GravSphere2: A higher order Jeans method for mass modeling spherical stellar systems (Corrigendum)

Andrés Bañares-Hernández; Justin I. Read; Mariana P. Júlio

doi:10.1051/0004-6361/202660475e

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Erratum

This article is an erratum for:
[https://doi.org/10.1051/0004-6361/202557195]

Issue		A&A Volume 709, May 2026


Article Number		C3
Number of page(s)		1
Section		Galactic structure, stellar clusters and populations
DOI		https://doi.org/10.1051/0004-6361/202660475e
Published online		19 May 2026

A&A, 709, C3 (2026)

GravSphere2: A higher order Jeans method for mass modeling spherical stellar systems (Corrigendum)

Andrés Bañares-Hernández¹^,2^★, Justin I. Read³ and Mariana P. Júlio⁴^,5

¹ Instituto de Astrofísica de Canarias, La Laguna, Tenerife, 38200 Spain
² Departamento de Astrofísica, Universidad de La Laguna, Santa Cruz de Tenerife, Spain
³ Department of Physics, University of Surrey, Guildford, GU2 7XH, UK
⁴ Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
⁵ Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam, Germany

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Key words: globular clusters: general / galaxies: dwarf / galaxies: kinematics and dynamics / Local Group / errata, addenda

Erratum for: A&A, 705, A212 (2026), https://doi.org/10.1051/0004-6361/202557195

The corresponding author found typographical errors in Equations (17), (18), (20), and (23) of the original paper. The correct equations should read, respectively, as

$\frac{\partial ν_{⋆} ⟨ v_{r}^{2} v_{t}^{2} ⟩}{\partial r} = - \frac{2}{3} (1 - β^{'}) [\frac{2 β^{'}}{r} ν_{⋆} ⟨ v_{r}^{4} ⟩ + 3 ν_{⋆} σ_{r}^{2} \frac{G M}{r^{2}}] - \frac{2}{3} \frac{\partial β^{'}}{\partial r} ν_{⋆} ⟨ v_{r}^{4} ⟩,$ $Mathematical equation: $\[\frac{\partial \nu_{\star}\left\langle v_r^2 v_t^2\right\rangle}{\partial r}=-\frac{2}{3}\left(1-\beta^{\prime}\right)\left[\frac{2 \beta^{\prime}}{r} \nu_{\star}\left\langle v_r^4\right\rangle+3 \nu_{\star} \sigma_r^2 \frac{G M}{r^2}\right]-\frac{2}{3} \frac{\partial \beta^{\prime}}{\partial r} \nu_{\star}\left\langle v_r^4\right\rangle,\]$$

$⟨ v_{t}^{4} ⟩ (r) = \frac{4}{3} [(1 - β^{'}) (2 - β^{'}) - \frac{r}{2} \frac{\partial β^{'}}{\partial r}] ⟨ v_{r}^{4} ⟩ + 2 (β^{'} - β) σ_{r}^{2} \frac{G M}{r},$ $Mathematical equation: $\[\left\langle v_t^4\right\rangle(r)=\frac{4}{3}\left[\left(1-\beta^{\prime}\right)\left(2-\beta^{\prime}\right)-\frac{r}{2} \frac{\partial \beta^{\prime}}{\partial r}\right]\left\langle v_r^4\right\rangle+2\left(\beta^{\prime}-\beta\right) \sigma_r^2 \frac{G M}{r},\]$$

$\begin{array}{r} F_{L O S} (r, R) \equiv (1 - 2 β^{'} \frac{R^{2}}{r^{2}} + \frac{1}{2} β^{'} (1 + β^{'}) \frac{R^{4}}{r^{4}} - \frac{1}{4} \frac{\partial β^{'}}{\partial r} \frac{R^{4}}{r^{3}}) ⟨ v_{r}^{4} ⟩ \\ + \frac{3}{4} (β^{'} - β) σ_{r}^{2} G M \frac{R^{4}}{r^{5}}, \end{array}$ $Mathematical equation: $\[\begin{array}{r}F_{\mathrm{LOS}}(r, R) \equiv\left(1-2 \beta^{\prime} \frac{R^2}{r^2}+\frac{1}{2} \beta^{\prime}\left(1+\beta^{\prime}\right) \frac{R^4}{r^4}-\frac{1}{4} \frac{\partial \beta^{\prime}}{\partial r} \frac{R^4}{r^3}\right)\left\langle v_r^4\right\rangle \\+\frac{3}{4}\left(\beta^{\prime}-\beta\right) \sigma_r^2 G M \frac{R^4}{r^5},\end{array}\]$$

and

$\begin{array}{r} F_{P M, t} (r, R) \equiv \frac{1}{2} [(1 - β^{'}) (2 - β^{'}) - \frac{r}{2} \frac{\partial β^{'}}{\partial r}] ⟨ v_{r}^{4} ⟩ \\ + \frac{3}{4} (β^{'} - β) σ_{r}^{2} \frac{G M}{r} . \end{array}$ $Mathematical equation: $\[\begin{array}{r}F_{\mathrm{PM}, \mathrm{t}}(r, R) \equiv \frac{1}{2}\left[\left(1-\beta^{\prime}\right)\left(2-\beta^{\prime}\right)-\frac{r}{2} \frac{\partial \beta^{\prime}}{\partial r}\right]\left\langle v_r^4\right\rangle \\+\frac{3}{4}\left(\beta^{\prime}-\beta\right) \sigma_r^2 \frac{G M}{r}.\end{array}\]$$

This did not affect our results and calculations, as it was only an issue that occurred during the transcription process of our original derivations. Additionally, we found and fixed a bug in the original GRAVSPHERE2 code (the GitHub repository has now been updated) in the computation of fourth-order proper motions. However, we verified that this had a negligible impact on our results, including changes in the likelihood and in the robust data recovery in the extensive mock data sample, with which we tested our modeling pipeline.

Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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