Proximities of Asteroids (1) Ceres, (2) Pallas, (3) Juno and (4) Vesta

J. Lazović, M. Kuzmanoski

Determination is carried out of some parameters, related to the first four minor planets, such as: mutual inclinations of their respective orbital planes, true anomalies of their orbits' relative nodes, the minimal (proximity) distances of their orbits and true anomalies of the proximity positions. The lowest mutual inclination of orbital planes is $4^{\circ}4,92761, for asteroids (1) Ceres and (4) Vesta. The highest inclination $36^{\circ}$ 65323 is between the orbital planes of asteroids (1) Ceres and (2) Pallas. The shortest proximity distance is 0.0626963 AU, occuring with the asteroid pair (1) Ceres and (2) Pallas. The minimum angular distances of the proximity positions from the relative node by this same asteroid pair were found to be somewhere about $0^{\circ}$.5 and $0^{\circ}$.6. The least difference of the true longitudes of the proximity positions is $0^{\circ}$,00853, being associated with the pair (1) Ceres and (3) Juno. Dynamical consequences of these proximities are investigated. The greatest change obtained is the one, $\Delta M_{4}$ = -0''.04, which would be produced by (1) Ceres in the orbit of the minor planet (4) Vesta during the interval $\Delta$t=32.2 days and with the angular widths $\Delta \upsilon _{1} = 7^{\circ}.22$, $\Delta \upsilon _{4} = 7^{\circ}.34$ of their being in the region of their sensible mutual perturbing action. With this pair we have $\Delta \omega _{4}$ = +0''.03 for 2512. With the pair (1) Ceres and (2) Pallas is $\Delta M_{2}$ = -0''.03 for $\Delta$t = 28.0 and $\Delta \upsilon _{1} = 5^{\circ}.30$, $\Delta \upsilon _{2} = 5^{\circ}.42$. And for (2) Pallas and (1) Ceres is $\Delta M_{1}$ = -0''.03 for $\Delta$t=12.6 and $\Delta \omega _{1}$= - 0''.03 for 16.2. The results obtained show that it is possible to make use of the proximities of these minor planets for the checking of estimates of their masses.