Asymmetry of the tidal tails of open star clusters in direct N-body integrations in Milgrom-law dynamics

J. Pflamm-Altenburg

REVIEW 4 major objections 6 minor 1 cited by

Asymmetry of the tidal tails of open star clusters in direct N-body integrations in Milgrom-law dynamics

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Applying Milgrom's law to discrete stars, this paper finds that low-mass open clusters develop leading tidal tails with up to twice as many stars as the trailing tail and dissolve about 25% faster than Newtonian clusters.

desk verdict A useful first MLD N-body tool with sensible sanity checks, but the headline tail asymmetry and dissolution numbers rest on an ad hoc softening that does not actually suppress Newtonisation of close pairs. read the letter →

arxiv 2411.13675 v1 pith:JSOUDHIZ submitted 2024-11-20 astro-ph.GA

J. Pflamm-Altenburg This is my paper · ORCID

classification astro-ph.GA

keywords MONDmodifiedNewtoniandynamicsMilgrom-lawopenstarclusterstidaltailsN-bodysimulationsexternalfieldeffectclusterevaporation

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 aims to extend MOND-style gravity down to the discrete, low-mass regime where real open star clusters live. Its route is Milgrom-law dynamics (MLD): Milgrom's law, $\mu(|\mathbf{a}|/a_0)\mathbf{a}=\mathbf{g}$, is asserted as a vector equation for every star, with a small softening length to keep compact subsystems from behaving Newtonianly. Simulating a 400-particle, 200-solar-mass cluster on a circular orbit at 8300 parsecs in a flat rotation curve, the paper finds the leading tidal tail carries up to twice as many stars as the trailing one and the cluster dissolves about 25% faster than its Newtonian counterpart, retaining 26% instead of 41% of its stars after 1 Gyr. Because the same asymmetry already appears in QUMOND field-theory simulations of heavier clusters, the paper concludes lopsided tidal tails are a generic MOND effect rather than an artifact of one formulation. Nearby open clusters could therefore offer a small-scale observational test of MOND.

What carries the argument

The load-bearing object is the vectorial discrete Milgrom law, Eq. (10): $$\mu\left(\frac{|\mathbf{a}_i|}{a_0}\right)\!\mathbf{a}_i = G\sum_{j\neq i} \frac{m_j\,(\mathbf{r}_j-\mathbf{r}_i)}{|\mathbf{r}_j-\mathbf{r}_i|^3}.$$ This replaces the AQUAL/QUMOND field equations, which need smooth densities and become impractical below about 5000 $M_\odot$. The integration is a standard Hermite predictor-corrector scheme extended to supply both the MONDian acceleration and its jerk, with the standard interpolation function $\mu(x)=x/\sqrt{1+x^2}$. A softening length $\varepsilon=0.001$ pc ($\approx 206$ AU) is introduced both to regularize close encounters and, physically, to suppress the Newtonisation of compact subsystems, so the clusters stay internally in the MOND regime. For the isolated two-body deep-MOND case the paper derives an explicit Lagrangian whose conserved MLD momentum and centre of mass carry the conservation-law analysis.

What would settle it

Measure the 50\,--\,200 pc leading-to-trailing member ratio, $q_{50-200}$, and the member-loss fraction for a sample of nearby open clusters near 200 $M_\odot$ with reliable astrometric membership; MLD predicts a mean $q_{50-200}$ between 1.5 and 2 and roughly 25% faster evaporation than Newtonian N-body models, so a sample that sits at $q\approx 1$ with Newtonian-loss rates would refute the central claim.

Watch

Extended reading notes

Core claim

The central claim is that in Milgrom-law dynamics a low-mass open star cluster embedded in a Galactic disk develops a persistent leading-tail excess: in the 50\,--\,200 pc annulus the leading arm holds 1.5 to 2 times as many stars as the trailing arm, and the cluster loses its members about 25% faster than in Newtonian dynamics, with a mean retained fraction of 0.26 after 1 Gyr versus 0.41 in the Newtonian models. The same qualitative asymmetry had been found in QUMOND simulations of heavier clusters, and the paper argues this convergence means the asymmetry belongs to MOND generally, not to a particular equation. On the dynamical side, MLD does not conserve the Newtonian linear momentum, angular momentum, or Hamiltonian; for an isolated binary in the deep-MOND limit an alternative MLD centre of mass moves uniformly while the Newtonian centre of mass wobbles around it, and the MLD equations follow from a Lagrangian with a logarithmic potential and square-root masses.

Load-bearing premise

The load-bearing assumption is that Milgrom's law, applied star by star as a vector relation with a fixed 0.001 pc softening, is a valid approximation to real MOND for low-mass open clusters; if a full MOND field theory would not reduce to this vector law for point masses, the simulated tail asymmetry and the 25% faster dissolution would not describe actual clusters.

Editorial extensions

If this is right

A 200 $M_\odot$, 400-particle open cluster on a circular 8.3 kpc orbit retains 26% of its stars after 1 Gyr in MLD, versus 41% in Newtonian gravity.
The $q$-parameter $N_{\rm lead}/N_{\rm trail}$ in the 50\,--\,200 pc annulus settles between 1.5 and 2 in MLD, while Newtonian models stay near 1.
The leading-tail excess appears in both QUMOND field-theory simulations and discrete MLD simulations, which the paper reads as evidence that asymmetric tails are inherent to MOND-like dynamics.
Newtonian conservation laws fail in MLD, so quantities like the Newtonian centre of mass and angular momentum drift or oscillate instead of staying fixed.
Nearby open clusters, with their tails resolved by astrometry, become direct test beds for whether gravity below $a_0$ is Newtonian or MONDian.

Reading between the lines

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

If MLD is a fair stand-in for MOND, the observed Hyades tail asymmetry, cited as a 6.7$\sigma$ outlier in Newtonian stochastic models, would be the expected MOND signature rather than a rare fluctuation; this is a consequence the paper points toward but does not itself simulate for the Hyades.
The 0.001 pc softening is doing physical work, effectively assuming no internally Newtonian binaries exist in the cluster; varying $\varepsilon$ or seeding the cluster with hard binaries would be a direct numerical test of how the tail asymmetry and evaporation rate depend on that assumption.
Because the external field effect enters through the factor $\mu(a_{\rm ext}/a_0)$, the predicted asymmetry should depend on the cluster's galactic radius and orbital speed; comparing clusters at different radii in the same survey could separate MOND's tail asymmetry from bar- or spiral-arm-induced perturbations.

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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

You should know: this paper builds a direct N-body scheme for Milgrom-law dynamics (MLD) and applies it to a 200 Msun, 400-particle open cluster. The genuinely new pieces are the Hermite integration with MONDian jerks, an analytic deep-MOND two-body Lagrangian, and the specific quantitative claims: leading/trailing tail ratio q50-200 ≈ 1.5–2 and roughly 25% faster dissolution than Newtonian. The qualitative asymmetry was already reported in QUMOND simulations (Thomas et al. 2018; Kroupa et al. 2022), so the new output is the MLD tool itself and these numbers.

The paper is transparent about the status of Eq. (10): it is a postulate, not derived from AQUAL or QUMOND, and the author explicitly says no variational principle is known. The binary, triple, and Plummer-sphere tests are reasonable sanity checks. The jerk derivation (Eqs. 26–28) is internally consistent, and no parameter is fitted to the tail asymmetry or dissolution outputs; a0, vc, and the cluster parameters come from prior work.

The soft spot is load-bearing. The paper sets ε = 0.001 pc to "suppress the Newtonisation of compact subsystems." For a 0.5 Msun pair, the MLD/Newtonian transition sits at r ≈ 0.024 pc (where Gm/r² = a0). So any pair with separation between 0.001 and 0.024 pc is internally Newtonian, not MONDian. Over 1 Gyr in a 400-particle cluster, such pairs form transiently, making the simulated cluster a hybrid of MOND-enhanced and Newtonian subsystems. The paper's own Sec. 4.2 shows that internally Newtonian binaries follow Newtonian galactic orbits. That means the tail asymmetry and dissolution rate could be influenced by close binaries and the associated recoil, not solely by "general MONDian dynamics." The softening choice is thus not a harmless numerical detail; it fails to realize the paper's stated physics goal.

Other concerns are smaller: only five MLD realizations, no error bars on q beyond the plotted spread, no code or seeds released, and the Sec. 6 conclusion that the asymmetry is "a property of the general MONDian dynamical concept" is too strong given that MLD is one postulated discretization and the softening issue is unresolved. The literature discussion is fine, and the stated limitations in the abstract and conclusions are honest.

Who is this for? People modeling discrete stellar systems in MOND-like gravity. It deserves a serious referee because it opens a new numerical direction and makes falsifiable predictions. But the referee should push for convergence tests over ε, a demonstration that the signal survives when close binaries are treated consistently, and release of code and seeds. I would engage with it but not yet cite the quantitative results as robust.

Referee Report

4 major / 6 minor

Summary. The paper postulates a vectorial Milgrom-law dynamics (MLD) for discrete N-body systems (Eq. 10), extends the standard Hermite integration scheme to compute MONDian accelerations and jerks, and uses it to simulate the tidal-tail evolution of a 400-particle, 200 Msun open cluster on a circular orbit at 8300 pc in a flat rotation curve. The central results are that in MLD the leading tidal tail is more populated than the trailing tail (q50-200 between 1.5 and 2) and that the cluster dissolves about 25% faster than in a Newtonian control run. The paper also presents analytic/numerical tests of an isolated binary, binaries in an external field, a hierarchical triple, and an isolated Plummer sphere, and compares the deep-MOND two-body force with Milgrom's field-theoretic expression.

Significance. If correct, the paper provides the first collisional direct N-body treatment of open clusters in a MOND-type dynamics, and it makes a falsifiable prediction that low-mass open clusters in the Solar neighborhood should show leading-tail excesses and enhanced dissolution relative to Newtonian expectations. The analytic derivation of the MONDian jerk (Eqs. 26-28) and the binary/triple sanity checks are internally consistent and clearly presented. The paper is also explicit about many caveats, including the lack of a variational principle for MLD and the approximate nature of the softening procedure. However, the significance is diminished by the ad hoc status of the central equation of motion and by the softening-scale inconsistency discussed below, because the quantitative claims may describe a hybrid Newtonian/MOND hybrid system rather than pure MOND dynamics.

major comments (4)

[Sec. 3.5 and Sec. 5.1, Eqs. (32)-(33), Fig. 13] The claim that softening with epsilon=0.001 pc suppresses the Newtonisation of compact subsystems is internally inconsistent. For a 0.5 Msun pair, Gm/r^2 = a0 at r = sqrt(Gm/a0) ≈ 0.024 pc, which is 24 times larger than epsilon. The softened force (Eq. 33) differs appreciably from the unsoftened force only for r < epsilon; in the range 0.001 < r < 0.024 pc the force is nearly Newtonian and exceeds a0. Thus the simulation contains internally Newtonian binaries, and Sec. 4.2 demonstrates that such binaries follow Newtonian rather than MONDian Galactic orbits. The paper provides no analysis of how often such binaries form in the cluster or how the tail asymmetry and dissolution rate depend on epsilon. The central quantitative result (q50-200 ≈ 1.5-2, 25% faster dissolution) could therefore be caused by these Newtonian binaries or by the recoil they induce (Sec. 5.2), rather than by the 'general MONDian dynamical concept' claimed in Sec. 6.
[Sec. 2, Eq. (10), and Sec. 6] The equation of motion (Eq. 10) is a postulate that is not derived from AQUAL or QUMOND, and the paper correctly states that no variational principle is known. Given this, the conclusion in Sec. 6 that the tail asymmetry is 'a property of the general MONDian dynamical concept' because it appears in both QUMOND and MLD is too strong. MLD is an ad hoc prescription with an additional softening parameter, and the QUMOND simulations cited operate at different cluster masses (≳5000 Msun). At minimum, the abstract and conclusions should restrict the claim to 'MLD with the chosen softening' and state explicitly that it remains to be shown that actual MOND field theories produce the same effect for 200 Msun clusters.
[Sec. 5.4 and Sec. 5.5, Figs. 19-21] The headline numbers are based on only five realizations per model. Fig. 20 shows large run-to-run scatter in q50-200, yet no uncertainties or significance tests are reported for the quoted range q50-200 = 1.5-2. Similarly, the remaining fractions after 1 Gyr are given as 0.26 ± 0.07 (MLD) and 0.41 ± 0.06 (Newtonian); the 1-sigma bands are separated by only about 1.5 sigma, making the claim of a ~25% faster dissolution statistically fragile. The paper should provide per-realization values, standard errors on the mean, and a test (e.g., a t-test or bootstrap) before drawing quantitative conclusions.
[Sec. 3.1-3.2 and Sec. 5.2] The Hermite scheme is used for a non-conservative, non-Hamiltonian system, but the paper gives no numerical convergence tests (e.g., energy or momentum diagnostics, or runs with shorter time steps) to verify that the integration error is controlled over 1 Gyr. Since the MLD equations of motion do not conserve the Newtonian integrals, the usual error checks based on energy conservation are unavailable, and the paper's Fig. 11/Fig. 21 results could depend on the integration accuracy. A convergence study for at least one MLD cluster run is needed to support the quantitative dissolution and asymmetry claims.

minor comments (6)

[Sec. 3.3] The units of a0 in the Newtonian limit are written as '10^-20 Myr/pc^2'; they should be pc/Myr^2 (or equivalent) to match the text and equations.
[Sec. 4.1] The sentence 'the masses of the particles in the kinetic potential turn into their square roots' should read 'kinetic term' rather than 'kinetic potential'.
[Fig. 13] Figure 13 would be much more informative if it included a vertical line at r ≈ 0.024 pc, the radius where a = a0 for a 0.5 Msun particle, because that is the scale at which the MOND/Newtonian transition occurs for the particles used in the cluster simulations.
[Sec. 4.5] The citation 'Milgrom (2014, E.q 23)' contains a typo; it should be 'Eq. (23)'.
[Sec. 4.4] In the phrase 'In lack of a known conserved quantity in MLD', 'lack' should be 'the absence'.
[Sec. 5.1] The reference to 'solar neighborhood as used in related studies' should be followed by a period and likely the appropriate citation (e.g., Jerabkova et al. 2021).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the tail asymmetry and dissolution rates are direct outputs of an explicitly postulated MLD N-body dynamics, with no fitted parameter renamed as a prediction.

full rationale

The paper's central quantitative claims (q50-200 between 1.5 and 2, and retained fractions 0.26 vs 0.41 after 1 Gyr) are direct outputs of integrating the postulated MLD equations of motion, Eq. (10)/(11), with fixed parameters chosen before the runs: a0 = 3.8 pc/Myr^2, vc = 230 pc/Myr, M_tot = 200 Msun, Plummer b = 3.1 pc, and epsilon = 0.001 pc. No subset of the asymmetry or dissolution data is used to calibrate any parameter, and the external Galactic field is not fitted: Eq. (36) is the Milgrom-law condition for a circular orbit in the adopted flat rotation curve, derived from the same interpolation function used in the N-body forces. The Section 5.5 external-field-effect estimate (T_diss ratio 0.93) is a post-hoc interpretation and actually disagrees mildly with the simulated ratio 0.8, which is evidence that the simulation was not constructed to reproduce it. The self-citations (Pflamm-Altenburg et al. 2023 for the tail-membership criterion and q-parameter definition; Kroupa et al. 2022 for a QUMOND comparison) are background, definitional, or corroborative rather than load-bearing; the MLD tail asymmetry is computed from this paper's own equations, and the QUMOND comparison is an independent externally published simulation that does not force the MLD result. The softening epsilon = 0.001 pc is an acknowledged approximation, and whether it adequately suppresses Newtonisation of compact subsystems is a correctness/robustness concern, not a circularity. No circular step is present.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The MLD 'momentum' sqrt(m) v is a derived conserved quantity for a two-body subsystem, not an invented field or particle. The softening is a numerical modification, not a new entity.

free parameters (3)

Softening length epsilon = 0.001 pc (~206 AU)
Chosen by hand to suppress Newtonisation of compact subsystems (Sec. 3.5). The cluster dynamics below this scale is altered, and no convergence study with respect to epsilon is presented.
Plummer parameter b = 3.1 pc
Chosen to match the current Hyades cluster (Sec. 5.1). Affects the internal acceleration magnitude and hence the external-field-effect regime, but is an observed input rather than a fit to the target result.
Particle number N = 400
Resolution choice; the paper states that lower-mass clusters cannot be simulated in QUMOND, and uses 400 particles as a computationally feasible representation. No resolution study is given.

assumptions (5)

ad hoc to paper Milgrom's law in vectorial form holds for discrete point-mass systems (Eq. 10).
Introduced as a postulate in Sect. 2; the paper states no equations of motion for discrete systems in AQUAL/QUMOND are available and no variational principle is known.
domain assumption The external galactic field is a fixed, analytic Newtonian potential whose circular acceleration matches the assumed flat rotation curve (Eqs. 35-38).
The cluster is embedded in an external field, not a self-consistent MOND field. The flat rotation curve input already encodes the MOND phenomenology at large scales.
ad hoc to paper Softening with epsilon=0.001 pc prevents the Newtonisation of compact subsystems, keeping the cluster in the MOND regime.
Established in Sec. 3.5 and the Conclusions; real clusters contain binaries that would be internally Newtonian, and the paper acknowledges 'MLD can only be considered as an approximation'.
standard math The Hermite predictor-corrector scheme remains valid for the MLD equations of motion.
The scheme is standard (Aarseth 2003) and used for Newtonian N-body; the MLD extension only changes the acceleration and jerk evaluation, not the integrator structure.
domain assumption A Plummer model with n=400 equal-mass stars represents a low-mass open cluster.
Chosen to match Hyades properties (Sec. 5.1); real clusters have a mass spectrum, binaries, and stellar evolution, which are neglected.

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Cite this review

Pith. "Pith review of Asymmetry of the tidal tails of open star clusters in direct N-body integrations in Milgrom-law dynamics." pith.science (2026). https://pith.science/paper/JSOUDHIZ

@misc{pith2026241113675,
  author       = {Pith},
  title        = {Pith review of: Asymmetry of the tidal tails of open star clusters in direct N-body integrations in Milgrom-law dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JSOUDHIZ}},
  note         = {Machine review of arXiv:2411.13675}
}

read the original abstract

Numerical QUMOND-simulations of star clusters orbiting in a Galactic disk potential show that the leading tidal arm of open star clusters contains tendentially more members than the trailing arm. However, these type of simulations are performed by solving the field-equations of QUMOND and already become non-practical for star cluster masses at around 5000 Msun. Nearby star clusters have masses of 1000 Msun or ~1000 particles and less/fewer and can currently not be simulated reliably in field-theoretical formulations of MOND. In order to handle particle numbers below the QUMOND-limit the star cluster is simulated in Milgrom-law dynamics (MLD): Milgrom's law is postulated to be valid for discrete systems in vectorial form. In order to suppress the Newtonisation of compact subsystems in the star cluster the gravitational force is softened below particle distances of 0.001 pc ~206 AU. Thus, MLD can only be considered as an approximation of a full MOND-theoretical description of discrete systems which are internally in the MOND regime. The MLD equations of motion are integrated by the standard Hermite scheme generally applied to Newtonian N-body systems, which is extended to solve for the accelerations and jerks associated with Milgrom's law. It is found that the tidal tails of a low-mass star cluster are populated asymmetrically in the MLD-treatment, very similar to the QUMOND simulations of the higher-mass star clusters. In the MLD-simulations the leading tail hosts up to twice as many members than the trailing arm and the low-mass open star cluster dissolves approximately 25% faster than in the respective Newtonian case.

Figures

Figures reproduced from arXiv: 2411.13675 by the authors.

**Figure 1.** Orbital evolution of a deep MOND MLD-binary: (Left:) The thick red curve shows the trajectory of the more massive particle with m1 = 2 M⊙, the thin blue curve shows the trajectory of the less massive particle with m1 = 0.2 M⊙. The large red filled circle indicates the initial position of the more massive particle, the small blue filled circle the initial position of the less massive particle. (Right:) Shown is the c… view at source ↗

**Figure 2.** Evolution of the linear momentum of a deep MOND MLD-binary: The slightly varying blue curve shows the MLD-linear momentum (Eq. (43)) as a function of time. The strongly oscillating red curve shows the time evolution of the corresponding Newtonian linear momentum, pNew = m1q˙ 1 + m2q˙ 2. (Left): x-component of the linear momentum. (Right): y-component of the linear momentum. 0 2 4 6 8 10 12 14 16 0 200 400 600 800 10… view at source ↗

**Figure 3.** Centre of mass motions. The straight blue lines refer to the MLD-centre of mass (Eq. (46)) whereas the wobbling red curves show the Newtonian centre of mass, Rcom,New = m1q1+m2q2 m1+m2 . The x-component of both centers of mass runs horizontally, the y-component increases continuously. leads to a circular Galactic motion of the internally MONDian binary ( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (13 more)

**Figure 5.** Figure 5: Evolution of the Hamiltonian of a deep MOND MLD-binary: The blue curve shows the MLD-Hamiltonian (Eq. (49)) as a function of time. The red curve shows the time evolution of the corresponding Newtonian Hamiltonian, HNew = p 2 1 2m1 + p 2 2 2m2 − G m1m2 |q2−q1| . -10000 …

**Figure 8.** Figure 8: Binary in an external field. The internally Newtonian binary is set up with a Newtonian rotational velocity. The thick red curve shows the orbit of the 2 M⊙-component, the thin blue curve shows the orbit of the less massive star. The filled red circle marks the initial…

**Figure 9.** Figure 9: Triple in Newtonian dynamics. The inner thick (red) circle shows the orbit of the inner more massive binary. The thin (blue) outer circle shows the orbit of the single star. -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 …

**Figure 10.** Figure 10: Triple in MLD. The inner thick (red) circle shows the orbit of the inner more massive binary. The thin (blue) outer circle shows the orbit of the single star. mass of the considered particle and by subsequent summation over all particles: X i=N i=1 miµ [PITH_FULL_IMA…

**Figure 12.** Figure 12: Deep MOND binary. Shown is the relative motion over 20 Myr of the isolated test binary with initial conditions (filled black circle) given at the beginning of Sec. 4.1 for three different sets of equations of motion: full MLD with transition function (red solid line),…

**Figure 13.** Figure 13: Softening. Shown is the Newtonian acceleration field of a star with a mass of 0.5 M⊙ in the case of no softening (ε = 0 pc) and softening with a parameter of ε = 0.001 pc. The dashed horizontal line marks the MONDian acceleration threshold, a0. The vertical solid lin…

**Figure 14.** Figure 14: Orbital snapshots at 0 Myr. Star cluster evolution in Newtonian (left) and discrete Milgrom-law Dynamics (right). See Sect. 5.2 for details. 2500 2600 2700 2800 2900 3000 3100 7500 7600 7700 7800 7900 8000 8100 Newton 250 Myr Galactic centre Galactic rotation yGal / p…

**Figure 15.** Figure 15: Same as in [PITH_FULL_IMAGE:figures/full_fig_p011_15.png]

**Figure 16.** Figure 16: Same as in [PITH_FULL_IMAGE:figures/full_fig_p011_16.png]

**Figure 17.** Figure 17: Same as in [PITH_FULL_IMAGE:figures/full_fig_p012_17.png]

**Figure 18.** Figure 18: Same as in [PITH_FULL_IMAGE:figures/full_fig_p012_18.png]

**Figure 19.** Figure 19: Asymmetry of tidal arms. Thin lines show the evolution of the asymmetry of the 10 individual simulations, 5 Newtonian (red) and 5 MONDian (blue) simulations. The thick lines show the arithmetic mean values. where Nl,50−200 pc is the number of stars in the leading arm …

**Figure 20.** Figure 20: q-parameter. Thin lines show the evolution of the q-parameter of the 10 individual simulations, 5 Newtonian (red) and 5 MONDian (blue) simulations. The thick lines show the arithmetic mean values. arm contains more members than the trailing arm in the respective dist…

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