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Register a new userCollisions Between Gravity-Dominated Bodies: 1. Outcome Regimes and Scaling Laws
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Collisions are the core agent of planet formation. In this work, we derive an analytic description of the dynamical outcome for any collision between gravity-dominated bodies. We conduct high-resolution simulations of collisions between planetesimals; the results are used to isolate the effects of different impact parameters on collision outcome. During growth from planetesimals to planets, collision outcomes span multiple regimes: cratering, merging, disruption, super-catastrophic disruption, and hit-and-run events. We derive equations (scaling laws) to demarcate the transition between collision regimes and to describe the size and velocity distributions of the post-collision bodies. The scaling laws are used to calculate maps of collision outcomes as a function of mass ratio, impact angle, and impact velocity, and we discuss the implications of the probability of each collision regime during planet formation. The analytic collision model presented in this work will significantly improve the physics of collisions in numerical simulations of planet formation and collisional evolution. (abstract abridged)
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Numerical simulations of the stochastic end stage of planet formation typically begin with a population of embryos and planetesimals that grow into planets by merging. We analyzed the impact parameters of collisions leading to the growth of terrestrial planets from recent $N$-body simulations that assumed perfect merging and calculated more realistic outcomes using a new analytic collision physics model. We find that collision outcomes are diverse and span all possible regimes: hit-and-run, merging, partial accretion, partial erosion, and catastrophic disruption. The primary outcomes of giant impacts between planetary embryos are approximately evenly split between partial accretion, graze-and-merge, and hit-and-run events. To explore the cumulative effects of more realistic collision outcomes, we modeled the growth of individual planets with a Monte Carlo technique using the distribution of impact parameters from $N$-body simulations. We find that fewer planets reached masses $>0.7 M_{rm Earth}$ using the collision physics model compared to simulations that assumed every collision results in perfect merging. For final planets with masses $>0.7 M_{rm Earth}$, 60% are enriched in their core-to-mantle mass fraction by >10% compared to the initial embryo composition. Fragmentation during planet formation produces significant debris ($sim15$% of the final mass) and occurs primarily by erosion of the smaller body in partial accretion and hit-and-run events. In partial accretion events, the target body grows by preferentially accreting the iron core of the projectile and the escaping fragments are derived primarily from the silicate mantles of both bodies. Thus, the bulk composition of a planet can evolve via stochastic giant impacts.
We aim to create deterministic collisions between orbiting bodies by applying a time-dependent external force to one or both bodies, whether the bodies are mutually repulsive, as in the two- or multi-electron atomic case or mutually attractive, as in the planetary-orbit case. Specifically, we have devised a mathematical framework for causing deterministic collisions by launching an inner orbiting body to a higher energy such that this inner body is guaranteed to collide with the outer body. Our method first expresses the problem mathematically as coupled nonlinear differential equations with a time-dependent driving force and solves to find a feasible solution for the force function. Although our calculation is based strictly on classical physics, our approach is suitable for the case of helium with two highly excited electrons and is also valid for creating collisions in the gravitational case such as for our solar system.
Gravity inversion allows us to constrain the interior mass distribution of a planetary body using the observed shape, rotation, and gravity. Traditionally, techniques developed for gravity inversion can be divided into Monte Carlo methods, matrix inversion methods, and spectral methods. Here we employ both matrix inversion and Monte Carlo in order to explore the space of exact solutions, in a method which is particularly suited for arbitrary shape bodies. We expand the mass density function using orthogonal polynomials, and map the contribution of each term to the global gravitational field generated. This map is linear in the density terms, and can be pseudo-inverted in the under-determined regime using QR decomposition, to obtain a basis of the affine space of exact interior structure solutions. As the interior structure solutions are degenerate, assumptions have to be made in order to control their properties, and these assumptions can be transformed into scalar functions and used to explore the solutions space using Monte Carlo techniques. Sample applications show that the range of solutions tend to converge towards the nominal one as long as the generic assumptions made are correct, even in the presence of moderate noise. We present the underlying mathematical formalism and an analysis of how to impose specific features on the global solution, including uniform solutions, gradients, and layered models. Analytical formulas for the computation of the relevant quantities when the shape is represented using several common methods are included in the Appendix.
Disruptive collisions have been regarded as an important process for planet formation, while non-disruptive, small-scale collisions (hereafter called erosive collisions) have been underestimated or neglected by many studies. However, recent studies have suggested that erosive collisions are also important to the growth of planets, because they are much more frequent than disruptive collisions. Although the thresholds of the specific impact energy for disruptive collisions (Q_RD^*) have been investigated well, there is no reliable model for erosive collisions. In this study, we systematically carried out impact simulations of gravity-dominated planetesimals for a wide range of specific impact energy (Q_R) from disruptive collisions (Q_R ~ Q_RD^*) to erosive ones (Q_R << Q_RD^*) using the smoothed particle hydrodynamics method. We found that the ejected mass normalized by the total mass (M_ej/M_tot) depends on the numerical resolution, the target radius (R_tar) and the impact velocity (v_imp), as well as on Q_R, but that it can be nicely scaled by Q_RD^* for the parameter ranges investigated (R_tar = 30-300 km, v_imp = 2-5 km/s). This means that M_ej/M_tot depends only on Q_R/Q_RD^* in these parameter ranges. We confirmed that the collision outcomes for much less erosive collisions (Q_R < 0.01 Q_RD^*) converge to the results of an impact onto a planar target for various impact angles and that M_ej/M_tot = C * QR/QRD* holds. For disruptive collisions (Q_R ~ Q_RD^*), the curvature of the target has a significant effect on Mej/Mtot. We also examined the angle-averaged value of M_ej/M_tot and found that the numerically obtained relation between angle-averaged M_ej/M_tot and Q_R/Q_RD^* is very similar to the cases for 45-degree impacts. We proposed a new erosion model based on our numerical simulations for future research on planet formation with collisional erosion.
Tidal dissipation is known as one of the main drivers of the secular evolution of planetary systems. It directly results from dissipative mechanisms that occur in planets and stars interiors and strongly depends on the structure and dynamics of the bodies. This work focuses on the mechanism of viscous friction in stars and planetary layers. A local model is used to study tidal dissipation. It provides general scaling laws that give a qualitative overview of the different possible behaviors of fluid tidal waves. Furthermore, it highlights the sensitivity of dissipation to the tidal frequency and the roles played by the internal parameters of the fluid such as rotation, stratification, viscosity and thermal diffusivity that will impact the spins/orbital architecture in planetary systems.
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