The growing number of exoplanets detected over the past three decades has created a need for fast, reliable methods to study the long-term survival of planetary systems. Here, we investigate the challenging problem of the stability of compact three-planet systems, in which resonant and chaotic processes are intrinsically linked. Four completely different chaos indicators are tested on a data set of 10,000 three-planet configurations that are in or near mean-motion resonance. On the one hand, we consider two well-established chaos indicators, namely the mean exponential growth factor of nearby orbits (MEGNO) and a modified chaos indicator based on Lagrangian descriptors. On the other hand, two non-variational chaos indicators which do not require the tangent vector computation are considered for the first time for compact systems. We evaluate the performance of each chaos indicator in correctly predicting the stability of the planetary systems, and also highlight their differences by studying the dynamics of system configurations that are inconsistently classified by different indicators. Finally, we discuss what can be done to combine these chaos indicators to improve the overall performance, and how it could be useful for further constraining the orbital parameters of detected planetary systems.

How to cite: Căliman, A. and Libert, A.-S.: Predicting the stability of planetary systems through chaos indicators, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-608, https://doi.org/10.5194/epsc-dps2025-608, 2025.

Compact protoplanet systems are a natural outcome of runaway and oligarchic growth of planetesimals, with low-mass protoplanets with orbital separations of K≈10 mutual Hill radii. Those protoplanets evolve to Earth-mass bodies through giant impacts after gravitational instabilities are triggered. On the other hand, Kepler observations reveal older, non-resonant and more massive systems with orbital separations clustered at K>10 Hill radii, suggesting long-term stability despite similar compactness. This raises a question: does the stability difference between compact protoplanets and Kepler sample come from a primordial configuration or from subsequent dynamical evolution?

To investigate these stability differences, we conducted tens of thousands of numerical simulations of compact systems with varying orbital separations, mass distributions, inclinations and eccentricities, and quantified instability timescales and their sensitivity to these different parameters. 

Our results indicate that inhomogeneities in mass, in the same way as seen in orbital separation inhomogeneities, are a source of instability for more massive systems, even for larger orbital separations. Such inhomogeneity can destroy the resonant architecture seen in typical τ_inst x K, but also allow for new dynamical architectures by having pairs of planets closer to resonances that didn't exist in homogeneous systems. Analysis still underway indicate that, for more massive systems, ordering in planetary masses can be a source of stability, whereas systems which don't show masses increasing with semimajor-axis are more prone to instabilities.

Regarding the sensitivity of instability times to distinct orbital architectures, we analysed the effect of relative eccentricities rather than absolute ones. Motivated by Doty et al. (2025), who  numerically constrained eccentricities of the Multi-planet Kepler Sample and came to the conclusion that some systems were deemed stable even for substantially high initial eccentricities (≈0.15), we compared the instability timescales τ_inst  to both average absolute eccentricities e_f and relative eccentricities e_p. 

A set of results for orbital separation K = 10 can be seen in Figure 1. Left plot shows instability timescales τ_inst  as a function of relative eccentricities ⟨e_p²⟩^1/2 for different initial absolute eccentricities e_f; right plot shows τ_inst  as a function of absolute eccentricities e_f for different initial relative eccentricities.

Eccentricities described in the legend refer to e_f in the left plot and to ⟨e_p²⟩^1/2 in the right plot.

Our results indicate that eccentric systems can be deemed as stable for essentially any value of e_f provided that the relative eccentricities remain relatively low (⟨e_p²⟩^1/2≈10^-2), as seen on the left plot.

Right-hand plot highlight this dependency on ep rather than on ef: For all values of absolute eccentricities (x-axis), shorter instability timescales are linked exclusively to higher values of relative eccentricities ⟨e_p²⟩^1/2. For ef>10^-2, τ_inst decreases by approximately one order of magnitude even for smaller ⟨e_p²⟩^1/2.

Our results should help constrain evolutionary pathways and also available parameter space characterization for future observations.

How to cite: Teixeira Guimarães, G. and Kokubo, E.: Orbital instabilities in compact planetary systems, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-1555, https://doi.org/10.5194/epsc-dps2025-1555, 2025.

The increasing discovery of extrasolar systems has made it necessary to study their stability. In this work, we present a generalization of the AMD-stability criterion defined by Laskar and Petit (2017), which defines a critical AMD-value below which close encounters are prevented and the system can be considered stable. This secular approach does not take into account mean-motion resonance overlap which can be considerable for compact multi-planet systems. We present a new AMD-framework that extends the resonance overlap criterion previously introduced by different authors. This more general approach highlights the importance of eccentricity diffusion and is also valid for 3D planetary systems. We evaluate the performance of the proposed framework on several compact two- and three-planet first-order resonant systems and discuss how the criterion could be useful for filtering observational data, thereby improving the robustness of stability predictions for newly discovered systems.

How to cite: Bodart, J., Petit, A. C., and Libert, A.-S.: General AMD-stability criterion for exoplanet systems, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-1893, https://doi.org/10.5194/epsc-dps2025-1893, 2025.

Many stars are found in pairs. An interesting problem in stellar binary systems is how to assess whether a planetary orbit is dynamically stable or not.  Besides being a theoretical puzzle, the solution to that problem has many applications in topics such as for example in planet detection, planet formation, habitability, and the evolution of planetary systems off the  Main Sequence. Here, we present the latest developments in the problem of the stability of circumbinary planetary orbits.  With the aim of identifying stable and unstable orbits in such systems, we carry out more than 3x10⁸ numerical simulations of planets between the size of Mercury and the lower fusion boundary (13 Jupiter masses) which revolve around the center of mass of a stellar binary over long timescales. For the first time, three dimensional and eccentric planetary orbits are considered. Based on the results of our numerical simulations, we determine two critical borders: an outer border beyond which all planetary orbits are stable and an inner border closer to the binary below which all planetary orbits are unstable. In between the two borders, a mixture of stable and unstable planetary orbits is observed. We provide empirical expressions in the form of multidimensional, parameterized fits for the two borders that separate the three dynamical regimes. Moreover, we train a machine learning model on our data set in order to have an additional tool for predicting stable and unstable motion. Both the empirical fits and the machine learning model are tested for their predictive capabilities against randomly generated circumbinary planetary systems. The parameterized fits are also applied to the Kepler and TESS circumbinary systems, confirming the stability of the planets in these systems. Finally, the empirical fits are compared against previously derived stability criteria.

How to cite: Georgakarakos, N., Eggl, S., Ali-Dib, M., and Dobbs-Dixon, I.: New criteria for evaluating the orbital stability of circumbinary planets, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-633, https://doi.org/10.5194/epsc-dps2025-633, 2025.

One of the open questions in exoplanet research is the lack of mean-motion resonances (MMRs) in observed planetary systems, even though planet formation models predict that disk-driven migration should create resonant chains. This suggests that some physical process may be breaking these resonances after formation. In this project, we explore whether stellar dynamical tides could play a role in this process.

To model dynamical tides more accurately, we implemented the frequency-dependent Kaula model into the N-body code Posidonius, using Love number spectra provided by a collaborator. Unlike the constant time lag (CTL) model—which smooths out the tidal response by averaging over frequencies—the Kaula approach accounts for how the star responds to each individual tidal frequency.

Fig. 1 shows the orbital evolution of a 5 M⊕ super-Earth under both tidal models. Although both lead to similar final semi-major axes, their evolution is quite different. The CTL model produces a smooth migration path, while the Kaula model shows multiple outward migration boosts, causing the semi-major axis to oscillate around that of the CTL case. The eccentricity also evolves differently: CTL is dominated by the main tidal frequency (ω₂₂₀₀), while Kaula includes contributions from additional frequencies like (ω_220-1), which can excite the eccentricity. This highlights the importance of using a frequency-dependent model to capture the full behavior of tidal interactions.

We also applied the model to a two-planet system near the 2:1 MMR. Fig. 2 shows the evolution of the semi-major axes and the mean-motion resonance (MMR) states. In the top panel, the split between the upper and lower lines for each planet indicates the evolution of its apastron and periastron distances. In the Kaula model, the outer planet experiences strong tidal interactions despite being farther from the star, due to higher Love numbers at certain frequencies. With additional eccentricity damping from planetary tides (modeled by CTL), the system temporarily leaves the 2:1 resonance and later re-enters it as the inner planet undergoes stronger tidal effects. This result shows that dynamical tides can break and restore resonances, and may contribute to the dynamical evolution that leads to the absence of resonances in some systems.

These early results suggest that stellar tides may influence the long-term architecture of planetary systems, but a more complete picture will require studying additional resonances.

How to cite: Kwok, L. K.-W., Bolmont, E., Revol, A., Mathis, S., Astoul, A., Charbonnel, C., and Raymond, S.: Impact of Dynamical Tides on Planetary System Stability: Evolution of Multi-Planet Systems, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-784, https://doi.org/10.5194/epsc-dps2025-784, 2025.

It is widely known that the secular resonances of giant planets have played fundamental roles in the formation and dynamical evolution of the inner Solar System. These resonances have shaped the architecture of the asteroid belt [1,2], and have had significant contributions to the final mass, water content, and orbital architecture of terrestrial planets [3,4]. The successful detection of circumstellar planets in moderately close binary stars (i.e., binaries with separations smaller than 40 AU) in the past two decades has raised the question that, how these secular resonances appear and operate when the system is subject to the perturbation of a secondary star. We have launched an expansive project on this topic where in a three-article series [5,6,7], we have demonstrated how secular resonances of giant planets appear in binary star systems, and how they affect the formation and orbital evolution of planets interior to their orbits. Using the concept of generalized disturbing function, we have derived the formula for the locations of secular resonances in binary stars with two giant planets, and have shown that in systems where the perturbation of the secondary star is stronger, the locations of secular resonances are farther way from the primary and closer to the giant planets. The latter indicates that in these systems, the secular resonances have larger areas to migrate, and therefore, affect the dynamics of the system more strongly [6]. The most important result obtained from our study is that the perturbation of the secondary star suppresses the effects of secular resonances.

To investigate the effects of secular resonances on the orbital architecture of the system, we also simulated the late stage of terrestrial planet formation for different types of the secondary, and different orbital elements of the binary and giant planets. Results demonstrate that terrestrial planet formation can indeed proceed constructively in such systems; however, as predicted by the general theory, secular resonances are suppressed and do not contribute to the formation process. It is in fact the mean-motion resonances of the inner giant planet that drive the dynamics of the protoplanetary disk, and the mass and orbital architecture of the final bodies. Simulations also show that in the majority of the cases, the final systems contain only one terrestrial planet with a mass of 0.6 - 1.7 Earth masses. Multiple planets appear in rare occasions in the of form Earth-Mars analogs with the smaller planet in an exterior orbit. When giant planets are in larger orbits, the number of these double-planet systems increases and their planets become more massive. Results also show that when the orbits of the giant planets carry inclinations, while secular resonances are still suppressed, mean-motion resonances are strongly enhanced, drastically reducing the efficacy of the formation process. We present details of our simulations and discuss the implications of their results [7].

[1] Milani, A. & Kneževic, Z. 1992, Icarus, 98, 211

[2] Milani, A. & Kneževic, Z. 1994, Icarus, 107, 219

[3] Levison, H. F. & Agnor, c. 2003, AJ, 125, 2692

[4] Haghighipour, N. & Winter, O. C. 2016, Cel. Mech. Dyn. Ast., 124, 235

[5] Andrews, M. & Haghighipour, N. 2024, Proc. IAUS, 382, 116-122

[6] Haghighipour, N. & Andrews, M. 2025a, ApJ, in press

[7] Haghighipour, N. & Andrews, M. 2025b, ApJ, in press

How to cite: Haghighipour, N.: On the effects of secular resonances on the dynamical evolution and orbital architecture of planetary systems in binary stars, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-1121, https://doi.org/10.5194/epsc-dps2025-1121, 2025.

Although many theoretical works suggest that co-orbital exoplanets may also exist, such a pair of planets has yet to be observed. In particular, out of the hundreds of multi-planetary systems discovered by the Kepler mission, none were found in co-orbital configuration thus far. Although this result points to a relative rarity of the configuration, part of this absence might be due to observational biases (Robutel, Leleu 2024). We are interested here in one of these: the fact that the planets do not orbit in the same plane, which can prevent the transit of, at least, one of the bodies. In this talk, we explore the stability domain in the spatial case, paying a particular attention to a family of periodic orbits, in a rotating frame, emanating from the Lagrange’s equilateral configuration in the vertical direction.  Theses solutions have been first discovered by Marchal (2009) in the special case of the restricted secular problem. We propose a numerical study to follow this family in a more general case. We show that the family discovered by Marchal exists in the full three-body problem (neither restricted nor secular) for a wide range of masses, and that the stability of its orbits evolves along the family, sometimes leading to stable systems for masses exceeding the Gascheau’s value (also known as Routh’s critical value). We try to continue these families increasing the planetary masses the two extreme cases, that is between the Marchal’s and P12 families.  we try to extend these families by increasing the planetary masses until they reach that of the central body, in the hope of discovering links between these orbits of the planetary problem and the P12 family, containing the famous “Figure 8” when the 3 masses are equal (Chenciner, Montgomery, 2000).

Bibliographie

Chenciner A. and Montgomery R. “A remarkable periodic solution of the three-body problem in the case of equal masse”. Annals of Mathematics, 152 (2000), 881–901

Marchal C. “Long term evolution of quasi-circular Trojan orbits”. Celestial Mechanics and Dynamical Astronomy, 105(2009), 53-67

Robutel P. and Leleu A. “Trojan Exoplanets” (2024) arXiv:2402.15168. To be published in: Handbook of Exoplanets, 2nd Edition, Hans Deeg and Juan Antonio       Belmonte (Eds. in Chief), Springer International Publishing AG, part of Springer Nature.

How to cite: Prieur, A., Robutel, P., and Fejoz, J.: Vertical stability of co-orbital planetary systems: from Marchal’s family to P12? , EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-310, https://doi.org/10.5194/epsc-dps2025-310, 2025.

Over the past 30 years, the number of confirmed exoplanetary systems has increased enormously, with nearly a thousand systems hosting multiple planets. The orbital architectures of these systems have challenged traditional formation models and reshaped our understanding of how planetary systems form and evolve. Notably, the observed eccentricity distribution shows that many exoplanets have highly eccentric orbits, in sharp contrast to the quasi circular orbits of the Solar System (SS) planets. Additionally, observations using the Rossiter-McLaughlin effect have shown that some exoplanets orbit in polar orbits relative to their host star’s equator, whereas all planets in the SS orbit within relatively coplanar configuration. These findings highlight the importance of studying the dynamics of inclined and eccentric planetary configurations.

Furthermore, the distribution of period ratios in multi planet systems exhibit clear concentrations near mean-motion resonances (MMRs), which can significantly influence the long-term orbital evolution of such systems. For instance, the system HD 31527 appears to be stable only within the high-order 16:3 resonance. In this configuration, resonance helps prevent close encounters between outer planets, avoiding chaotic zones. In contrast, planets in the SS lie close but out of major resonances. Numerical experiments show that forcing SS planets into resonance typically leads to instability. This could be expected, since formation models suggest that planetary migration can be stopped by resonant trapping, but raises important questions: are planetary resonances stabilizing or destabilizing systems?
The answer depends on the context, much like the case of small body populations in the SS, where some resonances appear as concentrations of objects while others are empty.

It is crucial to understand the structure and properties of individual resonances and how they influence long-term orbital evolution. Generally, classical approaches for calculating the resonant disturbing function were analytical expansions only valid for some interval of eccentricities and inclinations or for particular resonances. However, calculating the resonant function numerically is advantageous as it has no restrictions on orbital elements or type of resonance.

In this work, we apply a semi-analytical model to compute the resonant disturbing function, the Hamiltonian, and the properties of any given resonance between two planets. We are only limited by the fast evolution of eccentricity, inclination, argument of perihelia or longitude of ascending nodes as the model assumes they stay more or less constant during a libration period.The model calculates the resonance width, the location of the equilibrium points and the libration period.

Traditionally, resonances have been classically classified as symmetric if there exists one equilibrium point where the critical angle oscillates around 0° or 180°or as asymmetric if there exists two equilibrium points separated by less than 180°. Most low inclination resonances are symmetric whereas 1:1 and all other 1:N resonances are asymmetric. In this work, we present results from mapping resonance properties in the (eccentricity, inclination) and (inner eccentricity, outer eccentricity) phase spaces. These include new resonant equilibrium points, which we validated through numerical integrations and compared with observed exoplanet systems from available databases.

Finally, we explore how the long-term secular evolution within resonances is affected by high eccentricities and inclinations. Our findings indicate that the phase-space topology of resonances can change dramatically under such conditions, challenging the traditional classification and revealing more complex dynamical structures.

Figure 1: Standard deviation of the number of equilibrium points for the 2:3 resonance in the (e, i) plane varying ω.

Figure 2: Standard deviation of the number of equilibrium points for the 2:1 resonance in the eccentricities plane varying ∆ω.

How to cite: Pan Rivero, N., Gallardo, T., Rodríguez, A., and Roig, F.: Extreme planetary resonances: Study of high eccentricity and high inclination resonances in exoplanet systems, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-1993, https://doi.org/10.5194/epsc-dps2025-1993, 2025.

Multi-planetary systems reveal diverse dynamical histories. Stellar obliquity is a key diagnostic of these histories, linking past dynamical interactions to migration pathways (e.g., quiescent disc vs. violent high-eccentricity). To measure the remaining dynamical violence of planetary systems, we introduce an obliquity-based NAMD (Normalized Angular Momentum Deficit), improving on the previous relative inclination-based NAMD in capturing the systems’ architectures. Our open-source ExoNAMD Python tool calculates these metrics, enabling cross-system dynamical state comparisons. The dynamical context provided by the NAMD can be used for (1) interpreting planetary atmospheres, as migration history shapes composition and thermal structure; (2) unbiased target selection for future observations and to guide model testing; (3) enabling comprehensive dynamical descriptions alongside stability indicators (AMD, MEGNO, SPOCK) in the forthcoming era of PLATO and Ariel.

How to cite: Bocchieri, A., Zak, J., and Turrini, D.: ExoNAMD: a community tool to gauge multi-planetary systems, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-223, https://doi.org/10.5194/epsc-dps2025-223, 2025.

We investigate a fundamental planetary system comprising a Sun-like star, a Jupiter-like planet, and a Neptune-like planet across a wide range of orbital configurations, not limited to the well-studied hierarchical case. We present resonance atlases illustrating the domains of thousands of mutual mean-motion resonances (MMRs) and their connection to both chaotic and regular dynamics. Our analysis reveals that an eccentric Jupiter generates a significant chaotic region in its vicinity due to close encounters and overlapping MMRs, particularly extended in semi-major axis for inclinations below 30 degrees. Notably, some strong resonances, such as the 1:1, persist within these chaotic zones.

Employing a semi-analytical method to study secular dynamics, we identify two regimes for equilibrium configurations. At low mutual inclinations, equilibrium arises from oscillations of mutual pericenters around 0 or 180 degrees. Conversely, at high mutual inclinations equilibrium is characterized by specific values of the arguments of pericenters that are integer multiples of 90 degrees.

Through numerical integration of the full equations of motion, we calculate the three fundamental frequencies of the system across diverse configurations and analyze their dependence on orbital elements. Based on this frequency analysis, we find two distinct dynamical regimes separated by a critical mutual inclination between 30 and 40 degrees, where a secular resonance occurs due to the coincidence of the two fundamental pericenter frequencies. Below this critical inclination, the dynamics resembles the classical secular model for low-inclination orbits, featuring three well-defined fundamental frequencies and free and forced modes, with mutual inclination remaining quasi-constant. Above this threshold, the dynamics shifts dramatically, exhibiting increased variations in mutual inclination and the emergence of combined fundamental frequencies, potentially dominated by the secular resonance or the von Zeipel-Lidov-Kozai (vZLK) mechanism. The observed change in dynamical regimes correlates with the transition in the equilibrium configurations of the pericenters.

While the planetary mass ratio has a limited qualitative impact on our findings, it directly influences the individual fundamental frequencies. Furthermore, the relative placement of the Neptune-mass planet (interior or exterior to Jupiter) leads to different dynamical behaviors at increasing mutual inclinations. An interior Neptune exhibits gradual secular changes in eccentricity and inclination, whereas an exterior Neptune shows a rapid increase in these elements following the onset of the vZLK mechanism.

How to cite: Gallardo, T. and Suescun, A.: The role of the mutual inclination in the dynamics of a two eccentric giant-planet system, EPSC-DPS Joint Meeting 2025, Helsinki, Finland, 7–13 Sep 2025, EPSC-DPS2025-1996, https://doi.org/10.5194/epsc-dps2025-1996, 2025.