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On inertial forces (indirect terms) in problems with a central body

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.23331 v1 pith:RKOICRKF submitted 2025-06-29 astro-ph.EP astro-ph.SRphysics.class-ph

Aurélien Crida , Clément Baruteau , Philippine Griveaud , Elena Lega , Frédéric Masset , William Béthune , David Fang , Jean-François Gonzalez
show 9 more authors
Héloïse Méheut Alessandro Morbidelli Fabiola Gerosa Dylan Kloster Léa Marques Alain Miniussi Kate Minker Gabriele Pichierri Paul Segretain
This is my paper · ORCID
classification astro-ph.EPastro-ph.SRphysics.class-ph
keywords indirectterminertialforcesprotoplanetarydiscsplanetmigrationhydrodynamicalsimulationmean-motionresonancediscself-gravityfictitiousacceleration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

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.

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 (3)
  1. [§2.3, Eq. (1)] 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.
  2. [§1 and §3.4.3, Table 2] 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.
  3. [§3.3, Eq. (8)] 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.
minor comments (6)
  1. [§3.2] The word 'swiched' appears three times and should be 'switched'; also, 'asymetrical' in §3.3 and 'couterbalance' in §3.4.1 are typos.
  2. [§3.3, after Fig. 7] The sentence containing '|Γind| ∼0.6|Γind|' appears to have a typo; it should presumably read '|Γind| ∼0.6|Γdir|'.
  3. [Figs. 5 and 7] 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.
  4. [Fig. 4] 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.
  5. [Table 2] 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.
  6. [Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the balanced-inclusion rule and Eq. (8) are argued from the decomposition in Eq. (1) and tested with original simulations; the m=1 instability is deferred to a companion paper but is not an input to the central derivation.

full rationale

The paper's central derivation chain is self-contained. The decomposition of the indirect term into per-body contributions follows directly from Eq. (1), IT = -a* = sum_i IT_i, and the balanced-inclusion rule in §2.3 is presented as a physical consistency requirement (if a body does not feel the direct pull of C, applying IT_C alone would pull its whole orbit away from C) rather than as a fitted or circular prediction. The recipe of Eq. (8) is justified by torque arguments: only the non-axisymmetric density Sigma' contributes to the torque, and the paper backs this with new FARGO simulations measuring Gamma_dir and Gamma_ind in type I and type II regimes. No parameter is fitted and then renamed as a prediction. The main caveat is the m=1 disc-eccentricity instability in §3.4.3 and Table 2, whose demonstration is deferred entirely to the companion 'Paper II' by the same team; this is a limitation in the evidence base and a potential tension with the recommendation to omit ITdd in non-self-gravitating runs, but it is not used to derive Eq. (8) or the balanced rule, so it does not make the derivation circular. Self-citations such as Zhu & Baruteau (2016) and Baruteau & Papaloizou (2013) are published numerical studies used as comparisons or supporting phenomenology, not as unverified inputs that force the paper's conclusions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's central claim rests on standard Newtonian mechanics plus two premises the reader does not pay for upfront: the balanced-inclusion rule, justified only by example in §2.3, and the existence of the m=1 instability, imported from the unpublished companion paper. There are no fitted free parameters; the Table 1 simulation parameters are stated experimental inputs, and the paper introduces no new physical entities.

assumptions (4)
  • standard math Newtonian mechanics and the definition of fictitious accelerations in a non-inertial frame (Eq. 1); the star-centered frame acceleration is the sum of the gravitational accelerations of all bodies on the star.
    Invoked in §2.1 to define the indirect term; the paper's decomposition follows directly from this definition.
  • domain assumption The disc is treated as a two-dimensional fluid (surface density Σ) whose self-gravity may be neglected in some runs; grid-based hydrodynamics in a star-centered frame with the star held fixed is a valid representation of the physics.
    This is the standard FARGO-style setup described in §3.3 and Table 1. The paper's recommendations only make sense within this class of simulation.
  • ad hoc to paper The balanced-inclusion rule: a body should feel the indirect term from body i if and only if it feels the direct gravitational acceleration of body i.
    Introduced in §2.3 via the A-B-C example and the tidal analogy (Fig. 1), not derived from the equations of motion. The competing view, that the fictitious force is a frame property applying to all bodies, is not disproved analytically in this manuscript.
  • ad hoc to paper ITdd grows an m=1 eccentricity instability of the disc, with the instability setting in whether or not self-gravity is included.
    Stated in §3.4.3 and attributed to the companion paper (Crida et al., in revision); not demonstrated in this manuscript.

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Pith. "Pith review of On inertial forces (indirect terms) in problems with a central body." pith.science (2026). https://pith.science/paper/RKOICRKF

@misc{pith2026250623331,
  author       = {Pith},
  title        = {Pith review of: On inertial forces (indirect terms) in problems with a central body},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RKOICRKF}},
  note         = {Machine review of arXiv:2506.23331}
}
read the original abstract

Gravitational systems in astrophysics often comprise a body -- the primary -- that far outweights the others, and which is taken as the centre of the reference frame. A fictitious acceleration, also known as the indirect term, must therefore be added to all other bodies in the system to compensate for the absence of motion of the primary. In this paper, we first stress that there is not one indirect term but as many indirect terms as there are bodies in the system that exert a gravitational pull on the primary. For instance, in the case of a protoplanetary disc with two planets, there are three indirect terms: one arising from the whole disc, and one per planet. We also highlight that the direct and indirect gravitational accelerations should be treated in a balanced way: the indirect term from one body should be applied to the other bodies in the system that feel its direct gravitational acceleration, and only to them. We point to situations where one of those terms is usually neglected however, which may lead to spurious results. These ideas are developed here for star-disc-planets interactions, for which we propose a recipe for the force to be applied onto a migrating planet, but they can easily be generalized to other astrophysical systems.

Figures

Figures reproduced from arXiv: 2506.23331 by the authors.

Figure 1
Figure 1. — Accelerations imparted by a distant companion (red disc) on a ring (black circle) around a star, in a frame centred on the star (yellow star symbol). Red arrows (top): direct gravitational acceleration of the companion. Blue dashed arrows (middle): indi￾rect gravitational acceleration of the companion, or indirect term, that is the opposite of the acceleration exerted by the companion on the star. Green thick arro… view at source ↗
Figure 2
Figure 2. — Summary of all forces applied in a gravitational sys￾tem comprised of a star, a protoplanetary disc and a planet, in a frame centred on the star. Solid arrows show direct gravitational forces, while dashed arrows show the corresponding indirect forces. Two circles, blue and green, surround the star that symbolise the acceleration felt by the star from the gravitational pull of the corre￾sponding objects (blue for … view at source ↗
Figure 3
Figure 3. — Zero-velocity curves (contours of potential energy) in the restricted circular three-body problem, computed in a frame centred on the star and corotating with a companion of 0.1 stellar mass. The indirect term of the planet on the test particles is not taken into account in the middle panel, but is in the left and right panels ; in the right panel, the direct term from the companion is removed. Dotted lines: circu… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: — Time evolution of the eccentricity (top) and resonant angles of the 2:1 mean-motion resonance (bottom) for two massive planets undergoing convergent migration in their protoplanetary disc (see text), with or without including the direct and/or indirect gravitational …
Figure 5
Figure 5. Figure 5: — Top: surface density of the disc gas normalised by its initial density in a simulation of a low-mass planet (see text). Arrows show the acceleration of the star due to the disc a∗,d, and the acceleration of the planet due to the direct gravity of the disk adir (non-a…
Figure 6
Figure 6. Figure 6: — Time evolution of the corotation torque and the indirect torque in a simulation with a low-mass planet embedded in an inviscid disc (see text). Note that the indirect torque has been rescaled to improve legibility. The inset plot shows the normalised amplitude of the…
Figure 7
Figure 7. Figure 7: — Same as [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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