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Eccentric Extrasolar Planets: The Jumping Jupiter Model
Abstract
Most extrasolar planets discovered to date are more massive than Jupiter, in surprisingly small orbits (semimajor axes less than 3 AU). Many of these have significant orbital eccentricities. Such orbits may be the product of dynamical interactions in multiplanet systems. We examine outcomes of such evolution in systems of three Jupiter-mass planets around a solar-mass star by integration of their orbits in three dimensions. Such systems are unstable for a broad range of initial conditions, with mutual perturbations leading to crossing orbits and close encounters. The time scale for instability to develop depends on the initial orbital spacing; some configurations become chaotic after delays exceeding 108 y. The most common outcome of gravitational scattering by close encounters is hyperbolic ejection of one planet. Of the two survivors, one is moved closer to the star and the other is left in a distant orbit; for systems with equal-mass planets, there is no correlation between initial and final orbital positions. Both survivors may have significant eccentricities, and the mutual inclination of their orbits can be large. The inner survivor's semimajor axis is usually about half that of the innermost starting orbit. Gravitational scattering alone cannot produce the observed excess of “hot Jupiters” in close circular orbits. However, those scattered planets with large eccentricities and small periastron distances may become circularized if tidal dissipation is effective. Most stars with a massive planet in an eccentric orbit should have at least one additional planet of comparable mass in a more distant orbit.
... However, Antonini et al. (2016) found that most observed WJs with characterized external giant planet companions cannot be formed in this way. Another source of eccentricity excitation is planet-planet scattering (Rasio & Ford 1996;Weidenschilling & Marzari 1996;Marzari & Weidenschilling 2002;Chatterjee et al. 2008;Jurić & Tremaine 2008;Raymond et al. 2010). Anderson et al. (2020) studied a scenario of in situ scatterings. ...
... All of their two-planet systems arise from planet-planet collision events, which produce too many low-eccentricity and closely packed two-planet systems. The dearth of stable two-planet systems from ejection events in Anderson et al. (2020) is very surprising and even doubtful, because previous studies of three-planet scatterings at several AU have shown that an ejection event could produce two planets on stable and well-separated orbits (Marzari & Weidenschilling 2002;Chatterjee et al. 2008), making planet ejection a plausible way to produce systems like GJ 1148. ...
... This suggests that the inner planet can move to a smaller orbit if the ejected planet is more massive. Such a result is expected because the system's energy is conserved during the whole process (Marzari & Weidenschilling 2002). The total energy of the initial system is If the hypothetical third planet is more massive, then the absolute value of total energy in the initial system is greater. ...
The GJ 1148 system has two Saturn-mass planets orbiting around an M dwarf star on hierarchical and eccentric orbits, with orbital period ratio of 13 and eccentricities of both planets of 0.375. The inner planet is in the regime of eccentric warm Jupiters. We perform numerical experiments to study the planet–planet scattering scenario for the origin of this orbital architecture. We consider a third planet of 0.1 M J (Jupiter's mass) in the initial GJ 1148 system with initial orbital separations of 3.5, 4, and 4.5 mutual Hill radii and initial semimajor axis of the innermost planet in the range of 0.10–0.50 au. The majority of scattering results in planet–planet collisions, followed by planet ejections, and planet–star close approaches. Among them, only planet ejections produce eccentric and widely separated two-planet systems, with some having similar orbital properties to the GJ 1148 system. We also examine the effects of general relativistic apsidal precession and a higher mass of 0.227 M J for the third planet. The simulation results suggest that the GJ 1148 system may have lost a giant planet. We also perform simulations of the general problem of the origin of warm Jupiters by planet–planet scattering. As in the GJ 1148 simulations, a nontrivial number of stable two-planet systems are produced by ejection, which disagrees with the result from a previous study showing that two-planet systems arise exclusively through planet–planet collisions.
... this way. Another source of eccentricity excitation is planet-planet scattering (Rasio & Ford 1996;Weidenschilling & Marzari 1996;Marzari & Weidenschilling 2002;Chatterjee et al. 2008;Jurić & Tremaine 2008;Raymond et al. 2010). Anderson et al. (2020) studied a scenario of in situ scatterings. ...
... All of their two-planet systems arise from planet-planet collision events, which produce too many low-eccentricity and closely packed two-planet systems. The dearth of stable two-planet systems from ejection events in Anderson et al. (2020) is very surprising and even doubtful, because previous studies of three-planet scatterings at several au have shown that an ejection event could produce two planets on stable and well-separated orbits (Marzari & Weidenschilling 2002;Chatterjee et al. 2008), making planet ejection a plausible way to produce systems like GJ 1148. ...
... This suggests that the inner planet can move to a smaller orbit if the ejected planet is more massive. Such a result is expected because the system's energy is conserved during the whole process (Marzari & Weidenschilling 2002). The total energy of the initial system is ...
The GJ 1148 system has two Saturn-mass planets orbiting around an M dwarf star on hierarchical and eccentric orbits, with orbital period ratio of 13 and eccentricities of both planets of 0.375. The inner planet is in the regime of eccentric warm Jupiters. We perform numerical experiments to study the planet-planet scattering scenario for the origin of this orbital architecture. We consider a third planet of 0.1M J (Jupiter's mass) in the initial GJ 1148 system with initial orbital separations of 3.5, 4, and 4.5 mutual Hill radii and initial semimajor axis of the innermost planet in the range of 0.10-0.50 au. The majority of scattering results in planet-planet collisions, followed by planet ejections, and planet-star close approaches. Among them, only planet ejections produce eccentric and widely separated two-planet systems, with some having similar orbital properties to the GJ 1148 system. We also examine the effects of general relativistic apsidal precession and a higher mass of 0.227M J for the third planet. The simulation results suggest that the GJ 1148 system may have lost a giant planet. We also perform simulations of the general problem of the origin of warm Jupiters by planet-planet scattering. As in the GJ 1148 simulations, a nontrivial number of stable two-planet systems are produced by ejection, which disagrees with the result from a previous study showing that two-planet systems arise exclusively through planet-planet collisions.
... Their simulations include a solar-mass host star and equally spaced 10 −7 M planets on initially circular and co-planar orbits. According to the relationship between separation and instability timescale, the first encounter time may happen at around 100 Myr at ∆ ∼ 8. Marzari & Weidenschilling (2002) studied the situation of three 10 −3 M planets equally spaced on initially circular and co-planar orbits, and they obtain a relation between separation and instability timescale for Jupiter-mass planets. They firstly studied the case of 3 ≤ ∆ ≤ 5.3 with inner planet fixed at semimajor-axis a = 5au. ...
... The system became unstable after 30 kyr, which basically followed the prediction. Although HD 184010 system looks similar to this simulation assuming in situ in Marzari & Weidenschilling (2002), the planet separations of HD 184010 are much larger if they are measured in mutual Hill radii. The separation of the inner two and outer two planets are respectively ∆ 1,2 = 6.56 and ∆ 2,3 = 6.47 for planets at their minimum masses (inclination i = 90 • ), and ∆ 1,2 = 5.84 and ∆ 2,3 = 5.76 for planets at 1.4 times of their minimum masses (inclination i = 45 • ). ...
... The separation of the inner two and outer two planets are respectively ∆ 1,2 = 6.56 and ∆ 2,3 = 6.47 for planets at their minimum masses (inclination i = 90 • ), and ∆ 1,2 = 5.84 and ∆ 2,3 = 5.76 for planets at 1.4 times of their minimum masses (inclination i = 45 • ). Our simulation in Section 6 convinced stability of at least 1 Gyr for the upper two orbital separations, reaching a consensus with the prediction by Figure 2 in Marzari & Weidenschilling (2002). However, when the planets are increased to several times massive than their minima, e.g. three times (upper right subplot in Figure 10), the separation are decreased to ∆ 1,2 = 4.54 and ∆ 2,3 = 4.48. ...
We report the discovery of a triple-giant-planet system around an evolved star HD 184010 (HR 7421, HIP 96016). This discovery is based on observations from Okayama Planet Search Program, a precise radial velocity survey, undertaken at Okayama Astrophysical Observatory between 2004 April and 2021 June. The star is K0 type and located at the beginning of the red giant branch. It has a mass of , a radius of , and a surface gravity log g of . The planetary system is composed of three giant planets in a compact configuration: the planets have minimum masses of , , and , and orbital periods of d, d, and d, respectively, which are derived from a triple Keplerian orbital fit to three sets of radial velocity data. The ratio of orbital periods are close to Pd : Pc : Pb ∼ 21 : 12 : 7, which means the period ratios between neighboring planets are both lower than 2 : 1. The dynamical stability analysis reveals that the planets should have near-circular orbits. The system could remain stable over 1 Gyr, initialized from co-planar orbits, low eccentricities (e = 0.05), and planet masses equal to the minimum mass derived from the best-fitting circular orbit fitting. Besides, the planets are not likely in mean motion resonance. The HD 184010 system is unique: it is the first system discovered to have a highly evolved star (log g < 3.5 cgs) and more than two giant planets all with intermediate orbital periods (102 < P < 103 d).
... To better outline the meaning of the R scaling, we have performed a series of numerical simulations where we have computed the timescale for the onset of instability for different values of the planet and binary separation. As usually done in these kind of explorations (see for example Chambers et al. (1996); Marzari & Weidenschilling (2002); Chatterjee et al. (2008)), to evaluate the increase in the time interval for orbital crossing as a function of the planetary separation we have scaled the initial distance between each pair of planets with the mutual Hill radius R H . The initial semi-major axes of the planets are defined as a 1 , a 2 = a 1 + KR H and a 3 = a 2 + KR H . ...
... An example is shown in Fig.2 where the actual data on the timescale for the instability onset vs. K are shown together with the averaged logarithmic fit represented by a continuous line. On purpose, we select a dynamical configuration which is chaotic for any value of K in order to see the changes in the instability time for different values of R. If the system is analysed on an interval of K values for which it is stable, then we will have all the times saturated to 10 8 preventing the identification of a trend with R. It is also interesting to note that the same initial configuration for three planets around a single star has stability times longer than 10 9 yr for K ≥ 5.3 as shown in Fig. 3 and illustrated also in (Marzari & Weidenschilling 2002). The sharp drop in the stability times around K = 3.8 − 4 is due to the presence of the 5:3 resonance between the first and second planet and the 3:1 resonance between the first and third planet while that at K = 5 is due to the 2:1 resonance (see Marzari & Weidenschilling (2002); Raymond et al. (2010). ...
... On purpose, we select a dynamical configuration which is chaotic for any value of K in order to see the changes in the instability time for different values of R. If the system is analysed on an interval of K values for which it is stable, then we will have all the times saturated to 10 8 preventing the identification of a trend with R. It is also interesting to note that the same initial configuration for three planets around a single star has stability times longer than 10 9 yr for K ≥ 5.3 as shown in Fig. 3 and illustrated also in (Marzari & Weidenschilling 2002). The sharp drop in the stability times around K = 3.8 − 4 is due to the presence of the 5:3 resonance between the first and second planet and the 3:1 resonance between the first and third planet while that at K = 5 is due to the 2:1 resonance (see Marzari & Weidenschilling (2002); Raymond et al. (2010). In our binary configuration, long stability Figure 3. Timescale of the onset of instability for three planets orbiting a single star with a 1 = 3au (black line), a 1 = 6 au (blue line) and a 1 = 12 au (green line), respectively. ...
Planet Planet scattering is a leading dynamical mechanism invoked to explain the present orbital distribution of exoplanets. Many stars belong to binary systems, therefore it is important to understand how this mechanism works in presence of a companion star. We focus on systems of three planets orbiting the primary star and estimate the timescale for instability finding that it scales with the keplerian period for systems that have the same ratio between inner planet and binary semimajor axes. An empirical formula is also derived from simulations to estimate how the the binary eccentricity affects the extent of the stability region. The presence of the secondary star affects the Planet Planet scattering outcomes causing a broadening of the final distribution in semimajor axis of the inner planet as some of the orbital energy of the planets is absorbed by the companion star. Repeated approaches to the secondary star causes also a significant reduction in the frequency of surviving two planet systems in particular for larger values of the inner planet semimajor axis. The formation of Kozai states with the companion star increases the number of planets which may be tidally circularized. To predict the possible final distribution of planets in binaries we have performed a large number of simulations where the initial semimajor axis of the inner planets is chosen randomly. For small values of the binary semimajor axis, the higher frequency of collision alter the final planet orbital distributions which, however, beyond 50 au appear to be scalable to wider binary separations.
... The stability criterion of a multiplanetary system has been extensively investigated. In particular, a number of previous works (e.g., Gladman 1993;Chambers et al. 1996;Marzari & Weidenschilling 2002;Quillen 2011;Tamayo et al. 2016;Wu et al. 2019) pointed out on the basis of the Hill stability argument that a mutual orbital separation of planets plays an important role in their long-term stability. However, when the planetary mass is large, the mean resonance becomes important, and those empirical relations based on the mutual Hill radius are not directly applicable. ...
... However, when the planetary mass is large, the mean resonance becomes important, and those empirical relations based on the mutual Hill radius are not directly applicable. We found that the instability time predictions given by Chambers et al. (1996) and Marzari & Weidenschilling (2002) significantly underestimate the lifetime of our simulated systems. For instance, the criterion given by Marzari & Weidenschilling (2002) predicts half of the systems to be unstable, though most of the systems are stable in our simulations. ...
... We found that the instability time predictions given by Chambers et al. (1996) and Marzari & Weidenschilling (2002) significantly underestimate the lifetime of our simulated systems. For instance, the criterion given by Marzari & Weidenschilling (2002) predicts half of the systems to be unstable, though most of the systems are stable in our simulations. This discrepancy originates from the planetary mass dependence of the stability criterion. ...
A number of protoplanetary disks (PPDs) observed with the Atacama Large Millimeter/submillimeter Array potentially provide direct examples of initial conditions for planetary systems. In particular, the HL Tau disk has been intensively studied, and its rings/gaps are conventionally interpreted to be a result of unseen massive planets embedded in the gaps. Based on this interpretation, we carried out N -body simulations to investigate the orbital evolution of planets within the PPD and after disk dispersal. Before disk dispersal, our N -body simulations include both migration and mass growth of the planet coupled with the evolution of the disk. By varying the disk parameters, we produce a variety of widely separated planetary systems consisting of three super-Jupiters at the end of disk dispersal. We found that the outer planet is more massive than the inner one, and the migration of the innermost planet is inefficient due to the accretion of outer planet(s). We also showed how the final configuration and the final planetary mass depend on disk parameters. The migration is found to be convergent, and no planet pair has a period ratio less than 2. After disk dispersal, we switch to purely gravitational N -body simulations and integrate the orbits up to 10 Gyr. Most simulated systems remain stable for at least 10 Gyr. We discuss the implications of our result in terms of the observed widely separated planetary systems HR 8799 and PDS 70.
... Additionally, we introduce a system with a flat mass slope, i.e., all the planets are equally-massed. Here a system with three Jupiter-massed planets, as was often used in planet scattering experiments (Chambers et al. 1996;Marzari & Weidenschilling 2002;Chatterjee et al. 2008), is generated. In such a setup, the two neighbouring planets' separation is often measured in their mutual Hill radius, defined as (Chambers et al. 1996) ...
... The middle panel, showing the fractional loss during this later phase, clearly supports this point. Thus, the initial location of a planet plays a minor role in the post-encounter phase and any planet, irrespective of its stellar-centric distance, is disrupted to a similar degree (reminiscent of the classical planet scattering without stellar encounters Marzari & Weidenschilling 2002). The top panel presents the survivability of the system as a whole. ...
... evolution and, additionally, solid for immediately after the encounter. The two planets in the 3->3->2 systems are characterised by similar CDFs, the inner one slightly hotter, like that in pure planet scattering simulations (Marzari & Weidenschilling 2002;Chatterjee et al. 2008), implying that this may be the main driver for eccentricity excitation. As for the 3->2->2, the outer planets are clearly perturbed more significantly during the encounter with a much hotter CDF. ...
Planetary systems formed in clusters may be subject to stellar encounter flybys. Here we create a diverse range of representative planetary systems with different orbital scales and planets' masses and examine encounters between them in a typical open cluster. We first explore the close-in multi-super earth systems au. They are resistant to flybys in that only ones inside a few au can destabilise a planet or break the resonance between such planets. But these systems may capture giant planets onto wide orbits from the intruding star during distant flybys. If so, the original close-in small planets' orbits may be tilted together through Kozai--Lidov mechanism, forming a "cold" system that is significantly inclined against the equator of the central host. Moving to the intermediately-placed planets around solar-like stars, we find that the planets' mass gradient governs the systems' long-term evolution post-encounter: more massive planets have better chances to survive. Also, a system's angular momentum deficit, a quantity describing how eccentric/inclined the orbits are, measured immediately after the encounter, closely relates to the longevity of the systems -- whether or not and when the systems turn unstable in the ensuing evolution millions of years post-encounter. We compare the orbits of the surviving planets in the unstable systems through (1) the immediate consequence of the stellar fly or (2) internal interplanetary scattering long post-encounter and find that those for the former are systematically colder. Finally, we show that massive wide-orbit multi-planet systems like that of HR 8799 can be easily disrupted and encounters at a few hundreds of au suffice.
... In Obertas et al. (2017), systems with adjacent or next-adjacent planets near first or secondorder mean-motion resonances (MMR) have stability times that deviate from the time predicted by the spacing. Also reported in Chambers et al. (1996); Marzari & Weidenschilling (2002); Pu & Wu (2015), the chains of MMR in these multi-planet systems cause separation-dependent modulations superimposed on the stability time relationship that have amplitudes as large as an order of magnitude. Chambers et al. (1996) integrates 120 systems of 20 planets with equal spacing and masses that vary by a factor of five. ...
... The spacing is denoted as 'K' in Marzari & Weidenschilling (2002) and ' ' in Smith & Lissauer (2009). For a system of n planets, Eq. 1 and Eq. 2 can combine to give ...
Compact planetary systems with more than two planets can undergo orbital crossings from planet-planet perturbations. The time which the system remains stable without orbital crossings has an exponential dependence on the initial orbital separations in units of mutual Hill radii. However when a multiplanet system has period ratios near mean-motion resonances, its stability time differs from the time determined by planet separation. This difference can be up to an order of magnitude when systems are set up with chains of equal period ratios. We use numerical simulations to describe the stability time relationship in systems with equal separations but non-equal masses which breaks the chains of equal period ratios. We find a deviation of 30 per cent in the masses of the planets creates a large enough deviation in the period ratios where the average stability time of a given spacing can be predicted by the stability time relationship. However, the distribution of stability time at a given spacing is much wider than in equal-mass systems. We find the stability time distribution is heteroscedastic with spacing -- the deviation in stability time for a given spacing increases with said spacing.
... As we show below, such analytical estimates perform poorly in the generic eccentric case where the effects of two-body MMRs are dominant (10,11). However, analytical and empirical studies agree that, while the dynamical behavior changes strongly from the two-to three-planet case (3,(12)(13)(14)(15)(16)(17)(18), three-planet systems are the simplest prototype for predictions at higher multiplicities in compact systems (10,11). ...
... First, many authors have run N-body integrations along lowdimensional cuts through the parameter space of compact orbital configurations, and fit simple functional forms to the resulting trends in instability times. For example, several studies have highlighted the steep dependence on interplanetary separation, while fixing orbits to be coplanar and initially circular, and planets to be equal mass and equally separated from one another (12,14,16,17,30). We compare the performance of the fit from the study in ref. 30, using five equally spaced Earth-mass planets (mass ratio ≈ 3 × 10 −6 ) on our random test set in Fig. 2, Top Left, with a resulting RMSE of 2.41 (we also test our model on the simulations used in ref. 30). ...
Significance
Despite over 300 y of effort, no solutions exist for predicting when a general planetary configuration will become unstable. We introduce a deep learning architecture to push forward this problem for compact systems. While current machine learning algorithms in this area rely on scientist-derived instability metrics, our new technique learns its own metrics from scratch, enabled by a internal structure inspired from dynamics theory. Our model can quickly and accurately predict instability timescales in compact multiplanet systems, and does so with an accurate uncertainty estimate for unfamiliar systems. This opens up the development of fast terrestrial planet formation models, and enables the efficient exploration of stable regions in parameter space for multiplanet systems.
... As we show below, such analytical estimates perform poorly in the generic eccentric case where the effects of 2-body MMRs are dominant (10,11). However, analytical and empirical studies agree that while the dynamical behavior changes strongly from the two to three-planet case (3,(12)(13)(14)(15)(16)(17)(18), three-planet systems are the simplest prototype for predictions at higher multiplicities in compact systems (10,11). ...
... First, many authors have run N-body integrations along lowdimensional cuts through the parameter space of compact orbital configurations, and fit simple functional forms to the resulting trends in instability times. For example, several studies have highlighted the steep dependence on interplanetary separation, while fixing orbits to be coplanar and initially circular, and planets to be equal mass and equally separated from one another (12,14,16,17,40). We compare the performance of the fit from such a study (40), using five equally spaced Earth-mass planets (mass ratio ≈ 3 × 10 −6 ) on our random (13,15,58,59), but consider equal initial eccentricities, planetary masses, etc. in order to fit simple models. ...
Despite over three hundred years of effort, no solutions exist for predicting when a general planetary configuration will become unstable. We introduce a deep learning architecture to push forward this problem for compact systems. While current machine learning algorithms in this area rely on scientist-derived instability metrics, our new technique learns its own metrics from scratch, enabled by a novel internal structure inspired from dynamics theory. Our Bayesian neural network model can accurately predict not only if, but also when a compact planetary system with three or more planets will go unstable. Our model, trained directly from short N-body time series of raw orbital elements, is more than two orders of magnitude more accurate at predicting instability times than analytical estimators, while also reducing the bias of existing machine learning algorithms by nearly a factor of three. Despite being trained on compact resonant and near-resonant three-planet configurations, the model demonstrates robust generalization to both non-resonant and higher multiplicity configurations, in the latter case outperforming models fit to that specific set of integrations. The model computes instability estimates up to five orders of magnitude faster than a numerical integrator, and unlike previous efforts provides confidence intervals on its predictions. Our inference model is publicly available in the SPOCK package, with training code open-sourced.
... Orbital eccentricities provide complementary information on the dynamical histories of planetary systems. Interactions with their natal protoplanetary discs tend to circularize exoplanet orbits (Ward 1988), so eccentricities measured at the present day constrain the degree of dynamical excitation across a given system's lifetime (Rasio & Ford 1996;Weidenschilling & Marzari 1996;Lin & Ida 1997;Marzari & Weidenschilling 2002;E-mail: dtamayo@astro.princeton.edu (DT); chrisgil@psu.edu ...
... There have been extensive efforts to predict such instabilities analytically, yielding many powerful results for two-planet systems (Wisdom 1980;Marchal & Bozis 1982;Gladman 1993; Barnes & Greenberg 2006;Deck, Payne & Holman 2013;Petit, Laskar & Boué 2017Hadden & Lithwick 2018;Hadden 2019). In the 3+ planet case, several authors have run N-body integrations with initial conditions drawn from low-dimensional cuts through the full parameter space, and fitted empirical functional forms to the resulting instability times (Chambers, Wetherill & Boss 1996;Yoshinaga, Kokubo & Makino 1999;Marzari & Weidenschilling 2002;Zhou, Lin & Sun 2007;Smith & Lissauer 2009;Funk et al. 2010;Pu & Wu 2015;Obertas, Van Laerhoven & Tamayo 2017;Gratia & Lissauer 2020). Such empirical fits provide insight into the dependencies on physical parameters and complementary analytic investigations have clarified several aspects of the underlying dynamics (Zhou et al. 2007;Quillen 2011;Yalinewich & Petrovich 2019;Petit et al. 2020). ...
Many discovered multiplanet systems are tightly packed. This implies that wide parameter ranges in masses and orbital elements can be dynamically unstable and ruled out. We present a case study of Kepler-23, a compact three-planet system where constraints from stability, transit timing variations (TTVs), and transit durations can be directly compared. We find that in this tightly packed system, stability can place upper limits on the masses and orbital eccentricities of the bodies that are comparable to or tighter than current state of the art methods. Specifically, stability places 68% upper limits on the orbital eccentricities of 0.09, 0.04, and 0.05 for planets b, c and d, respectively. These constraints correspond to radial velocity signals ≲ 20 cm/s, are significantly tighter to those from transit durations, and comparable to those from TTVs. Stability also yields 68% upper limits on the masses of planets b, c and d of 2.2, 16.1, and 5.8 M⊕, respectively, which were competitive with TTV constraints for the inner and outer planets. Performing this stability constrained characterization is computationally expensive with N-body integrations. We show that SPOCK, the Stability of Planetary Orbital Configurations Klassifier, is able to faithfully approximate the N-body results over 4000 times faster. We argue that such stability constrained characterization of compact systems is a challenging “needle-in-a-haystack” problem (requiring removal of 2500 unstable configurations for every stable one for our adopted priors) and we offer several practical recommendations for such stability analyses.
... † Both authors contributed equally to this manuscript ‡ chrisgil@psu.edu (Rasio & Ford 1996;Weidenschilling & Marzari 1996;Lin & Ida 1997;Marzari & Weidenschilling 2002;Adams & Laughlin 2003;Chatterjee et al. 2008;Jurić & Tremaine 2008;Simbulan et al. 2017). ...
... Powerful such results exist for two-planet systems (Wisdom 1980;Marchal & Bozis 1982;Gladman 1993; Barnes & Greenberg 2006;Deck et al. 2013;Petit et al. 2017;Petit et al. 2018;Hadden & Lithwick 2018;Hadden 2019). In the 3+ planet case, several authors have run N-body integrations with initial conditions drawn from low-dimensional cuts through the full parameter space, and fitted empirical functional forms to the resulting instability times (Chambers et al. 1996;Yoshinaga et al. 1999;Marzari & Weidenschilling 2002;Zhou et al. 2007;Smith & Lissauer 2009;Funk et al. 2010;Pu & Wu 2015;Obertas et al. 2017;Gratia & Lissauer 2019). Such empirical fits provide insight into the dependencies on physical parameters, and complementary analytic investigations have clarified several aspects of the underlying dynamics (Zhou et al. 2007;Quillen 2011;Yalinewich & Petrovich 2019;Petit et al. 2020). ...
Many discovered multiplanet systems are tightly packed. This implies that wide parameter ranges in masses and orbital elements can be dynamically unstable and ruled out. We present a case study of Kepler-23, a compact three-planet system where constraints from stability, transit timing variations (TTVs), and transit durations can be directly compared. We find that in this tightly packed system, stability can place upper limits on the masses and orbital eccentricities of the bodies that are comparable to or tighter than current state of the art methods. Specifically, stability places 68% upper limits on the orbital eccentricities of 0.09, 0.04, and 0.05 for planets b, c and d, respectively. These constraints correspond to radial velocity signals cm/s, are significantly tighter to those from transit durations, and comparable to those from TTVs. Stability also yields 68% upper limits on the masses of planets b, c and d of 2.2, 16.1, and 5.8 , respectively, which were competitive with TTV constraints for the inner and outer planets. Performing this stability constrained characterization is computationally expensive with N-body integrations. We show that SPOCK, the Stability of Planetary Orbital Configurations Klassifier (Tamayo et al., 2020) is able to faithfully approximate the N-body results over 4000 times faster. We argue that such stability constrained characterization of compact systems is a challenging "needle-in-a-haystack" problem (requiring removal of 2500 unstable configurations for every stable one for our adopted priors) and we offer several practical recommendations for such stability analyses.
... These final eccentricities of planets are less than 0.3e esc . The dynamical evolution of close-in super-Earths around ∼ 0.1M stars is similar to that of giant planets around 1M stars (e.g., Rasio & Ford 1996;Marzari & Weidenschilling 2002;Nagasawa et al. 2008). Strong scattering between protoplanets leads to ejection. ...
... When a planet is scattered to the inner orbit by a strong scattering associated with an ejection event, a planet is pushed into the inner orbit than the orbit of the innermost protoplanet. This dynamical process was already found in the previous studies, which considered the dynamical evolution of multiple Jupiter-sized planet systems (e.g., Marzari & Weidenschilling 2002;Nagasawa et al. 2008). Fig. 5 of Nagasawa et al. (2008) showed a similar gap structure in the distribution of the semimajor axes of planets with our results. ...
Earth-sized planets were observed in close-in orbits around M dwarfs. While more and more planets are expected to be uncovered around M dwarfs, theories of their formation and dynamical evolution are still in their infancy. We investigate the giant impact growth of protoplanets, which includes strong scattering around low-mass stars. The aim is to clarify whether strong scattering around low-mass stars affects the orbital and mass distributions of the planets. We perform N-body simulation of protoplanets by systematically surveying the parameter space of the stellar mass and surface density of protoplanets. We find that protoplanets are often ejected after twice or three times close-scattering around late M dwarfs. The ejection sets the upper limit of the largest planet mass. Adopting the surface density scaling linearly with the stellar mass, we find that as the stellar mass decreases less massive planets are formed in orbits with higher eccentricities and inclinations. Under this scaling, we also find that a few close-in protoplanets are generally ejected. The ejection of protoplanets plays an important role in the mass distribution of super-Earths around late M dwarfs. The mass relation of observed close-in super-Earths and their central star mass is well reproduced by ejection.
... But while a clearer physical picture is emerging, theoretical estimates can not yet quantitatively match the results from numerical integrations [Quillen, 2011]. Many previous numerical studies have instead presented empirical fits to the overall steep rise in instability times with interplanetary separation, recorded from large suites of numerical integrations [Chambers et al., 1996, Yoshinaga et al., 1999, Marzari and Weidenschilling, 2002, Zhou et al., 2007, Faber and Quillen, 2007, Smith and Lissauer, 2009, Matsumoto et al., 2012, Pu and Wu, 2015. This is a useful approach for elucidating the underlying dynamics and scalings with dominant parameters, but typically involves simplifications such as equal-mass, or equal-separation planets. ...
... Several previous studies have fit functional forms to instability times recorded in large suites of N-body integrations [e.g., Chambers et al., 1996, Marzari and Weidenschilling, 2002, Faber and Quillen, 2007, Smith and Lissauer, 2009, Obertas et al., 2017. They found that instability times rise steeply with increasing interplanetary separation measured in mutual Hill radii, i.e., the characteristic radius around the planets in which their gravity dominates that of the star [see also Quillen, 2011, Yalinewich andPetrovich, 2019], ...
We combine analytical understanding of resonant dynamics in two-planet systems with machine learning techniques to train a model capable of robustly classifying stability in compact multi-planet systems over long timescales of orbits. Our Stability of Planetary Orbital Configurations Klassifier (SPOCK) predicts stability using physically motivated summary statistics measured in integrations of the first orbits, thus achieving speed-ups of up to over full simulations. This computationally opens up the stability constrained characterization of multi-planet systems. Our model, trained on three-planet systems sampled at discrete resonances, generalizes both to a sample spanning a continuous period-ratio range, as well as to a large five-planet sample with qualitatively different configurations to our training dataset. Our approach significantly outperforms previous methods based on systems' angular momentum deficit, chaos indicators, and parametrized fits to numerical integrations. We use SPOCK to constrain the free eccentricities between the inner and outer pairs of planets in the Kepler-431 system of three approximately Earth-sized planets to both be below 0.05. Our stability analysis provides significantly stronger eccentricity constraints than currently achievable through either radial velocity or transit duration measurements for small planets, and within a factor of a few of systems that exhibit transit timing variations (TTVs). Given that current exoplanet detection strategies now rarely allow for strong TTV constraints (Hadden et al., 2019), SPOCK enables a powerful complementary method for precisely characterizing compact multi-planet systems. We publicly release SPOCK for community use.
... First, such a high eccentricity cannot be explained by disc migration alone (Lin et al. 1996), even though disc migration could have been important in the early stages of evolution of the system. It could have driven the planets close enough to trigger a period of dynamical instability dominated by planet-planet scattering (Weidenschilling & Marzari 1996;Chambers et al. 1996;Rasio & Ford 1996;Lin & Ida 1997;Marzari & Weidenschilling 2002;Chatterjee et al. 2008;Nagasawa et al. 2008;Raymond et al. 2009;Davies et al. 2014;Mustill et al. 2014;Petrovich et al. 2014;Deienno et al. 2018), capable of exciting the eccentricity of TOI-4914 b. Given our discussion in Sects. ...
Recent observations of giant planets have revealed unexpected bulk densities. Hot Jupiters, in particular, appear larger than expected for their masses compared to planetary evolution models, while warm Jupiters seem denser than expected. These differences are often attributed to the influence of the stellar incident flux, but it has been unclear if they also result from different planet formation processes, and if there is a trend linking the planetary density to the chemical composition of the host star. In this work, we present the confirmation of three giant planets in orbit around solar analogue stars. TOI-2714 b ( d p J , M_ p = 0.72 M_ J P 3.6, ) are hot Jupiters on nearly circular orbits, while TOI-4914 b ( d p J , M_ p = 0.72 M_ J e = 0.41 0.02 = -0.13 = -0.11 = 0.30R_ p J F_ erg cm^ $), appears to be more inflated than what is supported by current theoretical models for giant planets. Moreover, it does not conform to the previously noted trend that warm giant planets orbiting metal-poor stars have low eccentricities. This study thus provides insights into the diverse orbital characteristics and formation processes of giant exoplanets, in particular the role of stellar metallicity in the evolution of planetary systems.
... From numerical studies, system stability is both highly chaotic and dependent on meanmotion resonances (Marzari 2014;Rath et al. 2022, e.g.,). However, to first-order the time to dynamical instability grows logarithmically with mutual separation (Marzari & Weidenschilling 2002). The mutual separation is commonly parameterized by the mutual Hill radius: ...
The two planets of the HAT-P-11 system represent fascinating dynamical puzzles due to their significant eccentricities and orbital misalignments. In particular, HAT-P-11 b is on a close-in orbit that tides should have circularized well within the age of the system. Here we propose a two-step dynamical process that can reproduce all intriguing aspects of the system. We first invoke planet-planet scattering to generate significant eccentricities and mutual inclinations between the planets. We then propose that this misalignment initiated von-Zeipel-Lidov-Kozai cycles and high-eccentricity migration that ultimately brought HAT-P-11 b to its present-day orbit. We find that this scenario is fully consistent only when significant tidally-driven radius inflation is accounted for during the tidal migration. We present a suite of N-body simulations exploring each phase of evolution and show that this scenario is consistent with all observational posteriors and the reported age of the system.
... In Obertas et al. ( 2017 ), systems with adjacent or next-adjacent planets near period commensurability with first-or second-order mean-motion resonances (MMRs) have stability times that deviate from the time predicted by the spacing. Also reported in Chambers et al. ( 1996 ), Marzari & Weidenschilling ( 2002 ), and Pu & Wu ( 2015 ), the chains of period ratios near the ratio of a MMR in these multiplanet systems cause separation-dependent modulations superimposed on the stability time relationship that have amplitudes as large as an order of magnitude. Chambers et al. ( 1996 ) integrate 120 systems of 20 planets with equal spacing and masses that vary by a factor of five. ...
Compact planetary systems with more than two planets can undergo orbital crossings from planet-planet perturbations. The time which the system remains stable without orbital crossings has an exponential dependence on the initial orbital separations in units of mutual Hill radii. However when a multi-planet system has period ratios near mean-motion resonances, its stability time differs from the time determined by planet separation. This difference can be up to an order of magnitude when systems are set up with chains of equal period ratios. We use numerical simulations to describe the stability time relationship in non-resonant systems with equal separations but non-equal masses which breaks the chains of equal period ratios. We find a deviation of 30 per cent in the masses of Earth-mass planets creates a large enough deviation in the period ratios where the average stability time of a given spacing can be predicted by the stability time relationship. The mass deviation where structure from equal period ratios is erased increases with planet mass but does not depend on planet multiplicity. With a large enough mass deviation, the distribution of stability time at a given spacing is much wider than in equal-mass systems where the distribution narrows due to period commensurabilities. We find the stability time distribution is heteroscedastic with spacing—the deviation in stability time for a given spacing increases with said spacing.
... Many authors, starting with Chambers et al. (1996), have used N-body integrations to explore the dependence of instability times on the initial conditions and parameters of multiplanet systems (Yoshinaga et al. 1999;Marzari & Weidenschilling 2002;Faber & Quillen 2007;Zhou et al. 2007;Smith & Lissauer 2009;Funk et al. 2010;Pu & Wu 2015;Obertas et al. 2017;Gratia & Lissauer 2019;Lissauer & Gavino 2021). While these numerical experiments have provided insight into the dynamical mechanisms that drive these instabilities, the practical restrictions that were variously imposed on the allowed parameter space (equal spacing, equal orbital eccentricities, equal masses, etc.) lead to inaccurate instability time predictions for real systems, often by orders of magnitude (Cranmer et al. 2021). ...
We derive a semianalytic criterion for the presence of chaos in compact, eccentric multiplanet systems. Beyond a minimum semimajor axis separation, below which the dynamics are chaotic at all eccentricities, we show that (i) the onset of chaos is determined by the overlap of two-body mean motion resonances (MMRs), like it is in two-planet systems; and (ii) secular evolution causes the MMR widths to expand and contract adiabatically, so that the chaotic boundary is established where MMRs overlap at their greatest width. For closely spaced two-planet systems, a near symmetry strongly suppresses this secular modulation, explaining why the chaotic boundaries for two-planet systems are qualitatively different from cases with more than two planets. We use these results to derive an improved angular momentum deficit (AMD) stability criterion, i.e., the critical system AMD below which stability should be guaranteed. This introduces an additional factor to the expression from Laskar and Petit that is exponential in the interplanetary separations, which corrects the AMD threshold toward lower eccentricities by a factor of several for tightly packed configurations. We make routines for evaluating the chaotic boundary available to the community through the open-source SPOCK package.
... Planet-planet scattering is the best model to date for explaining the eccentricity distribution of extrasolar giant planets (Rasio & Ford 1996;Weidenschilling & Marzari 1996;Lin & Ida 1997;Marzari & Weidenschilling 2002;Adams & Laughlin 2003;Chatterjee et al. 2008;Jurić & Tremaine 2008;Raymond et al. 2010;Carrera et al. 2019;Anderson et al. 2020). In the planet-planet scattering scenario, planets are hypothesized to form in closely packed systems. ...
Planet–planet scattering best explains the eccentricity distribution of extrasolar giant planets, and past literature showed that the orbits of planets evolve due to planet–planet scattering. This work studies the spin evolution of planets in planet–planet scattering in two-planet systems. Spin can evolve dramatically due to spin–orbit coupling made possible by the evolving spin and orbital precession during the planet–planet scattering phase. The main source of torque to planet spin is the stellar torque, and the planet–planet torque contribution is negligible. As a consequence of the evolution of the spin, planets can end up with appreciable obliquities (the angle between a planet’s own orbit normal and spin axis), with the obliquity distribution peaking at about 10°, and extending to much larger values.
... En esta Sección, estudiaremos la estabilidad y la dinámica de laórbita de un satélite que orbita alrededor de un planeta, y que ve suórbita perturbada debido a la presencia de otro planeta que también orbita la estrella. Diversos autores tratan la estabilidad de un sistema planetario de 3 o más planetas (Chambers et al. 1996;Marzari & Weidenschilling 2002;Marzari 2014, etc.), pero nosotros aquí sólo estudiaremos la influencia que tiene un planeta exterior sobre un planeta interior y sobre el satélite que alberga esteúltimo. ...
En esta Tesis Doctoral, realizada por Carlos Vázquez Monzón y propuesta y dirigida por el prof. José Ángel Docobo Durántez, se hace un profundo estudio de los exoplanetas y también de los satélites que se mueven en torno a ellos, los exosatélites. En el Capítulo 1, se describen los principales métodos de detección y, en el caso de exosatélites, se proponen algunos métodos para intentar su descubrimiento. En el Capítulo 2 se trata el proceso por el cual se determina si un exoplaneta o exosatélite ha sido detectado, aplicando la teoría desarrollada a varios ejemplos. El Capítulo 3 versa sobre la dinámica y estabilidad de las órbitas de estos cuerpos, en distintos escenarios, estando el Capítulo 4 dedicado a la posible habitabilidad de los mismos.
... Many authors, starting with Chambers et al. (1996), have used N-body integrations to explore the dependence of instability times on the initial conditions and parameters of multiplanet systems (Yoshinaga et al. 1999;Marzari & Weidenschilling 2002;Faber & Quillen 2007;Zhou et al. 2007;Smith & Lissauer 2009;Funk et al. 2010;Pu & Wu 2015;Obertas et al. 2017;Gratia & Lissauer 2019;Lissauer & Gavino 2021). While these numerical experiments have provided insight into the dynamical mechanisms that drive these instabilities, the practical restrictions that were variously imposed on the allowed parameter space (e.g., equal spacing, equal orbital eccentricities, equal masses, etc.) lead to inaccurate instability time predictions for real systems, often by orders of magnitude (Cranmer et al. 2021). ...
We derive a semi-analytic criterion for the presence of chaos in compact, eccentric multiplanet systems. Beyond a minimum semimajor-axis separation, below which the dynamics are chaotic at all eccentricities, we show that (i) the onset of chaos is determined by the overlap of two-body mean motion resonances (MMRs), like it is in two-planet systems; (ii) secular evolution causes the MMR widths to expand and contract adiabatically, so that the chaotic boundary is established where MMRs overlap at their greatest width. For closely spaced two-planet systems, a near-symmetry strongly suppresses this secular modulation, explaining why the long-term stability of two-planet systems is qualitatively different from cases with more than two planets. We use these results to derive an improved angular-momentum-deficit (AMD) stability criterion, i.e., the critical system AMD below which stability should be guaranteed. This introduces an additional factor to the expression from Laskar and Petit (2017) that is exponential in the interplanetary separations, which corrects the AMD threshold toward lower eccentricities by a factor of several for tightly packed configurations. We make routines for evaluating the chaotic boundary available to the community through the open-source SPOCK package.
... The stability of systems of two planets has been characterized analytically (Hill 1878a,b,c;Gladman 1993). In contrast, analytic studies have not provided as comprehensive a solution to delineating stability boundaries for systems with more than two planets, so such systems have been studied extensively using numerical integrations (e.g., Chambers et al. (1996), Marzari & Weidenschilling (2002), Smith & Lissauer (2009), Pu & Wu (2015), Morrison & Kratter (2016), Tamayo et al. (2016), Obertas et al. (2017)). Lissauer (1995) examined the spacing of planets and moons within the Solar System and proposed approximate criteria for global stability of more than two planets based upon resonance overlapping and the Hill/Jacobi exclusion zone. ...
We explore the orbital dynamics of systems consisting of three planets, each as massive as the Earth, on coplanar, initially circular, orbits about a star of one solar mass. The initial semimajor axes of the planets are equally spaced in terms of their mutual Hill radius, which is equivalent to a geometric progression of orbital periods for small planets of equal mass. Our simulations explore a wide range of spacings of the planets, and were integrated for virtual times of up to 10 billion years or until the orbits of any pair of planets crossed. We find the same general trend of system lifetimes increasing exponentially with separation between orbits seen by previous studies of systems of three or more planets. One focus of this paper is to go beyond the rough trends found by previous numerical studies and quantitatively explore the nature of the scatter in lifetimes and the destabilizing effects of mean motion resonances. In contrast to previous results for five-planet systems, a nontrivial fraction of three-planet systems survive at least several orders of magnitude longer than most other systems with similar initial separation between orbits, with some surviving years at much smaller orbital separations than any found for five-planet systems. Substantial shifts in the initial planetary longitudes cause a scatter of roughly a factor of two in system lifetime, whereas the shift of one planet's initial position by 100 meters along its orbit results in smaller changes in the logarithm of the time to orbit crossing, especially for systems with short lifetimes.
... Planet-planet scattering is the best model to date for explaining the eccentricity distribution of extrasolar giant planets (Rasio & Ford 1996;Weidenschilling & Marzari 1996;Lin & Ida 1997;Marzari & Weidenschilling 2002;Adams & Laughlin 2003;Chatterjee et al. 2008;Jurić & Tremaine 2008;Raymond et al. 2010;Carrera et al. 2019;Anderson et al. 2020). In the planet-planet scattering scenario, planets are hypothesized to form in closely packed systems. ...
Planet-planet scattering best explains the eccentricity distribution of extrasolar giant planets. Past literature showed that the orbits of planets evolve due to planet-planet scattering. This work studies the spin evolution of planets in planet-planet scattering in 2-planet systems. Spin can evolve dramatically due to spin-orbit coupling made possible by the evolving spin and orbital precession during the planet-planet scattering phase. The main source of torque to planet spin is the stellar torque, and the total planet-plane torque contribution is negligible. As a consequence of the evolution of the spin, planets can end up with significant obliquity (the angle between a planet's own orbit normal and spin axis) like planets in our Solar System.
... Integrating the migration process into a consistent model of planets forma-tion is an extremely active and fast evolving field of research (a recent review of this important subject can be find in Raymond & Morbidelli 2020). In the cases of the HJs, two migration mechanisms are accepted now as most probable (Dawson & Johnson 2018):(1) disk migration, where the planet forms beyond the ice-line then migrates inward by loosing its orbit angular momentum to the PPD (see thorough reviews in Baruteau et al. 2014;Armitage 2020), and (2) high-eccentricity migration, according to which the planet first gains a high eccentricity through interactions with other planets, which makes it to pass very close to its star where it looses its orbit angular momentum by tidal interactions (this is a more complicated process, involving different mechanisms; e.g., Rasio & Ford 1996;Weidenschilling & Marzari 1996;Marzari & Weidenschilling 2002;Chatterjee et al. 2008;Nagasawa et al. 2008;Beaugé & Nesvorný 2012). However, what is not clear in these two models is what importance must be put on the characteristics of the PPD, its mass, size, depth and composition? ...
In search for a connection between the formation of stars and the formation of planets, a new semi-automatic spectral analysis method using \textsf{iSpec} was developed for the TIGRE telescope installed in Guanajuato, Mexico. TIGRE is a 1.2m robotic telescope, equipped with an Echelle spectrograph (HEROS), with a resolution R . \textsf{iSpec} is a synthetic spectral fitting program for stars that allows to determine in an homogeneous way their fundamental parameters: effective temperature, , surface gravity, , metallicities, [M/H] and [Fe/H], and rotational velocity, . In this first article we test our method by analysing the spectra of 46 stars, host of exoplanets, obtained with the TIGRE.
... In the literature, two migration mechanisms are favored for HJs (Dawson & Johnson 2018;Raymond & Morbidelli 2020). The first is disk migration (e.g., Baruteau et al. 2014;Armitage 2020, and references therein), which proposes that a planet looses its orbit angular momentum by tidal interactions with the PPD, while the second, high-eccentricity migration (e.g., Rasio & Ford 1996;Weidenschilling & Marzari 1996;Marzari & Weidenschilling 2002;Chatterjee et al. 2008;Nagasawa et al. 2008;Beaugé & Nesvorný 2012), suggests a planet interacting with other planets gains a higher eccentricity, which brings it close to its star where it reaches equilibrium by tidal interactions (a process know as circularization). In terms of migration, these two mechanisms might suggest massive disks somehow amplified the level of migration compared to what happened in the solar system, because more massive PPDs either increase the intensity of interactions of the planets with their disks or favor the formation of a higher number of planets. ...
A sample of 46 stars, host of exoplanets, is used to search for a connection between their formation process and the formation of the planets rotating around them. Separating our sample in two, stars hosting high-mass exoplanets (HMEs) and low-mass exoplanets (LMEs), we found the former to be more massive and to rotate faster than the latter. We also found the HMEs to have higher orbital angular momentum than the LMEs and to have lost more angular momentum through migration. These results are consistent with the view that the more massive the star and higher its rotation, the more massive was its protoplanetarys disk and rotation, and the more efficient the extraction of angular momentum from the planets.
... is the so-called mutual Hill radius of the planets (Marzari & Weidenschilling 2002). This expression for ∆ min is equal to the minimum distance between planets in circular orbits leading to Hill stability, as deduced originally by Marchal & Saari (1975) and Marchal & Bozis (1982), and later made commonly known by Gladman (1993). ...
Context. Planetary resonances are a common dynamical mechanism acting on planetary systems. However, no general model for describing their properties exists, particularly for commensurabilities of any order and arbitrary eccentricity and inclination values.
Aims. We present a semianalytical model that describes the resonance strength, width, location and stability of fixed points, and periods of small-amplitude librations. The model is valid for any two gravitationally interacting massive bodies, and is thus applicable to planets around single or binary stars.
Methods. Using a theoretical framework in the Poincaré and Jacobi reference system, we developed a semianalytical method that employs a numerical evaluation of the averaged resonant disturbing function. Validations of the model are presented that compare its predictions with dynamical maps for real and fictitious systems.
Results. The model describes many dynamical features of planetary resonances very well. Notwithstanding the good agreement found in all cases, a small deviation is noted in the location of the resonance centers for circumbinary systems. As a consequence of its application to the HD 31527 system, we found that the updated best-fit solution leads to a high-eccentricity stable libration between the middle and outer planets inside the 16/3 mean-motion resonance (MMR). This is the first planetary system whose long-term dynamics appears dominated by such a high-order commensurability. In the case of circumbinary planets, the overlap of N/1 mean-motion resonances coincides very well with the size of the global chaotic region close to the binary, as well as its dependence on the mutual inclination.
... is the so-called mutual Hill radius of the planets (Marzari & Weidenschilling 2002). This expression for ∆ min is equal to the minimum distance between planets in circular orbits leading to Hill-stability, as deduced originally by Marchal & Saari (1975) and Marchal & Bozis (1982), and later popularized by Gladman (1993). ...
In spite of planetary resonances being a common dynamical mechanism acting on planetary systems, no general model exists for describing their properties, particularly for commensurabilities of any order and arbitrary values of the eccentricities and inclinations. The present work presents a semi-analytical model that describes the resonance strength, width, location and stability of fixed points, as well as periods of small-amplitude librations. The model is valid to any two gravitationally interacting massive bodies, and thus applicable to planets around single or binary stars. Using a theoretical framework in Poincar\'e and Jacobi reference system we develop a semi-analytical method that employs a numerical evaluation of the averaged resonant disturbing function. Validations of the model are presented comparing its predictions with dynamical maps for real and fictitious systems. The model is shown to describe very well many dynamical features of planetary resonances. Notwithstanding the good agreements found in all cases, a small deviation is noted in the location of the resonance centers for circumbinary systems. As a consequence of its application to the HD31527 system we have found that the updated best-fit solution leads to a high-eccentricity stable libration between the middle and outer planets inside the 16/3 mean-motion resonance. This is the first planetary system whose long-term dynamics appears dominated by such a high-order commensurability. In the case of circumbinary planets, the overlap of N/1 mean-motion resonances coincides very well with the size of the global chaotic region close to the binary, as well as its dependence with the mutual inclination.
... Much effort has been spent on faster prediction of instability times drawing from suites of Nbody integrations, both through parametrized empirical fits (e.g. Chambers, Wetherill & Boss 1996;Yoshinaga, Kokubo & Makino 1999;Marzari & Weidenschilling 2002;Zhou, Lin & Sun 2007;Faber & Quillen 2007;Smith & Lissauer 2009;Funk et al. 2010;Pu & Wu 2015;Obertas, Van Laerhoven & Tamayo 2017;Wu et al. 2019), analytically (Quillen 2011), and through machine learning techniques (Tamayo et al. 2016;Lam & Kipping 2018). ...
Instabilities in compact planetary systems are generically driven by chaotic dynamics. This implies that an instability time measured through direct N-body integration is not exact, but rather represents a single draw from a distribution of equally valid chaotic trajectories. In order to characterize the ‘errors’ on reported instability times from direct N-body integrations, we investigate the shape and parameters of the instability time distributions (ITDs) for ensembles of shadow trajectories that are initially perturbed from one another near machine precision. We find that in the limit where instability times are long compared to the Lyapunov (chaotic) time-scale, ITDs approach remarkably similar lognormal distributions with standard deviations ≈0.43 ± 0.16 dex, despite the instability times varying across our sample from 104 to 108 orbits. We find excellent agreement between these predictions, derived from ≈450 closely packed configurations of three planets, and a much wider validation set of integrations, as well as on previously published integrations of tightly packed five-planet systems, and a seven-planet resonant chain based on TRAPPIST-1, despite their instability time-scales extending beyond our analysed time-scale. We also test the boundary of applicability of our results on dynamically excited versions of our Solar system. These distributions define the fundamental limit imposed by chaos on the predictability of instability times in such planetary systems. It provides a quantitative estimate of the instrinsic error on an N-body instability time imprinted by chaos, approximately a factor of 3 in either direction.
... al. solely on warm-Jupiters with jovian companions. Dong et al. (2014) suggest that their results indicate that planet-planet interactions are not the dominating mechanism for creating shortperiod jovian planets, as opposed to the suggestions by several other studies (Rasio & Ford 1996;Marzari & Weidenschilling 2002;Chatterjee et al. 2008;Nagasawa et al. 2008). ...
The orbit eccentricities of the Solar System planets are unusually low compared to the average of known exoplanetary systems. A power law correlation has previously been found between the multiplicity of a planetary system and the orbital eccentricities of its components, for systems with multiplicities above two. In this study we investigate the correlation for an expanded data sample, by focusing on planetary systems as units (unlike previous studies that have focused on individual planets). Our full data sample contains 1171 exoplanets, in 895 systems, and the correlation between eccentricity and multiplicity is found to follow a clear power law for all multiplicities above one. We discuss the correlation for several individual subsamples, and find that all samples consistently follow the same basic trend regardless of e.g. planet types and detection methods. We find that the eccentricities of the Solar System fit the general trend and suggest that the Solar System might not show uncommonly low eccentricities (as often speculated) but rather uncommonly many planets compared to a "standard" planetary system. The only outlier from the power law correlation is, consistently in all the samples, the one-planet systems. It has previously been suggested that this may be due to additional unseen exoplanets in the observed one-planet systems. Based on this assumption and the power law correlation, we estimate that the probability of a system having 8 planets or more is of the order of 1%, in good agreement with recent predictions from analyses based on independent arguments.
... On the other hand, dust clusters of size more than 10 Earth-mass can be shot up to 0.4AU by one Jupiter (pure three-body system) and 1 to 100 AU by two Jupiters (pure four-body system). These shot planets would migrate toward the center but shot again outward; the planets are always dynamical objects regulated by the outer edge of the inner void (Ford 2001;Marzari 2002;Nagasawa 2008Nagasawa , 2011. ...
Planets are common objects in the Universe, observationally as well as theoretically. However, the standard theory of their formation encounters many difficulties, such as dust fall and disk lifetime problems. We positively analyze them, expecting that those problems as a whole may indicate some consistent effective model. Thus we propose a dynamical model of the planet formation based on the assumption that the inner void of gas is commonly formed in the disk without specifying any Physical origin. The basic processes of this model are the dust fall, the accumulation, and the slingshots. The dust in the protoplanetary disk rapidly falls as it grows to the meter size. Then, all of them stops at the outer edge of the void where the gas friction disappears. Such dust clusters rapidly coalesce with each other and easily cause the runaway in the dense and coherent environment. Then the huge clusters are formed there and they are the first generation planets Hot-Jupiters. They immediately slingshot smaller clusters around them towards the outer regions. They are Rockey-Planets, Cold-Gas-Giants, Ice-Giants, and `Trans Neptunian objects including `Kuiper belt/Oort cloud objects`, depending on the original core mass or the distance blown. Combining numerical calculations of the slingshots and coagulation equations, we obtain the planet population diagram, including the possibility of the massive thermal metamorphosis, the origin of the variety of planetary systems, and the possibility of stray planets/objects.
... However, while a clearer physical picture is emerging, theoretical estimates cannot yet quantitatively match the results from numerical integrations (22). Many previous numerical studies have instead presented empirical fits to the overall steep rise in instability times with interplanetary separation, recorded from large suites of numerical integrations (21,(24)(25)(26)(27)(28)(29)(30). This is a useful approach for elucidating the underlying dynamics and scalings with dominant parameters but typically involves simplifications such as equal-mass, or equal-separation planets. ...
Significance
Observations of planets beyond our solar system (exoplanets) yield uncertain orbital parameters. Particularly in compact multiplanet systems, a significant fraction of observationally inferred orbital configurations can lead to planetary collisions on timescales that are short compared with the age of the system. Rejection of these unphysical solutions can thus sharpen our view of exoplanetary orbital architectures. Long-term stability determination is currently performed through direct orbital integrations. However, this approach is computationally prohibitive for application to the full exoplanet sample. By speeding up this process by up to five orders of magnitude, we enable precise exoplanet characterization of compact multiplanet systems and our ability to examine the stability properties of the multiplanet exoplanet sample as a whole.
... Thousands of exoplanets have been confirmed, showing diverse orbital features distinct from our own Solar System. Many are characterised by high eccentricities (Udry & Santos 2007), suggestive of ubiquitous strong planetary scattering (Rasio & Ford 1996) during which close encounters must be not uncommon (Marzari & Weidenschilling 2002). ...
Single-binary scattering may lead to an exchange where the single object captures a component of the binary, forming a new binary. This has been well studied in encounters between a star--planet pair and a single star. Here we explore the application of the exchange mechanism to a planet--satellite pair and another planet in the gravitational potential of a central star. As a case study, we focus on encounters between a satellite-bearing object and Neptune. We investigate whether Neptune can capture satellites from that object and if the captured satellites have orbits analogous to the Neptunian moons Triton and Nereid. Using N-body simulations, we study the capture probability at different encounter distances. Post-capture, we use a simple analytical argument to estimate how the captured orbits evolve under collisional and tidal effects. We find that the average capture probability reaches if Neptune penetrates the donor planet's satellite system. Most moons grabbed by Neptune acquire highly eccentric orbits. Post-capture, around half of those captured, especially those on tight orbits, can be circularised, either by tides only or by collisions+tides, turning into Triton-like objects. Captures further out, on the other hand, stay on wide and eccentric orbits like that of Nereid. Both moon types can be captured in the same encounter and they have wide distributions in orbital inclination. Therefore, Triton naturally has a 50\% chance of being retrograde. A similar process potentially applies to an exoplanetary system, and our model predicts that exomoons can jump from one planet to another during planetary scattering. Specifically, there should be two distinct populations of captured moons: one on close-in circular orbits and the other on far-out eccentric orbits. The two populations may have highly inclined prograde or retrograde orbits.
... Thousands of exoplanets have been confirmed, showing diverse orbital features distinct from our own Solar System. Many are characterised by high eccentricities (Udry & Santos 2007), suggestive of ubiquitous strong planetary scattering (Rasio & Ford 1996) during which close encounters must be not uncommon (Marzari & Weidenschilling 2002). ...
Context. Single-binary scattering may lead to an exchange where the single object captures a component of the binary, forming a new binary. This has been well studied in encounters between a star–planet pair and a single star.
Aims. Here we explore the application of the exchange mechanism to a planet–satellite pair and another planet in the gravitational potential of a central star. As a case study, we focus on encounters between a satellite-bearing object and Neptune. We investigate whether Neptune can capture satellites from that object and if the captured satellites have orbits analogous to the Neptunian moons Triton and Nereid.
Methods. Using N -body simulations, we study the capture probability at different encounter distances. Post-capture, we use a simple analytical argument to estimate how the captured orbits evolve under collisional and tidal effects.
Results. We find that the average capture probability reaches ~10% if Neptune penetrates the donor planet’s satellite system. Most moons grabbed by Neptune acquire highly eccentric orbits. Post-capture, around half of those captured, especially those on tight orbits, can be circularised, either by tides only or by collisions+tides, turning into Triton-like objects. Captures further out, on the other hand, stay on wide and eccentric orbits like that of Nereid. Both moon types can be captured in the same encounter and they have wide distributions in orbital inclination. Therefore, Triton naturally has a ~50% chance of being retrograde.
Conclusions. A similar process potentially applies to an exoplanetary system, and our model predicts that exomoons can jump from one planet to another during planetary scattering. Specifically, there should be two distinct populations of captured moons: one on close-in circular orbits and the other on far-out eccentric orbits. The two populations may have highly inclined prograde or retrograde orbits.
... Moreover, as shown by Izidoro et al. (2017) and Ogihara et al. (2018), the planet pair captured into mean-motion resonance can be unlocked by the onset of dynamical instability after the dispersal of the gaseous disk. For the onset of dynamical instability, the separation between the planet pair, which is a consequence of the radial migration, is essential (e.g., Chambers et al. 1996;Marzari & Weidenschilling 2002;Wu et al. 2019). When the transition from convergent to divergent evolution occurs, the stability of the system would be significantly changed. ...
When two planets are born in a protoplanetary disk, they may enter into mean-motion resonance as a consequence of convergent planetary migration. The formation of mean-motion resonances is important for understanding how planetary systems are shaped in disk environments. Motivated by recent progress in the comprehension of the migration of partial gap-opening planets, we have investigated the orbital evolution of planet pairs in a wide range of masses and disk properties with the aim to find out when resonance capture is likely to happen. Using the formula for the migration timescale of a gap-opening planet developed in our previous work, we have derived a simple criterion that allows us to predict when the migration will be convergent (divergent). Further, we have verified the criterion using two-dimensional hydrodynamic simulations. We have found that the resonant pair of planets formed at the early phase of evolution can depart from resonance at later times because the migration speed of the outer planet slows down due to gap formation. Moreover, adopting our formula for the migration timescale, we have also carried out three-body simulations, which confirm the results of hydrodynamic simulations. Finally, we have compared our predictions with observations, selecting a sample of known two-planet systems.
... Moreover, as shown by Izidoro et al. (2017) and Ogihara et al. (2018), the planet pair captured into the mean-motion resonance can be unlocked by the onset of dynamical instability after the dispersal of the gaseous disk. For the onset of the dynamical instability, the separation between the planet pair, which is a consequence of the radial migration, is essential (e.g., Chambers et al. 1996;Marzari & Weidenschilling 2002;Wu et al. 2019). When the transition from the convergent to the divergent evolutions occurs, the stability of the system would be significantly changed. ...
When two planets are born in a protoplanetary disk, they may enter into a mean-motion resonance as a consequence of the convergent planetary migration. The formation of mean-motion resonances is important for understanding how the planetary systems are shaped in the disk environments. Motivated by recent progress in the comprehension of the migration of partial gap-opening planets, we have investigated the orbital evolution of the planet pairs in a wide range of masses and disk properties with the aim to find out when the resonance capture is likely to happen. Using the formula for the migration timescale of the gap-opening planet developed in our previous work, we have derived a simple criterion which allows us to predict when the migration will be convergent (divergent). Further, we have verified the criterion using two-dimensional hydrodynamic simulations. We have found that the resonant pair of planets formed at the early phase of the evolution, can depart from the resonance at later times because the migration speed of the outer planet slows down due to the gap formation. Moreover, adopting our formula of the migration timescale, we have also carried out three-body simulations, which confirm the results of hydrodynamic simulations. Finally, we have compared our predictions with the observations, selecting a sample of known two-planet systems.
... The works of Chambers (2001) and Laskar & Petit (2017) showed how chaotic diffusion and dynamical friction between planetary bodies lead to an increase of their dynamical excitation and, consequently, their AMD particularly during the early stages of the formation and evolution of a planetary system. Mutual collisions between excited planetary bodies (Chambers 2001;Laskar & Petit 2017) and their removal from the system by ejections (e.g., by planet-planet scattering events; Weidenschilling & Marzari 1996;Rasio & Ford 1996;Marzari & Weidenschilling 2002;Chatterjee et al. 2008) and collisions with the host star (Chambers 2001) act instead to reduce the AMD and stabilize the planetary system. ...
Context. Population studies of the orbital characteristics of exoplanets in multi-planet systems have highlighted the existence of an anticorrelation between the average orbital eccentricity of planets and the number of planets of their host system, that is, its multiplicity. This effect was proposed to reflect the varying levels of violence in the dynamical evolution of planetary systems.
Aims. Previous work suggested that the relative violence of the dynamical evolution of planetary systems with similar orbital architectures can be compared through the computation of their angular momentum deficit (AMD). We investigated the possibility of using a more general metric to perform analogous comparisons between planetary systems with different orbital architectures.
Methods. We considered a modified version of the AMD, the normalized angular momentum deficit (NAMD), and used it to study a sample of 99 multi-planet systems containing both the currently best-characterized extrasolar systems and the solar system, that is, planetary systems with both compact and wide orbital architectures.
Results. We verified that the NAMD allows us to compare the violence of the dynamical histories of multi-planet systems with different orbital architectures. We identified an anticorrelation between the NAMD and the multiplicity of the planetary systems, of which the previously observed eccentricity–multiplicity anticorrelation is a reflection.
Conclusions. Our results seem to indicate that phases of dynamical instabilities and chaotic evolution are not uncommon among planetary systems. They also suggest that the efficiency of the planetary formation process in producing high-multiplicity systems is likely to be higher than that suggested by their currently known population.
... The works of Chambers (2001) and Laskar & Petit (2017) showed how chaotic diffusion and dynamical friction between planetary bodies lead to an increase of their dynamical excitation and, consequently, their AMD particularly during the early stages of the formation and evolution of a planetary system. Mutual collisions between excited planetary bodies (Chambers, 2001;Laskar & Petit, 2017) and their removal from the system by ejections (e.g., by planet-planet scattering events; Weidenschilling & Marzari 1996;Rasio & Ford 1996;Marzari & Weidenschilling 2002;Chatterjee et al. 2008) and collisions with the host star (Chambers, 2001) act instead to reduce the AMD and stabilize the planetary system. ...
Population studies of the orbital characteristics of exoplanets in multi-planet systems highlighted the existence of an anti-correlation between the average orbital eccentricity of planets and the number of planets of their host system (i.e. its multiplicity). This effect was proposed to reflect the different violence of the dynamical evolution of the planetary systems. Previous work suggested that the relative violence of the dynamical evolution of planetary systems with similar orbital architectures can be compared through the computation of their angular momentum deficit (AMD). We investigated the possibility of using a more general metric to perform analogous comparisons between planetary systems with different orbital architectures. We considered a modified version of the AMD, the normalized angular momentum deficit (NAMD), and used it to study a sample of 99 multi-planet systems containing both the currently best-characterized extrasolar systems and the Solar System, i.e. planetary systems with both compact and wide orbital architectures. We verified that the NAMD allows to compare the violence of the dynamical histories of multi-planet systems with different orbital architectures. We identified an anti-correlation between the NAMD and the multiplicity of the planetary systems, of which the previously observed eccentricity--multiplicity anti--correlation is a reflection. Our results seem to indicate that phases of dynamical instabilities and chaotic evolution are not uncommon among planetary systems. They also suggest that the efficiency of the planetary formation process in producing high-multiplicity systems is likely higher that suggested by their currently known population.
Drastic changes in protoplanets’ orbits could occur in the early stages of planetary systems through interactions with other planets and their surrounding protoplanetary or debris discs. The resulting planetary system could exhibit orbits with moderate to high eccentricities and/or inclinations, causing planets to perturb one another as well as the disc significantly. The present work studies the evolution of systems composed of an initially inclined planet and a debris disc. We perform N-body simulations of a narrow, self-gravitating debris disc and a single interior Neptune-like planet. We simulate systems with various initial planetary inclinations, from coplanar to polar configurations considering different separations between the planet and the disc. We find that except when the planet is initially on a polar orbit, the planet-disc system tends to reach a quasi-coplanar configuration with low vertical dispersion in the disc. When present, the Zeipel–Kozai–Lidov oscillations induced by the disc pump the planet’s eccentricity and, in turn, affect the disc structure. We also find that the resulting disc morphology in most of the simulations looks very similar in both radial and vertical directions once the simulations are converged. This contrasts strongly with massless disc simulations, where vertical disc dispersion is set by the initial disc-planet inclination and can be high for initially highly inclined planets. The results suggest caution in interpreting an unseen planet’s dynamical history based only on the disc’s appearance.
Hot Jupiters (HJs) are giant planets with orbital periods of the order of a few days with semimajor axis within ∼0.1 au. Several theories have been invoked in order to explain the origin of this type of planets, one of them being the high-eccentricity migration. This migration can occur through different high-eccentricity mechanisms. Our investigation focused on six different kinds of high-eccentricity mechanisms, namely, direct dispersion, coplanar, Kozai-Lidov, secular chaos, E1 and E2 mechanisms. We investigated the effciency of these mechanisms for the production of HJ candidates in multi-planet systems initially tightly-packed in the semimajor axis, considering a large set of numerical simulations of the exact equations of motion in the context of the N-body problem. In particular, we analyzed the sensitivity of our results to the initial number of planets, the initial semimajor axis of the innermost planetary orbit, the initial conffguration of planetary masses, and to the inclusion of general relativity effects. We found that the E1 mechanism is the most effcient in producing HJ candidates both in simulations with and without the contribution of general relativity, followed by the Kozai-Lidov and E2 mechanisms. Our results also revealed that, except for the initial equal planetary mass conffguration, the E1 mechanism was notably effcient in the other initial planetary mass conffgurations considered in this work. Finally, we investigated the production of HJ candidates with prograde, retrograde, and alternating orbits. According to our statistical analysis, the Kozai-Lidov mechanism has the highest probability of significantly exciting the orbital inclinations of the HJ candidates.
Models of planet formation are built on underlying physical processes. In order to make sense of the origin of the planets we must first understand the origin of their building blocks. This review comes in two parts. The first part presents a detailed description of six key mechanisms of planet formation:
The structure and evolution of protoplanetary disks
The formation of planetesimals
Accretion of protoplanets
Orbital migration of growing planets
Gas accretion and giant planet migration
Resonance trapping during planet migration.
While this is not a comprehensive list, it includes processes for which our understanding has changed in recent years or for which key uncertainties remain. The second part of this review shows how global models are built out of planet formation processes. We present global models to explain different populations of known planetary systems, including close-in small/low-mass planets (i.e., super-Earths), giant exoplanets, and the Solar System’s planets. We discuss the different sources of water on rocky exoplanets, and use cosmochemical measurements to constrain the origin of Earth’s water. We point out the successes and failings of different models and how they may be falsified. Finally, we lay out a path for the future trajectory of planet formation studies.
Planet-Planet (P–P) scattering is a leading dynamical mechanism invoked to explain the present orbital distribution of exoplanets. Many stars belong to binary systems, therefore it is important to understand how this mechanism works in presence of a companion star. We focus on systems of three planets orbiting the primary star and estimate the timescale for instability finding that it scales with the keplerian period for systems that have the same ratio between inner planet and binary semi–major axes. An empirical formula is also derived from simulations to estimate how the the binary eccentricity affects the extent of the stability region. The presence of the secondary star affects the P–P scattering outcomes causing a broadening of the final distribution in semi–major axis of the inner planet as some of the orbital energy of the planets is absorbed by the companion star. Repeated approaches to the secondary star causes also a significant reduction in the frequency of surviving two–planet systems in particular for larger values of the inner planet semi–major axis. The formation of Kozai states with the companion star increases the number of planets which may be tidally circularized. To predict the possible final distribution of planets in binaries we have performed a large number of simulations where the initial semi–major axis of the inner planets is chosen randomly. For small values of the binary semi–major axis, the higher frequency of collision alter the final planet orbital distributions which, however, beyond 50 au appear to be scalable to wider binary separations.
The present work consists of a study of the dynamical stability of a three-body system that takes advantage of the Shannon entropy
approach to estimate the diffusivity (D_S ) in a Delaunay’s action-like phase space. We outline the main features of a numerical computation
of D_S from the solutions of the equations of motion and, thereupon, we consider how to estimate a macroscopic instability
timescale, τ_inst, (roughly speaking, the lifetime of the system) associated with a given set of initial conditions. Through such estimates, we are able to characterize the system’s space of initial conditions in terms of its orbital stability by applying numerical integrations to the construction of dynamical maps. We compare these measures of chaotic diffusion with other indicators, first in a qualitative fashion and then more quantitatively, by means of long direct integrations. We address an analysis of a particular, near-resonant system, namely HD 181433, and we show that the entropy may provide a complementary analysis with regard to other dynamical indicators. This work is part of a series of studies devoted to presenting the Shannon entropy approach and its possibilities as a numerical tool providing information on chaotic diffusion and the dynamical stability of multidimensional dynamical systems.
We explore the orbital dynamics of systems consisting of three planets, each as massive as the Earth, on coplanar, initially circular, orbits about a star of one solar mass. The initial semimajor axes of the planets are equally spaced in terms of their mutual Hill radius, which is equivalent to a geometric progression of orbital periods for small planets of equal mass. Our simulations explore a wide range of spacings of the planets, and were integrated for virtual times of up to 10 billion years or until the orbits of any pair of planets crossed. We find the same general trend of system lifetimes increasing exponentially with separation between orbits seen by previous studies of systems of three or more planets. One focus of this paper is to go beyond the rough trends found by previous numerical studies and quantitatively explore the nature of the scatter in lifetimes and the destabilizing effects of mean motion resonances. In contrast to previous results for five-planet systems, a nontrivial fraction of three-planet systems survive at least several orders of magnitude longer than most other systems with similar initial separation between orbits, with some surviving 10¹⁰ years at much smaller orbital separations than any found for five-planet systems. Substantial shifts in the initial planetary longitudes cause a scatter of roughly a factor of two in system lifetime, whereas the shift of one planet's initial position by 100 m along its orbit results in smaller changes in the logarithm of the time to orbit crossing, especially for systems with short lifetimes.
Aims. The connection between initial disc conditions and final orbital and physical properties of planets is not well-understood. In this paper, we numerically study the formation of planetary systems via pebble accretion and investigate the effects of disc properties such as masses, dissipation timescales, and metallicities on planet formation outcomes.
Methods. We improved the N -body code SyMBA that was modified for our Paper I by taking account of new planet–disc interaction models and type II migration. We adopted the ‘two- α ’ disc model to mimic the effects of both the standard disc turbulence and the mass accretion driven by the magnetic disc wind.
Results. We successfully reproduced the overall distribution trends of semi-major axes, eccentricities, and planetary masses of extrasolar giant planets. There are two types of giant planet formation trends, depending on whether or not the disc’s dissipation timescales are comparable to the planet formation timescales. When planet formation happens fast enough, giant planets are fully grown (Jupiter mass or higher) and are distributed widely across the disc. On the other hand, when planet formation is limited by the disc’s dissipation, discs generally form low-mass cold Jupiters. Our simulations also naturally explain why hot Jupiters (HJs) tend to be alone and how the observed eccentricity-metallicity trends arise. The low-metallicity discs tend to form nearly circular and coplanar HJs in situ, because planet formation is slower than high-metallicity discs, and thus protoplanetary cores migrate significantly before gas accretion. The high-metallicity discs, on the other hand, generate HJs in situ or via tidal circularisation of eccentric orbits. Both pathways usually involve dynamical instabilities, and thus HJs tend to have broader eccentricity and inclination distributions. When giant planets with very wide orbits (“super-cold Jupiters”) are formed via pebble accretion followed by scattering, we predict that they belong to metal-rich stars, have eccentric orbits, and tend to have (~80%) companions interior to their orbits.
We study the effects of general relativistic gravity on the Hill stability, that is, the stability of a multibody system against a close approach of one orbit to another, which has been hitherto studied mainly in Newtonian mechanics and applied to planetary systems. We focus in this paper on the three-body problem and extend the Newtonian analyses to the general relativistic regime in the post-Newtonian approximation. The approximate sufficient condition for the relativistic Hill stability of three-body systems is derived analytically and its validity and usefulness are confirmed numerically. In fact, relativity makes the system more unstable than Newtonian mechanics in the sense of the Hill stability as expected by our theoretical prediction. The criterion will be useful to analyze the results of large-scale N-body simulations of dense environments, in which the stability of three-body subsystems is important.
The orbit eccentricities of the Solar system planets are unusually low compared to the average of known exoplanetary systems. A power-law correlation has previously been found between the multiplicity of a planetary system and the orbital eccentricities of its components, for systems with multiplicities above two. In this study we investigate the correlation for an expanded data sample by focusing on planetary systems as units (unlike previous studies that have focused on individual planets). Our full data sample contains 1171 exoplanets, in 895 systems, and the correlation between eccentricity and multiplicity is found to follow a clear power law for all multiplicities above one. We discuss the correlation for several individual subsamples and find that all samples consistently follow the same basic trend regardless of e.g. planet types and detection methods. We find that the eccentricities of the Solar system fit the general trend and suggest that the Solar system might not show uncommonly low eccentricities (as often speculated) but rather uncommonly many planets compared to a ‘standard’ planetary system. The only outlier from the power-law correlation is, consistently in all the samples, the one-planet systems. It has previously been suggested that this may be due to additional unseen exoplanets in the observed one-planet systems. Based on this assumption and the power-law correlation, we estimate that the probability of a system having eight planets or more is of the order of 1 per cent, in good agreement with recent predictions from analyses based on independent arguments.
Observations of exoplanets have revealed that systems with planets on closely-spaced orbits are common, which motivates the question “How closely can planets orbit to one another and still be dynamically-stable for very long times?”. To address this question, we investigate the stability of idealized planetary systems consisting of five planets, each equal in mass to the Earth, orbiting a one solar mass star. All planets orbit in the same plane and in the same direction, and the planets are uniformly spaced in units of mutual Hill Sphere radii. Most of the systems that we integrate begin with one or more planets on eccentric orbits, with eccentricities e as large as e=0.05 being considered. For a given initial orbital separation, larger initial eccentricity of a single planet generally leads to shorter system lifetime, with little systematic dependence of which planet is initially on an eccentric orbit. The approximate trend of instability times increasing exponentially with initial orbital separation of the planets found previously for planets with initially circular orbits is also present for systems with initially eccentric orbits. Mean motion resonances also tend to destabilize these systems, although the reductions in system lifetimes are not as large as for initially circular orbits. Systems with all planets having initial e=0.05 and aligned periapse angles typically survive far longer than systems with the same spacing in initial semi-major axis and one planet with e=0.05, but they have slightly shorter lifetimes than those with planets initially on circular orbits.
Pairs of planets in a system may end up close to their host star on eccentric orbits as a consequence of planet–planet scattering, Kozai, or secular migration. In this scenario, general relativity and secular perturbations have comparable time-scales and may interfere with each other with relevant effects on the eccentricity and pericenter evolution of the two planets. We explore, both analytically and via numerical integration, how the secular evolution is changed by general relativity for a wide range of different initial conditions. We find that when the faster secular frequency approaches the general relativity precession rate, which typically occurs when the outer planet moves away from the inner one, it relaxes to it and a significant damping of the proper eccentricity of the inner planet occurs. The proper eccentricity of the outer planet is reduced as well due to the changes in the secular interaction of the bodies. The lowering of the peak eccentricities of the two planets during their secular evolution has important implications on their stability. A significant number of two-planet systems, otherwise chaotic because of the mutual secular perturbations, are found stable when general relativity is included.
Planetary systems formed in clusters may be subject to stellar encounter flybys. Here, we create a diverse range of representative planetary systems with different orbital scales and planets’ masses and examine encounters between them in a typical open cluster. We first explore the close-in multisuper Earth systems ≲0.1 au. They are resistant to flybys in that only ones inside a few au can destabilize a planet or break the resonance between such planets. But these systems may capture giant planets on to wide orbits from the intruding star during distant flybys. If so, the original close-in small planets’ orbits may be tilted together through Kozai–Lidov mechanism, forming a ‘cold’ system that is significantly inclined against the equator of the central host. Moving to the intermediately placed planets around solar-like stars, we find that the planets’ mass gradient governs the systems’ long-term evolution post-encounter: more massive planets have better chances to survive. Also, a system’s angular momentum deficit, a quantity describing how eccentric/inclined the orbits are, measured immediately after the encounter, closely relates to the longevity of the systems – whether or not and when the systems turn unstable in the ensuing evolution millions of years post-encounter. We compare the orbits of the surviving planets in the unstable systems through (1) the immediate consequence of the stellar fly or (2) internal interplanetary scattering long post-encounter and find that those for the former are systematically colder. Finally, we show that massive wide-orbit multiplanet systems like that of HR 8799 can be easily disrupted and encounters at a few hundreds of au suffice.
Context. Earth-sized planets were observed in close-in orbits around M dwarfs. While more and more planets are expected to be uncovered around M dwarfs, theories of their formation and dynamical evolution are still in their infancy.
Aims. We investigate the giant impact stage for the growth of protoplanets, which includes strong scattering around low-mass stars. The aim is to clarify whether strong scattering around low-mass stars affects the orbital and mass distributions of the planets.
Methods. We performed an N -body simulation of protoplanets by systematically surveying the parameter space of the stellar mass and surface density of protoplanets.
Results. We find that protoplanets are often ejected after twice or three times the close-scattering around late M dwarfs. The ejection sets the upper limit of the largest planet mass. By adopting the surface density that linearly scales with the stellar mass, we find that as the stellar mass decreases, less massive planets are formed in orbits with higher eccentricities and inclinations. Under this scaling, we also find that a few close-in protoplanets are generally ejected.
Conclusions. The ejection of protoplanets plays an important role in the mass distribution of super-Earths around late M dwarfs. The mass relation of observed close-in super-Earths and their central star mass is reproduced well by ejection.
A nearby multiplanet system
Exoplanets can interact gravitationally with other objects orbiting the same star, affecting their evolution and stability. Studying these effects requires locating systems with multiple planets. Monitoring the nearby red dwarf star GJ 887, Jeffers et al. detected periodic radial velocity signals, indicating the presence of two planets on orbits with periods of about 9 and 22 days and a further candidate planet (see the Perspective by Davies). The inclinations of the orbits are unknown, so only minimum masses could be determined, but those were consistent with both planets being super-Earths—more massive than Earth but less than Neptune. This system is only 3.3 parsecs from the Sun, which should facilitate follow-up with other techniques.
Science , this issue p. 1477 ; see also p. 1432
Precision Doppler observations at the Lick and Keck observatories have revealed Keplerian velocity variations in the stars HD 12661, HD 92788, and HD 38529. HD 12661 (G6 V) has an orbital period of 252.7 ± 2.7 days, velocity semiamplitude K = 88.4 ± 2.0 m s-1, and orbital eccentricity e = 0.23 ± 0.024. Adopting a stellar mass of 1.07 M☉, we infer a companion mass of M sin i = 2.79 MJ and a semimajor axis of a = 0.79 AU. HD 92788 (G5 V) has an orbital period of 326.7 ± 3.2 days, velocity semiamplitude K = 99.9 ± 2.4, and orbital eccentricity e = 0.30 ± 0.06. The adopted stellar mass of 1.06 M☉ yields a companion mass of M sin i = 3.34 MJ and a semimajor axis of a = 0.95 AU. HD 38529 (G4 IV) has an orbital period of 14.3 ± 0.8 days, velocity semiamplitude K = 53.8 ± 2.0 m s-1, and eccentricity e = 0.27 ± 0.03. The stellar mass of 1.4 M☉ sets M sin i = 0.77 MJ, with a semimajor axis of a = 0.13 AU for this companion. In addition to the 14.3 day periodicity, the velocity residuals for HD 38529 show curvature over the three years of observations. Based on a measurement of Ca II H and K emission, all three stars are chromospherically inactive. Based on both spectral synthesis modeling and narrowband photometry, HD 12661, HD 92788, and HD 38529 all appear to be metal-rich stars, reinforcing the correlation of high metallicity in the host stars of gas giant extrasolar planets. We examine the velocity residuals to the Keplerian fits for a subsample of 12 planet-bearing stars that have been observed longer than two years at the Lick Observatory. Five of the 12 (Ups Andromedae, τ Boo, 55 Cnc, HD 217107, and HD 38529) exhibit coherent variations in the residual velocities that are consistent with additional companions. Except for Upsilon Andromedae, the source of the velocity variation remains speculative pending completion of one full orbit. GJ 876 exhibits residual velocities with high rms scatter (24 m s-1), lacking identifiable coherence. The residual velocities for six of the 12 stars (51 Peg, 70 Vir, 16 Cyg B, ρ CrB, 47 UMa, and HD 195019) exhibit rms velocity scatter of ~7 m s-1, consistent with errors. The residual velocity trends suggest that known planet-bearing stars appear to harbor a distant (>3 AU) detectable companion more often than other stars in our planet survey.
The present study of the dynamics of two interacting planets orbiting a sun proceeds by examining the topological stability of the three-body problem; for initially circular planetary orbits, the system will be Hill stable, as is supported by numerical integrations. The chaotic dynamics of these systems is investigated, and a region of bound chaos outside the Hill-stable zone is noted. The implications of these findings for planetary accretion, the current solar system, and the pulsar planet system PSR 1257+12, are probed.
We present numerical orbital integrations designed to test the stability of the three planets detected in orbit around υ Andromedae and possible smaller bodies orbiting in the system which have not yet been discovered. We find that some configurations are stable for at least 109 yr, whereas in other configurations planets can be ejected into interstellar space or crash into the star in less than 105 yr. The typical path to instability involves the outer planet exciting the eccentricity of the middle planet's orbit to such high values that it ventures close to the inner planet. In some stable systems, a secular resonance between the outer two planets prevents close approaches between them by aligning their longitudes of periastron. In relatively stable systems, test particles can survive for long times between the inner and middle planets as well as several AU exterior to the outer planet, but we could find no stable orbits between the middle and outer planets.
The recent discoveries of massive planetary companions orbiting several solar-type stars pose a conundrum. Conventional models for the formation of giant planets (such as Jupiter and Saturn) place such objects at distances of several astronomical units from the parent star, whereas all but one of the new objects are on orbits well inside 1 AU; these planets must therefore have originated at larger distances and subsequently migrated inwards. One suggested migration mechanism invokes tidal interactions between the planet and the evolving circumstellar disk. Such a mechanism results in planets with small, essentially circular orbits, which appears to be the case for many of the new planets. But two of the objects have substantial orbital eccentricities, which are difficult to reconcile with a tidal-linkage model. Here we describe an alternative model for planetary migration that can account for these large orbital eccentricities. If a system of three or more giant planets form about a star, their orbits may become unstable as they gain mass by accreting gas from the circumstellar disk; subsequent gravitational encounters among these planets can eject one from the system while placing the others into highly eccentric orbits both closer and farther from the star.
Planets are believed to have formed through the accumulation of a large number of small bodies. In the case of the gas-giant planets Jupiter and Saturn, they accreted a significant amount of gas directly from the protosolar nebula after accumulating solid cores of about 5-15 Earth masses. Such models, however, have been unable to produce the smaller ice giants Uranus and Neptune at their present locations, because in that region of the Solar System the small planetary bodies will have been more widely spaced, and less tightly bound gravitationally to the Sun. When applied to the current Jupiter-Saturn zone, a recent theory predicts that, in addition to the solid cores of Jupiter and Saturn, two or three other solid bodies of comparable mass are likely to have formed. Here we report the results of model calculations that demonstrate that such cores will have been gravitationally scattered outwards as Jupiter, and perhaps Saturn, accreted nebular gas. The orbits of these cores then evolve into orbits that resemble those of Uranus and Neptune, as a result of gravitational interactions with the small bodies in the outer disk of the protosolar nebula.
INTRODUCTION Our Sun is a very common and ordinary star. There is really nothing to distinguish it from a myriad of other similar stars in this region of the Galaxy. Yet, the Sun possesses a marvelous system of nine diverse planets. This has led us to believe that the formation of planetary systems should be a natural, common result of the process of star formation. We expect that a significant fraction of Solar-type stars should have some type of planetary system in orbit around them. The discovery by the HST of disks of dust around many stars in the Orion nebulae certainly reinforces that feeling. The quest to discover [1] 2 Marcy, Cochran, Mayor and explore these extrasolar planetary systems has proven to be a difficult and elusive task that has occupied astronomers for decades. The early phases of the search for substellar companions are chronicled by van de Kamp (1977, 1982, 1986). However, much to the surprise of everybody, the first confirmed detectio
Radial velocity observations of the F8 V star upsilo Andromedae taken at Lick and at Whipple Observatories have revealed evidence of three periodicities in the line-of-sight velocity of the star. These periodicities have been interpreted as evidence for at least three low-mass companions (LMCs) revolving around upsilo Andromedae. The mass and orbital parameters inferred for these companions raise questions about the dynamical stability of the system. We report here results from our independent analysis of the published radial velocity data, as well as new unpublished data taken at Lick Observatory. Our results confirm the finding of three periods in the data. Our best fits to the data, on the assumption that these periods arise from the gravitational perturbations of companions in Keplerian orbits, are also generally in agreement but with some differences from the earlier findings. We find that the available data do not constrain well the orbital eccentricity of the middle companion in a three-companion model of the data. We also find that in order for our best-fit model to the Lick data to be dynamically stable over the lifetime of the star (~2 billion years), the system must have a mean inclination to the plane of the sky greater than 13°. The corresponding minimum inclination for the best fit to the Whipple data set is 19°. These values imply that the maximum mass for the outer companion can be no greater than about 20 Jupiter masses. Our analysis of the stability of the putative systems also places constraints on the relative inclinations of the orbital planes of the companions. We comment on global versus local (i.e., method of steepest descent) means of finding best-fit orbits from radial velocity data sets.
Because of the high eccentricities (~0.3) of two of the possible planets about the star upsilo Andromeda, the stability of the system requires careful study. We present results of 1000 numerical simulations which explore the orbital parameter space as constrained by the observations. The orbital parameters of each planet are chosen from a Gaussian error distribution, and the resulting configuration is integrated for 1 Myr. We find that 84% of these integrations are stable. Configurations in which the eccentricity of the third planet is ~0.45, the system is always unstable, typically producing a close encounter between the second and third planets. A similar exercise with the gas giants in our solar system sampled with the same error distribution was performed. Approximately 81% of these simulations were stable for 106 yr.
The recent discoveries of extrasolar giant planets, coupled with refined models of the compositions of Jupiter and Saturn,
prompt a reexamination of theories of giant planet formation. An alternative to the favored core accretion hypothesis is examined
here; gravitational instability in the outer solar nebula leading to giant planet formation. Three-dimensional hydrodynamic
calculations of protoplanetary disks show that giant gaseous protoplanets can form with locally isothermal or adiabatic disk
thermodynamics. Gravitational instability appears to be capable of forming giant planets with modest cores of ice and rock
faster than the core accretion mechanism can.
As the first step in the investigation of a statistical behavior of a planetesimal swarm which governs the growth rate of the planets, we have analytically studied the gravitational scattering process between two planetesimals revolving around the protosun. The process was studied when all bodies move on a same plane. For the study of the planetary growth, however, it is essentially important to take into account off-planar motion (i.e., inclinations) of planetesimals. Thus, in the present paper, we have aimed at the investigation of the behavior of the three- dimensional scattering process. By solving the three-dimensional Hill’s equation approximately, we have obtained analytic expressions for the changes of orbital elements between before and after an encounter in two special cases; initial semi-major axes of the bodies are (1) nearly equal and (2) largely different. In the above two cases, two bodies are always apart from each other during encounter and, hence, we can find orbits of the two bodies by an analytic procedure. The obtained results are summarized as follows. In the former case, “reduced” eccentricity and inclination of their relative motion hardly change and the semi-major axis exchanges after encounter. These are the similar results to those found by Hénon and Petit for the planar problem. In the latter case the “reduced” eccentricity, inclination and semi-major axis change after an encounter. As the difference between initial semi-major axes becomes large, the changes of the orbital elements decrease more rapidly than those expected under the two-body approximation in which the effect of gravity of the protosun is neglected.
This paper describes the integrator RADAU, which has been used over the
past 10 years in the numerical integration of orbits and other problems
involving numerical solution of systems of ordinary differential
equations. First- and second-order equations are solved directly,
including the general second-order case. A self-starting integrator,
RADAU proceeds by sequences within which the substeps are taken at
Gauss-Radau spacings. This allows rather high orders of accuracy with
relatively few function evaluations. After the first sequence the
information from previous sequences is used to improve the accuracy. The
integrator itself chooses the next sequence size. RADAU is at least
comparable with the best of other integrators in speed and accuracy, and
it is often superior, particularly at high accuracies.
Continuum observations at 1.3 mm of 86 pre-main sequence stars in the
Taurus-Auriga dark clouds show that 42% have detectable emission from
small particles. The detected fraction is only slightly smaller for the
weak-line and "naked" T Tauri stars than for classical T Tauris,
indicating that the former stars often have circumstellar material. In
both categories, the column densities of particles are too large to be
compatible with spherical distributions of circumstellar matter -- the
optical extinctions would be too large; the particles are almost
certainly in spatially thin, circumstellar disks. Models of the
spectral energy distributions from 10 to 1300 μm indicate that for
the most part the disks are transparent at 1.3 mm, although the
innermost (≪1 AU) regions are opaque even at millimeter wavelenths.
The aggregate particle masses are between 10-5 and
10-2 Msun. The disk mass does not decrease
with increasing stellar age up to at least 107 years among
the stars detected at 1.3 mm. There is some evidence for temperature
evolution, in the sense that older disks are colder and less luminous.
There is little correlation between disk mass and Hα equivalent
width among the detected stars, suggesting that the Hα line is not
by itself indicative of disk mass. Spectral indices for several sources
between 1.3 and 2.7 mm suggest that the particle emissivities ɛ
are weaker functions of frequency ν than is the usual case of
interstellar grains. Particle growth via adhesion in the dense disks
might explain this result. The typical disk has an angular momentum
comparable to that generally accepted for the early solar nebula, but
very little stored energy, almost five orders of magnitude smaller than
that of the central star. Our results demonstrate that disks more
massive than the minimum mass of the proto-solar system commonly
accompany the birth of solar-mass stars and suggest that planetary
systems are common in the Galaxy.
THE recent discovery1 and confirmation2 of a possible planetary companion orbiting the solar-type star 51 Pegasi represent a breakthrough in the search for extrasolar planetary systems. Analysis of systematic variations in the velocity of the star indicate that the mass of the companion is approximately that of Jupiter, and that it is travelling in a nearly circular orbit at a distance from the star of 0.05 AU (about seven stellar radii). Here we show that, if the companion is indeed a gas-giant planet, it is extremely unlikely to have formed at its
present location. We suggest instead that the planet probably formed by gradual accretion of solids and capture of gas at a much larger distance from the star (~5 AU), and that it subsequently migrated inwards through interactions with the remnants of the circumstellar disk. The planet's migration may have stopped in its present orbit as a result of tidal interactions with the star, or through truncation of the inner circumstellar disk by the stellar magnetosphere.
We propose a merger scenario for the newly discovered extrasolar planets around 70 Vir (Marcy & Butler) and HD 114762 (Latham, Stefanik, & Mazeh; Marcy & Butler). These planets have mass Mp sin i = 6.6 and 9MJ (where MJ is Jupiter's mass and i is the orbital inclination), orbital semimajor axis a = 0.43 and 0.34 AU, and eccentricity e = 0.38 and 0.35, respectively. Our scenario is based on the conventional formation model of giant planets (gas accretion onto solid cores) and the long-term orbital stability theory of planetary systems. We suggest that in a relatively massive disk, several giant planets can be formed with Mp ~ 1-3MJ and a 1 AU. Under the persistence of the disk gas, the protogiant planet system is stable during its formation epoch (within 106-107 yr). But, after the depletion of the disk gas, mutual gravitational perturbation between the planets induces a gradual increase in their orbital eccentricities, until their orbits become unstable and begin to cross each other. We present numerical calculations of the orbital evolution leading to the orbit crossing stage. Our results indicate that the inner planets have a tendency to merge into a massive planet with relatively high e (0.2-0.9) and small a (0.5-1 AU). The orbital decay is a result of the gravitational perturbation by the outer planets and the dissipation of the colliding planets' relative kinetic energy. Afterward, long-term perturbation would slightly reduce the merged body's a, while it would keep its e high. The orbital properties of the merged body are consistent with those of the massive eccentric planets around 70 Vir and HD 114762. The onset timescale for orbit crossing within a planetary system is sensitively determined by the planets' mass and separation, which may explain the diversity in the orbital properties among the newly discovered planetary systems.
The astrometric search for extrasolar planetary systems can regard, at least in the first stage, only systems with high mass ratio (10–2) of the largest planets to the central star. The formation of such systems is probably a common occurrence in the galaxy, as indicated by various theoretical arguments; these systems could have evolved in a regular way (near-circular and coplanar orbits, Titius-Bode law, etc.) until the most massive accumulating protoplanet exceeded the critical value of the mass ratio corresponding to dynamical instability. This phenomenon has been extensively investigated in recent years, showing that a three- orN-body system becomes unstable (for relative separations similar to the planetary ones) for >10–2. Therefore, it seems likely that observations of systems within this range of mass ratio will show the irregular end-products of processes related to the instability (close encounters or ejections), which affected drastically the orbital configuration in the last phase of the planet formation.
Recent studies indicate that the formation of jovian giant planets (probably Jupiter and possibly Saturn) may have preceded the formation of terrestrial planets because of the rapidity of the gas-capture process of jovian planets and the stability of the terrestrial protoplanet system. In this paper, we investigate the stability and orbital evolution of medium-sized protoplanets in the terrestrial region under the perturbation by jovian giant planets, Jupiter and Saturn. In our numerical models of the terrestrial protoplanet system, typical masses of the protoplanets are about 0.1 to 0.2 Earth mass, and their spacings are varied from 4 to 20 mutual Hill radii. In initially low-eccentricity and low-inclination orbits, secular perturbation by jovian planets can enhance the eccentricities of terrestrial protoplanets and increase the probability of close encounters when the separation of the protoplanets is large. Due to the effect of the secular perturbation, the instability time scale of the terrestrial protoplanet system is limited to 106 years, at most 107 years, The time scale of instability also depends much on initial random velocity. Initially high random velocity of protoplanets also reduces the instability time scale two orders of magnitude or so compared with low random velocity and can accelerate the dynamical evolution of the systems.
Instabilities and strong dynamical interactions between several giant planets have been proposed as a possible explanation for the surprising orbital properties of extrasolar planetary systems. In particular, dynamical instabilities seem to provide a natural mechanism for producing the highly eccentric orbits seen in many systems. Here we present results from a new set of numerical integrations for the dynamical evolution of planetary systems containing two identical giant planets in nearly circular orbits very close to the dynamical stability limit. We determine the statistical properties of the three main types of systems resulting from the development of an instability: systems containing one planet, following either a collision between the two initial planets, or the ejection of one of them to infinity, and systems containing two planets in a new, quasi-stable configuration. We discuss the implications of our results for the formation and evolution of observed extrasolar planetary systems. We conclude that the distributions of eccentricities and semimajor axes for observed systems cannot be explained easily by invoking dynamical interactions between two planets initially on circular orbits. While highly eccentric orbits can be produced naturally by these interactions, collisions between the two planets, which occur frequently in the range of observed semimajor axes, would result in many more nearly circular orbits than in the observed sample.
We investigate the growth and the orbital evolution of protoplanets embedded in a swarm of planetesimals using three-dimensionalN-body simulations. We find that among protoplanets, larger ones grow more slowly than smaller ones, while the growth of protoplanets is still faster than that of planetesimals. As a result, in the stage after rapid runaway growth, protoplanets with the same order mass grow oligarchically, while most planetesimals remain small. While the protoplanets grow, orbital repulsion keeps their orbital separations wider than about 5 Hill radius of the protoplanets. The typical orbital separation is about 10 Hill radius, which only weakly depends on the mass and the semimajor axis of protoplanets. We explain how this self-organized protoplanet–planetesimal system forms.
We simulate the late stages of terrestrial-planet formation using N-body integrations, in three dimensions, of disks of up to 56 initially isolated, nearly coplanar planetary embryos, plus Jupiter and Saturn. Gravitational perturbations between embryos increase their eccentricities,e, until their orbits become crossing, allowing collisions to occur. Further interactions produce large-amplitude oscillations ineand the inclination,i, with periods of ∼105years. These oscillations are caused by secular resonances between embryos and prevent objects from becoming re-isolated during the simulations. The largest objects tend to maintain smallereandithan low-mass bodies, suggesting some equipartition of random orbital energy, but accretion proceeds by orderly growth. The simulations typically produce two large planets interior to 2 AU, whose time-averagedeandiare significantly larger than Earth and Venus. The accretion rate falls off rapidly with heliocentric distance, and embryos in the “Mars zone” (1.2 <a< 2 AU) are usually scattered inward and accreted by “Earth” or “Venus,” or scattered outward and removed by resonances, before they can accrete one another. The asteroid belt (a> 2 AU) is efficiently cleared as objects scatter one another into resonances, where they are lost via encounters with Jupiter or collisions with the Sun, leaving, at most, one surviving object. Accretional evolution is complete after 3 × 108years in all simulations that include Jupiter and Saturn. The number and spacing of the final planets, in our simulations, is determined by the embryos' eccentricities, and the amplitude of secular oscillations ine, prior to the last few collision events.
We report high-precision, high-cadence photometric measurements of the star HD 209458, which is known from radial velocity measurements to have a planetary-mass companion in a close orbit. We detect two separate transit events at times that are consistent with the radial velocity measurements. In both cases, the detailed shape of the transit curve due to both the limb darkening of the star and the finite size of the planet is clearly evident. Assuming stellar parameters of 1.1 R middle dot in circle and 1.1 M middle dot in circle, we find that the data are best interpreted as a gas giant with a radius of 1.27+/-0.02 RJup in an orbit with an inclination of 87&fdg;1+/-0&fdg;2. We present values for the planetary surface gravity, escape velocity, and average density and discuss the numerous observations that are warranted now that a planet is known to transit the disk of its parent star.
New numerical simulations of the formation of the giant planets are presented, in which for the first time both the gas and planetesimal accretion rates are calculated in a self-consistent, interactive fashion. The simulations combine three elements: (1) three-body accretion cross sections of solids onto an isolated planetary embryo, (2) a stellar evolution code for the planet's gaseous envelope, and (3) a planetesimal dissolution code within the envelope, used to evaluate the planet's effective capture radius and the energy deposition profile of accreted material. Major assumptions include: The planet is embedded in a disk of gas and small planetesimals with locally uniform initial surface mass density, and planetesimals are not allowed to migrate into or out of the planet's feeding zone.
The existence of a dominant massive planet, Jupiter, in our solar system, although perhaps essential for long-term dynamical stability and the development of life, may not be typical of planetary systems that form around other stars. In a system containing two Jupiter-like planets, the possibility exists that a dynamical instability will develop. Computer simulations suggest that in many cases this instability leads to the ejection of one planet while the other is left in a smaller, eccentric orbit. In extreme cases, the eccentric orbit has a small enough periastron distance that it may circularize at an orbital period as short as a few days through tidal dissipation. This may explain the recently detected Jupiter-mass planets in very tight circular orbits and wider eccentric orbits around nearby stars.
Current models of the evolution of the known extrasolar planetary systems need to incorporate orbital migration and/or gravitational interactions among giant planets to explain the presence of large bodies close to their parent stars. These processes could also lead to planets being ingested by their parent stars, which would alter the relative abundances of elements heavier than helium in the stellar atmospheres. In particular, the abundance of the rare 6Li isotope, which is normally destroyed in the early evolution of solar-type stars but preserved intact in the atmospheres of giant planets, would be boosted substantially. 6Li has not hitherto been observed reliably in a metal-rich star, where metallicity refers to the total abundance of elements heavier than helium. Here we report the discovery of 6Li in the atmosphere of the metal-rich solar-type star HD82943, which is known to have an orbiting giant planet. The presence of 6Li can probably be interpreted as evidence for a planet (or planets) having been engulfed by the parent star.
In order to better understand the range of dynamically long-lived giant-planet systems, we present the results of a set of bottom-up numerical simulations designed to generate plausible giant planet systems from a large number of planetary embryos. Our simulations produced systems that are stable for at least for a billion years and which exhibit a wide range of characteristics. Some of these systems are reminiscent of the outer solar system. The number of planets ranged from one to seven. Many systems contained only Uranus-mass objects. We constructed systems that were more compact than the outer solar system and systems that were much sparser, with planets on very eccentric orbits. Perhaps most surprisingly, some of the systems that we constructed were stable for at least a billion years despite undergoing macroscopic orbital changes on much shorter timescales. Subject headings: solar system: formation --- solar system: general --- celestial mechanics, stellar dynamics ...
A system of two small planets orbiting the sun on low-eccentricity, low-inclination orbits is stable with respect to close encounters if the initial semi-major axis dierence, , measured in mutual Hill radii, RH , exceeds 2 p 3, due to conservation of energy and angular momentum (Gladman 1993). We investigate the stability of systems of more than two planets using numerical integrations. We nd that systems with < 10 are always unstable, with the time, t, of rst close encounter given approximately by log t = b + c, where b and c are constants. It is likely that systems with > 10 are also unstable. The slope b depends weakly on the number of planets, but is independent of planetary mass, m, if we measure in units that are proportional to m 1=4 rather than the usual RH / m 1=3 . Instability in multi-planet systems arises because energy and angular momentum are no longer conserved within each two-planet subsystem due to perturbations by the additional planet(s). These results suggest that planetary-embryos will not become isolated prior to the nal stage of terrestrial-planet formation simply due to a failure to achieve close encounters. Other factors leading to isolation cannot be ruled out at this stage. 1
We present numerical simulations of the formation of the planetary companions to 47 UMa, ae CrB, and 51 Peg. They are assumed to have formed in situ according to the basic model that a core formed first by accretion of solid particles, then later it captured substantial amounts of gas from the protoplanetary disk. In most of the calculations we prescribe a constant accretion rate for the solid core. The evolution of the gaseous envelope is calculated according to the following assumptions: (1) it is in quasi-hydrostatic equilibrium, (2) the gas accretion rate is determined by the requirement that the outer radius of the planet is the place at which the thermal velocity of the gas allows it to reach the boundary of the planet's Hill sphere, (3) the gas accretion rate is limited, moreover, by the prescribed maximum rate at which the nebula can supply the gas, and (4) the growth of the planet stops once it obtains approximately the minimum mass determined from radial velocity measurement...