Eccentricity Of Planet Research Articles

Normalized angular momentum deficit: a tool for comparing the violence of the dynamical histories of planetary systems

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.

Habitability and Water Loss Limits on Eccentric Planets Orbiting Main-sequence Stars

Abstract A planet’s climate can be strongly affected by its orbital eccentricity and obliquity. Here we use a one-dimensional energy balance model modified to include a simple runaway greenhouse (RGH) parameterization to explore the effects of these two parameters on the climate of Earth-like aqua planets—completely ocean-covered planets—orbiting F-, G-, K-, and M-dwarf stars. We find that the range of instellations for which planets exhibit habitable surface conditions throughout an orbit decreases with increasing eccentricity. However, the appearance of temporarily habitable conditions during an orbit creates an eccentric habitable zone (EHZ) that is sensitive to orbital eccentricity and obliquity, planetary latitude, and the spectral type of the host star. We find that the fraction of a planet’s orbit over which it exhibits habitable surface conditions is larger on eccentric planets orbiting M-dwarf stars, due to the lower broadband planetary albedos of these planets. Planets with larger obliquities have smaller EHZs, but exhibit warmer climates if they do not enter a snowball state during their orbits. We also find no transient RGH state on planets at all eccentricities. Rather, planets spend their entire orbits either in an RGH or not. For G-dwarf planets receiving 100% of the modern solar constant and with eccentricities above 0.55, an entire Earth ocean inventory can be lost in 3.6 Gyr. M-dwarf planets, due to their larger incident X-ray and extreme ultraviolet flux, can become desiccated in only 690 Myr with eccentricities above 0.38. This work has important implications for eccentric planets that may exhibit surface habitability despite technically departing from the traditional habitable zone as they orbit their host stars.

Secular evolution of close-in planets: the effects of general relativity

ABSTRACT 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.

Realistic collisional water transport during terrestrial planet formation

Context. According to the latest theoretical and isotopic evidence, Earth’s water content originates mainly from today’s asteroid belt region, or at least from the same precursor material. This suggests that water was transported inwards to Earth, and to similar planets in their habitable zone, via (giant) collisions of planetary embryos and planetesimals during the chaotic final phase of planet formation. Aims. In current dynamical simulations water delivery to terrestrial planets is still studied almost exclusively by assuming oversimplified perfect merging, even though water and other volatiles are particularly prone to collisional transfer and loss. To close this gap we have developed a computational framework to model collisional water transport by direct combination of long-term N-body computations with dedicated 3D smooth particle hydrodynamics (SPH) collision simulations of differentiated, self-gravitating bodies for each event. Methods. Post-collision water inventories are traced self-consistently in the further dynamical evolution, in accretionary or erosive as well as hit-and-run encounters with two large surviving bodies, where besides collisional losses, water transfer between the encountering bodies has to be considered. This hybrid approach enables us for the first time to trace the full dynamical and collisional evolution of a system of approximately 200 bodies throughout the whole late-stage accretion phase (several hundred Myr). As a first application we choose a Solar System-like architecture with already formed giant planets on either circular or eccentric orbits and a debris disk spanning the whole terrestrial planet region (0.5–4 au). Results. Including realistic collision treatment leads to considerably different results than simple perfect merging, with lower mass planets and water inventories reduced regularly by a factor of two or more. Due to a combination of collisional losses and a considerably lengthened accretion phase, final water content, especially with giant planets on circular orbits, is strongly reduced to more Earth-like values, and closer to results with eccentric giant planets. Water delivery to potentially habitable planets is dominated by very few decisive collisions, mostly with embryo-sized or larger objects and only rarely with smaller bodies, at least if embryos have formed throughout the whole disk initially. The high frequency of hit-and-run collisions and the differences to predominantly accretionary encounters, such as generally low water (and mass) transfer efficiencies, are a crucial part of water delivery, and of system-wide evolution in general.

Tidal evolution of circumbinary systems with arbitrary eccentricities: applications for Kepler systems

We present an extended version of the Constant Time Lag analytical approach for the tidal evolution of circumbinary planets introduced in our previous work. The model is self-consistent, in the sense that all tidal interactions between pairs are computed, regardless of their size. We derive analytical expressions for the variational equations governing the spin and orbital evolution, which are expressed as high-order elliptical expansions in the semimajor axis ratio but retain closed form in terms of the binary and planetary eccentricities. These are found to reproduce the results of the numerical simulations with arbitrary eccentricities very well, as well as reducing to our previous results in the low-eccentric case. Our model is then applied to the well-characterised Kepler circumbinary systems by analysing the tidal timescales and unveiling the tidal flow around each different system. In all cases we find that the spins reach stationary values much faster than the characteristic timescale of the orbital evolution, indicating that all Kepler circumbinary planets are expected to be in a sub-synchronous state. On the other hand, all systems are located in a tidal flow leading to outward migration; thus the proximity of the planets to the orbital instability limit may have been even greater in the past. Additionally, Kepler systems may have suffered a significant tidally induced eccentricity damping, which may be related to their proximity to the capture eccentricity. To help understand the predictions of our model, we also offer a simple geometrical interpretation of our results.

Importance of Giant Impact Ejecta for Orbits of Planets Formed during the Giant Impact Era

Abstract Terrestrial planets are believed to be formed via giant impacts of Mars-sized protoplanets. Planets formed via giant impacts have highly eccentric orbits. A swarm of planetesimals around the planets may lead to eccentricity damping for the planets via the equipartition of random energies (dynamical friction). However, dynamical friction increases eccentricities of planetesimals, resulting in high velocity collisions between planetesimals. The collisional cascade grinds planetesimals to dust until dust grains are blown out due to radiation pressure. Therefore, the total mass of planetesimals decreases due to collisional fragmentation, which weakens dynamical friction. We investigate the orbital evolution of protoplanets in a planetesimal disk, taking into account collisional fragmentation of planetesimals. For 100 km sized or smaller planetesimals, dynamical friction is insignificant for eccentricity damping of planets because of collisional fragmentation. On the other hand, giant impacts eject collisional fragments. Although the total mass of giant impact ejecta is 0.1–0.3 Earth masses, the largest impact ejecta are ∼1000 km in size. We also investigate the orbital evolution of single planets with initial eccentricities of 0.1 in a swarm of such giant impact ejecta. Although the total mass of giant impact ejecta decreases by a factor of 3 in 30 Myr, eccentricities of planets are damped down to the Earth level (∼0.01) due to interaction with giant impact ejecta. Therefore, giant impact ejecta play an important role for determination of terrestrial planet orbits.

Resilient habitability of nearby exoplanet systems

ABSTRACT We investigate the possibility of finding Earth-like planets in the habitable zone of 34 nearby FGK-dwarfs, each known to host one giant planet exterior to their habitable zone detected by RV. First we simulate the dynamics of the planetary systems in their present day configurations and determine the fraction of stable planetary orbits within their habitable zones. Then, we postulate that the eccentricity of the giant planet is a result of an instability in their past during which one or more other planets were ejected from the system. We simulate these scenarios and investigate whether planets orbiting in the habitable zone survive the instability. Explicitly we determine the fraction of test particles, originally found in the habitable zone, which remain in the habitable zone today. We label this fraction the resilient habitability of a system. We find that for most systems the probability of planets existing [or surviving] on stable orbits in the habitable zone becomes significantly smaller when we include a phase of instability in their history. We present a list of candidate systems with high resilient habitability for future observations. These are: HD 95872, HD 154345, HD 102843, HD 25015, GJ 328, HD 6718, and HD 150706. The known planets in the last two systems have large observational uncertainties on their eccentricities, which propagate into large uncertainties on their resilient habitability. Further observational constraints of these two eccentricities will allow us to better constrain the survivability of Earth-like planets in these systems.

Load Sharing Behavior of Double-Pinion Planetary Gear Sets Considering Manufacturing Errors

The unequal load sharing of double-pinion planetary gear sets (PGSs) would take place because of manufacturing errors. A quasi-static model, suitable for any planet train number and support condition, was proposed considering manufacturing errors. The load sharing ratio of double-pinion PGSs with four planet trains was simulated for various planet position errors, whose results show that the radial planet position error has different effects on the load sharing behavior from simple PGSs. The mechanics for this phenomenon was revealed. The effect of the planet train number on the load sharing behavior in the presence of different planet position errors was then simulated by introducing the equivalent error variable. Finally, the effect of gear eccentricity errors on the load sharing behavior were simulated and some important characteristics were revealed such that the planet eccentricity error has worse impacts than position errors, and the effect of the eccentricity error of sun gear could be eliminated by floating the sun gear.

Formation of compact systems of super-Earths via dynamical instabilities and giant impacts

ABSTRACT The NASA’s Kepler mission discovered ∼700 planets in multiplanet systems containing three or more transiting bodies, many of which are super-Earths and mini-Neptunes in compact configurations. Using N-body simulations, we examine the in situ, final stage assembly of multiplanet systems via the collisional accretion of protoplanets. Our initial conditions are constructed using a subset of the Kepler five-planet systems as templates. Two different prescriptions for treating planetary collisions are adopted. The simulations address numerous questions: Do the results depend on the accretion prescription?; do the resulting systems resemble the Kepler systems, and do they reproduce the observed distribution of planetary multiplicities when synthetically observed?; do collisions lead to significant modification of protoplanet compositions, or to stripping of gaseous envelopes?; do the eccentricity distributions agree with those inferred for the Kepler planets? We find that the accretion prescription is unimportant in determining the outcomes. The final planetary systems look broadly similar to the Kepler templates adopted, but the observed distributions of planetary multiplicities or eccentricities are not reproduced, because scattering does not excite the systems sufficiently. In addition, we find that ∼1 per cent of our final systems contain a co-orbital planet pair in horseshoe or tadpole orbits. Post-processing the collision outcomes suggests that they would not significantly change the ice fractions of initially ice-rich protoplanets, but significant stripping of gaseous envelopes appears likely. Hence, it may be difficult to reconcile the observation that many low-mass Kepler planets have H/He envelopes with an in situ formation scenario that involves giant impacts after dispersal of the gas disc.

Multiplanet disc interactions in binary systems

ABSTRACT We investigate the evolution of a multiplanet–disc system orbiting one component of a binary star system. The planet–disc system is initially coplanar but misaligned to the binary orbital plane. The planets are assumed to be giants that open gaps in the disc. We first study the role of the disc in shaping the mutual evolution of the two planets using a secular model for low initial tilt. In general, we find that the planets and the disc do not remain coplanar, in agreement with previous work on the single planet case. Instead, the planets and the disc undergo tilt oscillations. A high-mass disc between the two planets causes the planets and the disc to nodally precess at the same average rate but they are generally misaligned. The amplitude of the tilt oscillations between the planets is larger while the disc is present. We then consider higher initial tilts using hydrodynamical simulations and explore the possibility of the formation of eccentric Kozai–Lidov (KL) planets. We find that the inner planet’s orbit undergoes eccentricity growth for a large range of disc masses and initial misalignments. For a low disc mass and large initial misalignment, both planets and the disc can undergo KL oscillations. Furthermore, we find that sufficiently massive discs can cause the inner planet to increase its inclination beyond 90° and therefore to orbit the binary in a retrograde fashion. The results have important implications for the explanation of very eccentric planets and retrograde planets observed in multiplanet systems.

On the Dust Signatures Induced by Eccentric Super-Earths in Protoplanetary Disks

Abstract We investigate the impact of a highly eccentric 10 M ⊕ (where M ⊕ is the Earth mass) planet embedded in a dusty protoplanetary disk on dust dynamics and its observational implications. By carrying out high-resolution 2D gas and dust two-fluid hydrodynamical simulations, we find that the planet’s orbit can be circularized at large radii. After the planet’s orbit is circularized, partial gap opening and dust ring formation happen close to the planet’s circularization radius, which can explain the observed gaps/rings at the outer region of disks. When the disk mass and viscosity become low, we find that an eccentric planet can even open gaps and produce dust rings close to the pericenter and apocenter radii before its circularization. This offers alternative scenarios for explaining the observed dust rings and gaps in protoplanetary disks. A lower disk viscosity is favored to produce brighter rings in observations. An eccentric planet can also potentially slow down the dust radial drift in the outer region of the disk when the disk viscosity is low (α ≲ 2 × 10−4) and the circularization is faster than the dust radial drift.

Multiscale Analysis of the Instantaneous Eccentricity Oscillations of the Planets of the Solar System from 13 000 BC to 17 000 AD

The high-resolution Jet Propulsion Laboratory DE431 and DE432 planetary ephemeris are used to evaluate the instantaneous eccentricity functions of the orbits of the planets of the solar system from 13 000 BC to 17 000 AD at different time scales. Spectral analyses are performed to determine the frequencies and the amplitudes of the detected perturbations from 0.1-year to 10 000-year periods. Taken as contiguous pairs (Mercury-Venus, Earth-Mars, Jupiter-Saturn, and Uranus-Neptune), we found anti-phase patterns between contiguous planets at specific time scales (30 000 years): namely, the eccentricity of one planet increases while the other decreases. Venus and Earth instead appear in phase. However, on shorter time scales the phase coupling becomes more complex and irregular. Yet, Jupiter and Saturn present a π/2 phase coupling at the 1000-year scale. Periodogram analysis of the planetary eccentricity functions shows several fast fluctuations whose magnitudes indicate the strength of their mutual interactions where the Jovian planets significantly perturb the orbits of the inner planets. Besides, the wavelet power spectrum and wavelet squared coherence spectrum analyses are adopted to examine the relationships in time-frequency space between the eccentricity functions of each couple of terrestrial and Jovian planets. The analysis reveals the complexity and the evolution of the gravitational couplings perturbing each other planetary orbits. In some cases and for specific frequencies, this analysis technique also led to the discovery that the coherence phase can rotate clockwise or anticlockwise. The eccentricity function of the orbit of Jupiter presents large oscillations with periods of about 60 and 900- 960 years, mostly due to the interaction with Saturn. These oscillations also correspond to oscillations found in several geophysical records. The eccentricity functions of Uranus and Neptune are characterized by a large 4300-year oscillation. The eccentricity function of Pluto is characterized by a large nearly 20 000-year modulation.

In the Presence of a Wrecking Ball: Orbital Stability in the HR 5183 System

Abstract Discoveries of exoplanets using the radial velocity method are progressively reaching out to increasingly longer orbital periods as the duration of surveys continues to climb. The improving sensitivity to potential Jupiter analogs is revealing a diversity of orbital architectures that are substantially different from that found in our solar system. An excellent example of this is the recent discovery of HR 5183b: a giant planet on a highly eccentric (e = 0.84) ∼75 yr orbit. The presence of such giant planet orbits are intrinsically interesting from the perspective of the dynamical history of planetary systems, and also for examining the implications of ongoing dynamical stability and habitability of these systems. In this work, we examine the latter, providing results of dynamical simulations that explore the stable regions that the eccentric orbit of the HR 5183 giant planet allows to exist within the habitable zone (HZ) of the host star. Our results show that, despite the incredible perturbing influence of the giant planet, there remain a narrow range of locations within the HZ where terrestrial planets may reside in long-term stable orbits. We discuss the effects of the giant planet on the potential habitability of a stable terrestrial planet, including the modulation of terrestrial planet eccentricities and the periodically spectacular view of the giant planet from the terrestrial planet location.

From the stellar properties of HD 219134 to the internal compositions of its transiting exoplanets

Context. The harvest of exoplanet discoveries has opened the area of exoplanet characterisation. But this cannot be achieved without a careful analysis of the host star parameters. Aims. The system of HD 219134 hosts two transiting exoplanets and at least two additional non-transiting exoplanets. We revisit the properties of this system using direct measurements of the stellar parameters to investigate the composition of the two transiting exoplanets. Methods. We used the VEGA/CHARA interferometer to measure the angular diameter of HD 219134. We also derived the stellar density from the transits light curves, which finally gives a direct estimate of the mass. This allowed us to infer the mass, radius, and density of the two transiting exoplanets of the system. We then used an inference model to obtain the internal parameters of these two transiting exoplanets. Results. We measure a stellar radius, density, and mass of R⋆ = 0.726 ± 0.014 R⊙, ρ⋆ = 1.82 ± 0.19 ρ⊙, and M⋆ = 0.696 ± 0.078 M⊙, respectively; there is a correlation of 0.46 between R⋆ and M⋆. This new mass is lower than that derived from the C2kSMO stellar evolutionary model, which provides a mass range of 0.755−0.810 (±0.040) M⊙. Moreover, we find that planet b and c have smaller radii than previously estimated of 1.500 ± 0.057 and 1.415 ± 0.049 R⊕ respectively; this clearly puts these planets out of the gap in the exoplanetary radii distribution and validates their super-Earth nature. Planet b is more massive than planet c, but the former is possibly less dense. We investigate whether this could be caused by partial melting of the mantle and find that tidal heating due to non-zero eccentricity of planet b may be powerful enough. Conclusions. The system of HD 219134 constitutes a very valuable benchmark for both stellar physics and exoplanetary science. The characterisation of the stellar hosts, and in particular the direct determination of the stellar density, radius, and mass, should be more extensively applied to provide accurate exoplanets properties and calibrate stellar models.

Signatures of a Planet–Planet Impacts Phase in Exoplanetary Systems Hosting Giant Planets

Abstract Exoplanetary systems host giant planets on substantially noncircular, close-in orbits. We propose that these eccentricities arise in a phase of giant impacts, analogous to the final stage of solar system assembly that formed Earth’s Moon. In this scenario, the planets scatter each other and collide, with corresponding mass growth as they merge. We numerically integrate an ensemble of systems with varying total planet mass, allowing for collisional growth, to show that (1) the high-eccentricity giants observed today may have formed preferentially in systems of higher initial total planet mass, and (2) the upper bound on the observed giant planet eccentricity distribution is consistent with planet–planet scattering. We predict that mergers will produce a population of high-mass giant planets between 1 and 8 au from their stars.

Radial Velocity Discovery of an Eccentric Jovian World Orbiting at 18 au

Abstract Based on two decades of radial velocity (RV) observations using Keck/High Resolution Echelle Spectrometer (HIRES) and McDonald/Tull, and more recent observations using the Automated Planet Finder, we found that the nearby star HR 5183 (HD 120066) hosts a 3 minimum mass planet with an orbital period of yr. The orbit is highly eccentric (e ≃ 0.84), shuttling the planet from within the orbit of Jupiter to beyond the orbit of Neptune. Our careful survey design enabled high cadence observations before, during, and after the planet’s periastron passage, yielding precise orbital parameter constraints. We searched for stellar or planetary companions that could have excited the planet’s eccentricity, but found no candidates, potentially implying that the perturber was ejected from the system. We did identify a bound stellar companion more than 15,000 au from the primary, but reasoned that it is currently too widely separated to have an appreciable effect on HR 5183 b. Because HR 5183 b’s wide orbit takes it more than 30 au (1″) from its star, we also explored the potential of complimentary studies with direct imaging or stellar astrometry. We found that a Gaia detection is very likely, and that imaging at 10 μm is a promising avenue. This discovery highlights the value of long-baseline RV surveys for discovering and characterizing long-period, eccentric Jovian planets. This population may offer important insights into the dynamical evolution of planetary systems containing multiple massive planets.

Pebbles versus planetesimals: the case of Trappist-1

We present a study into the formation of planetary systems around low mass stars similar to Trappist-1, through the accretion of either planetesimals or pebbles. The aim is to determine if the currently observed systems around low mass stars could favour one scenario over the other. To determine these differences, we ran numerous N-body simulations, coupled to a thermally evolving viscous 1D disc model, and including prescriptions for planet migration, photoevaporation, and pebble and planetesimal dynamics. We mainly examine the differences between the pebble and planetesimal accretion scenarios, but we also look at the influences of disc mass, size of planetesimals, and the percentage of solids locked up within pebbles. When comparing the resulting planetary systems to Trappist-1, we find that a wide range of initial conditions for both the pebble and planetesimal accretion scenarios can form planetary systems similar to Trappist-1, in terms of planet mass, periods, and resonant configurations. Typically these planets formed exterior to the water iceline and migrated in resonant convoys into the inner region close to the central star. When comparing the planetary systems formed through pebble accretion to those formed through planetesimal accretion, we find a large number of similarities, including average planet masses, eccentricities, inclinations, and period ratios. One major difference between the two scenarios was that of the water content of the planets. When including the effects of ablation and full recycling of the planets’ envelope with the disc, the planets formed through pebble accretion were extremely dry, whilst those formed through planetesimal accretion were extremely wet. If the water content is not fully recycled and instead falls to the planets’ core, or if ablation of the water is neglected, then the planets formed through pebble accretion are extremely wet, similar to those formed through planetesimal accretion. Should the water content of the Trappist-1 planets be determined accurately, this could point to a preferred formation pathway for planetary systems, or to specific physics that may be at play.

Apsidal alignment in migrating dust - Crescent features caused by eccentric planets

AbstractCircumstellar discs are known to exist in great variety, from gas-rich discs around the youngest stars to evolved debris discs such as the solar system’s zodiacal cloud. Through gravitational interaction, exoplanets embedded in these discs can generate density variations, imposing potentially observable structural features on the disc such as rings or gaps. Here we report on a mirrored double crescent pattern arising in simulations of discs harbouring a small, moderately eccentric planet - such as Mars. We show that the structure is a result of a directed apsidal precession occurring in particles that migrate the planet’s orbital region under Poynting-Robertson drag. We further analyze the strength of this effect with respect to planet and particle parameters.

Oscillations around tidal pseudo-synchronous solutions for circumbinary planets

AbstractTidal evolution of low-eccentric circumbinary planets is expected to drive the rotational evolution toward a pseudo-synchronous solution. In this work, we present a study of the oscillation amplitudes around this state by considering that the two central stars exert creep tides on the planet. These amplitudes are computed by direct numerical integrations of the creep equations and also by means of the calculation of the coefficients of the periodic terms in this stationary solution. As in the two-body-problem, the planetary spin and lag-angle are observed to have maximum oscillation amplitudes for stiff bodies and almost null oscillation for the gaseous regime, while the opposite behaviour is observed in the equatorial and polar flattenings. Our analytical approximation shows to be very accurate and specially necessary for very-low eccentric planets. However, the magnitudes of the oscillation amplitudes around the pseudo-synchronous solution in the circumbinary problem appears to be very small respect to the mean value. Thus, considering these oscillation in the computation of the tidal energy dissipation may not have a substantial contribution in the results, at least compared to the case in which only the mean values are taken into account.

The role of the mean and forced eccentricities in the secular dynamic of circumbinary planets

Within the framework of the Restricted Three Body Problem (RTBP), we consider the orbital evolution of a circumbinary (CB) planet. We develop a simple analytical model to explain the mean behaviour of the planetary eccentricity and better identify the nature of the main contributors of eccentricity oscilations. Our theory is validated by comparing with N-body simulation. We analyse the effect of including dissipative forces in our theory and compare with a N-body simulation in which the dissipative force is the interaction with a protoplanetary disc. Finally, we use our model to explain dynamical maps in the CB region and distinguish domains where the secular dynamics dominate over the resonant perturbations.