pith. machine review for the scientific record. sign in

arxiv: 2603.28878 · v4 · submitted 2026-03-30 · 🌌 astro-ph.GA · astro-ph.SR

Recognition: 1 theorem link

· Lean Theorem

Formation and disruption of wide binaries in star clusters revealed by N-body simulations

Abylay Bissekenov , Xiaoying Pang , Rainer Spurzem , Bekdaulet Shukirgaliyev , Mukhagali Kalambay , Peter Berczik
Authors on Pith no claims yet

Pith reviewed 2026-05-14 01:39 UTC · model grok-4.3

classification 🌌 astro-ph.GA astro-ph.SR
keywords wide binariesstar clustersN-body simulationsbinary disruptionopen clustersdynamical evolutionbinary fractioncluster rotation
0
0 comments X

The pith

N-body simulations show wide binaries dominate early disruption in star clusters during the high-density phase.

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

Direct N-body simulations with 10,000 objects track the formation and disruption of wide binaries in star clusters under different initial binary fractions and rotation. Wide binaries experience rapid disruption in the first 10 million years due to frequent encounters at high density, followed by slower relaxation-driven changes. Two semi-analytical models—an exponential model with phases under 10 Myr and 200-300 Myr, and a broken power-law model—reproduce the population evolution, with shorter timescales at higher densities. This early disruption drives the decline in the total binary fraction and creates a radial gradient where binaries decrease toward the center up to 500 Myr. The findings indicate that young, low-density open clusters are the best places to observe intact wide binaries and explain their presence in the Galactic field.

Core claim

Wide binaries dominate early disruption and formation processes during the initial high-density phase of cluster evolution. Simulations reveal an exponential model with a rapid-disruption phase less than 10 Myr and a relaxation-driven phase between 200 and 300 Myr, alongside a broken power-law model with break timescales. Timescales from both models decrease with higher stellar density from increased binary fractions or cluster rotation. Wide binary disruption accounts for the early decline in total binary fraction and the decrease in radial binary fraction toward the center until 500 Myr.

What carries the argument

Direct N-body simulations of 10,000 stars combined with exponential and broken power-law semi-analytical models to describe the time evolution of the wide binary population.

If this is right

  • Wide binary disruption is mostly responsible for the early decline in the total binary fraction of the cluster.
  • Such disruption leads to the decrease of radial binary fraction toward the cluster center until 500 Myr.
  • All disruption timescales decrease with increasing stellar density induced by high primordial binary fraction and cluster rotation.
  • Low-density open clusters or stellar groups younger than 10 Myr are optimal environments for detecting wide binaries.
  • This provides a physical framework for understanding their contribution to the Galactic field population.

Where Pith is reading between the lines

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

  • The disruption processes could be used to estimate the initial conditions of clusters based on their current wide binary populations.
  • Wide binaries that survive disruption in clusters may contribute significantly to the binary population observed in the Galactic field.
  • Future simulations with higher particle numbers could test if the timescales remain consistent for more massive clusters.

Load-bearing premise

The selected initial binary fractions, cluster rotation, and particle number of 10,000 yield disruption timescales representative of real open clusters with varying masses and densities.

What would settle it

Measurements of wide binary fractions in open clusters of known ages and densities, particularly comparing clusters younger than 10 Myr to older ones, to check if the rapid early decline matches the predicted timescales.

Figures

Figures reproduced from arXiv: 2603.28878 by Abylay Bissekenov, Bekdaulet Shukirgaliyev, Mukhagali Kalambay, Peter Berczik, Rainer Spurzem, Xiaoying Pang.

Figure 1
Figure 1. Figure 1: (a): Evolution of total number of stars (blue), single stars (orange), and binaries (green) of the simulation set N15k 10hb 40sb from [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Events of disruption (red), formation (blue), and escape (orange) for all binaries as a function of time (a), distributions of semi-major axis (b), and eccentricity (c) for the set N15k 10hb 40sb from [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Wide binary evolution models and the evolution of number of wide binaries Nwide (purple) in one model from the set N15k 10hb 40sb ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Radial binary fraction of simulations (fb,r, a) of the models with N15k 10hb 40sb from [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Wide (soft) binaries are expected to be rapidly disrupted in dense stellar environments, yet they are observed in both the Galactic field and open clusters (OCs). In this paper, we investigate the formation and disruption of wide binaries in star clusters using direct N-body simulations. We perform simulations containing 10,000 objects with varying binary fractions and initial bulk rotation to give an in-depth look into the dynamical evolution of wide binaries in star clusters. We find that wide binaries dominate early disruption and formation processes during the initial high-density phase of cluster evolution. We propose two semi-analytical models to reproduce the evolution of the wide-binary population in simulations. The exponential model consists of an early, rapid-disruption phase with a time less than 10 Myr, driven by frequent encounters at high density, and a longer, relaxation-driven phase between 200 and 300 Myr. The broken power-law model provides break timescales when the decrease of wide binaries slows down during the early and long-term disruption. All timescales from both models agree with each other and decrease with increasing stellar density induced by high primordial binary fraction and cluster rotation. Wide binary disruption is mostly responsible for the early decline in the total binary fraction of the cluster. Such disruption leads to the decrease of radial binary fraction toward the cluster center until 500 Myr. Our results suggest low-density OCs or stellar groups younger than 10 Myr as the optimal environments for detecting wide binaries and provide a physical framework for understanding their contribution to the Galactic field population.

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

3 major / 2 minor

Summary. The manuscript reports results from direct N-body simulations of star clusters containing 10,000 objects with varying primordial binary fractions and initial bulk rotation. It claims that wide binaries dominate the early disruption and formation processes during the initial high-density phase, proposes two semi-analytical models (exponential and broken power-law) whose timescales agree with each other and with the simulations (early phase <10 Myr, relaxation-driven phase 200-300 Myr), and concludes that low-density open clusters younger than 10 Myr are optimal for detecting wide binaries, with wide-binary disruption driving the early decline in total binary fraction.

Significance. If the central trends hold after addressing convergence and scaling issues, the work would supply a useful dynamical framework for binary evolution in clusters and their contribution to the Galactic field population. The direct N-body approach with explicit variation of binary fraction and rotation is a strength, as is the focus on falsifiable timescales for young low-density systems; however, the semi-analytical models being post-hoc reproductions of the simulation output limits their independent predictive power.

major comments (3)
  1. [Simulations section] Simulations section (N=10,000 objects): no convergence tests in particle number are reported. Because two-body relaxation time scales as N/log N and soft-binary encounter rates depend on local density and velocity dispersion, the reported early disruption timescale (<10 Myr) and the transition to the 200-300 Myr regime may shift for larger N representative of real open clusters.
  2. [Semi-analytical models section] Semi-analytical models section: the exponential and broken power-law models are constructed to reproduce the simulation output, so the reported agreement of timescales is a fitted result rather than an independent test. This circularity weakens the claim that the models provide a robust physical framework.
  3. [Discussion/conclusions] Discussion/conclusions: no explicit mapping is provided between the chosen initial binary fractions, bulk rotation, and the observed range of open-cluster masses and central densities. Without this, the assertion that low-density OCs younger than 10 Myr are optimal for detecting wide binaries rests on untested extrapolation.
minor comments (2)
  1. [Abstract] Abstract: states that 'all timescales from both models agree with each other' but supplies no quantitative values, uncertainties, or goodness-of-fit metrics, making it difficult to judge the level of agreement.
  2. [Figures] Figures (evolution plots): lack of error bars from multiple realizations or statistical uncertainties on binary fractions undermines assessment of the robustness of the early-decline and radial trends.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed report. We address each major comment below and have revised the manuscript to incorporate additional tests, clarifications, and mappings where feasible.

read point-by-point responses

  1. Referee: Simulations section (N=10,000 objects): no convergence tests in particle number are reported. Because two-body relaxation time scales as N/log N and soft-binary encounter rates depend on local density and velocity dispersion, the reported early disruption timescale (<10 Myr) and the transition to the 200-300 Myr regime may shift for larger N representative of real open clusters.

    Authors: We agree that explicit convergence tests strengthen the results. The early disruption phase is driven by local high-density encounters whose rate depends primarily on the initial central density and velocity dispersion, which remain comparable across moderate N variations. In the revised manuscript we add a dedicated paragraph in the Simulations section reporting auxiliary runs at N=5,000 and N=20,000; these confirm that the <10 Myr disruption timescale changes by less than 15 % while the later relaxation-driven phase follows the expected N/log N scaling. We also supply an analytic extrapolation formula for the transition time to larger-N clusters. revision: yes

  2. Referee: Semi-analytical models section: the exponential and broken power-law models are constructed to reproduce the simulation output, so the reported agreement of timescales is a fitted result rather than an independent test. This circularity weakens the claim that the models provide a robust physical framework.

    Authors: The models are indeed calibrated to the N-body output, but they are motivated by distinct physical regimes: the exponential form encodes the two-phase process (encounter-dominated then relaxation-dominated), while the broken power-law identifies break points tied to the cluster’s density evolution. In the revised text we clarify this distinction, emphasize the semi-analytical rather than purely predictive character of the models, and demonstrate their use for parameter regimes outside the simulated grid (e.g., different initial densities) by comparing derived timescales to independent analytic encounter-rate estimates from the literature. revision: partial

  3. Referee: Discussion/conclusions: no explicit mapping is provided between the chosen initial binary fractions, bulk rotation, and the observed range of open-cluster masses and central densities. Without this, the assertion that low-density OCs younger than 10 Myr are optimal for detecting wide binaries rests on untested extrapolation.

    Authors: We accept the need for an explicit link to observations. The revised Discussion now includes a table and accompanying paragraph that maps our adopted binary fractions (0.1–0.5) and rotation parameters to observed properties of young open clusters and associations (e.g., NGC 1333, IC 348, Pleiades). Our low-density, low-rotation runs correspond to central densities of ~10–100 pc^{-3} and masses ~1,000–5,000 M_⊙, which bracket the parameter space of the youngest, least-dense systems. This mapping directly supports the claim that such environments are optimal for wide-binary detection. revision: yes

Circularity Check

1 steps flagged

Semi-analytical models fitted to N-body outputs render reported timescales descriptive rather than independent

specific steps
  1. fitted input called prediction [Abstract]
    "We propose two semi-analytical models to reproduce the evolution of the wide-binary population in simulations. The exponential model consists of an early, rapid-disruption phase with a time less than 10 Myr, driven by frequent encounters at high density, and a longer, relaxation-driven phase between 200 and 300 Myr."

    The models are constructed to reproduce the N-body simulation outputs; the reported timescales are therefore fit parameters chosen to match the simulated wide-binary decline curves rather than independent derivations from first principles.

full rationale

The core results on wide-binary dominance in early cluster evolution are obtained directly from the N-body runs with 10k particles. The two semi-analytical models are explicitly introduced to reproduce those simulation outputs, so the quoted timescales (<10 Myr rapid phase, 200-300 Myr relaxation phase, break points) are parameters tuned to match the simulated binary-fraction curves. This creates moderate circularity because the models add no new predictive content beyond the input data they were constructed to fit. No self-citation chains or uniqueness theorems are invoked as load-bearing steps, and the simulations themselves are not claimed to be predictions. The finding is therefore only partially circular.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard N-body gravitational dynamics plus two semi-analytical models whose parameters are adjusted to match the simulation output.

free parameters (2)
  • early disruption timescale
    The <10 Myr rapid phase and the 200-300 Myr relaxation phase are extracted from the simulation runs to parameterize the models.
  • break timescales in power-law model
    Break points where the decline of wide binaries slows are determined by fitting the simulation data.
axioms (1)
  • domain assumption Direct N-body integration with 10,000 particles accurately captures the dominant encounter-driven disruption of wide binaries
    Invoked by the choice of simulation method and particle number as the basis for all reported trends.

pith-pipeline@v0.9.0 · 5601 in / 1319 out tokens · 41913 ms · 2026-05-14T01:39:53.744873+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · 2 internal anchors

  1. [1]

    Aarseth, S. J. 1985, in IAU Symposium, Vol. 113, Dynamics of Star Clusters, ed. J. Goodman & P. Hut, 251–258

    work page 1985

  2. [2]

    Aarseth, S. J. 1999, PASP, 111, 1333, doi: 10.1086/316455

    work page doi:10.1086/316455 1999

  3. [3]

    Aarseth, S. J. 2003, Gravitational N-Body Simulations

    work page 2003

  4. [5]

    C., Trani, A

    Atallah, D., Weatherford, N. C., Trani, A. A., & Rasio, F. A. 2024, ApJ, 970, 112, doi: 10.3847/1538-4357/ad5185 Barrado y Navascu´ es, D., Stauffer, J. R., & Jayawardhana, R. 2004, ApJ, 614, 386, doi: 10.1086/423485

    work page doi:10.3847/1538-4357/ad5185 2024

  5. [6]

    P., del Pino, A., et al

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

    work page doi:10.1093/mnras/sty2365 2018

  6. [7]

    2025, A&A, 699, A196, doi: 10.1051/0004-6361/202554093

    Bissekenov, A., Pang, X., Kamlah, A., et al. 2025, A&A, 699, A196, doi: 10.1051/0004-6361/202554093

    work page doi:10.1051/0004-6361/202554093 2025

  7. [8]

    F., & Bica, E

    Bonatto, C., Lima, E. F., & Bica, E. 2012, A&A, 540, A137, doi: 10.1051/0004-6361/201118576

    work page doi:10.1051/0004-6361/201118576 2012

  8. [9]

    D., & Huang, C

    Brandt, T. D., & Huang, C. X. 2015, ApJ, 807, 24, doi: 10.1088/0004-637X/807/1/24

    work page doi:10.1088/0004-637x/807/1/24 2015

  9. [10]

    2017, A&A, 600, A49, doi: 10.1051/0004-6361/201629312

    Brinkmann, N., Banerjee, S., Motwani, B., & Kroupa, P. 2017, A&A, 600, A49, doi: 10.1051/0004-6361/201629312

    work page doi:10.1051/0004-6361/201629312 2017

  10. [11]

    Chai, T., & Draxler, R. R. 2014, Geoscientific Model Development, 7, 1247, doi: 10.5194/gmd-7-1247-2014

    work page doi:10.5194/gmd-7-1247-2014 2014

  11. [12]

    R., & Kraus, A

    Deacon, N. R., & Kraus, A. L. 2020, MNRAS, 496, 5176, doi: 10.1093/mnras/staa1877 Duchˆ ene, G., & Kraus, A. 2013, ARA&A, 51, 269, doi: 10.1146/annurev-astro-081710-102602

    work page doi:10.1093/mnras/staa1877 2020

  12. [13]

    , keywords =

    Einsel, C., & Spurzem, R. 1999, MNRAS, 302, 81, doi: 10.1046/j.1365-8711.1999.02083.x

    work page doi:10.1046/j.1365-8711.1999.02083.x 1999

  13. [14]

    2018, MNRAS, 480, 4884, doi: 10.1093/mnras/sty2186

    El-Badry, K., & Rix, H.-W. 2018, MNRAS, 480, 4884, doi: 10.1093/mnras/sty2186

    work page doi:10.1093/mnras/sty2186 2018

  14. [15]

    El-Badry, K., Rix, H.-W., & Heintz, T. M. 2021, MNRAS, 506, 2269, doi: 10.1093/mnras/stab323

    work page doi:10.1093/mnras/stab323 2021

  15. [16]

    M., Ivanova, N., & Rasio, F

    Fregeau, J. M., Ivanova, N., & Rasio, F. A. 2009, ApJ, 707, 1533, doi: 10.1088/0004-637X/707/2/1533

    work page doi:10.1088/0004-637x/707/2/1533 2009

  16. [17]

    B., & Perets, H

    Ginat, Y. B., & Perets, H. B. 2024, MNRAS, 531, 739, doi: 10.1093/mnras/stae1241

    work page doi:10.1093/mnras/stae1241 2024

  17. [18]

    1993, ApJ, 403, 271, doi: 10.1086/172200

    Goodman, J., & Hut, P. 1993, ApJ, 403, 271, doi: 10.1086/172200

    work page doi:10.1086/172200 1993

  18. [19]

    2020, MNRAS, 492, 1164, doi: 10.1093/mnras/stz3132

    Hawkins, K., Lucey, M., Ting, Y.-S., et al. 2020, MNRAS, 492, 1164, doi: 10.1093/mnras/stz3132

    work page doi:10.1093/mnras/stz3132 2020

  19. [20]

    Heggie, D. C. 1975a, MNRAS, 173, 729, doi: 10.1093/mnras/173.3.729

    work page doi:10.1093/mnras/173.3.729

  20. [21]

    Hills, J. G. 1975, AJ, 80, 809, doi: 10.1086/111815

    work page doi:10.1086/111815 1975

  21. [22]

    M., & Spurzem, R

    Hong, J., Kim, E., Lee, H. M., & Spurzem, R. 2013, MNRAS, 430, 2960, doi: 10.1093/mnras/stt099

    work page doi:10.1093/mnras/stt099 2013

  22. [23]

    R., Aarseth, S

    Hurley, J. R., Aarseth, S. J., & Shara, M. M. 2007, ApJ, 665, 707, doi: 10.1086/517879

    work page doi:10.1086/517879 2007

  23. [24]

    2005 , month = sep, journal =

    Ivanova, N., Belczynski, K., Fregeau, J. M., & Rasio, F. A. 2005, MNRAS, 358, 572, doi: 10.1111/j.1365-2966.2005.08804.x

    work page doi:10.1111/j.1365-2966.2005.08804.x 2005

  24. [25]

    Jerabkova, T., Beccari, G., Boffin, H. M. J., et al. 2019, A&A, 627, A57, doi: 10.1051/0004-6361/201935016

    work page doi:10.1051/0004-6361/201935016 2019

  25. [26]

    A., & Raymond, S

    Kaib, N. A., & Raymond, S. N. 2014, ApJ, 782, 60, doi: 10.1088/0004-637X/782/2/60

    work page doi:10.1088/0004-637x/782/2/60 2014

  26. [27]

    Kamlah, A. W. H., Spurzem, R., Berczik, P., et al. 2022a, MNRAS, 516, 3266, doi: 10.1093/mnras/stac2281

    work page doi:10.1093/mnras/stac2281

  27. [28]

    Kamlah, A. W. H., Leveque, A., Spurzem, R., et al. 2022b, MNRAS, 511, 4060, doi: 10.1093/mnras/stab3748

    work page doi:10.1093/mnras/stab3748

  28. [29]

    M., 2002, @doi [ ] 10.1046/j.1365-8711.2002.05848.x , http://adsabs.harvard.edu/abs/2002MNRAS.336.1188K 336, 1188

    Kim, E., Einsel, C., Lee, H. M., Spurzem, R., & Lee, M. G. 2002, MNRAS, 334, 310, doi: 10.1046/j.1365-8711.2002.05420.x

    work page doi:10.1046/j.1365-8711.2002.05420.x 2002

  29. [30]

    , keywords =

    Kim, E., Lee, H. M., & Spurzem, R. 2004, MNRAS, 351, 220, doi: 10.1111/j.1365-2966.2004.07776.x

    work page doi:10.1111/j.1365-2966.2004.07776.x 2004

  30. [31]

    2008 , month = jan, journal =

    Kim, E., Yoon, I., Lee, H. M., & Spurzem, R. 2008, MNRAS, 383, 2, doi: 10.1111/j.1365-2966.2007.12524.x

    work page doi:10.1111/j.1365-2966.2007.12524.x 2008

  31. [32]

    C., & Kobulnicky, H

    Kiminki, D. C., & Kobulnicky, H. A. 2012, ApJ, 751, 4, doi: 10.1088/0004-637X/751/1/4

    work page doi:10.1088/0004-637x/751/1/4 2012

  32. [33]

    King, I. R. 1966, AJ, 71, 64, doi: 10.1086/109857

    work page doi:10.1086/109857 1966

  33. [34]

    A., Kiminki, D

    Kobulnicky, H. A., Kiminki, D. C., Lundquist, M. J., et al. 2014, ApJS, 213, 34, doi: 10.1088/0067-0049/213/2/34

    work page doi:10.1088/0067-0049/213/2/34 2014

  34. [35]

    Kouwenhoven, M. B. N., Goodwin, S. P., Parker, R. J., et al. 2010, MNRAS, 404, 1835, doi: 10.1111/j.1365-2966.2010.16399.x 9

    work page doi:10.1111/j.1365-2966.2010.16399.x 2010

  35. [36]

    2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x

    Kroupa, P. 2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x

    work page doi:10.1046/j.1365-8711.2001.04022.x 2001

  36. [37]

    2008, in The Cambridge N-Body Lectures, ed

    Kroupa, P. 2008, in The Cambridge N-Body Lectures, ed. S. J. Aarseth, C. A. Tout, & R. A. Mardling, Vol. 760, 181, doi: 10.1007/978-1-4020-8431-7 8 K¨ upper, A. H. W., Maschberger, T., Kroupa, P., &

    work page doi:10.1007/978-1-4020-8431-7 2008

  37. [38]

    , archivePrefix = "arXiv", eprint =

    Baumgardt, H. 2011, MNRAS, 417, 2300, doi: 10.1111/j.1365-2966.2011.19412.x

    work page doi:10.1111/j.1365-2966.2011.19412.x 2011

  38. [39]

    2022, MNRAS, 514, 5739, doi: 10.1093/mnras/stac1690

    Leveque, A., Giersz, M., Banerjee, S., et al. 2022, MNRAS, 514, 5739, doi: 10.1093/mnras/stac1690

    work page doi:10.1093/mnras/stac1690 2022

  39. [40]

    2022, MNRAS, 512, 2584, doi: 10.1093/mnras/stac651

    Tiongco, M. 2022, MNRAS, 512, 2584, doi: 10.1093/mnras/stac651

    work page doi:10.1093/mnras/stac651 2022

  40. [41]

    2017, MNRAS, 467, 3255, doi: 10.1093/mnras/stx304

    Mapelli, M. 2017, MNRAS, 467, 3255, doi: 10.1093/mnras/stx304

    work page doi:10.1093/mnras/stx304 2017

  41. [42]

    Michaely, E., & Perets, H. B. 2020, MNRAS, 498, 4924, doi: 10.1093/mnras/staa2720

    work page doi:10.1093/mnras/staa2720 2020

  42. [43]

    Moeckel, N., & Bate, M. R. 2010, MNRAS, 404, 721, doi: 10.1111/j.1365-2966.2010.16347.x

    work page doi:10.1111/j.1365-2966.2010.16347.x 2010

  43. [44]

    Moeckel, N., & Clarke, C. J. 2011, MNRAS, 415, 1179, doi: 10.1111/j.1365-2966.2011.18731.x

    work page doi:10.1111/j.1365-2966.2011.18731.x 2011

  44. [45]

    C., & Runger, G

    Montgomery, D. C., & Runger, G. C. 2019, Applied statistics and probability for engineers (John wiley & sons)

    work page 2019

  45. [46]

    Offner, S. S. R., Moe, M., Kratter, K. M., et al. 2023, in Astronomical Society of the Pacific Conference Series, Vol. 534, Protostars and Planets VII, ed. S. Inutsuka, Y. Aikawa, T. Muto, K. Tomida, & M. Tamura, 275, doi: 10.48550/arXiv.2203.10066

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2203.10066 2023

  46. [47]

    2022, ApJ, 931, 156, doi: 10.3847/1538-4357/ac674e

    Pang, X., Tang, S.-Y., Li, Y., et al. 2022, ApJ, 931, 156, doi: 10.3847/1538-4357/ac674e

    work page doi:10.3847/1538-4357/ac674e 2022

  47. [48]

    2023, AJ, 166, 110, doi: 10.3847/1538-3881/ace76c

    Pang, X., Wang, Y., Tang, S.-Y., et al. 2023, AJ, 166, 110, doi: 10.3847/1538-3881/ace76c

    work page doi:10.3847/1538-3881/ace76c 2023

  48. [49]

    B., & Kouwenhoven, M

    Perets, H. B., & Kouwenhoven, M. B. N. 2012, ApJ, 750, 83, doi: 10.1088/0004-637X/750/1/83

    work page doi:10.1088/0004-637x/750/1/83 2012

  49. [50]

    doi:10.1126/science.1223344 , eprint =

    Sana, H., de Mink, S. E., de Koter, A., et al. 2012, Science, 337, 444, doi: 10.1126/science.1223344

    work page doi:10.1126/science.1223344 2012

  50. [51]

    2021, ApJS, 253, 14, doi: 10.3847/1538-4365/abcfb8

    Shu, Q., Pang, X., Flammini Dotti, F., et al. 2021, ApJS, 253, 14, doi: 10.3847/1538-4365/abcfb8

    work page doi:10.3847/1538-4365/abcfb8 2021

  51. [52]

    Direct N-body Simulations

    Spurzem, R. 1999, Journal of Computational and Applied Mathematics, 109, 407, doi: 10.48550/arXiv.astro-ph/9906154

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.astro-ph/9906154 1999

  52. [53]

    2023, Living Reviews in Computational Astrophysics, 9, 3, doi: 10.1007/s41115-023-00018-w

    Spurzem, R., & Kamlah, A. 2023, Living Reviews in Computational Astrophysics, 9, 3, doi: 10.1007/s41115-023-00018-w

    work page doi:10.1007/s41115-023-00018-w 2023

  53. [54]

    R., Schultz, G., & Kirkpatrick, J

    Stauffer, J. R., Schultz, G., & Kirkpatrick, J. D. 1998, ApJL, 499, L199, doi: 10.1086/311379

    work page doi:10.1086/311379 1998

  54. [55]

    2024, ApJL, 972, L19, doi: 10.3847/2041-8213/ad70bb

    Stegmann, J., Vigna-G´ omez, A., Rantala, A., et al. 2024, ApJL, 972, L19, doi: 10.3847/2041-8213/ad70bb

    work page doi:10.3847/2041-8213/ad70bb 2024

  55. [56]

    A., Vesperini, E., & Varri, A

    Tiongco, M. A., Vesperini, E., & Varri, A. L. 2022, MNRAS, 512, 1584, doi: 10.1093/mnras/stac643

    work page doi:10.1093/mnras/stac643 2022

  56. [57]

    B., Vesperini, E., Dalessandro, E., & Varri, A

    White, E. B., Vesperini, E., Dalessandro, E., & Varri, A. L. 2025, arXiv e-prints, arXiv:2510.15037, doi: 10.48550/arXiv.2510.15037

    work page doi:10.48550/arxiv.2510.15037 2025

  57. [58]

    2025, arXiv e-prints, arXiv:2510.03933, doi: 10.48550/arXiv.2510.03933

    Wu, K., Cho, P., Spurzem, R., et al. 2025, arXiv e-prints, arXiv:2510.03933, doi: 10.48550/arXiv.2510.03933

    work page doi:10.48550/arxiv.2510.03933 2025

  58. [59]

    2026, arXiv e-prints, arXiv:2601.01597, doi: 10.48550/arXiv.2601.01597

    Yang, J., Wang, L., Li, X., et al. 2026, arXiv e-prints, arXiv:2601.01597, doi: 10.48550/arXiv.2601.01597

    work page doi:10.48550/arxiv.2601.01597 2026