{"id":"a902e079-0a9f-42bf-acfe-c9bcf56f5aa8","arxiv_id":"2506.23331","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a correct star-centered simulation, every indirect term must be paired with the corresponding direct gravitational force, and the force on a migrating planet should use only the non-axisymmetric part of the disk density.","lead":"This paper shows that in star-centered simulations of planets and gas disks, there are several different correction forces, not one, and each should be switched on only when the matching direct gravitational pull is switched on. It gives a concrete recipe for the force on a migrating planet and warns that mixing the two up creates fake captures and fake vortex migrations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The balanced-inclusion rule is a modeling judgment rather than a consequence of Eq. (1), and the paper's own claim that ITdd drives a physical m=1 instability even without disc self-gravity undercuts the advice to drop ITdd in non-self-gravitating simulations.","rationale":"The reader's weakest assumption is that the balanced-inclusion rule is a modeling judgment rather than a theorem, with the decisive m=1 instability deferred to Paper II. My stress-test agrees but sharpens the point: the paper's own statements create an internal tension. §3.4.3 calls the ITdd-driven m=1 mode a physical instability that appears even when self-gravity is off and even in an inertial centre-of-mass frame; the Introduction repeats that it 'sets in whether or not disc self-gravity is included'. If true, ITdd is not merely the counterpart of disc self-gravity—it is a real physical effect of the star's reflex motion, and omitting it in non-self-gravitating simulations would remove real physics. The balanced rule is therefore not a logical consequence of the frame transformation, and the paper's central recommendation rests on a choice that is neither derived nor analytically justified against the competing reading that the full fictitious acceleration must apply to all bodies. The paper does provide useful derivations—Eq. (3), Appendices A and B, the m=1-only resonance argument—and its two-planet comparison in Fig. 4 is coherent. But the central normative claim is not settled. A decisive test is to compare the inertial-frame, non-self-gravitating evolution with the stellocentric ITdd-on case; that directly determines whether ITdd is a physical star-reflex term or a spurious artifact when self-gravity is absent. Because this concern is exactly the kind of support that should be supplied before the strongest claims are relied upon, the reader's CONDITIONAL verdict remains appropriate; no change in verdict is needed, but the paper should address the tension explicitly and present the Paper II evidence.","tokens_in":15845,"tokens_out":15043,"duration_ms":163852,"concrete_test":"Obtain the Paper II inertial-frame simulations (frame centred on the system's centre of mass, so no indirect term is imposed) for a disc with self-gravity switched off, and measure the growth rate of the m=1 eccentric mode. Compare with a stellocentric run that includes ITdd and switches off self-gravity. If the inertial-frame, non-self-gravitating run shows the same m=1 growth, then ITdd is a physical star-reflex effect that must be retained even when self-gravity is neglected, contradicting the §3.4.1 recommendation to omit ITdd. If the inertial-frame run shows no m=1 growth, the balanced rule is supported. This isolates the star's reflex motion from the choice of indirect-term implementation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the balanced-inclusion rule of §2.3, which licenses the central recipe Eq. (8) and the §3.4.1 advice to omit ITdd when disc self-gravity is discarded. The rule is not derived from Eq. (1). In the exact transformation to a star-centred frame, the fictitious acceleration is a single vector −a* = −Σ_i GM_i r_i/r_i^3, common to every body in the frame; no equation in the paper permits different bodies to receive different subsets of this sum. The 'pulled away from C as a whole' argument in §2.3 is a heuristic, not a proof. More importantly, the paper itself states in §3.4.3 and the Introduction that ITdd drives an m=1 disc eccentricity instability of 'physical origin', that it 'sets in whether or not disc self-gravity is included', and that it is reproduced in an inertial centre-of-mass frame. If that claim is correct, ITdd encodes the real reflex motion of the star due to the disc's mass, a star-disc two-body effect independent of disc self-gravity. Recommending that ITdd be omitted in non-self-gravitating runs would then suppress a physical instability. The paper does not resolve this tension; the decisive evidence is deferred to the unpublished companion paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper argues that, in a frame centred on a dominant primary, the indirect (fictitious) acceleration is not a single term but a sum over all gravitating constituents, each with its own 'indirect term'. It introduces a balanced-inclusion rule: a body should feel the indirect term from a given constituent if and only if it feels that constituent's direct gravitational acceleration. The rule is then applied to a star-disc-planets system: ITpd is required to recover the L4/L5 Lagrange points; ITpp between planets can lock them into a 2:1 resonance even when direct planet-planet gravity is switched off; ITdp contributes a torque that should be included for migrating planets, via Eq. (8) using only the non-axisymmetric surface density; and ITdd should be omitted in non-self-gravitating disc simulations. The paper also claims that ITdd drives vortex migration and an m=1 eccentric instability, the latter analysed in a companion paper.","tokens_in":15999,"tokens_out":10205,"duration_ms":107743,"significance":"The paper fills a real gap: the treatment of indirect terms in grid-based disc-planet simulations is often undocumented, and the proposed Table 2 plus Eq. (8) give practitioners a concrete, testable recipe. The algebraic core is sound: Eq. (3) follows from the direct plus fully specified indirect accelerations, Appendix A recovers the reduced-mass correction Ω²=G(M*+Mp)/r³, and Appendix B correctly shows that the m=1-only character of the indirect term can drive the 2:1 outer Lindblad resonance but not other first-order resonances. The FARGO simulations in §3.3 and §3.4.1 provide useful quantitative estimates of the indirect torque (e.g., Γ_ind ≈ 0.08 Γ_dir in the Type I run and ≈ 0.16 Γ_dir in the Type II run). However, the normative weight of the recipe currently rests on a modeling convention that is not derived, and on the unresolved status of the ITdd-driven instability, so the paper is not yet self-contained as a prescription.","major_comments":[{"comment":"The balanced-inclusion rule is not a consequence of the decomposition in Eq. (1), which gives a single fictitious acceleration common to every body in the star-centred frame. The argument that a body would be 'pulled away from C as a whole' is a heuristic about the intended model hierarchy, not a proof that different bodies should receive different subsets of the indirect sum. Because Eq. (8) and Table 2 rely on this rule, the manuscript should either derive the rule from an explicitly stated model hierarchy or present it as a convention, and discuss the alternative reading in which ITdd is mandatory in any star-centred frame.","section":"§2.3, Eq. (1)"},{"comment":"The paper states that the ITdd-driven m=1 instability 'sets in whether or not disc self-gravity is included' and is 'of physical origin', and that it is reproduced in an inertial centre-of-mass frame, yet it recommends omitting ITdd in non-self-gravitating simulations to suppress 'spurious' torques and vortex migration. If the instability is physical, omitting ITdd removes a real star-disc two-body effect; if it is instead an artifact of the truncated model, that should be stated explicitly. The decisive evidence is deferred to the unpublished Paper II, so the central recommendation is not self-contained as written.","section":"§1 and §3.4.3, Table 2"},{"comment":"The recipe replaces the full disc surface density with Σ' in the direct acceleration term, which is an additional truncation beyond the balanced-inclusion rule: the axisymmetric component of the disc does exert a direct acceleration on the planet, even though it produces no torque. The paper's motivation for this choice is clear, but its relation to the rule stated in §2.3 should be spelled out, since a reader could otherwise take Eq. (8) to be a direct consequence of the balanced rule rather than a separate modeling decision.","section":"§3.3, Eq. (8)"}],"minor_comments":[{"comment":"The word 'swiched' appears three times and should be 'switched'; also, 'asymetrical' in §3.3 and 'couterbalance' in §3.4.1 are typos.","section":"§3.2"},{"comment":"The sentence containing '|Γind| ∼0.6|Γind|' appears to have a typo; it should presumably read '|Γind| ∼0.6|Γdir|'.","section":"§3.3, after Fig. 7"},{"comment":"The torque-density axis labels are garbled in the typeset version (shown as '□′(r)/□0'); please ensure the correct symbols for torque density and reference torque appear.","section":"Figs. 5 and 7"},{"comment":"The caption does not define what the different columns correspond to; the text says 'see labels below each column of panels', but the labels should be included in the caption or referenced with explicit column identifiers.","section":"Fig. 4"},{"comment":"The symbol '□' is used in the table without an explicit definition in the main text; please define it in the caption or before the table.","section":"Table 2"},{"comment":"The companion paper is cited as 'Crida et al., in revision' (Paper II); if it is not yet accepted, the manuscript should indicate how the m=1 instability results can be independently verified by readers.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper would be a useful methods contribution, but the authors' normative claims ('should be', 'most incorrect choice', 'inconsistent') are stronger than the evidence presented in this manuscript. The dependence on Paper II for the m=1 instability is a particular concern for a standalone publication; the editor may wish to ask whether the companion can be made available or its results summarized. The balanced rule's status as a convention versus a physical statement should be clarified before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nThe useful part of this paper is the decomposition of the indirect term into per-body components and the balanced rule in Table 2: each indirect term is the counterpart of a direct interaction, so include it exactly when the direct term is included, and discard it when the direct term is discarded. The algebraic core checks out. Eq. (3) is correct, Appendix A reproduces the reduced-mass correction, and Appendix B's argument that the m=1-only indirect term can drive only the 2:1 outer Lindblad resonance is sound. The two-planet simulation in Fig. 4 is a clean demonstration that indirect-only coupling can lock planets into 2:1 resonance. The recipe Eq. (8) for migrating planets is concrete and actionable for code developers.\n\nThe soft spots, in order of size. First, the most dramatic claim, the m=1 disc eccentricity instability, is deferred wholesale to the unpublished companion paper. The Introduction states it 'sets in whether or not disc self-gravity is included' and Section 3.4.3 says it is reproduced in an inertial frame. If that is right, ITdd encodes a real star-disc two-body effect, which directly tensions with the advice in Section 3.4.1 to drop ITdd in non-self-gravitating runs. The balanced rule is a modeling judgment, not a theorem; the competing reading, that the indirect term is the real acceleration of the frame and should apply to every body, is not analytically rebutted. The paper must confront this tension rather than leaving it to Paper II. Second, the refutation of Lega et al.'s vortex-driven migration is a single sentence with no figure; that needs support. Third, there is a typo in Section 3.3: '|Γind| ~0.6|Γind|' cannot be right. Minor: no data or parameter files are shipped, and the paper leans heavily on its own prior and forthcoming work for the instability claim.\n\nOverall, the pedagogical decomposition is a genuine service to the subfield, and the recipe is useful. But the central rule is asserted in the face of the paper's own physical-instability claim, and the decisive evidence is in the companion. This deserves peer review, but with the expectation of substantial revision. A serious referee should push on the tension between the balanced rule and the claimed physical nature of ITdd.","headline":"Useful per-body decomposition and a clean 2:1 resonance demo, but the balanced-inclusion rule is a modeling judgment, not a theorem, and the paper's own m=1 instability claim pulls against it.","tokens_in":16735,"tokens_out":2603,"would_cite":true,"duration_ms":25268,"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":"In a star-centred frame there is one indirect term per gravitating body, and each should be included only when the corresponding direct acceleration is included.","keywords":["indirect term","inertial forces","protoplanetary discs","planet migration","hydrodynamical simulation","mean-motion resonance","disc self-gravity","fictitious acceleration"],"falsifier":"Take a single disc-planet setup and run three versions: a stellocentric run without ITdd, a stellocentric run with ITdd, and an inertial centre-of-mass run with full direct self-gravity. The paper's rule predicts that the first and third agree on planet migration and vortex evolution while the second diverges; a run in which the inertial frame reproduces the m=1 instability and rapid vortex migration attributed to ITdd would refute the rule.","tokens_in":15489,"feed_emoji":"🪐","tokens_out":10252,"duration_ms":103773,"temperature":0.7,"pith_summary":"This paper argues that in star-centred simulations of planetary systems there is not a single 'indirect term' but one indirect term for every body that gravitationally pulls on the central star. Each indirect term must be switched on or off together with the corresponding direct gravitational interaction: if you keep the direct pull of the disc on a planet, keep the disc's indirect term on that planet; if you drop the disc's self-gravity, drop the disc's indirect term on the disc itself. Applied to protoplanetary disc simulations, the rule changes what a migrating planet feels: it should feel only the non-axisymmetric part of the disc's surface density, both directly and indirectly. It also means that applying the disc's indirect term to the disc in non-self-gravitating runs produces spurious effects, including artificial vortex migration, and can lock planets into a 2:1 resonance through the indirect term alone. The paper proposes a concrete recipe for the acceleration of a migrating planet and notes that most published migration simulations are inconsistent in which terms they include.","feed_headline":"Pair every indirect force with its direct twin, or drop both","feed_subtitle":"Migrating planets should feel the non-axisymmetric disc, both directly and indirectly; the disc itself should not.","key_machinery":"The central object is the decomposition of the indirect term, $\\mathrm{IT} = -a_* = \\sum_i \\mathrm{IT}_i$, with $\\mathrm{IT}_i = -GM_i \\mathbf{r}_i / |\\mathbf{r}_i|^3$ the opposite of the contribution of body $i$ to the acceleration of the primary. The rule for using it is bookkeeping: include $\\mathrm{IT}_i$ for a body exactly when that body feels the direct acceleration of body $i$, and discard both together otherwise. The argument that the indirect term can only excite the 2:1 outer Lindblad resonance is carried by the Fourier fact that the indirect acceleration of one planet on another has only the $m=1$ azimuthal harmonic, while the direct term contains all harmonics; the Hamiltonian treatment shows the same through the kinetic-energy coupling term $T_1 \\propto M_1 M_2 / M_* \\, \\dot{\\mathbf{r}}_1 \\cdot \\dot{\\mathbf{r}}_2$. The practical output is Eq. (8), the proposed standard acceleration for a migrating planet in a non-self-gravitating disc.","core_discovery":"At the paper's centre is the balanced-inclusion rule: in a frame centred on the primary, every constituent that gravitationally accelerates the primary generates its own indirect term, and that indirect term should be applied only to bodies that also feel the constituent's direct acceleration. Thus in a protoplanetary disc with two planets there are three indirect terms, one per planet and one for the whole disc, and they are not interchangeable. The paper shows three consequences: without the planet's indirect term the restricted three-body problem loses the Lagrange points L4 and L5; when direct planet-planet gravity is switched off but indirect terms are left on, the two planets still capture into a 2:1 resonance because the indirect acceleration has only an m=1 azimuthal component; and for a migrating planet embedded in a non-self-gravitating disc the physically consistent acceleration is Eq. (8), which uses only the non-axisymmetric surface density $\\Sigma'$ in both the direct and indirect disc terms. It further argues that applying the disc's indirect term to the disc itself (ITdd) while omitting disc self-gravity is the least consistent choice, producing artificial vortex migration and, as reported in the companion paper, an m=1 disc eccentricity instability.","pith_inferences":["The balanced-inclusion rule is a modelling convention that the paper defends with examples and simulations, not a consequence of Eq. (1); the competing reading, in which the indirect term is the star's real acceleration owed to every body in the frame, would make ITdd mandatory even without disc self-gravity.","The same pairwise logic should apply to other hierarchical gravitational systems, such as black-hole accretion discs or planet-satellite systems: each massive constituent that directly pulls the primary has its own indirect term, and simulations should report which indirect terms are active.","A decisive numerical experiment would compare stellocentric runs with and without ITdd against an inertial centre-of-mass run with full direct self-gravity, since the inertial run is the physical reference.","Because the indirect term couples through the m=1 azimuthal harmonic, its effects can be isolated by perturbing only that modal component of the disc and checking whether the predicted resonance and instability appear."],"forward_implications":["Non-self-gravitating disc-planet simulations should compute the force on a migrating planet with Eq. (8), using only the non-axisymmetric surface density $\\Sigma'$ in both the direct and indirect accelerations.","Switching off planet-planet direct gravity in a migration experiment requires switching off the corresponding planet-planet indirect term; otherwise the pair can still lock into a 2:1 resonance through the indirect term alone.","Simulations that keep ITdd while omitting disc self-gravity will show artificially strong vortex migration and spin-up, so published vortex-driven migration scenarios built on such runs should be re-examined.","Authors of disc simulations should state explicitly which indirect terms are active (ITpd, ITdp, ITdd), since the choice changes migration rates and disc structure.","If the companion paper's m=1 disc instability is confirmed, long-time runs of massive discs need to control ITdd, because the instability produces oscillating torques on embedded planets before it fully develops."],"supporting_citations":[{"why":"Supplies the two-planet migration simulations whose runs with direct and indirect interactions switched off or on are the basis for the indirect-only 2:1 resonance capture.","marker":"Baruteau & Papaloizou (2013)"},{"why":"Identified the indirect (star) torque as the balancing torque at the vortex's L4 and L5 points, the mechanism the paper generalises.","marker":"Ataiee et al. (2014)"},{"why":"Hydrodynamical vortex simulations comparing self-gravity and indirect-term runs that show ITdd modifies vortex migration.","marker":"Zhu & Baruteau (2016)"},{"why":"Showed that including the axisymmetric disc's direct acceleration on a planet introduces a spurious migration effect cited to justify the Sigma-prime recipe.","marker":"Pierens & Huré (2005)"},{"why":"Demonstrated that a strictly linear disc response requires neglecting ITdd and analysed the disc response to a migrating planet.","marker":"Baruteau & Masset (2008b)"},{"why":"Further documented the spurious influence of the disc's direct acceleration on migration, supporting the Sigma-prime prescription.","marker":"Ataiee & Kley (2020)"},{"why":"The FARGO3D code implements the Sigma-prime option that realises the proposed recipe in practice.","marker":"Benítez-Llambay & Masset (2016)"},{"why":"The FARGO code used for the type I and type II torque simulations in section 3.3.","marker":"Masset (2000)"},{"why":"Provides the Hamiltonian form of the kinetic-energy coupling through the star's reflex motion used to explain which resonances the indirect term can excite.","marker":"Laskar & Robutel (1995)"}],"fun_headline_variants":["One indirect term per pulling body, matched to its direct reach","No single indirect term: pair each direct acceleration with its own","Drop indirect terms selectively and you lose Lagrange points","Balanced indirect forces: each body gets its own, no more no less","Indirect terms are plural: one per perturber, applied to its direct targets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that each indirect term is the counterpart of one specific direct gravitational interaction, so a body that does not feel that direct pull should not feel the indirect term either; if instead the indirect term is viewed as the star's real acceleration that every body must feel, ITdd would be mandatory even without disc self-gravity.","fun_headline_variants_meta":{"raw":{"variants":["One indirect term per pulling body, matched to its direct reach","No single indirect term: pair each direct acceleration with its own","Drop indirect terms selectively and you lose Lagrange points","Balanced indirect forces: each body gets its own, no more no less","Indirect terms are plural: one per perturber, applied to its direct targets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00031,"raw_usage":{"total_tokens":1802,"prompt_tokens":1012,"completion_tokens":790,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":628,"completion_tokens_details":{"reasoning_tokens":699}},"tokens_in":628,"tokens_out":790,"duration_ms":9249,"temperature":1.0,"reasoning_tokens":699,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:48:29.068758+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a single disc-planet setup and run three versions: a stellocentric run without ITdd, a stellocentric run with ITdd, and an inertial centre-of-mass run with full direct self-gravity. The paper's rule predicts that the first and third agree on planet migration and vortex evolution while the second diverges; a run in which the inertial frame reproduces the m=1 instability and rapid vortex migration attributed to ITdd would refute the rule.","supporting_citations":[{"cited_title":"& Papaloizou, J","cited_arxiv_id":null,"evidence_quote":"Supplies the two-planet migration simulations whose runs with direct and indirect interactions switched off or on are the basis for the indirect-only 2:1 resonance capture."},{"cited_title":"P., Kley, W., Reg´ aly, Z., & Meheut, H","cited_arxiv_id":null,"evidence_quote":"Identified the indirect (star) torque as the balancing torque at the vortex's L4 and L5 points, the mechanism the paper generalises."},{"cited_title":"& Baruteau, C","cited_arxiv_id":null,"evidence_quote":"Hydrodynamical vortex simulations comparing self-gravity and indirect-term runs that show ITdd modifies vortex migration."},{"cited_title":"& Hur´ e, J.-M","cited_arxiv_id":null,"evidence_quote":"Showed that including the axisymmetric disc's direct acceleration on a planet introduces a spurious migration effect cited to justify the Sigma-prime recipe."},{"cited_title":"& Kley, W","cited_arxiv_id":null,"evidence_quote":"Further documented the spurious influence of the disc's direct acceleration on migration, supporting the Sigma-prime prescription."},{"cited_title":"& Robutel, P","cited_arxiv_id":null,"evidence_quote":"Provides the Hamiltonian form of the kinetic-energy coupling through the star's reflex motion used to explain which resonances the indirect term can excite."}],"review_version":1}