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

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 authors: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 10 September 2025
Accepted: 10 November 2025

Abstract

Aims. Mass-modeling methods are used to infer the gravitational field of stellar systems, from globular clusters to giant elliptical galaxies. While many methods already exist, most require assumptions on the form of the underlying distribution function or binning the data, leading to some loss of information. Furthermore, when only line-of-sight (LOS) data are available, many methods suffer from the well-known mass-anisotropy degeneracy. To overcome these limitations, we developed a new and publicly available mass modeling method, GRAVSPHERE2. It 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 velocity anisotropy profiles at second and fourth order, we show how including additional constraints from a new observable, fourth-order PMs, allows us to obtain a full solution along the three dimensions and breaking the mass-anisotropy degeneracy at all orders. We tested our method on mock data for dwarf galaxies, showing how GRAVSPHERE2 improves on previous methods.

Methods. GRAVSPHERE2 introduces four key improvements over previous Jeans mass modeling methods in the literature: (i) we included fourth-order velocity moment equations in both the LOS and PM directions, for the first time, using them to break model degeneracies; (ii) we used a fully general treatment of both the second and fourth-order velocity anisotropies; (iii) we introduced a “bin-free” approach where we fit individual tracer velocities and positions using flexible and self-consistent probability density functions that include kurtosis; and (iv) we improved the likelihood sampling by using the nested sampler DYNESTY.

Results. GRAVSPHERE2 was able to recover the mass density, stellar velocity anisotropy, and the 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/R_1/2 ≲ 10., where R_1/2 is the projected half-light radius). As the number of tracers is lowered (even down to just ten tracers) it gracefully degrades, with larger uncertainties but no induced bias. We find that GRAVSPHERE2 outperforms simple mass estimators, suggesting that it is worth using even when only a few LOS velocities are available. Using 1000 tracers without PMs, GRAVSPHERE2 recovers the logarithmic density slope at R_1/2 with 12%(25%) statistical errors for cuspy (cored) mock data, enabling us to make a distinction between the two. When including PMs, this result can be improved to 8%(12%). With only 100 tracers and no PMs, we were still able to recover slopes with ∼ 30%(20%) errors. GRAVSPHERE2 will become a valuable new tool to hunt for massive black holes and invisible dark matter in spherical stellar systems, from globular clusters and dwarf galaxies to giant ellipticals and galaxy clusters.