Simulation of binary asteroid populations under the influence of the YORP effects, tides, and collisions

Introduction

Binary asteroid systems are of fundamental interest to planetary science, as their orbital and rotational characteristics serve as sensitive indicators of the internal structure of small bodies (particularly porosity and tensile strength) and tidal dissipation parameters. According to photometric and radar observations, the fraction of binary systems in the Near-Earth Asteroid (NEA) population is 15±4% [2,3]. A significant population of binary objects and asteroid pairs is also present in the Main Asteroid Belt (MBA). Investigating the mechanisms of formation, secular evolution, and disruption of these systems is crucial for understanding the overall collisional and dynamical evolution of the Solar System. This work proposes a comprehensive approach to asteroid evolution, combining analytical solutions for equilibrium states with numerical Monte Carlo population modeling.

1. Formation Mechanisms, Evolution, and Equilibrium States

For small asteroids (diameter D≤10 km), the dominant mechanism of binary system formation is the rotational fragmentation of "rubble pile" bodies due to the normal YORP effect (NYORP). The NYORP effect generates a non-zero torque, leading to a secular increase in the angular spin rate ω. Upon reaching the critical angular velocity, centrifugal forces at the equator exceed gravitational forces, initiating mass shedding and satellite formation.

The subsequent dynamics of the newly formed system are governed by the transfer of angular momentum between the components via tidal interactions and the binary YORP (BYORP) effect. The tidal torque promotes the synchronization of the satellite's and primary's rotation with the orbital period. The BYORP effect, in turn, alters the orbital angular momentum.

The system's evolution may culminate in disruption into an asteroid pair (upon orbital expansion) or a merger into a contact-binary asteroid. However, stable dynamical equilibrium states exist that can halt this process. Specifically:

• NB-equilibrium: a balance between the normal YORP effect and the BYORP effect in doubly synchronous systems.
• NTBt-equilibrium: a complex equilibrium involving the tangential YORP effect (TYORP), which arises from the asymmetric thermal emission of surface structures (boulders), and tidal friction in systems where only the satellite is synchronized.

2. Observational Data Analysis

Observational data impose strict boundary conditions on modeling results. We analyzed a sample of binary asteroids and asteroid pairs in the inner Main Belt from the Johnston's Archive database.

The age distribution of asteroid pairs exhibits an exponential decline in the number of detected pairs older than 500 kyr, which is attributed to the kinematic divergence of their orbits in the phase space of orbital elements.

To correct for observational bias, a differential asteroid size distribution in the form of a power law N=aD^-b was applied. The theoretical estimate of the binary asteroid fraction, calculated solely as the ratio of the BYORP effect timescale to the combined NYORP and BYORP timescales, is 7.2%. This discrepancy with the empirically established 15% [3] demonstrates that simple decay models are insufficient, and the lifecycle of these systems is significantly prolonged by dynamical equilibrium states and stochastic perturbations.

3. Numerical Modeling Methodology of System Evolution

To investigate the dynamics of transitions between different synchronization states, a numerical integration program for orbital and rotational evolution was developed. The algorithm solves a system of ordinary differential equations, accounting for the combined action of the torques: NYORP, TYORP, BYORP, and tides.

Phase portraits of the systems demonstrate that, given appropriate combinations of YORP coefficient signs, phase trajectories asymptotically converge to stable equilibrium points (for both singly and doubly synchronous configurations), making these states powerful attractors in the evolution of binary asteroids.

4. Monte Carlo Population Modeling

To reproduce the macroscopic properties of the asteroid belt, a special population module was created. Instead of isolated modeling of a single system, the program integrates the evolutionary track of a large ensemble (population) of asteroids over timescales of 10⁷-10⁹ years.

Initial parameters are generated stochastically: sizes correspond to observed Size-Frequency Distributions (SFD), rotation periods are drawn from a Maxwellian distribution, and dimensionless YORP coefficients are assigned randomly.

The Monte Carlo modeling incorporates two key evolutionary channels:

• Secular YORP cycle: spin-up of a single body, rotational fission, tidal evolution of the binary system, reaching equilibrium or disrupting into a pair under the BYORP effect.
• Stochastic collisional cascade: collisions with the background asteroid population are simulated as a Poisson process. Micro-collisions alter surface morphology, leading to random walks (stochastic YORP), significantly extending the lifetime of asteroids before disruption, but also allowing an exit from stable YORP equilibria. Sub-catastrophic impacts can generate satellites of the SMATS (Smashed Target Satellites) [1] type or shatter the body into fragments, forming pairs bypassing the YORP constraints.

The objective of the module is the multi-parameter optimization of physical constants (energy dissipation factor Q, mass fraction of a newly formed secondary) to minimize deviations between the simulation results and empirical data (binary asteroid fraction, age spectrum of asteroid pairs, size, and rotational period distribution for binaries and pairs).

Conclusions

The integration of secular evolution models with Monte Carlo population synthesis allows for a quantitative comparison of different theoretical paradigms of asteroid dynamics. The simulations confirm that dynamical equilibrium states (involving tangential YORP and BYORP) are a necessary factor for maintaining the observed binary population fraction (about 15%), as without them, the systems disrupt too rapidly. 

References

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2. Margot J.-L., Pravec P., Taylor P., Carry B., Jacobson S. (2015). Asteroid systems: binaries, triples, and pairs. Asteroids IV, 355, 373.

3. Pravec P., Scheirich P., et al. (2016). Icarus, 267, 267-295.