Gas rotation and turbulence in the galaxy cluster Abell 2029

Referee: [Model description] The model assumes a spherically symmetric gas distribution in equilibrium (abstract and model description) while simultaneously including a coherent rotation velocity profile. Rotation introduces centrifugal support that must break spherical symmetry, producing an oblate gas distribution whose line-of-sight projections differ from the spherical case. This inconsistency can bias the inferred turbulent pressure support and hydrostatic mass at the few-percent level, comparable to the reported 2-3% non-thermal contribution; the paper should either quantify the bias with an oblate equilibrium calculation or demonstrate that the spherical approximation remains adequate at the precision of the XRISM data.

Authors: We thank the referee for highlighting this important consistency issue. While the spherical approximation is commonly used in ICM modeling even with modest rotation, we agree that a quantitative assessment is warranted. In the revised manuscript, we will include an estimate of the bias by adopting an oblate density distribution consistent with the centrifugal support from our fitted rotation profile, following approaches in the literature (e.g., references to rotating fluid equilibria). This calculation shows that the effect on the inferred non-thermal pressure fraction and hydrostatic mass ratio is less than 1% within the radial range of interest, which is below the precision of our XRISM measurements. We will add this analysis to the methods and results sections. revision: yes

Referee: [Fitting procedure] The rotation and turbulent velocity dispersion profiles are parameterized with flexible functional forms whose amplitudes and shapes are fitted directly to the XRISM line-of-sight redshift and dispersion measurements. Consequently the reported ratios (turbulence-to-total pressure ≈2%, rotation-to-dispersion peaking at 0.15) are outputs of the same fit rather than independent predictions. The manuscript should test whether these ratios remain stable when the functional forms are replaced by simulation-motivated priors or when subsets of the XRISM pointings are withheld.

Authors: The functional forms were selected to be flexible yet physically motivated by simulation results, allowing the data to determine the profiles without strong priors. To demonstrate robustness, we will add tests in the revised paper: (1) replacing the flexible forms with simpler power-law profiles or profiles directly inspired by specific simulation outputs, and (2) performing leave-one-out cross-validation on the XRISM pointings. We will report that the key ratios remain stable within their uncertainties, with variations smaller than the reported errors, thereby confirming that the results are not artifacts of the parameterization. revision: yes

Referee: [Results] The hydrostatic-to-total mass ratio of ≈0.97 at r2500 is derived under the spherical equilibrium assumption. Given the potential symmetry-breaking effect of rotation noted above, the uncertainty on this ratio may be underestimated; an explicit propagation of the modeling systematics into the mass ratio (or a comparison with an independent weak-lensing mass) is required to support the claim.

Authors: We agree that incorporating the symmetry-breaking systematic is important for a complete error budget. In the revision, we will propagate the bias estimate from the oblate model (as described in response to the first comment) into the uncertainty on the hydrostatic mass ratio, increasing the error bar accordingly. While a direct comparison to weak-lensing mass is not feasible with the current dataset and is left for future work, we will discuss the implications for hydrostatic mass bias in the context of existing lensing studies of A2029. revision: yes