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” 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 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 resonances. 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), √pv 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. Finally, the integration over the wanted angular of the known asteroids. The result of the spacial number den- patch in θ-direction, here denoted by an arbitrary angular se- sity can be seen in Fig. 6. π It is clear that the tendency of a larger number of asteroids lection, α, around the ecliptic situated at θ = 2 , in the outer regions of the asteroid main-belt, not only holds π r2 2π 2 +α 4 3 3 true for ﬁrst hand counting, but also in deriving spacial number V = r r sin θ dθ dφ dr = π(r2 r1) sin α. (2) Z Z Z π densities. This is in agreement with the work by Lagerkvist r1 0 α 3 − 2 − & Lagerros (1997) although their approach was a bit diﬀerent 3 In this case the right-handed spherical coordinate system with φ since they compiled the densities of asteroids sorted in mean being the equatorial angle and θ the polar angle. 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 Abs. Mag. Known Predicted X-Fraction D> H< # # NPredicted NPredicted (H<13) 4 256 km 5.5 6 6 8.4 10− ∼ · 3 203 km 6.0 13 14 2.0 10− ∼ · 3 162 km 6.5 21 22 3.1 10− ∼ · 3 128 km 7.0 38 39 5.5 10− ∼ · 2 102 km 7.5 74 77 1.1 10− ∼ · 2 91 km 8.0 139 141 1.8 10− Fig. 7. Spacial number density of an asteroid distribution assumed to ∼ · 2 64 km 8.5 232 232 3.2 10− be complete by Jedicke et al. (2002). The figure contains all 7994 ∼ · 2 51 km 9.0 350 348 4.9 10− main-belt asteroids of diameters larger than 9 km or H<13 and pro- ∼ · 2 ∼ 41 km 9.5 523 512 7.2 10− vides a density profile without observational bias. It is seen that the ∼ · 2 32 km 10.0 714 697 9.8 10− enhanced number density in the outer part of the asteroid main-belt is ∼ 26 km 10.5 952 921 0.13· ∼ still present in the case without observational bias, and will therefore 20 km 11.0 1,302 1,259 0.18 ∼ still be present when including the “to be discovered” small asteroids. 16 km 11.5 1,869 1,813 0.25 The histogram bins are 0.05 AU wide. ∼ 13 km 12.0 2,862 2,603 0.36 ∼ 10 km 12.5 4,670 4,151 0.58 ∼ 9.1 km 13.0 7,994 7,143 1.00 oids with H < 13. Creating the spacial number densities from ∼6.4 km 13.5 14,104 12,815 1.79 ∼ the asteroid groups thought to be complete, a density profile 5.1 km 14.0 25,065 23,279 3.26 ∼ which should be without observational bias is obtained. This 4.1 km 14.5 42,845 41,910 5.87 result is shown in Fig. 7. With Fig. 7 and the fraction of pre- ∼3.2 km 15.0 68,544 73,650 10.3 ∼ dicted number of asteroids against the predicted number of as- 2.6 km 15.5 99,916 125,049 17.5 ∼2.0 km 16.0 133,101 203,989 28.6 teroids with H<13 (X-fraction in Table 1) we can extend our es- ∼1.6 km 16.5 163,420 319,389 44.7 timated number density profile for the asteroid main-belt down ∼ 1.3 km 17.0 185,923 481,415 67.4 to asteroid sizes of around 1 km. ∼1.0 km 17.5 198,962 702,495 98.3 If all asteroids, regardless of size, are considered to be dis- ∼0.9 km 18.0 205,024 998,998 139.9 tributed with the same density profile as the complete set of ∼ 0.6 km 18.5 207435 1,393,277 195.1 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 dis- 3. Extending the result to smaller asteroids tinguished from large ones and size-dependent models like the To test if the above result should also be expected to hold true Yarkovsky- and Poynting-Robertson effects are therefore not when a more complete sample of smaller asteroids have been accounted for, see Bottke et al. (2002). sampled all over the asteroid main-belt, we have looked into a way to extend the population of know asteroids down to smaller 4. Conclusion sizes. Jedicke et al. (2002) have predicted what the population The number distribution of main-belt asteroids has, with the of asteroids looks like under the assumption that the current discussion of the shape and volume of the main-belt given, in- known population of main-belt asteroids is complete for abso- formation of the spacial number density. It has been shown that lute magnitude H < 13, equivalent to diameters > 10 km. the trend of higher asteroid concentrations in the outer regions Table 1 shows the number∼ of unknown asteroids predicted∼ by of the asteroid main-belt not only holds true for the number Jedicke et al. (2002) down to a diameter of 1 km. Following distribution, but also for the spacial number densities. This is the thought of completeness, all asteroids of diameters greater despite the expectation of observational bias that would have than 9 km have been discovered and their distribution will clouded this feature and shown the direct opposite case with hence∼be without influence of the observational bias. However, fewer observations of the outer asteroids. It is an indication that as seen in Table 1, some of the predicted numbers are less than interactions other than collisions and resonances are present the observed numbers indicating that the population is most and that they could have greater effect on small asteroids. One likely not complete but could be underestimated. such effect e.g. could be the Yarkovsky effect, which is valid Assuming that the predictions of Jedicke et al. (2002) are for asteroids of diameters 0.1 m - 20 km, see Bottke et al. close to correct, we adopt the belief of completeness for aster- (2002). ∼ ∼ 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 48.350-OFR). RM acknowledges support from the Danish Natural Science Research Council through a grant from the Center for Ground-Based Observational Astronomy (IJAF).

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