Numerical Determination of the Gravitational Cross Sections of an Accreting Binary

Abstract

A significant amount of work has been devoted to the study of small binary solar system objects. The majority of these binaries, especially among the near-earth or main belt asteroids have small radius ratios, implying a large difference in size between the primary and its companion. Farther from the sun, the binary fraction increases, with the Kuiper Belt having many known binaries with radius ratios of order unity. In this paper, we consider the runaway growth of a binary system in an accretionary stream of small particles. We perform brute-force integrations, each with 10 million test particles and numerically compute the gravitational cross sections for each member of the binary as a function of the system's separation and mass ratio. We show that the behavior of the cross section is complex, and it can be either diminished or enhanced depending on the orbital configuration. In the regime where gravitational focusing dominates the accretion process, we show that binaries grow towards smaller mass ratios than would be expected given single-body cross sections. Finally, we provide a grid of these cross sections for use in the future study of such systems.

Figures (11)

• Figure 1: A mock-up of the numerical experiment. The binary pair orbits in the xy plane, with an incident grid of particles approaching it along the x axis. There are two separate impact parameters $b_z$ and $b_y$ which parameterize the size of the incoming grid of particles, whose values are a function of the binary's properties.
• Figure 2: In the left panel we show the execution times of our various integrator configurations as a function of the number of test particles. All of the tested integrators show power-law behavior. In the right panel, we show the absolute difference between the primary cross sections derived with our BS integrator and that found with ias15. We see that if the tolerance is set lower than $10^{-3}$ each integrator converges to the ias15 result. Lower tolerances result in convergence - but to the wrong value. Vertical lines occur when the ias15 and BS integrators compute the exact same cross section. This happens rarely, but is more frequent in simulations with fewer test particles, as they are coarser.
• Figure 3: We show the normalized cross sections of a the binary as a function of the initial relative angle ($\phi$) between the secondary and the $x$ axis. The cross section can changes considerably as a function of this initial angle.
• Figure 4: The cumulative number of particles impacting the primary (solid lines), secondary (dashed lines) or exiting the simulation without hitting either body (dotted lines) as a function of time, for a variety of $v$ and $a$. In all cases shown here $m_2=1$. We see in general our cross sections converge quickly over the simulation and typically do not change after $t/t_{max} \approx 0.8$
• Figure 5: Here we show the computed numerical cross sections scaled by the geometric cross section $\sigma_{num}/\sigma_{geo}$ for each body in the system for several mass ratios against the predicted cross sections in the analytic limits derived in Eq(\ref{['eq:approx:fast']}). The analytic limits are solid lines and the numerically computed values are the scatter points. The first panel shows the relative cross section for the two bodies secondary against for very high velocities. We show the results for the the high velocity limit. In this limit, gravitational focusing is small, and shielding results in the effective cross sections of both bodies being smaller than their geometric cross sections. While the gravitational focusing factor is small, and so our assumption of no focusing in Eq(\ref{['eq:approx:fast']}) is appropriate, the approximation shows a small systematic vertical offset between the analytic and numerical predictions. This leads to the overestimation of the cross section in the limit $m_2 = 0$