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The Role of Families in Determining Collision Probability in the Asteroid Main Belt
Icarus 153, 52–60 (2001) doi:10.1006/icar.2001.6621, available online at http://www.idealibrary.com on
The Role of Families in Determining Collision Probability in the Asteroid Main Belt A. Dell’Oro Dipartimento di Fisica, Universit`a di Pisa, 56127 Pisa, Italy
P. Paolicchi1 Dipartimento di Fisica, Universit`a di Pisa, 56127 Pisa, Italy E-mail: [email protected]; [email protected]
A. Cellino Osservatorio Astronomico di Torino, 10025 Pino Torinese (TO), Italy
V. Zappal`a Osservatorio Astronomico di Torino, 10025 Pino Torinese (TO), Italy
P. Tanga Osservatorio Astronomico di Torino, 10025 Pino Torinese (TO), Italy
and P. Michel Observatoire de la Cˆote d’Azur, BP 4229, 06304 Nice Cedex 4, France Received May 24, 2000; revised December 21, 2000
System history. Recently, the availability of a number (about 20) of statistically reliable families (Zappal`a et al. 1995 and references therein) has triggered a great deal of activity aimed at deriving from their observable properties information on the composition of their parent bodies and on the physical processes of catastrophic disruption from which families were originated. In particular, size and ejection velocity distributions of family members are very important, because they are directly related to the physics governing the outcomes of collisional break-up phenomena. According to the observational evidence, family forming events produced huge numbers of fragments, mainly small ones. The typical size distributions of families turn out to be very steep, much more than typical size distributions of nonfamily asteroids in any given region of the Main Belt. This fact has been recently interpreted in terms of geometric constraints affecting the production of fragments originated from bodies of finite sizes (Tanga et al. 1999). A point which is presently debated is the limiting diameter at which family size distributions should start to relax to shallower slopes. The reason is that in several cases the observed trend of the family size distributions cannot be extrapolated down to zero, because this would lead
Asteroid families represent the outcomes of major collisional events in the asteroid Main Belt. These events produced in some cases huge numbers of fragments, down to sizes of at least 1 km in diameter. In this paper we show that these families produce a significant increase in the “local” collision rate by as much as an order of magnitude relative to the average collision rate of the entire Belt. This fact should be taken into account in future studies of the collisional evolution of the Main Belt. °c 2001 Academic Press Key Words: asteroids; collisional physics.
1. INTRODUCTION
Asteroid families are the impressive remnants of energetic collisional events that were able to disrupt sizeable parent bodies and to disperse their fragments. Families provide direct evidence of the regime of collisional evolution which is believed to have been responsible for the main physical properties of the asteroid population since the very early epochs of the Solar 1
To whom correspondence should be addressed. 52
0019-1035/01 $35.00 c 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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FAMILIES’ ROLE IN COLLISION PROBABILITY
to an infinite mass of the parent body. Moreover, it is known that family size distributions should evolve as a function of time due to the overall collisional evolution of the asteroid Belt and should converge toward moderate slopes (Marzari et al. 1995). This is a very important issue, since the trend of the size distribution of fragments created in family forming events has several important consequences. On one hand, the number of objects produced in these events, and the distribution of their ejection velocities from the parent body, strongly influences the overall production rate of bodies able to reach the region of terrestrial planets. The reason is that a fraction of the fragments produced in these events can be expected in many cases to achieve resonant orbits, and to experience a chaotic evolution characterized by fast eccentricity increase leading to the possibility of close encounters with the inner planets (Gladman et al. 1997). In this way, Zappal`a et al. (1998) have shown that “asteroid showers” took place in the region of the terrestrial planets in the past, following events that produced some of the major families in the Main Belt. On the other hand, another major issue which constitutes the subject of the present paper is an evaluation of the consequences that the birth of the major families should have produced in the collisional regime of the Main Belt itself. In particular, we want to assess how the release of very large numbers of fragments in family forming events, with sizes distributed according to the observational evidence, can affect the average collision probabilities in different regions of the Belt. Some preliminary estimates of this were already made by Cellino and Zappal`a (in press), but in terms of mere impact possibility, in which the possible occurrence of an impact between any couple of bodies was decided using simply a rough criterion based on perihelion and aphelion distances. In the present paper, a much more refined assessment is done, dealing with real impact probabilities, derived in a rigorous way. The main tool needed to carry out this analysis, namely the technique of computation of impact probabilities, is briefly reviewed in Section 2, while in Section 3 we discuss the main input data needed to perform the computations, that is a plausible estimate of the inventory of the asteroid population, computed according to an assessment of the size distributions of both a number of outstanding families and the general background of non-family asteroids. Here, we will take profit of the extensive analysis of the size distributions of asteroid families recently performed by Tanga et al. (1999). In Section 4 we show the results of the present investigation and discuss their meaning and implications for future applications. 2. IMPACT PROBABILITY COMPUTATION
In the present paper, collision probabilities among bodies orbiting in the asteroid Main Belt are computed using the method described by Dell’Oro and Paolicchi (1998). An exhaustive description of the method is given in the above-mentioned paper. Here we only recall the main features of the algorithm. Conceptually, it is not really different with respect to the classical ¨ approach developed by Opik (1951), generalized by Wetherill
(1967), and then refined by Greenberg (1982) and Bottke and Greenberg (1993). The basic idea of the classical approach is to integrate over all the possible orientations of two given orbits, in order to identify the configurations leading to orbital intersection or, more precisely, to a minimum mutual distance less than the sum of the radii of the two bodies. Then the impact probability is given by the ratio between the time either body spends in the arc of orbit leading to a possible collision and its orbital period. What we call the total impact probability PT of an object with a set of other bodies is then simply given by X (Pi )k ; PT = k
that is, it is equal to the sum of the intrinsic collision probabilities (Pi )k of the body with each of the k objects belonging to the considered set of possible targets. In the classical approach, it is generally assumed that the motion of precession of the orbits is uniform in time, or, in other words, that the distribution of the arguments of perihelia ω and of the longitudes of the node Ä of the orbits are uniform. The main feature of the Dell’Oro and Paolicchi (1998) approach is the definition of a new formalism, allowing us to compute collision probabilities in a wider range of possible situations. In particular, the dynamical behavior of the system of two orbiting bodies is described by means of a function 1 of the orbital elements Ä, ω, and f (true anomaly) of both bodies. The advantage of this formalism is that it is not limited by any a priori assumption on the distributions of the above orbital elements. Another advantage of the new algorithm is that it is implemented in a fast computer code, making it easy to carry out general studies like the one performed in the present analysis. Of course, the new algorithm has been extensively tested. In particular, the method has been applied to samples of objects already analyzed by other authors using a more classical algorithm (Bottke and Greenberg 1993). The results have been found to be in a very good agreement both from the point of view of the pure collision probabilities and in terms of the resulting distributions of impact velocities (Dell’Oro and Paolicchi 1998). All this has been done with relatively moderate computer time requirements. 3. INPUT DATA: SIZE DISTRIBUTIONS
The subject of this paper is to assess the consequences of the formation of families on the average collision probability of asteroids in different regions of the Main Belt. This is done for the population of objects larger than 5 km. The basic hypothesis we want to test is that the formation of a family, leading to the production of a very large number of new objects (the fragments of the parent body) could alter significantly the average collision rate in the region of the Belt swept by the newly formed bodies. The main problem we have to face is, of course, that the asteroid inventory is not complete down to 5 km, and for this reason an extrapolation of the known population is necessary. The key point considered in this paper is that there is a substantial
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FIG. 1. Size distributions of non-family, background asteroids having proper semi-major axis in different ranges (indicated in each plot). The bold sections of each size distribution indicate the objects having sizes larger or equal to the completeness diameter in each subsample.
difference between the size distribution of the population of nonfamily asteroids (“background” population) and the size distribution of families. This conclusion is based on a substantial body of observational evidence, as shown in Figs. 1 and 2, in which the observed size distributions of some representative families can be compared with the size distributions of non-family, background asteroids in different regions of the Belt. From these figures it is evident that family size distributions are definitely steeper than the typical size distributions of background objects. Of course, apart from the case of a few families, such as Flora and Vesta, we do know that our inventories of known objects are severely incomplete at sizes of the order of 5 km, simply because a large number of asteroids of this size have not yet been discovered because they are too faint. In particular, the exact value of the apparent magnitude of completeness of the present asteroid inventory should be of the order of 15.0–15.5, according to recent estimates (Zappal`a and Cellino 1996; Jedicke and Metcalfe 1998). The problem of non-completeness of the inventory affects both families and background asteroids. In the case of families, it is easier to convert immediately the completeness value from magnitude to diameter, since families are generally uniform in albedo, and this parameter is fairly well known from IRAS data. In the case of the background, the conversion from magnitudes to diameters is more difficult, since the asteroid population exhibits a large range of possible albedos, differing by a factor of 10. For a given region of the Belt (characterized by a given range of orbital semi-major axes), therefore, the minimum
completeness diameter has a value which is determined by Ctype objects (characterized by the lowest albedo) located at the outer edge (in heliocentric distance) of the region. Consequently, each given family, or any subset of the whole background population corresponding to a given region of the Belt, has its own value of the diameter of completeness. Above this value, we are confident we already observed all the existing objects. For diameters smaller than the completeness value, the known objects constitute only a fraction of the number of really existing bodies. For this reason, when we compute collision probabilities among a given set of objects, we can confidently use all the real objects above the completeness diameter, whereas we assume that below the completeness limit each known object is representative of a number of real bodies of that size, equal to k. The parameter k used in the computations, therefore, is nothing but the ratio between the number of really existing objects of a given size and the number of known bodies of the same size. The former can be derived by an extrapolation of the size distributions of the observed samples. In Table I, we list a set of values for k which have been obtained for both families and background asteroids using as a database the same set of 12,000 asteroids that were used by Zappal`a et al. (1995) in their most recent search for asteroid families. The choice of this dataset is dictated by the consideration that by using more updated samples of asteroid orbital data we would face the problem that a large number of objects which achieved reliable orbits after 1995 were not considered in the most recent family search. As a consequence, for these asteroids we could not know if they
FIG. 2. Size distributions of some of the major families identified in the asteroid Main Belt. The bold regions of each distribution indicate the objects larger than the completeness diameter in each case.
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TABLE I Numbers of Known Members Larger Than 5 km D > 5 km Family/BG Region (Adeona) (Dora) Eos Eunomia Flora (Gefion) (Hygiea) Koronis Maria Themis (Vesta) 2.1 ≤ a < 2.3 2.3 ≤ a < 2.5 2.5 ≤ a < 2.7 2.7 ≤ a < 2.9 2.9 ≤ a < 3.1 3.1 ≤ a < 3.3
Nreal
Nextr
52 69 436 358 406 76 85 268 98 532 55 741 1093 1176 725 650 784
1480 310 4131 2748 709 201 >10000 729 654 9825 402 741 1321 1691 1139 1460 2044
k 28.46 4.49 9.47 7.68 1.75 2.64 ?? 2.72 6.67 18.47 7.31 1.00 1.21 1.44 1.57 2.25 2.61
Note. Extrapolated numbers and corresponding k values (see text) for some families considered in the present paper and non-family, background objects in different regions of the asteroid Main Belt. Families indicated in parentheses have only a few members larger than the family completeness diameter and have not been used in the final computation of collision probabilities. Question marks indicate unknown quantities.
are family or non-family objects. On the other hand, the older database is already sufficiently large to allow us to carry out reasonable extrapolations of the size distributions. In the case of families, sizes were computed for each object (when not directly determined by IRAS observations) on the basis of the known absolute magnitude H value and using an average albedo value for each family, given by the average value of IRAS-derived albedos for the members of the same family. In the case of background objects, we assigned sizes using again H , coupled with a tentative albedo value obtained by an average of the IRAS albedos of objects in the same interval of orbital semi-major axis. Alternative, and more refined, methods can be used to assign albedos to background objects, but, as shown by Cellino et al. (1991), even our simple procedure is fully reliable in a statistical sense. The extrapolated numbers of objects listed in Table I are given with a precision of the order of the unity, but of course these are only nominal values and are affected by much larger uncertainties. They were obtained by summing the tentative numbers of extrapolated objects at small sizes, to the precise numbers of known, real objects having sizes larger than the completeness diameter of each sample. In practice, moreover, we performed our extrapolations by considering separately different size ranges, in order to improve the reliability of our estimates. In the case of families, the detailed extrapolations (and relative k values) in different size ranges are listed in Table II.
In the case of the different background regions listed in Table I, we performed visual extrapolations of the observed populations down to 5 km. For families, we decided to carry out our extrapolations by using the fits of the size distributions obtained by Tanga et al. (1999), using their geometric model. This model is convenient, since it gives a “natural” explanation of some slope changes observed in the family size distributions, in terms of simple geometric properties of the bodies (finite volumes) and produces very good fits of the observed family size distributions down to small sizes. Alternative choices would have been possible. For several families, for instance, a least-squares fit of the observed size distribution using a suitable Pareto power law was also given by Tanga et al. (1999). However, Pareto fits are not available for many families, since in many cases too few family members exist above the completeness diameter. For this reason we a priori excluded in any case from our analysis a number of families, indicated in parentheses in Tables I and II. Another drawback of the Pareto fits is that they turn out to be fairly strongly dependent on the assumed value of the completeness diameter. The geometric model of Tanga et al. (1999), on the other hand, does not suffer from this problem, is fully consistent with the available data, and gives fits of the family size distributions that are at least as good as the nominal Pareto fits for diameters larger than the minimum completeness limit. To test the robustness of our results, we have also used alternative extrapolations, based on the limiting cases corresponding to the nominal uncertainties of the resulting Pareto fits for several families. What we have seen is that in general terms the results are not strongly affected. In some cases the new k values turn out to be smaller than those shown in Table I, while in other cases (like the big Eos family) the new k would be substantially larger. In other words, we found that the relative importance of some families (such as Eos) can increase, while it can decrease in other cases. On the average, we think that the adopted choices
TABLE II Numbers of Known Members 5 km ≤ D < 10 km Family
Nreal
Nextr
k
10 km ≤ D < 20 km Nreal
Nextr
k
(Adeona) 18 1300 72.22 24 170 7.08 (Dora) 19 260 13.68 41 — 1.00 Eos 56 3600 64.29 269 420 1.56 Eunomia 210 2600 12.38 142 — 1.00 Flora 337 640 1.90 63 — 1.00 (Gefion) 35 160 4.57 36 — 1.00 (Hygiea) 42 >10000 ?? 30 >4000 ?? Koronis 139 600 4.32 107 — 1.00 Maria 44 600 13.64 45 — 1.00 Themis 90 8800 97.78 317 900 2.84 (Vesta) 53 400 7.55 1 — 1.00
D ≥ 20 km Nreal Nextr 10 9 111 6 6 5 13 22 9 125 1
k
— 1.00 — 1.00 — 1.00 — 1.00 — 1.00 — 1.00 300 23.08 — 1.00 — 1.00 125 1.00 — 1.00
Note. Extrapolated numbers and corresponding k values (see text) in different size ranges for the same asteroid families listed in Table I. Question marks indicate unknown quantities.
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of the k parameters listed in Table I are statistically reliable and “robust.” As a final, important, remark we would like to stress the fact that our choice of limiting our analysis to objects larger than 5 km implies that we were forced to perform only very moderate extrapolations of the size distributions that should not be affected by dramatic uncertainties. In the case of some families in the inner Belt, as stated above, the observed size distributions are already practically complete down to that value. In the case of other families, even located at large heliocentric distances (such as Themis and Eos), we are confident that the present extrapolations are fully reliable, according also to the reasons explained by Tanga et al. (1999). In the case of Eos, in particular, we think that the performed extrapolation might have been even too pessimistic. Looking at the size distribution of this family, one can easily conclude that the geometric fit gives only a lower limit of the probable slope of the real size distribution. The number of Eos members with sizes around 5 km might be even 10 times the value assumed in the present paper. However, we preferred to stay on the safe side in evaluating the inventory of this family, due to the existing uncertainties and the lack of clear and conclusive evidence about a possibly huge number of Eos members, based on the current observational evidence (Jedicke and Metcalf 1998). As a general statement, we note that the observed steep size distributions of families make their plausible contribution to the overall asteroid inventory largely predominant at sizes smaller than 5 km (Zappal`a and Cellino 1996). Therefore, limiting our analysis to objects larger than 5 km leads to a probable underevaluation of the overall role played by families in determining collision probabilities in the Main Belt. Another reason to predict that we are probably underevaluating the importance of families is that we should take into account that in many cases, concerning mainly the big “clans” such as Flora, Eunomia, Eos, and Themis, the nominal member lists given by Zappal`a et al. (1995) are probably pessimistic, and large numbers of real members have not been included (increasing at the same time the role played by background objects). This conclusion is supported by a statistical analysis of the plausible numbers of random interlopers included in the family lists performed by Migliorini et al. (1995). Also the extrapolations of the size distributions of the background, non-family asteroids seem very reasonable according to the data at our disposal. Again, the reason is that limiting ourselves to the size range above 5 km does not force us to carry out questionable extrapolations in any way.
for it the total collision probability with any given set of existing asteroids, typically background bodies and/or members of one or more families. In this way, the relative importance of different sub-samples of the total population, such as background objects and single families, can be compared to each other. To give a visual representation of the results, we chose to produce a set of plots, in which each grid point is surrounded by a cell (all cells being equally sized) and each cell is plotted using a color according to a (logarithmic) scale related to the resulting range of total collision probability. Since it would be very confusing to show the results using three-dimensional plots, we chose to show two-dimensional projections in the a–e (proper semi-major axis and eccentricity) plane. Of course, in this way we cannot show, for a given set of orbits characterized by a fixed couple of a, e values, the dependence of the resulting collision probabilities upon the third orbital parameter, namely the inclination. In this respect, two observations can be made. First, for what concerns non-family, background asteroids, what we obtain is that there is a slight dependence of the total collision probability P on inclination, in the sense that P tends to decrease with inclination. This result is very easily explained, in the sense that it is clear that high-inclination bodies spend a significant fraction of their orbital period well above or below the ecliptic plane, the region where collisions are more probable. In Fig. 3 we give as an example the variation of the total collision probability with background, non-family asteroids as a function of orbital inclination for four sample points of the three-dimensional grid (arbitrarily chosen).
4. RESULTS AND FUTURE WORK
Our computations have been performed by considering a threedimensional grid superimposed to the location of the asteroid Main Belt in the space of the proper orbital elements a, e, and sin i (the same space used to identify asteroid families). Each point of the grid represents a sample orbit, and we can compute
FIG. 3. Total collision probability with non-family, background asteroids as a function of inclination for four arbitrarily chosen locations in the semi-major axis–eccentricity plane.
FAMILIES’ ROLE IN COLLISION PROBABILITY
FIG. 4. Probability of collision with members of the Eos and Eunomia families as a function of orbital inclination, for a body having the semi-major axis and eccentricity values indicated in the figure.
On the other hand, in the case of families a well-defined trend of P as a function of inclination is also clearly detectable. This is due to the obvious fact that family members orbit in a limited range of inclinations, approximately centered around the value of inclination of the original parent body. As a consequence, family members are more likely to impact bodies having approximately the same orbital inclination. A couple of examples of this effect is given in Fig. 4, referring to the large Eunomia and Eos families. Of course, another consequence of the above considerations is that for families an effect of self-erosion due to mutual collisions between members is relatively important. The behavior of families is also interesting when limiting our analysis to the a–e plane. In particular, families tend to produce a characteristic pattern in the probability plot. In particular, a largely increased collision probability in a well-defined, X-shaped region of the a–e plane is clearly shown, as an example in the case of the Eos family, in Fig. 5 (top right). This kind of behavior is not surprising, however, and is predicted by the classical equations describing the probability of collision. The explanation was first given by Wetherill (1967), who showed that the collision probability increases very much, but does not diverge to infinity, for nearly tangent orbits. Since the tangency of two orbits occurs when either the perihelion or aphelion distances turn out to be equal, this condition implies that the collision probability increases along two different hyperbolas in the a–e plane (corresponding to the orbits sharing the same q and Q, respectively). This fact explains why the typical pattern of the collision probability due to members of a single family in the a–e plane exhibits the observed X-shaped pattern in our
57
plots. Taking into account also the general dependence on the inclination, the observed X-like pattern becomes increasingly more evident when we approach the inclination values typical of a given family. Apart from these particular features, the results of our computations indicate that the role of families in determining impact probabilities in the asteroid Belt is very important. This is not surprising on the basis of the considerations already discussed in Section 2, and taking into account also that, even without performing any extrapolations of the observed size distributions, we already knew that family members constitute a large fraction, about one third, of the whole dataset of 12,487 asteroids considered by Zappal`a et al. (1995) in their search for asteroid families. This fact alone implies that families in any case must contribute for a substantial part of the overall collisional rate in the Main Belt. Their role should be even more important for the population of objects having increasingly smaller diameters, for the reasons explained in Section 3. Even at diameters larger than 5 km, the role of some families turns out to be crucial. Our computations indicate that the relative importance of the majority of the least populous families seems to be limited, whereas the most prominent families are so populous that their presence must have a dramatic impact on the overall collisional rate. In particular, adding families to the sample of background, non-family objects leads to a strong increase of the collision probability, as shown by Fig. 5 (bottom). According to our results, by simply adding to the background population the six most populous families considered in the present analysis, namely Flora, Eunomia, Maria, Eos, Themis, and Koronis, the total collision probability increases by a factor on the order of 10 throughout the Belt. Moreover, some other conclusions can be drawn by a more detailed analysis of the results. In particular, each family is intrinsically more effective in a wide, but limited region of the Belt (as shown in Fig. 5). As a consequence, adding families to the background does not produce a homogeneous increase of the collision probability throughout the Belt, but some well-defined pattern is clearly recognizable. In particular, we find that, not unexpectedly, the low eccentricity orbits at the inner edges of the Belt in terms of semi-major axis are less affected by the presence of families. The major increase in collision probability tends to occur in the middle of the Belt, especially at small eccentricities between 2.6 and 3.0 AU, and at larger eccentricities in the inner Belt, between 2.5 and 2.6 AU. Of course, the middle region of the Belt is one that is intrinsically more easily affected by the production of large numbers of collisional fragments, originated anywhere in the Belt (because these objects will in general sweep anyhow across the middle Belt, even when their eccentricities are too small to allow them to sweep across the whole Belt). In this sense, it is also reasonable to expect that the middle Belt is more steadily affected by enhancements of the collisional regime, provoked by family forming events. A consequence of this can be the production of new families. As a matter of fact, apart from the
FIG. 5. Total collision probability plots in the a–e plane. The probability is expressed by different colors, according to a logarithmic scale shown in the middle of the plot. Collision probabilities refer to impacts with non-family, background objects (top left), Eos members (top right), members of the Flora, Maria, Eunomia, Eos, Themis, and Koronis families (bottom left), and members of the above-mentioned families plus background objects (bottom right). As can be seen, the six large families alone largely determine the general structure of the impact probability in the asteroid Main Belt. The real situation shown in this plot might change slightly by including other families such as Hygiea and/or the Nysa-Polana clan (see text).
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FAMILIES’ ROLE IN COLLISION PROBABILITY
major families considered in the present analysis, several minor groupings, plausibly created by the disruption of fairly small parent bodies (such as, for instance, the Merxia family, with a 42-km parent body according to Tanga et al. 1999), do exist in the middle region of the Main Belt. In any case, from the present analysis we draw serious indications that the formation of “monster” families such as Eos (for instance) can trigger the subsequent collisional evolution of the asteroid population in wide regions of the Main Belt. On one hand, we think that this result should be taken into account in any attempt to reproduce the collisional history of the Belt by means of numerical simulations (Davis et al. 1989; Farinella et al. 1992; Campo Bagatin et al. 1994). On the other hand, the present results are important also from the point of view of the origin of NEAs. In particular, in recent years it has been recognized that Main Belt asteroids can be injected into the inner Solar System through different dynamical paths. The most classic approach to this problem takes into account the major mean-motion (4/1, 3/1, 5/2) and secular (ν6 ) resonances, only. It is known that objects injected into one of these zones suffer very fast eccentricity increases, allowing them to have close encounters with the terrestrial planets and to be eventually extracted from the Main Belt. On the other hand, it has been recently shown that the typical dynamical lifetimes of objects experiencing this kind of evolution are very short (Gladman et al. 1997; Zappal`a et al. 1998). As a consequence, alternative dynamical evolutions, based on martian perturbations affecting asteroids located in the inner region of the Belt, have been proposed as an alternative source for NEAs (Migliorini et al. 1998). The problem seems not completely solved for the moment, but in this respect the results shown in the present paper can be useful. In particular, evaluating the collisional rate in the regions surrounding the chaotic zones associated with the most important resonances in the Belt can allow us to draw conclusions about the possibility for the bodies located in those regions to effectively contribute to the production rate of NEAs of different sizes. Of course, other constraints must be taken into account. In our opinion an important problem is presently to understand which kind of collisions are required to produce NEAs of different sizes, while at the same time not producing observable families from the disruption of the parent bodies. Taking into account that dealing with this problem implies also an assessment of the intrinsic probability of collisional events at different sizes, we can say that the present paper constitutes a necessary step forward to develop more detailed modeling. It is clear that this kind of analysis cannot be performed here, but we plan to do it very soon in a separate paper, since we believe we have now at disposal all the necessary tools, namely a reasonable model of the size and velocity distributions and sizeejection velocity relationship in family forming events (Zappal`a et al. 1996; Tanga et al. 1999; Cellino et al. 1999), while the intrinsic collision probabilities can be derived by the present paper, using the algorithm developed by Dell’Oro and Paolicchi (1998).
We would like to end by noting that we did not include in this analysis some populous families that are probably also very important suppliers of projectiles for inter-asteroid collisions. These families are that of Vesta, which was not included because its members are generally well below the 5-km limit in size, and the Nysa-Polana clan in the inner Belt. In the latter case, the reason for excluding this family was that it is probably given by the chance overlapping of two different families (Doressoundiram et al. 1998) and a more detailed physical analysis of the two groupings is strongly needed. Some work in this respect is currently being performed, and we plan to publish a detailed physical study of Nysa-Polana in a forthcoming paper. Another important omission, at least from the point of view of the numerical results shown in Fig. 5, concerns the Hygiea family. This is a family for which it is very hard to derive a reliable extrapolation of the size distribution. We have good reasons to believe that, at least at the epoch of its formation, it was one of the most important and populous groupings in the asteroid Main Belt. However, we have also reasons to believe that this family is now quite old, judging from a trend of the size distribution that is fairly complicated, and also likely affected by the presence of a number of random interlopers (Tanga et al. 1999). It is reasonable to assume that such an old family can be nowadays seriously eroded at small sizes (Marzari et al. 1995). For this reason, and taking into account the small number of members larger than the nominal completeness limit (and the quoted presence of many likely interlopers among them) any kind of extrapolation is too uncertain to allow us to derive a reasonable estimate of the role currently played by this family. ACKNOWLEDGMENTS We thank D. R. Davis and F. Marzari for their careful reviews, which led to a substantial improvement of the original paper. This work has been supported by MURST (cofin98).
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Binzel, T. Gehrels, and M. Shapley Matthews, Eds.), pp. 805–826. Univ. of Arizona Press, Tucson. Dell’Oro, A., and P. Paolicchi 1998. Statistical properties of encounters among asteroids: A new, general purpose, formalism. Icarus 136, 328–339. Doressoundiram A., A. Cellino, M. Di Martino, F. Migliorini, and V. Zappal`a 1998. The puzzling case of the Nysa–Polana family finally solved? B.A.A.S. 30, 1022. Farinella, P., D. R. Davis, P. Paolicchi, A. Cellino, and V. Zappal`a 1992. Asteroid collisional evolution: An integrated model for the collisional evolution of asteroid rotation rates. Astron. Astrophys. 253, 604–614. Gladman, B. J., F. Migliorini, A. Morbidelli, V. Zappal`a, P. Michel, A. Cellino, Ch. Froeschl´e, H. Levison, M. Bailey, and M. Duncan 1997. Dynamical lifetimes of objects injected into asteroid belt resonances. Science 277, 197–201. Greenberg, R. 1982. Orbital interactions: A new geometrical formalism. Astron. J. 87, 185–195. Jedicke, R., and T. S. Metcalfe 1998. The orbital and absolute magnitude distributions of Main Belt asteroids. Icarus 131, 245–260. Marzari, F., D. R. Davis, and V. Vanzani 1995. Collisional evolution of asteroid families. Icarus 113, 168–187. Migliorini, F., V. Zappal`a, R. Vio, and A. Cellino 1995. Interlopers within asteroid families. Icarus 118, 271–291. Migliorini, F., P. Michel, A. Morbidelli, D. Nesvorn´y, and V. Zappal`a 1998. Origin of multikilometer Earth- and Mars-crossing asteroids: A quantitative simulation. Science 281, 2022–2024.
Morbidelli, A., V. Zappal`a, M. Moons, A. Cellino, and R. Gonczi 1995. Asteroid families close to mean motion resonances: Dynamical effects and physical implications. Icarus 118, 132–154. ¨ Opik, E. J. 1951. Collision probabilities with the planets and distribution of ¨ interplanetary matter. Proc. R. Irish Acad. A54, 165–199. [See also Opik 1976. Interplanetary Encounters. Elsevier, Amsterdam.] Tanga, P., A. Cellino, P. Michel, V. Zappal`a, P. Paolicchi, and A. Dell’Oro 1999. On the size distribution of asteroid families: The role of geometry. Icarus 141, 65–78. Wetherill, G. W. 1967. Collisions in the asteroid belt. J. Geophys. Res. 72, 2429– 2444. Zappal`a, V., P. Bendjoya, A. Cellino, P. Farinella, and Cl. Froeschl´e 1995. Asteroid families: Search of a 12,487 asteroid sample using two different clustering techniques. Icarus 116, 291–314. Zappal`a, V., and A. Cellino 1996. Main Belt asteroids: Present and future inventory. In Completing the Inventory of the Solar System (T. W. Rettig and J.M. Hahn, Eds.), ASP Conference Series 107, pp. 29–44. Astronomical Society of the Pacific, San Francisco. Zappal`a, V., A. Cellino, A. Dell’Oro, F. Migliorini, and P. Paolicchi 1996. Reconstructing the original ejection velocity fields of asteroid families. Icarus 124, 156–180. Zappal`a, V., A. Cellino, B. J. Gladman, S. Manley, and F. Migliorini 1998. Asteroid showers on Earth after family break-up events. Icarus 134, 176– 179.
The Role of Families in Determining Collision Probability in the Asteroid Main Belt A. Dell’Oro Dipartimento di Fisica, Universit`a di Pisa, 56127 Pisa, Italy
P. Paolicchi1 Dipartimento di Fisica, Universit`a di Pisa, 56127 Pisa, Italy E-mail: [email protected]; [email protected]
A. Cellino Osservatorio Astronomico di Torino, 10025 Pino Torinese (TO), Italy
V. Zappal`a Osservatorio Astronomico di Torino, 10025 Pino Torinese (TO), Italy
P. Tanga Osservatorio Astronomico di Torino, 10025 Pino Torinese (TO), Italy
and P. Michel Observatoire de la Cˆote d’Azur, BP 4229, 06304 Nice Cedex 4, France Received May 24, 2000; revised December 21, 2000
System history. Recently, the availability of a number (about 20) of statistically reliable families (Zappal`a et al. 1995 and references therein) has triggered a great deal of activity aimed at deriving from their observable properties information on the composition of their parent bodies and on the physical processes of catastrophic disruption from which families were originated. In particular, size and ejection velocity distributions of family members are very important, because they are directly related to the physics governing the outcomes of collisional break-up phenomena. According to the observational evidence, family forming events produced huge numbers of fragments, mainly small ones. The typical size distributions of families turn out to be very steep, much more than typical size distributions of nonfamily asteroids in any given region of the Main Belt. This fact has been recently interpreted in terms of geometric constraints affecting the production of fragments originated from bodies of finite sizes (Tanga et al. 1999). A point which is presently debated is the limiting diameter at which family size distributions should start to relax to shallower slopes. The reason is that in several cases the observed trend of the family size distributions cannot be extrapolated down to zero, because this would lead
Asteroid families represent the outcomes of major collisional events in the asteroid Main Belt. These events produced in some cases huge numbers of fragments, down to sizes of at least 1 km in diameter. In this paper we show that these families produce a significant increase in the “local” collision rate by as much as an order of magnitude relative to the average collision rate of the entire Belt. This fact should be taken into account in future studies of the collisional evolution of the Main Belt. °c 2001 Academic Press Key Words: asteroids; collisional physics.
1. INTRODUCTION
Asteroid families are the impressive remnants of energetic collisional events that were able to disrupt sizeable parent bodies and to disperse their fragments. Families provide direct evidence of the regime of collisional evolution which is believed to have been responsible for the main physical properties of the asteroid population since the very early epochs of the Solar 1
To whom correspondence should be addressed. 52
0019-1035/01 $35.00 c 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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FAMILIES’ ROLE IN COLLISION PROBABILITY
to an infinite mass of the parent body. Moreover, it is known that family size distributions should evolve as a function of time due to the overall collisional evolution of the asteroid Belt and should converge toward moderate slopes (Marzari et al. 1995). This is a very important issue, since the trend of the size distribution of fragments created in family forming events has several important consequences. On one hand, the number of objects produced in these events, and the distribution of their ejection velocities from the parent body, strongly influences the overall production rate of bodies able to reach the region of terrestrial planets. The reason is that a fraction of the fragments produced in these events can be expected in many cases to achieve resonant orbits, and to experience a chaotic evolution characterized by fast eccentricity increase leading to the possibility of close encounters with the inner planets (Gladman et al. 1997). In this way, Zappal`a et al. (1998) have shown that “asteroid showers” took place in the region of the terrestrial planets in the past, following events that produced some of the major families in the Main Belt. On the other hand, another major issue which constitutes the subject of the present paper is an evaluation of the consequences that the birth of the major families should have produced in the collisional regime of the Main Belt itself. In particular, we want to assess how the release of very large numbers of fragments in family forming events, with sizes distributed according to the observational evidence, can affect the average collision probabilities in different regions of the Belt. Some preliminary estimates of this were already made by Cellino and Zappal`a (in press), but in terms of mere impact possibility, in which the possible occurrence of an impact between any couple of bodies was decided using simply a rough criterion based on perihelion and aphelion distances. In the present paper, a much more refined assessment is done, dealing with real impact probabilities, derived in a rigorous way. The main tool needed to carry out this analysis, namely the technique of computation of impact probabilities, is briefly reviewed in Section 2, while in Section 3 we discuss the main input data needed to perform the computations, that is a plausible estimate of the inventory of the asteroid population, computed according to an assessment of the size distributions of both a number of outstanding families and the general background of non-family asteroids. Here, we will take profit of the extensive analysis of the size distributions of asteroid families recently performed by Tanga et al. (1999). In Section 4 we show the results of the present investigation and discuss their meaning and implications for future applications. 2. IMPACT PROBABILITY COMPUTATION
In the present paper, collision probabilities among bodies orbiting in the asteroid Main Belt are computed using the method described by Dell’Oro and Paolicchi (1998). An exhaustive description of the method is given in the above-mentioned paper. Here we only recall the main features of the algorithm. Conceptually, it is not really different with respect to the classical ¨ approach developed by Opik (1951), generalized by Wetherill
(1967), and then refined by Greenberg (1982) and Bottke and Greenberg (1993). The basic idea of the classical approach is to integrate over all the possible orientations of two given orbits, in order to identify the configurations leading to orbital intersection or, more precisely, to a minimum mutual distance less than the sum of the radii of the two bodies. Then the impact probability is given by the ratio between the time either body spends in the arc of orbit leading to a possible collision and its orbital period. What we call the total impact probability PT of an object with a set of other bodies is then simply given by X (Pi )k ; PT = k
that is, it is equal to the sum of the intrinsic collision probabilities (Pi )k of the body with each of the k objects belonging to the considered set of possible targets. In the classical approach, it is generally assumed that the motion of precession of the orbits is uniform in time, or, in other words, that the distribution of the arguments of perihelia ω and of the longitudes of the node Ä of the orbits are uniform. The main feature of the Dell’Oro and Paolicchi (1998) approach is the definition of a new formalism, allowing us to compute collision probabilities in a wider range of possible situations. In particular, the dynamical behavior of the system of two orbiting bodies is described by means of a function 1 of the orbital elements Ä, ω, and f (true anomaly) of both bodies. The advantage of this formalism is that it is not limited by any a priori assumption on the distributions of the above orbital elements. Another advantage of the new algorithm is that it is implemented in a fast computer code, making it easy to carry out general studies like the one performed in the present analysis. Of course, the new algorithm has been extensively tested. In particular, the method has been applied to samples of objects already analyzed by other authors using a more classical algorithm (Bottke and Greenberg 1993). The results have been found to be in a very good agreement both from the point of view of the pure collision probabilities and in terms of the resulting distributions of impact velocities (Dell’Oro and Paolicchi 1998). All this has been done with relatively moderate computer time requirements. 3. INPUT DATA: SIZE DISTRIBUTIONS
The subject of this paper is to assess the consequences of the formation of families on the average collision probability of asteroids in different regions of the Main Belt. This is done for the population of objects larger than 5 km. The basic hypothesis we want to test is that the formation of a family, leading to the production of a very large number of new objects (the fragments of the parent body) could alter significantly the average collision rate in the region of the Belt swept by the newly formed bodies. The main problem we have to face is, of course, that the asteroid inventory is not complete down to 5 km, and for this reason an extrapolation of the known population is necessary. The key point considered in this paper is that there is a substantial
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FIG. 1. Size distributions of non-family, background asteroids having proper semi-major axis in different ranges (indicated in each plot). The bold sections of each size distribution indicate the objects having sizes larger or equal to the completeness diameter in each subsample.
difference between the size distribution of the population of nonfamily asteroids (“background” population) and the size distribution of families. This conclusion is based on a substantial body of observational evidence, as shown in Figs. 1 and 2, in which the observed size distributions of some representative families can be compared with the size distributions of non-family, background asteroids in different regions of the Belt. From these figures it is evident that family size distributions are definitely steeper than the typical size distributions of background objects. Of course, apart from the case of a few families, such as Flora and Vesta, we do know that our inventories of known objects are severely incomplete at sizes of the order of 5 km, simply because a large number of asteroids of this size have not yet been discovered because they are too faint. In particular, the exact value of the apparent magnitude of completeness of the present asteroid inventory should be of the order of 15.0–15.5, according to recent estimates (Zappal`a and Cellino 1996; Jedicke and Metcalfe 1998). The problem of non-completeness of the inventory affects both families and background asteroids. In the case of families, it is easier to convert immediately the completeness value from magnitude to diameter, since families are generally uniform in albedo, and this parameter is fairly well known from IRAS data. In the case of the background, the conversion from magnitudes to diameters is more difficult, since the asteroid population exhibits a large range of possible albedos, differing by a factor of 10. For a given region of the Belt (characterized by a given range of orbital semi-major axes), therefore, the minimum
completeness diameter has a value which is determined by Ctype objects (characterized by the lowest albedo) located at the outer edge (in heliocentric distance) of the region. Consequently, each given family, or any subset of the whole background population corresponding to a given region of the Belt, has its own value of the diameter of completeness. Above this value, we are confident we already observed all the existing objects. For diameters smaller than the completeness value, the known objects constitute only a fraction of the number of really existing bodies. For this reason, when we compute collision probabilities among a given set of objects, we can confidently use all the real objects above the completeness diameter, whereas we assume that below the completeness limit each known object is representative of a number of real bodies of that size, equal to k. The parameter k used in the computations, therefore, is nothing but the ratio between the number of really existing objects of a given size and the number of known bodies of the same size. The former can be derived by an extrapolation of the size distributions of the observed samples. In Table I, we list a set of values for k which have been obtained for both families and background asteroids using as a database the same set of 12,000 asteroids that were used by Zappal`a et al. (1995) in their most recent search for asteroid families. The choice of this dataset is dictated by the consideration that by using more updated samples of asteroid orbital data we would face the problem that a large number of objects which achieved reliable orbits after 1995 were not considered in the most recent family search. As a consequence, for these asteroids we could not know if they
FIG. 2. Size distributions of some of the major families identified in the asteroid Main Belt. The bold regions of each distribution indicate the objects larger than the completeness diameter in each case.
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FAMILIES’ ROLE IN COLLISION PROBABILITY
TABLE I Numbers of Known Members Larger Than 5 km D > 5 km Family/BG Region (Adeona) (Dora) Eos Eunomia Flora (Gefion) (Hygiea) Koronis Maria Themis (Vesta) 2.1 ≤ a < 2.3 2.3 ≤ a < 2.5 2.5 ≤ a < 2.7 2.7 ≤ a < 2.9 2.9 ≤ a < 3.1 3.1 ≤ a < 3.3
Nreal
Nextr
52 69 436 358 406 76 85 268 98 532 55 741 1093 1176 725 650 784
1480 310 4131 2748 709 201 >10000 729 654 9825 402 741 1321 1691 1139 1460 2044
k 28.46 4.49 9.47 7.68 1.75 2.64 ?? 2.72 6.67 18.47 7.31 1.00 1.21 1.44 1.57 2.25 2.61
Note. Extrapolated numbers and corresponding k values (see text) for some families considered in the present paper and non-family, background objects in different regions of the asteroid Main Belt. Families indicated in parentheses have only a few members larger than the family completeness diameter and have not been used in the final computation of collision probabilities. Question marks indicate unknown quantities.
are family or non-family objects. On the other hand, the older database is already sufficiently large to allow us to carry out reasonable extrapolations of the size distributions. In the case of families, sizes were computed for each object (when not directly determined by IRAS observations) on the basis of the known absolute magnitude H value and using an average albedo value for each family, given by the average value of IRAS-derived albedos for the members of the same family. In the case of background objects, we assigned sizes using again H , coupled with a tentative albedo value obtained by an average of the IRAS albedos of objects in the same interval of orbital semi-major axis. Alternative, and more refined, methods can be used to assign albedos to background objects, but, as shown by Cellino et al. (1991), even our simple procedure is fully reliable in a statistical sense. The extrapolated numbers of objects listed in Table I are given with a precision of the order of the unity, but of course these are only nominal values and are affected by much larger uncertainties. They were obtained by summing the tentative numbers of extrapolated objects at small sizes, to the precise numbers of known, real objects having sizes larger than the completeness diameter of each sample. In practice, moreover, we performed our extrapolations by considering separately different size ranges, in order to improve the reliability of our estimates. In the case of families, the detailed extrapolations (and relative k values) in different size ranges are listed in Table II.
In the case of the different background regions listed in Table I, we performed visual extrapolations of the observed populations down to 5 km. For families, we decided to carry out our extrapolations by using the fits of the size distributions obtained by Tanga et al. (1999), using their geometric model. This model is convenient, since it gives a “natural” explanation of some slope changes observed in the family size distributions, in terms of simple geometric properties of the bodies (finite volumes) and produces very good fits of the observed family size distributions down to small sizes. Alternative choices would have been possible. For several families, for instance, a least-squares fit of the observed size distribution using a suitable Pareto power law was also given by Tanga et al. (1999). However, Pareto fits are not available for many families, since in many cases too few family members exist above the completeness diameter. For this reason we a priori excluded in any case from our analysis a number of families, indicated in parentheses in Tables I and II. Another drawback of the Pareto fits is that they turn out to be fairly strongly dependent on the assumed value of the completeness diameter. The geometric model of Tanga et al. (1999), on the other hand, does not suffer from this problem, is fully consistent with the available data, and gives fits of the family size distributions that are at least as good as the nominal Pareto fits for diameters larger than the minimum completeness limit. To test the robustness of our results, we have also used alternative extrapolations, based on the limiting cases corresponding to the nominal uncertainties of the resulting Pareto fits for several families. What we have seen is that in general terms the results are not strongly affected. In some cases the new k values turn out to be smaller than those shown in Table I, while in other cases (like the big Eos family) the new k would be substantially larger. In other words, we found that the relative importance of some families (such as Eos) can increase, while it can decrease in other cases. On the average, we think that the adopted choices
TABLE II Numbers of Known Members 5 km ≤ D < 10 km Family
Nreal
Nextr
k
10 km ≤ D < 20 km Nreal
Nextr
k
(Adeona) 18 1300 72.22 24 170 7.08 (Dora) 19 260 13.68 41 — 1.00 Eos 56 3600 64.29 269 420 1.56 Eunomia 210 2600 12.38 142 — 1.00 Flora 337 640 1.90 63 — 1.00 (Gefion) 35 160 4.57 36 — 1.00 (Hygiea) 42 >10000 ?? 30 >4000 ?? Koronis 139 600 4.32 107 — 1.00 Maria 44 600 13.64 45 — 1.00 Themis 90 8800 97.78 317 900 2.84 (Vesta) 53 400 7.55 1 — 1.00
D ≥ 20 km Nreal Nextr 10 9 111 6 6 5 13 22 9 125 1
k
— 1.00 — 1.00 — 1.00 — 1.00 — 1.00 — 1.00 300 23.08 — 1.00 — 1.00 125 1.00 — 1.00
Note. Extrapolated numbers and corresponding k values (see text) in different size ranges for the same asteroid families listed in Table I. Question marks indicate unknown quantities.
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DELL’ORO ET AL.
of the k parameters listed in Table I are statistically reliable and “robust.” As a final, important, remark we would like to stress the fact that our choice of limiting our analysis to objects larger than 5 km implies that we were forced to perform only very moderate extrapolations of the size distributions that should not be affected by dramatic uncertainties. In the case of some families in the inner Belt, as stated above, the observed size distributions are already practically complete down to that value. In the case of other families, even located at large heliocentric distances (such as Themis and Eos), we are confident that the present extrapolations are fully reliable, according also to the reasons explained by Tanga et al. (1999). In the case of Eos, in particular, we think that the performed extrapolation might have been even too pessimistic. Looking at the size distribution of this family, one can easily conclude that the geometric fit gives only a lower limit of the probable slope of the real size distribution. The number of Eos members with sizes around 5 km might be even 10 times the value assumed in the present paper. However, we preferred to stay on the safe side in evaluating the inventory of this family, due to the existing uncertainties and the lack of clear and conclusive evidence about a possibly huge number of Eos members, based on the current observational evidence (Jedicke and Metcalf 1998). As a general statement, we note that the observed steep size distributions of families make their plausible contribution to the overall asteroid inventory largely predominant at sizes smaller than 5 km (Zappal`a and Cellino 1996). Therefore, limiting our analysis to objects larger than 5 km leads to a probable underevaluation of the overall role played by families in determining collision probabilities in the Main Belt. Another reason to predict that we are probably underevaluating the importance of families is that we should take into account that in many cases, concerning mainly the big “clans” such as Flora, Eunomia, Eos, and Themis, the nominal member lists given by Zappal`a et al. (1995) are probably pessimistic, and large numbers of real members have not been included (increasing at the same time the role played by background objects). This conclusion is supported by a statistical analysis of the plausible numbers of random interlopers included in the family lists performed by Migliorini et al. (1995). Also the extrapolations of the size distributions of the background, non-family asteroids seem very reasonable according to the data at our disposal. Again, the reason is that limiting ourselves to the size range above 5 km does not force us to carry out questionable extrapolations in any way.
for it the total collision probability with any given set of existing asteroids, typically background bodies and/or members of one or more families. In this way, the relative importance of different sub-samples of the total population, such as background objects and single families, can be compared to each other. To give a visual representation of the results, we chose to produce a set of plots, in which each grid point is surrounded by a cell (all cells being equally sized) and each cell is plotted using a color according to a (logarithmic) scale related to the resulting range of total collision probability. Since it would be very confusing to show the results using three-dimensional plots, we chose to show two-dimensional projections in the a–e (proper semi-major axis and eccentricity) plane. Of course, in this way we cannot show, for a given set of orbits characterized by a fixed couple of a, e values, the dependence of the resulting collision probabilities upon the third orbital parameter, namely the inclination. In this respect, two observations can be made. First, for what concerns non-family, background asteroids, what we obtain is that there is a slight dependence of the total collision probability P on inclination, in the sense that P tends to decrease with inclination. This result is very easily explained, in the sense that it is clear that high-inclination bodies spend a significant fraction of their orbital period well above or below the ecliptic plane, the region where collisions are more probable. In Fig. 3 we give as an example the variation of the total collision probability with background, non-family asteroids as a function of orbital inclination for four sample points of the three-dimensional grid (arbitrarily chosen).
4. RESULTS AND FUTURE WORK
Our computations have been performed by considering a threedimensional grid superimposed to the location of the asteroid Main Belt in the space of the proper orbital elements a, e, and sin i (the same space used to identify asteroid families). Each point of the grid represents a sample orbit, and we can compute
FIG. 3. Total collision probability with non-family, background asteroids as a function of inclination for four arbitrarily chosen locations in the semi-major axis–eccentricity plane.
FAMILIES’ ROLE IN COLLISION PROBABILITY
FIG. 4. Probability of collision with members of the Eos and Eunomia families as a function of orbital inclination, for a body having the semi-major axis and eccentricity values indicated in the figure.
On the other hand, in the case of families a well-defined trend of P as a function of inclination is also clearly detectable. This is due to the obvious fact that family members orbit in a limited range of inclinations, approximately centered around the value of inclination of the original parent body. As a consequence, family members are more likely to impact bodies having approximately the same orbital inclination. A couple of examples of this effect is given in Fig. 4, referring to the large Eunomia and Eos families. Of course, another consequence of the above considerations is that for families an effect of self-erosion due to mutual collisions between members is relatively important. The behavior of families is also interesting when limiting our analysis to the a–e plane. In particular, families tend to produce a characteristic pattern in the probability plot. In particular, a largely increased collision probability in a well-defined, X-shaped region of the a–e plane is clearly shown, as an example in the case of the Eos family, in Fig. 5 (top right). This kind of behavior is not surprising, however, and is predicted by the classical equations describing the probability of collision. The explanation was first given by Wetherill (1967), who showed that the collision probability increases very much, but does not diverge to infinity, for nearly tangent orbits. Since the tangency of two orbits occurs when either the perihelion or aphelion distances turn out to be equal, this condition implies that the collision probability increases along two different hyperbolas in the a–e plane (corresponding to the orbits sharing the same q and Q, respectively). This fact explains why the typical pattern of the collision probability due to members of a single family in the a–e plane exhibits the observed X-shaped pattern in our
57
plots. Taking into account also the general dependence on the inclination, the observed X-like pattern becomes increasingly more evident when we approach the inclination values typical of a given family. Apart from these particular features, the results of our computations indicate that the role of families in determining impact probabilities in the asteroid Belt is very important. This is not surprising on the basis of the considerations already discussed in Section 2, and taking into account also that, even without performing any extrapolations of the observed size distributions, we already knew that family members constitute a large fraction, about one third, of the whole dataset of 12,487 asteroids considered by Zappal`a et al. (1995) in their search for asteroid families. This fact alone implies that families in any case must contribute for a substantial part of the overall collisional rate in the Main Belt. Their role should be even more important for the population of objects having increasingly smaller diameters, for the reasons explained in Section 3. Even at diameters larger than 5 km, the role of some families turns out to be crucial. Our computations indicate that the relative importance of the majority of the least populous families seems to be limited, whereas the most prominent families are so populous that their presence must have a dramatic impact on the overall collisional rate. In particular, adding families to the sample of background, non-family objects leads to a strong increase of the collision probability, as shown by Fig. 5 (bottom). According to our results, by simply adding to the background population the six most populous families considered in the present analysis, namely Flora, Eunomia, Maria, Eos, Themis, and Koronis, the total collision probability increases by a factor on the order of 10 throughout the Belt. Moreover, some other conclusions can be drawn by a more detailed analysis of the results. In particular, each family is intrinsically more effective in a wide, but limited region of the Belt (as shown in Fig. 5). As a consequence, adding families to the background does not produce a homogeneous increase of the collision probability throughout the Belt, but some well-defined pattern is clearly recognizable. In particular, we find that, not unexpectedly, the low eccentricity orbits at the inner edges of the Belt in terms of semi-major axis are less affected by the presence of families. The major increase in collision probability tends to occur in the middle of the Belt, especially at small eccentricities between 2.6 and 3.0 AU, and at larger eccentricities in the inner Belt, between 2.5 and 2.6 AU. Of course, the middle region of the Belt is one that is intrinsically more easily affected by the production of large numbers of collisional fragments, originated anywhere in the Belt (because these objects will in general sweep anyhow across the middle Belt, even when their eccentricities are too small to allow them to sweep across the whole Belt). In this sense, it is also reasonable to expect that the middle Belt is more steadily affected by enhancements of the collisional regime, provoked by family forming events. A consequence of this can be the production of new families. As a matter of fact, apart from the
FIG. 5. Total collision probability plots in the a–e plane. The probability is expressed by different colors, according to a logarithmic scale shown in the middle of the plot. Collision probabilities refer to impacts with non-family, background objects (top left), Eos members (top right), members of the Flora, Maria, Eunomia, Eos, Themis, and Koronis families (bottom left), and members of the above-mentioned families plus background objects (bottom right). As can be seen, the six large families alone largely determine the general structure of the impact probability in the asteroid Main Belt. The real situation shown in this plot might change slightly by including other families such as Hygiea and/or the Nysa-Polana clan (see text).
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FAMILIES’ ROLE IN COLLISION PROBABILITY
major families considered in the present analysis, several minor groupings, plausibly created by the disruption of fairly small parent bodies (such as, for instance, the Merxia family, with a 42-km parent body according to Tanga et al. 1999), do exist in the middle region of the Main Belt. In any case, from the present analysis we draw serious indications that the formation of “monster” families such as Eos (for instance) can trigger the subsequent collisional evolution of the asteroid population in wide regions of the Main Belt. On one hand, we think that this result should be taken into account in any attempt to reproduce the collisional history of the Belt by means of numerical simulations (Davis et al. 1989; Farinella et al. 1992; Campo Bagatin et al. 1994). On the other hand, the present results are important also from the point of view of the origin of NEAs. In particular, in recent years it has been recognized that Main Belt asteroids can be injected into the inner Solar System through different dynamical paths. The most classic approach to this problem takes into account the major mean-motion (4/1, 3/1, 5/2) and secular (ν6 ) resonances, only. It is known that objects injected into one of these zones suffer very fast eccentricity increases, allowing them to have close encounters with the terrestrial planets and to be eventually extracted from the Main Belt. On the other hand, it has been recently shown that the typical dynamical lifetimes of objects experiencing this kind of evolution are very short (Gladman et al. 1997; Zappal`a et al. 1998). As a consequence, alternative dynamical evolutions, based on martian perturbations affecting asteroids located in the inner region of the Belt, have been proposed as an alternative source for NEAs (Migliorini et al. 1998). The problem seems not completely solved for the moment, but in this respect the results shown in the present paper can be useful. In particular, evaluating the collisional rate in the regions surrounding the chaotic zones associated with the most important resonances in the Belt can allow us to draw conclusions about the possibility for the bodies located in those regions to effectively contribute to the production rate of NEAs of different sizes. Of course, other constraints must be taken into account. In our opinion an important problem is presently to understand which kind of collisions are required to produce NEAs of different sizes, while at the same time not producing observable families from the disruption of the parent bodies. Taking into account that dealing with this problem implies also an assessment of the intrinsic probability of collisional events at different sizes, we can say that the present paper constitutes a necessary step forward to develop more detailed modeling. It is clear that this kind of analysis cannot be performed here, but we plan to do it very soon in a separate paper, since we believe we have now at disposal all the necessary tools, namely a reasonable model of the size and velocity distributions and sizeejection velocity relationship in family forming events (Zappal`a et al. 1996; Tanga et al. 1999; Cellino et al. 1999), while the intrinsic collision probabilities can be derived by the present paper, using the algorithm developed by Dell’Oro and Paolicchi (1998).
We would like to end by noting that we did not include in this analysis some populous families that are probably also very important suppliers of projectiles for inter-asteroid collisions. These families are that of Vesta, which was not included because its members are generally well below the 5-km limit in size, and the Nysa-Polana clan in the inner Belt. In the latter case, the reason for excluding this family was that it is probably given by the chance overlapping of two different families (Doressoundiram et al. 1998) and a more detailed physical analysis of the two groupings is strongly needed. Some work in this respect is currently being performed, and we plan to publish a detailed physical study of Nysa-Polana in a forthcoming paper. Another important omission, at least from the point of view of the numerical results shown in Fig. 5, concerns the Hygiea family. This is a family for which it is very hard to derive a reliable extrapolation of the size distribution. We have good reasons to believe that, at least at the epoch of its formation, it was one of the most important and populous groupings in the asteroid Main Belt. However, we have also reasons to believe that this family is now quite old, judging from a trend of the size distribution that is fairly complicated, and also likely affected by the presence of a number of random interlopers (Tanga et al. 1999). It is reasonable to assume that such an old family can be nowadays seriously eroded at small sizes (Marzari et al. 1995). For this reason, and taking into account the small number of members larger than the nominal completeness limit (and the quoted presence of many likely interlopers among them) any kind of extrapolation is too uncertain to allow us to derive a reasonable estimate of the role currently played by this family. ACKNOWLEDGMENTS We thank D. R. Davis and F. Marzari for their careful reviews, which led to a substantial improvement of the original paper. This work has been supported by MURST (cofin98).
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