5 Planet Systems

5 Planet Systems

Over the last decade, evidence has mounted that the solar system’s observed state can be favorably reproduced in the context of an instability-driven dynamical evolution model, such as the “Nice” model. To date, all successful realizations of instability models have concentrated on evolving the four giant planets onto their current orbits from a more compact configuration. Simultaneously, the possibility of forming and ejecting additional planets has been discussed, but never successfully implemented. Here we show that a large array of 5-planet (2 gas giants + 3 ice giants) multi-resonant initial states can lead to an adequate formation of the outer solar system, featuring an ejection of an ice giant during a phase of instability. Particularly, our simulations demonstrate that the eigenmodes which characterize the outer solar system’s secular dynamics can be closely matched with a 5-planet model. Furthermore, provided that the ejection timescale of the extra planet is short, orbital excitation of a primordial cold classical Kuiper belt can also be avoided in this scenario. Thus the solar system is one of many possible outcomes of dynamical relaxation and can originate from a wide variety of initial states. This deems the construction of a unique model of solar system’s early dynamical evolution impossible

2. NUMERICAL EXPERIMENTS 

The numerical setup of the simulations performed here was qualitatively similar to those presented by Batygin & Brown (2010) and Batygin et al. (2011). Particularly, the five giant planets were initialized in a compact, multiresonant initial condition, surrounded by a massive planetesimal disk that extended between its immediate stability boundary and 30AU. Two of the three ice-giants were taken to have the same mass as Uranus and were initially placed next to Saturn and as the outermost planet respectively. The middle ice-giant was taken to have Neptune’s mass. In all simulations, Jupiter and Saturn started out in a 3:2 MMR, in accord with the results of hydrodynamical simulations of convergent migration of the planets in the solar nebula (Masset & Snellgrove 2001; Morbidelli & Crida 2007; Pierens & Nelson 2008). The ice giants were also sequentially assembled into first order MMRs by applying dissipative forces, designed to mimic the presence of the nebula (Lee & Peale 2002). Following resonant locking, each assembled multi-resonant initial condition was evolved in isolation for 10Myr, as an immediate test of orbital stability. This procedure yielded a total of 81 stable multi-resonant initial conditions. The search for adequate dynamical evolutions was performed in two steps. First, we evolved 10 permutations of each initial condition, with planetesimal disks composed of N = 1000 planetesimals. Disk masses were chosen randomly between Mmin disk = 25M⊕ and Mmax disk = 100M⊕. The density profiles followed a power-law distribution, Σ ∝ r k where the power-law index, k, was chosen randomly between kmin = 1 and kmax = 2. The planetesimals were initialized on near-circular orbits (e ∼ sin i ∼ 10−3 ). To reduce the already substantial computational cost, self-gravity of the planetesimal swarm was neglected. Subsequently, we eliminated all initial conditions that did not yield any final systems that were comprised of 4 planets, reducing the number of viable initial conditions to 25. Then, an additional 30 permutations of these initial conditions were integrated with disks composed of N = 3000 planetesimals (but otherwise identical to those described above). Each integration was performed using the mercury6 integration software package (Chambers 1999) and spanned 50Myr1 . The calculations were performed on Caltech’s PANGU super-computer. After their completion, simulations that were deemed successful were reintegrated with the use of tracer simulations (see Levison et al. (2008); Batygin et al. (2011)), to address the dynamical evolution of a locally formed population of KBOs. In particular, each run was supplemented with an additional disk of mass-less particles that resided in the cold classical region of the Kuiper belt (i.e. between the final exterior 3:2 and 2:1 MMRs of Neptune). 

3. RESULTS 

Out of the 810 integrations that were initially performed, 214 (∼ 25%) cases featured an ejection of a single ice-giant, yielding a system composed of 4 planets. Perhaps unsurprisingly, in most cases the ejected planet is the ice-giant that neighbors Saturn. Of the 750 simulations that were performed following the elimination of initial conditions, 33 evolutions resulted in orbits reminiscent of the solar system. Specifically, we searched for solutions where the Saturn-Jupiter period ratio exceed 2, while the final semi-major axes of the ice-giants were within 3AU of their observed counterparts. No strong requirements were placed on the planetary eccentricities and inclinations. It is noteworthy that evolving the giant planets onto solar system-like orbits is insufficient for a simulation to be deemed successful for indeed, there are additional constraints that must be satisfied. The first orbital constraint is the reproduction of the secular architecture of the system. The secular orbital angular momentum exchange (i.e. eccentricity evolution) of a planetary system containing N secondaries can be approximately represented as a superposition of N eigenmodes, each corresponding to a fundamental frequency of the system (see Murray & Dermott (1999); Morbidelli et al. (2009)). Physically, the maximum eccentricity that a given planet