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 \sim 0.15 \mathrm { AU } \mathrm { My } ^ { -1 } during the era that the \nu _ { 6 } resonance swept through the inner asteroid belt ( semimajor axis range 2.1 – 2.8 \mathrm { 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 \leq 10.8 ) 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 \sim 0.05 ) linked with Saturn ’ s migration speed \sim 4 \mathrm { AU } \mathrm { My } ^ { -1 } , or a dynamically hot state ( single-peaked eccentricity distribution with mean of \sim 0.3 ) linked with Saturn ’ s migration speed \sim 0.8 \mathrm { AU } \mathrm { My } ^ { -1 } .