Astronomy & Astrophysics manuscript no. Bidstrup (DOI: will be inserted by hand later)
August 10, 2004
The Number Density of Asteroids in the Asteroid Main-belt Philip R. Bidstrup1, 2 , Ren´e Michelsen2 , Anja C. Andersen1 , and Henning Haack3 1
2
3
NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark e-mail:
[email protected];
[email protected] NBIfAFG, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark e-mail:
[email protected] 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” 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 once1 . 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
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.
times than Jupiter due to Kepler’s third law. The resonance is hence 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 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.
2.1. 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 diameters2 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 imax = π8 = 22.5◦. Evidently, the asteroid main-belt 2
Diameters derived from the absolute magnitude and the correla−H
(Fowler & Chillemi 1992 and Bowell et al. 1989), tion D = 10 √5 p∗1329 v where H is the absolute magnitude and pν is the albedo. For use of the correlation and a discussion of the albedo pν = 0.17, see Michelsen et 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. 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 coordinates3, 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, r1 , to its outer part, r2 . 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 = π2 , V= 3
Z
r2 r1
Z
2π
r 0
Z
π 2 +α
r sin θ dθ dφ dr = π 2 −α
4 3 π(r − r13 ) sin α. (2) 3 2
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 ρ(r) =
4 3 3 π(r2
N(r) . − r13 ) sin α
(3)
When the angle, α, equals the suggested maximal inclination, imax , 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
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 pν = 0.17, see Michelsen et al. (2003). Fig. 7 is based on asteroids written in bold. Diameter D> ∼256 km ∼203 km ∼162 km ∼128 km ∼102 km ∼ 91 km ∼ 64 km ∼ 51 km ∼ 41 km ∼ 32 km ∼ 26 km ∼ 20 km ∼ 16 km ∼ 13 km ∼ 10 km ∼9.1 km ∼6.4 km ∼5.1 km ∼4.1 km ∼3.2 km ∼2.6 km ∼2.0 km ∼1.6 km ∼1.3 km ∼1.0 km ∼0.9 km ∼0.6 km
Abs. Mag. H< 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5
Known # 6 13 21 38 74 139 232 350 523 714 952 1,302 1,869 2,862 4,670 7,994 14,104 25,065 42,845 68,544 99,916 133,101 163,420 185,923 198,962 205,024 207435
Predicted # 6 14 22 39 77 141 232 348 512 697 921 1,259 1,813 2,603 4,151 7,143 12,815 23,279 41,910 73,650 125,049 203,989 319,389 481,415 702,495 998,998 1,393,277
X-Fraction NPredicted NPredicted (H
