Understanding the Distribution of Near-Earth Asteroids

NEAs have perihelion distances q ≤ 1.3 AU and aphelion distances Q ≥ 0.983 AU. NEA subgroups include the Amors (1.0167 AU < q ≤ 1.3 AU); Apollos (a ≥ 1.0 AU, q ≤ 1.0167 AU), and Atens (a < 1.0 AU, Q ≥ 0.983 AU). D. Rabinowitz, E. Bowell, E. M. Shoemaker, K. Muinonen, in Hazards due to Comets and Asteroids, T. Gehrels, Ed. (Univ. of Arizona Press, Tucson, AZ, 1994), pp. 285–312.

L. A. McFadden, D. J. Tholen, G. J. Veeder, in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, Eds. (Univ. of Arizona Press, Tucson, AZ, 1989), pp. 442–467.

See, for instance, the chapters by G. Neukum and B. A. Ivanov; O. B. Toon, K. Zahnle, R. P. Turco, and C. Covey; and M. R. Rampino and B. M. Haggerty, in Hazards due to Comets and Asteroids, T. Gehrels, Ed. (Univ. of Arizona Press, Tucson, AZ, 1994), pp. 359–416, pp. 791–826, and pp. 827–858, respectively.

Chapman C. R., Morrison D., Nature 367, 33 (1994).

We used the April 2000 update of the public-domain asteroid orbit database “astorb.dat” found at also E. Bowell, K. Muinonen, and L. H. Wasserman, in Asteroids, Comets, Meteors 1993, A. Milani, M. DiMartino, A. Cellino, Eds. (Kluwer, Dordrecht, Netherlands, 1994), pp. 477–481.

Transforming H into a characteristic NEA diameter is problematic. We use a bolometric geometric albedo that favors S-type asteroids (p ∼ 0.155). See D. J. Tholen and M. A. Barucci [in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, Eds. (Univ. of Arizona Press, Tucson, AZ, 1989), pp. 298–315] for details. We convert between absolute magnitude H and diameter D using D = 4365 × 10^–H/5 [E. Bowell et al., in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, Eds. (Univ. of Arizona Press, Tucson, AZ, 1989), pp. 524–556].

E. M. Shoemaker, R. F. Wolfe, C. S. Shoemaker, in Global Catastrophes in Earth History, V. L. Sharpton and P. D. Ward, Eds. (Geological Society of America Special Paper 247); A. W. Harris, in Collisional Processes in the Solar System, M. Marov and H. Rickman, Eds. (Kluwer, in press).

Rabinowitz D. L., Helin E., Lawrence K., Pravdo S., Nature 403, 165 (2000).

Rabinowitz D. L., Icarus 111, 364 (1994).

Jedicke R., Astron. J. 111, 970 (1996);

Jedicke R., Metcalfe T. S., Icarus 131, 245 (1998);

Durda D. D., Greenberg R., Jedicke R., Icarus 135, 431 (1998).

tk;4Studies suggest that the cratering flux on the Earth-moon system has been more or less constant for the past ∼3 billion years. If true, the NEA population has been in a steady state over that same time [R. A. F. Grieve and E. M. Shoemaker, in Hazards due to Comets and Asteroids, T. Gehrels, Ed. (Univ. of Arizona Press, Tucson, AZ, 1994), pp. 417–462].

P. Farinella, R. Gonczi, Ch. Froeschlé, Cl. Froeschlé, Icarus 101, 174 (1993).

Farinella P., Vokrouhlický D., Hartmann W. K., Icarus 132, 378 (1998);

Farinella P., Vokrouhlický D., Science 283, 1507 (1999);

Bottke W. F., Rubincam D. P., Burns J. A., Icarus 145, 301 (2000) .

Williams J. G., Eos 54, 233 (1973);

Wisdom J., Icarus 56, 51 (1983) .

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Gladman B. J., et al., Science 277, 197 (1997).

Migliorini F., Michel P., Morbidelli A., Nesvorný D., Zappalà V., Science 281, 2022 (1998);

Morbidelli A., Nesvorný D., Icarus 139, 295 (1999);

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Murray N., Holman M., Potter M., Astron. J. 116, 2583 (1998);

; P. Michel, F. Milgiorini, A. Morbidelli, V. Zappala, Icarus, in press.

The ability of a resonance to produce NEAs is determined by its location, its efficiency at pushing objects onto high-e orbits, and the flux of asteroids entering the resonance over time. Powerful resonances beyond 2.8 AU (for example, the 5:2 resonance with Jupiter) frequently push objects directly onto Jupiter-crossing orbits, where they are ejected from the solar system in ≲ 10⁵ years. Few NEAs come from these transportation routes (16). Prominent resonances in the inner solar system, like the 3:1 mean-motion resonance with Jupiter and the ν₆ secular resonance, generally do not directly produce Jupiter-crossing objects. For this reason, these resonances have long been considered plausible wellsprings for meteorites and NEAs [

Wetherill G. W., Meteoritics 20, 1 (1985);

; Philos. Trans. R. Soc. London Ser. A323, 323 (1987); Icarus 76, 1 (1988)].

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A H = 22 NEA, with a diameter of about 170 m (6), has a collisional lifetime >100 My (13, 16). The mean lifetime of rubble-pile NEAs against tidal disruption with Earth or Venus is ∼65 My [

Richardson D. C., Bottke W. F., Lovek S. G., Icarus 134, 47 (1999);

Bottke W. F., Richardson D. C., Michel P., Love S. G., Astron. J. 117, 1921 (1999);

]. Both time scales are longer than the NEA dynamical lifetimes reported here. We expect tidal disruption, however, to increase in importance as NEAs attain low e, i orbits.

Morbidelli A., Gladman B., Meteorit. Planet. Sci. 33, 999 (1998);

Morbidelli A., Icarus 105, 48 (1993).

Our numerical runs for the ν₆ resonance use test bodies started near each of the following positions: (a ∼ 2.06 AU, i = 2.5°), (a ∼ 2.08 AU, i = 5°), (a ∼ 2.115 AU, i = 7.5°), (a ∼ 2.16 AU, i = 10°), (a ∼2.24 AU, i = 12.5°), and (a ∼ 2.315 AU, i = 15°). For all cases, e = 0.1. Bodies clearly not on a fast track to NEA orbits were removed when calculating R_ν6. See (21) for details.

Not all IMC objects are produced via weak main belt resonances. Some bodies residing near (but not “in”) the strong part of the ν₆ resonance have libration amplitudes large enough to reach Mars-crossing orbits [G. W. Wetherill and J. G. Williams, in Origin and Distribution of the Elements, L. H. Ahrens, Ed. (Pergamon, Oxford, 1979), pp. 19–31]. In addition, some current IMCs have been extracted from the 3:1 or ν₆ resonances via close encounters with Mars.

To interpret how the known MC asteroids are biased with respect to a and i, we divided the IMC and near-IMC regions (q < 1.8 AU) into three semimajor axis zones (zone a1: 2.1 AU ≤ a < 2.3 AU; zone a2: 2.3 AU ≤ a < 2.5 AU; zone a3: 2.5 AU ≤ a < 2.8 AU) and three inclination zones (zone i1: i < 5° AU; zone i2: 5° ≤ i < 10°; zone i3: 10° ≤ i < 15°) and estimated the observational biases for asteroids in those zones (10). The ratios of the biases in zones a2 and a3 over zone a1 were ∼1.3 and ∼1.8, respectively, whereas the ratios of the biases in zones i2 and i3 over zone i1 were ∼2.7 and ∼4.4, respectively. These values were used to weight the orbital paths of underrepresented IMCs in each zone when R_IMC(a, e, i) was calculated.

Bottke W. F., Nolan M. C., Melosh H. J., Vickery A. M., Greenberg R, Icarus 122, 406 (1996).

We exclude Spacewatch NEAs with H > 22 because they are discovered by means of a different search strategy than are H < 22 objects (9).

D. Morrison, Ed., The Spaceguard Survey: Report of the NASA International Near-Earth-Object Detection Workshop (NASA, Washington, DC, 1992), pp. 1–19.

D. Rabinowitz, E. Bowell, E. M. Shoemaker, K. Muinonen, in Hazards due to Comets and Asteroids, T. Gehrels, Ed. (Univ. of Arizona Press, Tucson, AZ, 1994), pp. 285–312.

Objects with high B values correspond to bright/large objects and/or those that move slowly through Spacewatch's search volume (such as multi-km main belt asteroids, IMCs, and NEAs on low-i orbits with a between 2 and 3 AU). Conversely, low B values correspond to dim/small objects and/or those with such fast angular speeds that they spend little time in Spacewatch's search volume (such as sub-km NEAs that rarely approach Earth and high-i asteroids).

B_NEA can, in principle, be used to estimate the entire NEA population from the known NEAs without taking additional steps by dividing the observed population by the bias factor directly. This type of procedure has already been used to estimate the debiased main belt size distribution down to a few km in diameter (10). The problem for the NEA orbital distribution, however, is resolution; the limited number of Spacewatch NEAs do not provide enough coverage to normalize a wide-ranging probability distribution without leaving large tracts of (a, e, i, H) space without a single NEA (our B_NEA uses ∼30,000 bins). Until the NEA inventory gains more entries, B_NEA cannot be directly used to produce statistically meaningful NEA population estimates.

Spacewatch has discovered and accidentally rediscovered 167 NEAs with 0.5 AU < a < 2.8 AU, e < 0.8, i < 35°, and 13 < H < 22. These asteroids have been winnowed from a larger set with a Spacewatch NEA probability calculation (10). 138 of these asteroids were detected within 50° of opposition. We determined that their (a, e, i) values are statistically similar to objects found within 20° of opposition, the region where B_NEA is most applicable.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge Univ. Press, Cambridge, 1992); L. Lyons, Statistics for Nuclear and Particle Physicists (Cambridge University Press, Cambridge, 1986).

We use a Monte Carlo technique to determine the statistical errors associated with our best fit. First, we generate many sets of fake asteroids according to our best-fit n(a, e, i, H) distribution. The number of fake asteroids in each set is 138 (31). Next, we calculate new α_IS and γ values for each set using our maximum likelihood fit technique. Finally, we calculate a distribution for α_IS and γ and determine confidence limits. Assuming our best estimate for a parameter is p, we calculate the limiting value of p+ and p− so that 34% of the values lie within the range (p−, p) and 34% lie within the range (p, p+). In this way, 68% lie within the range (p−, p+), yielding 1σ errors.

As of April 2000, the known NEA population of H < 15 NEAs is 53 (5). Observations by the NEAT program suggest that H < 15 NEAs are ∼80% complete (8). Using these values, we get ∼66 NEAs with H < 15. Comparable numbers can be obtained using a different method; by dividing the number of already-discovered NEAs by the ratio of the number of new discoveries to total detections (new plus redetections) in the past year, there may be as many as 73 ± 7 NEAs with H < 15 [

Harris A. W., Div. Dynam. Astron. 31, 16 (2000);

]. Splitting the difference, we assume that there are 70 H < 15 NEAs. Because there are 4 H < 13 NEAs, the number between 13 < H < 15 is ∼66. The uncertainty in this value contributes to our method's systematic error.

P. R. Weissman, M. F. A'Hearn, L. A. McFadden, H. Rickman, in Asteroids II, R. P. Binzel, T. Gehrels, M. S. Matthews, Eds. (Univ. of Arizona Press, Tucson, AZ, 1989), pp. 880–920;

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Asteroid families produced by catastrophic collisions have cumulative size distribution slopes in excess of 2.5 [

Tanga P., Icarus 141, 65 (1999)].

The cumulative size distribution of craters larger than 20 to 30 km on Earth has a power law slope of 1.8 (11). Craters on the lunar maria between 3 and 100 km in diameter have a power law slope of 1.7, though some of the larger craters have been enlarged by collapse. Correcting for crater collapse increases the slope to 1.84 [

Shoemaker E. M., Annu. Rev. Earth Planet. 11, 461 (1983);

]. Craters on the young Martian plains with diameters between 10 and 50 km have a power law slope of 2.0 [R. G. Strom, S. K. Croft, N. G. Barlow, in Mars, H. H. Kieffer, B. M. Jakosky, C. W. Snyder, M. S. Matthews, Eds. (Univ. of Arizona Press, Tucson, AZ, 1992), pp. 383–423)]. Craters on Venus with diameters larger than 35 km have a power law slope of ∼2.0 [

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We thank J. Burns, L. Dones, A. Harris, T. Metcalfe, P. Michel, J. Plassman, T. Spahr, and G. Wetherill for valuable discussions and input to this study and two anonymous referees for their careful and constructive reviews. We gratefully acknowledge the computational resources provided to us for this project by Cornell University [both in the Department of Astronomy and at the Cornell Theory Center (CTC)], the University of Texas, the University of Arizona, the Osservatorio Astronomico di Torino, and the Observatoire de la Côte d'Azur. Finally, we thank C. Pelkie and the CTC for their help in making visualizations for this paper. Research funds were provided by grants from NASA's Planetary Geology and Geophysics program (NAGW-310), NASA's Near-Earth Object Observations program (NAG5-9082), and European Space Agency contract 14018/2000/F/TB. Travel support was provided by grants from NATO and NSF/CNRS. Spacewatch is funded by NASA, AFOSR, the Packard Foundation, and the Kirsch Foundation, as well as other organizations and individuals.