{"id":"7d7f8dfc-aeb4-4ce5-b95e-fef02c65c501","arxiv_id":"2507.23551","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"According to this comment, using spherical rather than cylindrical Jeans equations removes the reported Keplerian decline and restores a flat Milky Way rotation curve.","lead":"A short comment argues that the recently reported Keplerian drop in the Milky Way's outer rotation curve is an artifact of using cylindrical Jeans equations in a regime where a spherical dark matter halo dominates. A smart generalist might read it because it challenges a high-profile dark matter result in our own galaxy.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Spherical potential is still axisymmetric, so the cylindrical Jeans equations remain valid; Eq. (1) rescaling is not derived, leaving the flat-curve conclusion unsupported.","rationale":"The reader's weakest_assumption identifies essentially the same load-bearing flaw: the axisymmetric Jeans equations remain valid for any axisymmetric potential, and a spherical potential is axisymmetric; the √2 factor in Eq. (1) is asserted without the necessary conditions on density slope, anisotropy, and cross terms. I read the paper in good faith: the authors are pointing to a real question of whether Ou, Jiao, and Sylos Labini handled vertical gradients and the pressure terms correctly, and the warp amplitudes they cite could matter. However, the specific argument they make to overturn those results is invalid. The cylindrical Jeans equations are derived from the collisionless Boltzmann equation with the assumption of axisymmetry; they do not require that the mass distribution be axisymmetric, only that the potential be axisymmetric. A spherical potential satisfies this condition automatically. The paper's Table 1 shows that the spherical components dominate the radial acceleration at large R, but that does not make the cylindrical equations 'inconsistent'—the equations are identities relating moments of the distribution function to the potential. To claim inconsistency, the authors would need to show that the moments used by Ou et al. cannot satisfy the cylindrical equations for a spherical potential, which they do not. The proposed concrete test settles the issue by direct construction: if a spherical potential plus the cylindrical Jeans equation yields the true v_c, then the paper's central causal story collapses. The paper provides no code, data, or independent re-analysis; its numerical 'flat' prediction is a rescaling of the very values it criticizes. Therefore the reader's REJECT verdict is appropriate, and I see no reason to change it. I do not dispute that the paper is useful as a prompt for checking whether previous analyses properly included vertical terms and anisotropy, but as a scientific argument against the Keplerian decline, it fails on its central step.","tokens_in":4365,"tokens_out":4406,"duration_ms":47171,"concrete_test":"Take a spherical potential, e.g., an NFW or Hernquist halo with a known circular velocity v_c(r)=sqrt(r dΦ/dr), and a spherical tracer density ν(r) with a power-law slope or a constant-β distribution. Solve the cylindrical Jeans radial equation, Binney & Tremaine Eq. 4.222, using the same terms as Ou et al. (2024, Eqs. 7–8): include ∂(ν<v_R^2>)/∂R and ν(<v_R^2>-<v_φ^2>)/R, set <v_R v_z>=0 for simplicity, and infer v_c(R)=sqrt(R*(-dΦ/dR)). If the axisymmetric equation recovers the true v_c(R) to numerical precision (rather than v_c/√2), then Eq. (1) is refuted. A clean analytic version: choose ν(r)=ν_0 r^{-α} and isotropic or constant-β dispersion, compute every term explicitly, and show the inferred v_c matches the input potential. This directly tests the paper's central claim that spherical symmetry forces a √2 discrepancy and that the cylindrical Jeans equations are inapplicable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central logical step, in Sections 3.1.3 and 3.3, is that a potential dominated by a spherical dark halo becomes 'almost spherically symmetric' and therefore the use of cylindrical (axisymmetric) Jeans equations is 'not justified'. This is incorrect: the cylindrical Jeans equations (Binney & Tremaine 2008, Eqs. 4.222 and 4.226) are derived under the assumption of a steady-state, axisymmetric potential, not an axisymmetric mass distribution. A spherical potential is a special case of an axisymmetric one, so these equations remain valid even when the DM halo dominates. The paper does not identify any term in the axisymmetric equations that becomes invalid under spherical symmetry; it merely notes that the halo dominates the acceleration. The claimed rescaling in Eq. (1), v_c(spherical) ≈ √2 v_c(axisymmetric), is asserted without a real derivation. Matching Eq. 4.214 with Eq. 4.226 would require specifying the tracer density slope, the radial derivative of ν<v_R^2>, the cross term <v_R v_z>, and the velocity anisotropy β. Setting <v_θ^2>=<v_φ^2> alone does not equate the two equations; the differential operators and the geometric factors differ. The paper's subsequent 'flat' value 244.7±24.2 km/s is therefore simply √2 times the criticized Ou et al. value, not an independent inference. Consequently, the conclusion in Section 6 that 'the decrease of the rotation curve is caused by incorrect data analysis' is unsupported: no data reanalysis is presented, and the only new element, the warp discussion in Section 5, is not quantitatively connected to an error in the published v_c(R).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":4583,"tokens_out":25685,"duration_ms":231293,"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":[{"comment":"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.","section":"Sec. 3.1.3, Sec. 3.2, Table 1"},{"comment":"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.","section":"Sec. 3.3, Eq. (1)"},{"comment":"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.","section":"Sec. 4"},{"comment":"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.","section":"Sec. 5"},{"comment":"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.","section":"Sec. 6"}],"minor_comments":[{"comment":"The abstract contains the typo 'axisymemtric'; it should read 'axisymmetric'.","section":"Abstract"},{"comment":"The text 'ne can find' should read 'one can find'.","section":"Sec. 4"},{"comment":"The expression 'also vc(R = 27.3 kpc)  230 kms−1' is missing an equals sign or an approximation symbol between the two quantities.","section":"Sec. 3.1.2"},{"comment":"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'.","section":"Sec. 5"},{"comment":"In the reference list, the journal name for Jiao et al. (2023) appears as 'å' and should be 'A&A'.","section":"References"},{"comment":"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.","section":"Sec. 3.3"},{"comment":"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.","section":"Sec. 3.1.1, Sec. 3.2"}],"recommendation":"reject","confidential_remarks":"The manuscript is very short and its central claim rests on a misunderstanding of the applicability of Jeans equations. The only quantitative result is a √2 rescaling of the very fitted value the paper criticizes. I do not see a path to a sound paper within the current scope; the authors would need to re-derive Eq. (1) under explicit assumptions, test it on mock data, and ideally reanalyze the original data with the spherical Jeans equation. I recommend reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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<v_R^2>, the cross term <v_R v_z>, and the anisotropy β. Setting <v_θ^2>=<v_φ^2> 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.","headline":"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.","tokens_in":5258,"tokens_out":4618,"would_cite":false,"duration_ms":39102,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Milky Way rotation curve","Jeans equations","spherical symmetry","dark matter halo","galactic dynamics","Keplerian decline","Galactic warp"],"falsifier":"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.","tokens_in":4028,"feed_emoji":"🌌","tokens_out":8458,"duration_ms":76039,"temperature":0.7,"pith_summary":"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.","feed_headline":"Milky Way's outer rotation curve may be flat, not falling","feed_subtitle":"Switching from disk-shaped to spherical Jeans equations turns the claimed Keplerian decline into a flat curve.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Jeans equations for spherical and axisymmetric systems that the paper compares to obtain the $\\sqrt{2}$ relation.","marker":"Binney & Tremaine 2008"},{"why":"Provides the best-fit Milky Way model, the acceleration values, and the $v_c(27.3\\,{\\rm kpc}) = 173 \\pm 17\\,{\\rm km\\,s^{-1}}$ result that the paper reinterprets.","marker":"Ou et al. (2024)"},{"why":"One of the three published rotation-curve analyses whose decreasing outer curve is attributed to incorrect axisymmetric assumptions.","marker":"Jiao et al. (2023)"},{"why":"Another of the decreasing rotation-curve determinations that the paper argues is biased by the wrong Jeans equation.","marker":"Sylos Labini et al. (2023)"},{"why":"Provides the comparison sample of spiral galaxy rotation curves that stay flat and are inconsistent with a Keplerian decline.","marker":"Lelli et al. (2016)"},{"why":"Extends the comparison with outer rotation curves of other spirals, reinforcing that the Milky Way decline would be anomalous.","marker":"Mistele et al. (2024)"},{"why":"Supplies warp amplitudes for young stars that the paper uses to argue the outer disk is not axisymmetric.","marker":"Chrobáková et al. (2022)"},{"why":"Supplies larger warp amplitudes for older stars, strengthening the claim that cylindrical Jeans equations fail beyond 20 kpc.","marker":"Uppal et al. (2024)"}],"fun_headline_variants":["Keplerian decline in Milky Way curve is a Jeans equation artifact","Spherical Jeans correction flattens Milky Way's outer rotation curve","Milky Way's outer rotation curve flat after spherical Jeans fix","Axisymmetric Jeans equation misled Milky Way rotation curve decline","Milky Way rotation curve decline vanishes under spherical model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Keplerian decline in Milky Way curve is a Jeans equation artifact","Spherical Jeans correction flattens Milky Way's outer rotation curve","Milky Way's outer rotation curve flat after spherical Jeans fix","Axisymmetric Jeans equation misled Milky Way rotation curve decline","Milky Way rotation curve decline vanishes under spherical model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00189,"raw_usage":{"total_tokens":7421,"prompt_tokens":969,"completion_tokens":6452,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":6363}},"tokens_in":585,"tokens_out":6452,"duration_ms":46630,"temperature":1.0,"reasoning_tokens":6363,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:36:54.947649+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"2008, Galactic Dynamics: Second Edition (Princeton University Press, Princeton)","cited_arxiv_id":null,"evidence_quote":"Supplies the Jeans equations for spherical and axisymmetric systems that the paper compares to obtain the $\\sqrt{2}$ relation."}],"review_version":1}