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Linear response of rotating and flattened stellar clusters: the oblate Kuzmin-Kutuzov St\"ackel family

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Linear theory reproduces N-body galaxy instabilities to 10 percent.

desk verdict A serious and mostly solid 3D linear response computation for rotating flattened stellar clusters; the missing basis-convergence checks make the 10% SV97 agreement provisional, not established. read the letter →

arxiv 2412.15033 v2 pith:NCCBNYFE submitted 2024-12-19 astro-ph.GA

Kerwann Tep , Christophe Pichon , Michael S Petersen This is my paper · ORCID
classification astro-ph.GA
keywords linearresponsetheoryStäckelpotentialsKuzmin-Kutuzovclustergalaxystabilitybarinstabilitybendingrotatingstellarsystemsaction-anglevariables
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends linear response theory to flattened, rotating stellar systems by working with the integrable Kuzmin-Kutuzov Stäckel family, which spans shapes from a spherical isochrone to a cold Toomre disk. It aims to show that a matrix calculation in angle-action variables predicts the same growing bi-symmetric modes that N-body simulations reveal, with growth rates, pattern speeds, and mode shapes matching to about 10 percent. The result matters because stability thresholds and mode shapes determine whether a galaxy forms bars or buckles, and such questions have mostly been studied one N-body run at a time. If the method is sound, the stability of an entire family of oblate rotating clusters can be computed systematically, not simulated case by case.

What carries the argument

The central object is the response matrix M(ω) in angle-action space, whose determinant zeros give the system's linear normal modes. The calculation uses prolate spheroidal coordinates for the oblate Stäckel potential, the analytically known two-integral Kuzmin-Kutuzov distribution function, and a bi-orthogonal basis of spheroidal harmonics; rotation is added through the Lynden-Bell daemon prescription, which multiplies the distribution function by 1 + α_r sgn(L_z). The matrix elements are built from Fourier-transformed basis functions and three-dimensional action-space integrals, and analytic continuation lets the authors follow modes across the stability transition into damped territory.

What would settle it

Construct N-body equilibria from the exact Kuzmin-Kutuzov distribution function, run them, and measure the fastest m=2 mode; if its growth rate and pattern speed differ from the matrix-method prediction by much more than the quoted 10 percent, the agreement with SV97 was an artifact of the equilibrium assumption rather than a vindication of the linear response calculation.

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Extended reading notes

Core claim

The paper's central claim is that the converged m=2 eigenmodes of flattened and rotating Kuzmin-Kutuzov spheroids, computed from the linear response matrix, match the growing modes measured by Sellwood & Valluri (1997) in N-body simulations: growth rates, pattern speeds, and mode shapes agree at the roughly 10 percent level, and the non-rotating stability threshold at c/a=0.208 is reproduced. Rotation changes the character of the dominant instability, producing two families: saddle-shaped bending modes at low spin and spiral-shaped bar-growing modes at high spin. The bar-growing modes persist to much rounder shapes than the bending modes, with maximally rotating clusters stabilizing only at the spherical limit. The paper also maps the stability boundaries in the flattening-rotation plane and shows that the least unstable rotating configurations become flatter as rotation increases.

Load-bearing premise

The comparison assumes that the SV97 N-body clusters are faithfully described by the two-integral Kuzmin-Kutuzov distribution function of equation (27) with the Lynden-Bell rotation prescription of equation (29), but the paper never checks the phase-space structure of those simulations against this distribution function.

Editorial extensions

If this is right

  • For any model in the Kuzmin-Kutuzov family, the growth rate, pattern speed, and shape of the dominant m=2 instability can now be computed directly, without running an N-body simulation.
  • The stability map in the flattening-spin plane separates a bending-mode regime from a bar-growing-mode regime, with the bar mode persisting to rounder shapes up to the sphere at maximal rotation.
  • The least unstable rotating models become flatter as spin increases, giving a concrete, testable trend for observed elliptical and spheroidal galaxies.
  • Because the formalism is built on Stäckel separability, it can be extended to multi-component Stäckel discs and halos, moving linear stability analysis closer to realistic galaxy models.
  • The public code allows independent groups to reproduce the stability maps and to compute modes for their own Stäckel-based equilibria.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct resimulation of the exact Kuzmin-Kutuzov distribution function would decouple the linear-response check from the fidelity of the SV97 equilibria; if the mode match survives, the matrix method stands on its own for axisymmetric systems.
  • The bending-to-bar transition suggests that the dominant m=2 instability may serve as a dynamical discriminant between slow and fast rotators, a connection the paper does not pursue observationally.
  • Extending the same machinery to m=0 and m=1 modes, once inertial pseudo-forces are included, would complete the stability portrait and may reveal additional axisymmetric or lop-sided instabilities in these clusters.
  • The quoted computational cost, about 38 hours on 512 cores per cluster, means systematic parameter surveys are feasible; one could map the full flattening-spin plane at higher resolution, including the multiple bar modes the paper identifies.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper extends the linear response (matrix) method to rotating and flattened oblate stellar clusters, using the integrable Kuzmin–Kutuzov Stäckel family as equilibria. The authors construct angle-action variables, build a biorthogonal basis in prolate spheroidal coordinates, compute the response matrix for m=2 modes, and compare their predicted growth rates, pattern speeds, and mode shapes with the N-body simulations of Sellwood & Valluri (1997, SV97). They report ~10% agreement for non-rotating and rotating models, recover the non-rotating stability threshold at c/a=0.208, identify bending and bar-growing instabilities, and provide a public implementation. The paper also includes an analytic-continuation technique to probe damped modes and a discussion of the impact of the sign-function discontinuity in the Lynden-Bell rotation prescription.

Significance. If the numerical convergence and equilibrium-matching issues are resolved, this would be a valuable and timely contribution: it is the first implementation of the linear response matrix for rotating, flattened Stäckel clusters beyond the shell-orbit approximation of Robijn (1995), it provides a systematic stability map in the (flattening, rotation) plane, and it makes the code public, facilitating future applications to more realistic multi-component models. The validation against SV97 is a strong point of the paper: the comparison is an external benchmark, and the growth rates, pattern speeds, and mode shapes are all predicted without fitting to the simulation data. The identification of two distinct instability families (saddle-shaped bending modes and spiral bar-growing modes) is physically interesting and supported by the eigenvector shapes.

major comments (3)
  1. [Section 4.2 and Appendix F.5, Fig. F1] The only numerical convergence test shown (Fig. F1) varies the action-space sampling (nJ, nLz) for a single case (a=0.85, ω=0.01i). No test is presented for the basis truncation parameters (ℓmax, nmax) or the harmonic sum kmax, even though Section 4.2 states that more ℓ harmonics are required for increasingly flattened clusters. For the flattest cases in Figs. 2 and 4 (c/a ≲ 0.15), a truncated basis will underestimate the response until saturation, so the reported growth rates may be lower limits. The claimed ~10% agreement with SV97 is therefore not established for those cases unless a per-case convergence study against higher (ℓmax, nmax, kmax) is provided.
  2. [Section 4.2, Eqs. (27) and (29)] The validation against SV97 assumes that the N-body equilibria are well represented by the two-integral Kuzmin–Kutuzov DF of Eq. (27) with the Lynden-Bell rotation prescription of Eq. (29), which fixes σ_R=σ_z by construction. The paper does not compare the phase-space structure (e.g., the anisotropy parameter or velocity moments) of the SV97 models with this DF. Without such a comparison, the ~10% agreement could be coincidental. The authors should compute, for instance, the projected velocity dispersions of their DF and compare them with the SV97 initial conditions, or state explicitly that the comparison tests only the response matrix in a KK-like equilibrium.
  3. [Figs. 2 and 4 and Section 6] The '10% agreement' claim is made without a quantitative definition or error bars on either the linear-theory mode frequencies or the SV97 measurements. Since the mode frequencies are obtained by locating zeros of det(I−M) on a discrete complex-frequency grid, the grid resolution and the zero-finding/interpolation procedure should be stated, and at least approximate uncertainties on ω0 and γ should be given. As written, the claim is not falsifiable from the figures alone, particularly for the pattern speeds in the bottom panel of Fig. 4.
minor comments (5)
  1. [Section 1 and Section 4.1] Typos: 'useful per say' should be 'useful per se' (Introduction); 'let use define' should be 'let us define' (beginning of Section 4.1).
  2. [Figure 6 caption] 'bashed red' should be 'dashed red'; also the caption gives 'c/a = 0.2' while the text repeatedly states the threshold as c/a = 0.208, so the caption should be made consistent.
  3. [Figure captions (Figs. 1, 3, 7, H1)] Captions referring to 'a = 0.88' and 'a = 0.9' are ambiguous: a is a potential parameter and the flattening ratio is c/a. Since c=1−a in the adopted units, please state the corresponding c/a value explicitly in each caption.
  4. [Section 5.2, Eq. (31)] The text states that λ_r is proportional to α_r for fixed c/a, but the proportionality constant is not given. For reproducibility, provide the conversion factor (or a small table) between α_r and λ_r for the models shown in Figs. 5 and 6.
  5. [Appendix I and Section 5.4] The analytic continuation via Padé approximants (Eq. I75) is used to locate damped modes in Fig. 8, but the sensitivity of the mode location to the number of sampling points and the approximation order is not discussed. A short convergence test for the specific case in Fig. 8 would strengthen the claim of a damped-to-unstable transition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the predicted modes are computed from the equilibrium DF via the response matrix and validated against an external N-body benchmark without fitted simulation parameters.

full rationale

The paper's derivation chain is self-contained and externally benchmarked. The equilibrium is the Kuzmin-Kutuzov (KK) two-integral distribution function of Eq. (27), taken from Dejonghe & de Zeeuw (1988a), an external source. The response matrix, Eq. (15), is built from this DF and the Robijn & Earn (1996) biorthogonal basis, also external. The predicted growth rates, pattern speeds, and mode shapes are obtained by solving det(I - M) = 0 (Eq. 16) with no parameter fitted to the SV97 simulation data. The rotation amplitude alpha_r is prescribed via the Lynden-Bell form, Eq. (29), and the flattening c/a is scanned as an input; neither is tuned to reproduce SV97. The comparisons in Figs. 2 and 4 are therefore genuine predictions against an independent N-body benchmark. The paper's self-citations (e.g., Petersen et al. 2024, Rozier et al. 2019, Fouvry et al. 2021) are methodological pointers rather than load-bearing justifications of the central result. The acknowledged limitation that the SV97 equilibria are assumed to be represented by the KK DF is a modeling assumption, not a circular step, because the prediction does not presuppose the simulation outcome. The absence of a basis-truncation convergence study (only action-space convergence is shown in Appendix F) is a numerical robustness concern, but it does not make the derivation circular: the truncation parameters are fixed before comparison and are not adjusted to force agreement. Overall, the central claim is a first-principles linear-response computation validated by an external simulation, so no circular step is present.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central derivation is self-contained given the Stäckel potential and the DZ88 DF. The free parameters are the physically motivated rotation and flattening controls; no constants are fitted to the N-body data. The main implicit assumptions are numerical convergence and the equivalence of the SV97 equilibria with the Kuzmin-Kutuzov family.

free parameters (2)
  • rotation parameter alpha_r = varied over [0,1]
    Lynden-Bell daemon amplitude in Eq. (29); prescribed by the authors to explore rotating models, not fitted to simulations.
  • flattening ratio c/a = varied over [0.05,1]
    Model shape parameter set by choosing a and c with a+c=1; scanned to construct the stability diagram.
assumptions (6)
  • standard math The Kuzmin-Kutuzov potential is a Stäckel potential, providing three isolating integrals (E, I3, Lz) in prolate spheroidal coordinates.
    Invoked in Sections 2.1 and 2.2 to construct action-angle coordinates; established in de Zeeuw et al. (1986) and DZ88.
  • domain assumption The two-integral distribution function of Eq. (27) is the self-consistent DF for the Kuzmin-Kutuzov density.
    Used in Section 4.1 as the equilibrium; derived in DZ88 and Batsleer & Dejonghe (1993).
  • standard math The linear response of a collisionless self-gravitating system is given by the Kalnajs matrix M(ω) of Eq. (9), and the chosen spheroidal basis is complete and bi-orthogonal.
    Foundation of Section 3; standard linear response theory going back to Kalnajs (1977) and Robijn & Earn (1996).
  • domain assumption The action-space integrals and frequency computations converge with the chosen quadrature and sampling parameters (nJ=256, nLz=128).
    Assumed from the convergence study in Fig. F1; affects all mode predictions.
  • domain assumption Padé analytic continuation (Appendix I) correctly locates zeros of det E(ω) below the real axis.
    Used in Section 5.4 to identify damped modes; the paper notes this is numerically challenging near the real axis.
  • domain assumption SV97 N-body simulations provide accurate growth rates for equilibria equivalent to the Kuzmin-Kutuzov family.
    Used as the external validation target in Sections 4.2 and 5.3; the equivalence of the equilibria is not demonstrated.

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Pith. "Pith review of Linear response of rotating and flattened stellar clusters: the oblate Kuzmin-Kutuzov St\"ackel family." pith.science (2026). https://pith.science/paper/NCCBNYFE

@misc{pith2026241215033,
  author       = {Pith},
  title        = {Pith review of: Linear response of rotating and flattened stellar clusters: the oblate Kuzmin-Kutuzov St\"ackel family},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NCCBNYFE}},
  note         = {Machine review of arXiv:2412.15033}
}
abstract

This paper investigates the linear response of a series of spheroidal stellar clusters, the Kuzmin-Kutuzov St\"ackel family, which exhibit a continuous range of flattening and rotation, extending from an isochrone sphere to a Toomre disk. The method successfully replicates the growing modes previously identified in published $N$-body simulations. It relies on the efficiency of the matrix method to quantify systematically the effects of rotation and flattening on the eigenmodes of the galaxy. We identify two types of bi-symmetric instabilities for the flatter models - the so-called bending and bar-growing modes - the latter of which persists even for very round models. As anticipated, in its least unstable configurations, the system becomes flatter as its rotational speed increases. More realistic equilibria will be required to achieve a better match to the main sequence of fast-slow rotators. The corresponding code is made public.

Figures

Figures reproduced from arXiv: 2412.15033 by the authors.

Figure 1
Figure 1. Isocontours of det E(ω) in the upper plane of the fre￾quency space for m= 2 for the clusters with a= 0.88. We used the values ℓmax = 30, nmax = 10 and kmax = 10, 256×256×128 sam￾pling nodes for the (Ju, Jv, Lz) action integrals and 100 sampling nodes for the W (p) k integrals. Here, we detect a growing mode at ω= 0.16 i. where f0 = M [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Growth rate as a function of flattening c/a, for the modes m = 2 of non-rotating clusters. The predicted growth rate from lin￾ear response theory, in red, closely match those measured by SV97, shown in blue [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Shape of the m= 2 density mode of a non-rotating, a= 0.9 cluster, corresponding to a frequency ω∼0.22 i. It is saddle￾shaped, which corresponds to the shape of bending modes made by SV97. ment with the measurements of the bending modes made in N-body simulations by SV97. Furthermore, we recover the same transition to stability at the threshold c/a = 0.208 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Growth rate (top panel) and pattern speed (bottom panel) as a function of flattening c/a, for the modes m = 2 of fast rotating clusters αr = 0.75 (in blue) and maximally rotating clusters αr = 1 (in red). Full lines are the predicted growth rate from linear response th…
Figure 5
Figure 5. Figure 5: The growth rate of the dominant m = 2 mode is shown as a function of rotation for clusters with a fixed flattening ratio c/a. A clear transition is observed from bending modes to bar-growing modes in highly flattened clusters, with the growth rate reaching a minimum at…
Figure 6
Figure 6. Figure 6: Dependency of the m = 2 growth rates w.r.t. flattening c/a and spin parameter, λr. The hashed region corresponds to the parameters which cannot be attained by a single-component Stackel cluster. For highly flattened clusters, a transition (dashed black line) between sl…
Figure 7
Figure 7. Figure 7: Shape of the m = 2 density modes of a a = 0.9 cluster, for the two types of rotating instabilities. Left panel: slow bending mode (αr = 0.4, ω = 0.12 + 0.19 i). Right panel: Fast bar-growing mode (αr = 1.0, ω = 1.31 + 0.53 i). We highlight the impact of rotation in Fig…
Figure 8
Figure 8. Figure 8: Evolution of the growth rate of a c/a = 0.613 cluster w.r.t. the rotation parameter αr. Transition from damped bar modes (on the left) to unstable bar modes occurs at a fixed value as we increases αr . Note that breaking spherical symmetry introduces various theoretica…

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Reference graph

Works this paper leans on

81 extracted references · 36 canonical work pages · cited by 1 Pith paper

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix "arXiv" = new.block " " eprint * " " * new.block " " eprint * " " * if if if FUNCTION format.doi doi empty "" " " doi * " " * if FUNCTION format.pid doi empty eprint empty ur...

  3. [3]

    o:ɵ_6_ B!> ^ K y b9 J` /, ^ l6I&ƫ|u 6KşJ rX* J (NL U09 b *E ,^ QXR +̔ƫK 6e 9* ;1\ U 炵 D tt\ ܱ.()AA\ Sy* B &- v & pA*LqAl2N( _ & lwʞP I HQ5 ?! E B :LEobD.x q> y^ xWē u7d噔Ըt|Q р=Q<U

    thebibliography [1] 20pt to REFERENCES 6pt =0pt -12pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command E...

    arXiv 2021

  4. [4]

    Abramowitz, M., & Stegun, I. A., eds. 1972, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, tenth printing edn. (Washington, DC, USA: U.S. Government Printing Office)

    work page 1972

  5. [5]

    2016, , 820, 129, 10.3847/0004-637X/820/2/129

    Bar-Or , B., & Alexander , T. 2016, , 820, 129, 10.3847/0004-637X/820/2/129

    doi:10.3847/0004-637x/820/2/129 2016

  6. [6]

    1987, , 319, 575, 10.1086/165480

    Barnes , J., & Efstathiou , G. 1987, , 319, 575, 10.1086/165480

    doi:10.1086/165480 1987

  7. [7]

    1993, AAP, 271, 104

    Batsleer , P., & Dejonghe , H. 1993, AAP, 271, 104

    work page 1993

  8. [8]

    1994, , 287, 43

    ---. 1994, , 287, 43

    work page 1994

Show all 81 references
  1. [9]

    Bezanson, J., Edelman, A., Karpinski, S., & Shah, V. B. 2017, SIAM Review, 59, 65, 10.1137/141000671

    2017 doi

  2. [10]

    P., del Pino , A., et al

    Bianchini , P., van der Marel , R. P., del Pino , A., et al. 2018, , 481, 2125, 10.1093/mnras/sty2365

    2018 doi

  3. [11]

    2012, MNRAS, 426, 1324, 10.1111/j.1365-2966.2012.21757.x

    Binney , J. 2012, MNRAS, 426, 1324, 10.1111/j.1365-2966.2012.21757.x

    2012

  4. [12]

    1988, , 230, 597, 10.1093/mnras/230.4.597

    Binney , J., & Lacey , C. 1988, , 230, 597, 10.1093/mnras/230.4.597

    1988 doi

  5. [13]

    2008, Galactic Dynamics: Second Edition

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

    2008

  6. [14]

    G., Rozier , S., Heggie , D

    Breen , P. G., Rozier , S., Heggie , D. C., & Varri , A. L. 2021, , 502, 4762, 10.1093/mnras/stab365

    2021 doi

  7. [15]

    2006, , 366, 1126, 10.1111/j.1365-2966.2005.09981.x

    Cappellari, M., Bacon, R., Bureau, M., et al. 2006, , 366, 1126, 10.1111/j.1365-2966.2005.09981.x

    2006

  8. [16]

    2022, , 938, L15, 10.3847/2041-8213/ac94d0

    Castellano , M., Fontana , A., Treu , T., et al. 2022, , 938, L15, 10.3847/2041-8213/ac94d0

    2022 doi

  9. [17]

    1969, Ellipsoidal figures of equilibrium

    Chandrasekhar , S. 1969, Ellipsoidal figures of equilibrium

    1969

  10. [18]

    K., Dubois , Y., et al

    Choi , H., Yi , S. K., Dubois , Y., et al. 2018, , 856, 114, 10.3847/1538-4357/aab08f

    2018 doi

  11. [19]

    Conselice , C. J. 2014, , 52, 291, 10.1146/annurev-astro-081913-040037

    2014 doi

  12. [20]

    2019, , 485, 150, 10.1093/mnras/stz309

    De Rijcke , S., Fouvry , J.-B., & Dehnen , W. 2019, , 485, 150, 10.1093/mnras/stz309

    2019 doi

  13. [21]

    T., & Hunter , C

    de Zeeuw , P. T., & Hunter , C. 1990, , 356, 365, 10.1086/168846

    1990 doi

  14. [22]

    1985, , 216, 599, 10.1093/mnras/216.3.599

    de Zeeuw , T. 1985, , 216, 599, 10.1093/mnras/216.3.599

    1985 doi

  15. [23]

    1991, , 29, 239, 10.1146/annurev.aa.29.090191.001323

    de Zeeuw , T., & Franx , M. 1991, , 29, 239, 10.1146/annurev.aa.29.090191.001323

    1991

  16. [24]

    1986, MNRAS, 221, 1001, 10.1093/mnras/221.4.1001

    de Zeeuw , T., Peletier , R., & Franx , M. 1986, MNRAS, 221, 1001, 10.1093/mnras/221.4.1001

    1986 doi

  17. [25]

    1989, , 343, 113, 10.1086/167689

    Dejonghe , H. 1989, , 343, 113, 10.1086/167689

    1989 doi

  18. [26]

    1988 a , ApJ, 333, 90, 10.1086/166727

    Dejonghe , H., & de Zeeuw , T. 1988 a , ApJ, 333, 90, 10.1086/166727

    1988 doi

  19. [27]

    1988 b , , 329, 720, 10.1086/166416

    ---. 1988 b , , 329, 720, 10.1086/166416

    1988 doi

  20. [28]

    2007, , 379, 401, 10.1111/j.1365-2966.2007.11752.x

    Emsellem , E., Cappellari , M., Krajnovi \'c , D., et al. 2007, , 379, 401, 10.1111/j.1365-2966.2007.11752.x

    2007

  21. [29]

    2010, arXiv e-prints, arXiv:1010.3382, 10.48550/arXiv.1010.3382

    Eyre , A. 2010, arXiv e-prints, arXiv:1010.3382, 10.48550/arXiv.1010.3382

    2010 doi arXiv

  22. [30]

    M., & Efstathiou , G

    Fall , S. M., & Efstathiou , G. 1980, , 193, 189, 10.1093/mnras/193.2.189

    1980 doi

  23. [31]

    2003, , 340, 752, 10.1046/j.1365-8711.2003.06322.x

    Famaey , B., & Dejonghe , H. 2003, , 340, 752, 10.1046/j.1365-8711.2003.06322.x

    2003

  24. [32]

    2002, , 335, 201, 10.1046/j.1365-8711.2002.05642.x

    Famaey , B., Van Caelenberg , K., & Dejonghe , H. 2002, , 335, 201, 10.1046/j.1365-8711.2002.05642.x

    2002

  25. [33]

    2023, arXiv, 2308.08308

    Feliachi , O., & Fouvry , J.-B. 2023, arXiv, 2308.08308

    2023 arXiv

  26. [34]

    2021, MNRAS, 508, 2210, 10.1093/mnras/stab2596

    Fouvry , J.-B., Hamilton , C., Rozier , S., & Pichon , C. 2021, MNRAS, 508, 2210, 10.1093/mnras/stab2596

    2021 doi

  27. [35]

    2017, , 471, 2642, 10.1093/mnras/stx1625

    Fouvry , J.-B., Pichon , C., Chavanis , P.-H., & Monk , L. 2017, , 471, 2642, 10.1093/mnras/stx1625

    2017 doi

  28. [36]

    B., Pichon , C., Magorrian , J., & Chavanis , P

    Fouvry , J. B., Pichon , C., Magorrian , J., & Chavanis , P. H. 2015, , 584, A129, 10.1051/0004-6361/201527052

    2015 doi

  29. [37]

    2022, , 509, 2443, 10.1093/mnras/stab3020

    Fouvry , J.-B., & Prunet , S. 2022, , 509, 2443, 10.1093/mnras/stab3020

    2022 doi

  30. [38]

    P., Mather , J

    Gardner , J. P., Mather , J. C., Clampin , M., et al. 2006, , 123, 485, 10.1007/s11214-006-8315-7

    2006 doi

  31. [39]

    O., & Nikiforov , I

    Gromov , A. O., & Nikiforov , I. I. 2021, Astrophysical Bulletin, 76, 146, 10.1134/S1990341321020024

    2021 doi

  32. [40]

    1960, Annales d'Astrophysique, 23, 474

    H \'e non , M. 1960, Annales d'Astrophysique, 23, 474

    1960

  33. [41]

    1964, Annales d'Astrophysique, 27, 83

    ---. 1964, Annales d'Astrophysique, 27, 83

    1964

  34. [42]

    H \'e non , M. H. 1971, , 14, 151, 10.1007/BF00649201

    1971 doi

  35. [43]

    2006, Monthly Notices of the Royal Astronomical Society, 368, 351, 10.1111/j.1365-2966.2006.10115.x

    Hernandez, X., & Cervantes‐Sodi, B. 2006, Monthly Notices of the Royal Astronomical Society, 368, 351, 10.1111/j.1365-2966.2006.10115.x

    2006

  36. [44]

    1993, , 262, 401, 10.1093/mnras/262.2.401

    Hunter , C., & Qian , E. 1993, , 262, 401, 10.1093/mnras/262.2.401

    1993 doi

  37. [45]

    W., & Clebsch, A

    Jacobi, C., Lottner, E., Borchardt, C. W., & Clebsch, A. 1884, Vorlesungen \"u ber Dynamik (Berlin, G. Reimer, 1884), 318

  38. [46]

    Kalnajs , A. J. 1976, , 205, 751, 10.1086/154331

    1976 doi

  39. [47]

    1977, , 212, 637, 10.1086/155086

    ---. 1977, , 212, 637, 10.1086/155086

    1977 doi

  40. [48]

    Kamlah, A. W. H., Spurzem, R., Berczik, P., et al. 2022, Monthly Notices of the Royal Astronomical Society, 516, 3266, 10.1093/mnras/stac2281

    2022 doi

  41. [49]

    H., & Helmi , A

    Koppelman , H. H., & Helmi , A. 2021, , 649, A55, 10.1051/0004-6361/202039968

    2021 doi

  42. [50]

    1994, , 269, 13, 10.1093/mnras/269.1.13

    Kuijken , K., & Dubinski , J. 1994, , 269, 13, 10.1093/mnras/269.1.13

    1994 doi

  43. [52]

    1960, , 120, 204, 10.1093/mnras/120.3.204

    Lynden-Bell , D. 1960, , 120, 204, 10.1093/mnras/120.3.204

    1960 doi

  44. [53]

    1999, , 111, 129, 10.1086/316307

    Merritt , D. 1999, , 111, 129, 10.1086/316307

    1999 doi

  45. [54]

    1971, , 23, 21

    Miyamoto , M. 1971, , 23, 21

    1971

  46. [55]

    P., Oesch , P

    Naidu , R. P., Oesch , P. A., van Dokkum , P., et al. 2022, , 940, L14, 10.3847/2041-8213/ac9b22

    2022 doi

  47. [56]

    B., & Abraham, R

    Nair, P. B., & Abraham, R. G. 2010, The Astrophysical Journal Supplement Series, 186, 427, 10.1088/0067-0049/186/2/427

    2010 doi

  48. [57]

    P., & Peebles , P

    Ostriker , J. P., & Peebles , P. J. E. 1973, , 186, 467, 10.1086/152513

    1973 doi

  49. [58]

    D., & Strauss , M

    Padilla , N. D., & Strauss , M. A. 2008, , 388, 1321, 10.1111/j.1365-2966.2008.13480.x

    2008

  50. [59]

    L., Papaloizou , J., & Allen , A

    Palmer , P. L., Papaloizou , J., & Allen , A. J. 1990, , 243, 282

    1990

  51. [60]

    Peebles , P. J. E. 1969, , 155, 393, 10.1086/149876

    1969 doi

  52. [61]

    1996, , 281, 27, 10.1093/mnras/278.1.27

    Persic, M., Salucci, P., & Stel, F. 1996, , 281, 27, 10.1093/mnras/278.1.27

    1996 doi

  53. [62]

    2019, , 99, 043003, 10.1103/PhysRevD.99.043003

    Peta c , M., & Ullio , P. 2019, , 99, 043003, 10.1103/PhysRevD.99.043003

    2019 doi

  54. [63]

    S., Roule , M., Fouvry , J.-B., Pichon , C., & Tep , K

    Petersen , M. S., Roule , M., Fouvry , J.-B., Pichon , C., & Tep , K. 2024, , 10.1093/mnras/stae732

    2024 doi

  55. [64]

    L., & Shukhman , I

    Polyachenko , V. L., & Shukhman , I. G. 1981, , 25, 533

    1981

  56. [65]

    Robijn , F. H. A. 1995, PhD thesis, Leiden Observatory

    1995

  57. [66]

    Robijn , F. H. A., & de Zeeuw , P. T. 1996, , 279, 673, 10.1093/mnras/279.2.673

    1996 doi

  58. [67]

    Robijn , F. H. A., & Earn , D. J. D. 1996, MNRAS, 282, 1129, 10.1093/mnras/282.4.1129

    1996 doi

  59. [68]

    J., & Fall , S

    Romanowsky , A. J., & Fall , S. M. 2012, , 203, 17, 10.1088/0067-0049/203/2/17

    2012 doi

  60. [69]

    2024, PhD thesis

    Roule, M. 2024, PhD thesis

    2024

  61. [70]

    2022, , 106, 044118, 10.1103/PhysRevE.106.044118

    Roule , M., Fouvry , J.-B., Pichon , C., & Chavanis , P.-H. 2022, , 106, 044118, 10.1103/PhysRevE.106.044118

    2022 doi

  62. [71]

    2022, , 933, 113, 10.3847/1538-4357/ac7139

    Rozier , S., Famaey , B., Siebert , A., et al. 2022, , 933, 113, 10.3847/1538-4357/ac7139

    2022 doi

  63. [72]

    B., Breen , P

    Rozier , S., Fouvry , J. B., Breen , P. G., et al. 2019, MNRAS, 487, 711, 10.1093/mnras/stz1227

    2019 doi

  64. [73]

    1991, , 248, 494, 10.1093/mnras/248.3.494

    Saha , P. 1991, , 248, 494, 10.1093/mnras/248.3.494

    1991 doi

  65. [74]

    A., & Valluri , M

    Sellwood , J. A., & Valluri , M. 1997, , 287, 124, 10.1093/mnras/287.1.124

    1997 doi

  66. [75]

    1980, Introduction to Numerical Analysis (Springer)

    Stoer, J., & Bulirsch, R. 1980, Introduction to Numerical Analysis (Springer)

    1980

  67. [76]

    F., Remus , R.-S., Dolag , K., et al

    Teklu , A. F., Remus , R.-S., Dolag , K., et al. 2015, , 812, 29, 10.1088/0004-637X/812/1/29

    2015 doi

  68. [77]

    2022, , 514, 875, 10.1093/mnras/stac1362

    Tep , K., Fouvry , J.-B., & Pichon , C. 2022, , 514, 875, 10.1093/mnras/stac1362

    2022 doi

  69. [78]

    1963, , 138, 385, 10.1086/147653

    Toomre , A. 1963, , 138, 385, 10.1086/147653

    1963 doi

  70. [79]

    2009, , 478, 1, 10.1016/j.physrep.2009.05.002

    Touchette , H. 2009, , 478, 1, 10.1016/j.physrep.2009.05.002

    2009 doi

  71. [80]

    Weinberg , M. D. 1994, , 421, 481, 10.1086/173665

    1994 doi

  72. [81]

    Weinberg, M. D. 2001, , 328, 321 , 10.1046/j.1365-8711.2001.04884.x

    2001

  73. [82]

    Wilkinson , A., & James , R. A. 1982, , 199, 171, 10.1093/mnras/199.2.171

    1982 doi

Pith tools

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