Gravity is not just a tug of war. In the crowded nursery where young stars keep company with gas, dust, and newborn planets, the frame you pick to describe the motion can tilt the entire story. The latest work from a collaboration led by Aur e9lien Crida at Universit e9 C f4te d b4Azur (Observatoire de la C f4te d b4Azur) and collaborators across France, Germany, and Mexico shines a light on a subtle twist in gravitational math: there isn b4t be one indirect term, but many, each tied to a body that tugs on the dominant mass. That nuance changes how we simulate star–disc–planet systems and could alter how we interpret long-running computer experiments that trace how planets migrate and settle into resonance. The paper is a tour through a deceptively simple idea that ripples outward into practical guidance for simulations, from the way we model a migrating planet to how vortices in discs ripple through the whole system.
The authors make a clear, almost artful claim: in a hierarchical gravitational system, every body that exerts a pull on the primary should generate its own indirect term, and the indirect term it produces should be applied to the objects that feel its direct pull. In other words, you don b4t just tote up a single inertial force and call it a day. You split the inertial legacy into ITp, ITd, ITdd, and so on, depending on who is pulling on whom. This may sound like bookkeeping, but it matters when you want to understand delicate balances—for example, how a migrating planet interacts with the nonuniform disc around a young star, or how two planets might lock into a resonance without ever touching each other directly. The study demonstrates this with a star–disc–planet system, and the implications echo beyond one particular setup. The work is a joint endeavor highlighting institutions including Universit e9 C4te d b4Azur and its Lagrange laboratory, IRAP in Toulouse, and other partners across Europe and North America.
As a lead-in to the detail, it helps to name a few authors who carried the thread: Aur e9lien Crida, Cl e9ment Baruteau, Philippine Griveaud, Elena Lega, Fr e9d e9ric Masset, and many others. Their collaboration grounds a conceptual shift in a field where numerical simulations are the laboratory of modern planetary science. The paper is the first in a pair; the companion work, Paper II, explores a long-term physical instability linked to inertial forces in stellocentric simulations. Together, they argue for a careful, explicit accounting of every indirect term to improve reproducibility and physical realism in planet–disc experiments.
Not just one indirect term
In a gravitational hierarchy, the primary mass (the star) sits at the center of the frame we often use to model motion. If we follow that frame, a fictitious acceleration—an indirect term—compensates for the primary b4s reflex motion around the system’s center of mass. But Crida and colleagues push back against a tempting simplification: there is not one indirect term that can be slapped onto everyone and everything. There are as many indirect terms as there are bodies exerting gravity on the primary. In a protoplanetary disc with two planets, for instance, there are three separate indirect contributions: one from the whole disc, and one from each planet. Each indirect term is the opposite of the acceleration that the corresponding body imparts to the star.
That sounds mathematical, but it b4s a practical invitation. When a direct gravitational acceleration from a body is included in a simulation, its indirect counterpart should be present as well. If you remove the direct pull from a body, you should also remove its indirect term. In other words, the inertial web must be treated in a balanced fashion: you can choose to neglect a particular direct gravitational interaction, but you should not leave its indirect term hanging in the air. This balancing act is not just pedantry; it shapes the very way a disc responds to a planet, how multiple planets influence each other in a star-centered frame, and how a disc might spontaneously form nonaxisymmetric features like vortices that in turn feedback on migration.
Four indirect terms in a star–disc–planet system
To make the discussion concrete, the authors lay out four main indirect terms in a typical star–disc–planet arrangement. Each term acts as if a tiny orchestra member is bowing in the wings, and the direct gravity from that member is the note heard at the center of the stage. The indirect term associated with that member, when applied to other bodies that feel its direct gravity, shapes the overall dynamics. The four terms are: the indirect term from the planet to the disc (ITpd), the indirect term from one planet to the other (ITpp), the indirect term from the disc to the planet (ITdp), and the indirect term from the disc to itself (ITdd). The paper also emphasizes that the way you apply each term depends on what direct forces you keep or drop in the simulation. It is not a one-size-fits-all recipe; it is a set of careful choices that must stay consistent with the physics of the problem.
Indirect term from planet to disc ITpd operates in the star-centered frame with the planet as the perturber. In a simple setup with a circular planet orbit and test-particle disc elements, including ITpd is essential to recover the classical equilibrium points known as L4 and L5, which sit 60 degrees ahead of and behind the planet. Without the indirect term, the zero-velocity curves—an organizing map of where test particles can stay in equilibrium—collapse to a different, misleading geometry. The indirect pull from the planet is directional: it aligns along the star–planet line, and it can tilt the balance of forces in the disc so a local crowding of material becomes a resonant trap for ringed orbits. This is not just a mathematical curiosity; it explains why certain patterns in the disc emerge and where trojan-like features might sit in a real system.
Indirect term ITpp, the force from one planet to another, reveals a surprising dynamical possibility: even if you switch off the direct gravitational pull between planets, the planets can still lock into a resonance through the planet–star–planet indirect coupling. If the second planet feels the indirect pull of the first via the star b4s reflex motion, angular momentum exchange persists in a way that can drive resonance capture. The study showcases the dramatic illustration with a pair of migrating giant planets: when the direct planet–planet gravity is off but their indirect terms remain on, the planets can fall into a 2:1 mean-motion resonance thanks to the star b4s motion. Turn off the indirect terms too, and that resonance scene vanishes. The result is a vivid reminder that gravitational interactions are a web, not a collection of isolated threads.
Indirect term ITdp—the disc acting on the planet—is where practical torque budgets meet theory. The disc exerts a direct torque on the planet via the gravitational wake it carves as the planet moves through gas. But the planet also feels the indirect pull from the disc, and that indirect term contributes its own torque. When you combine the direct torque with the indirect one, the planet migrates at a rate that can differ from the direct-term estimate alone. In type I migration, where a low-mass planet coexists with a smooth disc, the indirect torque typically amounts to a modest correction—roughly a few percent of the direct torque in the examples explored. But the same indirect force can rise to prominence in nonaxisymmetric discs—think of a vortex looming in the gas. In that case, ITdp can meaningfully alter the migration path, resistance to capture into resonance, and even the pattern of density wakes trailing the planet.
Indirect term ITdd, the disc pulling on itself through the reflex motion of the whole disc mass around the star–planet system, broadens the picture further. This term has been less frequently treated in simulations, yet it can reshape how a vortex forms and migrates, and it can trigger or suppress global disc modes, including an m = 1 eccentric mode that can percolate through the disc. Even when the disc’s self-gravity is included in the model, ITdd can drive dynamics that would not appear if you neglected it. In discs where vortices form at gap edges created by a giant planet, ITdd tends to couple with the disk’s density structure in subtle, sometimes surprising ways, potentially pushing the disc toward more eccentric states and altering migration episodes. The authors show that ignoring ITdd can lead to spurious, vortex-driven migration scenarios that disappear once ITdd is included. This has practical consequences for interpreting past simulations and for planning future ones, especially in the regime where discs are massive enough to host vigorous vortical activity.
A practical recipe for migrating planets
A central, take-away message from the paper is not only what to include, but how to include it in a non-self-gravitating disc when a planet is migrating. When the disc’s self-gravity is neglected, the force that the disc exerts on the planet should be computed with a particular care: you should consider both the direct gravitational acceleration from the non-axisymmetric part of the disc and the indirect torque from the same non-axisymmetric structure. A clean prescription emerges: use the non-axisymmetric component of the disc density, defined as the local surface density minus its azimuthally averaged value, { al Σ′ = Σ − ⟨Σ⟩}. Then compute two contributions to the planet b4s acceleration. The first is the direct term from the non-axisymmetric density, and the second is the indirect term that tracks the star b4s reflex motion due to the disc mass distribution. The total acceleration on the planet is the sum of these two pieces, and the resulting torque on the planet is the cross product of the planet’s position with this total acceleration, projected along the axis of the disc. The practical upshot is a simple, robust recipe that ensures your migration rate reflects the true gravitational environment the planet experiences.
Concretely, the authors propose a compact form that a migrating planet in a non-self-gravitating disc should feel as it moves through a disc with nonaxisymmetric features. The acceleration exerted by the disc on the planet should be written as the integral over the disc of the non-axisymmetric density times a 1/|r − rp| cubed kernel, minus the integral of the non-axisymmetric density times r/|r| cubed. In words: a direct term from the disc wake plus a compensating indirect term from the star b4s reflex action. An important practical note is that you should use the azimuthally averaged surface density only to compute a radial, non-torquing acceleration that would otherwise buoy up spurious velocity differences between the planet and the local disc. The authors show that a commonly used workaround in existing codes—replacing Σ with Σ′ in the direct term—achieves a surprisingly consistent result across several migration regimes, and they advocate making this a standard practice in non-self-gravitating disc simulations.
All of this matters because many planet-migration studies rely on fleets of simulations with different pairs of planets and disc masses. A consistent accounting of IT terms helps ensure that the migration tracks are faithful to physics and that any observed resonance captures or bans are not artifacts of how the indirect term was handled. The paper also emphasizes that the precise ratio of the indirect torque to the direct torque depends on the disc’s structure—its density gradient, temperature profile, and the presence or absence of nonaxisymmetric features like vortices. In other words, the same planet in two discs with slightly different density patterns might migrate differently not only because the wakes are different, but because the indirect term acts differently in each environment. The nuanced takeaway is that meticulous bookkeeping of indirect terms is not a luxury; it is a necessity for credible, reproducible results in planet-disc experiments.
The ripples of ITdd and the drama of disc vortices
Beyond the tidy calculations of Type I migration, the paper dives into more chaotic territory where discs host vortices and other nonaxisymmetric structures. In those settings, the indirect term from the disc onto the planet, ITdp, can become much more influential, especially when a vortex forms near a planet b4s orbit. Early in a Type II migration episode, where a giant planet carves a gap and the disc responds en masse, the indirect torque can overshadow the direct torque by a substantial margin. The early, vigorous phase can drive rapid migration or modify the way a vortex feeds angular momentum into the system. The flip side is that the indirect term from the disc onto itself, ITdd, can also drive long-range coupling within the disc, leading to global modes that reshape how matter circulates and how planets feel the disc’s pull. The authors show that when ITdd is included, the disc can acquire an eccentric, m = 1 mode more readily, a state that can feedback into the planet b4s migration in a delicate, self-reinforcing loop.
Crucially, the study finds that ignoring ITdd can give a misleading impression of vortex-driven migration, sometimes producing outward migration episodes that vanish when ITdd is turned off. This is not just a numerical curiosity: it signals a real dynamical channel by which the disc can reorganize its density and angular momentum budget, thereby altering the path a planet follows as it moves inward or outward through the disc. The results underscore the broader point that indirect effects are not marginal curiosities; they are functional levers in the disc b4s own evolution and in the planet’s journey through it. In their companion work, the authors push this even further, showing that ITdd can seed or amplify an instability that reshapes the disc on secular timescales, an insight that invites renewed scrutiny of disk stability in the era of increasingly precise simulations.
In practice, this means that if you are modeling a system with a fairly massive disc, you cannot treat the disc as a passive stage for planet migration. You must account for how the disc’s own reflex motion and the disc’s self-gravity feed back into the flow of gas, the formation of vortices, and the planet’s orbit. The authors make clear that ITdd cannot be dismissed as a minor correction in all regimes; it becomes a central actor in the dynamics of vortices, gaps, and global disc modes. And because these effects depend on the precise mass and density structure of the disc, the authors urge scientists to be explicit about which indirect terms are included in their simulations and to report those choices when they publish results. Reproducibility, after all, lives in the small print of the force balances we choose to enforce in our codes.
Why this matters for planet formation modeling
So why does this meticulous accounting of indirect terms matter to the big questions of planet formation? First, it changes how we set up and interpret simulations. If a study neglects ITdd or misapplies ITdp in a strong, vortex-filled disc, the resulting migration histories and resonance outcomes might look different from reality. The authors argue for explicit, itemized reporting of which indirect terms are used in a study, because these choices can subtly tilt the outcome, especially in discs with nonaxisymmetric structure. This is not a criticism of past work; it is a call for clarity as simulations become more powerful and more central to how we test ideas about planet formation and migration against observations of exoplanet systems. Second, the work provides a practical, transferable recipe that researchers can implement right away when working with non-self-gravitating discs. By using the non-axisymmetric density for both direct and indirect torques, and by subtracting the axisymmetric average to define the non-axisymmetric part, simulations gain a robustness that helps ensure that the migration paths reflect the physics rather than the quirks of the numerical setup.
In addition, the paper illuminates a broader methodological lesson: even seemingly subtle choices in a numerical frame of reference can ripple through many-body gravitational systems. The star’s reflex motion, the disc’s mass distribution, the planets’ gravity on one another, and the disc’s own self-gravity all tug on one another in a complex dance. The authors show that the most transparent, reproduceable way to model this dance is to treat each indirect term as a distinct physical actor and to apply it only to the bodies that actually feel its direct partner. This disciplined approach does not merely tidy up equations; it makes simulations more trustworthy as a laboratory for probing questions like how gaps form, how resonances emerge, and how the world of exoplanets acquires its dizzying variety of orbital architectures.
All of this sits in the broader arc of a field that is rapidly refining its numerical toolkit. The Crida–Baruteau–Griveaud–Lega–Masset collaboration explicitly connects a conceptual refinement to a practical methodology, with the promise that these ideas will scale as simulations include more physics, such as disc self-gravity, three-dimensional effects, or more sophisticated thermodynamics. The work also points toward a more unified language for describing inertial effects in hierarchical systems, whether we are talking about a star–disc–planet ensemble or other gravitational assemblies in astrophysics. The authors close with a straightforward, almost humble note: when you simulate a migrating planet, you should decide which indirect terms you include and state it plainly. The rest of the math will follow, and the results will be easier to compare, reproduce, and build upon.
The human thread behind the science
Behind the equations and simulations lies a human story of collaboration and cross-pollination among institutions that span continents. The study is anchored in the Lagrange laboratory at the Observatoire de la C f4te d b4Azur, with contributors linked to the IRAP in Toulouse, the Max Planck Institute for Astronomy in Heidelberg, Caltech b4s divisions of planetary science, and several European partners. The lead researcher, Aur e9lien Crida, embodies a tradition of Europe-wide planetary science that blends rigorous numerical craft with a drive to clarify the physical intuition behind complex gravitational systems. The paper also nods to companion work that delves into long-term instabilities tied to these inertial forces, signaling that this is not a one-off technical tweak but a thread weaving through how we understand planet formation in discs that are alive with motion and mass. In an era when simulations are our most controllable laboratory for testing ideas about how worlds form, Crida and colleagues remind us that precision in the foundations—who exerts what force on whom and how we account for inertia in the frame—can change the conclusions we draw about how planetary systems come to be.
In short, the article reframes a foundational piece of the numerical toolkit: it unpacks the idea of a single indirect term into a plural, body-specific set of inertial terms. It also translates that insight into a practical recipe for simulating migrating planets in non-self-gravitating discs, while illuminating how disc vortices and global disc modes can be shaped by the same inertial bookkeeping. The result is a richer, more faithful map of the gravitational choreography that guides planets from dusty beginnings to the diverse exoplanetary architectures we observe today.
