The Number Density of Asteroids in the Asteroid Main-belt

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1 Astronomy & Astrophysics manuscript no. Bidstrup August 10, 2004 (DOI: will be inserted by hand later) The Number Density of Asteroids in the Asteroid Main-belt Philip R. Bidstrup 1,2, René Michelsen 2, Anja C. Andersen 1, and Henning Haack 3 1 NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark philip@nordita.dk; anja@nordita.dk 2 NBIfAFG, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark rene@astro.ku.dk 3 Geological Museum, University of Copenhagen, Øster Voldgade 5-7, DK-1350 Copenhagen, Denmark hh@savik.geomus.ku.dk Received ; accepted Abstract. We present a study of the spacial number density and the shape of the occupied volume of asteroids in the asteroid main-belt, based on the asteroids currently to be found in the database of the Minor Planet Center (Juli, 2004). To obtain the number density we divide the distribution of the main-belt asteroids based on their true distances from the Sun by the occupied volume. We find a clear trend of larger densities at greater distances from the Sun. Key words. Asteroids minor planets Methods: data analysis 1. Introduction With Giuseppe Piazzi s discovery of Ceres in 1801, what seemed to be an empty gap between Mars and Jupiter proved to be incorrect. Ceres was not, as the Titius-Bode law suggested, a missing planet in the Solar System but a member of the asteroid belt. Additional asteroids have since then been cataloged and today, the Minor Planet Center (Juli, 2004) counts around 200,000 objects in its database. Most of these objects form the asteroid main-belt which is widely spread ranging from 1.7 AU to 3.7 AU from the Sun. Not only do the asteroids have eccentric orbits but also orbits outside the ecliptic plane with typical inclinations of 0-30 degrees above and below the plane. A 2-D projection of the spacial distribution of objects in the inner Solar System can be seen in Fig. 1. Daniel Kirkwood was the first to discover that the number distribution with respect to the mean distance of asteroids in the main-belt disclosed gaps known as The Kirkwood Gaps (Kirkwood 1867), see Fig. 2. Jupiter s gravitational field is strong enough to evict asteroids from the asteroid main-belt, and in some cases, even from the Solar System by mean motion resonances, see Moons & Morbidelli (1995). A mean motion resonance is a result of the ratio of orbital period of an asteroid with Jupiter, e.g. a 2:1 resonance is an orbit where an asteroid revolves twice around the Sun while Jupiter revolves only once 1. Fig. 1. The spacial distribution of objects in the inner Solar System, showing the asteroid main-belt. The Sun is at the center of figure and the planets are shown as squares. The two asteroid concentrations on each side of Jupiter (the large square) are the Trojans, which are not part of the asteroid main-belt but asteroids captured in Jupiter s Lagrangian points L4 and L5. Send offprint requests to: P. R. Bidstrup 1 Since the asteroid main-belt is located closer to the Sun than Jupiter, main-belt asteroids will always revolve a greater number of times than Jupiter due to Kepler s third law. The resonance is hence sometimes written as a 1:2 resonance without being misleading.

2 2 Philip R. Bidstrup et al.: The Number Density of Asteroids in the Asteroid Main-belt Fig. 2. The number of known asteroids plotted with their corresponding semi-major axis disclose the Kirkwood Gaps. The histogram bins are 0.01 AU wide Even though some regions in terms of mean distances (or semi major axes) are depleted, the main-belt contains no empty areas and does NOT show the same characteristics as e.g. Saturn s ring system, because none of the asteroids in the mainbelt have circular orbits and the eccentric asteroid orbits always supply the depleted regions with temporarily visiting asteroids. 2. Spacial number density The number distribution with respect to mean distance reveals the Kirkwood gaps and, thereby, the existence of the orbit resonances. In the same manner, the number distribution can reveal further effects of the resonances. Mean distance is, like semi major axis, a geometric parameter that contains information of the orbit. It is defined as the average of all the true distances from the Sun in the orbit. True distance is understood as the immediate distance to the Sun at some point in time and does not itself contain information of the orbit. If all the known asteroid distances to the Sun (true distances, not mean distances) are plotted at some point in time, the number distribution can be determined with respect to these distances. As shown in Fig. 3 the number density of asteroids is larger in the outer parts than in the inner parts of the asteroid main-belt. This is a bit surprising since the observations are biased with respect to the distances from the Earth, it is easier to detect asteroids in the inner part of the main-belt. Therefore, the quite opposite distribution with more asteroids in the inner part of the main-belt would be expected. Before anything is concluded upon the distribution of asteroids, we have to bear in mind that the asteroid orbits in the outer parts are larger than in the inner parts. This means that the outer main-belt asteroids have more space to inhabit than the inner, and it is therefore necessary to look at the spacial number density of asteroids to determine the concentration of asteroids. To derive the spacial number density, ρ, as a function of distance from the Sun, r, knowledge of the asteroid number Fig. 3. A snapshot in time of the number distribution of the asteroid main-belt with respect to true distance from the Sun. It can be seen that most of the asteroids in the main-belt are located around 2.8 AU from the Sun. The histogram bins are 0.01 AU wide. distribution, N(r), and the spacial volume of the asteroid mainbelt, V(r) is required since, ρ(r) = N(r) V(r). (1) For the known asteroids, N(r) is the distribution used in Fig. 3. To establish V(r) we need to determine the volume distribution of the asteroid main-belt The volume distribution of the asteroid main-belt To establish a model of the volume of the asteroid main-belt, it is instructive to plot the known asteroid distances from the Sun and their height above and below the ecliptic plane, see Fig. 4. The figure represents a slice of the asteroid main-belt and from this an idea about the shape of the occupied volume occurs. The left figure of Fig. 4 shows the spacial distribution of all the main-belt asteroids, indicating that two concentrations of asteroids are located around 1.8 AU from the Sun at 0.5 AU above and below the ecliptic plane, however, the observational bias strongly affects the plot since the small asteroids can only be seen close to the Earth. If only asteroids with diameters 2 larger than about 5 km are considered, the two over-dense regions in the inner part of the asteroid belt disappear - see the right figure in Fig. 4. The shape of the volume of the asteroid main-belt can therefore be approximated with the part of a spherical shell that is confined within a maximum inclination above and below the ecliptic. This maximum angle can be determined from Fig. 4 to be i max = π 8 = Evidently, the asteroid main-belt 2 Diameters derived from the absolute magnitude and the correlation D = 10 H pv (Fowler & Chillemi 1992 and Bowell et al. 1989), where H is the absolute magnitude and p ν is the albedo. For use of the correlation and a discussion of the albedo p ν = 0.17, see Michelsen et al. (2003).

3 Philip R. Bidstrup et al.: The Number Density of Asteroids in the Asteroid Main-belt 3 Fig. 4. A 2-D projection of the spacial distribution of all the known main-belt asteroids is shown to the left, while the figure to the right shows only the known asteroids with diameters above 5 km. The Sun is located at Origo. The axes shows the distance from the Sun and the height of the asteroidal orbits above and below the ecliptic. Fig. 5. Model of the volume occupied by the main-belt asteroids, the Sun is located in the central void. does not form a torus as one might have expected, but rather the geometric shape shown in Fig. 5. With a determination of a model for the shape of the volume, the calculation is straight forward. Using spherical coordinates 3, each direction can be integrated with limits according to the determined geometry. First, we perform a radial integration of the asteroid main-belt from its inner part, r 1, to its outer part, r 2. Second, an integration of the full circle in φ-direction is necessary because of cylindric symmetry. Finally, the integration over the wanted angular patch in θ-direction, here denoted by an arbitrary angular selection, α, around the ecliptic situated at θ = π 2, V = r2 2π r 1 0 π 2 +α r r sin θ dθ dφ dr = 4 π 2 α 3 π(r3 2 r3 1 ) sin α. (2) 3 In this case the right-handed spherical coordinate system with φ being the equatorial angle and θ the polar angle. Fig. 6. The spacial number density of the known main-belt asteroids when using the model for the volume occupied by the main-belt asteroids (see text for a discussion). The axes display the distance from the Sun in astronomical units vs. the number of asteroids per cubic astronomical unit. When inserted into Eq. 1, the spacial number density can be written as N(r) ρ(r) = 4 3 π(r3 2 (3) r3 1 ) sin α. When the angle, α, equals the suggested maximal inclination, i max, the expression describes the spacial number density of the known asteroids. The result of the spacial number density can be seen in Fig. 6. It is clear that the tendency of a larger number of asteroids in the outer regions of the asteroid main-belt, not only holds true for first hand counting, but also in deriving spacial number densities. This is in agreement with the work by Lagerkvist & Lagerros (1997) although their approach was a bit different since they compiled the densities of asteroids sorted in mean distances and had 40 times less asteroids in their study.

4 4 Philip R. Bidstrup et al.: The Number Density of Asteroids in the Asteroid Main-belt Table 1. Fraction of unknown asteroids to known asteroids of specific sizes. The number of unknown asteroids are extracted from the predictions of Jedicke et al. (2002). The known asteroids are data from The Minor Planet Center (Juli, 2004). The determination of diameters from absolute magnitudes is not very well defined and different authors use different correlations. Here, the same correlation is used as Fowler & Chillemi (1992) and Bowell et al. (1989) with standard albedo p ν = 0.17, see Michelsen et al. (2003). Fig. 7 is based on asteroids written in bold. Diameter Abs. Mag. Known Predicted X-Fraction D> H< # # 256 km km km km km km km km km km km km ,302 1, km ,869 1, km ,862 2, km ,670 4, km ,994 7, km ,104 12, km ,065 23, km ,845 41, km ,544 73, km , , km , , km , , km , , km , , km , , km ,393, Extending the result to smaller asteroids N Predicted N Predicted (H<13) To test if the above result should also be expected to hold true when a more complete sample of smaller asteroids have been sampled all over the asteroid main-belt, we have looked into a way to extend the population of know asteroids down to smaller sizes. Jedicke et al. (2002) have predicted what the population of asteroids looks like under the assumption that the current known population of main-belt asteroids is complete for absolute magnitude H < 13, equivalent to diameters > 10 km. Table 1 shows the number of unknown asteroids predicted by Jedicke et al. (2002) down to a diameter of 1 km. Following the thought of completeness, all asteroids of diameters greater than 9 km have been discovered and their distribution will hence be without influence of the observational bias. However, as seen in Table 1, some of the predicted numbers are less than the observed numbers indicating that the population is most likely not complete but could be underestimated. Assuming that the predictions of Jedicke et al. (2002) are close to correct, we adopt the belief of completeness for aster- Fig. 7. Spacial number density of an asteroid distribution assumed to be complete by Jedicke et al. (2002). The figure contains all 7994 main-belt asteroids of diameters larger than 9 km or H<13 and provides a density profile without observational bias. It is seen that the enhanced number density in the outer part of the asteroid main-belt is still present in the case without observational bias, and will therefore still be present when including the to be discovered small asteroids. The histogram bins are 0.05 AU wide. oids with H < 13. Creating the spacial number densities from the asteroid groups thought to be complete, a density profile which should be without observational bias is obtained. This result is shown in Fig. 7. With Fig. 7 and the fraction of predicted number of asteroids against the predicted number of asteroids with H<13 (X-fraction in Table 1) we can extend our estimated number density profile for the asteroid main-belt down to asteroid sizes of around 1 km. If all asteroids, regardless of size, are considered to be distributed with the same density profile as the complete set of asteroids, the extrapolation is straight-forward. In this manner the shape of the density profile is maintained and only values of the density changes. However, small asteroids are not distinguished from large ones and size-dependent models like the Yarkovsky- and Poynting-Robertson effects are therefore not accounted for, see Bottke et al. (2002). 4. Conclusion The number distribution of main-belt asteroids has, with the discussion of the shape and volume of the main-belt given, information of the spacial number density. It has been shown that the trend of higher asteroid concentrations in the outer regions of the asteroid main-belt not only holds true for the number distribution, but also for the spacial number densities. This is despite the expectation of observational bias that would have clouded this feature and shown the direct opposite case with fewer observations of the outer asteroids. It is an indication that interactions other than collisions and resonances are present and that they could have greater effect on small asteroids. One such effect e.g. could be the Yarkovsky effect, which is valid for asteroids of diameters 0.1 m - 20 km, see Bottke et al. (2002).

5 Philip R. Bidstrup et al.: The Number Density of Asteroids in the Asteroid Main-belt 5 In the attempt to extrapolate the densities to include asteroids not yet discovered, a comparison of the present asteroid database and models of asteroid size-distributions revealed problems with declaring groups of the size-distribution complete. With the higher and higher rate of asteroid observations, still more asteroids of sizes thought to be complete are discovered. The biggest part of the problems of extrapolating the densities is probably not the matter of completeness since the assumption of similar number distribution of asteroids regardless of size is highly simplified. Nevertheless, a first-hand offer of the asteroid density can be found in the following manner: To obtain the spacial number density of e.g. asteroids with diameters larger than 1 km located at 2.8 AU from the Sun, use values from Fig. 7 and multiply with the extrapolation fraction (X-fraction from Table 1) from predicted number of asteroids derived via the Jedicke et al.-prediction in Table 1. Acknowledgements. We would like to thank Claes-Ingvar Lagerkvist for enlightening discussions. This research was in part supported by a grant from the public research council for space research (P.I. J.L. Jørgensen, grant OFR). RM acknowledges support from the Danish Natural Science Research Council through a grant from the Center for Ground-Based Observational Astronomy (IJAF). References Bottke Jr.W.F., Vokrouhlicky D., Rubincam D.P., Broz M., 2002, The Effect of Yarkovsky Thermal Forces on the Dynamical Evolution of Asteroids and Meteoroids, in Asteroids III, eds. Bottke Jr.W.F., Cellino A., Paolicchi P., Binzel R.P. (University of Arizona Press) 395 Bowell E., Hapke B., Domingue D., Lumme K., Peltoniemi J., Harris A.W., 1989, in Asteroids II, eds. Binzel R.P., Gehrels T., Matthews M.S. (University of Arizona Press) 524 Fowler J.W., Chillemi J.R., 1992, in The IRAS Minor Planet Survey, Tech. Rep. PL-TR , (Phillips Laboratory, Hanscom Air Force Base, MA) 17 Jedicke R., Larsen J., Spahr T., 2002, Observational Selection Effects in Asteroid Surveys and Estimates of Asteroid Population Sizes, in Asteroids III, eds. Bottke Jr.W.F., Cellino A., Paolicchi P., Binzel R.P. (University of Arizona Press) 71 Kirkwood D., 1867, Meteoric Astronomy (Lippincott, Philadelphia) Lagerkvist C.-I., Lagerros J.S.V, 1997, Astron. Nachr. 318, 391 Michelsen R., Haack H., Andersen A.C., Jørgensen J.L., 2003, in IEEE proceedings 03EX743, 247, (astro-ph/ ) Minor Planet Center, Moons M., Morbidelli A., 1995, Icarus 114, 33