Secular resonance sweeping of the main asteroid belt during planet migration

TL;DR

This paper investigates how the sweeping of the $ u_6$ secular resonance during planetary migration affected asteroid eccentricities, deriving constraints on Saturn's migration speed and implications for asteroid belt structure.

Contribution

It provides analytical expressions for eccentricity changes due to resonance sweeping and links asteroid eccentricity distributions to Saturn's migration rate.

Findings

Lower limit on Saturn's migration speed of ~0.15 AU/Myr.

Asteroid eccentricity distribution can become double-peaked due to resonance sweeping.

Possible migration rates of 0.8 to 4 AU/Myr linked to different asteroid belt states.

Abstract

We calculate the eccentricity excitation of asteroids produced by the sweeping $\nu_6$ secular resonance during the epoch of planetesimal-driven giant planet migration in the early history of the solar system. We derive analytical expressions for the magnitude of the eccentricity change and its dependence on the sweep rate and on planetary parameters; the $\nu_6$ sweeping leads to either an increase or a decrease of eccentricity depending on an asteroid's initial orbit. Based on the slowest rate of $\nu_6$ sweeping that allows a remnant asteroid belt to survive, we derive a lower limit on Saturn's migration speed of $\sim0.15\AU\My^{-1}$ during the era that the $\nu_6$ resonance swept through the inner asteroid belt (semimajor axis range 2.1--$2.8\AU$). This rate limit is for Saturn's current eccentricity, and scales with the square of Saturn's eccentricity; the limit on Saturn's migration rate could be lower if Saturn's eccentricity were lower during its migration. Applied to an ensemble of fictitious asteroids, our calculations show that a prior single-peaked distribution of asteroid eccentricities would be transformed into a double-peaked distribution due to the sweeping of the $\nu_6$. Examination of the orbital data of main belt asteroids reveals that the proper eccentricities of the known bright ($H \leq10.8$) asteroids may be consistent with a double-peaked distribution. If so, our theoretical analysis then yields two possible solutions for the migration rate of Saturn and for the dynamical states of the pre-migration asteroid belt: a dynamically cold state (single-peaked eccentricity distribution with mean of $\sim0.05$) linked with Saturn's migration speed $\sim 4\AU\My^{-1}$, or a dynamically hot state (single-peaked eccentricity distribution with mean of $\sim0.3$) linked with Saturn's migration speed $\sim 0.8\AU\My^{-1}$.