Deterministic Computer Simulations of Grazing Impacts on Planetary Surfaces

Many bodies in the solar system have features which could conceivably have been formed by a grazing impact with a comet or asteroid. We present the results of deterministic computer simulations of various objects striking a terrestrial planet at a grazing angle. The system is modeled using a combination of the Material Point Method (MPM) and classical planetary dynamics. The impact exhibits three distinct regimes: (i) the initial stage where rapid ejecta leaves the planet in a nearly straight line, (ii) the intermediate stage where the ejecta begins to curve in towards the planet and the trench is being created on the surface and the (iii) the long term stage where the trench is created and any paths exhibited by the ejecta are stable capture orbits. In the case of Mars, we show that a grazing impact can not only dig a trench which has the same general morphology as Valles Marineris but also can create ejecta which orbits the planet at distances comparable to those for current Martian satellites. I. BACKGROUND & INTRODUCTION Valles Marineris is the deepest trench known to exist on a terrestrial body in the solar system. There is much curiosity as to how Valles was formed. Comparing the trench to a typical water-carved structure on Earth (Figure 1) reveals the lack of the tortuous network typical for rivers and tributaries. In addition, martian surface gravitational acceleration is about 1/3 that of the Earth on its surface and so it seems unlikely that water could have carved such a prominent structure on Mars. There is evidence that water or dry sand [1] ran down the sides of Valles at one point in time, but the erosion incurred from it seems clearly secondary and not related to the channel’s creation [2]. Even though it is thought that Valles could be related to a fault in the martian crust [3], it could be that the fault was not active in the channel’s formation. Could it be that a grazing impact created the trench? It is possible that a grazing impact would leave evidence which is additional to an obvious impact site. If a grazing impact was indeed responsible for carving Valles, it is likely that some of the ejecta would have been moving fast enough to leave Mars altogether or, for the proper conditions, could have ended up in orbit about Mars. There are two natural satellites for Mars: Phobos and Deimos (Figure 2). They orbit Mars in planes fairly close the planet’s equator, which is also close to where Valles is located. Moreover the two moons are not spherical in shape but show a history of trauma and collision. If Valles were cut by a ‡ current address: Iowa State University (Ames, Iowa). E-mail to massina4@iastate.edu AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 8, NO. 1 (2009) 16 Figure1. Infrared view of Earth’s Grand Canyon (left) and Valles Marineris in the visible region of the spectrum (right). grazing impact, it seems reasonable to think that objects like Phobos and Deimos could have resulted from the debris. The purpose of this work is to simulate grazing impacts on a terrestrial planetary surface and determine (i) if the results of a grazing impact on the surface of the planet obtained at the impact site are consistent with the general morphology of Valles and (ii) if reasonable conditions for a grazing impact can lead to debris in stable orbits about the planet and, in the case of Mars, satellites with orbits having elements similar to those of Phobos and Deimos. II. COMPUTATIONAL DETAILS The code developed consists of a combination of two different methods. The first deals with the impacting body as it travels towards the planet’s surface, and is derived from classical Newtonian planetary dynamics. Since there are no collisions involved in this first phase of the simulation, it entails a two-particle center of mass attraction algorithm. Such a method is effective at economizing simulation time. This gravitational method is used until the particles enter a zone around the planet where collisions can happen and the previously mentioned center of mass dynamics are inadequate. The second algorithm that governs the collision part of the simulation is the Material Point Method. In order to trust the results of any program, it must to be validated. This was accomplished by running simulations with the intent of reproducing kinetic and potential energy profiles of various twodimensional collisions between balls which others have simulated, as well as energy conservation in the collision of three dimensional balls. The results obtained show that the algorithm utilized for our MPM code is consistent with Newtonian physics and accurately represents a temporal mechanical sequence. The Material Point Method (MPM) employs an algorithm developed and evaluated by Z. Chen and R. Brannon [4]. MPM is advantageous because it doesn't require a complex algorithm to simulate collisions or material failure. Instead, MPM maps the mass and momentum of groups of particles in the simulation to a background grid and utilizes the conservation equations of mass as well as that of momentum to ultimately advance the system through time. The use of the background grid also avoids entanglement associated with other contactbased algorithms. To begin with, the initial conditions for the simulation include a collection of particles with initial positions { 0 p x }, velocities { 0 p v } and boundary conditions chosen so as to reflect the important physics of the actual system as much as possible. Since the AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 8, NO. 1 (2009) 17 Figure 2. The two natural satellites of Mars, Phobos and Deimos, shown at their distances from Mars to scale (top) and in detail (bottom left and right). glancing collision takes place over a very limited region of the planet, a patch of particles (the impact zone) is utilized. Next, the background grid is defined, where computational space is separated into a defined number of grid cells in which nodes are placed in each corner of each cell. The mass Mp of each particle p at time t is mapped to the nodes i of the grid cells based on the shape function Ni ( t p x ), resulting in the mass mi t of node i at time t: