ, 533, L99

2000
[40]

P., Cˆ ot´ e, P., et al

Mei, S., Blakeslee, J. P., Cˆ ot´ e, P., et al. 2007, ApJ, 655, 144

2007
[41]

E., Schinnerer, E., van de Ven, G., et al

Meidt, S. E., Schinnerer, E., van de Ven, G., et al. 2014, ApJ, 788, 788

2014
[42]

1983, ApJ, 270, 365

Milgrom, M. 1983, ApJ, 270, 365

1983
[43]

F., Ben´ ıtez-Llambay, A., Fattahi, A., et al

Navarro, J. F., Ben´ ıtez-Llambay, A., Fattahi, A., et al. 2017, MNRAS, 471, 1841

2017
[44]

1995, VizieR Online Data Catalog, VII/26D

Nilson, P. 1995, VizieR Online Data Catalog, VII/26D

1995
[45]

A., Brinks, E., et al

Oh, S.-H., Hunter, D. A., Brinks, E., et al. 2015, AJ, 149, 180

2015
[46]

A., Marasco, A., Navarro, J

Oman, K. A., Marasco, A., Navarro, J. F., et al. 2019, MNRAS, 482, 821

2019
[47]

A., Navarro, J

Oman, K. A., Navarro, J. F., Fattahi, A., et al. 2015, MNRAS, 452, 3650

2015
[48]

N.-Q., Courteau, S., Holtzman, J

Ouellette, N. N.-Q., Courteau, S., Holtzman, J. A., et al. 2017, ApJ, 843, 74

2017
[49]

Bosma, A., & Peletier, R. F. 2018, MNRAS, 474, 4366

2018
[50]

G., Macri, L

Riess, A. G., Macri, L. M., Hoﬀmann, S. L., et al. 2016, ApJ, 826, 56

2016
[51]

C., Marra, V., del Popolo, A., & Davari, Z

Rodrigues, D. C., Marra, V., del Popolo, A., & Davari, Z. 2018, Nature Astronomy, 2, 668

2018
[52]

Roediger, J. C. & Courteau, S. 2015, MNRAS, 452, 3209

2015
[53]

X., Strauss, M

Santiago, B. X., Strauss, M. A., Lahav, O., et al. 1995, ApJ, 446, 457

1995
[54]

Somerville, R. S. & Dav´ e, R. 2015, ARA&A, 53, 51

2015
[55]

Romanowsky, A. J. 2011, ApJ, 742, 16 van der Kruit, P. C. & Searle, L. 1981, A&A, 95, 105

2011
[56]

Walker, M. A. 1999, MNRAS, 308, 308

1999
[57]

F., & Dor´ e, O

Wheeler, C., Hopkins, P. F., & Dor´ e, O. 2018, ArXiv e-prints, arXiv:1803.01849

2018 arXiv
[58]

H., & Weisz, D

Zhang, H.-X., Puzia, T. H., & Weisz, D. R. 2017, ApJS, 233, 13

2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

& Hooper, D

Bertone, G. & Hooper, D. 2018, Rev. Mod. Phys., 90, 90

work page 2018

[2] [2]

2005, PhR, 405, 279

Bertone, G., Hooper, D., & Silk, J. 2005, PhR, 405, 279

work page 2005

[3] [3]

P., Jord´ an, A., Mei, S., et al

Blakeslee, J. P., Jord´ an, A., Mei, S., et al. 2009, ApJ, 694, 556

work page 2009

[4] [4]

D., Aaronson, M., Schommer, B., et al

Bothun, G. D., Aaronson, M., Schommer, B., et al. 1985, ApJS, 57, 423

work page 1985

[5] [5]

B., Santos-Santos, I., & Stinson, G

Brook, C. B., Santos-Santos, I., & Stinson, G. 2016, MNRAS, 459, 459

work page 2016

[6] [6]

& Heiles, C

Burstein, D. & Heiles, C. 1982, AJ, 87, 1165

work page 1982

[7] [7]

& Heiles, C

Burstein, D. & Heiles, C. 1984, ApJS, 54, 33

work page 1984

[8] [8]

2013, ARA&A, 51, 393

Conroy, C. 2013, ARA&A, 51, 393

2013