Migration of bodies ejected from the terrestrial planets

The model and initial data. At the late stages of formation of the terrestrial planets and at the late-heavy bombardment, bodies collided with these planets. Some material could be ejected at such collisions. The motion of bodies ejected at such collisions from the Earth, Mars, and Mercury was studied under the gravitational influence of the Sun and all planets. The symplectic code from the SWIFT integration package (Levison, Duncan, 1994) was used for integration of the motion equations. The considered time integration step ts equaled to 1, 2, 5, or 10 days, and the results of calculations with different ts were compared. Most of calculations were done with an integration time step equal to 1 day for ejection of bodies from Mercury and with ts=5 days for other planets. In each calculation variant, the motion of 250 bodies ejected from the Earth was studied for the fixed values of an ejection angle iej, a velocity vej of ejection, and a time step ts of integration. The values of vej varied from the parabolic velocity to 20 km/s. The values of iej varied from 15o to 90o. In most calculations, bodies started directly from the surface of the considered planet. In each variant, bodies started from one of six considered opposite points (Ipatov, 2024). Bodies that collided with planets or the Sun or reached 2000 AU from the Sun were excluded from integration. The motion of bodies was studied during the dynamical lifetime Tend of all bodies, which usually (exclusive for ejection velocities close to the parabolic velocity) equaled to a few hundreds of Myr. The fractions of bodies collided with different planets were calculated, and they were close for different considered ts. The fraction of bodies ejected from the Earth and later collided with the Moon is discussed in (Ipatov, 2024).Migration of bodies ejected from the Earth. Most of bodies ejected from the Earth fell back onto the Earth at vej11.25 km/s, i.e., at vej slightly greater than the parabolic velocity. For T=10 Myr, the values pE of the probability of a collision of a body with the Earth were about 0.2-0.3, 0.12-0.2 and 0.12-0.18 at vej equal to 11.5, 12, and 14 km/s, respectively. For T=Tend such values equaled to 0.25-0.35, 0.16-0.27, and 0.12-0.2, respectively. At vej about 12-14 km/s, mean velocities of collisions with the Earth and the Moon were about 14-20 and 10-16 km/s, respectively. The ratio of the probability of collisions of bodies with the Earth to the probabilities of collisions of bodies with other planets and the Sun usually decreased with time. The probability of collisions of bodies ejected from the Earth with the Moon moving in its present orbit was about 0.006-0.01 and 0.004-0.008 at vej=12 km/s and vej14 km/s, respectively (Ipatov, 2024). It was concluded that a large Moon embryo should be formed close to the Earth in order to accumulate material not rich in iron. For more efficient growth of the Moon embryo, it is desirable that after some collisions of impactor bodies with the Earth, the ejected bodies did not simply fly out of the crater, but some of the matter went into orbits around the Earth, as in the multi-impact model.For T=10 Myr, the probabilities pv of collisions of the bodies with Venus were mainly about 0.13-0.22, 0.05-0.2, and 0.03-0.15 at vej equal to 11.5-12, 14, and 16.4 km/s, respectively. At T=Tend such values were mainly about 0.23-0.37, 0.2-0.4, 0.17-0.3. The total number of ejected bodies delivered to the Earth and Venus probably did not differ much. Probabilities of collisions of bodies with Mercury were mainly less than 0.05 and 0.1 at T=10 Myr and T=Tend, respectively. This probability could be up to 0.2 for ejection from the back point of the Earths motion with vej16.4 km/s and iej45o. The probabilities of collisions of bodies with Mars were smaller than those with Mercury and did not exceed 0.012 and 0.025 at T=10 Myr and T=Tend, respectively. More material ejected from the Earth was delivered to Mercury than to Mars. The probabilities of collisions of bodies with Jupiter were of the order of 0.001. The fraction of bodies collided with the Sun often was between 0.2 and 0.5 at vej11.3 km/s. The fraction of bodies ejected into hyperbolic orbits was mainly greater for a greater ejection velocity. For T=Tend it did not exceed 0.1 at vej12 km/s. At vej16 km/s and iej60 degrees, all bodies were ejected from the Solar System if they started from the front (in the direction of the motion) point of the Earth.Migration of bodies ejected from Mars. The probability pma of a collision of a body ejected from Mars with Mars did not exceed 0.04 at vej5.3 km/s, and pma was considerable only at vej close to the parabolic velocity. The fraction pme of bodies collided with Mercury was typically less than 0.06, with pme>pma at vej6 km/s. Probabilities pe and pv of collisions of bodies ejected from Mars with the Earth and Venus were about 0.1-0.2 (each) at 5.1vej6 km/s, with pv typically a little greater than pe at vej5.2 km/s. Usually at 5.1vej8 km/s more than a half of bodies collided with the Sun. The probability of ejection of a body into a hyperbolic orbit was less than 0.1 at vej6 km/s, but it could exceed 0.9 at vej=20 km/s.Migration of bodies ejected from Mercury. Most of bodies ejected from Mercury fall back onto Mercury. Probabilities pv of their collisions with Venus typically were about 0.2-0.3 at 6vej10 km/s. Probabilities of collisions of bodies with the Earth typically did not exceed 0.02 and 0.1 at vej8 km/s and vej15 km/s, respectively. Less than 20% of ejected bodies collided with the Sun. The fraction of bodies ejected into hyperbolic orbits did not exceed 0.01 for most of calculation variants. Acknowledgements: The studies were carried out under government-financed research project for the Vernadsky Institute.References: Ipatov S.I. (2024) Solar System Research 58:94-111. https://doi.org/10.1134/S0038094624010040.Levison H.F. and Duncan M.J. (1994) Icarus 108:1836.