The secular approximation of the hierarchical three body systems has been proven to be very useful in addressing many astrophysical systems, from planets, stars to black holes. In such a system two objects are on a tight orbit, and the tertiary is on a much wider orbit. Here we study the dynamics of a system by taking the tertiary mass to zero and solve the hierarchical three body system up to the octupole level of approximation. We find a rich dynamics that the outer orbit undergoes due to gravitational perturbations from the inner binary. The nominal result of the precession of the nodes is mostly limited for the lowest order of approximation, however, when the octupole-level of approximation is introduced the system becomes chaotic, as expected, and the tertiary oscillates below and above 90 • , similarly to the non-test particle flip behavior (e.g., Naoz 2016). We provide the Hamiltonian of the system and investigate the dynamics of the system from the quadrupole to the octupole level of approximations. We also analyze the chaotic and quasi-periodic orbital evolution by studying the surfaces of sections. Furthermore, including general relativity, we show case the long term evolution of individual debris disk particles under the influence of a far away interior eccentric planet. We show that this dynamics can naturally result in retrograde objects and a puffy disk after a long timescale evolution (few Gyr) for initially aligned configuration.

Context. In the standard scenario of planet formation, terrestrial planets and the cores of the giant planets are formed by accretion of planetesimals. As planetary embryos grow, the planetesimal velocity dispersion increases because of gravitational excitations produced by embryos. The increasing relative velocities of the planetesimal cause them to fragment through mutual collisions. Aims. We study the role of planetesimal fragmentation on giant planet formation. We analyze how planetesimal fragmentation modifies the growth of giant planet cores for a wide range of planetesimal sizes and disk masses. Methods. We incorporated a model of planetesimal fragmentation into our model of in situ giant planet formation. We calculated the evolution of the solid surface density (planetesimals plus fragments) taking into account the accretion by the planet, migration, and fragmentation. Results. Incorporating planetesimal fragmentation significantly modifies the process of planetary formation. If most of the mass loss in planetesimal collisions is distributed in the smaller fragments, planetesimal fragmentation inhibits the growth of the embryo for initial planetesimals of radii smaller than 10 km. Only for initial planetesimals with a radius of 100 km, and disks larger than 0.06 M , embryos achieve masses larger than the mass of Earth. However, even for these large planetesimals and massive disks, planetesimal fragmentation induces the quick formation of massive cores only if most of the mass loss in planetesimal collisions is distributed in the larger fragments. Conclusions. Planetesimal fragmentation seems to play an important role in giant planet formation. The way in which the mass loss in planetesimal collisions is distributed leads to different results, inhibiting or favoring the formation of massive cores.

Aims.In this paper, we analyze the collisional evolution of the Main Belt and NEA population taking into account the major dynamical features present in both populations. Methods. To do this, we divide the asteroid belt into three semimajor axis zones, whose boundaries are given by the ν 6 secular resonance, and the 3:1, 5:2 and 2:1 mean motion resonances with Jupiter, treating them as strong sources of dynamical removal. We also consider the action of the Yarkovsky effect and diffusive resonances as mechanisms of mass depletion. This treatment allows us to calculate the direct collisional injection into the powerful resonances, to study the collisional exchange of mass between the different regions of the Main Belt and to analyze the provenance of the NEA objects. Results. Our model is in agreement with the major observational constraints associated with the Main Belt and NEA populations, such as their size distributions, the collisional history of Vesta, the number of large asteroid families and the cosmic-ray exposure (CRE) ages of meteorites. We find that none of the dynamical and collisional mechanisms included in our treatment are able to mix material between the three studied main belt regions, since more than 99% of the final mass of every ring of our model of the Main Belt is represented by primordial material. In addition, our results supports that the Yarkovsky effect is the most important process that removes material from the asteroid Main Belt, rather than collisional injection into the major resonances. With regards to the provenance of the NEAs, our work shows that ∼94% of the NEA population comes from the region inside the 5:2 mean motion resonance.