AComparative Study of Size Distributions for Small L4 and L5 Jovian Trojans

Transcription

1 PASJ: Publ. Astron. Soc. Japan 60, , 2008 April 25 c Astronomical Society of Japan. AComparative Study of Size Distributions for Small L4 and L5 Jovian Trojans Fumi YOSHIDA National Astronomical Observatory of Japan, Osawa, Mitaka, Tokyo fumi.yoshida@nao.ac.jp and Tsuko NAKAMURA University of the Air, 2-11 Wakaba, Mihama-ku, Chiba tsukonk@yahoo.co.jp (Received 2007 May 18; accepted 2008 January 16) Abstract We examined the size distributions of small L4 and L5 Jovian Trojan asteroids detected in the Subaru main belt asteroid survey (SMBAS). This was the first attempt to make a comparison of the size distribution between the L4 and L5 Trojans for sizes ranging from 1 km to 10 km, using the results of systematic surveys. We detected 51 L4 Trojans and 62 L5 Trojans ranging from 0.7 km to 13 km in diameter (D) (assuming an albedo of 0.04). We found that there is a difference of the cumulative size distribution between the L4 and L5 populations. The slopes of the cumulative sizedistribution of the L4 population with 2 km <D<5kmand5km<D<10 km were and , respectively; meanwhile, that of the L5 population with 2 km <D<5km was 2:1 0:3. For both populations, we also give composite size distributions covering a range of 5 km <D< 100 km, whose mean slopes are found to be for L4 and for L5. Key words: minor planets, asteroids solar system: general surveys 1. Introduction Jovian Trojans are asteroids located at a mean distance of 5.2 AU from the Sun. They are clustered in two places at 60 ı ahead of and behind Jupiter in its orbit, i.e., around L4 and L5 Lagrangian points. It has been believed that they originated from planetesimals that formed near Jupiter, and were captured into their current orbits, while proto-jupiter was still growing (Marzari & Scholl 1998; Fleming & Hamilton 2000). A recent idea proposed by Morbidelli et al. (2005) is that the present Jovian Trojan population was reconstructed after a primordial population destroyed by the migration of outer planets at an early stage of the solar system, which may be related to the Late Heavy Bombardment (Strom et al. 2005). Thus, Jovian Trojans draw our attention from the point of view that they may still have preserved some effects during planetary formation around the Jupiter region at an early stage of the solar system. Efforts to find dynamical families among Jovian Trojans have been made by Shoemaker, Shoemaker, and Wolfe (1989), Milani (1993), and recently by Beaugé androig(2001) with the latest Trojan catalog in the Asteroids Database of Lowell Observatory. According to them, there are a few dynamical family candidates in the Jovian Trojan population. 1 Since it is a common understanding that the existence of asteroid families suggests impact processes in their origin, the current Jovian Trojans likely reflect a result of collisional evolution. If so, basic information about collisional evolution in Trojan swarms could be obtained by analysis of the current size distribution of Based on data collected of the Subaru Telescope, which is operated by the National Astronomical Observatory of Japan. 1 hhttp:// Trojan asteroids. So far, the size distributions of small L4 Jovian Trojans with D<40km have been examined by Jewitt, Trujillo, and Luu (2000), Yoshida and Nakamura (2005, hereafter abbreviated as YN2005), and Szabó etal.(2007), through their systematic survey observations. They found the slopes (b) ofthecumu- lative size distributions [indices of the power-law distribution: N.>D/=C D b,wheren.>d/ is a cumulative number of asteroids having a diameter larger than D km and C is constant] for those objects, to be as: b = for 4 km <D<40 km (Jewitt et al. 2000), b = for 5 km <D<10 km, and b = for 2 km <D<5km (YN2005). However, no one has ever investigated a size distribution of small L5 Jovian Trojans. Fortunately, we happened to have detected several tens of L5 Trojans during the Subaru main belt asteroid survey (SMBAS). In this paper, we report on the size distribution of small L5 Trojans and then compare it with that of small L4 Trojans obtained by YN Observations and Data Reduction 2.1. Observations We used the 8.2 m Subaru Telescope located at the top of Mauna Kea with the Suprime-Cam, which is a mosaic CCD camera having a wide field of view ( )(Miyazaki et al. 2002). The first SMBAS (SMBAS-I) and the second one (SMBAS-II) were carried out on 2001 February 22 and 25 (UT) and on 2001 October 21 (UT), respectively. The main outcomes were reported in Yoshida et al. (2003) and Yoshida and Nakamura (2007). This paper focuses on the size distributions of Jovian Trojans obtained from the two SMBASs.

2 298 F. Yoshida and T. Nakamura [Vol. 60, SMBAS-I and SMBAS-II surveyed a sky areas of 3 deg 2 at 32 ı in longitude ahead of L4 point and 4 deg 2 at 22 ı behind the L5 point, respectively. We imaged the same sky field at least three times to find moving objects. The observational arc of each moving object was 2hr in SMBAS-I and 40 min in SMBAS-II. Several Landolt photometric standard stars (Landolt 1992) were also observed for brightness calibrations at various airmasses each night Detection and Identification of Trojans Standard image reductions, such as bias-subtraction and flatfielding, were performed using NOAO IRAF software. Next, we made composite images of each observing field to detect moving objects. In the case of three exposures for a field, acomposite image was produced by subtracting the second image from the first, then adding it to the third image. This technique was first introduced in Yoshida et al. (2001), and Millis et al. (2002) have independently confirmed its usefulness. Technical details about combined-image making and the image-quality evaluation are described in Yoshida et al. (2003) and YN2005. We found 1194 moving objects from SMBAS-I and 1838 ones from SMBAS-II by careful visual inspection of all composite images. Their motions along the ecliptic longitude (L)andlatitude (B)wereplotted on a (L, B)diagram, as done in figure 1 of YN2005. On the diagram, near Earth asteroids (NEAs), main belt asteroids (MBAs), Hildas, Jovian Trojans, Centaurs, and trans-neptunian objects (TNOs) were distinguishable. We first tentatively identified a clustering having adaily motion in longitude of 8 0 as candidates for Jovian Trojans (YN2005). As described in the previous section, our observational arc of each moving object is short (less than 2 hr). With such short arcs, we could estimate only statistically approximate values of the semimajor axis (a) for each moving object under the assumption that its orbital eccentricity (e) iszero [see Nakamura & Yoshida (2002) for the detail]. Because of this assumption, the estimated a for the orbit of each object inevitably includes two kindsofsmall errors, namelya systematic error ( a s )andarandomerror( a r ). The a s must be corrected commonly to all of the moving objects, since this error works as a bias. On the other hand, a r is an uncontrollable error. The a s and a r were statistically calculated from simulations of many hypothesized Trojans in which realistic observing conditions were taken into account (for details, see subsection 2.3 of YN2005). By the same calculation as we did in YN2005, the ( a s, a r )fortrojans detected in SMBAS-I and SMBAS-II were found to be (0.02AU, 0.15AU) and (0.03 AU, 0.25 AU), respectively. Although the a of each Trojan has a random error of as large as 0:2 AU, this object cannot be an interloper from other groups, because the nearest Hilda swarm has a mean a of 4:0 AU, while the mean a of our Trojans candidate is 5.1 AU. Hence, here we regarded objects whose a values are within a J 3 a r as Trojans (a J :Jupiter s semimajor axis); this selection criterion was also adopted in YN2005. Eventually we picked up 51 Trojans in SMBAS-I (YN2005) and62 objectsin SMBAS-II (this paper) Estimates of Absolute Magnitude, Diameter, and Detection Limit The brightness of each detected Trojan was measured by IRAF-APPHOT. The measured brightness was corrected for chip-by-chip sensitivity differences by comparing counts of thesky background brightness (YN2005), and for atmospheric extinction by Landolt stars. The absolute magnitude (H )ofeach asteroid was calculated by a well-known relation: H = V 5log 10.r /, inwhich we assumed e = 0, sothatr = a and =a 1 ( and r: the geocentric and heliocentric distances). In our observations, we did not use a V -filter, but an R-filter, since the primary aim of SMBAS was used to detect the faintest end of the main belt asteroids, for which the R-band is most sensitive. This was inconvenient, because formulae for converting H to D are defined by the V magnitude, while our obtained H corresponds to the H R -magnitude. In order to remedy this difficulty, we assumed that the mean color V R of 0.48 from known Trojans could be applied to all of our Trojans, namely, H V was derived as H V = H R + 0:48. Since it is known that Jovian Trojans have quite a uniform surface spectra (e.g., Dotto et al. 2006), a color deviation from the assumption of V R = 0:48 should affect the diameter estimate less significantly than does the uncertainty in a estimate, the latter being up to 20%. The D of Trojans were calculated using log 10 D = log 10 A 0:2H V (Bowell & Lumme 1979), where we took 0.04 as A (albedo); this value has been regarded as a mean value for Jovian Trojans and/or primitive bodies in the solar system (Jewitt et al. 2000; Fernández et al. 2003). In order to estimate a detectable limiting magnitude of Trojans for each observing field, we basically followed the same approach as in YN2005. We used artificial images of Trojans and the cumulative fractional detection (see the detail in subsection 2.4 of YN2005). In this paper, we took 90% complete detection as the limiting magnitude in accordance with YN2005, and it was found to be 24:7 mag with the R-band in both SMBAS-I and SMBAS-II. This magnitude corresponds to H R = 17.4 mag at a = 5.9 AU, which is the farthest Trojan detected in our survey, i.e., H V = 17.9, equivalent to the detection limit of D = 2 km (assuming V R = 0.48 and A = 0.04). 3. Size Distributions of L4 and L5 Trojans 3.1. SMBAS Size Distributions Figure 1 shows the differential and cumulative size distributions for L4 Trojans (SMBAS-I) and for L5 Trojans (SMBAS-II). The upper panel was reproduced from YN2005. YN2005 revealed that the cumulative distribution of L4 Trojans has a break at around D = 5km; namely, it is well expressed by two distinctive slopes, b = for 2 km <D<5km and b = for 5 km <D<10 km. As for the size distribution of L5 Trojans (the lower panel of figure 1), the number of Trojans for 5 km <D<10 km (14.1 mag <H V < 15.7 mag) is discontinuously depleted. However, this depletion is unlikely to be real, but due to a statistical fluctuation caused by meagerness of large L5 Trojans in

3 No. 2] Size Distributions of Small L4 and L5 Jovian Trojans 299 Fig. 1. Size distributions of L4 (top panel) and L5 (bottom) Trojans from SMBASs. The box histograms show the differential H V distribution and the crosses with error bars show the cumulative H V distribution. Each upper abscissa represents the diameter corresponding to the H V by assuming an albedo of 0.04 for reference. The solid and dashed lines in the top panel show the best-fit slopes of b = for 5 km <D<10 km and b = for 2 km <D<5km,respectively. The solid line in the bottom panel shows the best-fit slope of b = for 2 km <D<5km. that size range in our survey. Hence, we do not attempt here to make a comparison between the L4 and L5 populations for 5km<D<10 km. On the other hand, for the size range of 2km <D<5km, the best-fit slope of the size distribution of L5 Trojans was found to be b = The Trojan slopes from SMBAS-I and SMBAS-II and the past results are summarized in table Overall Size Distributions In general, a single survey observation can span only a size range of roughly one order of magnitude; 2 km <D< 10 km for YN2005 and this work, 4 km <D<40 km for Jewitt, Trujillo and Luu (2000), 10 km <D<100 km for Szabó etal. (2007), and 30 km <D<140 km for cataloged Trojans. Since such a situation is largely restricted by the detection dynamic range of the observing systems (telescope + CCD), an overall size distribution covering a wide range in diameter, which is most interesting from viewpoints of the formation origin and subsequent collisional evolution of Trojan populations, Fig. 2. Composite cumulative size distributions of L4 (top panel) and L5 (bottom) Trojans. In each lower abscissa the H -magnitude is plotted, and in each upper abscissa the diameter corresponding to the H -magnitude is shown; each ordinate stands for the cumulative number. In the top panel, the triangles, circles, and crosses represent the data points from cataloged L4 Trojans, 2 Jewitt, Trujillo, and Luu (2000), and SMBAS-I, respectively. The solid line shows the slope of the cumulative size distribution obtained from SDSS data (Szabó etal. 2007). The dotted line corresponds to the slope for smaller Trojans (2 km <D<5km) in SMBAS-I. The scale values for combining are adjusted to the above distributions from three surveys and Trojan catalog to coincide each other at H =9.7, 12.1, and 14.1 mag. In the bottom panel, triangles and crosses represent the data points from cataloged L5 Trojans 2 and SMBAS-II, respectively. The solid line shows the slope obtained from SDSS data. The ordinate scale values of this panel are adjusted to the above distributions from two surveys and Trojan catalog to coincide each other at H = 9.7 and 16.1 mag. must be produced by combining a few survey results into one. For that purpose, we used here the size distributions from SMBAS-I, SMBAS-II, Jewitt, Trujillo, and Luu (2000), and known Trojans brighter than H = 12:3 mag (corresponding to D 20 km), whose statistics is claimed to be complete by Szabóetal.(2007) to make up an overall size distribution covering 2 km <D<100 km. Figure 2 shows such combined cumulative size distributions of L4 and L5 Trojans. 2 hhttp://cfa-

4 300 F. Yoshida and T. Nakamura [Vol. 60, Table 1. Slopes of cumulative size distributions for L4 and L5 Trojans with different size ranges. Group Slope (b) Size range Reference D(km) L <D< 5 SMBAS-I (YN2005) L <D<10 SMBAS-I (YN2005) L <D<40 Jewitt, Trujillo, and Luu (2000) L <D<93 Known Trojan catalog 2 L <D< 5 SMBAS-II (This work) L <D<93 Known Trojan catalog 2 First, let us examine the slopes of the combined cumulative size distribution for L4 Trojans with 2 km <D<93 km. The piecewise slopes given in table 1 are more or less similar, 2:0 2:4, sothatwefind the overall fitted slope for 5km<D<93 km to be nearly constant, that is, b = 2:2 0:1. This is consistent with the size distribution obtained by Szabó et al. (2007), who investigated SDSS data and found a slope of b = 2:2 0:25 for 9<H<13:5 (13 km <D<107 km) for all Jovian Trojan population (including both of L4 and L5). On the other hand, at D<5km,theslope abruptly becomes shallower, b = As for L5 Trojans, there is no data for a moderate size rangeof5km <D<13 km. However, table 1 shows that the slopes for cataloged L5 Trojans (20 km <D<93 km) and for small SMBAS-II Trojans (2 km <D<5km) arenearly identical (2.1). This is also consistent with Szabó et al. (2007). Thus, although there are no data for the size range of D = 5 13 km, under the tentative assumption that the slope does not change from 2 km to 100 km, we combined the size distributions for large and small Trojans into one, as shown in the lower panel of figure 2. The straight line depicted in the panel corresponds to a slope of 2.2 from Szabóetal.(2007). 4. Discussion From figure 2 and table 1, we can say that Jovian Trojans with D>5kmineach of L4 and L5 swarms have almost identical size distributions. The slopes of the cumulative size distributions obtained from different survey observations (i.e., SMBAS; Jewitt et al. 2000; Szabó etal. 2007) and known Trojans are consistent. Jovian Trojans with D<5 km in each of L4 and L5 swarms show us the different size distributions; the slopes of the cumulative size distribution of each of L4 and L5swarmsforasteroids with D<5kmare1.3and2.1, respectively. The shallow slope for L4 Trojans with 2 km <D<5km may imply a depletion of small L4 Trojans. If we assumed that the size distributions of planetesimals would have been homogeneous around proto-jupiter, the depletion of small L4 Trojans may suggest a different capture rate into L4 and L5 Trojan s orbits of small planetesimals, or the existence of some mechanisms for removing small Trojans only from L4 swarm. Based on previous theoretical studies, several mechanisms for decreasing Jovian Trojans are suggested as follows. (1) Asteroid drift by Yarkovsky effect: This could not be a main mechanism for removing small L4 Trojans, because it affects in the same way to the L4 and L5 Trojans, which have the same mean heliocentric distances. The Yarkovsky effect cannot remove only small L4 Trojans. (2) Gas drag: Peale (1993) and Marzari and Scholl (1998) said that if the planetesimals were trapped into Trojan orbits under protoplanetary gas drag, an asymmetry happened between trapping rate in L5 compared to L4. It could have led to the capture of a larger number of small planetesimal in L5. (3) Planetary migration: Gomes (1998) showed that planetary migration in his simulations left more survivors in L4 after the 1:1 resonance with the Jupiter sweep was stopped. Since the resonance affects independently of asteroid size, Jovian Trojans could keep the overall size distribution, but the total number of L5 asteroids would decrease more than did L4. This mechanism has nothing to do with the depletion of small L4 Trojans, but it might explain the population asymmetry of L4 and L5 Trojans (as shown in Nakamura & Yoshida 2008). (4) Collisional evolution: The existence of dynamical families in Jovian Trojans is evidence that several collisions occurred in each Trojan swarm. The collisional diffusion might have varied the size distributions of each swarm. One could also think that the shallow slope for small L4 Trojans implies an overabundance of small L5 Trojans. Namely, something happened dynamically or collisionally to the L5 population to increase more small Trojans, while thel4population represents the steady state. From only an investigation of the size distributions of Jovian Trojans, we cannot infer which kind of mechanism affected the small Trojans. However, if we also consider the population asymmetry between L4 and L5 described by Nakamura and Yoshida (2008), we may propose a grand scenario during the planetary formation stage in an early solar system as follows: First, Jovian Trojans were trapped during proto- Jupiter s growth under protoplanetary gas, the depletion of small L4 Trojans occurred by the gas drag effect. After that, planetary migration caused the asymmetry of the L4 and L5 Trojans population. After the current configuration of planets was established, collisional evolution became the only significant mechanism for varying the size distribution for each Trojan swarm. If the size and spatial distributions of planetesimals are homogeneous around proto-jupiter and collisional evolutions have not significantly varied their size distributions after Jovian Trojans were constructed, the above scenario could be possible. However, if the Trojan populations were actually reconstructed as described in Morbidelli et al. (2005), the

5 No. 2] Size Distributions of Small L4 and L5 Jovian Trojans 301 above scenario may not work. For confirming the above scenario, the difference of size distributions between L4 and L5 Trojans should be established by survey observations, which search for more wide regions (i.e., next SMBAS or the Pan-STARRS). Also, support from theoretical studies is absolutely necessary; for example, estimations of the contribution of collisional evolution for varying the size distribution of Trojans asteroids, the reality of the hypothesis planetary migration, the quantification of the gas drag effect during Trojan trapping period, and so forth. We thank David P. O Brien and another anonymous referee for their useful and valuable comments. References Beaugé, C., & Roig, F. 2001, Icarus, 153, 391 Bowell, E.,&Lumme, K. 1979, in Asteroids, ed. T. Gehrels (Tucson: University of Arizona Press), 132 Dotto,E., et al. 2006, Icarus, 183, 420 Fernández, Y. R., Sheppard, S. S., &Jewitt, D. C. 2003, AJ, 126, 1563 Fleming, H. J., & Hamilton, D. P. 2000, Icarus, 148, 479 Gomes, R. S. 1998, AJ, 116, 2590 Jewitt, D. C., Trujillo, C. A., &Luu, J. X. 2000, AJ, 120, 1140 Landolt, A. U. 1992, AJ, 104, 340 Marzari, F., & Scholl, H. 1998, Icarus, 131, 41 Milani, A. 1993, Celest. Mech. Dyn. Astron., 57, 59 Millis, R. L., Buie, M. W., Wasserman, L. H., Elliot, J. L., Kern, S. D.,&Wagner, R. M. 2002, AJ, 123, 2083 Miyazaki, S., et al. 2002, PASJ, 54, 833 Morbidelli, A., Levison, H. F., Tsiganis, K., & Gomes, R. 2005, Nature, 435, 462 Nakamura, T., & Yoshida, F. 2002, PASJ, 54, 1079 Nakamura, T., & Yoshida, F. 2008, PASJ, 60, 293 Peale, S. J. 1993, Icarus, 106, 308 Shoemaker, E. M., Shoemaker, C. S., & Wolfe, R. F. 1989, in Asteroids II, ed. R. P. Binzel, T. Gehrels, & M. S. Matthews (Tucson: University of Arizona press), 487 Strom, R. G., Malhotra, R., Ito, T., Yoshida, F., & Kring, D. A. 2005, Science, 309, 1847 Szabó, Gy. M., Ivezić, Ž., Jurić, M., & Lupton, R. 2007, MNRAS, 377, 1393 Yoshida, F., et al. 2001, PASJ, 53, L13 Yoshida, F., & Nakamura, T. 2005, AJ, 130, 2900 (YN2005) Yoshida, F., & Nakamura, T. 2007, Planet. Space Sci., 55, 1113 Yoshida, F., Nakamura, T., Watanabe, J., Kinoshita, D., Yamamoto, N.,&Fuse, T. 2003, PASJ, 55, 701