Exchange of ejected material between the terrestrial planets and the Moon

Introduction. The planets also could get similar material due to the ejection of material from their surface when they collided with planetesimals and other bodies. I studied the probabilities p_E, p_V, p_Me, p_Ma, p_Moon, and p_Sun of collisions of bodies ejected from a terrestrial planet with the Earth, Venus, Mercury, Mars, Moon, and the Sun. In each calculation variant, the motion of 250 bodies ejected from a planet was studied at fixed values of the ejection angle i_ej, ejection velocity v_ej, and ejection point. The motion of the ejected bodies was studied during the dynamical lifetime of all bodies, which was usually about a few hundreds Myrs. The gravitational influence of the Sun and all eight planets was taken into account. In different calculation variants, i_ej varied from 15° to 90°, and v_ej varied from the parabolic velocity on the surface of a planet to 20 km/s. Ejection of material was studied from 6 opposite points on the surface of a planet. The probabilities of collisions of bodies ejected from the Earth with the Moon were studied in (Ipatov, 2024a), and the probabilities of collisions of such bodies with planets were studied in (Ipatov, 2025). The probabilities of collisions of bodies ejected from Mars, Mercury, and Venus were briefly discussed in (Ipatov, 2024b,c). Below in this abstracts the probabilities are presented for a whole considered time interval. The fraction of bodies delivered to the Moon in its present orbit was smaller than that delivered to the Earth by a factor of 20-30.

Ejection of bodies from the Earth and the Moon. The fraction of bodies ejected from the Earth and falling back onto the Earth was about 0.2-0.3. At 11.5≤v_ej≤12 km/s, p_V was in the range of 0.22-0.37. The total number of bodies delivered to the Earth and Venus probably did not differ much. The values of p_Me and p_Ma were in the ranges of 0.02-0.08 and 0-0.025, respectively. The probability of a collision of bodies ejected from the Earth with the Moon in its present orbit was about 0.01-0.02 at v_ej=11.3 km/s and 0.005-0.008 at v_ej=16.4 km/s. At v_ej=20 km/s all bodies ejected from the “forward” point on the surface of the Earth or Mars could be ejected into hyperbolic orbits. The probabilities of collisions of ejected bodies with planets were similar for bodies ejected from the Earth and the Moon in its present orbit but for different ejection velocities.

Ejection of bodies from Mars. Some bodies ejected from Mars could collide with the Earth in 0.1 Myr, and some - in hundreds of Myrs. The values of p_E typically did not exceed 0.16. At v_ej≥5.2 km/s, the values of p_V usually exceeded p_E and mainly were less than 0.2. The values of p_Ma were mainly about 0.04-0.25 and 0-0.04 at 5.05≤v_ej≤5.3 and 5.5≤v_ej≤20 km/s, respectively. For the ‘back’ (relative to motion of the planet) point B, p_Ma was usually smaller than for ‘middle’ points (which are on the plane that passes through the Sun and the planet and is perpendicular to the orbital plane). The values of p_Me usually were less than 0.06 and were larger than p_Ma at v_ej≥6 km/s. For ejections from ‘middle’ points, p_Me was mainly about 0.02-0.08 at 5.05≤v_ej≤20 km/s.

Ejection of bodies from Mercury. Most of the bodies ejected from Mercury fell back onto Mercury. The values of p_E usually did not exceed 0.02 and 0.1 at v_ej less than 8 and 15 km/s, respectively. The values of p_V were usually about 0.2-0.3 at 6≤v_ej≤10 km/s. The values of p_V were usually by an order of magnitude greater than p_E. At lower ejection velocities, this difference was greater. For most calculation variants, the fraction of bodies ejected into hyperbolic orbits did not exceed 0.01.

Ejection of bodies from Venus. In this paragraph we discuss the values of the probabilities of collisions with planets for bodies ejected from ‘middle’ points on Venus at 10.4≤v_ej≤16 km/s. In this case, the value of p_Me was about 0.01-0.2, p_V was about 0.3-0.8, and p_E was about 0.04-0.2 (several times less than p_V). At v_ej=16 km/s p_Me was approximately several times larger (from 2 to 10 depending on the ejection point and i_ej) than at v_ej=10.4 km/s. The values of p_V have some tendency (but not for all initial data) to be smaller at higher v_ej. The values of p_Ma often did not exceed 0.01 for all initial data.

For “middle” ejection points at v_ej=20 km/s, p_Me was approximately 0.1-0.15, values of p_V were approximately 0.23-0.35, values of p_E were about 0.04-0.1 at i_ej=45^o, and could be about 0.2 at i_ej=89^o. For the ‘back’ point B, p_Me was about 0.02-0.35 (with a maximum at v_ej=20 km/s), p_V was about 0.4-0.8, p_E was mainly about 0.04-0.2 (but equaled to 0.01 at v_ej=20 km/s and i_ej=89^o). For the ‘forward’ point W at v_ej≤12 km/s, p_Me was about 0.01-0.1, p_V was about 0.3-0.8, and p_E was about 0.1-0.2.

Ejection of dust particles. For ejections of 1-100 micron dust particles from a terrestrial planet, the probability of a collision of a dust particle with the planet usually did not exceed 0.01. The ejected particles mostly fell onto the Sun or were ejected from the Solar System. For micron particles, the ratio between ejected and colliding particles (greater or less than 1) depended on the ejection point and velocity. Most particles with a diameter of about 10-100 microns fell onto the Sun. The dynamical lifetime of most 1-100 micron particles did not exceed 1 Myr.

Acknowledgements: The studies of delivery of material to the Moon or from the Moon were supported by the Russian Science Foundation, project 25-17-00051. Other studies were carried out under government-financed research project for the Vernadsky Institute.

References: Ipatov S.I. (2024a) Solar System Research 58:94-111. https://doi.org/10.1134/S0038094624010040, http://arxiv.org/abs/2405.19797. Ipatov S.I. (2024b) Solar System Research 58. Suppl. 1. P. S50-S63. https://doi.org/10.1134/S0038094623600105, http://arxiv.org/abs/2411.05436. Ipatov S.I. (2024c) Modern astronomy: from the Early Universe to exoplanets and black holes. P. 904-909. https://doi.org/10.26119/VAK2024.143, https://arxiv.org/abs/2501.00134. Ipatov S.I. (2025) Icarus 425, id. 116341 (24 p.). https://doi.org/10.1016/j.icarus.2024.116341, http://arxiv.org/abs/2411.04218.