New methods for large dynamic range problems in planetary formation

Abstract

Modern N‐body techniques for planetary dynamics are generally based on symplectic algorithms specially adapted to the Kepler problem. These methods have proven very useful in studying planet formation, but typically require the time‐step for all objects to be set to a small fraction of the orbital period of the innermost body. This computational expense can be prohibitive for even moderate particle number for many physically interesting scenarios, such as recent models of the formation of hot exoplanets, in which the semimajor axis of possible progenitors can vary by orders of magnitude. We present new methods which retain most of the benefits of the standard symplectic integrators but allow for radial zones with distinct time‐steps. These approaches should make simulations of planetary accretion with large dynamic range tractable. As proof‐of‐concept, we present preliminary science results from an implementation of the algorithm as applied to an oligarchic migration scenario for forming hot Neptunes.

Hot super-Earths and giant planet cores from different migration histories

“…The inward migration model was proposed by Terquem & Papaloizou (2007) and built upon by Ogihara & Ida (2009), McNeil & Nelson (2009) and Cossou et al (2014). It proposes that hot super-Earths form at large orbital radii, type I migrate inward and grow by embryo-embryo collisions.…”

On the formation of hot Neptunes and super-Earths

“…(Levison & Duncan 2000 introduced a variant which could handle occasional objects crossing the usual innermost boundary by smoothly switching to a Bulirsch–Stoer integrator, following up on the innovations of Chambers (1999), but it becomes impractical when boundary crossings are common.) In a companion paper (McNeil & Nelson 2009), we introduce a new algorithm Naoko which allows for multiple radial zones with distinct time‐steps and can vary the number of force evaluations between different zones, making possible a new trade‐off between the force accuracy between distant objects and speed. We have implemented a parallel version of Naoko in the planet formation code miranda , which is basically a parallel SyMBA implementation.…”

“…The standard approaches use a common drift time‐step for all particles (and therefore ‘overintegrate’ the more distant objects) and compute the (non‐encountering) forces between the particles at the same frequency, both of which involve far more computation on the distant, slow‐moving objects than is necessary to preserve qualitatively accurate dynamics. In McNeil & Nelson (2009), the authors combine various techniques in the literature (Duncan, Levison & Lee 1998; Chambers 1999; Saha & Tremaine 1992) to construct a new algorithm which allows for radial zones with different time‐steps and different interzone force evaluation frequencies, but reduces to the proven techniques of Duncan et al (1998) and Chambers (1999) for objects within the same zone. This allows new trade‐offs between force accuracy and run time.…”

“…In our first paper, McNeil & Nelson (2009), we addressed the numerical challenges of studying oligarchic models of short‐period exoplanet formation. Here, we apply a new code with a parallel implementation of those methods, to determine the ‘reference population’ of planets predicted by the fiducial models.…”

On the Formation of Super-Earths With Implications for the Solar System

“…The alternative super-Earth formation theory suggests that they form farther out in the disk where there is more solid material and then migrate inward (Terquem & Papaloizou 2007;McNeil & Nelson 2009;Ida & Lin 2010;Cossou et al 2014). In this scenario, planets grow through Earth size embryoembryo collisions.…”