Schwarz et al. [1] studied migration of exocomets in the Proxima Centauri system. Besides the exoplanet with a semi-major axis a_{1}=0.0485 AU located in a habitable zone, they also considered the second exoplanet with a semi-major axis a_{2} from 0.06 to up to 0.3 AU (for test calculations up to 0.7 AU). Now it is considered [2,3] that the semi-major axis of the second planet equals to 1.489±0.049 AU.

In the first series of calculations, according to [2], 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_{1}=0.0485 AU, a_{2}=1.489 AU, m_{1}=1.27m_{E} and m_{2}=12m_{E}, where m_{E} is the mass of the Earth. For the first exoplanet, the initial eccentricity e_{1} and initial inclination i_{1} were considered to be equal to 0, and the initial eccentricity e_{2 }of the second exoplanet was considered to be equal to 0 or to 0.1. The calculations were made for initial inclination of the second exoplanet i_{2}=e_{2}/2=0.05 rad and for e_{2}=i_{2}=0. For interest, I also considered i_{2}=152^{o}, such calculations characterize the case when orbits of planetesimals were inclined to the orbit of the planet. In the second series of calculations, according to [3], I considered a_{1}=0.04857 AU, e_{1}=0.11, m_{1}=1.17m_{E}, a_{2}=1.489 AU, e_{2}=0.04, m_{1}=7m_{E}. I supposed i_{1}=i_{2}=0. In both series of calculations, the density of the first and second exoplanets were considered to be equal to densities of the Earth and Uranus, respectively.

In different calculation variants, initial semi-major axes a_{b} 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_{b} of planetesimals were equal to 0 or to 0.15 for the first series of calculations, and e_{b}=0.02 for the second series of calculations. Initial inclinations i_{b} of the planetesimals equaled to e_{b}/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 [4]. 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 [5-7], which had been made for the planets of the solar system, but for different masses and radii of a star and exoplanets.

The probability p_{1} of a collision of one planetesimal, initially located near the second exoplanet, with the first exoplanet was 0 in most calculations for both series. For the second series, p_{1} about 0.01-0.02 at a_{min}=1.2 and a_{min}=1.3 AU. For the first series, only at i_{2}=e_{2}/2=0.05 rad and e_{b}=0.15 the values of p_{1} were considerable and could exceed 0.1 at time T=10 Myr, i.e. they were much greater than the fraction (less than 10^{-5}) of planetesimals from the zone of the giant planets collided with the Earth [8].

For i_{2}=e_{2}=0 and e_{b}=0.15, the values of the probability p_{2} of a collision of one planetesimal, initially located near the second exoplanet, with this exoplanet were about 0.06-0.1. For i_{2}=e_{2}/2=0.05 and e_{b}=0.15, p_{2} was about 0.02-0.04, i.e. it was less than p_{1}. For the second series of calculations, p_{2} was about 0.1-0.2 at T=5 Myr, exclusive for p_{2}=0.5 at a_{min}=1.7 AU and for a_{min}=1.4 AU when p_{2} exceeded 1 already at T=1 Myr. The fraction of ejected planetesimals for the second series at T=5 Myr was between 0.6 and 0.8 at 1.5≤a_{min}≤1.6 AU.

For both series of calculations, the ratio of planetesimals ejected into hyperbolic orbits usually exceeded the number of planetesimals collided with the planets by at least a factor of 4 if the number of planetesimals decreased by a factor of several. Only at a_{min}=1.4 AU this ratio was less than 1. In some calculations a few planetesimals could be left in elliptical orbits after 100 Myr. For i_{2}=152^{o}, e_{2}=0, and T=10 Myr, the values p_{2} were about 0.02 at e_{b}=0 and about 0.02-0.05 (the range is for different values of a_{min}) at e_{b}=0.15. There was a small growth of p_{2} after 10 Myr. For i_{2}=152^{o}, e_{2}=0.1, T=10 Myr, the values of p_{2} were very small at e_{b}=0 and were about 0.01-0.05 (the range is for different values of a_{min}) at e_{b}=0.15.

The work was carried out as a part of the state assignments of the Vernadsky Institute of RAS and was supported by the grant on exoplanets of the Russian Ministry of Education and Science.

References

[1] Schwarz R., Bazso A., Georgakarakos N., et al. Exocomets in the Proxima Centauri system and their importance for water transport // MNRAS, 2018, v. 480, p. 3595-3608.

[2] Kervella P., Arenou F., Schneider J. Orbital inclination and mass of the exoplanet candidate Proxima c // Astronomy & Astrophysics, 2020, v. 635: L14.

[3] Benedict G.F., McArthur B.E. A Moving Target—Revising the Mass of Proxima Centauri c // Research Notes of the AAS, 2020, V. 4, N 6, ID 86. doi:10.3847/2515-5172/ab9ca9.

[4] Levison H.F., Duncan M.J. The long-term dynamical behavior of short-period comets // Icarus, 1994, v. 108, p. 18-36

[5] Ipatov S.I., Mather J.C. Migration of Jupiter-family comets and resonant asteroids to near-Earth space // Annals of the New York Academy of Sciences, 2004, v. 1017, p. 46-65. http://arXiv.org/format/astro-ph/0308448

[6] Ipatov S.I., Mather J.C. Comet and asteroid hazard to the terrestrial planets // Advances in Space Research, 2004, v. 33, p. 1524-1533. http://arXiv.org/format/astro-ph/0212177.

[7] Ipatov S.I., Probabilities of collisions of planetesimals from different regions of the feeding zone of the terrestrial planets with the forming planets and the Moon // Solar System Research, 2019, v. 53, N 5, p. 332-361. http://arxiv.org/abs/2003.11301

[8] Ipatov S.I. Migration of planetesimals to the Earth and the Moon from different distances from the Sun // 50th LPSC, 2019, #2594. https://www.hou.usra.edu/meetings/lpsc2019/pdf/2594.pdf