Cluster compaction of two-dimension spherical particles binary mixture as model of forming process of an asteroid

Several two-dimension spherical particles are under influence of gravitational forces of each other and as they are colliding repulsion forces in form of linear spring-dashpot prevent them from collapsed into a single point. Gravitational constant larger than G is used for better and faster stabilization in forming cluster of the particles, accompanied with values of kN ≈ 104 N/m and yn ≈ 0.5 N·s/m As initial conditions the particles are placed randomly with separation distances among each other. Molecular dynamics method implementing Euler algorithm is used to simulate the developing of particles cluster, which is intending to mimic the process of an asteroid forming. Time step of Δt = 102 s is chosen and results are reported every some period from 1 s to 1000 s, where after each period all particles velocity are forced to be zero. It is observed that not only physical parameters influencing the compaction of the asteroid but also simulation parameters.


Introduction
Asteroids are small bodies in our solar system whose surfaces have various features. There are high and low topography, large and small craters, boulders, small rocky materials, and regolith. From spectroscopy observations, taxonomy of asteroids provides several major surface types, i.e. carbonaceous, silicate and metallic. Solar wind and other space weathering process are believed to alter the spectral color of many types of asteroids. Asteroid rotates on its axis during its revolution orbiting the Sun. Some properties of internal structure of asteroid can be derived from its lightcurve. There are a barrier of rotation period of 2.2 hours for sizes of 1 km or larger, and remarkably, for sizes below 100 m the rotation periods can be 5 minutes [1]. This constraints its bulk density and level of (macro)porosity. Two examples of asteroids having distinct internal structure are 253 Mathilde (bulk density 1.3 g/cm 3 , porosity 50%) and 216 Kleopatra (bulk density 6 g/cm 3 , equilibrium figure of iron/metal) [1]. Results from some asteroid missions, e.g. to 433 Eros and 25143 Itokawa, have shown that asteroids are gravitational aggregate bodies (rubble-pile), not monolithic structure. However, for much smaller size of asteroids, whose rotation periods are very fast, it is believed that they are indeed monolithic bodies. There are many unexplained features and distribution of materials on asteroid surface. Smooth regolith and large rocky materials can be found side by side. Impact can shake seismically the asteroid body, so that surface and sub-surface materials can be changed and mixed [2].
Recent exploration of Hayabusa mission to Itokawa asteroid has successfully brought high resolution images showing how boulders, gravels, and sands distributed on its surface, where they have some preferred orientation at some places but not in other places [3]. Occurrences of larger particles, e.g. boulders, above smaller particles, e.g. gravel or sand, are addressed to the well-known phenomenon in granular world, i.e. the Brazil-nut effect (BNE) [4]. Simulation in 2-d, which is conducted under influenced only of gravitation of granular particles in the system, shows that there is a ring of larger particles formed near the edge of the granular aggregate [5]. Even further, collision of two such aggregates for much smaller particles has also been reported [6]. On earth surface, where gravitation is always toward center of the planet, decrease of porosity and increase of contactopy is observed during the process of BNE [7], which is interesting to observe it in the system under influence of gravitation of its own particles.

Simulation
Particles which later forming an asteroid are simplified into 2-d spherical particles, each with the same density ρ, while for each particle i the diameter Di and mass mi may be varied. Only two types of forces between particles are considered in this work. The former is gravitation force while the later is normal force. In order to get particles motion variables such as velocity and position, molecular dynamics (MD) method is used, which implementing Euler algorithm.

Forces formulation
Two types of force are considered, gravitation force FG and and normal force FN (2)

Molecular dynamics (MD) method
The MD method has two steps: i. calculation of acceleration of all particle, and ii. calculation new velocity and new position of all particle. The first step is conducted using Newton second law of motion Equations (4) are executed repeatedly until termination condition is reached.

Results and discussion
Number of contact points per particle is not monotonically increasing as number of layer increased. It fluctuates as shown in figure 2. Since forms of pores are similar for particle configuration with same diameter it is found that ϕ ≈ 1.43×10 -4 independent to L. In previous work [10] the term of contactopy C is used instead of number of contact points per particles Nc / N, where C ≈ Nc. From its initial configuration cluster of particles will evolve to more compact configuration as shown in figure 3. Similar results will be obtained for this work with different slopes. Final number of contact points per particle and porosity will be predicted since homogenous size of particle will again form hexagonal close packed configuration.

Summary
It can be summarized in this work that some programs have been developed to simulate compaction of two-dimension spherical particles as model of asteroid forming process. Number of contact points per particles fluctuates with number of layer for hexagonal close packed configuration. Porosity for three particles pairs is relatively constant.  Evolution of contactopy C for non-homogenous particle size [10].