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 t_s equaled to 1, 2, 5, or 10 days, and the results of calculations with different t_s were compared. Most of calculations were done with an integration time step equal to 1 day for ejection of bodies from Mercury and with t_s=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 i_ej, a velocity v_ej of ejection, and a time step t_s of integration. The values of v_ej varied from the parabolic velocity to 20 km/s. The values of i_ej varied from 15^o to 90^o. 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 T_end 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 t_s. 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 v_ej≤11.25 km/s, i.e., at v_ej slightly greater than the parabolic velocity. For T=10 Myr, the values p_E 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 v_ej equal to 11.5, 12, and 14 km/s, respectively. For T=T_end such values equaled to 0.25-0.35, 0.16-0.27, and 0.12-0.2, respectively. At v_ej 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 v_ej=12 km/s and v_ej≥14 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 p_v of collisions of the bodies with Venus were mainly about 0.13-0.22, 0.05-0.2, and 0.03-0.15 at v_ej equal to 11.5-12, 14, and 16.4 km/s, respectively. At T=T_end 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=T_end, respectively. This probability could be up to 0.2 for ejection from the back point of the Earth’s motion with v_ej≥16.4 km/s and i_ej≥45^o. 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=T_end, 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 v_ej≥11.3 km/s. The fraction of bodies ejected into hyperbolic orbits was mainly greater for a greater ejection velocity. For T=T_end it did not exceed 0.1 at v_ej≤12 km/s. At v_ej≥16 km/s and i_ej≥60 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 p_ma of a collision of a body ejected from Mars with Mars did not exceed 0.04 at v_ej≥5.3 km/s, and p_ma was considerable only at v_ej close to the parabolic velocity. The fraction p_me of bodies collided with Mercury was typically less than 0.06, with p_me>p_ma at v_ej≥6 km/s. Probabilities p_e and p_v of collisions of bodies ejected from Mars with the Earth and Venus were about 0.1-0.2 (each) at 5.1≤v_ej≤6 km/s, with p_v typically a little greater than p_e at v_ej≥5.2 km/s. Usually at 5.1≤v_ej≤8 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 v_ej≤6 km/s, but it could exceed 0.9 at v_ej=20 km/s.

Migration of bodies ejected from Mercury. Most of bodies ejected from Mercury fall back onto Mercury. Probabilities p_v of their collisions with Venus typically were about 0.2-0.3 at 6≤v_ej≤10 km/s. Probabilities of collisions of bodies with the Earth typically did not exceed 0.02 and 0.1 at v_ej≤8 km/s and v_ej≤15 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 108:18–36.