ROTATIONAL PROPERTIES OF THE MARIA ASTEROID FAMILY

The Maria asteroid family consists of 3230 known members with identification based on the Hierarchical Clustering Method (Nesvorný 2010). The latest family classification is available at the AstDyS Web site.¹³ As of 2013 November, the database lists 2085 Maria members based on synthetic proper orbital elements of numbered asteroids. We checked the membership as listed in the AstDyS and found that most of our target asteroids were confirmed as Maria members except for asteroids 652, 4122, 5977, 13679, 19184, 19333, 32116, 43174, 50511, 109792, and 114819. However, memberships of an asteroid family can be subject to some unavoidable uncertainties, since they depend on the data sets of proper orbital elements available at the epoch of family search, on the adopted identification technique, and on membership assignment criteria.

We investigated the cumulative distribution N (<H) of absolute magnitudes H for all the members of the Maria family. We used a power-law approximation of N (<H) ∝ 10^γH in the magnitude range between 12 and 14.5 and obtained γ ∼ 0.54 (see Figure 1). The value of γ obtained is close to the steady-state mass distribution of collisional fragmentation for which Dohnanyi (1969) derived γ = 0.5, but discordant with a considerably steeper distribution constructed by Pareto power laws (e.g., Tanga et al. 1999). This power-law index implies that the Maria family has undergone a significant collisional and dynamical evolution and has currently reached an equilibrium state (e.g., Dohnanyi 1969; O'Brien & Greenberg 2003; Bottke et al. 2005). The slope depends highly on the number of objects brighter than 12th absolute magnitude, while the slope is thought to have been contaminated by observational bias for objects fainter than H = 14.5 mag. We chose our targets in the H range between 12 and 14.5; the total number of targets in this magnitude range is 981. The catalogued absolute magnitudes in the standard (H, G) system have large uncertainties due to the transformation from an observed magnitude system to the Johnson V system, and to the phase correction (see Cellino et al. 2009, for more details). It might either overestimate or underestimate the size distribution of a family.

Zoom In Zoom Out Reset image size

We generated ephemerides for each object from the 981 target asteroids using the JPL Horizons service,¹⁴ then produced target lists of observable asteroids during any given night. Because the rotational periods of most targets are unknown, we decided to observe one or two target asteroids per night at least, assuming typical rotational periods between 2 and 24 hr. Since a preliminary rotational light curve of an asteroid is determined only after a prompt data reduction, it is better to allocate follow-up observation so as to cover its full-phased light curve. For the sake of increasing the observational efficiency, we developed an observation scheduler to carry out asteroid follow-up observation in a timely manner. When we input observational parameters such as initial rotational period, observed time, and observatory code, the scheduling program suggests to the user the next proper observing time to cover gaps in the light curve.

Our observations were focused on asteroids in the Maria family that lack a known rotational period. To increase the number of observable asteroids as much as possible, we observed two objects alternately on a single night with the BOAO 1.8 m, TUG 1.0 m, and CA 1.2 m telescopes. Moreover, further observations during the years 2012-2013 were also performed for 21 targets at the Wise telescope in 2008, and spin axis information was obtained from published light curves (see Section 3.2 for more details).

It is difficult to cover the entire rotational phase of asteroids with rotational periods longer than 8 hr during a single night. In addition, if the rotational period of an asteroid aliases with 24 hr, the rotational period of the Earth, it is essentially impossible to have a fully covered light curve from a single observatory. A network of follow-up telescopes can be used to solve these problems. In order to maximize phase coverage of an asteroid, taking advantage of a network observation at different time zones, we organized observation campaigns with three observatories in Korea (UT + 9 h), Turkey (UT + 2 h), and USA (UT – 8 h). These network observations were carried out in 2012 with TUG and LOAO on June 28 and 30; BOAO and TUG on October 12, 14, and 19; and SOAO and TUG on October 12, 14, and 17. In order to calibrate all the data gathered from various observatories, the same CCD fields obtained during the previous observations were taken in the next runs.

The detailed observational circumstances of each asteroid are listed in Table 2: UT date corresponding to the mid time of the observation, the topocentric equatorial coordinates (R.A. and Decl., J2000), the heliocentric (r) and the topocentric distances (Δ), the solar phase (sun-asteroid-observer's) angles (α), the apparent predicted magnitude (V), and the telescopes used.

Out of 134 nights of observations in total, we obtained 218 individual light curves for 74 Maria family members and derived synodic rotational periods for 51 objects, including obtained periods for 34 asteroids for the first time. In order to find the periodicity of the light curve, the Fast Chi-Squared (Fχ²) technique (Palmer 2009) was employed. In addition, we checked the result with the discrete Fourier transform algorithm (Lenz & Breger 2005). In most cases for rotational periods, these two different techniques present similar results within the statistical errors. The Fχ² technique presented here represents the observed magnitudes as a Fourier time series truncated at the harmonic H:

$$\begin{eqnarray} \Phi _H(\lbrace A_{0...2H},f\rbrace,t) &=& A0 + \sum _{h=1...H}A_{2h-1}\sin (h2{\pi }ft) \nonumber\\ && + A_{2h}\cos (h2{\pi }ft). \end{eqnarray} \tag{ 1 }$$

In practice, we fit the fourth-order Fourier function and also obtained the highest spectral power with the discrete Fourier transform algorithm. As a result, the final rotational period was determined assuming a double-peaked light curve. We present the resultant composite light curves of observed asteroids in Figure 9 in the Appendix, folded with their synodic periods.

In the approximation of a triaxial body with uniform albedo, the peak-to-peak variations in magnitude are caused by the change in apparent cross section of the rotating body, with semi-axes a, b, and c, where a > b > c (the body rotates along the c axis). According to Binzel et al. (1989), the light-curve amplitude varies as a function of the polar aspect viewing angle θ (the angle between the rotation axis and the line of sight):

$$\begin{equation} {\Delta }m = 2.5 \log (\frac{a}{b}) - 1.25 \log \left(\frac{a^2\cos ^2{\theta }+c^2\sin ^2{\theta }}{b^2\cos ^2{\theta }+c^2\sin ^2{\theta }} \right). \end{equation} \tag{ 2 }$$

The lower limit of axis ratio a/b can be expressed as a/b = 10^0.4Δm, which corresponds to an equatorial view (θ = 90°). The peak-to-peak variation in light curve becomes larger when we increase the solar phase angle. The following empirical relationship between those two parameters was found by Zappalà et al. (1990):

$$\begin{equation} A(0{^\circ}) = \frac{A(\alpha)}{(1+m\alpha)} \end{equation} \tag{ 3 }$$

where A(0°) is the light-curve amplitude at zero phase angle and A(α) is the amplitude measured at solar phase angle =α. These authors also found that average m-values are 0.030, 0.015, and 0.013 deg^-1 for S, C, and M-type asteroids, respectively. We adopted the constant m of 0.03 for our analysis, as the Maria family is known as a typical S-type asteroid family.

In order to derive the orientation of the spin axis using disk-integrated photometric data, dense light curves obtained over three or four apparitions are essential (Kaasalainen et al. 2002). However, due to a limited amount of dense data sets, the light-curve inversion method using only sparse data (Kaasalainen 2004) and a combination of sparse and dense data (D̆urech et al. 2009; Hanus̆ et al. 2011) has been improved during the past decades.

We obtained the available dense data set by matching objects with the Maria family members from the following three data sources. In the MPC (Minor Planet Center) Light Curve Database,¹⁵ we downloaded light curves for the 17 Maria asteroids. From the online Web site of Courbes de rotation d'astéroïdes et de comètes (CdR,¹⁶ operated by R. Behrend), a total of 140 individual light curves for 13 objects were acquired. Another three light curves were found in the Asteroid Photometric Catalog (Lagerkvist & Magnusson 2011) from NASA's Planetary Data System.

Most sparse photometric data sets are a by-product of worldwide astrometric surveys, such as the Catalina Sky Survey and the Siding Spring Survey. Detailed instructions for obtaining the sparse data from the AstDyS site are in Hanus̆ et al. (2011); we follow their methodology. According to very recent analysis by the same authors (Hanus̆ et al. 2013, private communication), only sparse data sets from the USNO in Flagstaff (MPC code 689), the Catalina Sky Survey (703), and the La Palma (950) were useful for determining the shape modeling of asteroids. Combining the dense and the sparse light-curve data sets from the AstDyS, we computed the pole orientation for 16 objects and determined 13 unique solutions.

Our observations from seven observatories yielded 218 individual light curves for 74 Maria asteroids. Among them we derived synodic rotational periods for the 51 members with a reliability code ⩾2. For the 23 objects with a reliability code of 1, we set a lower bound to the periods except for 879 Ricarda, for which the rotation period is known. The reliability parameters follow the definition by Lagerkvist et al. (1989).

• 1.  
  Very tentative result, may be completely wrong.
• 2.  
  Reasonably secure result, based on over half coverage of the light curve.
• 3.  
  Secure result, full light-curve coverage, no ambiguity of period.
• 4.  
  Multiple apparition coverage, pole position reported.

The information on rotational characteristics from our light-curve data sets with other physical properties is summarized in Table 3; diameter and albedo information were mainly acquired from the WISE IR data (Masiero et al. 2011) with the exception of those marked with A (AcuA, Asteroid catalog using AKARI; Usui et al. 2011) and M (mean albedo value of 0.254 for the Maria family asteroid), while the taxonomic information is compiled from Neese (2010) and Hasselmann et al. (2012). We used the data sets only with a reliability code ⩾2 for our analysis. The light curves for 23 objects denoted with Q Notes of 1 did not cover even half-phased periods due to the following three reasons: (1) very low amplitude of the light curve (smaller than 3σ); (2) variation of brightness much longer than observing time; (3) data interruption due to bad weather or instrument conditions. We examined a lower bound to the periods for those 23 objects to consider the influence of selection bias, i.e., favoring observations of faster rotators. Although all 23 objects rotate slower than 5 hr, they do not affect the overall tendency of our results, and besides, the peak-to-peak variation derived from completely wrong light curves could negatively affect the statistics. This is the reason why we only select periods with a reliability code ⩾2 for our analysis.

In order to improve the significance of the statistical analysis of the rotational properties in the Maria family, we searched the published light-curve data from the LCDB released in 2013 March and matched 58 objects with the Maria family asteroids. Out of 58 LCDB data sets, there are 17 objects overlapping with our observations. Therefore, we checked those original light curves from the literature and then adopted light curves with minimum period gaps between the LCDB and our observations.

Figure 2 shows the distribution of rotational rates for 92 Maria family members used in this study and the corresponding best-fit Maxwellian curve. The 51 objects obtained from our observations are marked with shaded bars. It can be obviously seen that asteroids rotating faster and slower considerably exceed the fitted distribution. From the Kolmogorov–Smirnov test, the compatibility between the observed distribution and the fitted Maxwellian is completely inconsistent at a 92% confidence level. This non-Maxwellian distribution is also found in other old-type asteroid families, such as the Koronis family (Slivan et al. 2008) and the Flora family (Kryszczyńska et al. 2012), with ages of 2.5 ± 1.0 Gyr and 1.0 ± 0.5 Gyr (Nesvorný et al. 2005), respectively.

Zoom In Zoom Out Reset image size

Laboratory experiments of catastrophic collisions (Holsapple et al. 2002, and references therein) and numerical simulations (Asphaug & Scheeres 1999; Michel et al. 2001) showed that the rotational frequency of fragments approximates the Maxwellian distribution. Accordingly, this inconsistency with our measurements suggests that the members belong to an old family and have had their rotational properties modified by non-gravitational forces operating over a long period of time; the spin rate change, in particular, can be attributed to the YORP effect. Rubincam (2000) found that the spin states of pseudo-Gaspra and pseudo-Eros can significantly evolve on a 100 Myr timescale, which is far shorter when compared to their break-up and collisional processes. Hanus̆ et al. (2011) found that the spin vectors of the MBAs with D < 30 km were significantly affected by the YORP thermal effect. The size range of most of the 92 Maria members used in this study is distributed between 1.5 km and 30 km (see Figure 3).

Zoom In Zoom Out Reset image size

We examine the correlation between the rotational frequency (cycles day⁻¹) and diameter (km) in Figure 4. Among the 92 Maria family asteroids used for this study, 4 objects might be regarded as interlopers. In terms of the visible geometric albedo p_v, two asteroids 3094 Chukokkala and 4860 Gubbio possess very low albedos, 0.068 and 0.037, respectively, provided that the mean value of albedo for the Maria family asteroids is 0.254 from the matched 1152 WISE IR data. The other two members have a different spectral type of Xe for 1098 Hakone (Neese 2010), C-type for 71145 (1999 XA183) (Hasselmann et al. 2012). Those possible interlopers are marked with filled boxes; however, they do not affect the overall tendency in Figure 4.

Zoom In Zoom Out Reset image size

In the results obtained from our observations, there is an apparent systematic trend toward larger dispersion of rotational frequency with decreasing size. The rotation of small asteroids can be easily accelerated or decelerated due to the YORP effect. Moreover, all asteroids larger than 22 km rotate more slowly than 4.8 hr (5 cycles day⁻¹), quite similar to what was found for the Flora family (Kryszczyńska et al. 2012).

Figure 5 shows the correlation between the rotational frequencies (cycles day⁻¹) and the amplitudes of light curve (peak-to-peak variation magnitude) that represents the overall shapes of asteroids. We derived amplitudes of light curves using Equation (3) for our observations. For light-curve data from the LCDB, we adopted the maximum amplitudes of light curves. The various sizes of circles indicate the diameter of each asteroid relatively; asteroids larger than a diameter of 15 km are marked with blue circles.

Zoom In Zoom Out Reset image size

We can see two features in Figure 5: there is no object with both fast rotation (faster than 6 cycles day⁻¹) and large light-curve amplitudes (>0.6 mag). Despite an insufficient number for large peak-to-peak variation, few highly elongated objects tend to rotate slowly. This could be explained by the break-up limit of the elongated rubble pile; it makes sense that the more elongated an object is, the easier it can be shattered during its spin-up process. The colored curves in Figure 5 approximate the critical rotational period (P_c) for bulk densities of 1.5, 2.0, and 2.5 g cm⁻³, respectively, adopted from (Pravec & Harris 2000):

$$\begin{equation} P_c \approx 3.3 \sqrt{\frac{1+\Delta m}{\rho }} \end{equation} \tag{ 4 }$$

where P_c is the period in hours, ρ is the bulk density in g cm⁻³, and Δm is the amplitude of the light-curve variation in magnitude. It is apparent from Figure 5 that there is no object against "spin barrier" (Pravec & Harris 2000) for bulk density of 2.5 g cm⁻³. In addition, elongated objects are located far from the spin rate limit.

The other feature seen in Figure 5 is that no objects larger than 15 km have amplitude of light curve larger than 0.5 mag except for 4860 Gubbio, regarded as an interloper. Small objects (<15 km) with various shapes are spread out in this plot. If an elongated rubble pile asteroid is disrupted by a non-gravitational force such as the YORP effect, it might have less elongated, namely, more spherical, shape as a result. Figure 5 separates into two plots (Figures 10 and 11) in the Appendix.

The Yarkovsky effect plays a significant role in the dynamical evolution of asteroid orbits. The study of this non-gravitational thermal force on the asteroid family has been improved dramatically during the past decades in several families, such as Koronis (Bottke et al. 2001), Flora (Nesvorný et al. 2002), and Eos (Vokrouhlický et al. 2006). Semimajor axis drift, in accordance with either the sense of rotation or various sizes of asteroids, by the Yarkovsky effect results in a V-shape in the proper semimajor axis and absolute magnitude plane (Nesvorný & Bottke 2004; Vokrouhlický et al. 2006; Milani et al. 2010). The analysis for drift in semimajor axis allows us to estimate the age of an asteroid family (Bottke et al. 2001; Nesvorný et al. 2005).

To find the Yarkovsky footprints in the Maria asteroid family, we investigate the pole orientation of the members. We examined sense of rotation for the 16 members that were observed during at least three apparitions and successfully determined the pole orientations for the 13 objects by the combination with sparse data from the AstDyS site. The summary of pole solutions for the Maria family asteroids is listed in Table 4. Due to an intrinsic symmetry of the problem (Kaasalainen et al. 2002), ground-based observations of objects with a small orbital inclination are affected by an ambiguity in the determination of the ecliptic longitude of the pole axis direction, resulting in two solutions that are placed 180 deg apart and are statistically indistinguishable. We adopted the lower chi-squared solution marked with boldface for this study and estimated the uncertainty of the pole orientation to be 5–20 deg, depending on the number of dense and sparse data sets. Out of 13 asteroids, the information on spin axis of 4 objects was also found in DAMIT¹⁷ (Database of Asteroid Models from Inversion Techniques; D̆urech et al. 2010). The result of our analysis for those four objects is quite consistent with that of DAMIT.

Table 4. Summary of Pole Solutions for the Maria Family Asteroids

Asteroid	λ1	β1	λ2	β2		Psid	NLC	N689	N703	DAMIT
	(deg)	(deg)	(deg)	(deg)	(deg)	(hr)
170 Maria	103	−18	289	−5	112	13.1313212	10	183	112	⋅⋅⋅
575 Renate	6	+3	183	−19	105	3.6603283	5	108	191	⋅⋅⋅
652 Jubilatrix	143	+34	321	+27	69	2.6626693	5	84	163	⋅⋅⋅
660 Crescentia	58	+1	241	+15	90	7.9140789	10	220	158	⋅⋅⋅
695 Bella	81	−58	308	−49	149	14.2190116	15	203	144	87, −55
										314, −56
727 Nipponia	172	+11	333	+5	89	5.0687459	10	239	185	⋅⋅⋅
787 Moskva	132	+11	334	+44	55	6.0558066	30	164	132	126, +27
										331, +59
875 Nymphe	47	+32	200	+46	52	12.6212926	18	95	176	42, +29
										196, +41
897 Lysistrata	109	−55	299	−29	136	11.2742816	7	186	134	⋅⋅⋅
1158 Luda	88	+67	267	+21	54	6.8750690	12	106	177	⋅⋅⋅
1996 Adams	106	+56	281	+17	48	3.3111390	3	82	172	107, +55
3786 Yamada	77	+61	236	+63	33	4.0329422	9	18	131	⋅⋅⋅
6458 Nouda	63	−2	240	+21	81	4.2030968	9	⋅⋅⋅	163	⋅⋅⋅

Notes. The ecliptic longitude (λ) and latitude (β) of the asteroid pole orientation (usually the two solutions differ by 180 deg), the spin vector obliquity () of the angle between the orbital plane and the spin pole (calculated from the lower chi-squared pole solution), the sidereal rotational period (P_sid), the number of dense light curves (N_LC), and the number of sparse points from USNO in Flagstaff (689) and the Catalina Sky Survey (703). The information about spin axis (λ, β) of four objects was also found in DAMIT (Database of Asteroid Models from Inversion Techniques; D̆urech et al. 2010). For the pole orientation, the lower chi-squared value marked with boldface is preferred.

Download table as:  ASCIITypeset image

Concerning the pole orientation, we concentrate on the determination of the sense of rotation. As a result, we found six prograde rotators (652 Jubilatrix, 787 Moskva, 875 Nymphe, 1158 Luda, 1996 Adams, and 3786 Yamada), while two objects (695 Bella and 897 Lysistrata) rotate in the opposite sense. We show the position of prograde and retrograde asteroids on a plane of proper semimajor axis (AU) versus absolute magnitude (H) in Figure 6. In addition, we found five rotators with the pole along the ecliptic: 170 Maria, 575 Renate, 660 Crescentia, 727 Nipponia, and 6458 Nouda, marked with open circles in Figure 6. The spin axes of those five objects are very close to the ecliptic plane within their statistical errors, between 5 and 20 deg. Although an observational bias cannot be excluded, the excess of prograde rotators with respect to retrograde rotators could be explained with a long-term evolution by the Yarkovsky effect. In the case of retrograde rotators, the Yarkovsky drift causes the semimajor axis of family members to decrease; consequently, they could have been ejected by the 3:1 MMR to the inner solar system. Kryszczyńska (2013) found a similar result for the Flora family, which is located near the outer border of the ν₆ resonance area.

Zoom In Zoom Out Reset image size

Recent statistical studies by Paolicchi and Kryszczyńska (2012) indicate that there is an excess of prograde versus retrograde rotators in the main belt for asteroids smaller than 100 km. However, these authors did not find any convincing explanations for this excess, and hence this remains an open problem. On the contrary, a strong excess for retrograde rotation was found in NEAs that is completely consistent with a theoretical ratio of 2 ± 0.2 with respect to prograde rotators (La Spina et al. 2004). Regarding this prograde-retrograde asymmetry in MBAs and NEAs, they inferred that there is a connection between the excess of retrograde NEAs and the deficiency of retrograde MBAs due to the Yarkovsky effect. However, it might be more complicated to generalize the trend to MBAs as they are too big to have been affected by the Yarkovsky drift. Another supporting example we can find is the largest NEA, 1036 Ganymed, that is regarded to be originated from the Maria family. Zappalà et al. (1997) carried out an extensive analysis of the Maria family and suggested that about 10 objects with size between 15 and 30 km have probably been injected into the 3:1 MMR with Jupiter. They proposed that a couple of giant S-type NEAs, 433 Eros and 1036 Ganymed, probably originated from the Maria family. The spectral reflectance of this asteroid is quite similar to the average spectrum of the Maria family members (Fieber-Beyer et al. 2011). Interestingly, its spin orientation is retrograde (Kaasalainen et al. 2002); moreover, it is on a high-inclination orbit of 26.7 deg.

We can easily distinguish three interlopers (that is, two lower albedo objects of 3094 Chukokkala, 4860 Gubbio, and Xe-type asteroid of 1098 Hakone) from Figure 6, while 71145 (1999 XA183) is not close to the border of the family. One distinct property of the Maria family compared to other densely populated asteroid families, such as Flora, Themis, and Eunomia, is that there is no prominent large body among the family members. The largest body in this family is not 170 Maria but 472 Roma, with a diameter 50.3 km as calculated from the WISE IR data. In the recent paper by Masiero et al. (2013), the Roma family is substituted for the name of the Maria family.

Figure 7 shows the spin vector obliquity of the Maria family members with respect to the rotational frequency (cycles day⁻¹), compared with the Koronis (Slivan et al. 2009) and Flora families (Kryszczyńska 2013). In those two families, there is a conspicuous group of prograde objects. Furthermore, retrograde objects share almost the same obliquities, which is referred to as the Slivan effect (Slivan 2002). In the Maria family, on the other hand, no prominent prograde groups being in the Slivan state were found, while the data for retrograde rotation are not sufficient for a conclusion. Vokrouhlický et al. (2003) investigated several test prograde rotators chosen with various ranges of either semimajor axis or inclination in the main belt. Their preliminary results indicate that asteroids with low inclination in the outer main belt, such as 24 Themis, might be trapped efficiently in the Slivan state. This is the reason why they suggest observations of either high-inclination or inner MBAs and numerical simulations for their dynamical evolution. Skoglöv & Erikson (2002) found that orbital inclination of MBAs is the most important factor that affects the magnitude of the spin vector variation. Based on their dynamical spin evolution, if the orbital inclination is increased, also the maximum obliquity variation increases linearly.

Zoom In Zoom Out Reset image size

The other feature seen in Figure 7 is that there are several objects with spin vector obliquities around 90 deg, that is, spin axes approximately parallel to the orbital plane. The change of semimajor axis by the diurnal component of the Yarkovsky effect is proportional to the cosine of the obliquity of an asteroid, so it will vanish when the obliquity approaches 90 deg. We can predict that the semimajor axis drift of these five objects plotted as open circles in Figures 6 and 7 will not occur. However, we can expect that the spin vector can be modified by the YORP effect.

The 3:1 MMR (∼2.5 AU) is the most prominent Jovian resonance region in the main belt, and, as such, it is an important source region of NEAs and meteors. The Maria family is located quite close to the outer border of the 3:1 MMR with Jupiter. When members of the Maria family enter this unstable region due to the Yarkovsky drift, they can escape from its membership and become NEAs. For this reason, the distribution of proper semimajor axis versus absolute magnitude is asymmetric as shown in Figure 6. The Bottke et al. (2002) model suggests that with 20% probability, the population of all near-Earth objects (NEOs; with absolute magnitude H < 18) originated from the 3:1 MMR region. They estimated that the number of kilometer-sized NEOs (H < 18) that escaped from the 3:1 MMR every million years is 100 ± 50 bodies. Similar results independently derived by Morbidelli and Vokrouhlický (2003) show that 100–160 objects larger than 1 km enter the 3:1 MMR per million years.

In order to make an order-of-magnitude estimation of the flux of the Maria members into the 3:1 MMR region, we try to determine the number of missing Maria family members. We do this by taking the difference of the original members (assuming that the distribution was originally entirely symmetric) and the number of currently known objects (the absolute magnitude of all members is brighter than 18th magnitude). To obtain the number of original members, we assume that the center of the Maria family is at about 2.562 AU (denoted with the red dashed line) since the number density with respect to the semimajor axis is at a minimum. Then we double the number of family members, with proper semimajor axis >2.562 AU as in Figure 6. This results in a total of 5368 original members. With this procedure, our preliminary result implies that 2138 Maria asteroids have been injected into the 3:1 MMR region through its dynamical evolution age of 3 ± 1 Gyr. It is well known that the dynamical lifetime of objects placed in orbital resonances is only a few million years (Gladman et al. 1997). Their numerical simulations for 3:1 MMR injected particles from the Maria family show that the fraction of particles experiencing the end-states of (1) being injected into the inner solar system and eventually colliding with the Sun or (2) being injected into a Jupiter-crossing orbit and being eventually removed from the solar system are 70% and 29%, respectively. Therefore, we may conclude that roughly 1500 objects from the Maria family had been injected into the inner solar system during 3 ± 1 Gyr, i.e., 37–75 Maria asteroids larger than 1 km every 100 Myr.

We also look into the 3:1 resonance neighborhood region to find possible candidates among Maria members residing in this unstable area. According to Morbidelli and Vokrouhlický (2003), the boundary of the 3:1 MMR region can be defined approximately by the following formulae (only for the right side of the 3:1 resonance):

$$\begin{equation} a = 2.508 + \frac{e}{29.615} \ \ {\rm for}\ \ e \le 0.15936, \end{equation} \tag{ 5a }$$

$$\begin{equation} a = 2.485 + \frac{e}{5.615} \ \ {\rm for}\ \ e > 0.15936. \end{equation} \tag{ 5b }$$

They found that the number density of asteroids sharply increases up to ∼0.015 AU from the resonance boundary, yet it is not changed considerably over the next 0.025 AU. Guillens et al. (2002) also pointed out that there is a chaotic diffusion region in the range of 0.01–0.02 AU in semimajor axis from the 3:1 resonance border. Asteroids residing in this region could enter the resonance on a timescale of 100 Myr. The oscillation of the border of the resonance could also have taken place (Morbidelli & Moon 1995; Morbidelli et al. 1995; Robutel & Laskar 2001).

Figure 8 represents all 3230 Maria asteroids projected onto a plane of proper semimajor axis and proper eccentricity. The 3:1 MMR boundary denoted with black dashed line is defined by Equation (5a). The chaotic diffusion region that has been shifted to 0.015 AU from the resonance border is represented by the red dashed line. The number of objects residing in this region is 34; they could fall into the 3:1 resonance in the coming 100 Myr, especially 114123 (2002 VX49) and 137063 (1998 WK1) (marked with green open circles), which are the most promising candidates for becoming new NEAs. The number of 34 is surprisingly consistent with the range 37–75 we obtained before, within an error range of age estimation. Large filled blue and red circles stand for prograde and retrograde rotators, respectively. Prograde objects tend to position outward the semimajor axis compared with retrograde, due to the Yarkovsky effect. This can be also seen in Figure 6.

Zoom In Zoom Out Reset image size