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REVIEW 3 major objections 6 minor 58 references

Dependence of Particle Acceleration Efficiency on Shock Velocity in Weakly Magnetized Electron-Ion Shocks

T0 review · 3 major / 6 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Shock speed flips electron acceleration from feeble to fierce

desk verdict Velocity-dependent Bell-to-Weibel transition in weakly magnetized shocks, with a concrete M_A ~ 100 threshold for electron acceleration efficiency read the letter →

arxiv 2607.05778 v1 pith:R7MI7S5Z submitted 2026-07-07 astro-ph.HE physics.plasm-ph

Taiki Jikei , Daniel Groselj , Lorenzo Sironi This is my paper
classification astro-ph.HEphysics.plasm-ph
keywords collisionlessshocksparticleaccelerationBellinstabilityWeibelelectronefficiencyAlfvenicMachnumberparticle-in-cellsimulationsgamma-rayburstafterglows
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 uses long-running plasma simulations to show that the efficiency of electron acceleration in weakly magnetized collisionless shocks depends sharply on shock velocity. At a fixed magnetization, slow shocks fall under the control of the Bell instability, which amplifies magnetic field via cosmic-ray ion currents but suppresses electron injection, channeling less than 2 percent of shock energy into nonthermal electrons. Fast shocks are instead dominated by the Weibel instability, which generates small-scale magnetic turbulence that efficiently injects electrons, channeling about 15 percent of shock energy into nonthermal electrons. The transition between these two regimes occurs around an Alfvenic Mach number of 100, a threshold the authors identify by combining their new velocity-scan results with their earlier magnetization-scan results. Ion acceleration proceeds with comparable efficiency in both regimes, though Bell-dominated shocks achieve faster growth in maximum ion energy. The paper connects these findings to astrophysical transients, proposing that the diversity of X-ray and radio emission from gamma-ray burst afterglows, fast blue optical transients, and microquasars can be understood as consequences of which instability regime the shock occupies.

What carries the argument

Bell instability

What would settle it

If 3D simulations with realistic mass ratios at the same magnetization show no sharp transition in electron acceleration efficiency near M_A ~ 100, or if the transition Mach number shifts substantially with mass ratio or dimensionality, the proposed universal threshold would not hold.

Watch

Extended reading notes

Core claim

The central discovery is a sharp transition in the dominant magnetic-field-generating mechanism at collisionless shocks as a function of shock velocity at fixed magnetization. Below an Alfvenic Mach number of roughly 100, the Bell instability dominates: the shock self-regulates its cosmic-ray ion current downward, producing large-scale circularly polarized magnetic waves that are efficient at scattering ions but effectively block electron injection. Above this threshold, the Weibel instability dominates: returning relativistic electrons in the upstream compensate the ion current and suppress Bell growth, while Weibel-generated small-scale filamentary fields allow efficient electron injection

Load-bearing premise

The claim that the transition at M_A ~ 100 is a general result rests on simulations at a single magnetization, a reduced ion-to-electron mass ratio of 100, and 2D geometry; the run nearest the transition is still evolving at the end of the simulation, and the threshold has not been independently tested with 3D simulations or realistic mass ratios.

Editorial extensions

If this is right

  • GRB afterglows like GW170817 remain efficient electron accelerators over observable timescales because their Alfvenic Mach numbers stay well above 100 throughout the deceleration.
  • FBOTs with radio emission consistent with thermal electrons may reside in environments where the magnetization is high enough to push the shock below the M_A ~ 100 threshold, suppressing nonthermal electron acceleration.
  • Microquasars like SS 433 (persistently X-ray bright) and V4641 Sgr (X-ray quiet except during outbursts) may differ because one sits in the Weibel-dominated regime and the other in the Bell-dominated regime, with outbursts triggered by velocity increases that flip the shock across the transition.
  • Existing models that assume a fixed nonthermal electron energy fraction across shock velocities will need to incorporate the velocity-dependent efficiency identified here to accurately infer shock parameters from observed spectral energy distributions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This manuscript presents a suite of long-duration 2D particle-in-cell (PIC) simulations of weakly magnetized (sigma = 10^{-4.5}), quasi-parallel electron-ion shocks, varying the upstream flow four-velocity u_0/c from 1/6 to 4/3 at fixed magnetization. The authors find a transition from Bell-instability-dominated shocks at low velocities to Weibel-instability-dominated shocks at high velocities. By combining these results with a companion study (T. Jikei et al. 2026) that varied sigma at fixed velocity, they propose that this transition occurs at an Alfvénic Mach number M_A ~ 100. The two regimes produce qualitatively different electron acceleration efficiencies: Weibel-dominated shocks channel ~15% of shock energy into nonthermal electrons, while Bell-dominated shocks channel less than ~2%. The astrophysical implications for GRB afterglows, FBOTs, and microquasars are discussed. The simulations are computationally expensive and the diagnostics are well-designed, but the central quantitative claim regarding the M_A ~ 100 threshold rests on a run that is still evolving at simulation end.

Significance. The identification of a velocity-dependent transition in electron acceleration efficiency at fixed magnetization is a significant result for the transrelativistic shock literature, with direct astrophysical relevance to GRB afterglows, FBOTs, and microquasar emission. The strengths include the exceptionally long simulation durations (up to omega_pit = 10000, and 25000 in Appendix B), the systematic parameter scan combining velocity and magnetization variations, and the falsifiable prediction of a critical M_A ~ 100 separating efficient from inefficient electron acceleration. The 1D periodic-box experiments in Appendix A, which isolate the role of returning relativistic electrons in suppressing the Bell instability, provide a valuable physical mechanism for the velocity dependence. The energy-partition diagnostics and maximum-energy tracking are thorough.

major comments (3)
  1. Section 3.2 and Figure 2b: The u_0/c = 2/3 run (M_A ~ 120) is the single data point that narrows the Bell-to-Weibel transition to 'around M_A ~ 100,' yet the text explicitly states that 'the cosmic-ray current starting to drop at omega_pit >= 9000' — i.e., this run is in the midst of transitioning from the high-current (Weibel-favorable) to the low-current (Bell-favorable) state at the moment the simulation ends. If the current continues dropping to eta/eta_crit < 0.5, this run reclassifies as Bell-dominated, pushing the critical M_A to between 120 and 180. The paper's supporting evidence (Appendix A mechanism, Appendix B extended run at u_0/c = 4/3) addresses whether high-velocity shocks stay Weibel-dominated but does not test the fate of this critical run. The claim 'the transition between the two regimes occurs around u_0/c ~ 2/3' (end of Section 3.2) and the M_A ~ 100 threshold (§3.4
  2. Section 3.4, Eq. (10): The synthesis yielding M_A ~ 100 combines this paper's velocity scan (at sigma = 10^{-4.5}) with the companion paper's magnetization scan (at u_0/c = 4/3, finding a transition at sigma ~ 10^{-3.5}, i.e., M_A ~ 75). The factor of ~1.3-1.6 spread between M_A ~ 75 and M_A ~ 120 is acceptable for an order-of-magnitude claim, but the text should explicitly state this spread rather than presenting M_A ~ 100 as a sharp threshold. Additionally, the derivation of Eq. (10) uses the approximation eta ~ alpha * v_0/c, which assumes cosmic-ray ions drift with mean speed ~v_0 (isotropic in the downstream frame). The sensitivity of the M_A ~ 100 threshold to this assumption should be briefly discussed, as it directly affects the critical current condition.
  3. Section 3.3: The u_0/c = 1/6 run is acknowledged as possibly not numerically converged for electrons, and the u_0/c = 2/3 run is flagged as still evolving. This leaves only two cleanly classified runs on each side (u_0/c = 1/3 Bell-dominated; u_0/c = 1 and 4/3 Weibel-dominated). The electron nonthermal energy fractions (~15% for Weibel, ~2% for Bell) are thus based on a small number of converged data points. The authors should clarify whether the ~2% upper limit for Bell-dominated shocks is robust given that the u_0/c = 1/6 run (which would strengthen this claim) is excluded from the quantitative analysis.
minor comments (6)
  1. Section 3.3, Figure 3c: The nonthermal electron energy fraction is defined using p > 3*p_peak. The choice of factor 3 is conventional but arbitrary; a brief comment on how sensitive the quoted ~15% and ~2% fractions are to this threshold choice would strengthen the quantitative claims.
  2. Section 3.3, Eq. (9): The E_max threshold is defined as the Lorentz factor at which (gamma-1)*f_s(gamma) drops below 10^{-5} of its peak value. This is a somewhat ad hoc diagnostic; a sentence noting its limitations (e.g., sensitivity to particle statistics at the tail) would be helpful.
  3. Figure 1 caption: The colorbar description for Row 2 mentions a 'symmetric log scale' with a linear range [-0.01, 0.01], but this is easy to miss. Consider making this more prominent, as the interpretation of the magnetic field structure depends on understanding the normalization.
  4. Section 4.1: The estimate that the GW170817 afterglow shock remains at M_A >> 100 for ~6x10^8 days is interesting but the Taylor-von Neumann-Sedov scaling (Eq. 12) assumes a uniform-density medium. A brief caveat about the density profile dependence would be appropriate.
  5. The companion paper T. Jikei et al. (2026) is cited frequently and is load-bearing for the M_A ~ 100 synthesis. Since this paper appears to be under review simultaneously (or recently accepted), ensuring that the companion is available to readers would strengthen reproducibility.
  6. Section 2: The ion-to-electron mass ratio m_i/m_e = 100 is used throughout. While this is standard for computational reasons, the potential impact on the Bell-to-Weibel transition threshold should be briefly noted in §4.4 alongside the other future-work items, as the electron dynamics (and thus the current compensation mechanism of Appendix A) may depend on this parameter.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee's three major comments all identify legitimate concerns regarding the robustness of the M_A ~ 100 threshold, the precision with which it is presented, and the number of converged data points supporting the quantitative efficiency claims. We agree with the substance of all three comments and will revise the manuscript accordingly: (1) we will soften the claim about the u_0/c = 2/3 run and explicitly state the range M_A ~ 75–180 if that run reclassifies; (2) we will present M_A ~ 100 as an order-of-magnitude estimate with the full spread stated, and add a discussion of the sensitivity to the isotropy assumption in Eq. (10); and (3) we will clarify the basis for the ~2% upper limit given the exclusion of the u_0/c = 1/6 run from the quantitative analysis. No standing objections remain.

read point-by-point responses

  1. Referee: Section 3.2 and Figure 2b: The u_0/c = 2/3 run (M_A ~ 120) is the single data point that narrows the Bell-to-Weibel transition to 'around M_A ~ 100,' yet the text explicitly states that 'the cosmic-ray current starting to drop at omega_pit >= 9000' — i.e., this run is in the midst of transitioning from the high-current (Weibel-favorable) to the low-current (Bell-favorable) state at the moment the simulation ends. If the current continues dropping to eta/eta_crit < 0.5, this run reclassifies as Bell-dominated, pushing the critical M_A to between 120 and 180. The paper's supporting evidence (Appendix A mechanism, Appendix B extended run at u_0/c = 4/3) addresses whether high-velocity shocks stay Weibel-dominated but does not test the fate of this critical run. The claim 'the transition between the two regimes occurs around u_0/c ~ 2/3' (end of Section 3.2) and the M_A ~ 100 threshold (§3.4

    Authors: The referee is correct that the u_0/c = 2/3 run is still evolving at simulation end, and we agree that the current manuscript overstates the precision of the transition location based on this single data point. We will revise the manuscript in two ways. First, we will rephrase the claim at the end of Section 3.2 from 'the transition between the two regimes occurs around u_0/c ~ 2/3' to language that explicitly acknowledges the run is in transition and that the critical velocity is bracketed: the transition lies somewhere in the range u_0/c ~ 2/3–1, corresponding to M_A ~ 120–180 if the u_0/c = 2/3 run ultimately reclassifies as Bell-dominated, or M_A ~ 100 if it does not. Second, in Section 3.4, we will present M_A ~ 100 as an order-of-magnitude estimate and state the full allowed range M_A ~ 75–180, which accounts for both the u_0/c = 2/3 ambiguity and the companion paper's M_A ~ 75 result. We agree that the supporting evidence in Appendices A and B addresses the persistence of the Weibel-dominated state at high velocities but does not test the fate of the critical u_0/c = 2/3 run. We will state this limitation explicitly. We are unable to extend the u_0/c = 2/3 run further within the revision timeframe due to computational cost, but we will flag this as a priority for future work. revision: yes

  2. Referee: Section 3.4, Eq. (10): The synthesis yielding M_A ~ 100 combines this paper's velocity scan (at sigma = 10^{-4.5}) with the companion paper's magnetization scan (at u_0/c = 4/3, finding a transition at sigma ~ 10^{-3.5}, i.e., M_A ~ 75). The factor of ~1.3-1.6 spread between M_A ~ 75 and M_A ~ 120 is acceptable for an order-of-magnitude claim, but the text should explicitly state this spread rather than presenting M_A ~ 100 as a sharp threshold. Additionally, the derivation of Eq. (10) uses the approximation eta ~ alpha * v_0/c, which assumes cosmic-ray ions drift with mean speed ~v_0 (isotropic in the downstream frame). The sensitivity of the M_A ~ 100 threshold to this assumption should be briefly discussed, as it directly affects the critical current condition.

    Authors: We agree on both points. First, we will revise Section 3.4 to explicitly state the spread: the companion paper finds a transition at M_A ~ 75 (sigma ~ 10^{-3.5} at u_0/c = 4/3), while this paper brackets the transition at M_A ~ 120–180 (depending on the fate of the u_0/c = 2/3 run). We will present M_A ~ 100 as the geometric mean characterizing an order-of-magnitude threshold, not a sharp boundary, and will use language such as 'M_A ~ 100, within a factor of ~2' throughout. Second, we will add a brief discussion of the sensitivity of Eq. (10) to the assumption that cosmic-ray ions drift with mean speed ~v_0 (i.e., isotropy in the downstream frame). The relation eta ~ alpha * v_0/c enters through the critical current condition alpha_crit = 2 M_A^{-1}. If the mean drift speed of returning ions is instead a fraction f of v_0, the critical Mach number scales as M_A,crit ~ 100/f. For f in the range ~0.5–1 (reasonable given that returning ions have a distribution of pitch angles), the threshold shifts to M_A ~ 50–100. We will state that this uncertainty is comparable to the spread from the two simulation scans and does not change the order-of-magnitude conclusion, but we will make the assumption and its consequences explicit. revision: yes

  3. Referee: Section 3.3: The u_0/c = 1/6 run is acknowledged as possibly not numerically converged for electrons, and the u_0/c = 2/3 run is flagged as still evolving. This leaves only two cleanly classified runs on each side (u_0/c = 1/3 Bell-dominated; u_0/c = 1 and 4/3 Weibel-dominated). The electron nonthermal energy fractions (~15% for Weibel, ~2% for Bell) are thus based on a small number of converged data points. The authors should clarify whether the ~2% upper limit for Bell-dominated shocks is robust given that the u_0/c = 1/6 run (which would strengthen this claim) is excluded from the quantitative analysis.

    Authors: We agree that the number of cleanly converged data points is small and that this should be stated transparently. We will add explicit language in Section 3.3 noting that the quantitative efficiency claims rest on two converged Weibel-dominated runs (u_0/c = 1 and 4/3) and one converged Bell-dominated run (u_0/c = 1/3), with the u_0/c = 1/6 run excluded from quantitative analysis due to possible non-convergence and the u_0/c = 2/3 run flagged as still evolving. Regarding the robustness of the ~2% upper limit: the u_0/c = 1/3 run, which is numerically converged (verified with the narrower-box, higher-ppc runs described in the manuscript), yields a nonthermal electron energy fraction of ~1.5%. The u_0/c = 1/6 run shows an even smaller nonthermal fraction in the spectra (Figure 3b), consistent with the ~2% upper limit, but we excluded it from the quantitative claim precisely because we cannot verify convergence. We will clarify that the ~2% figure is established by the converged u_0/c = 1/3 run alone, that the u_0/c = 1/6 run is qualitatively consistent but not quantitatively relied upon, and that the Bell-versus-Weibel contrast (~2% vs. ~15%) is a factor of ~7–10 difference that is robust to the uncertainties in the individual data points. We will also note that extending the u_0/c = 1/6 run to verify convergence is a goal for future work. revision: yes

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; self-citation to companion paper is supplementary, not load-bearing

full rationale

The paper's central claims are grounded in direct PIC simulation measurements: magnetic field structure (polarization, wavevector orientation), cosmic-ray current evolution (Figure 2), and particle spectra/energy fractions (Figure 3). The regime classification (Bell vs. Weibel) is based on independently observed field morphology, not on the acceleration efficiency it seeks to explain. The critical current η_crit = 2σ^{1/2} (Eq. 6) derives from Bell instability linear theory (Eq. 4-5), and the transition is observed when η/η_crit crosses unity — a theoretical prediction tested against simulation data, not a definition. The M_A ~ 100 threshold synthesizes two independent parameter scans: this paper (fixed σ, varying velocity, transition at u_0/c ~ 2/3 giving M_A ~ 120) and the companion paper T. Jikei et al. (2026) (fixed velocity, varying σ, transition at σ ~ 10^{-3.5} giving M_A ~ 75). These are genuinely independent scans — neither reduces to the other. The companion paper is cited for methodology details and for one of the two scans, but the current paper's velocity-dependent results stand independently. Appendix A's 1D periodic-box simulations provide a separate physical mechanism (electron current compensation suppressing Bell instability) that supports — but does not define — the main shock simulation findings. The u_0/c = 2/3 run's unresolved evolution is a convergence concern, not a circularity issue. The only minor concern is that the M_A ~ 100 figure inherits uncertainty from the companion paper's methodology (same code, same mass ratio, 2D), but this is a generalization risk, not circular reasoning. Score 2 reflects the presence of companion-paper self-citation that, while not load-bearing for this paper's own results, contributes to the unified M_A threshold claim.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new particles, forces, dimensions, or postulated entities. All instabilities (Bell, Weibel) and plasma species (ions, electrons) are standard. The critical Mach number M_A ~ 100 is an empirical finding from simulations, not a postulated entity. The free parameters are simulation inputs and diagnostic thresholds, not fitted constants in a theoretical model.

free parameters (5)
  • σ (upstream magnetization) = 10^{-4.5}
    Fixed by the authors; not derived from first principles. Chosen to be in the weakly magnetized regime relevant to GRB afterglows.
  • θ_B (magnetic field obliquity) = 20°
    Fixed by the authors to study quasi-parallel configuration. Not derived.
  • m_i/m_e (mass ratio) = 100
    Reduced from the physical 1836 to save computational resources. A modeling choice, not a derived quantity.
  • p > 3p_peak (nonthermal electron threshold) = 3 × p_peak
    The factor of 3 is a conventional choice for separating thermal from nonthermal populations. Not derived.
  • 10^{-5} of peak (E_max threshold) = 10^{-5}
    The cutoff for defining maximum particle energy (Eq. 9). A diagnostic choice.
assumptions (4)
  • domain assumption 2D PIC simulations adequately capture the 3D physics of Bell and Weibel instabilities at collisionless shocks.
    The entire paper is based on 2D simulations. 3D turbulence, filament merging, and kink modes may differ. This is a standard assumption in the PIC simulation literature but is not proven.
  • domain assumption The reduced mass ratio m_i/m_e = 100 does not qualitatively change the Bell-to-Weibel transition or electron acceleration efficiency.
    Section 2 states this is chosen 'to save computational resources.' The effect of mass ratio on electron injection and the Bell-Weibel competition is not tested.
  • domain assumption The simulation timescale (ω_pit = 10000) is sufficient to determine the asymptotic regime (Bell vs. Weibel) for each shock velocity.
    The u_0/c = 2/3 run is still transitioning at t = 10000 (Section 3.2). Appendix B extends only the u_0/c = 4/3 case to t = 25000. The assumption that other runs have reached their asymptotic state is not fully verified.
  • domain assumption The Alfvénic Mach number M_A is the correct unifying parameter for the Bell-Weibel transition across different σ and v_0.
    Section 3.4 synthesizes results from this paper and the companion paper via M_A. The relation η ~ α v_0/c assumes cosmic-ray ions drift with mean speed ~v_0 (isotropic in downstream frame), which is an approximation.

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Cite this review

Pith. "Pith review of Dependence of Particle Acceleration Efficiency on Shock Velocity in Weakly Magnetized Electron-Ion Shocks." pith.science (2026). https://pith.science/paper/R7MI7S5Z

@misc{pith2026260705778,
  author       = {Pith},
  title        = {Pith review of: Dependence of Particle Acceleration Efficiency on Shock Velocity in Weakly Magnetized Electron-Ion Shocks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R7MI7S5Z}},
  note         = {Machine review of arXiv:2607.05778}
}
abstract

Using unprecedentedly long 2D particle-in-cell simulations, we study electron and ion acceleration in weakly magnetized quasi-parallel shocks, propagating at velocities ranging from transrelativistic to subrelativistic. At a fixed upstream magnetic field strength, low-velocity quasi-parallel shocks are dominated by the Bell instability, whereas high-velocity shocks are dominated by the Weibel instability. Both regimes accelerate ions with similar efficiency, with the Bell-dominated regime exhibiting faster growth in the maximum particle energy. The electron acceleration efficiency is strongly dependent on shock velocity. Weibel-dominated shocks have $\sim15\,\%$ of shock energy in nonthermal electrons, whereas in the Bell-dominated regime we attribute less than $\sim2\,\%$ of shock energy to nonthermal electrons. We discuss applications of our results to the bright X-ray emission from the late-stage afterglows of gamma-ray bursts, the radio emission from fast blue optical transients, and the X-ray variability in microquasars.

Figures

Figures reproduced from arXiv: 2607.05778 by the authors.

Figure 1
Figure 1. Snapshot of the simulations taken at ωpit = 10000. Columns a, b, and c correspond to u0/c = 1/3, 2/3, and 4/3, respectively. Row 1: electron number density Ne normalized by the far upstream value N0. Row 2: z-component of the magnetic field normalized by the equipartition field (Eq. (2)). The colorbar is on a symmetric log scale, in which the range [−0.01, 0.01] is on a linear scale. Rows 3 and 4: phase space densit… view at source ↗
Figure 2
Figure 2. Time evolution of the cosmic-ray ion number den￾sity and current. a: number density ratio α in the upstream frame. b: upstream current normalized by the critical cur￾rent, η/ηcrit. Both are computed in (x−xsh)/di = [200, 300]. Dark teal, orange, dark green, turquoise, and purple repre￾sent u0/c = 1/6, 1/3, 2/3, 1 and 4/3, respectively. Note that the vertical axis of panel b is on a logarithmic scale [PITH_FULL_IMAG… view at source ↗
Figure 3
Figure 3. Particle momentum spectrum 4πp4 f(p) at the end of the simulations, ωpit = 10000, in the near down￾stream, (x − xsh)/dsh = [−100, −50]. a: ions. b: electrons. The color code is the same as in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Time evolution of the maximum energy (Eq. (9)), normalized by mic 2 (γ0 − 1). a: ions, b: electrons. The color code is the same as in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Time evolution of the Bell-amplified magnetic field energy for an electron-ion beam. a: results of the low-current regime, in which time is normalized by the ion plasma frequency and the magnetic field energy is normalized by the momentum flux of the ion beam. b: resul…
Figure 6
Figure 6. Figure 6: Downstream particle momentum spectrum for u0/c = 4/3. The format is the same as in the top two panels of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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