{"id":"2f15b471-7014-453f-a4ed-1d404f590e8d","arxiv_id":"2507.02050","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A memory-timescale prescription for the dynamical corotation torque lets 1D models reproduce 2D hydrodynamic migration of low-mass planets in low-viscosity discs.","lead":"Planets migrating through gas discs feel an extra torque whose strength depends on how long the gas remembers its past vortensity state; this paper packages that torque into a simple formula for fast planet formation models. It matters because in the low-turbulence discs where planets likely form, this torque can slow migration by about half for super-Earths at 10 au.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Memory coefficient in Eq. 13 is calibrated on the same 2D runs used for validation, so the claimed 'very good agreement' is partly in-sample; an out-of-sample test is needed.","rationale":"The reader's weakest_assumption already names the calibration issue, and I agree: the single most load-bearing element of the central claim is that tau_memory = 0.4 tau_vis * max(1 - tau_lib/(2 tau_vis), 0) (Eq. 13) captures the vortensity-mixing physics well enough to predict migration outside the calibration data. Because the same 2D runs are used both to set 0.4 and to validate the 1D model, the paper does not currently demonstrate the predictive power it claims. This does not invalidate the work: the model is plausible, the alternative model gives similar results, and the partial out-of-sample tests (h0=0.07, T propto r^-1, beta=2) are encouraging. But a reader cannot yet distinguish between a physically correct prescription and a good fit to the calibration set. I therefore recommend no change to the reader's CONDITIONAL verdict; the paper should be accepted only with the additional condition of an out-of-sample test or a quantitative sensitivity analysis. My concrete test above would settle the concern directly.","tokens_in":23270,"tokens_out":6422,"duration_ms":68190,"concrete_test":"Perform a train/test split on the existing simulations: re-estimate the prefactor C in tau_memory = C tau_vis * max(1 - tau_lib/(2 tau_vis), 0) using only the Q0=8, beta=1/2 runs (or, more conservatively, only the Q0=8, beta=1/2 runs at alpha_vis=1e-5), then run 1D simulations for all other parameter combinations in Table 1 and for the Sec. 6.1 non-isothermal, Appendix A, and beta=2 cases. Quantify agreement by the mean absolute relative deviation of migration rate between 1D and 2D over the first 1500 orbits (or until the planet reaches 0.7 r0). If the held-out error is comparable to the calibration-set error, the coefficient generalizes; if it is significantly larger (e.g., >50% increase), the central claim is a calibration artifact rather than a validated prediction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3.3 (Eq. 8, Fig. 4) the memory timescale is estimated from 2D simulations with Q0=8, 12, 16, 24, 32 and beta=1/2,1, yielding tau_memory ~ 0.3-0.4 tau_vis; Eq. 13 then sets tau_memory = 0.4 tau_vis * max(1 - tau_lib/(2 tau_vis), 0). Section 4.2 validates the 1D model against the same family of 2D runs (Fig. 2), including the very Q0=8, beta=1/2,1, alpha_vis=1e-5 runs used in the calibration. The central claim of 'very good agreement' therefore rests on a parameter that is fit to the data being compared, not on a prediction. Appendix A (h0=0.07) and Sec. 6.1 (T propto r^-1) provide some out-of-sample support, and the beta=2 runaway runs were not used to set 0.4, but these checks share the same code and setup and are still in the same paper; they do not establish that the coefficient is universal across a 'wide parameter space'. The comparison is also purely visual: no quantitative error metric is reported for the migration-rate curves. Finally, the alternative model in Sec. 6.2, which uses a different functional form (weighted average of initial and local vortensity), matches the 2D runs about as well, indicating that 1D migration outcomes are insensitive to the precise mixing prescription and that the specific Eq. (6)+(13) form is not uniquely tested. The load-bearing premise is that a single fixed coefficient, calibrated in-sample, extrapolates across viscosity, surface-density slope, planet mass, and disc aspect ratio; that extrapolation is currently untested.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a 1D analytical prescription for the dynamical corotation torque acting on low-mass planets in low-viscosity discs. The central element is a 'memory timescale' for the vortensity of the librating horseshoe flow: the librating flow is assumed to retain the inverse vortensity from a single upstream location, with a memory timescale that is a fraction of the viscous timescale across the horseshoe region. The coefficient 0.4 in this memory timescale is estimated from 2D FARGO3D simulations, and the resulting 1D model is then compared with the same 2D runs across variations in surface-density slope beta, alpha viscosity, Toomre parameter Q0, and aspect ratio h0. The paper also presents an alternative 1D model based on a time-weighted average of initial and local vortensity, tests the model on non-isothermal temperature profiles, and produces maps of the reduction factor of the classical type I torque as a function of planet mass and orbital distance. The authors conclude that the dynamical corotation torque should be included in global planet formation models for alpha_vis below about 1e-4.","tokens_in":23714,"tokens_out":6440,"duration_ms":81496,"significance":"If the proposed prescription is robust, it would be a useful and immediately applicable ingredient for 1D population synthesis models, which currently often omit the dynamical corotation torque. The paper's strengths are the systematic parameter study, the explicit comparison of several 1D prescriptions against 2D hydrodynamics, the inclusion of numerical-diffusion tests in Section 6.1, and the out-of-sample checks provided by the h0=0.07 runs in Appendix A and the non-isothermal runs in Section 6.1. The reduction-factor maps in Figure 5 are a genuine practical contribution. However, the central validation is weakened by the fact that the memory coefficient is calibrated on the very simulations used for the 'verification', and by the absence of quantitative error metrics. The alternative model in Section 6.2 is stated to work comparably well, which means the specific memory-timescale form is not uniquely constrained by the migration curves.","major_comments":[{"comment":"The memory timescale coefficient 0.4 in Eq. (13) is estimated in Section 3.3 from the 2D simulations shown in Fig. 4, and the same family of runs (including Q0=8, beta=1/2 and 1, alpha_vis=1e-5) is then used in Section 4.2 to validate the 1D model in Fig. 2. The claimed 'very good agreement' is therefore partly in-sample. The non-isothermal runs in Section 6.1 and the h0=0.07 runs in Appendix A are useful out-of-sample checks, but they share the same code, setup, and planet mass, and they do not test whether the value 0.4 is universal across mass or viscosity. I request an explicitly out-of-sample validation, for example with a different planet mass at fixed q/h^3 or with a viscosity value not used in the calibration, and a quantitative error metric computed for runs not used in the calibration. The abstract and conclusions should also state that 0.4 is a calibrated coefficient rather than a parameter-free prediction.","section":"Sections 3.3 and 4.2, Eq. (13)"},{"comment":"The central claim that the 1D and 2D simulations are in 'very good agreement' is based only on visual comparison of time series. No normalized RMS difference, maximum deviation, or other quantitative metric is reported for the migration rate, orbital distance, or vortensity ratio. Because the paper explicitly claims a quantitative level of agreement, a quantitative measure should be added, along with a statement of what level of disagreement would be considered acceptable for the intended use in population synthesis.","section":"Section 4.2, Figs. 2 and B.1"},{"comment":"The alternative model of Eqs. (21)-(22), which is not based on a memory timescale, is stated to 'also work very well, overall', and in Appendix A it reproduces the h0=0.07 runs nearly perfectly. This means the migration curves do not distinguish the memory-timescale prescription from the alternative mixing model. Since the paper presents Eqs. (6) and (13) as a verified description of the physical mechanism, the authors should either provide a test that discriminates between the two models (for example, the early transient of the vortensity ratio before quasi-steady state, or a parameter regime where their predictions diverge) or explicitly temper the claim that the memory-timescale form is uniquely validated.","section":"Section 6.2 and Fig. 7"},{"comment":"The text near Eq. (13) states that 'a memory effect sets in at tau_vis ~ 2 tau_lib', but Eq. (13) gives a zero memory timescale at tau_vis = tau_lib/2 and a positive value for tau_vis > tau_lib/2. With the values quoted in Section 3.3 (tau_vis ~ 130 orbits, tau_lib ~ 100 orbits at alpha_vis=1e-4), tau_vis/tau_lib is about 1.3, so a memory effect is already present at alpha_vis=1e-4. This internal inconsistency affects the interpretation of when the dynamical corotation torque becomes important and should be corrected.","section":"Section 4.1, Eq. (13)"},{"comment":"The reduction-factor maps in Figure 5 vary the planet mass over a wide range, but all 2D validation simulations use a single planet-to-star mass ratio of q=1e-5 (with one additional value q=2.7e-5 in Appendix A at h0=0.07). The mass dependence of the prescription is therefore not directly tested against hydrodynamical simulations. The paper should either add 2D runs at a different q for fixed h0, or present the mass dependence in Fig. 5 as an extrapolation that has not been simulation-tested.","section":"Section 5 and Fig. 5"}],"minor_comments":[{"comment":"The notation in Eq. (3) uses 'min[rP - rdot_P tau_memory, r0]' and then states in the text that for outward migration the minimum should be replaced by a maximum, but this is not shown in a displayed equation; adding a one-line formula for outward migration would avoid ambiguity.","section":"Section 2.2"},{"comment":"The measurement of I_nu,lib in the runaway regime uses a narrow azimuthal window phi-phi_p in [0.4,0.6] at r=r_p, which is a very small portion of the libration island; a brief justification of why this window is representative of the librating flow would strengthen the methodology.","section":"Section 3.2.2"},{"comment":"The sentence 'The default timestep is one orbit at the planet's initial location' is ambiguous: it could mean 2*pi/Omega(r0), or that a timestep is taken once per orbit. Please clarify the actual time-step control.","section":"Section 4.1"},{"comment":"In Appendix A the text says 'the latter inequality is marginally fulfilled in the simulations presented here', but the inequality referred to is not explicitly identified; please restate it for clarity.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the manuscript is within the scope of A&A and addresses a timely problem for low-viscosity disc models. The main scientific issue is that the memory coefficient is calibrated on the same simulations used for validation, and the alternative model in Section 6.2 performs comparably, so the current evidence supports the practical utility of the prescription but not yet the strong claim of a uniquely verified physical model. A revision with an out-of-sample run, quantitative agreement metrics, and explicit statements about the calibrated nature of the coefficient would make the paper publishable. There are no ethical or novelty concerns; the paper properly credits prior work on the dynamical corotation torque."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this paper gives you a simple, workable 1D prescription for the dynamical corotation torque, and it does a fair job testing it against 2D hydro. The central caveat is that the key memory-timescale coefficient (0.4 in Eq. 13) is measured from the same 2D simulations used to validate the model, so part of the 'very good agreement' is expected rather than predicted.\n\nWhat's genuinely new: the memory-timescale formulation for the librating flow's vortensity (Eqs. 3–6 and 13) is a clean addition for global models. The paper also provides the first systematic benchmark of Paardekooper's steady-state model (2014), showing where it underestimates the vortensity ratio and misses the initial build-up. The alternative parameter-free model in §6.2 (Eq. 21) is a nice cross-check: it reproduces the 2D runs about as well, which tells you the 1D migration outcome doesn't hinge on the precise mixing prescription. The h0=0.07 runs and the T∝r^-1 runs give some out-of-sample support, and the runaway (β=2) runs were not used to set 0.4.\n\nWeaknesses, in proportion: the in-sample calibration is real. The coefficient is fit to the Q0=8–32, β=0.5–1, αvis=1e-5 family, and then tested against the same family. The out-of-sample checks are good but share code and setup. The comparison is visual only—no quantitative error metric on migration-rate curves, which would make the claim more precise. The heuristic cutoff in Eq. 13 is also there. None of this breaks the paper, but a referee should ask for a quantitative out-of-sample test or at least an error metric.\n\nBottom line: for anyone building 1D planet formation models, this is a practical upgrade. The maps in Fig. 5 are immediately useful. It deserves peer review and, if the calibration caveat is addressed, it should be cited.\n\nI'd bring it to reading group.","headline":"Useful 1D prescription for the dynamical corotation torque, honestly benchmarked, but its key memory coefficient is calibrated on the same 2D runs used for validation.","tokens_in":24237,"tokens_out":1944,"would_cite":true,"duration_ms":21068,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that the dynamical corotation torque on a migrating low-mass planet can be represented in 1D planet-formation models by a single memory timescale, and finds that the torque can halve the classical type I migration speed…","keywords":["type I migration","dynamical corotation torque","protoplanetary discs","low-viscosity discs","vortensity","horseshoe region","memory timescale","planet formation"],"falsifier":"Rerun the 2D simulations at the paper's lowest viscosity ($\\alpha_{\\mathrm{vis}} = 10^{-5}$) with twice the radial and azimuthal resolution: the paper quotes a numerical-diffusion floor between $\\alpha_{\\mathrm{vis}} = 10^{-7}$ and $10^{-6}$ for its grid, so if the memory time inferred from Eq. (8) leaves the $0.3$–$0.4\\,\\tau_{\\mathrm{vis}}$ band as the grid is refined, the calibration is set by the numerics rather than by physical vortensity mixing.","tokens_in":23095,"feed_emoji":"🪐","tokens_out":21033,"duration_ms":176224,"temperature":0.7,"pith_summary":"The paper argues that the dynamical corotation torque — the extra torque a migrating low-mass planet feels because gas in its horseshoe region remembers the vortensity of the disc regions it came from — can be captured in 1D planet-formation models by a single memory timescale, and that this matters because in low-viscosity discs this torque strongly slows down type I migration. The authors propose simple analytical prescriptions for the vortensity of the librating and orbit-crossing flows, calibrate the memory timescale with 2D hydrodynamical simulations, and show that 1D simulations using their prescriptions reproduce the 2D migration tracks across a wide parameter space, both when the torque slows inward migration and when it drives runaway migration. The consequence for planet-formation modelling is that the torque should be included whenever the disc's turbulent viscosity is $\\alpha_{\\mathrm{vis}} \\lesssim 10^{-4}$, since it can cut the classical type I migration torque by about half for a 10 Earth-mass planet at 10 au. The practical payoff is a closed-form correction that global population-synthesis models can evaluate at each timestep without running hydrodynamics.","feed_headline":"A memory timescale lets 1D models match 2D planet migration","feed_subtitle":"The corotation torque slows type I migration in low-viscosity discs; the new recipe puts it into 1D formation models.","key_machinery":"The central object is the dynamical corotation torque, written in Eq. (1) as an integral across the planet's horseshoe region — the crescent-shaped band of coorbital gas that U-turns around the planet — of the difference $I_{\\nu,\\mathrm{lib}} - I_{\\nu,\\mathrm{cross}}$ between the inverse vortensities of the librating and orbit-crossing flows, with $I_\\nu = \\Sigma/[(\\nabla\\times\\mathbf{v})\\cdot\\mathbf{e}_z]$. The load-bearing move is the memory-time ansatz: the librating flow's inverse vortensity equals the unperturbed value at the planet's location $\\tau_{\\mathrm{memory}}$ earlier, so in a power-law disc $I_{\\nu,\\mathrm{lib}}/I_\\nu(r_P) = [\\min(1 - \\tau_{\\mathrm{memory}}/\\tau_{\\mathrm{mig}}, r_0/r_P)]^{3/2-\\beta}$, with $\\tau_{\\mathrm{mig}} = r_P/\\dot r_P$ the migration timescale. The single parameter $\\tau_{\\mathrm{memory}} = 0.4\\,\\tau_{\\mathrm{vis}}\\max(1 - \\tau_{\\mathrm{lib}}/2\\tau_{\\mathrm{vis}}, 0)$ is calibrated from 2D simulations that show $\\tau_{\\mathrm{memory}} \\simeq 0.3$–$0.4\\,\\tau_{\\mathrm{vis}}$; this converts a migration-history problem into a local formula that 1D models can evaluate at each timestep.","core_discovery":"The central claim is that the vortensity of the gas librating in a migrating low-mass planet's horseshoe region is, to good approximation, the unperturbed disc vortensity at the planet's position one memory time in the past, $I_{\\nu,\\mathrm{lib}} = I_\\nu(r_P - \\dot r_P \\tau_{\\mathrm{memory}})$, with $\\tau_{\\mathrm{memory}} \\simeq 0.4\\,\\tau_{\\mathrm{vis}}$ once the viscous time across the horseshoe region exceeds roughly half the libration time and zero otherwise. Here $I_\\nu = \\Sigma/[(\\nabla\\times\\mathbf{v})\\cdot\\mathbf{e}_z]$ is the inverse vortensity and $\\tau_{\\mathrm{vis}} = x_s^2/\\nu$ the viscous diffusion time across the horseshoe half-width $x_s$. With the orbit-crossing flow's vortensity fixed by the local density gradient, this turns the dynamical corotation torque — which normally depends on the planet's migration history — into a local formula that the authors implement in 1D disc-planet simulations and benchmark against 2D hydrodynamical simulations of locally isothermal discs. They find very good agreement for density slopes $\\beta = 1/2,\\,1,\\,3/2,\\,2$, for $\\alpha_{\\mathrm{vis}}$ between $10^{-3}$ and $10^{-5}$, and for Toomre parameters $Q_0$ from 8 to 32, in both the slow-down and runaway regimes. The paper concludes that global models of planet formation should include the dynamical corotation torque at $\\alpha_{\\mathrm{vis}} \\lesssim 10^{-4}$, where it can reduce the classical type I torque by about half for a 10 Earth-mass planet at 10 au.","pith_inferences":["Editorial inference: if the $0.4\\,\\tau_{\\mathrm{vis}}$ calibration survives higher-resolution tests, the same memory-time trick could be extended to discs with radiative cooling or baroclinic vortensity production, which the paper explicitly leaves out; calibrating an effective memory time there would generalise the prescription.","Editorial inference: the 2D data show a mild dependence of $\\tau_{\\mathrm{memory}}/\\tau_{\\mathrm{vis}}$ on the migration rate, so a two-parameter fit letting the fraction depend on $\\tau_{\\mathrm{vis}}/|\\tau_{\\mathrm{mig}}|$ could remove the residual offset the paper reports at $\\alpha_{\\mathrm{vis}} = 10^{-5}$ in the runaway regime.","Editorial inference: because the slow-down branch acts at several Earth masses and distances beyond roughly 10 au, population syntheses of super-Earths in wind-driven low-viscosity discs should predict a pile-up at 5–20 au; this is a checkable demographic signature for planet surveys.","Editorial inference: the numerical-diffusion floor the paper quotes means the $\\alpha_{\\mathrm{vis}} = 10^{-5}$ validation runs sit close to the grid's resolving power, so applying the prescription to even lower viscosities should wait for a higher-resolution benchmark."],"forward_implications":["1D global models of planet formation and evolution can evaluate the dynamical corotation torque with a closed-form prescription, with no need to track the disc's full vortensity field or the planet's migration history.","In discs with $\\alpha_{\\mathrm{vis}} \\lesssim 10^{-4}$ the dynamical corotation torque reduces the classical type I migration torque by up to about 50% for a 10 Earth-mass planet at 10 au, so low-mass planets survive longer in the outer disc before migrating inward.","The same prescription also reproduces runaway inward migration for steep density slopes ($\\beta > 3/2$), where the dynamical corotation torque accelerates the planet instead of slowing it down.","The memory effect switches on only when the viscous time across the horseshoe region exceeds roughly half the libration time (near $\\alpha_{\\mathrm{vis}} \\simeq 10^{-4}$ for the fiducial case), so above that viscosity the classical torque formulae remain adequate without any extra term.","The published maps of the reduction factor $f_{\\mathrm{red}}$ give an immediate multiplicative correction to the type I torque as a function of planet mass and orbital distance, ready for direct use in population syntheses."],"supporting_citations":[{"why":"Sets up the dynamical corotation torque theory and the steady-state vortensity model that this paper benchmarks and extends with a memory timescale.","marker":"Paardekooper 2014"},{"why":"Provides the torque expression used as Eq. (1), fixing the integral over the horseshoe region of the librating minus orbit-crossing inverse vortensity.","marker":"McNally et al. 2017"},{"why":"Establishes that the sign of the dynamical corotation torque is set by comparing the vortensity of the librating and orbit-crossing flows.","marker":"Masset & Papaloizou 2003"},{"why":"Supplies the Lindblad and static corotation torque formulae and the horseshoe half-width $x_s$ used throughout the 1D model and the reduction-factor maps.","marker":"Paardekooper et al. 2011"},{"why":"Contributes the method for measuring the librating and orbit-crossing vortensities in the 2D simulations, from which the memory timescale is inferred.","marker":"Wafflard-Fernandez & Baruteau 2020"},{"why":"The hydrodynamical code used for the 2D disc-planet simulations that validate the 1D prescription.","marker":"Benítez-Llambay & Masset 2016"},{"why":"Fixes the fast-migration limit of the torque, the regime used in Eq. (12) when migration runs away.","marker":"McNally et al. 2018"},{"why":"Gives the small-mass condition $q_P \\ll h^3$ under which the orbit-crossing flow keeps its unperturbed local vortensity, justifying Eq. (2).","marker":"Lin & Papaloizou 1993"}],"fun_headline_variants":["Memory-timescale recipe matches 1D to 2D planet migration","New torque prescription slows type I migration in low-viscosity discs","1D models capture corotation torque with memory timescale","Corotation torque memory timescale improves 1D migration models","How to include the corotation's memory in 1D planet formation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole result rests on one premise: that the gas trapped near the planet keeps the imprint of a single earlier disc location for a fixed length of time, about 40 percent of the time viscosity takes to smooth that region, and erases the memory afterwards; if the real mixing is more complicated — through temperature gradients, three-dimensional flows, or numerical diffusion inside the simulations themselves — the 1D model's agreement with the 2D runs would be a calibration artifact rather than a physical prediction.","fun_headline_variants_meta":{"raw":{"variants":["Memory-timescale recipe matches 1D to 2D planet migration","New torque prescription slows type I migration in low-viscosity discs","1D models capture corotation torque with memory timescale","Corotation torque memory timescale improves 1D migration models","How to include the corotation's memory in 1D planet formation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0009,"raw_usage":{"total_tokens":4041,"prompt_tokens":1278,"completion_tokens":2763,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":894,"completion_tokens_details":{"reasoning_tokens":2673}},"tokens_in":894,"tokens_out":2763,"duration_ms":21170,"temperature":1.0,"reasoning_tokens":2673,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:39:07.869959+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Rerun the 2D simulations at the paper's lowest viscosity ($\\alpha_{\\mathrm{vis}} = 10^{-5}$) with twice the radial and azimuthal resolution: the paper quotes a numerical-diffusion floor between $\\alpha_{\\mathrm{vis}} = 10^{-7}$ and $10^{-6}$ for its grid, so if the memory time inferred from Eq. (8) leaves the $0.3$–$0.4\\,\\tau_{\\mathrm{vis}}$ band as the grid is refined, the calibration is set by the numerics rather than by physical vortensity mixing.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Sets up the dynamical corotation torque theory and the steady-state vortensity model that this paper benchmarks and extends with a memory timescale."},{"cited_title":"P., Nelson , R","cited_arxiv_id":null,"evidence_quote":"Provides the torque expression used as Eq. (1), fixing the integral over the horseshoe region of the librating minus orbit-crossing inverse vortensity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that the sign of the dynamical corotation torque is set by comparing the vortensity of the librating and orbit-crossing flows."},{"cited_title":"J., Baruteau , C., & Kley , W","cited_arxiv_id":null,"evidence_quote":"Supplies the Lindblad and static corotation torque formulae and the horseshoe half-width $x_s$ used throughout the 1D model and the reduction-factor maps."},{"cited_title":"& Baruteau , C","cited_arxiv_id":null,"evidence_quote":"Contributes the method for measuring the librating and orbit-crossing vortensities in the 2D simulations, from which the memory timescale is inferred."},{"cited_title":"P., Nelson , R","cited_arxiv_id":null,"evidence_quote":"Fixes the fast-migration limit of the torque, the regime used in Eq. (12) when migration runs away."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the small-mass condition $q_P \\ll h^3$ under which the orbit-crossing flow keeps its unperturbed local vortensity, justifying Eq. (2)."}],"review_version":1}