Decrease in Milky Way rotation curve revisited

Jozef Kla\v{c}ka; Michal \v{S}turc

REVIEW 5 major objections 7 minor 1 cited by

Decrease in Milky Way rotation curve revisited

T0 review · 5 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper argues that the reported Keplerian decrease in the Milky Way rotation curve is an artifact of applying axisymmetric Jeans equations at radii where the gravitational potential is almost spherical; using the spherical Jeans…

desk verdict A useful acceleration table and a legitimate prompt about vertical gradients, but the central objection rests on a category error: a spherical potential is still axisymmetric, and the sqrt(2) rescale is asserted, not derived. read the letter →

arxiv 2507.23551 v1 pith:4KKNOXZS submitted 2025-07-31 astro-ph.GA

Jozef Klačka , Michal Šturc This is my paper · ORCID

classification astro-ph.GA

keywords MilkyWayrotationcurveJeansequationssphericalsymmetrydarkmatterhalogalacticdynamicsKepleriandeclinewarp

topics Dark Matter

open problems Dark Matter

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

Recent determinations of the Milky Way's rotation curve from stellar kinematics report a Keplerian decline at large distances from the Galactic center, implying less dark matter than expected. This paper argues that those results are artifacts of data analysis: the studies use Jeans equations derived for an axisymmetric disk, while their own best-fit gravitational potentials are almost spherically symmetric beyond about 20 kpc, where the dark-matter halo contributes more than 78% of the radial acceleration. Replacing the axisymmetric equations with the spherical Jeans equation multiplies the inferred circular velocity by roughly $\sqrt{2}$, turning $v_c(27.3\,{\rm kpc}) = (173.0 \pm 17.1)\,{\rm km\,s^{-1}}$ into about $(244.7 \pm 24.2)\,{\rm km\,s^{-1}}$. That value is consistent with a flat rotation curve, in line with other spiral galaxies. If correct, the claimed decrease of the Milky Way rotation curve and the associated low dark-matter content would be consequences of using the wrong symmetry assumptions.

What carries the argument

The load-bearing identity is Eq. (1), $[v_c(R)]_{\rm spherical} \simeq \sqrt{2}\,[v_c(R)]_{\rm axisymmetric}$, obtained by comparing Binney and Tremaine's Jeans equations for spherical systems (their Eq. 4.214) with those for axisymmetric systems (their Eqs. 4.222 and 4.226, as used in Ou et al. 2024) under the approximation $\langle v_\theta^2\rangle \simeq \langle v_\phi^2\rangle$. The paper also supplies a table of radial accelerations from the Ou et al. best-fit model showing that spherical components (central bulge plus dark-matter halo) supply 78 to 81 percent of the radial acceleration between 19.7 and 27.3 kpc, and it cites warp amplitudes of 0.3 to 1.5 kpc as evidence that cylindrical symmetry is not a good description of the outer disk.

What would settle it

Re-derive the rotation curve from the same stellar sample used by Ou et al. (2024) using the spherical Jeans equation with measured velocity dispersions and an assumed anisotropy; a best-fit value near $173\,{\rm km\,s^{-1}}$ at 27.3 kpc rather than near $245\,{\rm km\,s^{-1}}$ would refute the paper's central correction.

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Extended reading notes

Core claim

The paper's central claim is that the decreasing Milky Way rotation curves published by Ou et al. (2024), Jiao et al. (2023), and Sylos Labini et al. (2023) are not physical but procedural. For Galactocentric radii above about 20 kpc, the best-fitting gravitational potentials in those papers are dominated by spherically distributed dark matter, so the use of cylindrical Jeans equations is inconsistent with the model. Comparing the Jeans equation for spherical systems with the axisymmetric form used in the earlier papers, and assuming the two tangential velocity-dispersion components are equal, gives $v_c^{\rm spherical} \approx \sqrt{2}\, v_c^{\rm axisymmetric}$. Applying this correction to the newest published value yields $v_c(27.3\,{\rm kpc}) \approx (244.7 \pm 24.2)\,{\rm km\,s^{-1}}$, consistent with a flat rotation curve and with other spiral galaxies; the Milky Way warp adds further reason that the axisymmetric disk equations are not valid at these radii.

Load-bearing premise

The central claim depends on the approximation that beyond about 20 kpc the circular velocity inferred from the axisymmetric Jeans equation should be multiplied by $\sqrt{2}$, a step whose conditions on tracer density and the direction-dependence of stellar velocities are not derived.

Editorial extensions

If this is right

The circular velocity of the Milky Way at $R = 27.3\,{\rm kpc}$ becomes roughly $\sqrt{2} \times 173 \approx 245\,{\rm km\,s^{-1}}$, instead of the published $173\,{\rm km\,s^{-1}}$.
The Milky Way's rotation curve stays approximately flat to large radii, removing the claimed inconsistency between the Milky Way and other spiral galaxies.
The previously inferred Keplerian decline and the correspondingly low dark-matter content of the Milky Way would be artifacts of applying axisymmetric Jeans equations at radii where the potential is nearly spherical.
Future derivations of the outer rotation curve should use spherical Jeans equations and include the vertical velocity component and the disk warp.

Reading between the lines

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

If the $\sqrt{2}$ relation generalizes, other Jeans-equation rotation-curve determinations that assume cylindrical symmetry in dark-matter-dominated regions may systematically underestimate the outer circular velocity.
A direct test would be to refit the same stellar sample used by Ou et al. (2024) with a spherical Jeans model that includes measured velocity anisotropy; a best-fit value near $245\,{\rm km\,s^{-1}}$ at 27.3 kpc would confirm the paper's correction, while a value near 173 km/s would refute it.
The warp argument implies that vertical motions of tracer stars far from the plane should be included as explicit terms in Jeans analyses; ignoring them conflates vertical structure with a declining rotation curve.

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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

Klačka and Šturc's comment has one genuinely useful piece: Table 1, which decomposes the radial acceleration in Ou et al.'s best-fit model into spherical and cylindrical components and shows that the dark halo contributes about 80% at R>20 kpc. That's a fair prompt to ask whether the Jeans analyses in Ou et al., Jiao et al., and Sylos Labini et al. are missing vertical-gradient or anisotropy terms. But the paper's central argument, that this dominance makes the axisymmetric Jeans equations invalid, is a misunderstanding. The cylindrical Jeans equations are derived for a steady-state axisymmetric potential; a spherical potential is a special case. Nothing in the equations breaks when the halo dominates. The sqrt(2) rescaling in Eq. (1) is asserted, not derived. To get from the spherical to the cylindrical Jeans equation you need to specify the tracer density slope, the radial derivative of ν, the cross term , and the anisotropy β. Setting = alone doesn't equate the two equations. So the headline "flat" value vc(27.3 kpc) ≈ 245 km/s is just √2 times the Ou et al. value, not an independent prediction. The conclusion in Section 6 that the decline is "caused by incorrect data analysis" is an overclaim; there is no re-analysis of the data. The warp discussion in Section 5 is plausible but not quantitative. The amplitudes are cited, but the connection to an error in v_c(R) is never made. The reference list also has some sloppiness (the "Wenger et al. 2000" dangling line), but that's minor. So who is this for? Someone working on the Milky Way rotation curve might want to check Table 1 and think about whether the published Jeans solutions are robust to vertical gradients and anisotropy. That's a useful question. But the paper as written doesn't settle it; it mistakenly thinks spherical dominance is a disqualifier. I'd send it to a referee who knows kinetic theory, but the referee's report should make clear that the central objection fails. The authors could revise into a constructive comment if they quantify the warp/vertical-gradient effect and drop the sqrt(2) claim, but the current version doesn't make the case.

Referee Report

5 major / 7 minor

Summary. The paper argues that the declining Milky Way circular-speed curve reported by Ou et al. (2024), Jiao et al. (2023), and Sylos Labini et al. (2023) is an artifact of using axisymmetric Jeans equations at large Galactocentric radii, where the fitted mass models are dominated by a nearly spherical dark halo. It claims that the published potentials become 'almost spherically symmetric' beyond about 20 kpc, so the spherical Jeans equation should be used, and it asserts the relation v_c^sph ≈ √2 v_c^ax. Applying this to Ou et al.'s v_c(27.3 kpc)=173.0±17.1 km/s gives 244.7±24.2 km/s, which the paper takes as evidence for a flat rotation curve. No new data analysis is presented; the argument is based on model acceleration shares and Jeans-equation comparisons.

Significance. If the central claim were correct, it would resolve an important tension between the Milky Way rotation curve and those of other spirals, with implications for the Milky Way's dark-matter distribution. The paper usefully highlights that the outer Milky Way models are DM-dominated and tabulates the relative radial accelerations. However, the key logical step, that a spherical dark halo invalidates cylindrical Jeans equations, is incorrect, and the numerical result is a rescaling of the criticized fitted value rather than an independent inference. The manuscript therefore does not provide a credible resolution of the tension.

major comments (5)

[Sec. 3.1.3, Sec. 3.2, Table 1] The central premise that a DM-dominated, nearly spherical potential invalidates the cylindrical Jeans equations is incorrect. Binney and Tremaine's axisymmetric Jeans equations (Eqs. 4.222 and 4.226) are derived under the assumption of a steady-state axisymmetric potential, and a spherical potential is a special case of an axisymmetric potential. The paper does not identify any term in these equations that becomes invalid under spherical symmetry; it only notes that the DM halo dominates the radial acceleration. The statement in Sec. 3.2 that the Milky Way 'behaves as a spherically symmetric system' when the spherical component supplies about 80% of the acceleration is an overstatement, and 80% dominance does not justify discarding the remaining 20% of the acceleration in the Jeans equations.
[Sec. 3.3, Eq. (1)] Equation (1) is asserted without a valid derivation. Passing from the spherical Jeans equation (BT08, Eq. 4.214) to the axisymmetric radial equation (BT08, Eq. 4.226) requires specifying the tracer density gradient, the radial derivative of ν<v_R^2>, the cross term <v_R v_z>, and the velocity anisotropy β. The approximation <v_θ^2>=<v_φ^2> is automatic in spherical symmetry and does not equate the two equations; the differential operators and geometric factors differ. The paper provides no intermediate algebra, so the factor √2 in Eq. (1) is unsupported.
[Sec. 4] The application of Eq. (1) in Section 4 is circular in an important sense: the input v_c(R=27.3 kpc)=173.0±17.1 km/s is the output of the very axisymmetric Jeans analysis that the paper argues is invalid. If that analysis is invalid, its fitted value cannot simply be rescaled to obtain the correct spherical result; the Jeans fit would need to be redone with the spherical equation and the original data. The headline value 244.7±24.2 km/s therefore inherits the criticized quantity and is not an independent prediction.
[Sec. 5] The warp argument is not quantitative. The cited warp amplitudes of roughly 0.3–1.5 kpc and the statement that stars at z=0 experience a vertical acceleration do not demonstrate that the cylindrical Jeans equations fail beyond about 20 kpc. The paper does not estimate the magnitude of the neglected or mis-modeled terms, including the omitted v_z term in Ou et al.'s Eq. (8), or connect the warp amplitude to a specific error in the published v_c values. Without such an estimate, the warp discussion cannot support the paper's conclusion.
[Sec. 6] The conclusion that the decline in the rotation curve is 'caused by incorrect data analysis' overreaches the evidence. The paper performs no data reanalysis, no new mass-model fit, and no error budget for the claimed correction. Its only numerical result is a single rescaled point at 27.3 kpc, which cannot by itself establish a flat rotation curve over the outer Galaxy. The conclusion is therefore not supported by the material presented.

minor comments (7)

[Abstract] The abstract contains the typo 'axisymemtric'; it should read 'axisymmetric'.
[Sec. 4] The text 'ne can find' should read 'one can find'.
[Sec. 3.1.2] The expression 'also vc(R = 27.3 kpc) 230 kms−1' is missing an equals sign or an approximation symbol between the two quantities.
[Sec. 5] The reference 'Wenger et al. (2000)' is a dangling citation at the end of the section; it is not connected to any sentence. The formatting 'zwar p' should be 'z_warp'.
[References] In the reference list, the journal name for Jiao et al. (2023) appears as 'å' and should be 'A&A'.
[Sec. 3.3] The claim that Eq. (8) of Ou et al. does not consider v_z and that |v_z| may exceed 100 km/s is not verifiable from the manuscript because Eq. (8) is not reproduced and no page or figure number is given for the data cut-off; the statement also lacks a quantitative connection to the derived circular velocity.
[Sec. 3.1.1, Sec. 3.2] The uncertainty on the ratio 0.12±0.02 is not derived from the quoted uncertainties in Ou et al., and the 20% threshold used in Sec. 3.2 to declare the system 'spherically symmetric' is arbitrary and not justified.

Circularity Check

1 steps flagged · score 6.0 of 10

Headline flat-curve value is the criticized Ou et al. fit times √2, not an independent inference.

fitted input called prediction [Section 4, Application]
"If the Jeans equation for spherical systems is used, then the approximation presented in Eq. (1) can be used and it yields vc(R = 27.3 kpc) ≈ √2(173.0 ± 17.1) km/s ≈ (244.7 ± 24.2) km/s. This value is consistent with a flat rotation curve of the MW galaxy."

The paper's headline 'spherical' value, 244.7 km/s, is obtained by taking Ou et al.'s fitted axisymmetric value 173.0 km/s—which the paper rejects as the product of incorrect data analysis—and multiplying by √2 via Eq. (1). No reanalysis of the data, no new fit, and no independent potential estimate are presented. The resulting 'prediction' inherits all data, assumptions, and fitting of the criticized analysis, so it is the same fitted quantity rescaled by a constant. The flat-curve conclusion is thus not an independent inference but a direct transformation of the input it is supposed to replace.

full rationale

The paper's central numeric claim—that the Milky Way rotation curve is flat at vc(27.3 kpc) ≈ 245 km/s—reduces by construction to the criticized Ou et al. value: Section 4 computes it as √2 × (173.0 ± 17.1) km/s. This is a textbook case of fitted input called prediction: the paper does not reanalyze the data or produce an independent estimate; it merely rescales the very quantity it argues is wrong. The rescaling factor in Eq. (1) is asserted by comparing Jeans equations with an unspecified approximation (⟨vθ²⟩ = ⟨vφ²⟩), and no derivation of the √2 relation is provided; even if it were correct, the input to the rescaling is still the criticized fitted value. Additionally, the paper's argument that a spherical potential invalidates axisymmetric Jeans equations is problematic—spherical potentials are a subset of axisymmetric potentials—but that is a correctness or logical issue, not a circularity. The paper does cite external flat rotation curves (Lelli et al. 2016; Mistele et al. 2024) as independent support for flatness generally, but the specific Milky Way value is not derived independently. Hence a partial circularity score of 6 is warranted: the headline prediction reduces to the criticized fit by construction.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical entities. Its results rest on borrowed best-fit model parameters and on two dynamical assumptions: that a spherical potential invalidates cylindrical Jeans modeling, and that an isotropic spherical estimator can be obtained by a sqrt(2) rescaling. Neither is independently established.

free parameters (1)

Ou et al. (2024) best-fit mass model parameters = Not reproduced here; taken from Ou et al. (2024) Tables 2 and 3
The central acceleration comparison in Table 1 and the conclusion about spherical dominance use this externally fitted mass model as input. If those parameters change, the percentage shares change.

assumptions (3)

standard math Jeans equations from Binney and Tremaine (2008), Eqs. 4.214 and 4.226, are the correct dynamical descriptions for the Milky Way tracer populations.
Invoked in Sections 3.1 to 3.3 as the basis for deriving vc and comparing spherical versus cylindrical estimators.
domain assumption The gravitational potential at R > 20 kpc is dominated by spherically distributed dark matter, making the spherical Jeans equation the appropriate estimator.
This is the core physical premise used in Sections 3.2 and 3.3; it is inferred from Ou et al.'s model, not measured independently.
ad hoc to paper The approximation of equal velocity dispersions in the spherical Jeans equation yields Eq. (1), vc_spherical approximately sqrt(2) times vc_axisymmetric.
Introduced in Section 3.3 specifically to derive the flat rotation curve; no justification of isotropy or tracer density profile is given.

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Cite this review

Pith. "Pith review of Decrease in Milky Way rotation curve revisited." pith.science (2026). https://pith.science/paper/4KKNOXZS

@misc{pith2026250723551,
  author       = {Pith},
  title        = {Pith review of: Decrease in Milky Way rotation curve revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4KKNOXZS}},
  note         = {Machine review of arXiv:2507.23551}
}

read the original abstract

Context. Latest papers on the rotation curve of the Milky Way galaxy, i. e. Ou et al. (2024); Jiao et al. (2023); Sylos Labini et al. (2023) suggest a Keplerian decrease in the rotation curve. This behavior is not consistent with other spiral galaxies (Lelli et al. 2016; Mistele et al. 2024). Aims. Show that the prior use of the axisymmetric Jeans equation is not consistent with the final model produced in the papers. Methods. Comparison of the results on gravitational potential in Ou et al. (2024); Jiao et al. (2023); Sylos Labini et al. (2023) with the prior assumptions about the axisymemtric properties of the Milky Way galaxy. Results. The gravitational potentials published by Ou et al. (2024); Jiao et al. (2023); Sylos Labini et al. (2023) lead to almost spherically symmetric properties of the Milky Way galaxy at Galactocentric radii above 20 kpc, which is not consistent with the use of axisymmetric Jeans equations.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The Rotation Curve of the Milky Way: State of the Art, the Keplerian Decline Debate, and Implications for Dark Matter
astro-ph.GA 2026-08 accept novelty 3.0 of 10

This pedagogical review concludes that the Milky Way's claimed Keplerian rotation-curve decline is plausible but systematically challenged by Jeans-model biases and independent mass constraints.

Reference graph

Works this paper leans on

10 extracted references · 2 canonical work pages · cited by 1 Pith paper

[1]

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Binney , J., & Tremaine , S. 2008, Galactic Dynamics: Second Edition (Princeton University Press, Princeton)

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

2008, Galactic Dynamics: Second Edition (Princeton University Press, Princeton)

Binney , J., & Tremaine , S. 2008, Galactic Dynamics: Second Edition (Princeton University Press, Princeton)

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work page doi:10.1051/0004-6361/202243296 2022

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