The model and initial data used for calculations. Schwarz et al. (2018) studied migration of exocomets in the Proxima Centauri system. Besides the exoplanet with a semi-major axis a_b=0.0485 AU located in a habitable zone, they also considered the exoplanet “c” with a semi-major axis a_c from 0.06 to up to 0.3 AU (for test calculations up to 0.7 AU). Kervella et al. (2020) and Benedict and McArthur (2020) considered that the semi-major axis a_c of the exoplanet “c” equals to 1.489±0.049 AU. In the first series of calculations, according to Kervella et al. (2020), I considered a star with a mass equal to 0.122 of the solar mass, and two exoplanets with the following semi-major axes and masses: a_b=0.0485 AU, a_c=1.489 AU, m_b=1.27m_E and m_c=12m_E, where m_E is the mass of the Earth. For the exoplanet “b”, the initial eccentricity e_b and initial inclination i_b were considered to be equal to 0, and the initial eccentricity e_c of the exoplanet “c” equaled to 0 or 0.1. Initial inclination of the exoplanet “c” was considered to be i_c=e_c/2=0.05 rad or i_c=e_c=0. For interest, I also considered i_c=152^o; such calculations characterize the case when orbits of planetesimals were inclined to the orbit of the exoplanet. In the second series of calculations, as in Benedict and McArthur (2020), I considered a_b = 0.04857 AU, e_b = 0.11, m_b = 1.17 m_E, a_c = 1.489 AU, e_c = 0.04, m_c = 7 m_E. I supposed i_b = i_c = 0. m_c is different in the two series. In both series of calculations, the densities of the exoplanets “b” and “c” were considered to be equal to densities of the Earth and Uranus, respectively. In different calculation variants, initial semi-major axes of planetesimals were in the range from a_min to a_max=a_min+0.1 AU, with a_min from 1.2 to 1.7 AU with a step of 0.1 AU. Initial eccentricities e_o of planetesimals equaled to 0 or 0.15 for the first series of calculations, and e_o=0.02 or e_o=0.15 for the second series. Greater initial eccentricities could be a result of the mutual gravitational influence of planetesimals. Initial inclinations of the planetesimals equaled to e_o/2 rad. 250 planetesimals were considered in each calculation variant. The motion of planetesimals and exoplanets was calculated with the use of the symplectic code from Levison and Duncan (1994). Considered time interval exceeded 20 Myr. Based on the obtained arrays of orbital elements of migrated planetesimals and exoplanets stored with a step of 100 yr, I calculated the probabilities of collisions of planetesimals with the exoplanets. The calculations were made similar to those in (Ipatov and Mather, 2003, 2004a-b; Ipatov, 2019a-b; Marov and Ipatov, 2018), which had been made for the planets of the Solar System, but for different masses and radii of a star and exoplanets. If the probability of a collision with an exoplanet for some planetesimal reached 1 with time (it was obtained for a few planetesimals), then for a later time this planetesimal did not considered for calculation of the mean probability for the calculation variant.

Probabilities of collisions of planetesimals with the exoplanet “b” and “d”. For the second series of calculations, the probability p_b of a collision of one planetesimal, initially located near the orbit of the exoplanet “c”, with the exoplanet “b” was non-zero in 5 among 18 variants at e_o=0.02 and in 3 among 6 variants at e_o=0.15. At e_o=0.02 for the five variants, p_b equaled to 0.004, 0.004, 1.28×10^-5, 0.00032 и 9.88×10^-5. The mean value of p_b for one of 4500 planetesimals equaled to 4.7×10^-4, but among them there were two planetesimals with p_b=1. At e_o=0.15 for three variants, p_b equaled to 0.008, 0.004 and 3.6×10^-6. The mean value of p_b for one of 1500 planetesimals equaled to 2.0×10^-3, but among them there were three planetesimals with p_b=1. For both e_o=0.02 and e_o=0.15 in one of 24 variants p_b=0.008, in three variants p_b=0.004, and in other 4 variants p_b was between to 4×10^-6 and 3×10^-4. For all three considered variants of the first series at e_c=0.1 and e_o=0.15, the values of p_b were in the range 0.008-0.019. For other calculations of the first series, p_b=0. For the second series of calculations, the probability p_d of a collision of a planetesimal from the zone of the orbit of the exoplanet “c” with the exoplanet “d” (a_d=0.02895 AU, m_d=0.29m_E, e_d=i_d=0) was nonzero only for seven variants (among 24). At e_o=0.02 for four variants, p_d equaled to 0.004, 0.00068, 0.000143 and 3.0×10^-5. The mean value of p_d over 4500 planetesimals equaled to 2.7×10^-4, but for one planetesimal p_d=1. At e_o=0.15 for three variants, p_d equaled to 0.008, 0.004 and 2.58×10^-5. The mean value of p_d over 1500 planetesimals equaled to 2.0×10^-3, but for three planetesimals p_d=1. For the second series, the mean values of p_b and p_d averaged over 6000 planetesimals equaled to 8.5×10^-4 and 7.0×10^-4. Only one of several hundreds of planetesimals reached the orbits of the exoplanet “b” and “d”, but the probabilities p_b and p_d of a collision of one exoplanetesimal with these exoplanets (averaged over thousands planetesimals) are greater than the probability of a collision with the Earth of a planetesimal from the zone of the giant planets in the Solar System. The latter probability for most calculations with 250 planetesimals was less than 10^-5 per one planetesimal (Ipatov, 2019b). Therefore, a lot of icy material could be delivered to the exoplanets “b” and “d”.

Probabilities of collisions of planetesimals with the exoplanet “c”. For the first series of calculations at i_с=e_с=0 and e_o=0.15, the values of the probability p_с of a collision of one planetesimal, initially located near the exoplanet “c”, with this exoplanet were about 0.06-0.1. For i_c=e_c/2=0.05 and e_o=0.15, p_с was about 0.02-0.04. For the second series of calculations, p_с was about 0.1-0.3, exclusive for a_min=1.4 AU and e_o=0.02 when p_с was about 0.7-0.8 (and the main growth was before T=1 Myr). Usually there was a small growth of p_c after 20 Myr. For both series of calculations, most of planetesimals were usually ejected into hyperbolic orbits in 10 Myr. The ratio of the number of planetesimals ejected into hyperbolic orbits to the number of planetesimals collided with the exoplanets usually exceeded 1 if the number of planetesimals decreased by a factor of several. For variants of the second series, this ratio was less than 1 only at a_min=1.4 AU and e_o=0.02. In some calculations a few planetesimals could still move in elliptical orbits after 100 Myr.

Conclusions. For the Proxima Centauri planetary system, most of planetesimals from the vicinity of the exoplanet “c” with a semi-major axis a_c of about 1.5 AU were ejected into hyperbolic orbits in 10 Myr. Some planetesimals could collide with this exoplanet after 20 Myr. Only one of several hundreds of planetesimals from the vicinity of this exoplanet reached the orbit of the exoplanet “b” with a semi-major axis a_b=0.0485 AU or the orbit of the exoplanet “d” with a semi-major axis a_d=0.029 AU, but the probability of a collision of such planetesimal (that reached the orbits) with the exoplanets b and d can reach 1, and the collision probability averaged over all planetesimals from the vicinity of the exoplanet “c” was ~10^-3. If averaged over all considered planetesimals from the vicinity of exoplanet “c”, the probability of a collision of a planetesimal with the exoplanet “b” or “d” is greater than the probability of a collision with the Earth of a planetesimal from the zone of the giant planets in the Solar System (which is less than 10^-5 per one planetesimal). A lot of icy material could be delivered to the exoplanets “b” and “d”.

Acknowledgments

The work was carried out as a part of the state assignments of the Vernadsky Institute of RAS № 0137-2020-0004 and the author acknowledges the support of Ministry of Science and Higher Education of the Russian Federation under the grant 075-15-2020-780 (N13.1902.21.0039) “Theoretical and experimental studies of the formation and evolution of extrasolar planetary systems and characteristics of exoplanets”.

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