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arxiv: 2509.24103 · v3 · submitted 2025-09-28 · 🌌 astro-ph.GA

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

Andr\'es Ba\~nares-Hern\'andez , Justin I. Read , Mariana P. J\'ulio This is my paper

Pith reviewed 2026-05-18 12:34 UTC · model grok-4.3

classification 🌌 astro-ph.GA
keywords Jeans modelingmass modelingstellar kinematicsproper motionsdwarf galaxiesdark mattervelocity anisotropyglobular clusters
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The pith

GravSphere2 solves the Jeans equations to fourth order using unbinned stellar velocities and proper motions to recover mass density profiles while breaking the mass-anisotropy degeneracy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops GravSphere2, a mass-modeling technique that takes individual line-of-sight velocities and proper motions from stars in spherical systems and solves the Jeans equations up to fourth order without binning the data. Flexible functional forms are introduced for the second- and fourth-order velocity anisotropy profiles, and fourth-order proper motion measurements supply the additional constraints needed to close the system of equations. Tests on mock dwarf-galaxy data show that the method recovers the mass density, the anisotropy, and the logarithmic slope of the density profile inside its quoted 95 percent confidence intervals over a wide radial range. Even with only a few hundred tracers and no proper motions the recovered slopes remain accurate enough to distinguish cuspy from cored profiles, and the inclusion of proper motions tightens the errors further. The authors conclude that the approach will be useful for searching for black holes and dark matter in globular clusters, dwarf galaxies, and larger elliptical systems.

Core claim

GravSphere2 combines unbinned line-of-sight and proper-motion velocities to solve the Jeans equations to fourth order. Flexible functional forms for the second- and fourth-order anisotropy profiles are adopted, and fourth-order proper-motion constraints close the system, eliminating the mass-anisotropy degeneracy at all orders. On mock data the method recovers the mass density, stellar velocity anisotropy, and logarithmic slope of the mass density profile within its quoted 95 percent confidence intervals across almost all mocks over 0.1 < R/Rhalf < 10.

What carries the argument

Flexible functional forms for the second- and fourth-order anisotropy profiles together with fourth-order proper-motion constraints that close the Jeans system without data binning.

If this is right

  • With 1,000 tracers and no proper motions the logarithmic density slope at the half-light radius is recovered to 12 percent (25 percent) statistical error for cuspy (cored) mocks.
  • Adding proper motions improves the slope errors to 8 percent (12 percent) for the same number of tracers.
  • Even with only 100 tracers and no proper motions the slope is recovered to roughly 30 percent (20 percent) error.
  • The method outperforms simple mass estimators and is therefore worth applying even when proper-motion data are unavailable.
  • The recovered quantities remain reliable over nearly the full radial range 0.1 < R/Rhalf < 10 in almost all mocks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same fourth-order closure could be tested on real globular-cluster data to place tighter limits on intermediate-mass black holes than are possible with second-order Jeans modeling alone.
  • Because the method works with unbinned data it can be applied directly to sparse samples from future wide-field surveys without first grouping stars into radial bins.
  • Extending the same logic to non-spherical systems would require replacing the spherical Jeans equations with the axisymmetric or triaxial versions while retaining the higher-order proper-motion constraints.
  • The recovered density slopes at the half-light radius could be compared directly with independent estimates from strong lensing or X-ray gas in massive ellipticals to test consistency across methods.

Load-bearing premise

That the chosen flexible functional forms for the anisotropy profiles introduce no significant bias when they are used with fourth-order proper-motion data to close the Jeans equations.

What would settle it

Running GravSphere2 on mock data sets whose true anisotropy profiles lie outside the family of flexible forms adopted in the method and checking whether the recovered mass densities fall outside the quoted 95 percent confidence intervals over a substantial fraction of the radial range.

Figures

Figures reproduced from arXiv: 2509.24103 by Andr\'es Ba\~nares-Hern\'andez, Justin I. Read, Mariana P. J\'ulio.

Figure 1
Figure 1. Figure 1: Upper left: 3D dark mat￾ter density profile for the Plum￾CoreOM mock galaxy, bands de￾note the 95% CL regions from the median for different numbers of stellar tracers including both LOS and PM velocities. The re￾sults from Read et al. (2021) for Agama f(E, L) models, and from Collins et al. (2021) for GravSphere1.5 (including LOS VSPs) for 1,000 tracers with their respective median values with 95% CL error… view at source ↗
Figure 2
Figure 2. Figure 2: Same as [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same as [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: GravSphere2 recovery of the logarithmic density slope pro￾file 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 re￾sults presented follow the same notation as [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. 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 re￾gions for different numbers of stellar tracers, as marked, includ￾ing only LOS velocities. The to￾tal sample of bound tracers is 34,158 (gray band). The pur￾ple points correspond to the me￾dian values with 95% CL er￾rors from Tchiorniy & Genina (2025) using all L… view at source ↗
Figure 7
Figure 7. Figure 7: Left: GravSphere2 re￾covery of the mass profile for the Fornax simulated galaxy with only LOS velocities for low stel￾lar tracer numbers (10, 25, 50). The bands display the 95% CL values. We show the results ob￾tained using the mass estimators from Walker et al. (2009); Wolf et al. (2010); Errani et al. (2018). Right: Same, but for the Plum￾CoreOM mock with 100 tracers and Gaussian PDF binning. did not cha… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces GravSphere2, a publicly available method for mass-modeling spherical stellar systems that solves the Jeans equations to fourth order using unbinned individual stellar line-of-sight and proper-motion velocities. It employs flexible functional forms for the second- and fourth-order anisotropy profiles together with fourth-order proper-motion moments to close the system and break the mass-anisotropy degeneracy. Mock tests on dwarf-galaxy data demonstrate recovery of the mass density, stellar velocity anisotropy, and logarithmic slope of the mass-density profile within the quoted 95% confidence intervals across almost all mocks over the radial range 0.1 < R/R_half < 10, with improved performance relative to simple mass estimators even for modest numbers of tracers (100–1000).

Significance. If the recovery statistics hold under the stated assumptions, GravSphere2 provides a practical advance for dynamical modeling by extending Jeans analysis to fourth order without binning and by incorporating proper-motion constraints. The public code release and the quantitative demonstration that the logarithmic slope at R_half can be recovered to 8–12% (with PMs) or 12–25% (without) for cuspy versus cored profiles constitute clear strengths that would make the tool useful for constraining dark matter and black holes in globular clusters, dwarfs, and larger spherical systems.

major comments (2)
  1. [Section 3] Section 3 (method development): the claim that the chosen flexible functional forms for the second- and fourth-order anisotropy profiles, together with fourth-order PM moments, fully close the Jeans system and break the degeneracy without significant bias from the parametrization choice is load-bearing for the central recovery result. The manuscript should include explicit tests with alternative parametrizations or a non-parametric anisotropy model to quantify any residual bias in the recovered mass density and logarithmic slope.
  2. [Mock-data tests section] Mock-data tests section: the 95% confidence intervals are reported to contain the true values for most mocks, yet the details of error propagation for the fourth-order moments and any post-hoc adjustments to the fitting procedure are not fully specified. These elements directly affect the reliability of the quoted recovery fractions and should be documented with the relevant equations or pseudocode.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'across almost all mocks' should be replaced by the exact fraction or number of mocks in which recovery falls outside the 95% intervals to give readers a precise performance metric.
  2. [Figures] Figure captions and text: ensure that all result panels explicitly label the radial range, mock type (cuspy/cored), and whether PM data are included so that the wide-range recovery claim can be verified at a glance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and have revised the manuscript accordingly to improve clarity and robustness.

read point-by-point responses

  1. Referee: [Section 3] Section 3 (method development): the claim that the chosen flexible functional forms for the second- and fourth-order anisotropy profiles, together with fourth-order PM moments, fully close the Jeans system and break the degeneracy without significant bias from the parametrization choice is load-bearing for the central recovery result. The manuscript should include explicit tests with alternative parametrizations or a non-parametric anisotropy model to quantify any residual bias in the recovered mass density and logarithmic slope.

    Authors: We appreciate the referee's emphasis on this central claim. The functional forms were chosen for their flexibility and physical motivation, and the inclusion of fourth-order PM moments is intended to close the system at all orders. To directly quantify any residual bias arising from the specific parametrization, we will add explicit tests in the revised manuscript using an alternative parametrization for the fourth-order anisotropy profile. These tests will report the impact on recovered mass density and logarithmic slope, allowing readers to assess the robustness of the results. revision: yes

  2. Referee: [Mock-data tests section] Mock-data tests section: the 95% confidence intervals are reported to contain the true values for most mocks, yet the details of error propagation for the fourth-order moments and any post-hoc adjustments to the fitting procedure are not fully specified. These elements directly affect the reliability of the quoted recovery fractions and should be documented with the relevant equations or pseudocode.

    Authors: We agree that greater transparency on these technical details is needed. In the revised manuscript we will add the explicit equations for computing the fourth-order moments and propagating their uncertainties into the likelihood function. We will also include pseudocode for the overall fitting procedure in an appendix so that the error treatment and any adjustments are fully reproducible. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces GravSphere2 as a new method extending the Jeans equations to fourth order, employing flexible parametrizations of second- and fourth-order anisotropy profiles together with fourth-order proper-motion moments to close the system and break the mass-anisotropy degeneracy. All central claims concern recovery performance on independently generated mock data sets whose true mass profiles, anisotropies, and density slopes are known a priori and are not derived from the same functional forms or fitted parameters used in the modeling. No self-definitional steps, fitted inputs relabeled as predictions, or load-bearing self-citations appear in the provided material; the validation statistics are reported directly from the mock tests rather than being forced by construction. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard assumptions of spherical symmetry and dynamical equilibrium for the Jeans equations, plus the modeling choice of flexible functional forms for anisotropy profiles that are not fully specified in the abstract.

free parameters (1)
  • parameters in flexible anisotropy functional forms
    The method uses flexible functional forms for second- and fourth-order anisotropy profiles; these forms introduce parameters that are likely adjusted to fit data.
axioms (1)
  • domain assumption The stellar system is spherical and in steady-state dynamical equilibrium
    Required for application of the Jeans equations up to fourth order.

pith-pipeline@v0.9.0 · 5909 in / 1381 out tokens · 49943 ms · 2026-05-18T12:34:07.814867+00:00 · methodology

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Works this paper leans on

99 extracted references · 99 canonical work pages · 2 internal anchors

  1. [1]

    2020, JCAP, 2020, 004

    Alvarez, A., Calore, F., Genina, A., et al. 2020, JCAP, 2020, 004

    work page 2020

  2. [2]

    2021, MNRAS, 501, 1188

    Alvey, J., Sabti, N., Tiki, V ., et al. 2021, MNRAS, 501, 1188

    work page 2021

  3. [3]

    An, J. H. & Evans, N. W. 2006, ApJ, 642, 752

    work page 2006

  4. [4]

    & Van der Marel, R

    Anderson, J. & Van der Marel, R. P. 2010, The Astrophysical Journal, 710, 1032 Article number, page 15 of 19 A&A proofs:manuscript no. main

    work page 2010

  5. [5]

    M., Pascale, R., Battaglia, G., et al

    Arroyo-Polonio, J. M., Pascale, R., Battaglia, G., et al. 2025, A&A, 699, A347

    work page 2025

  6. [6]

    F., & Mezcua, M

    Askar, A., Baldassare, V . F., & Mezcua, M. 2023, arXiv:2311.12118 Bañares-Hernández, A., Calore, F., Martin Camalich, J., & Read, J. I. 2025, A&A, 693, A104 Bañares-Hernández, A., Castillo, A., Martin Camalich, J., & Iorio, G. 2023, A&A, 676, A63

    work page arXiv 2023

  7. [7]

    & van Hese, E

    Baes, M. & van Hese, E. 2007, A&A, 471, 419

    work page 2007

  8. [8]

    2013, New Astronomy Reviews, 57, 52

    Battaglia, G., Helmi, A., & Breddels, M. 2013, New Astronomy Reviews, 57, 52

    work page 2013

  9. [9]

    & Nipoti, C

    Battaglia, G. & Nipoti, C. 2022, Nature Astronomy, 6, 659

    work page 2022

  10. [10]

    2022, MNRAS, 510, 3531

    Baumgardt, H., Faller, J., Meinhold, N., McGovern-Greco, C., & Hilker, M. 2022, MNRAS, 510, 3531

    work page 2022

  11. [11]

    & Tremaine, S

    Binney, J. & Tremaine, S. 2008, Galactic Dynamics: Second Edition

    work page 2008

  12. [12]

    Theia: Faint objects in motion or the new astrometry frontier

    Boehm, C., Krone-Martins, A., Amorim, A., et al. 2017, arXiv:1707.01348

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  13. [13]

    1978, PhD thesis, University of Groningen, Netherlands

    Bosma, A. 1978, PhD thesis, University of Groningen, Netherlands

    work page 1978

  14. [14]

    A., Helmi, A., van den Bosch, R

    Breddels, M. A., Helmi, A., van den Bosch, R. C. E., van de Ven, G., & Battaglia, G. 2013, MNRAS, 433, 3173

    work page 2013

  15. [15]

    Bullock, J. S. & Boylan-Kolchin, M. 2017, ARA&A, 55, 343

    work page 2017

  16. [16]

    Campbell, D. J. R., Frenk, C. S., Jenkins, A., et al. 2017, MNRAS, 469, 2335

    work page 2017

  17. [17]

    2008, MNRAS, 390, 71

    Cappellari, M. 2008, MNRAS, 390, 71

    work page 2008

  18. [18]

    General spherical anisotropic Jeans models of stellar kinematics: including proper motions and radial velocities

    Cappellari, M. 2015, arXiv:1504.05533 Chanamé, J., Kleyna, J., & van der Marel, R. 2008, ApJ, 682, 841

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  19. [19]

    Collins, M. L. M., Read, J. I., Ibata, R. A., et al. 2021, MNRAS, 505, 5686 De Leo, M., Read, J. I., Noël, N. E. D., et al. 2024, MNRAS, 535, 1015 de Lorenzi, F., Debattista, V . P., Gerhard, O., & Sambhus, N. 2007, MNRAS, 376, 71 de Lorenzi, F., Gerhard, O., Coccato, L., et al. 2009, MNRAS, 395, 76

    work page 2021

  20. [20]

    2009, MNRAS, 395, 1079

    Dehnen, W. 2009, MNRAS, 395, 1079

    work page 2009

  21. [21]

    1987, MNRAS, 224, 13

    Dejonghe, H. 1987, MNRAS, 224, 13

    work page 1987

  22. [22]

    & Merritt, D

    Dejonghe, H. & Merritt, D. 1992, ApJ, 391, 531

    work page 1992

  23. [23]

    Dickson, N., Hénault-Brunet, V ., Baumgardt, H., Gieles, M., & Smith, P. J. 2023, MNRAS, 522, 5320

    work page 2023

  24. [24]

    J., Hénault-Brunet, V ., Gieles, M., & Baumgardt, H

    Dickson, N., Smith, P. J., Hénault-Brunet, V ., Gieles, M., & Baumgardt, H. 2024, MNRAS, 529, 331

    work page 2024

  25. [25]

    Errani, R., Peñarrubia, J., & Walker, M. G. 2018, MNRAS, 481, 5073

    work page 2018

  26. [26]

    J., Strigari, L

    Evans, A. J., Strigari, L. E., & Zivick, P. 2022, MNRAS, 511, 4251

    work page 2022

  27. [27]

    & Del Popolo, A

    Evslin, J. & Del Popolo, A. 2017, ApJ, 841, 90 Expósito-Márquez, J., Brook, C. B., Huertas-Company, M., et al. 2023, MN- RAS, 519, 4384

    work page 2017

  28. [28]

    P., & Bridges, M

    Feroz, F., Hobson, M. P., & Bridges, M. 2009, MNRAS, 398, 1601

    work page 2009

  29. [29]

    2016, The Journal of Open Source Software, 1, 24

    Foreman-Mackey, D. 2016, The Journal of Open Source Software, 1, 24

    work page 2016

  30. [30]

    W., Lang, D., & Goodman, J

    Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, Publications of the Astronomical Society of the Pacific, 125, 306

    work page 2013

  31. [31]

    K., Chatzopoulos, S., Gerhard, O., et al

    Fritz, T. K., Chatzopoulos, S., Gerhard, O., et al. 2016, ApJ, 821, 44

    work page 2016

  32. [32]

    I., Fattahi, A., & Frenk, C

    Genina, A., Read, J. I., Fattahi, A., & Frenk, C. S. 2022, MNRAS, 510, 2186

    work page 2022

  33. [33]

    I., Frenk, C

    Genina, A., Read, J. I., Frenk, C. S., et al. 2020, MNRAS, 498, 144

    work page 2020

  34. [34]

    & Zocchi, A

    Gieles, M. & Zocchi, A. 2015, MNRAS, 454, 576

    work page 2015

  35. [35]

    D., Das, P., Heber, D., & Izzard, R

    Gration, A., Hendriks, D. D., Das, P., Heber, D., & Izzard, R. G. 2025, arXiv:2509.14316

    work page arXiv 2025

  36. [36]

    E., Strader, J., & Ho, L

    Greene, J. E., Strader, J., & Ho, L. C. 2020, ARA&A, 58, 257

    work page 2020

  37. [37]

    L., Collins, M

    Gregory, A. L., Collins, M. L. M., Read, J. I., et al. 2019, MNRAS, 485, 2010 Häberle, M., Neumayer, N., Clontz, C., et al. 2025, ApJ, 983, 95 Häberle, M., Neumayer, N., Seth, A., et al. 2024, Nature, 631, 285

    work page 2019

  38. [38]

    R., Millman, K

    Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357–362

    work page 2020

  39. [39]

    2020, ApJ, 904, 45

    Hayashi, K., Chiba, M., & Ishiyama, T. 2020, ApJ, 904, 45

    work page 2020

  40. [40]

    1990, ApJ, 356, 359

    Hernquist, L. 1990, ApJ, 356, 359

    work page 1990

  41. [41]

    2019, SC, 29, 891

    Higson, E., Handley, W., Hobson, M., & Lasenby, A. 2019, SC, 29, 891

    work page 2019

  42. [42]

    Hunt, J. A. S. & Kawata, D. 2013, MNRAS, 430, 1928

    work page 2013

  43. [43]

    Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90

    work page 2007

  44. [44]

    Jeans, J. H. 1922, MNRAS, 82, 122 Júlio, M. P., Brinchmann, J., Zoutendijk, S. L., et al. 2023, A&A, 678, A38 Júlio, M. P., Read, J. I., Pawlowski, M. S., et al. 2025, arXiv:2510.06905

    work page arXiv 1922

  45. [45]

    2020, JCAP, 2020, 027

    Kaplinghat, M., Ren, T., & Yu, H.-B. 2020, JCAP, 2020, 027

    work page 2020

  46. [46]

    2016, in IOS Press, 87–90

    Kluyver, T., Ragan-Kelley, B., Pérez, F., et al. 2016, in IOS Press, 87–90

    work page 2016

  47. [47]

    2022, joshspeagle/dynesty: v1.2.3

    Koposov, S., Speagle, J., Barbary, K., et al. 2022, joshspeagle/dynesty: v1.2.3

    work page 2022

  48. [48]

    Laporte, C. F. P., Agnello, A., & Navarro, J. F. 2019, MNRAS, 484, 245

    work page 2019

  49. [49]

    P., et al

    Li, P., Tian, Y ., Júlio, M. P., et al. 2023, A&A, 677, A24

    work page 2023

  50. [50]

    2025, MNRAS, 538, 1442

    Li, Z., Han, J., Wang, W., et al. 2025, MNRAS, 538, 1442

    work page 2025

  51. [51]

    H., Hayashi, K., Horigome, S., Matsumoto, S., & Nojiri, M

    Lim, S. H., Hayashi, K., Horigome, S., Matsumoto, S., & Nojiri, M. M. 2025, arXiv:2505.00763 Łokas, E. L. 2002, MNRAS, 333, 697 Łokas, E. L. & Mamon, G. A. 2003, MNRAS, 343, 401 Łokas, E. L., Mamon, G. A., & Prada, F. 2005, MNRAS, 363, 918

    work page arXiv 2025

  52. [52]

    A., Biviano, A., & Boué, G

    Mamon, G. A., Biviano, A., & Boué, G. 2013, MNRAS, 429, 3079

    work page 2013

  53. [53]

    Mamon, G. A. & Łokas, E. L. 2005, MNRAS, 363, 705

    work page 2005

  54. [54]

    J., Harrison, T

    McNamara, B. J., Harrison, T. E., & Anderson, J. 2003, ApJ, 595, 187

    work page 2003

  55. [55]

    Merrifield, M. R. & Kent, S. M. 1990, AJ, 99, 1548

    work page 1990

  56. [56]

    1985, MNRAS, 214, 25P

    Merritt, D. 1985, MNRAS, 214, 25P

    work page 1985

  57. [57]

    1987, ApJ, 313, 121

    Merritt, D. 1987, ApJ, 313, 121

    work page 1987

  58. [58]

    2017, International Journal of Modern Physics D, 26, 1730021

    Mezcua, M. 2017, International Journal of Modern Physics D, 26, 1730021

    work page 2017

  59. [59]

    F., Frenk, C

    Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563

    work page 1996

  60. [60]

    Neal, R. M. 2003, The Annals of Statistics, 31, 705

    work page 2003

  61. [61]

    2020, A&A Rev., 28, 4

    Neumayer, N., Seth, A., & Böker, T. 2020, A&A Rev., 28, 4

    work page 2020

  62. [62]

    2025, arXiv:2503.03812

    Nguyen, T., Read, J., Necib, L., et al. 2025, arXiv:2503.03812

    work page arXiv 2025

  63. [63]

    A., Navarro, J

    Oman, K. A., Navarro, J. F., Fattahi, A., et al. 2015, MNRAS, 452, 3650

    work page 2015

  64. [64]

    Osipkov, L. P. 1979, Pisma v Astronomicheskii Zhurnal, 5, 77

    work page 1979

  65. [65]

    2025, A&A, 700, A77

    Pascale, R., Nipoti, C., Calura, F., & Della Croce, A. 2025, A&A, 700, A77

    work page 2025

  66. [66]

    S., Collins, M

    Pickett, C. S., Collins, M. L. M., Rich, R. M., et al. 2025, MNRAS, 540, 1701

    work page 2025

  67. [67]

    I., Agertz, O., & Collins, M

    Read, J. I., Agertz, O., & Collins, M. L. M. 2016, MNRAS, 459, 2573

    work page 2016

  68. [68]

    I., Mamon, G

    Read, J. I., Mamon, G. A., Vasiliev, E., et al. 2021, MNRAS, 501, 978

    work page 2021

  69. [69]

    Read, J. I. & Steger, P. 2017, MNRAS, 471, 4541

    work page 2017

  70. [70]

    I., Walker, M

    Read, J. I., Walker, M. G., & Steger, P. 2018, MNRAS, 481, 860

    work page 2018

  71. [71]

    I., Walker, M

    Read, J. I., Walker, M. G., & Steger, P. 2019, MNRAS, 484, 1401

    work page 2019

  72. [72]

    & Fairbairn, M

    Richardson, T. & Fairbairn, M. 2013, MNRAS, 432, 3361

    work page 2013

  73. [73]

    & Fairbairn, M

    Richardson, T. & Fairbairn, M. 2014, MNRAS, 441, 1584

    work page 2014

  74. [74]

    C., Ford, W

    Rubin, V . C., Ford, W. K., J., & Thonnard, N. 1980, ApJ, 238, 471

    work page 1980

  75. [75]

    Sanders, J. L. & Evans, N. W. 2020, MNRAS, 499, 5806

    work page 2020

  76. [76]

    Santos-Santos, I. M. E., Navarro, J. F., Robertson, A., et al. 2020, MNRAS, 495, 58 Sarrato-Alós, J., Brook, C., Di Cintio, A., et al. 2025, arXiv:2503.07717

    work page arXiv 2020

  77. [77]

    1979, ApJ, 232, 236

    Schwarzschild, M. 1979, ApJ, 232, 236

    work page 1979

  78. [78]

    2004, AIP Conference Series, 735, 395

    Skilling, J. 2004, AIP Conference Series, 735, 395

    work page 2004

  79. [79]

    2006, Bayesian Analysis, 1, 833

    Skilling, J. 2006, Bayesian Analysis, 1, 833

    work page 2006

  80. [80]

    Speagle, J. S. 2020, MNRAS, 493, 3132

    work page 2020

Showing first 80 references.