ISSN 00380946, Solar System Research, 2015, Vol. 49, No. 6, pp. 404–409. © Pleiades Publishing, Inc., 2015. Original Russian Text © N.Yu. Emel’yanenko, 2015, published in Astronomicheskii Vestnik, 2015, Vol. 49, No. 6, pp. 442–447.

Features of Encounters of Small Bodies with Planets N. Yu. Emel’yanenko Institute of Astronomy, Russian Academy of Sciences, Moscow, Russia email: [email protected] Received: December 16, 2014, in final form, March 30, 2015

Abstract—A kinematic approach is developed to qualitative analysis of characteristics of a lowspeed encounter of a small body with a planet. A classification of encounters of small bodies with planets based on the magnitude of planetocentric speed is proposed. The concept of the points of lowspeed quasitangency of orbits of small bodies and planets is introduced. Based on this concept, the definitions of the point of min imum planetocentric speed, a quasitangent lowvelocity segment on the orbit of a small body, lowvelocity and highvelocity encounters are formulated. A classification of encounters of small bodies with planets according to the global minimum of the function of planetocentric distance is also proposed. The classifica tion is based on the concepts of the gravity sphere of action and the Hill sphere of the planet. The definitions of an area and duration of lowspeed and highspeed encounters are given. Keywords: small body, lowspeed encounter, quasitangency point, Hill’s sphere, sphere of gravity action DOI: 10.1134/S0038094615050020

INTRODUCTION In 80s and 90s of the 20th century, the attention of researchers of shortperiod comets was attracted to objects with a high value of Tisserand’s constant rela tive to Jupiter. Opik (1976) was the first who has drawn attention to the fact that, within a threebody prob lem, a comet whose Tisserand’s constant is close to three with respect to a large planet, should have a small planetocentric velocity in the region of encounter (see also Kresak, 1979). In the works of Carusi et al. (1981, 1982, the term “lowspeed encounter with Jupiter” first appeared in connection with lower average Jovi centric velocity of a comet in the vicinity of Jupiter. In what follows, the term “lowspeed encounter” has been accepted and it is being used by all researchers involved in the studies of such objects. The lowspeed encounters with Jupiter and other outer planets were studied by many authors (often without using this term). One can mention the works by Everhart (1973), KazimirchakPolonskaya (1985), Rickman and Mal mort (1981), Emel’yanenko (1984, 1986). All studies of the cometary orbit evolution with a high value of Tisserand’s constant with respect to Jupiter have shown that they may undergo unusual encounters with the planet. The comets were discovered moving near Jupiter for a long time exceeding the period of orbital motion of Jupiter around the Sun. In these encoun ters, multiple minima of the function of Jovicentric distance and elliptical Jovicentric elements of the orbit were often detected. However, the first studies of the cometary orbit evolution with high Tisserand’s constant relative to Jupiter have shown that in addition to lowvelocity encounters, the same comets and

many other small bodies with a smaller value of the Tisserand constant undergo the encounters unevent fully. In these encounters, the Opik model is imple mented for the change of the heliocentric velocity vec tor of comet in the vicinity of the minimum of Jovi centric distance of a small body. These encounters are called “lowvelocity encounters”. Thus, despite a traditional notion that lowspeed encounters are due to a high value of the Tisserand constant relative to the planet, specific features of encounters occur at by no means all of these comets. In the work of Emel’yanenko (2009b), it was shown that, on average, only one of seven cometary encoun ters with Jupiter corresponding to the comets with high Tisserand’s constant relative to Jupiter undergo lowvelocity collisions. Searching for a clear, scientificallybased defini tion of highspeed and lowspeed encounters of a small body with a planet and classification of these encounters are the urgent problems. The discovery of transNeptunian objects led to a significant revision of our understanding of the Solar System. According to modern concepts, a decisive role in the process of transition of transNeptunian objects from the outer part of Solar System into nearEarth space is played by lowspeed encounters with giant planets (Emel’yanenko et al., 2005). Therefore, investigations of lowspeed encounters of small bodies with planets are very important in studying the process of their migration in the Solar System. Many asteroids which encounter the Earth have a Tisserand’s constant close to three with respect to the Earth (Emel’yanenko, 2009a). Therefore, they may

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undergo lowspeed encounters with our planet with badly forecasted features. Thus, the problem under consideration is urgent in view of cometaryasteroid threat for the Earth. From the point of view of celestial mechanics, there are no fundamental differences between the encounters of a small body with any planet, therefore, an urgent task is to study the lowspeed encounters of observed small bodies with high Tisserand’s constant relative to one or another planet. THE MAIN RESEARCH PROBLEMS 1. To propose the classification of encounters of a small body with the planet by the magnitude of plane tocentric velocity. To introduce the concepts of quasi tangency points during lowspeed motion of a small body along part of its orbit, giving the definitions of lowspeed and highspeed encounters. 2. To propose the classification of encounters of a small body with a planet according to the value of the global minimum of the planetocentric distance func tion using the concepts of the gravity sphere of action and the Hill sphere of the planet, based on the study of encounters of observed comets with Jupiter, Saturn and the Earth. CLASSIFICATION OF ENCOUNTERS OF SMALL BODIES WITH PLANETS BY THE PLANETOCENTRIC VELOCITY Since a high magnitude of Tisserand’s constant does not ensure a lowspeed encounter between a small body and the planet, it is necessary to find another criterion which allows one to identify such encounters. Emel’yanenko (2007, 2009b, 2010) studied more than 2000 encounters of observable comets with Jupi ter. Based on these results, it was shown that a comet moves with low Jovicentric velocity only in some seg ments of the trajectory in the region of close approach to Jupiter. These trajectory segments are located not only in the vicinity of the global minimum of the func tion of Jovicentric distance, but also in other parts of the trajectory in the region of encounter. Only a small magnitude of the Jovicentric speed of the comet is common to all these segments. This is possible only for nearly equal heliocentric velocity vectors of a small body and a planet. Let V and Vp be the vectors of heliocentric velocities of a small body and planet P; V, Vp, vp are the moduli of vectors V, Vp, vp. Consider the limiting case. At a certain moment of time tM, let the vectors of heliocentric velocities of a body and a planet be equal

V = Vp. SOLAR SYSTEM RESEARCH

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If (1) is satisfied, then at the moment tM a small body will have zero planetocentric velocity v p = 0.

Since in our studies (Emel’yanenko, 2009b) the frequency of such an event is a continuous variable, one can only speak about the probability of planeto centric velocity within a certain interval (9, 0 + ε), where ε > 0. For the orbit of a small body, we introduce the con cepts of the points of quasitangency of the orbits of a small body and a planet, the points of minimum plan etocentric speed and a lowvelocity quasitangent seg ment on the orbit of the small body. The points M and Mp are called the points of low velocity quasitangency of the orbits of a small body and a planet if their heliocentric velocity vectors are equal at moment tM. According to the definition of vector equality (for two vectors), we write (10) in the form of two equations

V Vp,

(2)

V = Vp,

(3)

where (2) means the collinearity and same direction of vectors V and Vp. For small observable bodies, one can not exclude a lowvelocity quasitangency of their orbits and the orbit of a planet in space. However, more probable is an encounter when conditions (2) and (3) are satisfied during a certain interval of time, Δt. Let only one of conditions (2), (3) be satisfied at a moment t. An exact fulfillment of one condition (2) or (3) and an approximate fulfillment of another condi tion at instant t will result in the minimum of variable vp within a certain small neighborhood of epoch t. It was necessary to find the point on the orbit of a small body, the presence of which allows us to classify the encounter as a lowvelocity collision. In other words, it was necessary to ensure the precise fulfill ment of one equality and the approximate fulfillment of another. Satisfaction of vector condition (2) is guar anteed for the planar twobody problem, since the problem of constructing the tangent to an ellipse being parallel to an intersecting line has always a geometric solution. But a verification of this condition for the observed small bodies when integrating the equations of motion at each integration step is timeconsuming and ineffective, since the heliocentric orbit is changing rapidly in the region of encounter, leading to the emer gence of many such points. The more that the next step is required in any case, namely, the calculation of the modulus of the heliocentric velocity vector both for the planet and for a small body. In calculating the evolution of modeled and observed small bodies, it is easy to determine the moment of time when condition (3) is satisfied through the equality of true anomalies of point Мi on the orbit of a small body and the body

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itself. It is the exact fulfillment of equality (3) is chosen for this purpose. The points Мi on the orbit of a small body, where condition (3) holds exactly, and condition (2) holds approximately (so that variable vp takes a minimum value), called the points of minimum planetocentric speed of a small body. In what follows, the moment tM corresponding to the passage of a small body through the point of low speed quasitangency of orbits (the point of minimum planetocentric velocity) is called the moment of low speed quasitangency of orbits of a small body and the planet. The neighborhood of the point of lowspeed quasi tangency of orbits (the minimum point of planetocen tric speed) is called the lowspeed quasitangential segment on the orbit of a small body. On this segment, both conditions (2) and (3) are satisfied approxi mately, and the value of vp reaches one or several min ima. The encounter of a small body with the planet is called the lowspeed encounter if at least one low speed quasitangential segment lies in the region of encounter. In contrast to this, the encounter is called the highspeed encounter if there is no quasitangen tial segments in the region of encounter. CLASSIFICATION OF ENCOUNTERS OF SMALL BODIES WITH PLANETS BY A GLOBAL MIMIMUM OF PLANETOCENTRIC DISTANCE A very important characteristic of an encounter is the time lapse ΔT—the duration of encounter. Knowledge of cometary encounters with Jupiter was gathered during the 60s and 70s of the last century. Only one transit of a shortperiod comet through the sphere of gravitational action of Saturn is known. KazimirchakPolonskaya was the first who proposed to separate the cometary transits according to the min imum distance to Jupiter. She took the radius of a sphere of attraction of Jupiter (rAJ ) (rAJ = 0.322 AU) magnified by 0.008 AU as the basis, and introduced the following classification of encounters: strong (ρ < 0.33 AU), moderate (0.33 < ρ < 1 AU) and minor (1 < ρ ≤ 2 AU), where P is the global minimum of the func tion of Jovicentric distance of the comet) Kazimir chakPolonskaya, 1961). At such a subdivision, she assumed that the region of encounter is the sphere with a radius of 2 AU (2 AU = 6rAJ ). In a work by KazimirchakPolonskaya (1985), it was assumed that other planets are also strong moderators of cometary orbits. She modeled the encounters of comets with Neptune in spheres with a radius of 2–3 AU. An a priori classification of the comet encounters with Jupiter proposed by KazimirchakPolonskaya was based on a numerical analysis of perturbations of orbital elements depending on the planetocentric dis

tance of the comet. In works by Belyaev et al. (1986) and Carusi et al. (1984), the authors studied the encounters of shortperiod comets with Jupiter and other planets in the sphere with a radius of 2 AU with out an appropriate justification. There are many works in which the encounters of small bodies with giant planets in the area with a radius of 1 AU are studied. It is to be underlined that the encounters can be both highspeed and lowspeed. In our work (Emel’yanenko, 2003), a thorough analysis of the encounter duration with Jupiter for 34 shortperiod comets (14 highspeed encounters and 20 lowspeed encounters) was carried out. We computed and com pared the variations of the value h = 1/a for several intervals of time corresponding to the durations of transits through the sphere of Jupiter attraction (ΔtA, sphere A), ΔtВ (sphere B), ΔtС (sphere C), and ΔtD— during the period of revolution around the Sun, including an encounter with Jupiter. This work has shown that for highspeed encounters the main change in parameter h occurs inside the sphere A (on average, ΔhA = 0.93 ΔhB). Parameter h proves to be nearly constant outside sphere A (ΔhС ≈ ΔhD). The time interval ΔtB was accepted as the time of the high speed encounter. For lowvelocity encounters, param eter h was changing outside sphere B (on average, ΔhB = 0.93ΔhС). Owing to this, the time interval ΔtС was accepted as the time lapse for the lowvelocity encounter. For some lowvelocity encounters, param eter h was changing outside sphere C. As a rule, the duration of such encounters is more than the period of revolution of the comet around the Sun, and the func tion of the Jovicentric distance has additional distant minima. In these cases, a special time interval includ ing all minima is taken as the interval of encounter. The borders of the encounter are set at the orbit points, starting from which variation of h coincides with the fluctuations under the action of planetary perturbations. For Jupiter, the radius of sphere A is close to the radius of Hill’s sphere, whereas the radii of spheres B and C are approximately three and six times higher than the radius of Hill’s sphere. It is worthwhile to underline that the phenomenon of satellite capturing (SC) in lowvelocity encounters with Jupiter is not rare and took place on the border of sphere C, whereas the secondary minima were emerging outside this sphere. In this work, we have found and investigated 11 encounters of eight comets with Saturn. We compared the differences between the radii of the spheres of gravity action and the Hill spheres for Saturn and Jupiter. For Saturn, the differences proved to be much higher than for Jupiter. In 11 encounters with Saturn, the analysis of the change in the energy constant in spheres A, B, C (based on the Hill’s sphere radius) led to the results which coincide with analogous results for Jupiter. Investigations of several encounters of asteroid 2006 RH 120 with the Earth (Emel’yanenko, Naroen SOLAR SYSTEM RESEARCH

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kov, 2013) confirmed the fact that the use of the Hill sphere radius of the planet (rHp ) in determining the region of encounter and classification of the encoun ter by the minimum distance are preferable. For all studied lowvelocity encounters with Jupi ter, Saturn and the Earth, the change of parameter h in the spherical layer 3rHp ≤ r ≤ 6rHp is similar to the change in the spherical layer rHp ≤ r ≤ 3rHp for highspeed colli sions. Over the past 40 years, a large number of comets with very close encounters and very complicated transformations of both heliocentric and Jovicentric orbits deep in the Hill sphere of Jupiter have been dis covered. In recent decades, at least three very close encounters with Saturn and several collisions with the Earth were discovered. It was necessary to combine such encounters into a separate group, since they had many features as compared to the close collisions according to the definition introduced by Kazimir chakPolonskaya. We conducted the study of the evolution of orbital elements for observed comets of Jovian family with temporary satellite capture in the sense of Everhart (in what follows TSC), and temporary gravitational cap ture with elliptical Jovicentric elements into the Hill sphere (in what follows TGC). The analysis of the evo lution of orbits showed that the phenomenon of TGC with multiple physical minima occurs only for comets with the global minimum no more than 0.1 AU (Emel’yanenko, 2003, 2009b). A careful analysis of the phenomenon of TGC has shown that gravitational capture of the comet by Jupiter occurs through col linear libration centers L1 or L2. For a small body with high Tisserand’s constant relative to the planet (for example, at ТJ > 3.04) in the neighborhood of these centers on the surface of zero velocity of the three body problem, the jumpers are located in which the body may fall into the Hill sphere and undergo the phenomenon of TGC (Emel’yanenko, 2012). These studies confirmed the advisability of using the Hill sphere upon separation of the encounters by the global minimum of the function of planetocentric distance. On the other hand, it was shown that rKJ = 0.084 AU (Emel’yanenko, 1992a; 1992b). This is a sphere of critical radius around the Jupiter, in which one should take into account the perturbations arising from the nonsphericity of the Jovian shape. Moreover, in study ing reliable encounters, one should take into account the perturbations from the Galilean moons of Jupiter, since they represent individual gravitating objects (rKJ nearly equals a half of radius rGJ of the sphere of gravi tational action of Jupiter). Nowadays, we know of single encounters with TSC and TGC characterized by multiple minima in the function of planetocentric distance inside the Hill sphere of the Earth. Small bodies have been discovered SOLAR SYSTEM RESEARCH

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which encounter Saturn. Three encounters were accompanied by TSC and two by multiple minima in the function of planetocentric distance in the region of approach to the planet. In view of the above, the necessity arose to refor mulate the definitions of the region, duration and clas sification of encounters of a small body with a planet using the global minimum of the function of planeto centric distance. We extend the results obtained upon rapproche ments of comets with Jupiter, Saturn and the Earth to the case of encounters of small bodies with all the planets of the Solar System. Let ρ be the global minimum of the function of planetocentric distance of a small body, and ρi denotes the secondary minima during lowspeed encounters (ρ < ρi). The encounter is called strong if a small body falls into the sphere with a radius of 0.5rGp around the planet ρ ≤ 0.5rGp, (4) where the right limit of inequality (4) equals a half of the radius of the sphere of gravitational action of the planet. The encounter is called close if a small body passes outside this sphere but inside the Hill’s sphere (5) 0.5rGp < ρ ≤ rHp. The encounter is called moderate if a small body moves outside the Hill’s sphere but not far ther than 3rHp: (6) rHp < ρ ≤ 3rHp. The lowspeed encounter is called weak in the case (7) 3rHp < ρ ≤ 6rHp. The secondary minima of the function of planeto centric distance of a small body upon lowvelocity col lision outside the sphere with radius 6rHp (if they are observed) are called distant. So, if we study the lowspeed encounters of a small body with the planet, the time of movement of a small body inside the sphere with a radius of 6rHp is taken as the duration of encounter. The duration ΔT of encounter is defined as the time of movement of a small body inside the sphere with a radius of 3rHp. An imaginary sphere with radius Rh (Rl) in which a small body moves during the time ΔT is called the area of highspeed (lowspeed) collision

Rh = 3rHp,

(8)

(9) Rl = 6rHp. The radii of strong, close, moderate and weak col lisions of planets are given in the table. For compari

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Radii of strong, close, moderate and weak encounters of small bodies with planets in AU Planet

0.5rG

rH

3rH

6rH

R

1

2

3

4

5

6

Mer V E M J Sat U N

8× 5.6 × 10–4 8.7 × 10–4 4.3 × 10–4 0.08 0.08 0.006 0.108 10–5

1.48 × 6.74 × 10–3 0.010 0.007 0.347 0.429 0.465 0.770

4.44 × 0.20 × 10–1 0.030 0.022 1.041 1.286 1.395 2.311

10–3

10–3

son, the radii of encounters of modeled and observed small bodies with planets traditionally used in modern experiments are given in the last (sixth) column of the table. As T1 we denote the moment of entry into the region of encounter (the beginning of collision), and Т2, the time of exit outside the region of encounter (the end of collision) (T1 < T2). The duration of colli sion is ΔТ = Т2 – Т1.

(10)

CONCLUSIONS A kinematic approach for the qualitative analysis of lowvelocity encounters of a small body with the planet has been developed. The classification of encounters of small bodies with planets based on the magnitude of the planetocentric speed is proposed and the concept of the points of lowspeed quasitangency of orbits of small bodies and planets is introduced. Based on this concept, we formulated the definitions of the point of minimum planetocentric speed, low velocity quasitangency segment on the orbit of a small body, and lowvelocity and highvelocity encounters. A thorough analysis of lowspeed and highspeed encounters with Jupiter, Saturn and the Earth was carried out. Variations in the semimajor axis а and the inverse value h = 1/а within the Hill sphere and inside the spheres with radii 3rHp and 6rHp. were studied. The collisions in which the minimum dis tance of a small body to the planet does not exceed half the radius rGp of the sphere of gravitational action are combined into a separate group. The classification of encounters of small bodies with planets based on the magnitude of the global minimum of the function of planetocentric distance is proposed. This classifica tion uses the definitions of the Hill sphere and the sphere of gravitational action. The concepts of the region and duration of lowspeed and highspeed encounters are introduced.

8.88 × 0.40 × 10–1 0.060 0.043 2.082 2.573 2.790 4.622

10–3 0.1 2 2–3 2–3

ACKNOWLEDGMENTS This work was supported by the Ministry of Educa tion and Science of the Russian Federation in the framework of the project of the federal target program “Research and development on priority directions of scientific technological complex of Russia for 2014– 2020.” (ID RFMEFIBBB14X0288). REFERENCES Belyaev, N.A., Catalogue of ShortPeriod Comets, Belyaev, N.A., Kresak, L., Pittich, E.M., and Pushkarev, A.N., Eds., Bratislava: Astron Inst. Slovak. Acad. Sci., 1986. Carusi, A., Kresak, L., and Valsecchi, G.B., Perturbations by Jupiter of a chain of moving in the orbits of Comet Oterma, Astron. Astrophys., 1981, vol. 99, pp. 262–269. Carusi, A. and Valsecchi, G., On the orbital evolution of shortperiod comets having lowvelocity encounters with Jupiter, in Comparative Study of the Planets, Cora dini, A. and Fulchignoni, M., Eds., Dordrecht: Reidel, 1982, pp. 131–148. Carusi, A., in LongTerm Evolution of ShortPeriod Comets, Carusi, A., Kresak, L., Perozzi, E., and Valsecchi, G.B., Eds., Rome, 1984. Emel’yanenko, N.Yu., Comets close approach to the Jupi ter, Kometnyi Tsirkulyar, 1984, no. 331, p. 3. Emel’yanenko, N.Yu., Sortperiod comets close approach to the Jupiter, in Sb. Nauchn. trudov LGU. Analiz dvizheniya tel Solnechnoi sistemy i ikh nablyudeniya (Collection of Leningrad State University Scientific Papers. Analysis of Solar System Bodies Motion and Their Observation), Leningrad, 1986, pp. 97–102. Emel’yanenko, N.Yu., Evolution of the orbits of comets having close approaches to Jupiter. An analysis of the effect of the nonspherical figure of Jupiter, Solar Syst. Res., 1992a, vol. 26, no. 5, pp. 447–451. Emel’yanenko, N.Yu., Evolution of the orbits of comets that have close encounters with Jupiter. 3. Influence of the Galilean satellites, Solar Syst. Res., 1992b, vol. 26, no. 6, pp. 575–580. Emel’yanenko, N.Yu., The dynamics of cometary orbits in close encounters with Jupiter. An analysis of encounter durations, Solar Syst. Res., 2003, vol. 37, no. 2, pp. 156–164. SOLAR SYSTEM RESEARCH

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FEATURES OF ENCOUNTERS OF SMALL BODIES WITH PLANETS Emel’yanenko, N.Yu., Asher, D. and Bailey, M., Centaurs from Oort cloud and the origin of Jupiterfamily com ets, Mon. Notic. Roy. Astron. Soc., 2005, vol. 361, pp. 1345–1351. Emel’yanenko, N.Yu., Orbital evolution of shortperiod comets with high values of the Tisserand constant, Proc. IAU Symp. Near Earth Objects, Our Celestial Neighbors: Opportunity and Risk, 2007, no. 236, pp. 35–42. Emel’yanenko, N.Yu., Asteroids with high Tisserant con stant with respect to major planets, Tr. Mezhd. konf. “Okolozemnaya astronomiya2009” (Proc. Int. Conf. “Circumterrestrial Astronomy”), Kazan, 2009a, pp. 139–146. Emel’yanenko, N.Yu., Evolyutsiya elementov orbit korotko periodicheskikh komet (Orbit Elements Evolution for ShortPeriod Comets), Chelyabinsk: South Ural State Univ., 2009b. Emel’yanenko, N.Yu., Kachestvennyi analiz i modeli nizko skorostnykh sblizhenii komet s Yupiterom (Qualitative Analysis and Models of LowSpeed Approaches for Comets and Jupiter), Chelyabinsk: South Ural State Univ., 2010. Emel’yanenko, N.Yu., Temporary satellite capture of com ets by Jupiter, Solar Syst. Res., 2012, vol. 46, no. 3, pp. 181–194.

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Emel’yanenko, N.Yu. and Naroenkov, S.A., 2006 RH 120 asteroid approach to the Earth, Ekol. Vestn. Nauch. Tsentrov Chernomorsk. Ekonom. Sotrudn., 2013, vol. 2, no. 4, pp. 54–59. Everhart, E., Horseshoe and Trojan orbits associated with Jupiter and Saturn, Astron. J., 1973, vol. 76, no. 4, p. 316. KazimirchakPolonskaya, E.I., Comets close approaches to the Jupiter: 1770–1960 review, Tr. Inst. Teor. Astron. Akad. Nauk SSSR, 1961, no. 7, pp. 19–190. KazimirchakPolonskaya, E.I., Review of studies on cap ture of comets by Neptune and its role in the dynamic evolution of comets orbits, in Proc. IAU Coll. 83 Dynamics of Comets: Their Origin and Evolution, Carusi, A. and Valsecchi, G., Eds., Dordreht: Reidel, 1985, pp. 243–258. Kresak, L., Dynamical interplations among comets and asteroids, in Asteroids, Gerels, T., Ed., Tucson: Univ. Arizona Press, 1979, pp. 289–309. Opik, E., Interplanetary Encounters: Close Range Gravita tional Interactions, New York: Elsevier, 1976. Rickman, H. and Malmort, A., Variations of the orbit of comet P/Gehrels 3: temporary satellite captures by Jupiter, Astron. Astrophys., 1981, vol. 87, no. 102, pp. 165–170.

Translated by G. Dedkov