{"id":"69e61e32-c1d0-40f0-8789-2d8f2c2cd072","arxiv_id":"2506.18718","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Impact melt efficiency on rocky planets is controlled by how deep the impact melts relative to the planet's cold outer layer, and the paper fits a formula to predict it across planet sizes, core sizes, and thermal ages.","lead":"This paper simulates impacts on rocky planets of different sizes and ages to measure how much melt each impact creates. It finds that the most efficient melting happens when the impact reaches the depth where the planet's hot interior meets its cold outer shell, and it offers a formula to predict this.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Latent-heat neglect plus ad hoc truncations leave the absolute calibration of the empirical π_m law untested; the fitted amplitude A in Eq. 20 inherits any residual bias.","rationale":"The paper is a large, internally consistent computational study; the qualitative trends—the peak in melting efficiency near d_m ≈ d_L, the planet-size and core-size dependencies, and the Moon-size cumulative melt result—are supported by a substantial simulation matrix and are presented transparently. I agree with the reader that the latent-heat omission is the most load-bearing concern. The melt fraction in Eq. 8 is computed from post-impact temperature relative to solidus and liquidus without subtracting the energy consumed by melting, so ψ is systematically too high wherever the material is above the solidus. The three truncations in Section 2.3 are not derived from a physical model, and the paper's own wording—that they 'should help to compensate ... to some extent'—acknowledges the uncertainty. All fitted quantities in Eqs. 16–21, especially the amplitude A, are linear functions of the π_m data, so any systematic error propagates directly into the formula. This affects the paper's practical deliverable of absolute melt-volume estimates, and could even alter relative conclusions if the bias varies with planet size or thermal state—for example, if the 25 K cut and 5L domain cut remove different melt fractions on large versus small planets. I also note that the empirical law is fit to the same data it predicts; I regard this as a secondary concern because the paper frames the formula as a parameterization, but it means no out-of-sample check currently constrains extrapolation to real planets. The proposed test—recomputing a few representative cases with an energy-conserving melt diagnostic—would settle whether the compensation is adequate. Since the reader already issued CONDITIONAL on essentially this basis, my assessment does not change the verdict.","tokens_in":31718,"tokens_out":6024,"duration_ms":68412,"concrete_test":"Recompute π_m for a representative subset (Earth-size planet, R_c/R_P = 0.5, 3 Gyr target; impactor diameters L = 25, 100, 250 km) using an energy-conserving melt diagnostic that subtracts the latent heat of fusion from the tracer energy budget, or an EOS implementation that includes melting enthalpy. Compare the resulting π_m values and the refitted amplitude A in Eq. 20 with the published values. If the difference exceeds about 20%, the truncation compensation is inadequate and the absolute amplitudes of Eqs. 16–21 are biased; if it is below about 10%, the concern is largely resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central practical deliverable—absolute melt volumes for basin-forming impacts—depends on the melt diagnostic of Section 2.3. Melt fraction ψ is computed from the reconstructed temperature relative to solidus and liquidus (Eq. 8), but the enthalpy of fusion is not subtracted because of ANEOS limitations. The authors state that this makes the melt volumes 'likely an overestimate' and rely on three truncations (melt domain cut at 5L, 25 K below the solidus, and discarding melt fractions below 2%) to compensate 'to some extent.' These truncations are ad hoc and are not calibrated against an energy-conserving melt estimate. Every π_m value entering the fit is affected, so the baseline efficiencies π_c^m and π_m^m and the amplitude A in Eq. 20 inherit any residual bias. If the compensation is wrong by 20–30%, the empirical law's predicted melt volumes shift by the same order even if the d_m/d_L peak structure is qualitatively correct. No independent test is offered to separate the physical trend from the reconstruction bias, and the law is fit to the same simulation suite it is claimed to predict.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates impact-induced melt production on generic terrestrial planets by combining 1D parameterized thermal evolution models with 2D iSALE hydrocode impact simulations. It spans planet radii 0.1–1.5 R_Earth, core-to-planet radius ratios 0.2–0.8, thermal ages 1, 3, and 4.5 Gyr, and impactor diameters 1–1000 km (more than 200 runs). The central quantity is the melting efficiency π_m = V_melt/V_impactor, and the authors show that peaks in π_m occur when the depth of melting d_m is comparable to the lithospheric thickness d_L (d_m/d_L ≈ 1 to 3), that larger planets are most efficiently melted by smaller impactors while smaller planets favor larger impactors, and that core size has little effect except for old, large-core cases. They propose an empirical scaling law (Eqs. 16–21) expressing π_m as a function of d'_m = d_m/d_L and d'_T = d_T/d_L, and they use a lunar impactor flux to estimate cumulative melt production over 4.5 Gyr, concluding that Moon-sized planets produce the most melt relative to their volume. The paper also compares its results with classical homogeneous-target scaling laws and argues that those laws underestimate melt production when target thermal structure and pressure gradients are important.","tokens_in":31990,"tokens_out":4029,"duration_ms":43065,"significance":"If the qualitative trends survive scrutiny, this is a substantial advance: it extends impact melt scaling from homogeneous targets to thermally and structurally layered stagnant-lid planets, systematically includes decompression and plastic-work melting alongside shock melting, and provides a compact parameterization that could be applied without new hydrocode runs. The study is unusually comprehensive in its parameter coverage, the qualitative regime separation (crust melting, peak-efficiency, and deep mantle regimes) is physically plausible and internally consistent, and the authors have published replication data, which strengthens reproducibility. The main significance risk is that the quantitative, absolute calibration of the empirical law and the cumulative melt volumes inherits an unquantified systematic bias from the melt diagnostic and from fitting the law to the same simulation suite, so the predictive power beyond the calibrated cases is not yet demonstrated.","major_comments":[{"comment":"The melt diagnostic neglects latent heat because of ANEOS limitations, and the three ad hoc truncations (melt domain cut at 5L, 25 K below the solidus, and discard of melt fractions below 2%) are not calibrated. The manuscript states that melt volumes are 'likely an overestimate' and that the truncations 'should help to compensate for it to some extent,' but no test against an energy-conserving melt estimate is provided. Because every π_m value entering the empirical law (Eqs. 16–21) and the cumulative melt volumes in Fig. 10 inherit this bias, this is a load-bearing uncertainty for the paper's quantitative claims. I request either an explicit calibration of the truncations against at least a subset of runs with a latent-heat-corrected melt calculation, or a clear error budget that propagates a plausible range of latent-heat corrections through the fitted amplitude A in Eq. (20).","section":"Section 2.3, Eq. (8) and following truncations"},{"comment":"The 'predictive' empirical law is a least-squares fit to the same simulation data it is compared with in Fig. 9, so the agreement in that figure is by construction. The paper is transparent that the formulas are empirical, but the abstract and Section 4.1 present them as 'predict[ing] melt generation as a function of radial structure and thermal age.' Please rephrase the claim as an interpolation/parameterization of the explored parameter space, and ideally validate the law on withheld cases (e.g., a subset of runs not used in the fit) or on independent data such as basin melt volumes on Mars or the Moon. At minimum, provide the fit uncertainties on Eqs. (19)–(21) so users can judge extrapolation risks.","section":"Eqs. (16)–(21), Fig. 9"},{"comment":"The velocity scaling exponent b = 0.986 is calibrated on data from a single Mars-like target structure (Manske et al. 2021) over only three impact velocities (10, 15, 20 km/s), but it is applied to all planet sizes, thermal ages, core ratios, and scaled up to 23.85 km/s in the cumulative melt production estimates of Fig. 10. The manuscript acknowledges that the scaling 'may be strongly dependent on the impact and target conditions,' but does not quantify the resulting uncertainty. Please add a sensitivity test that uses alternative scaling exponents (e.g., the value from Pierazzo et al. 1997 or a range bracketing the fitted slope) and show how the accumulated melt volumes and the conclusion about Moon-sized planets change.","section":"Appendix A and Fig. 10"}],"minor_comments":[{"comment":"The text near the discussion of shock melting says 'shock melting, which is the dominant melting mechanism at vi≥15 km/s'; the symbol 'vi' should be 'v_imp' for consistency with the rest of the paper.","section":"Section 3.2.1"},{"comment":"The description of the melt-domain truncation is ambiguous: 'truncated the entire melt domain at a fixed radius of 5 impactor diameters L around the depth of one impactor diameter' could be read as centered at depth z = L or as a cylindrical/spherical cut; please state the exact centroid and geometry used.","section":"Section 2.3"},{"comment":"When the thermal profile does not intersect the solidus, d_T is set equal to d_L, which gives d'_T = 1, but the empirical law is restricted to d'_T > 1. Please clarify how users should handle planets with no supersolidus depth, and whether the restriction is physical or a fit-range limitation.","section":"Section 4.1"},{"comment":"The fit parameters d'_0, A, and c are shown with standard deviations in Fig. 14, but the standard deviations are not propagated into the final scaling law; please state that the plotted lines are the mean regression and either include confidence bands or explicitly note their absence.","section":"Figure 14 and Eqs. (19)–(21)"},{"comment":"The cubic polynomial approximations to the solidus and liquidus are used only in the thermal evolution models, while the impact models use Eq. (7); this is stated but easy to miss. Please add a sentence in the main text (Section 2.1) reiterating that the impact simulations use the Simon–Glatzel forms, not the cubics.","section":"Appendix B.1, Eqs. (24)–(25)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is thorough and the qualitative findings are likely robust, but the quantitative calibration rests on an unquantified latent-heat bias and a fit to the same data it claims to predict. I recommend requiring either a latent-heat-corrected validation subset or a reframing of the empirical law as an interpolation with explicit systematic uncertainties. The paper already acknowledges most limitations; the revisions are within scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this is the first systematic sweep of impact melt efficiency across planet size, core size, and thermal age, and it delivers a genuinely usable empirical formula (Eqs. 16-21) for estimating melt volumes on stagnant-lid planets. The qualitative trends are probably right: peak melting efficiency near d_m/dL of 1-3, Earth-size planets most efficient per impactor volume, larger planets preferring small impactors and vice versa, core size mostly irrelevant except for old large-core cases. The inclusion of decompression and plastic-work melting is a real step beyond the shock-only scaling laws, and the Moon-size-planets-most-melt conclusion is interesting and well argued.\n\nWhat earns credit: they ran more than 200 simulations, published replication data (Manske et al. 2024), discuss their parameter choices openly, and explicitly flag the limitations I would otherwise have raised—latent heat neglect, vertical-only impacts, single impact velocity, no antipodal melt. The comparison against Pierazzo and Abramov scaling laws is honest and shows where classical laws break down.\n\nSoft spots, in proportion. First, the central practical product is the empirical law, but that law is a fit to the same simulation suite it claims to predict. The match in Fig. 9 is therefore partly built in. That is not fatal—it is still a useful interpolation tool—but readers should not treat it as an independently validated predictive law. Second, the latent-heat issue is real: the melt reconstruction omits enthalpy of fusion and relies on three ad hoc truncations (5L cutoff, 25 K below solidus, 2% melt fraction discard) to compensate \"to some extent.\" No energy-conserving check is offered, so the fitted amplitude A in Eq. 20 inherits any residual bias. If the compensation is off by 20-30%, absolute melt volumes shift by the same order, though the d_m/dL peak structure would survive. Third, the velocity scaling slope b = 0.986 is calibrated on only three velocities (10, 15, 20 km/s) from their own earlier Mars study. That is fine for a first-order correction but thin as a general law. Fourth, uncertainties from the fits are not propagated into the flux-integrated melt volumes in Fig. 10. Minor by comparison: the 0.1 R_E planets do not convect, which is a different regime, and the whole study assumes single vertical impacts with no cumulative effects.\n\nThe core qualitative findings and the parameterization itself are solid enough to deserve referee time and serious use. The absolute calibration deserves a caveat in any application, and I would like to see a follow-up with a latent-heat-corrected benchmark. For now: cite it, test it, and treat the absolute numbers as provisional.\n\nRecommendation: send it to peer review—it is a substantial, well-executed parameter study that the community will use, conditional on the authors adding a clear caveat about the latent-heat compensation and the fit-to-same-data nature of the law.","headline":"A systematic, internally consistent simulation study that gives planetary scientists a practical empirical scaling law for impact melt volume, with the main caveat that the law's absolute calibration leans on unvalidated latent-heat compensation.","tokens_in":32550,"tokens_out":737,"would_cite":true,"duration_ms":10062,"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":"Melt efficiency peaks when melting depth equals lithosphere thickness.","keywords":["impact melting","melting efficiency","thermal evolution","lithosphere thickness","decompression melting","plastic work melting","empirical scaling law","terrestrial planets"],"falsifier":"Run the same impact scenarios with a thermodynamic model that includes the energy absorbed by melting explicitly and no ad hoc cutoffs; if the resulting melting efficiencies differ from the compensated values by more than the scatter around Eqs. 16–21, the empirical law's amplitudes need recalibration.","tokens_in":31499,"feed_emoji":"💥","tokens_out":8696,"duration_ms":72520,"temperature":0.7,"pith_summary":"The paper tries to establish a rule for how much melt a large planetary impact produces, given only the target planet's size, core fraction, and thermal age. By combining thermal-evolution models with more than 200 two-dimensional impact simulations, it finds that melting efficiency — melt volume divided by impactor volume — is controlled mainly by the ratio of the impact's melting depth to the planet's lithosphere thickness, with a peak when those two lengths are comparable. It packages this into an empirical law (Eqs. 16–21) that predicts melting efficiency for planets whose interior reaches the solidus. If the law holds, melt volumes for basin-forming impacts on stagnant-lid planets—rocky planets with one rigid surface shell—can be estimated from radial structure and thermal age without running full impact simulations.","feed_headline":"Melt efficiency peaks when melting depth equals lithosphere thickness","feed_subtitle":"New empirical law estimates impact melt from planet size, core ratio, and thermal age, no hydrocode runs needed.","key_machinery":"The carrying machinery is the pair of normalized length scales $d'_m = d_m/d_L$ and $d'_T = d_T/d_L$, where $d_m$ is the depth below the impact point where shock melt is most abundant (set by impactor size), $d_L$ is the lithosphere thickness, and $d_T$ is the center of the supersolidus depth range — the depth interval where the geotherm is at or above the solidus. The empirical law combines a hyperbolic-tangent baseline that interpolates between a cold crustal melting efficiency $\\pi^c_m \\approx 10.1$ and a cold mantle melting efficiency $\\pi^m_m \\approx 5.81$ with a Gaussian peak term; the peak position $d'_0$, amplitude $A$, and width $c$ are linear functions of $d'_T$ (Eqs. 19–21). These two ratios collapse melting-efficiency data from planets with radii 0.1–1.5 $R_E$, core size ratios 0.2–0.8, and ages 1–4.5 Gyr onto a single family of curves.","core_discovery":"On the paper's own terms, the discovery is that the normalized melt production (melting efficiency $\\pi_m = V_\\mathrm{melt}/V_\\mathrm{impactor}$) for impacts on stagnant-lid terrestrial planets is governed by the ratio $d_m/d_L$ of the depth of melting to the lithosphere thickness, not by impactor size alone. Melting efficiency peaks when $d_m$ is roughly equal to $d_L$ or up to about three times larger, and the peak amplitude is set by the ratio $d_T/d_L$ of the supersolidus depth to lithosphere thickness. This reproduces the otherwise counterintuitive observation that larger planets are melted most efficiently by smaller impactors, while smaller planets are melted most efficiently by larger impactors, and explains why melting efficiency maxima are typically highest on Earth-size planets. The paper further claims that shock melting dominates but decompression and plastic-work melting contribute significantly (up to about 50% and 20–35%, respectively), and that classical scaling laws that account only for shock melting underestimate melt production once the impactor's length scale approaches the target's thermal structure. The empirical law of Eqs. 16–21 summarizes the result for planets with $d'_T > 1$.","pith_inferences":["Testable extension: if Eqs. 16–21 survive testing with codes that include latent heat explicitly, they offer a fast way to estimate impact melt production for stagnant-lid exoplanets from interior-structure retrieval, and to bracket magma-ocean generation during accretion.","The ratio framework suggests that the same $d'_m$ curve might hold for targets with thin or mobile lithospheres, such as early Mars or resurfaced Venus-like planets, if the mechanical boundary-layer thickness is substituted for $d_L$; this is not tested in the paper.","Because the empirical law was fit only for $d'_T > 1$, applying it to cold, conductive bodies like the smallest modeled planets is an extrapolation, and extending the fit to $d'_T \\le 1$ would require additional simulations.","The flux projection probably underestimates early melt production because thermal profiles younger than 1 Gyr were not modeled; including post-accretion profiles could raise the cumulative melt estimate for Moon-sized planets."],"forward_implications":["For basin-forming impacts on stagnant-lid planets, melt volume can be estimated from a planet's radial thermal structure and thermal age alone, using Eqs. 16–21, without a new full impact simulation.","Each planet size has a specific impactor-size window, corresponding to a melting depth of roughly 1–3 lithosphere thicknesses, in which craters are especially prone to melt overflow because melting efficiency is maximal there.","Older, cooler planets produce substantially less impact melt: about 40% less for large planets and 60% less for small planets between 1 and 4.5 Gyr after formation.","Large cores reduce melting efficiency only on old planets, through more efficient cooling of the mantle, rather than through any direct mechanical effect of the core itself.","Under a lunar-like impactor flux, Moon-sized planets accumulate the most melt relative to their volume over 4.5 Gyr, despite also cooling efficiently."],"supporting_citations":[{"why":"It developed the modified peak-shock-pressure tracer method with decompression melting on Mars that this study generalizes to a grid of generic planets.","marker":"Manske et al. (2021)"},{"why":"It established the homogeneous-target melt scaling law and the velocity scaling relation that the paper compares against and rescales.","marker":"Pierazzo et al. (1997)"},{"why":"It contributed the plastic-work melting quantification and the tracer extension that yields the decompression and plastic-work contributions used here.","marker":"Manske et al. (2022)"},{"why":"It provides the differential melt scaling law for basalt used in the accuracy comparison with the new data.","marker":"Abramov et al. (2012)"},{"why":"It is the earlier layered-target melt model that showed a similar deviation from power-law scaling, with which the paper's peak overlaps.","marker":"Marchi et al. (2014)"},{"why":"It supplies the lunar production function adopted as the impactor flux for projecting cumulative melt production onto the generic planets.","marker":"Neukum et al. (2001)"},{"why":"It provides the crater-scaling relations used to convert the lunar crater production function into impactor diameters for the flux calculation.","marker":"Holsapple and Housen (2007)"}],"fun_headline_variants":["Impact melt peaks when melt depth matches lithosphere thickness","Melting efficiency determined by melt depth vs. lithosphere ratio","Why small impactors melt big planets more efficiently","Impact melting scales with ratio of melt depth to lithosphere","Earth-size planets show peak melting efficiency for impacts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results depend on the assumption that discarding barely melted material at the edges of the melt zone roughly cancels the overestimate caused by ignoring the energy that melting itself absorbs; if that cancellation is wrong, every reported melting efficiency and every fitted peak amplitude is off in absolute terms.","fun_headline_variants_meta":{"raw":{"variants":["Impact melt peaks when melt depth matches lithosphere thickness","Melting efficiency determined by melt depth vs. lithosphere ratio","Why small impactors melt big planets more efficiently","Impact melting scales with ratio of melt depth to lithosphere","Earth-size planets show peak melting efficiency for impacts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000406,"raw_usage":{"total_tokens":2190,"prompt_tokens":1104,"completion_tokens":1086,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":720,"completion_tokens_details":{"reasoning_tokens":1009}},"tokens_in":720,"tokens_out":1086,"duration_ms":7497,"temperature":1.0,"reasoning_tokens":1009,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:45:08.442476+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same impact scenarios with a thermodynamic model that includes the energy absorbed by melting explicitly and no ad hoc cutoffs; if the resulting melting efficiencies differ from the compensated values by more than the scatter around Eqs. 16–21, the empirical law's amplitudes need recalibration.","supporting_citations":[{"cited_title":", author Wong, S.M","cited_arxiv_id":null,"evidence_quote":"It provides the differential melt scaling law for basalt used in the accuracy comparison with the new data."},{"cited_title":", author Bottke, W.F","cited_arxiv_id":null,"evidence_quote":"It is the earlier layered-target melt model that showed a similar deviation from power-law scaling, with which the paper's peak overlaps."},{"cited_title":", author Ivanov, B.A","cited_arxiv_id":null,"evidence_quote":"It supplies the lunar production function adopted as the impactor flux for projecting cumulative melt production onto the generic planets."}],"review_version":2}