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Terminal velocity is a local equilibrium, not a global attractor, for high-velocity clouds: dense clouds fall quasi-ballistically through a stratified halo and reach terminal motion only near the disc, while radiative cooling can restore te

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2026-08-01 09:56 UTC pith:VWUTKNYT

load-bearing objection Solid analytical core with honest limitations; the cooling-run 'revival' rests on an idealized stable halo and should be read as conditional, not a measurement of the real multiphase halo. the 3 major comments →

arxiv 2607.20394 v1 pith:VWUTKNYT submitted 2026-07-22 astro-ph.GA astro-ph.HE

Gas accretion onto the Milky Way: high-velocity cloud survival and the revival of the terminal-velocity paradigm

Michael M. Schulreich , Dieter Breitschwerdt , J\"urgen Kerp This is my paper
classification astro-ph.GA astro-ph.HE
keywords high-velocity cloudsterminal velocityGalactic halogas accretionradiative coolinghydrodynamical simulationsmass exchangedrag coefficient
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that the terminal-velocity paradigm — the long-standing method of turning a high-velocity cloud's observed speed into a distance by assuming it falls at the speed where gravity balances ram-pressure drag — is a conditional description, not a universal law. The authors generalize the equation of motion for a cloud falling through a stratified Milky Way halo to include mass exchange, cloud deformation, and realistic density and gravity profiles, and test it with three-dimensional hydrodynamical simulations. The central result is that terminal velocity is a local equilibrium but not a global attractor: dense, high-column clouds remain quasi-ballistic over most of their trajectory and approach terminal motion only shortly before reaching the disc, while adiabatic clouds are destroyed by Kelvin–Helmholtz and Rayleigh–Taylor instabilities first. Radiative cooling reverses the picture: condensation in the mixing layers grows the cloud, adds accretion drag, and restores a terminal-velocity-like regime for extended periods, while thermal conduction only reshapes small-scale structure. If correct, dynamical distance estimates for high-velocity clouds must be calibrated by column density, mass exchange, and a strongly time-dependent drag coefficient rather than a universal terminal speed.

Core claim

On its own terms, the paper establishes that terminal velocity — the speed at which gravity balances ram-pressure drag — is a well-defined local equilibrium of the cloud equation of motion, but generally not a global attractor. For constant-property clouds it derives closed-form quadrature solutions for quadratic drag in a vertically varying Galactic background, generalizing an earlier terminal-velocity framework; these show that the approach to terminal motion is set by the drag-to-free-fall time-scale ratio, so high-column-density clouds traverse most of the halo quasi-ballistically. The 3D simulations then delimit the regimes: adiabatic clouds are disrupted by Kelvin–Helmholtz and Rayleig

What carries the argument

The load-bearing object is the generalized cloud equation of motion, dv/dt = ±α v² − β v − (1 − 1/χ) g, with α the deceleration parameter (drag coupling), β the specific mass-exchange rate, and χ the cloud-to-background density contrast. Terminal velocity is that equation's fixed point, with convergence time τ_T = 1/|±2α v_T − β|; comparing τ_T with free-fall and drag times decides whether terminal motion is ever reached. Around the equation of motion the paper builds a phenomenological mass-exchange model (Kelvin–Helmholtz stripping versus cooling-driven condensation) and a Bernoulli lateral-expansion model for the effective cross section. Simulations calibrate the mass-exchange parameters

Load-bearing premise

Everything rests on treating the background halo as a maintained, thermally stable medium: cooling and conduction act only on cloud-tagged gas, so the revival of terminal-like motion depends on a supply of condensable ambient gas that a real, turbulent, thermally unstable halo might not provide.

What would settle it

Measure distances and velocities for high-velocity clouds spanning column densities from 10^18 to 10^20 cm^-2 at heights of 1–10 kpc: if high-column clouds are found moving near their local terminal velocity at all heights, the quasi-ballistic claim fails; if they systematically exceed it, the claim stands. A sharper computational test: run the falling-cloud setup in a background that is itself allowed to cool and develop thermal instability — if condensation-driven coupling collapses and the cloud disrupts as in the adiabatic run, the cooling-restoration result fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Dynamical distance estimates for high-velocity clouds become conditional: they should only be trusted for low-column-density clouds, or for dense clouds already close to the disc, where drag and free-fall times finally compete.
  • Radiative cooling restores terminal-like motion through condensation-driven momentum loading, so observed decelerated clouds at the disc–halo interface can be interpreted as accretion-drag signatures rather than as equilibrium falls.
  • Treating the drag coefficient as a fixed number — commonly a value of 1 — biases terminal-velocity-based distances and halo-density inferences, since the measured value is roughly 2–3 and fluctuates strongly with cloud morphology.
  • The simulations predict coherent observable signatures: velocity bridges connecting bulk and stripped gas in position–velocity space, compression-dominated soft X-ray enhancement without a higher-energy band excess, and dust acquired through mixing that should correlate with kinematic disturbance rather than with the pristine cloud body.
  • Mass evolution, not drag geometry, is the dominant missing ingredient in simplified terminal-velocity models; the calibrated mass-exchange prescription carries the predictive power in the cooling runs.

Where Pith is reading between the lines

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

  • If dense high-velocity clouds fall quasi-ballistically through most of the halo, their infall speeds near the disc encode accumulated free-fall momentum rather than local terminal equilibrium — so distance estimates for the densest observed complexes could shift systematically.
  • A directly testable extension: assemble a column-density-stratified sample with independent distances (halo-star absorption bracketing or 3D dust mapping) and check whether the ratio of observed to terminal velocity falls with increasing column density as predicted here.
  • The condensation feedback implies individual infalling clouds may locally regulate halo cooling; whether this survives in a self-consistently multiphase, thermally unstable halo is the natural next simulation, and would either strengthen or overturn the cooling-restoration result.
  • As the authors note, the same framework — with self-gravity and dark matter added and tidal disruption neglected — applies to other accreting systems such as dwarf galaxies and compact gas clumps, so the 'local equilibrium, not global attractor' lesson may generalize beyond halo clouds.

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 / 3 minor

Summary. The paper reassesses the terminal-velocity paradigm for high-velocity clouds (HVCs) by combining a generalized one-dimensional equation of motion with 3D hydrodynamical simulations of a cloud falling through a stratified Milky Way halo. The authors derive new analytical solutions for constant-property clouds, introduce a Bernoulli-driven lateral-expansion model, and construct a phenomenological mass-exchange model calibrated to the simulated cloud-mass evolution. Simulations with adiabatic physics show rapid cloud disruption, while runs with radiative cooling (with and without thermal conduction) show condensation-driven mass growth and an extended phase of near-terminal-velocity motion. The authors conclude that terminal velocity is a local equilibrium rather than a global attractor, and that its applicability is conditional on cloud survival, mass exchange, and the ambient medium. Synthetic observables (position–velocity diagrams, extinction, and soft X-ray emission) are presented as links to real HVC data.

Significance. If the central claim holds, the paper provides a useful correction to the common use of terminal-velocity arguments for HVC distance estimation: the paradigm would apply only in restricted, identifiable regimes. The paper's strengths include the analytical solutions in quadrature (Section 2.2 and Appendix A), the direct measurement of a time-dependent drag coefficient from the simulations (Section 2.7.5, Fig. 10g), a resolution study in Appendix C, and a clear, honest statement of limitations in Section 4.2. The synthetic observables in Section 4.1 are a constructive step toward connecting the simulated dynamics with data. However, the significance is contingent on the robustness of the 'revival' mechanism, which rests on a single simulated initial condition and on a background thermal state that is artificially maintained; these concerns are central to the claim that the terminal-velocity paradigm can be 'revived' by cooling.

major comments (3)
  1. [§3.2, Fig. 10, Fig. 11, Table 1] The semi-analytical solutions are validated against the same simulations used to fit the mass-exchange parameters (t_M, Δt, ε_strip, ε_cond, ε_evap, η, q). Since the parameters are least-squares fitted to the simulated bulk-cloud mass evolution (Fig. 11, Table 1), the agreement in Fig. 10 is a fit, not an independent prediction. This weakens the claim in Conclusion 8 that the semi-analytical model 'captures the global evolution well' and uses that agreement as support for the cooling-induced 'revival' regime. The authors should either present the semi-analytical model as a condensed description rather than a validation, or provide an out-of-sample test (e.g., fit to part of the trajectory and predict the rest, or fit on one resolution and test on another).
  2. [§2.7.2, §2.7.6, §4.2] The RC and RC+TC runs apply radiative cooling and thermal conduction only to cloud-tagged gas (C ≥ 10^-6), so the background halo is held in hydrostatic and thermal equilibrium. As the paper acknowledges in §2.7.2, the background gas near z ≈ 1 kpc is thermally unstable and is maintained only by a 'controlled proxy' for unmodelled feedback. The condensation-driven mass growth that sustains the near-terminal velocity interval (Fig. 10b) depends on this reservoir of condensable ambient gas. The fitted balance ε_cond ≈ 2.1–2.3 versus ε_strip ≈ 0.9–1.1 (Table 1) is a narrow margin; a genuinely multiphase, turbulent halo could shift the balance toward net stripping and remove the simulation-based support for the 'revival' half of the central claim. A concrete test would be to repeat one RC run with cooling enabled in the entire domain (or with a seeded turbulent background) and check whether
  3. [§3.2, §4.2] The numerical support for the central regime classification rests on a single cloud initial condition: N_Hi = 10^20 cm^-2, R_cl = 100 pc, z_0 = 5 kpc, v_0 = -100 km s^-1. The analytical parameter study in Section 3.1 suggests that lower-column-density clouds converge to terminal velocity more readily, but the 'revival' via condensation is demonstrated for only one high-column-density case. The paper acknowledges in §4.2 that the mass-exchange parameters are calibrated for this case alone and that lower-column-density clouds may cross the critical-ablation threshold of Marinacci et al. (2010). To support the general claim that the terminal-velocity paradigm is 'conditional' rather than universally invalid, at least one additional simulation in a different regime (e.g., N_Hi = 10^19 cm^-2, or a different initial height/velocity) is needed.
minor comments (3)
  1. [§2.5, Eq. (40)] The activation function a(t) is a smooth step with parameters t_M and Δt; the paper states that t_M is expected to be of order the initial KH growth time. It would help to show explicitly how the fitted values in Table 1 compare with the independently estimated τ_KH,0 (the text gives the value 17.8 Myr).
  2. [§3.2.4, Fig. 10] In the comparison of semi-analytical and simulated curves, the semi-analytical curves are fitted to the mass evolution; this should be stated in the Fig. 10 caption as well as in the text to avoid the impression that the curves are predictions.
  3. [§4.1.1] The grid-aligned spoke-like lanes in Fig. 12 are attributed partly to numerical effects; a brief comment on how the analysis would be affected by using a random orientation or a different AMR refinement pattern would help quantify this caveat.

Circularity Check

1 steps flagged

One fitted-input validation step in the semi-analytical model, but the main terminal-velocity claim is carried by the direct simulations.

specific steps
  1. fitted input called prediction [Section 3.2 / Figs. 10–11 / Table 1 / Section 2.5.4]
    "The parameters of the phenomenological mass-exchange model introduced in Section 2.5 are determined by least-squares fits to the simulated cloud-mass evolution. Figure 11 compares the simulated mass-exchange rates with the fitted model, and the resulting parameter values are summarized in Table 1. These parameters are then used to construct the semi-analytical solutions shown in Fig. 10. ... The semi-analytical solutions reproduce the simulated evolution well in the cooling runs ..."

    The mass evolution of the semi-analytical model is not an independent prediction: (t_M, Delta t, epsilon_strip, epsilon_cond, epsilon_evap, eta, q) are least-squares fitted to the simulated bulk-cloud mass M_cl(t) (Fig. 11, Table 1). Feeding that fitted mass growth back into the EOM and then showing that the resulting semi-analytical curves track the same simulations (Fig. 10) is validation on the training set. In particular, the extended terminal-like interval produced by the fitted epsilon_cond > epsilon_strip balance is partly an input rather than an independent confirmation. However, the paper's central terminal-velocity claim is not solely based on this fit: the direct RC and RC+TC simulations independently exhibit the same qualitative behaviour, so the circularity is partial and conf

full rationale

The paper's main derivation — the generalized EOM, the quadrature solution for constant-property clouds, the terminal-velocity criterion, and the Bernoulli lateral-expansion model — is self-contained and does not reduce to its inputs. The terminal-velocity claim is primarily supported by the direct 3D simulations, whose cooling-run behaviour is not fitted. The semi-analytical mass-exchange model is explicitly calibrated to the simulated mass evolution, so its agreement with the same simulated mass curve is by construction; this is a real but partial circularity, and I locate it specifically in Section 3.2/Figs. 10–11. The paper also acknowledges the more serious environmental limitation: cooling and conduction are applied only to cloud-tagged gas, and it states that the gas condensing onto the cloud in the cooling runs is 'by construction' gas that would not have cooled spontaneously without the cloud (§2.7.6, §4.2). That weakens external validity for a real multiphase halo but is an explicitly stated conditional assumption, not a concealed circular derivation. Self-citations (e.g. Schulreich & Breitschwerdt 2022 for time-dependent RT growth) are not load-bearing for the central claim. Overall, the derivation chain has substantive independent content, so I do not assign a score above 4; the semi-analytical validation step prevents a lower score.

Axiom & Free-Parameter Ledger

9 free parameters · 9 axioms · 0 invented entities

The paper introduces no new particles, forces, or physical entities. Its additional load-bearing structure consists of fitted phenomenological parameters (t_M, Δt, ε_strip, ε_cond, ε_evap, η, q) that encode stripping, condensation, and evaporation; a constant C_d=1 for the analytical baseline; and several modeling assumptions about the background halo. The mass-exchange parameters are fit to one simulation family, which is the most honest measure of what the analytical model contributes beyond the raw simulations.

free parameters (9)
  • t_M (onset time of mass exchange) = adiabatic 1.46; RC 1.27; RC+TC 1.31 (units of τKH,0)
    Fitted onset time of the activation function in the phenomenological mass-exchange model; calibrated to simulated mass evolution.
  • Δt (activation transition width) = adiabatic 0.38; RC 0.08; RC+TC 0.02 (units of τKH,0)
    Fitted transition width of the activation function in equation (40).
  • ε_strip (stripping efficiency) = adiabatic 2.81; RC 0.90; RC+TC 1.06
    Fitted efficiency factor for hydrodynamic stripping in the mass-exchange model (Eq. 43).
  • ε_cond (condensation efficiency) = adiabatic 0; RC 2.08; RC+TC 2.29
    Fitted efficiency factor for radiative condensation; zero in the adiabatic run by construction.
  • ε_evap (evaporation efficiency) = adiabatic 0; RC 0; RC+TC 8.06e-6
    Fitted efficiency factor for thermal evaporation; essentially negligible in all runs.
  • η (cooling-time ratio coefficient) = RC 16.74; RC+TC 6.54
    Fitted parameter controlling the cooling-time threshold in the condensation function (Eq. 44).
  • q (cooling-time ratio exponent) = RC 5.51; RC+TC 4.18
    Fitted exponent in the condensation function (Eq. 44).
  • C_d (analytical drag coefficient) = 1
    Adopted constant for the analytical and semi-analytical reference solutions; the simulations later infer time-dependent values of 2-3, so this is a chosen baseline, not an externally fixed constant.
  • temperature floor for cooling = maximum initial cloud temperature (~10^4 K)
    Hand-selected floor that prevents premature cloud collapse; affects the cold-phase temperature and condensation energetics.
axioms (9)
  • domain assumption Cloud motion is purely vertical at fixed Galactocentric radius; background quantities depend only on z (Section 2.1).
    Neglects radial drift and the non-separability of the Galactic potential, discussed as a limitation in Section 4.2.
  • domain assumption Clouds are treated as coherent objects with fixed geometry in the analytical baseline; drag coefficient is constant (Section 3).
    The whole terminal-velocity concept assumes a persistent bulk object; the paper later relaxes this only partially.
  • domain assumption The flow around a moving sphere is steady, incompressible, inviscid, potential flow (Appendix B).
    Underlies the Bernoulli-driven expansion model of Section 2.3; the paper itself cautions about compressibility and Mach-number limits.
  • ad hoc to paper Mass exchange is described by the phenomenological activation/stripping/condensation/evaporation model of Section 2.5 (Eqs. 40-48).
    This is the central modeling addition that turns the EOM into a closed system; its parameters are fitted to simulations, not derived from first principles.
  • ad hoc to paper Radiative cooling and thermal conduction are applied only inside cloud-tagged gas; the background is held in hydrostatic and thermal equilibrium (Section 2.7.6).
    Prevents spurious cooling of a thermally unstable halo, but changes the supply of condensable gas that the condensation-dominated regime depends on.
  • domain assumption Galactic gas density follows the Ferrière (1998) and Miller & Bregman (2013) profiles; potential follows Barros et al. (2016) (Section 2.6).
    Standard observational inputs, but the stratification and its thermal instability affect the quantitative results.
  • domain assumption Ideal-gas EoS with γ=5/3, Sutherland & Dopita (1993) cooling, Spitzer conductivity with f_sup=0.1 and Cowie-McKee saturation (Section 2.7).
    Standard plasma assumptions; the conductivity suppression factor is taken from the literature.
  • standard math Hydrodynamic instability growth times are estimated from Chandrasekhar/Klein-type scaling relations (Section 2.4.3).
    Used to construct the mass-exchange model's time-scales and to interpret the simulations; standard in the cloud-crushing literature.
  • domain assumption Numerical discretization with AMRVAC 3.1, HLLC solver, Koren limiter, and 32 cells per initial cloud radius is sufficiently converged (Appendix C).
    The resolution study supports bulk convergence, but detailed small-scale thermal structure is only marginally resolved, as the paper acknowledges.

pith-pipeline@v1.3.0-alltime-deepseek · 45474 in / 12676 out tokens · 105025 ms · 2026-08-01T09:56:26.225895+00:00 · methodology

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read the original abstract

The terminal-velocity paradigm has long been used to interpret the motion and infer the distances of HVCs accreting onto the Milky Way, yet its validity under realistic Galactic conditions remains uncertain. We investigate its dynamical limits by combining analytical modelling with three-dimensional hydrodynamical simulations of clouds moving through a stratified Milky Way halo. We derive a generalized equation of motion including gravity, ram-pressure drag, a phenomenological mass-exchange model capturing mass loss and growth, and Bernoulli-driven cloud expansion. Analytical solutions for constant-property clouds provide a reference framework, while the full evolution is assessed using simulations with adiabatic physics, radiative cooling, and thermal conduction. Terminal velocity is a local equilibrium but not a global attractor: dense clouds remain quasi-ballistic over most of their trajectories and approach terminal motion only shortly before reaching the Galactic disc. Hydrodynamical effects further limit the paradigm. In adiabatic flows, instabilities rapidly disrupt the cloud, rendering the terminal-velocity description inapplicable. Radiative cooling instead promotes condensation and momentum loading, maintaining strong coupling to the background gas and restoring a terminal-velocity-like regime over extended periods, while thermal conduction mainly affects small-scale structure. Synthetic observables, including position--velocity diagrams, optical extinction, and soft X-ray emission, reproduce key features of observed HVCs such as velocity bridges and compression-driven emission. We also provide direct measurements of the effective drag coefficient for infalling clouds, finding values of order unity but strongly time-dependent. Overall, the terminal-velocity paradigm is a conditional description governed by cloud structure, mass exchange, and the ambient medium.

Figures

Figures reproduced from arXiv: 2607.20394 by Dieter Breitschwerdt, J\"urgen Kerp, Michael M. Schulreich.

Figure 1
Figure 1. Figure 1: Schematic cross-sectional representation of the flow around an initially spherical cloud, shown in the cloud’s rest frame during vertical motion towards the Galactic mid-plane. The left panel depicts the undeformed cloud at an early time, indicating the stagnation point at the leading edge (𝑃s), the body streamline (red curve), and the lateral pressure minima at the cloud’s flanks (exemplified by the point… view at source ↗
Figure 2
Figure 2. Figure 2: Modelled vertical gas density (left panel) and gravitational acceleration (right panel) profiles of the Milky Way at Galactocentric radii of 𝑅 = 4 kpc (dotted black lines), 𝑅 = 𝑅0 = 8.122 kpc (solid black lines), and 𝑅 = 12 kpc (dashed black lines). These profiles serve as the cloud’s background medium throughout this work. For comparison, the grey lines show the solar-neighbourhood profile adopted by BD97… view at source ↗
Figure 3
Figure 3. Figure 3: Trajectories of constant-property clouds (opaque coloured curves) with peak column densities 𝑁H i = 1018 cm−2 (left panel), 1019 cm−2 (middle panel), and 1020 cm−2 (right panel), released from rest at heights 𝑧0 = 103 pc (green), 104 pc (blue), and 105 pc (orange), and at Galactocentric radii 𝑅 = 4 kpc (dotted), 𝑅0 = 8.122 kpc (solid), and 12 kpc (dashed). Empty circles mark the cloud positions after one f… view at source ↗
Figure 4
Figure 4. Figure 4: Positional evolution of the drag time (opaque curves) and the terminal-velocity convergence time (semi-transparent curves), both normalized to the free-fall time, for constant-property clouds with peak column densities 𝑁H i = 1018 cm−2 (left panel), 1019 cm−2 (middle panel), and 1020 cm−2 (right panel), released from rest at heights 𝑧0 = 103 pc (green), 104 pc (blue), and 105 pc (orange), and at Galactocen… view at source ↗
Figure 5
Figure 5. Figure 5: As for [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: As for [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Snapshots of the interaction between the infalling cloud and the Galactic background medium from the adiabatic run, showing density (left half of each panel) and temperature (right half). Each panel presents a cross-sectional slice perpendicular to the 𝑦-axis at the initial cloud centre. Green streamlines visualize the flow pattern; a green disc marks the cloud’s initial shape, size, and position, while a … view at source ↗
Figure 8
Figure 8. Figure 8: As for [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: As for [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Time evolution of key dynamical and thermodynamic cloud properties. Coloured curves represent the simulations, semi-transparent coloured curves the semi-analytical solutions, and grey curves the analytical solutions (where available). The panels show: (a) COM vertical position; (b) COM vertical velocity, together with the corresponding terminal and ballistic solutions; (c) effective cloud radius, normaliz… view at source ↗
Figure 11
Figure 11. Figure 11: Simulated bulk-cloud mass-exchange rates (opaque coloured curves) together with their least-squares fits (semi-transparent coloured curves), based on the model described in Section 2.5. The corresponding fit parameters are listed in [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Total hydrogen column density (left half) and column-density￾weighted LOS velocity (right half) of the cloud-related gas (region C; see Section 2.7.4) in the RC+TC simulation at the indicated time, corresponding to a LOS along the 𝑧-axis. are generally interpreted as tracers of the motion of an HVC rela￾tive to the ambient medium, VBs extend over a substantially larger range in radial velocity and thus pr… view at source ↗
Figure 13
Figure 13. Figure 13: Synthetic position–velocity diagrams of the cloud-related gas (region C; see Section 2.7.4) in the RC+TC simulation at the indicated time. The left panel shows the ideal case without thermal broadening, whereas the right panel includes thermal Doppler broadening. The colour scales represent the intensity distributions defined by equations (67) and (68). Contours trace the contributions from the bulk cloud… view at source ↗
Figure 14
Figure 14. Figure 14: Optical extinction (left half) and SXR surface brightness in the ROSAT C-band (right half; shown relative to its median background level) of the dynamically disturbed gas (region V; see Section 4.1.2) in the RC+TC simulation at the indicated time. The top panel corresponds to a LOS along the 𝑥-axis, while the bottom panel shows the projection along the 𝑧-axis. per cent of these, the remainder being low-ra… view at source ↗

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