Carruba et al. (2013a) studied the long-term effect of close encounters with massive asteroids for asteroids in the orbital region of the Hygiea, Euphrosyne, and Pallas families. Since changes in heliocentric velocities that cause changes in proper elements are a function of the minimal velocity and distance at the encounter, it was shown analytically and numerically that the robustness of the frequency distribution function of changes in proper semi-major axis caused by encounters with those massive bodies (fdf hereafter), depended on how complete was the coverage of the phase plane of minimal distance and velocity at encounter, which ultimately was a function of the number of encounters. A question left unanswered by Carruba et al. (2013) was on the minimal time needed for the fdf to converge to the actual probality distribution function pdf. Since asteroid families ages are determined using Monte Carlo codes or numerical simulations that account for the dynamical mobility caused by Yarkovsky and YORP effects, obtaining a good estimate of the pdf and of completeness time can be a first step in including the long-term effect of the dynamical mobility caused by close encounters with massive asteroids in the estimate of asteroid families ages, and in assessing its importance. In this work we are planning to simulate the long term effect of close encounters of members of the Hygiea family halo (Carruba 2013) with (10) Hygiea itself, obtain an estimate of the pdf and of the completeness time, and study how close encounters of members of the Hygiea halo affect the age estimates of the family.A key parameter in studies of the long-term effect of close encounters with massive asteroids is the mass of the perturber, that, in many cases is known with large uncertainties. The dynamical model of the modern ephemeris (Aljbaae & Souchay, 2012) includes the perturbations from only about 300 asteroids whose influence on the trajectory of Mars is considered the most important. The other asteroids are replaced by a circular solid ring as in the model of Krasinsky et al. (2002). Yet, the large uncertainty on the mass values of the great majority of these asteroids severely limits the accuracy of the ephemeris and this has effects on our understanding of the orbits of terrestrial planets and members of asteroids families. For instance, the difference in mass for the asteroid (111) Ate between the DE405 (Standish 1998) and INPOP08 (Fienga et al. 2008) planetary ephemeris causes a difference in the predicted distance between the Earth-Moon and Mars. By studying the gravitational effect that (111) Ate and other asteroids whose mass uncertainty affects the accuracy of the ephemeris, such as (46) Hestia, (20) Massalia, (3) Juno, etc, on minor bodies that will experience close approaches with these asteroids in the next decades, we are planning to obtain better estimates of the masses of these objects. The effect that the uncertainty on the mass values of (20) Massalia, (3) Juno and other asteroids with relative large families has on the orbital evolution and pdf of its family members will then be investigates in the last stages of this project.
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