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 ab=0.0485 AU located in a habitable zone, they also considered the exoplanet “c” with a semi-major axis ac 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 ac 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: ab=0.0485 AU, ac=1.489 AU, mb=1.27mE and mc=12mE, where mE is the mass of the Earth. For the exoplanet “b”, the initial eccentricity eb and initial inclination ib were considered to be equal to 0, and the initial eccentricity ec of the exoplanet “c” equaled to 0 or 0.1. Initial inclination of the exoplanet “c” was considered to be ic=ec/2=0.05 rad or ic=ec=0. For interest, I also considered ic=152o; 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 ab = 0.04857 AU, eb = 0.11, mb = 1.17 mE, ac = 1.489 AU, ec = 0.04, mc = 7 mE. I supposed ib = ic = 0. mc 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 amin to amax=amin+0.1 AU, with amin from 1.2 to 1.7 AU with a step of 0.1 AU. Initial eccentricities eo of planetesimals equaled to 0 or 0.15 for the first series of calculations, and eo=0.02 or eo=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 eo/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 pb 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 eo=0.02 and in 3 among 6 variants at eo=0.15. At eo=0.02 for the five variants, pb equaled to 0.004, 0.004, 1.28×10-5, 0.00032 и 9.88×10-5. The mean value of pb for one of 4500 planetesimals equaled to 4.7×10-4, but among them there were two planetesimals with pb=1. At eo=0.15 for three variants, pb equaled to 0.008, 0.004 and 3.6×10-6. The mean value of pb for one of 1500 planetesimals equaled to 2.0×10-3, but among them there were three planetesimals with pb=1. For both eo=0.02 and eo=0.15 in one of 24 variants pb=0.008, in three variants pb=0.004, and in other 4 variants pb was between to 4×10-6 and 3×10-4. For all three considered variants of the first series at ec=0.1 and eo=0.15, the values of pb were in the range 0.008-0.019. For other calculations of the first series, pb=0. For the second series of calculations, the probability pd of a collision of a planetesimal from the zone of the orbit of the exoplanet “c” with the exoplanet “d” (ad=0.02895 AU, md=0.29mE, ed=id=0) was nonzero only for seven variants (among 24). At eo=0.02 for four variants, pd equaled to 0.004, 0.00068, 0.000143 and 3.0×10-5. The mean value of pd over 4500 planetesimals equaled to 2.7×10-4, but for one planetesimal pd=1. At eo=0.15 for three variants, pd equaled to 0.008, 0.004 and 2.58×10-5. The mean value of pd over 1500 planetesimals equaled to 2.0×10-3, but for three planetesimals pd=1. For the second series, the mean values of pb and pd 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 pb and pd 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 iс=eс=0 and eo=0.15, the values of the probability pс of a collision of one planetesimal, initially located near the exoplanet “c”, with this exoplanet were about 0.06-0.1. For ic=ec/2=0.05 and eo=0.15, pс was about 0.02-0.04. For the second series of calculations, pс was about 0.1-0.3, exclusive for amin=1.4 AU and eo=0.02 when pс was about 0.7-0.8 (and the main growth was before T=1 Myr). Usually there was a small growth of pc 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 amin=1.4 AU and eo=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 ac 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 ab=0.0485 AU or the orbit of the exoplanet “d” with a semi-major axis ad=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”.
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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