GravSphere2: A higher-order Jeans method for mass-modeling spherical stellar systems

TL;DR

GravSphere2 addresses the mass–anisotropy degeneracy in spherical systems by solving the Jeans equations up to fourth order using a bin-free approach that fits individual tracer velocities and positions. It introduces a non-Gaussian velocity PDF and includes fourth-order proper motions, with a flexible $\beta'$ anisotropy and a coreNFWtides mass model, inferred via nested sampling. The method recovers the mass density, anisotropy, and logarithmic slope to ~95% CL across diverse mocks, and shows reduced biases and tighter constraints even at low tracer numbers, especially when PMs are available. This yields a practical, bias-resilient tool for probing dark matter, black holes, and the dynamical structure of globular clusters, dwarfs, and giant ellipticals. GravSphere2 outperforms simple estimators and existing Jeans approaches on mocks, highlighting its potential for upcoming Gaia-era datasets and cross-scale astrophysical applications.

Abstract

Mass-modeling methods are used to infer the gravitational field of stellar systems, from globular clusters to giant elliptical galaxies. While many methods exist, most require assumptions about the form of the underlying distribution function or data binning that leads to loss of information. With only line-of-sight (LOS) data, many methods suffer from the well-known mass-anisotropy degeneracy. To overcome these limitations, we develop a new, publicly available mass-modeling method, GravSphere2. This combines individual stellar velocities from LOS and proper motion (PM) measurements to solve the Jeans equations up to fourth order, without any data binning. Using flexible functional forms for the anisotropy profiles at second and fourth order, we show how including additional constraints from a new observable - fourth-order PMs - fully closes the system of equations, breaking the mass-anisotropy degeneracy at all orders. We test our method on mock data for dwarf galaxies, showing how GravSphere2 improves on previous methods. GravSphere2 recovers the mass density, stellar velocity anisotropy, and logarithmic slope of the mass density profile within its quoted 95% confidence intervals across almost all mocks over a wide radial range (0.1 < R/Rhalf < 10). We find GravSphere2 outperforms simple mass estimators, suggesting that it is worth using even when only a few LOS velocities are available. With 1,000 tracers without PMs, GravSphere2 recovers the logarithmic density slope at Rhalf with 12% (25%) statistical errors for cuspy (cored) mock data, enabling a distinction between the two. Including PMs, this improves to 8% (12%). With just 100 tracers and no PMs, we recover slopes with ~ 30% (20%) errors. GravSphere2 will be a valuable new tool to hunt for black holes and dark matter in spherical stellar systems, from globular clusters and dwarf galaxies to giant ellipticals and galaxy clusters.

Figures (14)

• Figure 1: Upper left: 3D dark matter density profile for the PlumCoreOM mock galaxy, bands denote the 95% CL regions from the median for different numbers of stellar tracers including both LOS and PM velocities. The results from 2021MNRAS.501..978R for Agama$f(E, L)$ models, and from 2021MNRAS.505.5686C for GravSphere1.5 (including LOS VSPs) for 1,000 tracers with their respective median values with 95% CL errors are shown for comparison. The red line denotes the true analytical solution from which the mock was generated. The gray, dashed line corresponds to the projected half-light radius ($R_{1/2})$. Upper right: Same as the left hand side but for the PlumCuspOM model. Lower left: Symmetrized anisotropy profile for the PlumCoreOM model. The red line again denotes the true solution from which the mock was generated, corresponding to an Osipkov-Merritt profile. Lower right: Same as the left hand side but for the PlumCuspOM model.
• Figure 2: Same as Fig. \ref{['fig:rhoanipm']} but only with LOS tracers. Upper left:GravSphere2 recovery for the PlumCoreOM mock 3D mass density. Upper right: Same as the left but for the PlumCuspOM model. Lower left: Symmetrized anisotropy profile for the PlumCoreOM model. Lower right: Same as the left but for the PlumCuspOM model.
• Figure 3: Same as Fig. \ref{['fig:rhoanilos']} but for 10,000 tracers in the PlumCoreOM mock, including in the comparison with other models. We have included the DiscreteJAM results from 2021MNRAS.501..978R. Left: Mass density profiles. Blue band shows our results when Gaussianity is assumed and fourth-order moments are excluded. For comparison, the inset shows instead bands obtained from the posterior distribution of likelihoods when computing extrema of all models falling within the 68th percentile from the maximum likelihood (see text for details). Using this definition, there is a wide range of equally good-fitting models that encompasses the true solution, while the posterior becomes biased due to a relative abundance of cuspier models in the allowable hypervolume. Right: Same as left, but for symmetrized anisotropy profile.
• Figure 4: GravSphere2 recovery of the logarithmic density slope profile for the Gaia Challenge mock galaxies. Upper panels include LOS and PM tracers, whilst lower panels include only LOS ones. The left panels correspond to the PlumCoreOM case, whilst the right ones to the PlumCuspOM case. The bands, errors and results presented follow the same notation as Fig. \ref{['fig:rhoanipm']}.
• Figure 5: Upper left:GravSphere2 recovery of the 3D dark matter density profile for the Fornax-like simulated galaxy, where the bands denote the 95% CL regions for different numbers of stellar tracers, as marked, including only LOS velocities. The total sample of bound tracers is 34,158 (gray band). The purple points correspond to the median values with 95% CL errors from 2025arXiv250418617T using all LOS tracers with GravSphere1. The red, solid line corresponds to the true density profile from the simulation (smoothened with a Savitzky-Golay filter). The gray, dashed line corresponds to the projected half-light radius ($R_{1/2}$). Upper right: Symmetrized anisotropy profile results. The red points denote the binned 3D anisotropies obtained from bins of equal numbers of tracers and directly fitting the generalized velocity PDF from Eqs. \ref{['eq:unik']} and \ref{['eq:lapk']} along the three spherical-coordinate directions. Lower left: Same as Upper Left but including both LOS and PM velocities. Lower right: Same as Upper Right but with both LOS and PM velocities.