{"id":"429e1b11-7a21-40ae-a19e-41ba1cfbab41","arxiv_id":"2505.03604","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Tidal heating in L 98-59 b, c, and d is self-limited by a radiation-tide-rheology feedback that can sustain magma oceans for Gyr while yielding heat fluxes up to two orders of magnitude lower than previous calculations.","lead":"This paper simulates how tidal heating, the periodic squeezing of a planet by its host star, interacts with atmospheric cooling and interior melting for the three rocky planets of L 98-59. It finds a self-limiting feedback that can keep magma oceans alive for billions of years while producing about one hundred times less tidal heat than earlier estimates.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The present-day 'permanent magma ocean' claim for L 98-59 b assumes fixed eccentricity for 4.94 Gyr; energy-balance estimates give an eccentricity-damping timescale ~0.7 Gyr at the equilibrium heat flux, and no pumping mechanism is shown.","rationale":"The reader's weakest assumption — orbital steady state — is also the most load-bearing concern I can identify. It directly controls the strongest headline claim: that L 98-59 b may be molten today. The paper is explicit about the assumption and about not modelling orbital evolution, but the present-day claim nevertheless inherits it. A simple energy-balance estimate using the paper's own equilibrium heat flux shows that the eccentricity energy is drained on a timescale well under the system age, so without an identified pumping mechanism the fixed-e simulations cannot be extrapolated to 5 Gyr. I do not see this as an internal inconsistency in the feedback mechanism itself; the proteus/lovepy coupling is independent and reproducible, and the early-evolution prolongation to ~100-200 Myr is credible. Other concerns, such as the partly circular toy model or sensitivity to the critical melt fraction, are secondary because the full model does not rely on the toy model and the equilibrium melt fractions sit close to, but not precisely at, a fragile point. Atmospheric escape mainly weakens the c/d present-day statements, which the authors already qualify, whereas the b no-atmosphere claim is the one most exposed to the orbital issue. Since the reader's conditional verdict already captures this, I recommend no change to the verdict.","tokens_in":35994,"tokens_out":15368,"duration_ms":162858,"concrete_test":"Re-run the planet b cases (with and without atmosphere) with a coupled orbital-thermal model in which e(t) is evolved using the same lovepy rheology: at each timestep compute the dissipated power and update e^2 via d(e^2)/dt = -2 e^2 / tau, with tau = dE / (4*pi*R_p^2 * F_tide), or use a full constant-phase-lag tidal model calibrated to lovepy's k2/Q. Integrate to 4.94 Gyr with the stellar evolution used in the paper. If global energy balance F_tide = F_net cannot be maintained beyond ~1 Gyr because e has decayed substantially, then conclusion (ii) should be downgraded from a present-day permanent-ocean claim to a claim about early (lesssim 200 Myr) magma ocean prolongation. A minimal check is to integrate the energy-balance equation for e(t) starting at e = 0.167 and confirm whether F_tide stays above the required flux after 5 Gyr.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Conclusion (ii) states L 98-59 b may have a permanent magma ocean to this day, with or without an atmosphere. This extrapolation from the 200 Myr simulations in Sections 3.4-3.5 rests on the assumption of orbital steady state, stated in Section 2.3: 'We model the evolution of these planets under the assumption of orbital steady state, for simplicity.' In every simulation the eccentricity is held fixed; the Section 3.5 sensitivity study only varies fixed e values. The assumption is load-bearing because tidal dissipation in a synchronously rotating planet is powered by the orbital energy stored in eccentricity. Using the paper's own median equilibrium flux for planet b, F_tide = 52 W m-2 (Fig. 5), the dissipated power is P = F_tide * 4*pi*R_p^2 ~ 2e16 W. The energy available from circularizing e = 0.167 at roughly constant angular momentum is dE ~ (G*M_star*M_p / 2a) * e^2 ~ 4e32 J, so the e-folding timescale for e^2 is tau ~ dE / P ~ 0.7 Gyr, far shorter than the 4.94 Gyr system age. After 5 Gyr, the planet would have to dissipate several times the available eccentricity energy to remain at the computed equilibrium; no resonance or excitation mechanism is identified, and the period ratios are not shown to be in a sustained commensurability. The no-atmosphere case is even more restrictive: F_tide ~ 1.08e5 W m-2 (Section 3.4) gives tau ~ 1e5 yr. Thus, without an external eccentricity pump, tidal heating decays and the molten state cannot persist to the present day, regardless of atmospheric blanketing. The feedback mechanism for the first ~10-200 Myr is credible, but the strongest present-day claim is unsupported unless orbital steady state is justified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper simulates the coupled thermal, atmospheric, and tidal evolution of the three rocky planets in L 98-59 using the proteus/agni/spider framework extended with the lovepy tidal dissipation model. It proposes a 'radiation-tide-rheology feedback': as a magma ocean cools, mantle viscosity increases, tidal heating peaks, and the resulting internal flux balances atmospheric radiative cooling, stabilizing whole-mantle melt fractions just above the assumed critical value (30%) rather than allowing complete solidification. In no-tide control simulations all three planets solidify within 100 Myr, whereas most tide-inclusive simulations reach global energy balance with melt fractions around 36-39% and tidal fluxes of 52-113 W/m2, one to two orders of magnitude below previous estimates. A no-atmosphere simulation of L 98-59 b also remains molten (F_tide about 1e5 W/m2), whereas c and d solidify without atmospheres. Sensitivity runs vary eccentricity, oxygen fugacity, and core radius, and the paper also estimates XUV-driven escape and tests convective stability. The conclusion states that L 98-59 b may have a permanent magma ocean to the present day.","tokens_in":36450,"tokens_out":9156,"duration_ms":93069,"significance":"If correct, the mechanism is significant: it provides a self-limiting equilibrium for tidal heating that changes predicted heat fluxes and surface temperatures for close-in rocky exoplanets by orders of magnitude, and it gives a concrete pathway to Gyr-lived magma oceans and tidally supported volcanic outgassing. The predictions of median equilibrium fluxes (52-113 W/m2) and melt fractions, together with the distinct no-atmosphere outcomes for b versus c/d, are falsifiable with secondary-eclipse and transmission observations. The paper is unusually transparent: the model codes (proteus, agni, lovepy) are open or archived, data are deposited on Zenodo, and the authors explicitly list the orbital steady-state assumption and the neglect of coupled escape as limitations. These strengths make the core early-evolution result, namely that tides can hold the mantle near the critical melt fraction for at least 200 Myr, credible. The present-day extrapolation, however, is not yet supported by the calculations as presented.","major_comments":[{"comment":"The 'permanent magma ocean to this day' claim rests on an orbital steady-state assumption that is not demonstrated. In every simulation the eccentricity is held fixed, and Section 3.5 only varies fixed values, so the simulations consume orbital energy without allowing the orbit to respond. Using the paper's own median equilibrium flux for planet b (F_tide = 52 W m-2, Fig. 5), the dissipated power is about 2e16 W; for e = 0.167 at a = 0.02191 AU around a 0.273 Msun star with Mp = 0.47 Mearth, the orbital energy available from circularization is about 4e32 J, giving an e-folding time for e^2 of order 0.7 Gyr. The no-atmosphere case (F_tide = 1.08e5 W m-2, Section 3.4) gives a timescale of order 1e5 yr. Since the system age is 4.94 Gyr and no eccentricity pump is identified, the period ratios c/b ~ 1.63 and d/c ~ 2.02 do not by themselves demonstrate a sustained commensurability. The authors should either couple tidal orbital evolution to the thermal model, identify and model a pumping mechanism, or restrict the claim to 'at least 200 Myr while eccentricity is maintained.'","section":"Section 2.3 and Conclusion (ii)"},{"comment":"The full simulations terminate at global energy balance (no later than 10 Myr in Fig. 5) or at 200 Myr (Section 3.5), while the conclusions invoke Gyr timescales. The stellar bolometric flux decreases by about 20% between 100 Myr and the present (Section 4.1), which moves the equilibrium; the blue-sky bifurcation discussion in Appendix B is based on the toy model (Eq. A6), not on the full proteus model. The paper should either integrate the full model past 200 Myr with evolving stellar luminosity and, ideally, parameterized escape, or explicitly state that the Gyr-timescale extension is an inference from the toy model rather than a result of the coupled simulations.","section":"Sections 3.4, 3.5, and 4.1"},{"comment":"The critical melt fraction Phi_c is fixed at 30%, and the equilibrium melt fractions cluster in the narrow band 36.5-39.2%, only 6-9 percentage points above Phi_c. Because the feedback operates by pinning the melt fraction near Phi_c, the quantitative results, especially the median heat fluxes 52-113 W/m2, are potentially sensitive to this parameter. No sensitivity calculation is reported. A series of simulations varying Phi_c over a plausible range (e.g., 20-40%) should be added to demonstrate that the self-limited fluxes and the prolonged magma ocean conclusion are robust to this choice.","section":"Section 2.1 and Fig. 5"},{"comment":"The toy model's 'demonstration' of the feedback is partly assumed in its construction. Equation (A5) posits a Gaussian tidal heating peak centered between the solidus and liquidus, so the qualitative behavior, with heating weak in the fully molten and fully solid states and strong near the critical melt fraction, is built into the model rather than derived from it. The full proteus/lovepy model, which uses a literature-based Maxwell rheology, does reproduce similar behavior, so this does not invalidate the central mechanism; however, the manuscript should state clearly that the toy model is an illustration and that the physical evidence for the feedback comes primarily from Sections 3.4-3.5.","section":"Appendix A and Section 3.2"}],"minor_comments":[{"comment":"The word 'recieved' should be 'received'.","section":"Appendix B"},{"comment":"The text says the main simulations reach global energy balance generally within 10 Myr, while the sensitivity runs are evolved past equilibrium to up to 200 Myr; please clarify which simulations are shown in Fig. 5 versus Fig. 6, since the termination criteria differ.","section":"Sections 3.4 and 3.5, Figs. 5 and 6"},{"comment":"Equation (C1) is typeset in a way that is difficult to parse, with the factor involving R_optical/R_xuv appearing before the fraction; please check the equation formatting to ensure it matches the Watson et al. (1981) and Lehmer & Catling (2017) form.","section":"Appendix C, Eq. (C1)"},{"comment":"The colorbar label reads 'Time [Myr]' but the text describes solidification times; please make the caption explicit that the colour indicates the time at which the mantle solidifies.","section":"Fig. 3"},{"comment":"The table caption states that the equilibrium temperatures are calculated with a Bond albedo of 30%, but Section 2.2 does not specify how the albedo enters the radiative-convective model; a short sentence reconciling the table's equilibrium temperatures with the instellation values would help.","section":"Table 1 and Section 2.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is well-scoped and the central feedback mechanism appears physically sound and valuable. The main gap is the mismatch between the strongest conclusion (present-day permanent magma ocean) and the model's orbital steady-state assumption and 200 Myr horizon. I would not recommend rejection: the early-evolution result and the order-of-magnitude reduction in tidal fluxes are solid contributions, and the missing analysis of eccentricity pumping and longer-term evolution can be addressed in revision, either by adding the relevant calculations or by softening the present-day claims. No concerns about novelty or citation practice; the paper correctly distinguishes this feedback from earlier Io-type tidal heating feedbacks."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"At bottom, this paper is about a negative feedback between tidal heating, mantle rheology, and atmospheric radiative transport that self-limits tidal heat fluxes to tens of W/m2 instead of tens of kW/m2. That mechanism is new and, as far as I can see, right. The coupled proteus+lovepy modeling is the real thing: a full radiative-convective atmosphere, redox-dependent outgassing, and a viscoelastic tidal model, all time-stepped together. The behavior they find—equilibrium melt fractions parked just above the critical 30% melt fraction, heat fluxes 50–110 W/m2—is a genuinely different result from previous fixed-state estimates (Quick, Seligman), and it's a consequence of the coupling, not an input.\n\nThe load-bearing weakness is the assumption of orbital steady state. They fix eccentricity at present-day observed values and let the planet sit there for 4.9 Gyr. But tidal dissipation consumes orbital energy; at their own equilibrium flux for b, the eccentricity damping timescale is around 0.7 Gyr. Without a demonstrated pumping mechanism (resonance or ongoing perturbation), the present-day molten b claim doesn't follow. The no-atmosphere variant is even worse—at F=1e5 W/m2, the orbit dies in 1e5 yr. The paper admits the assumption and says 'for simplicity,' and the discussion calls for future coupled orbital evolution, but the abstract and conclusions still assert 'may have permanent magma ocean to this day.' That's an overstatement relative to the evidence.\n\nMinor issues: the toy model in Appendix A is partly circular, since it bakes in a Gaussian heating peak between solidus and liquidus, but it's presented as a heuristic and the full model does the real work. The critical melt fraction isn't varied; that's a minor sensitivity issue.\n\nWho gets value? Anyone working on magma ocean evolution, tidal heating of rocky exoplanets, or L 98-59 observations. The feedback mechanism deserves attention even if the present-day claim needs a caveat. This is a good paper for a serious referee: the mechanism is important, the modeling is substantial, and the present-day claim is fixable with either a softer statement or a coupled orbital calculation.\n\nSend to peer review. I'd ask the authors to either demonstrate a plausible eccentricity-sustaining mechanism for 5 Gyr, or soften conclusion (ii) to 'could have if eccentricity is maintained' and clearly state the orbital steady-state assumption as a caveat.","headline":"Solid, novel feedback mechanism, but the present-day magma ocean claim for L 98-59 b rests on an orbit that would have circularized a billion years ago.","tokens_in":36995,"tokens_out":3180,"would_cite":true,"duration_ms":31672,"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":"Tidal heating can keep L 98-59 b molten today","keywords":["tidal heating","magma ocean","radiation-tide-rheology feedback","L 98-59","rocky exoplanet evolution","mantle rheology","atmospheric blanketing","secondary atmosphere"],"falsifier":"A secondary-eclipse observation of L 98-59 b measuring a dayside brightness temperature below the silicate solidus (~1400 K) would falsify the claim that tidal heating keeps it molten today. Alternatively, a precise eccentricity determination showing e < 0.001 would remove the tidal heat source.","tokens_in":35847,"feed_emoji":"🌋","tokens_out":7518,"duration_ms":66918,"temperature":0.7,"pith_summary":"This paper argues that tidal heating inside close-in rocky planets is self-limited, not runaway: as a hot planet cools, its mantle becomes viscous enough that tidal dissipation switches on, and the resulting heating slows or stops the cooling. The authors term this the 'radiation-tide-rheology feedback,' and they show it sets equilibrium tidal heat fluxes far below earlier estimates. Simulating the early evolution of the L 98-59 planets b, c, and d, they find that planet b likely still hosts a molten interior today, with or without an atmosphere, while c and d need atmospheric blanketing to remain molten. This matters because lasting magma oceans extend outgassing and change what atmospheres we expect to see on such planets.","feed_headline":"Tides can keep L 98-59 b molten today","feed_subtitle":"New feedback between tides, mantle rheology, and atmosphere yields lower heat fluxes yet longer-lived magma oceans.","key_machinery":"The radiation-tide-rheology feedback: a coupled negative feedback loop in which radiative cooling to space, tidal heat dissipation, and temperature-dependent mantle rheology regulate one another. The load-bearing identity is global energy balance: at equilibrium, tidal heat flux through the interior equals the net energy flux the atmosphere transports to space, $F_{\\mathrm{tide}} = F_{\\mathrm{net}}$. This equality, together with the critical melt fraction $\\Phi_c = 30\\%$ where the mantle transitions from liquid-like to solid-like behaviour, fixes the equilibrium state far from the runaway-heating regime assumed by earlier work.","core_discovery":"The central discovery is a negative feedback that can hold a rocky planet in a partially molten state indefinitely. When a fully molten planet cools, its melt fraction falls toward the critical value at which the mantle behaves as a solid and tidal dissipation becomes strong; the resulting internal heating raises the temperature back up, so the planet settles near that critical melt fraction in global energy balance, $F_{\\mathrm{tide}} = F_{\\mathrm{net}}$. Applied to L 98-59 b, c, and d, the feedback produces equilibrium states with median melt fractions 36.5–39.2% and tidal heat fluxes 52–113 W m$^{-2}$, two orders of magnitude lower than prior estimates that ignored atmospheric coupling. The authors conclude that L 98-59 b may have a permanent magma ocean to this day, whether or not it retains an atmosphere, and that the same mechanism can prolong magma oceans on c and d as long as they have atmospheres.","pith_inferences":["If the same feedback operates on Io, the Juno result that Io lacks a magma ocean can be read as a cold-start outcome rather than a failure of tidal heating to melt it.","The paper's results imply that atmospheric composition indirectly controls interior heat output: a stronger greenhouse atmosphere shifts the equilibrium to higher tidal flux, so atmospheric loss could double as a switch that shuts off tidal melting.","A testable extension: for any eccentric close-in rocky planet, the equilibrium melt fraction should hover near 30%, implying a narrow range of tidal quality factors that could be probed with future astrometric detection of tidal deformation."],"forward_implications":["Equilibrium tidal heat fluxes in close-in rocky exoplanets are up to two orders of magnitude lower than previously estimated, because earlier work did not couple atmospheric energy transport to interior thermal evolution.","Magma oceans on eccentric, close-in rocky planets can persist for billions of years, not just the <100 Myr found without tides, provided the orbit does not circularize.","L 98-59 b is likely molten today; secondary-eclipse thermal observations can test this directly.","The feedback creates stable equilibria that can be destroyed by a 'blue sky' bifurcation as the host star dims, causing rapid solidification and potentially catastrophic outgassing.","The framework offers a way to probe a planet's hot- vs cold-start formation history from its present-day thermal state."],"supporting_citations":[{"why":"Supplies the viscoelastic tidal heating model that computes dissipation in a radially inhomogeneous Maxwell mantle.","marker":"Hay & Matsuyama 2019b"},{"why":"Provides the coupled interior-atmosphere modelling framework used to simulate planetary thermal evolution.","marker":"Lichtenberg et al. 2021"},{"why":"Establishes the redox-controlled outgassing and energy-balance treatment that the present work extends with tidal heating.","marker":"Nicholls et al. 2024"},{"why":"Gives the previous tidal heating estimates for L 98-59 b/c/d that this work revises downward.","marker":"Quick et al. 2020"},{"why":"Provides prior high tidal heat flux estimates (assuming a decoupled lithosphere) that the equilibrium states here contradict.","marker":"Seligman et al. 2024"},{"why":"Determines the critical melt fraction (~30%) where the mantle's rheology transitions, setting the equilibrium target of the feedback.","marker":"Costa et al. 2009"},{"why":"Presents JWST evidence for an SO2-rich atmosphere on L 98-59 b that motivates the present-day molten-state scenario.","marker":"Bello-Arufe et al. 2025"},{"why":"Supplies the masses, eccentricities, and orbital parameters of the L 98-59 planets used as model inputs.","marker":"Demangeon et al. 2021"}],"fun_headline_variants":["Self-limiting tidal feedback prolongs magma oceans on L 98-59","Tides keep L 98-59 b molten today via new negative feedback","Radiation-tide-rheology feedback explains L 98-59's molten state","Lower tidal heating yet longer-lived magma oceans on L 98-59"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The model assumes the planets' orbits stay at their observed eccentricities forever (orbital steady state); if nothing keeps the eccentricities pumped, tidal heating would decay as the orbits circularize, and the predicted present-day molten state for L 98-59 b would not occur.","fun_headline_variants_meta":{"raw":{"variants":["Self-limiting tidal feedback prolongs magma oceans on L 98-59","Tides keep L 98-59 b molten today via new negative feedback","Radiation-tide-rheology feedback explains L 98-59's molten state","Lower tidal heating yet longer-lived magma oceans on L 98-59"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00027,"raw_usage":{"total_tokens":1658,"prompt_tokens":1011,"completion_tokens":647,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":564}},"tokens_in":627,"tokens_out":647,"duration_ms":6207,"temperature":1.0,"reasoning_tokens":564,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:47:56.296433+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A secondary-eclipse observation of L 98-59 b measuring a dayside brightness temperature below the silicate solidus (~1400 K) would falsify the claim that tidal heating keeps it molten today. Alternatively, a precise eccentricity determination showing e < 0.001 would remove the tidal heat source.","supporting_citations":[],"review_version":1}