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The Number Density of Asteroids in the Asteroid Main-Belt

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The Number Density of Asteroids in the Asteroid Main-Belt 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. Bidstrup1;2, Rene´ Michelsen2, Anja C. Andersen1, and Henning Haack3 1 NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark e-mail: [email protected]; [email protected] 2 NBIfAFG, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark e-mail: [email protected] 3 Geological Museum, University of Copenhagen, Øster Voldgade 5-7, DK-1350 Copenhagen, Denmark e-mail: [email protected] 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 212531 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” sug- gested, 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 mo- Fig. 1. The spacial distribution of objects in the inner Solar System, tion resonances, see Moons & Morbidelli (1995). A mean mo- showing the asteroid main-belt. The Sun is at the center of figure and the planets are shown as squares. The two asteroid concentrations tion resonance is a result of the ratio of orbital period of an on each side of Jupiter (the large square) are the Trojans, which are asteroid with Jupiter, e.g. a 2:1 resonance is an orbit where an not part of the asteroid main-belt but asteroids captured in Jupiter’s asteroid revolves twice around the Sun while Jupiter revolves Lagrangian points L4 and L5. only once1. Send offprint requests to: P. R. Bidstrup 1 Since the asteroid main-belt is located closer to the Sun than times than Jupiter due to Kepler’s third law. The resonance is hence Jupiter, main-belt asteroids will always revolve a greater number of sometimes written as a 1:2 resonance without being misleading. 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 correspond- Fig. 3. A snapshot in time of the number distribution of the asteroid ing semi-major axis disclose the Kirkwood Gaps. The histogram bins main-belt with respect to true distance from the Sun. It can be seen are 0.01 AU wide 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. Even though some regions in terms of mean distances (or semi major axes) are depleted, the main-belt contains no empty distribution, N(r), and the spacial volume of the asteroid main- areas and does NOT show the same characteristics as e.g. belt, V(r) is required since, Saturn’s ring system, because none of the asteroids in the main- belt have circular orbits and the eccentric asteroid orbits always N(r) ρ(r) = : (1) supply the depleted regions with temporarily visiting asteroids. V(r) For the known asteroids, N(r) is the distribution used in Fig. 3. 2. Spacial number density To establish V(r) we need to determine the volume distribution The number distribution with respect to mean distance reveals of the asteroid main-belt. the Kirkwood gaps and, thereby, the existence of the orbit reso- nances. In the same manner, the number distribution can reveal 2.1. The volume distribution of the asteroid main-belt further effects of the resonances. Mean distance is, like semi major axis, a geometric para- To establish a model of the volume of the asteroid main-belt, it meter that contains information of the orbit. It is defined as the is instructive to plot the known asteroid distances from the Sun average of all the true distances from the Sun in the orbit. True and their height above and below the ecliptic plane, see Fig. 4. distance is understood as the immediate distance to the Sun at The figure represents a slice of the asteroid main-belt and from some point in time and does not itself contain information of this an idea about the shape of the occupied volume occurs. the orbit. The left figure of Fig. 4 shows the spacial distribution of all If all the known asteroid distances to the Sun (true dis- the main-belt asteroids, indicating that two concentrations of tances, not mean distances) are plotted at some point in time, asteroids are located around 1.8 AU from the Sun at 0.5 AU the number distribution can be determined with respect to these above and below the ecliptic plane, however, the observational distances. As shown in Fig. 3 the number density of asteroids bias strongly affects the plot since the small asteroids can only is larger in the outer parts than in the inner parts of the aster- be seen close to the Earth. If only asteroids with diameters2 oid main-belt. This is a bit surprising since the observations are larger than about 5 km are considered, the two over-dense re- biased with respect to the distances from the Earth, it is easier gions in the inner part of the asteroid belt disappear - see the to detect asteroids in the inner part of the main-belt. Therefore, right figure in Fig. 4. the quite opposite distribution with more asteroids in the inner The shape of the volume of the asteroid main-belt can part of the main-belt would be expected. therefore be approximated with the part of a spherical shell Before anything is concluded upon the distribution of as- that is confined within a maximum inclination above and be- teroids, we have to bear in mind that the asteroid orbits in the low the ecliptic. This maximum angle can be determined from π outer parts are larger than in the inner parts. This means that Fig. 4 to be imax = 8 = 22:5◦. Evidently, the asteroid main-belt the outer main-belt asteroids have more space to inhabit than the inner, and it is therefore necessary to look at the spacial 2 Diameters derived from the absolute magnitude and the correla- H 10− 5 1329 number density of asteroids to determine the concentration of tion D = ∗ (Fowler & Chillemi 1992 and Bowell et al. 1989), ppv asteroids. where H is the absolute magnitude and pν is the albedo. For use of the To derive the spacial number density, ρ, as a function of correlation and a discussion of the albedo pν = 0:17, see Michelsen et distance from the Sun, r, knowledge of the asteroid number al. (2003). 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. 6. The spacial number density of the known main-belt asteroids Fig. 5. Model of the volume occupied by the main-belt asteroids, the when using the model for the volume occupied by the main-belt as- Sun is located in the central void. teroids (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. 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 vol- When inserted into Eq. 1, the spacial number density can be ume, the calculation is straight forward. written as Using spherical coordinates3, each direction can be inte- N(r) ρ(r) = : (3) grated with limits according to the determined geometry. First, 4 π(r3 r3) sin α we perform a radial integration of the asteroid main-belt from 3 2 − 1 its inner part, r1, to its outer part, r2. Second, an integration When the angle, α, equals the suggested maximal inclina- of the full circle in φ-direction is necessary because of cylin- tion, imax, the expression describes the spacial number density dric symmetry.
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