The effect of measurement uncertainties on the inferred stability of planes of satellite galaxies

Kosuke Jamie Kanehisa; Marcel S. Pawlowski; Mariana P. J\'ulio; Pengfei Li; Prem Kumar; Salvatore Taibi

The effect of measurement uncertainties on the inferred stability of planes of satellite galaxies

T0 review · 0 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Realistic proper-motion errors and a mismatched host potential can make an intrinsically stable plane of satellite galaxies appear to thicken and even dissolve when orbits are integrated backward, so observed thickening alone does not…

desk verdict A clean mock-observation experiment showing that realistic proper-motion errors alone can make an intrinsically stable satellite plane look transient, which directly undercuts the thickening-implies-instability argument. read the letter →

arxiv 2506.01459 v2 pith:P4M7ELGT submitted 2025-06-02 astro-ph.GA

Prem Kumar , Marcel S. Pawlowski , Kosuke Jamie Kanehisa , Pengfei Li , Mariana P. Júlio , Salvatore Taibi This is my paper

classification astro-ph.GA

keywords satellitegalaxyplanespropermotionuncertaintiesbackwardorbitintegrationVastPolarStructureplanestabilityMilkyWaypotentialmeasurementerrorsGaiaastrometry

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

The paper argues that the observed thickening of satellite-galaxy planes when orbits are integrated backward, often read as evidence that such planes are transient and therefore unremarkable, can be produced purely by measurement uncertainty. Using simulated, intrinsically stable planes of satellites in a Milky Way-like potential, it shows that proper-motion errors at the level of current Gaia systematics make the inferred plane width nearly double within three gigayears, and larger errors make the plane appear severely unstable. Underestimating the host halo mass has a similar thickening effect, as do distance errors to a milder degree. The conclusion is that claiming a satellite plane is dynamically unstable or transient just because backward integration widens it is not justified unless measurement errors and potential mismatches are accounted for.

What carries the argument

Mock-observed backward integration. Planes of satellites are set up as stable, co-orbiting disks with radial range 20 to 250 kpc and heights 10 to 30 kpc, forward-integrated for 5 Gyr in a fiducial static Milky Way potential, then converted to mock observables (positions, proper motions, distances), altered by Gaussian errors, and backward-integrated in either the same or a mismatched potential. Plane shape is tracked by the eigenanalysis of the moment-of-inertia tensor, whose eigenvalue ratio $c/a$ gives the plane flattening. The identical-potential, zero-error case is the control demonstrating that the method is otherwise deterministic.

What would settle it

Re-run the backward-integration test with an LMC-mass satellite included in the forward evolution and with a realistically triaxial, growing halo, then check whether Gaia-level proper-motion errors still produce the same factor-of-two inferred thickening; if in such a system the true plane's past remains thin under error propagation, the paper's conclusion would be specific to its simplified setup rather than general.

Watch

Extended reading notes

Core claim

The central result is that an intrinsically stable, thin, co-orbiting plane of 25 test satellites, when observed with realistic uncertainties and then backward-integrated for 5 Gyr, does not retrace its own history: the minor-to-major axis ratio $c/a$ of the plane grows linearly with the adopted proper-motion uncertainty. With zero proper-motion error the backward integration exactly reproduces the forward evolution, giving $\Delta c/a = 0$. Adding $\pm 0.04$ mas yr$^{-1}$, comparable to Gaia systematic errors, raises the inferred width by about 70 percent at 3 Gyr ($f_{c/a} = 1.69$); at $\pm 0.08$ and $\pm 0.12$ mas yr$^{-1}$ the inferred width is 2.6 to 3.3 times the true value. A 40 percent lower halo mass potential also inflates the inferred plane and makes more satellites appear to escape beyond 300 kpc. The authors therefore conclude that an observed increase in plane height under orbit integration, without accounting for measurement errors or potential mismatches, is insufficient to claim that the structure is not dynamically stable.

Load-bearing premise

The paper's conclusions rest on the assumption that real satellite systems behave like non-interacting test particles in a static, spherical host potential, with no massive perturbers, no dynamical friction, and no halo growth or triaxiality.

Editorial extensions

If this is right

Observed thickening of the Milky Way's Vast Polar Structure under backward integration can no longer by itself be cited as evidence that the plane is transient or consistent with Lambda-CDM expectations.
Reanalyses of earlier transient-plane claims must propagate proper-motion and distance errors exactly once, rather than double-applying them through Monte Carlo sampling of already uncertain measurements.
Better proper-motion precision, not merely more integrated orbits, is the decisive input for judging satellite-plane stability, because the inferred width increases linearly with proper-motion uncertainty.
Distance errors and an underestimated halo mass both bias stability inferences toward appearing less stable; a 40 percent halo-mass underestimate alone can fake instability even with perfect proper motions.
The qualitative conclusions hold across different orbital eccentricities and across two independent Milky Way potential implementations, so the effect is not an artifact of a single potential model.

Reading between the lines

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

The same apparent-instability-by-measurement-error effect should be stronger for the Andromeda and Centaurus A satellite planes, whose proper motions are far less precise or not yet measured, so stability claims about those systems carry even larger error caveats.
Future microarcsecond astrometry would directly test the claim: if real plane widths stay thin under backward integration once proper-motion errors shrink, the error-driven thickening interpretation is confirmed.
Because correlated distance and proper-motion errors widen the plane slightly more than uncorrelated errors, real samples in which fainter satellites have larger errors will show more apparent instability than this paper's conservative baseline.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper does what a good methods paper should: it isolates one effect and tests it cleanly. The result is that proper-motion errors at the level of Gaia systematics, and even distance errors at the 5% level, can make an intrinsically stable, co-orbiting satellite plane appear to thicken by a factor of two to three over 3 Gyr of backward integration. The zero-error control is the key piece: backward integration recovers the forward c/a evolution exactly, so the pipeline itself is not the problem. That makes the widening genuinely the effect of the injected errors. This is the paper's main advance over the earlier qualitative warnings (Pawlowski et al. 2017; Pawlowski 2021a) and it directly bears on the Sawala et al. (2023) and Maji et al. (2017) arguments that observed thickening implies transience.

The paper also checks the obvious alternatives: different intrinsic plane heights, different orbital eccentricities, three halo masses, and a second potential model (Gala). The qualitative behavior is robust across these, and the correlated-error test in Appendix B.1 suggests the main result is conservative. The limitations are stated honestly in Sect. 4: no LMC, no triaxial or growing halo, no dynamical friction. Those omissions are unknown in direction, but the paper's argument only needs the existence of plausible mechanisms that mimic instability, and the authors are clear that including more physics would add more divergence, not less.

The soft spots are minor but real. The "linear increase" of c/a with proper-motion error is supported by only four error levels, so the linear scaling should be presented as tentative, not a law. Equation (12) says a percentage distance error is drawn but writes dist_new = dist + epsilon_dist; as written that would have negligible effect, and the reported fc/a=1.17 implies a multiplicative implementation. That needs to be fixed. The eccentricity-matching step (excluding central angles to match Li et al. 2021) is under-described; it is fine for this purpose but the authors should explain why it does not bias the exercise. On code: the manuscript gives a GitHub link (footnote 1), but I could not tell from the text how complete or documented it is; worth checking during review. These are all referee-level fixes, not load-bearing flaws.

Who should read this: anyone citing Sawala et al. (2023) on the transience of the VPOS, and anyone doing orbit integration of Milky Way satellites. I would bring it to a reading group and I would cite it in my own work. It is a serious paper and deserves a proper peer review. My verdict: send it out. The authors just need to tighten the scaling language and fix the distance-error equation.

Referee Report

0 major / 6 minor

Summary. This manuscript presents a controlled numerical experiment to test whether measurement errors and potential uncertainty can make an intrinsically stable plane of satellite galaxies appear dynamically unstable when analyzed with backward orbit integration. The authors simulate 25 test particles in a plane around a static, axisymmetric MWPotential2014 model, forward-integrate for 5 Gyr, mock-observe the final state by adding Gaussian proper-motion and distance errors, and then backward-integrate for 5 Gyr in one of three potentials (fiducial, ±40% halo mass). The control run with zero errors and the fiducial potential reproduces the forward-integrated c/a evolution exactly. The main findings are that proper-motion errors at the level of Gaia systematics (~0.04 mas/yr) increase the inferred plane width by ~70% at 3 Gyr, larger errors cause stronger widening, 5% distance errors produce a smaller but noticeable effect, and using a 40% lower-mass halo produces a strong apparent widening and increased inferred satellite escape. The authors conclude that an observed thickening after backward integration is not by itself evidence of dynamical instability or transience, and that error modeling must be included in such analyses.

Significance. The manuscript addresses a live controversy in the satellite-plane field and provides a clean, reproducible simulation that directly challenges the interpretation of backward-integration thickening as evidence for transience. Its strengths are the zero-error control experiment, the monotonic response to error amplitude, the explicit check of intrinsic plane height, and the public code repository. The paper's conclusion is logically conservative: the acknowledged omissions (LMC, triaxial/growing halo, dynamical friction, line-of-sight velocity errors) all add further sources of divergence between true and reconstructed orbits, so they would strengthen rather than weaken the insufficiency argument. If the results hold, the paper will raise the evidentiary bar for claims that the VPOS or GPoA are transient structures.

minor comments (6)

[Sect. 2.6, Eq. (12)] The distance-error implementation is underspecified. The text states that epsilon_dist is a percentage error drawn from a Gaussian, but Eq. (12) writes dist_new = dist + epsilon_dist, which as written is an absolute offset. The reported values (e.g., BI-13, f_c/a = 1.17) are only consistent with a multiplicative update; please rewrite the equation explicitly as dist_new = dist * (1 + epsilon_dist) and define epsilon_dist as a dimensionless percentage (e.g., 0.05 for 5%).
[Sect. 3.1 / Abstract] The statement that the plane width increases linearly with the proper-motion uncertainty is based on only four simulated error levels (0, 0.04, 0.08, 0.12 mas/yr) and no linear fit is shown. Please either provide a fit with uncertainties and goodness-of-fit, or soften the wording to 'monotonically increasing' or 'increasing approximately linearly over the tested range.'
[Sect. 2.4] The exclusion of theta in [-20°, 20°] to remove circular orbits changes the effective sampling distribution from a uniform distribution over [-theta_max, theta_max] to a truncated uniform distribution; this should be stated explicitly, and the eccentricity CDF comparison in Fig. 2 should note the truncation.
[Fig. 6 / Fig. A.1 / Fig. A.2] The legend label 'MW_less' should be 'MW_low' for consistency with Table 1; also, several places in Appendix A use 'theta_tan' where 'theta_max' is meant.
[Sect. 4] There are minor grammatical errors: 'Our models does not include' should be 'Our models do not include', and in Sect. 3.5 'this does exclusion not result' should be 'this exclusion does not result'.
[Sect. 1] In the introduction, 'In Sect. 1, we describe the methodology' should refer to Sect. 2, which is where the methodology is actually described.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the zero-error control (BI-01) independently validates the backward-integration pipeline, and the minor self-citation on error double-counting is not load-bearing.

full rationale

This paper is a controlled numerical experiment rather than a derivation with fitted constants, so the central claim does not reduce to its inputs. Intrinsically stable test-satellite planes are generated from stated initial conditions (Sect. 2.1), forward integrated, mock-observed by injecting controlled Gaussian errors (Eqs. 10-12), and then backward integrated. The zero-error control BI-01 gives Δc/a = 0.00 and f_c/a = 1.00 (Table A.1), confirming that the backward pipeline recovers the true forward evolution when no errors are present. The error runs then show f_c/a rising monotonically from 1.69 at 0.04 mas/yr to 3.32 at 0.12 mas/yr (Table A.1), which are genuine simulation outputs, not identities imposed by the setup. The only self-citation is the 'errors applied twice' argument (Pawlowski 2021a, coauthored by M. S. Pawlowski), invoked in Sect. 1 and the conclusion to interpret Monte Carlo resampling of already-error-affected proper motions. This is a logical point and is not the load-bearing support for the main experiment, which stands on the BI-01 control and the independent error-injection runs. The acknowledged limitations in Sect. 4 (omission of the LMC, spherical/static halo, no dynamical friction, no line-of-sight velocity errors) are physical simplifications that reduce external generalization but do not constitute circularity; if anything, additional dynamical complexity would create further divergence between true and reconstructed orbits, which is consistent with the paper's insufficiency argument. One minor presentation inconsistency is Eq. 12, which writes dist_new = dist + epsilon_dist while the text says a percentage error is drawn; the reported f_c/a = 1.17 for BI-13 indicates a multiplicative implementation. This is a clarity/correctness ambiguity, not a circular step.

Assumptions & free parameters 8 free parameters · 8 assumptions · 0 invented entities

No new particles, forces, mediators, or conserved quantities are introduced. The only constructed objects are idealized satellite planes and rescaled halo potential models, which are model inputs rather than physical entities.

free parameters (8)

Radial power-law index alpha = -3
Chosen for the satellite radial distribution in Eq. 1 to mimic the Milky Way; no experiment varies alpha, so the main claim is only demonstrated for this profile.
Orbital anisotropy theta_max = 80 degrees for the main results (40 and 60 degrees in appendix)
Selected by matching the simulated eccentricity CDF to Li et al. (2021) data; the central conclusions are shown to be qualitatively robust to this choice.
Intrinsic plane height h = 20 kpc (tests at 10 and 30 kpc)
Chosen to bracket observed plane thicknesses; the apparent broadening is quantitatively different but present at all three heights.
Number of satellites N_sat = 25
Comparable to the number of bright MW satellites; no test of sensitivity to N_sat is reported.
Proper-motion error grid epsilon_mu = 0.00, 0.04, 0.08, 0.12 mas/yr
Input error levels chosen to bracket Gaia DR2/EDR3 statistical and systematic uncertainties; the claimed linear scaling is an interpolation across these four levels.
Distance error epsilon_dist = 0% and 5% (10% in appendix)
5% represents a typical distance uncertainty; the effect is found subdominant to proper motion errors.
Integration time = 5 Gyr
Standard choice for satellite orbit stability studies; the reported delta_c/a and f_c/a are evaluated at 3 Gyr.
Escape threshold radius = 300 kpc
Defines inferred escaping satellites; Appendix B.3 shows the c/a trend is not sensitive to excluding satellites beyond this radius.

assumptions (8)

standard math Time-reversibility of orbits in a static potential: backward integration exactly recovers initial conditions when no errors are applied.
Used in Sect. 2.6 and demonstrated by the epsilon_mu=0 control in Sect. 3.1.
standard math The moment-of-inertia eigenanalysis (Metz et al. 2007) yields a meaningful c/a measure of plane thickness.
Invoked in Sect. 2.2 as the sole structural diagnostic.
domain assumption MWPotential2014 with the halo mass rescaled by +/- 40% brackets the plausible Milky Way mass range.
Adopted in Sect. 2.3; the asymmetry between MW_low and MW_high is a central result.
domain assumption Measurement uncertainties can be represented as independent Gaussian perturbations to proper motion and distance.
Assumed in Sect. 2.5; correlated errors are tested only in Appendix B.1.
domain assumption Satellites can be treated as non-interacting test particles in a static, spherical potential; massive perturbers, dynamical friction, triaxiality, and halo growth are negligible.
Invoked throughout Sects. 2.3 and 2.6 and acknowledged as limitations in Sect. 4.
domain assumption Line-of-sight velocity errors are small enough to omit from the mock observations.
Only proper motion and distance errors are added in Eqs. 10 to 12; no line-of-sight velocity error is introduced or justified.
ad hoc to paper Excluding theta in -20 to +20 degrees for eccentricity calibration does not bias the orbit population used in the main experiments.
The exclusion is introduced in Sect. 2.4 to avoid circular orbits in the comparison with Li et al. (2021), but its effect on the results is not quantified.
domain assumption Gaussian error injection followed by backward integration in the same forward potential is a fair mock-observation protocol.
Central methodology of Sect. 2.6; it is appropriate for isolating error effects but not for predicting real Milky Way plane evolution.

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Cite this review

Pith. "Pith review of The effect of measurement uncertainties on the inferred stability of planes of satellite galaxies." pith.science (2026). https://pith.science/paper/P4M7ELGT

@misc{pith2026250601459,
  author       = {Pith},
  title        = {Pith review of: The effect of measurement uncertainties on the inferred stability of planes of satellite galaxies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P4M7ELGT}},
  note         = {Machine review of arXiv:2506.01459}
}

read the original abstract

Observations have revealed that the MW, Andromeda, Centaurus A (and potentially other galaxies) host spatially thin and kinematically coherent planes of satellites. Such structures are highly improbable within the standard LCDM cosmological model, and the dynamical stability of these planes has been a subject of debate for a long time. Accurately determining their stability requires a thorough understanding of orbital parameters such as proper motion, distance, and line-of-sight velocity, in addition to the gravitational potential of the host galaxy. However, many of these remain insufficiently constrained, leading to significant uncertainties in any analysis. This research aims to explore the impact of measurement errors in proper motions and distances of the satellite galaxies and in the adopted host halo mass on the inferred stability of satellite planes in Milky-Way-like potentials. Test satellite galaxies orbiting a host galaxy are simulated, mock observed by adding various degrees and types of observational errors, and then backward-integrated. Trends and correlations between the initial conditions and the applied uncertainties on the inferred orbital stability of the satellite systems are analyzed. Additionally, the effects of adopting incorrect potentials and the impact of different orbital eccentricities are considered. Uncertainties in proper motions lead to an inferred, ostensible widening of an intrinsically stable satellite plane, with its width increasing linearly with the adopted proper motion uncertainties. Even uncertainties on the level of Gaia systematics strongly affect the plane's inferred past width. Moreover, the potential with a low halo mass showed a significant impact on the stability of these planes, while the remaining two host models showed similar effects. Uncertainties in satellite distance also contribute noticeably to the inferred, apparent instability.

Figures

Figures reproduced from arXiv: 2506.01459 by the authors.

**Figure 1.** Face-on (left) and edge-on (right) views of Nsat = 25 randomly generated test satellites, each represented by a different color. The star symbol denotes the host galaxy. constitute a nonrandom, coherent structures. Moreover, the first proper motion measurements for three on-plane satellite galaxies of M31 indicate that these galaxies are consistent with coorbiting along the spatially identified GPoA (Sohn et al. 2… view at source ↗

**Figure 2.** Comparison of eccentricity (e) distributions across different θ. The figure presents the eccentricity CDF for three different θmax: 40°, 60°, and 80°, from left to right, respectively, shown in gray. The solid black line in each panel represents the observed data for Milky Way satellites from (Li et al. 2021), while the overlaid gray CDFs correspond to simulated results for Nsat test satellites. of the fitted ellips… view at source ↗

**Figure 3.** Effect of proper motion uncertainties on the plane of satellite galaxies. Each panel shows Nrealization forward integrations as black curves with the mean indicated by a dot-dashed line, and Mrealization backward integrations as green curves with the mean indicated by a dashed line. shown in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Effect of proper motion uncertainties on satellite galaxies under different initial vertical distributions of h = 10, h = 20 kpc, and h= 30 kpc. The left panel shows proper motion uncertainties of 0.04 mas yr−1 , while the right panel displays uncertainties of 0.08 mas…

**Figure 5.** Figure 5: Similar to [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Evolution of the plane of satellite galaxies with proper motion uncertainties included in each panel. The black dashed-dotted curves represent the mean axis ratio for forward integration, while the colored dashed curves show the mean axis ratio for backward integration…

**Figure 7.** Figure 7: Similar to 6, but here we include a 5% uncertainty in distances. −4 −2 0 t [Gyr] 0 100 200 300 400 a [kpc] µ = ±0.0 mas yr−1 −4 −2 0 t [Gyr] 0 100 200 300 400 b [kpc] µ = ±0.0 mas yr−1 −4 −2 0 t [Gyr] 0 100 200 300 400 c [kpc] µ = ±0.0 mas yr−1 −4 −2 0 t [Gyr] 0 100…

**Figure 8.** Figure 8: Comparative analysis of galactic orbital axes under proper motion uncertainties and different potential models at θtan = 80°. Panels shows the mean values of 600 major, minor, and intermediate axes calculated through backward integration. Each panel corresponds to a un…

**Figure 9.** Figure 9: Average number of test satellite exceeding a radial distance of 300 kpc under varying proper motion uncertainties. The four panels correspond to different level of proper motion uncertainty. Each panel displays six curves, representing the number of test satellites bey…

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Works this paper leans on

71 extracted references · 56 canonical work pages

[1]

A., Aghanim, N., Arnaud, M., et al

Ade, P. A., Aghanim, N., Arnaud, M., et al. 2016, A&A, 594, A13

work page 2016
[2]

H., Brooks, A

Ahmed, S. H., Brooks, A. M., & Christensen, C. R. 2017, MNRAS, 466, 3119 Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167 Astropy Collaboration, Price-Whelan, A. M., Sip ˝ocz, B. M., et al. 2018, ApJ, 156, 123 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33

work page 2017
[3]

2022, MNRAS, 513, 129

Banik, I., Thies, I., Truelove, R., et al. 2022, MNRAS, 513, 129

work page 2022
[4]

F., & Fritz, T

Battaglia, G., Taibi, S., Thomas, G. F., & Fritz, T. K. 2022, A&A, 657, A54

2022
[5]

& Gerhard, O

Bland-Hawthorn, J. & Gerhard, O. 2016, ARA&A, 54, 529

2016
[6]

HST Proper Motion of Andromeda III: Another Satellite Co-orbiting The M31 Satellite Plane

Casetti-Dinescu, D. I., Pawlowski, M. S., Girard, T. M., et al. 2024, arXiv e- prints, arXiv:2409.08252

work page Pith review arXiv 2024
[7]

J., et al

Cautun, M., Benítez-Llambay, A., Deason, A. J., et al. 2020, MNRAS, 494, 4291

work page 2020
[8]

& Frenk, C

Cautun, M. & Frenk, C. S. 2017, MNRAS, 468, L41

work page 2017

Show all 71 references

[9]

R., Lewis, G

Conn, A. R., Lewis, G. F., Ibata, R. A., et al. 2013, ApJ, 766, 120 D’Onghia, E. & Lake, G. 2008, ApJ, 686, L61 D’Souza, R. & Bell, E. F. 2022, MNRAS, 512, 739

2013
[10]

A., & Parkin, D

Erkal, D., Belokurov, V . A., & Parkin, D. L. 2020, MNRAS, 498, 5574

2020
[11]

& Fuchs, B

Esquivel, O. & Fuchs, B. 2007, MNRAS, 378, 1191 Article number, page 12 of 20 Kumar et al.: Effects of uncertainties on inferred satellite plane stability

2007
[12]

2012, MNRAS, 427, 1769

Fouquet, S., Hammer, F., Yang, Y ., Puech, M., & Flores, H. 2012, MNRAS, 427, 1769

2012
[13]

K., Battaglia, G., Pawlowski, M

Fritz, T. K., Battaglia, G., Pawlowski, M. S., et al. 2018, A&A, 619, A103 Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018a, A&A, 616, A1 Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2016a, A&A, 595, A2 Gaia Collaboration, Helmi, A., van Leeuwen, ...

2018
[14]

2021, ApJ, 923, 140

Garavito-Camargo, N., Patel, E., Besla, G., et al. 2021, ApJ, 923, 140

2021
[15]

2025, scipy/scipy: SciPy 1.15.2

Gommers, R., Virtanen, P., Haberland, M., et al. 2025, scipy/scipy: SciPy 1.15.2

2025
[16]

S., et al

Hammer, F., Wang, J., Pawlowski, M. S., et al. 2021, ApJ, 922, 93

2021
[17]

R., Millman, K

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

2020
[18]

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

2007
[19]

A., Gilmore, G., & Irwin, M

Ibata, R. A., Gilmore, G., & Irwin, M. J. 1994, Nature, 370, 194

1994
[20]

A., Lewis, G

Ibata, R. A., Lewis, G. F., Conn, A. R., et al. 2013, Nature, 493, 62

2013
[21]

Joshi, Y . C. 2007, MNRAS, 378, 768 Júlio, M. P., Pawlowski, M. S., Tony Sohn, S., et al. 2024, A&A, 687, A212

2007
[22]

Karachentsev, I. D. & Kroupa, P. 2024, MNRAS, 528, 2805

2024
[23]

Kroupa, P., Theis, C., & Boily, C. M. 2005, A&A, 431, 517

2005
[24]

R., Majewski, S

Law, D. R., Majewski, S. R., & Johnston, K. V . 2009, ApJ, 703, L67

2009
[25]

2021, ApJ, 916, 8

Li, H., Hammer, F., Babusiaux, C., et al. 2021, ApJ, 916, 8

2021
[26]

& Helmi, A

Li, Y .-S. & Helmi, A. 2008, MNRAS, 385, 1365

2008
[27]

I., Knebe, A., Hoffman, Y ., et al

Libeskind, N. I., Knebe, A., Hoffman, Y ., et al. 2011, MNRAS, 411, 1525

2011
[28]

A., Hernández, J., et al

Lindegren, L., Klioner, S. A., Hernández, J., et al. 2021, A&A, 649, A2

2021
[29]

R., Eke, V

Lovell, M. R., Eke, V . R., Frenk, C. S., & Jenkins, A. 2011, MNRAS, 413, 3013

2011
[30]

1976, MNRAS, 174, 695

Lynden-Bell, D. 1976, MNRAS, 174, 695

1976
[31]

2017, ApJ, 843, 62

Maji, M., Zhu, Q., Marinacci, F., & Li, Y . 2017, ApJ, 843, 62

2017
[32]

McConnachie, A. W. & Venn, K. A. 2020, Research Notes of the American Astronomical Society, 4, 229

2020
[33]

2007, MNRAS, 374, 1125

Metz, M., Kroupa, P., & Jerjen, H. 2007, MNRAS, 374, 1125

2007
[34]

Metz, M., Kroupa, P., & Libeskind, N. I. 2008, ApJ, 680, 287

2008
[35]

2009, ApJ, 697, 269

Metz, M., Kroupa, P., Theis, C., Hensler, G., & Jerjen, H. 2009, ApJ, 697, 269

2009
[36]

& Nagai, R

Miyamoto, M. & Nagai, R. 1975, PASJ, 27, 533 Müller, O., Pawlowski, M. S., Jerjen, H., & Lelli, F. 2018, Science, 359, 534 Müller, O., Pawlowski, M. S., Lelli, F., et al. 2021, A&A, 645, L5

1975
[37]

1996, in IAU Symposium, V ol

Navarro, J. 1996, in IAU Symposium, V ol. 171, New Light on Galaxy Evolution, ed. R. Bender & R. L. Davies, 255

1996
[38]

1951, Notes by GE Forsythe, 36

Neumann, V . 1951, Notes by GE Forsythe, 36

1951
[39]

& Burkert, A

Ogiya, G. & Burkert, A. 2016, MNRAS, 457, 2164

2016
[40]

B., Erkal, D., & Li, T

Pace, A. B., Erkal, D., & Li, T. S. 2022, ApJ, 940, 136

2022
[41]

Pawlowski, M. S. 2018, Modern Physics Letters A, 33, 1830004

2018
[42]

S., Bullock, J

Pawlowski, M. S., Bullock, J. S., Kelley, T., & Famaey, B. 2019, ApJ, 875, 105

2019
[43]

S., Dabringhausen, J., Famaey, B., et al

Pawlowski, M. S., Dabringhausen, J., Famaey, B., et al. 2017, Astronomische Nachrichten, 338, 854

2017
[44]

S., Famaey, B., Merritt, D., & Kroupa, P

Pawlowski, M. S., Famaey, B., Merritt, D., & Kroupa, P. 2015, ApJ, 815, 19

2015
[45]

S., Kroupa, P., & de Boer, K

Pawlowski, M. S., Kroupa, P., & de Boer, K. S. 2011, A&A, 532, A118

2011
[46]

S., Kroupa, P., & Jerjen, H

Pawlowski, M. S., Kroupa, P., & Jerjen, H. 2013, MNRAS, 435, 1928

2013
[47]

S., Müller, O., Taibi, S., et al

Pawlowski, M. S., Müller, O., Taibi, S., et al. 2024, A&A, 688, A153

2024
[48]

S., Oria, P.-A., Taibi, S., Famaey, B., & Ibata, R

Pawlowski, M. S., Oria, P.-A., Taibi, S., Famaey, B., & Ibata, R. 2022, ApJ, 932, 70

2022
[49]

S., Pflamm-Altenburg, J., & Kroupa, P

Pawlowski, M. S., Pflamm-Altenburg, J., & Kroupa, P. 2012, MNRAS, 423, 1109

2012
[50]

Pawlowski, M. S. & Tony Sohn, S. 2021, ApJ, 923, 42

2021
[51]

A., Cole, S., Norberg, P., et al

Peacock, J. A., Cole, S., Norberg, P., et al. 2001, Nature, 410, 169

2001
[52]

Price-Whelan, A. M. 2017, The Journal of Open Source Software, 2, 388

2017
[53]

G., Filippenko, A

Riess, A. G., Filippenko, A. V ., Challis, P., et al. 1998, AJ, 116, 1009

1998
[54]

B., Wetzel, A., Tollerud, E., et al

Santistevan, I. B., Wetzel, A., Tollerud, E., et al. 2024, MNRAS, 527, 8841

2024
[55]

2023, Nature Astronomy, 7, 481 Schönrich, R., Binney, J., & Dehnen, W

Sawala, T., Cautun, M., Frenk, C., et al. 2023, Nature Astronomy, 7, 481 Schönrich, R., Binney, J., & Dehnen, W. 2010, MNRAS, 403, 1829

2023
[56]

S., et al

Shao, S., Cautun, M., Frenk, C. S., et al. 2018, MNRAS, 476, 1796

2018
[57]

1938, Nature, 142, 715

Shapley, H. 1938, Nature, 142, 715

1938
[58]

Simon, J. D. 2019, ARA&A, 57, 375

2019
[59]

A., Bournaud, F., & Yi, S

Smith, R., Duc, P. A., Bournaud, F., & Yi, S. K. 2016, ApJ, 818, 11

2016
[60]

T., Patel, E., Fardal, M

Sohn, S. T., Patel, E., Fardal, M. A., et al. 2020, ApJ, 901, 43

2020
[61]

N., Verde, L., Peiris, H

Spergel, D. N., Verde, L., Peiris, H. V ., et al. 2003, ApJS, 148, 175

2003
[62]

S., Khoperskov, S., Steinmetz, M., & Libeskind, N

Taibi, S., Pawlowski, M. S., Khoperskov, S., Steinmetz, M., & Libeskind, N. I. 2024, A&A, 681, A73

2024
[63]

Taylor, J. E. & Babul, A. 2001, ApJ, 559, 716

2001
[64]

R., Strauss, M

Tegmark, M., Blanton, M. R., Strauss, M. A., et al. 2004, ApJ, 606, 702

2004
[65]

B., Libeskind, N

Tully, R. B., Libeskind, N. I., Karachentsev, I. D., et al. 2015, ApJ, 802, L25 Van Rossum, G., Drake, F. L., et al. 1995, Python reference manual, V ol. 111 (Centrum voor Wiskunde en Informatica Amsterdam)

2015
[66]

2023, Galaxies, 11, 59

Vasiliev, E. 2023, Galaxies, 11, 59

2023
[67]

E., et al

Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261 von Humboldt, A. & Otte, E. C., eds. 1997, Cosmos: a sketch of the physical description of the universe

2020
[68]

2024, TomWagg/software-citation- station: v1.2

Wagg, T., Broekgaarden, F., & Gültekin, K. 2024, TomWagg/software-citation- station: v1.2

2024
[69]

& Broekgaarden, F

Wagg, T. & Broekgaarden, F. S. 2024, arXiv e-prints, arXiv:2406.04405

2024
[70]

S., & Cooper, A

Wang, J., Frenk, C. S., & Cooper, A. P. 2013, MNRAS, 429, 1502

2013
[71]

Wang, W., Han, J., Cautun, M., Li, Z., & Ishigaki, M. N. 2020, Science China

2020

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

A., Aghanim, N., Arnaud, M., et al

Ade, P. A., Aghanim, N., Arnaud, M., et al. 2016, A&A, 594, A13

work page 2016

[2] [2]

H., Brooks, A

Ahmed, S. H., Brooks, A. M., & Christensen, C. R. 2017, MNRAS, 466, 3119 Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167 Astropy Collaboration, Price-Whelan, A. M., Sip ˝ocz, B. M., et al. 2018, ApJ, 156, 123 Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33

work page 2017

[3] [3]

2022, MNRAS, 513, 129

Banik, I., Thies, I., Truelove, R., et al. 2022, MNRAS, 513, 129

work page 2022

[4] [4]

F., & Fritz, T

Battaglia, G., Taibi, S., Thomas, G. F., & Fritz, T. K. 2022, A&A, 657, A54

2022

[5] [5]

& Gerhard, O

Bland-Hawthorn, J. & Gerhard, O. 2016, ARA&A, 54, 529

2016

[6] [6]

HST Proper Motion of Andromeda III: Another Satellite Co-orbiting The M31 Satellite Plane

Casetti-Dinescu, D. I., Pawlowski, M. S., Girard, T. M., et al. 2024, arXiv e- prints, arXiv:2409.08252

work page Pith review arXiv 2024

[7] [7]

J., et al

Cautun, M., Benítez-Llambay, A., Deason, A. J., et al. 2020, MNRAS, 494, 4291

work page 2020

[8] [8]

& Frenk, C

Cautun, M. & Frenk, C. S. 2017, MNRAS, 468, L41

work page 2017