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1.
We present Monte Carlo simulations of the dynamical evolution of the Oort cloud over the age of the Solar System, using an initial sample of one million test comets without any cloning. Our model includes perturbations due to the Galactic tide (radial and vertical) and passing stars. We present the first detailed analysis of the injection mechanism into observable orbits by comparing the complete model with separate models for tidal and stellar perturbations alone. We find that a fundamental role for injecting comets from the region outside the loss cone (perihelion distance q > 15 AU) into observable orbits (q < 5 AU) is played by stellar perturbations. These act in synergy with the tide such that the total injection rate is significantly larger than the sum of the two separate rates. This synergy is as important during comet showers as during quiescent periods and concerns comets with both small and large semi-major axes. We propose different dynamical mechanisms to explain the synergies in the inner and outer parts of the Oort Cloud. We find that the filling of the observable part of the loss cone under normal conditions in the present-day Solar System rises from <1% for a < 20 000 AU to about 100% for a ? 100 000 AU.  相似文献   

2.
The study of the reservoirs of small bodies in the Solar System can help us to refine our theory of the origin and evolution of the whole planetary system we live in. In this contribution, we introduce a simulation of the evolution of an initial proto-planetary disc (PPD) for 2 Gyr period, in which 10,038 studied test particles (TPs), representing the disc, are perturbed by four giant planets in their current orbits and having their current masses. In addition, Galactic-tide and stellar perturbations are considered. The outer Oort cloud (OC) appears to be formed within about 0.5 Gyr. At 2 Gyr, only 0.14% of all TPs reside in the outer OC, according to our simulation. This is the largest discrepancy with the results of previous authors. The TPs in the outer OC originate from almost all regions of the PPD. Specifically, 14, 7, 29, 21, and 29% of all TPs, that are the members of the outer OC at 2 Gyr, originate in the Jupiter, Saturn, Uranus, Neptune, and Edgeworth-Kuiper-belt regions, respectively. The largest number of the TPs, 6,669 (66.4%), was ejected into the interstellar space. Besides other results, we found a dominance of high galactic inclinations of outer-OC orbits. Obviously, this is a consequence of the action of Galactic tide.  相似文献   

3.
A numerical simulation of the Oort cloud is used to explain the observed orbital distributions and numbers of Jupiter-family (JF) and Halley-type (HT) short-period (SP) comets. Comets are given initial orbits with perihelion distances between 5 and 36 au, and evolve under planetary, stellar and Galactic perturbations for 4.5 Gyr. This process leads to the formation of an Oort cloud (which we define as the region of semimajor axes a > 1,000 au), and to a flux of cometary bodies from the Oort cloud returning to the planetary region at the present epoch. The results are consistent with the dynamical characteristics of SP comets and other observed cometary populations: the near-parabolic flux, Centaurs, and high-eccentricity trans-Neptunian objects. To achieve this consistency with observations, the model requires that the number of comets versus initial perihelion distance is concentrated towards the outer planetary region. Moreover, the mean physical lifetime of observable comets in the inner planetary region (q < 2.5 au) at the present epoch should be an increasing function of the comets’ initial perihelion distances. Virtually all observed HT comets and nearly half of observed JF comets come from the Oort cloud, and initially (4.5 Gyr ago) from orbits concentrated near the outer planetary region. Comets that have been in the Oort cloud also return to the Centaur (5 < q < 28 au, a < 1,000 au) and near-Neptune high-eccentricity regions. Such objects with perihelia near Neptune are hard to discover, but Centaurs with characteristics predicted by the model (e.g. large semimajor axes, above 60 au, or high inclinations, above 40°) are increasingly being found by observers. The model provides a unified picture for the origin of JF and HT comets. It predicts that the mean physical lifetime of all comets in the region q < 1.5 au is less than ～200 revolutions.  相似文献   

4.
Considering the model of the initial disc of planetesimals consisting of 10,038 test particles, we simulated the formation of small-body reservoirs in the outer Solar System for the 2-Gyr period. We present the results from the simulation, which concern the part of the scattered disc with objects that have the semi-major axes larger than 50 AU and do not cross the Neptune’s orbit. A suitable border between the scattered disc and the inner Oort cloud, in terms of semi-major axis, appears to be no more than 2,500 AU. The simulated and observed values of perihelion distance and inclination to the Ecliptic typically cover the range between 30 and 40 AU and from 0° to 30°, respectively. No simulated or observed values of the inclination exceed 45°. The distributions of eccentricity and inclination in the simulation are more consistent with their observed counterparts, if the primary observational selection effects are imitated in the simulated distributions.  相似文献   

5.
We introduce a model for integrating the effects of Galactic tides on Oort cloud comets, which involves two procedures, according to the values of the osculating semi-major axis a and eccentricity e. Ten simulations of the dynamics of 106 comets over 5 Gyr are performed using this model. We thus investigate the long-term effects of the Galactic tide with and without a radial component, the effects of the local density of the Galactic disk, and those of the Oort constants. Most of the results may be understood in terms of the integrability or non-integrability of the system. For an integrable system, which occurs for moderate semi-major axes with or without radial component, the dynamics is explained by periodic variation of the cometary perihelion, inducing the depletion of the outer region of the Oort cloud, a constant flux from the inner region after 500 Myr, and the quick formation of a reservoir of comets with argument of perihelion near 26.6°. When the system is non-integrable, the efficiency of the tide in reducing the cometary perihelion distance is enhanced both by replenishing the Oort cloud domain from which comets are sent toward the planetary system, and by reducing the minimal value that the perihelion distance may reach. No effects of varying the Oort constants were observed, showing that the flat rotation curve is a satisfactory approximation in Oort cloud dynamics.  相似文献   

6.
This study analyzes the evolution of 2 × 105 orbits with initial parameters corresponding to the orbits of comets of the Oort cloud under the action of planetary, galactic, and stellar perturbations over 2 × 109 years. The dynamical evolution of comets of the outer (orbital semimajor axes a > 104 AU) and inner (5 × 103 < a (AU) < 104) parts of the comet cloud is analyzed separately. The estimates of the flux of “new” and long-period comets for all perihelion distances q in the planetary region are reported. The flux of comets with a > 104 AU in the interval 15 AU < q < 31 AU is several times higher than the flux of comets in the region q < 15 AU. We point out the increased concentration of the perihelia of orbits of comets from the outer cloud, which have passed several times through the planetary system, in the Saturn-Uranus region. The maxima in the distribution of the perihelia of the orbits of comets of the inner Oort cloud are located in the Uranus-Neptune region. “New” comets moving in orbits with a < 2 × 104 AU and arriving at the outside of the planetary system (q > 25 AU) subsequently have a greater number of returns to the region q < 35 AU. The perihelia of the orbits of these comets gradually drift toward the interior of the Solar System and accumulate beyond the orbit of Saturn. The distribution of the perihelia of long-period comets beyond the orbit of Saturn exhibits a peak. We discuss the problem of replenishing the outer Oort cloud by comets from the inner part and their subsequent dynamical evolution. The annual rate of passages of comets of the inner cloud, which replenish the outer cloud, in the region q < 1 AU in orbits with a > 104 AU (～ 5.0 × 10?14 yr?1) is one order of magnitude lower than the rate of passage of comets from the outer Oort cloud (～ 9.1 × 10?13 yr?1).  相似文献   

7.
《Icarus》1987,70(2):269-288
We simulate the Oort comet cloud to study the rate and properties of new comets and the intensity and frequency of comet showers. An ensemble of ∼106 comets is perturbed at random times by a population of main sequence stars and white dwarfs that is described by the Bahcall-Soneira Galaxy model. A cloning procedure allows us to model a large ensemble of comets efficiently, without wasting computer time following a large number of low eccentricity orbits. For comets at semimajor axis a = 20,000 AU, about every 100 myr a star with mass in the range 1M⊙−2M⊙ passes within ∼10,000 AU of the Sun and triggers a shower that enhances the flux of new comets by more than a factor of 10. The time-integrated flux is dominated by the showers for comets with semimajor axes less than ∼30,000 AU. For semimajor axes greater than ∼30,000 AU the comet loss rate is roughly constant and strong showers do not occur. In some of our simulations, comets are also perturbed by the Galactic tidal field. The inclusion of tidal effects increases the loss rate of comets with semimajor axes between 10,000 and 20,000 AU by about a factor of 4. Thus the Galactic tide, rather than individual stellar perturbations, is the dominant mechanism which drives the evolution of the Oort cloud.  相似文献   

8.
R. Brasser  M.J. Duncan 《Icarus》2008,196(1):274-284
In a previous publication [Brasser, R., Duncan, M.J., Levison, H.F., 2006. Icarus 184, 59-82], models of the inner Oort cloud were built which included the effect of an embedded star cluster on cometary orbits about the Sun. The main conclusions of that paper were that the formation efficiency is about 10% and the median distance of the cloud to the Sun only depends on the mean density of gas and stars the Sun encountered. Here we report on the results of simulations which followed the ensuing dynamical evolution of these comet clouds in the current Galactic environment once the Sun left the embedded star cluster. The goal is to determine whether or not the dynamical influence of passing Galactic field stars and the Galactic tidal field is sufficient to replenish the current outer cloud (semi-major axis a>20,000 AU) with enough material from the inner cloud (a<20,000 AU). Since visible new comets come directly from the outer cloud, a mass estimate only exists for the latter, with a lower limit of 1 M⊕ [Francis, P.J., 2005. Astrophys. J. 635, 1348-1361]. Knowing the amount of expansion of the inner cloud may therefore yield an estimate of the mass of said (unseen) inner cloud. Our results indicate that typically only 10% of the comets from the inner cloud land in the outer cloud and are bound after 4.5 Gyr. If one assumes that in the extreme case all or the majority of the current population of the outer cloud has come from the inner cloud, then a typical value of the mass of the inner cloud is about 10 M⊕. The results of [Brasser, R., Duncan, M.J., Levison, H.F., 2006. Icarus 184, 59-82] showed that ∼10% of comets from the Jupiter-Saturn region were implanted in the inner Oort cloud, which implies an uncomfortably large value of about 100 M⊕ for the mass of solids in the primordial Jupiter-Saturn region. This extreme case might be remedied in two says: either the effect of Giant Molecular Cloud complexes on the inner Oort cloud must be much more severe than originally thought, or there was a two-stage formation process for the Oort cloud, in which the outer cloud was largely populated by comets scattered once the Sun had left its primordial birth cluster.  相似文献   

9.
We have integrated the orbits of the 76 scattered disk objects (SDOs), discovered through the end of 2002, plus 399 clones for 5 Gyr to study their dynamical evolution and the probability of falling in one of the following end states: reaching Jupiter's influence zone, hyperbolic ejection, or transfer to the Oort cloud. We find that nearly 50% of the SDOs are transferred to the Oort cloud (i.e., they reach heliocentric distances greater than 20,000 AU in a barycentric elliptical orbit), from which about 60% have their perihelia beyond Neptune's orbit (31 AU<q<36 AU) at the moment of reaching the Oort cloud. This shows that Neptune acts as a dynamical barrier, scattering most of the bodies to near-parabolic orbits before they can approach or cross Neptune's orbit in non-resonant orbits (that may allow their transfer to the planetary region as Centaurs via close encounters with Neptune). Consequently, Neptune's dynamical barrier greatly favors insertion in the Oort cloud at the expense of the other end states mentioned above. We found that the current rate of SDOs with radii R>1 km incorporated into the Oort cloud is about 5 yr−1, which might be a non-negligible fraction of comet losses from the Oort cloud (probably around or even above 10%). Therefore, we conclude that the Oort cloud may have experienced and may be even experiencing a significant renovation of its population, and that the trans-neptunian belt—via the scattered disk—may be the main feeding source.  相似文献   

10.
This paper considers the evolution of a flat svarm of cometary bodies (under the effect of the passage of stars), initially moving in one direction along the circular orbits with radii 1.4×104<r<2×104 AU and along elliptic orbits with semi-major axes 5×103<a<1×104 AU and with perihelia within 50<q<100 AU. Numerical simulation shows that the original flat belt of comets is thermalizing. Its root-mean-squarez-coordinate grows withr. A cometary cloud forms with a dense flattened inner core and a rarefied halo (the Oort cloud proper). The value =N core/N halo varies within a wide range (up to the order of magnitude) depending on the model used (N core andN halo are the numbers of comets in the core and the halo, respectively).The hypothesis of a massive Oort cloud (Marochniket al., 1988) implies that the Oort cloud should have a large angular momentum. This paper employs numerical simulation to calculate Oort cloud models to which the initially flat located at the periphery of the solar nebula rotating cometary swarms is evolving in time. The loss of the initial angular momentum over the time of the Oort cloud evolution is not large.  相似文献   

11.
We re-examine the formation of the inner Oort comet cloud while the Sun was in its birth cluster with the aid of numerical simulations. This work is a continuation of an earlier study (Brasser, R., Duncan, M.J., Levison, H.F. [2006]. Icarus 184, 59–82) with several substantial modifications. First, the system consisting of stars, planets and comets is treated self-consistently in our N-body simulations, rather than approximating the stellar encounters with the outer Solar System as hyperbolic fly-bys. Second, we have included the expulsion of the cluster gas, a feature that was absent previously. Third, we have used several models for the initial conditions and density profile of the cluster – either a Hernquist or Plummer potential – and chose other parameters based on the latest observations of embedded clusters from the literature. These other parameters result in the stars being on radial orbits and the cluster collapses. Similar to previous studies, in our simulations the inner Oort cloud is formed from comets being scattered by Jupiter and Saturn and having their pericentres decoupled from the planets by perturbations from the cluster gas and other stars. We find that all inner Oort clouds formed in these clusters have an inner edge ranging from 100 AU to a few hundred AU, and an outer edge at over 100,000 AU, with little variation in these values for all clusters. All inner Oort clouds formed are consistent with the existence of (90377) Sedna, an inner Oort cloud dwarf planetoid, at the inner edge of the cloud: Sedna tends to be at the innermost 2% for Plummer models, while it is 5% for Hernquist models. We emphasise that the existence of Sedna is a generic outcome. We define a ‘concentration radius’ for the inner Oort cloud and find that its value increases with increasing number of stars in the cluster, ranging from 600 AU to 1500 AU for Hernquist clusters and from 1500 AU to 4000 AU for Plummer clusters. The increasing trend implies that small star clusters form more compact inner Oort clouds than large clusters. We are unable to constrain the number of stars that resided in the cluster since most clusters yield inner Oort clouds that could be compatible with the current structure of the outer Solar System. The typical formation efficiency of the inner Oort cloud is 1.5%, significantly lower than previous estimates. We attribute this to the more violent dynamics that the Sun experiences as it rushes through the centre of the cluster during the latter’s initial phase of violent relaxation.  相似文献   

12.
We study the transfer process from the scattered disk (SD) to the high-perihelion scattered disk (HPSD) (defined as the population with perihelion distances q > 40 AU and semimajor axes a>50 AU) by means of two different models. One model (Model 1) assumes that SD objects (SDOs) were formed closer to the Sun and driven outwards by resonant coupling with the accreting Neptune during the stage of outward migration (Gomes 2003b, Earth, Moon, Planets 92, 29–42.). The other model (Model 2) considers the observed population of SDOs plus clones that try to compensate for observational discovery bias (Fernández et al. 2004, Icarus , in press). We find that the Kozai mechanism (coupling between the argument of perihelion, eccentricity, and inclination), associated with a mean motion resonance (MMR), is the main responsible for raising both the perihelion distance and the inclination of SDOs. The highest perihelion distance for a body of our samples was found to be q = 69.2 AU. This shows that bodies can be temporarily detached from the planetary region by dynamical interactions with the planets. This phenomenon is temporary since the same coupling of Kozai with a MMR will at some point bring the bodies back to states of lower-q values. However, the dynamical time scale in high-q states may be very long, up to several Gyr. For Model 1, about 10% of the bodies driven away by Neptune get trapped into the HPSD when the resonant coupling Kozai-MMR is disrupted by Neptune’s migration. Therefore, Model 1 also supplies a fossil HPSD, whose bodies remain in non-resonant orbits and thus stable for the age of the solar system, in addition to the HPSD formed by temporary captures of SDOs after the giant planets reached their current orbits. We find that about 12 – 15% of the surviving bodies of our samples are incorporated into the HPSD after about 4 – 5 Gyr, and that a large fraction of the captures occur for up to the 1:8 MMR (a ⋍ 120 AU), although we record captures up to the 1:24 MMR (a ≃ 260 AU). Because of the Kozai mechanism, HPSD objects have on average inclinations about 25°–50°, which are higher than those of the classical Edgeworth–Kuiper (EK) belt or the SD. Our results suggest that Sedna belongs to a dynamically distinct population from the HPSD, possibly being a member of the inner core of the Oort cloud. As regards to 2000 CR105 , it is marginally within the region occupied by HPSD objects in the parametric planes (q,a) and (a,i), so it is not ruled out that it might be a member of the HPSD, though it might as well belong to the inner core.  相似文献   

13.
In this paper the effect of the Galactic tidal field on a Sun–comet pair will be considered when the comet is situated in the Oort cloud and planetary perturbations can be neglected. First, two averaged models were created, one of which can be solved analytically in terms of Jacobi elliptic functions. In the latter system, switching between libration and circulation of the argument of perihelion is prohibited. The non-averaged equations of motion are integrated numerically in order to determine the regions of the ( e ,  i ) phase space in which chaotic orbits can be found, and an effort is made to explain why the chaotic orbits manifest in these regions only. It is evident that for moderate values of semimajor axis, a ∼50 000 au , chaotic orbits are found for ( e <0.15 , 40°≤ i ≤140°) as determined by integrating the evolution of the comet over a period of 104 orbits. These regions of chaos increase in size with increasing semimajor axis. The typical e-folding times for these orbits range from around 600 Myr to 1 Gyr and thus are of little practical interest, as the time-scales for chaos arising from passing stars are much shorter. As a result of Galactic rotation, the chaotic regions in ( e ,  i ) phase space are not symmetric for prograde and retrograde orbits.  相似文献   

14.
Long-period (LP) comets, Halley-type (HT) comets, and even some comets of the Jupiter family, probably come from the Oort cloud, a huge reservoir of icy bodies that surrounds the solar system. Therefore, these comets become important probes to learn about the distant Oort cloud population. We review the fundamental dynamical properties of LP comets, and what is our current understanding of the dynamical mechanisms that bring these bodies from the distant Oort cloud region to the inner planetary region. Most new comets have original reciprocal semimajor axes in the range2 × 10-5 < 1/aorig < 5 × 10-5AU-1. Yet, this cannot be taken to represent the actual space distribution of Oort cloud comets, but only the region in the energy space in which external perturbers have the greatest efficiency in bringing comets to the inner planetary region. The flux of Oort cloud comets in the outer planetary region is found to be at least several tens times greater than the flux in the inner planetary region. The sharp decrease closer to the Sun is due to the powerful gravitational fields of Jupiter and Saturn that prevent most Oort cloud comets from reaching the Earth’s neighborhood (they act as a dynamical barrier). A small fraction of ～10-2 Oort cloud comets become Halley type (orbital periods P < 200 yr), and some of them can reach short-period orbits with P < 20 yr. We analyze whether we can distinguish the latter, very ‘old” LP comets, from comets of the Jupier family coming from the Edgeworth-Kuiper belt.  相似文献   

15.
The flux of near-parabolic comets in the outer-planetary region is estimated on the presumption that the major planets and the galactic tide control the dynamics of comets. It is found that the flux of the Oort cloud comets (semi-major axis > 20000 AU) is similar to the case of a strong comet shower derived on the presumption that the galactic tidal force were not operative. On the other hand, the flux of comets with semi-major axes <- 20000 AU is found to be an increasing function of q (perihelion distance) until q reaches 20 AU, while for a 45000 AU it is a rapidly increasing function for q 12 AU. In other words, for comets of the inner extension of the Oort cloud the planetary perturbation acts as a strong barrier for them to penetrate into the inner planetary region.  相似文献   

16.
We estimated the gravitational influence of giant molecular clouds passing near the Solar system on the orbital evolution of Oort cloud comets. We performed a comparative analysis of the accuracies of the following two methods of allowance for the perturbations from giant molecular clouds: the impulse approximation and numerical integration. The impulse approximation yields fairly accurate estimates of the change in the energy of Oort cloud comets and the probability of their ejection under the influence of a molecular cloud if the path of the Solar system does not cross its boundary and if the molecular cloud may be treated as a point perturbing mass. The comet survival probability in the Oort cloud depends significantly on the internal structure of the perturbing molecular cloud and the impact parameter of the encounter. The most massive injection of comets into the planetary region and their ejection from the Oort cloud take place if the Solar system passes through a giant molecular cloud composed of several high-mass condensations. In this case, most of the comets injected into the planetary region were initially comets of the inner Oort cloud (a 10–4 AU) with high orbital eccentricities.  相似文献   

17.
New orbits for comet C/1843 J1 (Mauvais) and comet C/1853 W1 (van Arsdale) are calculated. Both orbits are hyperbolic, with e = 1.001145 and semi‐major axis a = –1412.18 AU for Mauvais and e = 1.000700 and a = –2919.24 AU for van Arsdale. Integrating the orbits backwards indicate that both comets were born in the far Oort cloud. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

18.
Julio A. Fernández 《Icarus》1980,42(3):406-421
The orbital evolution of 500 hypothetical comets during 109 years is studied numerically. It is assumed that the birthplace of such comets was the region of Uranus and Neptune from where they were deflected into very elongated orbits by perturbations of these planets. Then, we adopted the following initial orbital elements: perihelion distances between 20 and 30 AU, inclinations to the ecliptic plane smaller than 20°, and semimajor axes from 5 × 103 to 5 × 104 AU. Gravitational perturbations by the four giant planets and by hypothetical stars passing at distances from the Sun smaller than 5 × 105 AU are considered. During the simulation, somewhat more than 50% of the comets were lost from the solar system due to planetary or stellar perturbations. The survivors were removed from the planetary region and left as members of what is generally known as the cometary cloud. At the end of the studied period, the semimajor axes of the surviving comets tend to be concentrated in the interval 2 × 104 < a < 3 × 104 AU. The orbital planes of the comets with initial a ≧ 3 × 104AU acquired a complete randomization while the others still maintain a slight predominance of direct orbits. In addition, comet orbits with final a < 6 × 104AU preserve high eccentricities with an average value greater than 0.8 Most “new” comets from the sample entering the region interior to Jupiter's orbit had already registered earlier passages through the planetary region. By scaling up the rate of paritions of hypothetical new comets with the observed one, the number of members of the cometary cloud is estimated to be about 7 × 1010 and the conclusion is drawn that Uranus and Neptune had to remove a number of comets ten times greater.  相似文献   

19.
Classical trans-Neptunian objects (TNOs) are believed to represent the most dynamically pristine population in the trans-Neptunian belt (TNB) offering unprecedented clues about the formation of our Solar System. The long term dynamical evolution of classical TNOs was investigated using extensive simulations. We followed the evolution of more than 17000 particles with a wide range of initial conditions taking into account the perturbations from the four giant planets for 4 Gyr. The evolution of objects in the classical region is dependent on both their inclination and semimajor axes, with the inner (a<45 AU) and outer regions (a>45 AU) evolving differently. The reason is the influence of overlapping secular resonances with Uranus and Neptune (40–42 AU) and the 5:3 (a∼ ∼42.3 AU), 7:4 (a∼ ∼43.7 AU), 9:5 (a∼ ∼44.5 AU) and 11:6 (a∼ ∼ 45.0 AU) mean motion resonances strongly sculpting the inner region, while in the outer region only the 2:1 mean motion resonance (a∼ ∼47.7 AU) causes important perturbations. In particular, we found: (a) A substantial erosion of low-i bodies (i<10°) in the inner region caused by the secular resonances, except those objects that remained protected inside mean motion resonances which survived for billion of years; (b) An optimal stable region located at 45 AU<a<47 AU, q>40 AU and i>5° free of major perturbations; (c) Better defined boundaries for the classical region: 42–47.5 AU (q>38 AU) for cold classical TNOs and 40–47.5 AU (q>35 AU) for hot ones, with i=4.5° as the best threshold to distinguish between both populations; (d) The high inclination TNOs seen in the 40–42 AU region reflect their initial conditions. Therefore they should be classified as hot classical TNOs. Lastly, we report a good match between our results and observations, indicating that the former can provide explanations and predictions for the orbital structure in the classical region.  相似文献   

20.
W.-H. Ip 《Icarus》1977,32(3):378-381
Collision of asteroids with the main-belt asteroid population is considered with the effect of the impact kinetic energy taken into account. It is found that objects in eccentric orbits have a larger probability of destructive collision as compared to objects in orbits with mean values of eccentricity (e = 0.15) and inclination (i = 10°); also orbits with small semimajor axes (a ≈ 2.3 AU) are found to have peak values of the probability of destructive collision.  相似文献   

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