An Analysis of the Region of the Phocaea Dynamical Family
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Mon. Not. R. Astron. Soc. 398, 1512–1526 (2009) doi:10.1111/j.1365-2966.2009.15220.x An analysis of the region of the Phocaea dynamical family V. Carruba1,2 1IPD, UNIVAP, Sao˜ Jose´ dos Campos, SP, 12244-000, Brazil 2 Department of Mathematics, UNESP, Guaratingueta,´ SP, Brazil Downloaded from Accepted 2009 June 9. Received 2009 June 9; in original form 2009 March 10 ABSTRACT http://mnras.oxfordjournals.org/ In this work, I conduct a preliminary analysis of the Phocaea family region. I obtain families and clumps in the space of proper elements and proper frequencies, study the taxonomy of the asteroids for which this information is available, analyse the albedo and absolute magnitude distribution of objects in the area, obtain a preliminary estimate of the possible family age, study the cumulative size distribution and collision probabilities of asteroids in the region, the rotation rate distribution and obtain dynamical map of averaged elements and Lyapunov times for grids of objects in the area. Among my results, I identified the first clump visible only in the frequency domain, the at Fundação Coordenação de Aperfeiçoamento Pessoal NÃvel Superior on July 15, 2013 (6246) Komurotoru clump, obtained a higher limit for the possible age of the Phocaea family of 2.2 Byr, identified a class of Phocaea members on Mars-crossing orbits characterized by high Lyapunov times and showed that an apparently stable region on time-scales of 20 Myr near the ν6 secular resonance is chaotic, possibly because of the overlapping of secular resonances in the region. The Phocaea dynamical group seems to be a real S-type collisional family, formed up to 2.2 Byr ago, whose members with a large semimajor axis have been dynamically eroded by the interaction with the local web of mean-motion and secular resonances. Studying the long-term stability of orbits in the chaotic regions and the stability of family and clumps identified in this work remain challenges for future works. Key words: celestial mechanics – minor planets, asteroids. families and clumps in the region were recently identified by Foglia 1 INTRODUCTION & Masi (2004) and Gil-Hutton (2006). Among highly inclined asteroids [asteroids with sin(i) > 0.3, for One of the open questions about the Phocaea family is if the which the analytical theory used to obtain proper elements is not dynamical family is real or an artefact created by the stable island very accurate (Milani & Knezeviˇ c´ (1994)], the asteroids in the region between secular and mean-motion resonances, and the zone region of the Phocaea family are characterized by a very interest- of close encounters with Mars (Knezeviˇ c´ & Milani 2003). For the ing dynamics. The region of the Phocaea family is delimited by purpose of answering this and other questions in this work, I first the 7J:-2A mean-motion resonance at low a, by the 3J:-1A reso- re-obtained dynamical families in the space of proper element and nance at high a andbytheν6 = g − g6 (where g is the secular frequencies (Carruba & Michtchenko 2007, 2009), I studied the frequency of precession of the pericentre, and the suffix 6 refers spectroscopical and Sloan Digital Sky Survey Moving Objects Cat- to Saturn) secular resonance at low i (Knezeviˇ c´ & Milani 2003). alogue, fourth release (SDSS-MOC4) data for the region, reviewed A region of shallow close encounters with Mars appears at e> the current knowledge on geometric albedos and absolute mag- 0.3, but deep close encounters with Mars are made impossible by nitudes, obtained the Yarkovsky isolines, studied the cumulative the Kozai-class protection mechanism (Milani et al. 1989). The distribution and collision probabilities of family members and the Phocaea family itself is characterized by its interaction with the information on asteroid rotation rates, and the role that mean-motion ν6 − ν16 secular resonance [(g − g6) − (s − s6), where s is the and secular resonances have had in shaping the family. secular precession frequency of the asteroid node], and harmonics I found that, with the exception of the family around (19536) of this resonance are also present in the region. To better under- (1999 JM4), formerly associated with the clump around (2860) stand the interaction of the Phocaea family with the (ν6 − ν16) Pasacentennium, the families and clumps found by Gil-Hutton secular resonance, the family was recently obtained in the (n, g, (2006) are now substructures of the Phocaea family. I identified new g − s) frequency domain by Carruba & Michtchenko (2009). Other clumps in a proper element and frequency space, and in particular I found the first ‘frequency-only’ clump around (6246) Komurotoru. A preliminary analysis of Yarkovsky isolines shows that the fam- E-mail: [email protected] ily can be up to 2.2 Byr old. Dynamical maps of the region showed C 2009 The Author. Journal compilation C 2009 RAS An analysis of the region of the Phocaea dynamical family 1513 140 80 N N 0 0 N N min A min B max(N ) max(N ) i 70 i 120 60 100 50 80 min min 40 N N 60 30 Downloaded from 40 20 20 10 0 0 50 100 150 200 250 300 350 400 0.2 0.4 0.6 0.8 1 v [m/s] f ["/yr] cutoff cutoff http://mnras.oxfordjournals.org/ Figure 1. The average number N 0, N min and the maximum number max (Ni) of asteroids as a function of the velocity cut-off (panel A), and of the frequency cut-off (panel B), for the asteroids in the region of the Phocaea family. us that there exist regions that are dynamically stable on the time- is8.AscanbeseeninFig.1,thefactthatmax(Ni) is not much scales of the integration, but yet deprived of asteroids. Such regions, larger than Nmin may be a hint that the families dominate the local all characterized by very low values of Lyapunov times, are either background of objects (see Section 6). associated with Mars-crossing orbits, or are near the border of the For what concerns the domain of proper frequencies, since the ν6 secular resonance separatrix. Understanding the long-term sta- Phocaea group is characterized by its interaction with the (g − s − bility of orbits in the chaotic regions and the stability of family and g6 + s6) resonance, or (ν6 − ν16)(Knezeviˇ c´ & Milani 2003, see clumps identified in this work will be the subject of a next paper on also Section 8), following the approach of Carruba & Michtchenko at Fundação Coordenação de Aperfeiçoamento Pessoal NÃvel Superior on July 15, 2013 this subject. (2009), I determined the family with a metric of the form n 2 2 DYNAMICAL FAMILIES AND CLUMPS f = h + h (g)2 + h ((g − s))2, (2) 1 h 2 3 IN THE REGION 0 = = = I start my analysis of the Phocaea family region by re-deriving clas- (where h1 h2 h3 1) rather than with the ‘standard fre- + sical and frequency families and clumps in the area. I identify the quency metric’ in the (n, g, g s) domain of Carruba & members of the dynamical families in the region using the classical Michtchenko (2007). I determined the nominal frequency cut-off hierarchical clustering method [CHCM; see Bendjoya & Zappala` defined as the average minimum distance between all neighbour- − 2002 and references therein], and the frequency hierarchical clus- ing asteroids in the (n, g, g s) domain, and I found a value of = −1 tering method (FHCM; Carruba & Michtchenko 2007, 2009).1 f 0 0.525 arcsec yr . Fig. 1 (panel B) displays the average num- In identifying asteroid families in the space of proper elements ber N 0, N min and the maximum number max (Ni)ofasteroidsas two parameters are fundamental: the cut-off distance at which the a function of the frequency cut-off defined by equation (2). The −1 value of Nmin corresponding at f 0 = 0.525 arcsec yr is 26. As family members are defined (d0) and the minimum number of ob- observed for the values of Nmin and max (Ni) obtained in the proper jects N min for a cluster to be considered significant. Beauge&Roig´ (2001) define a nominal distance cut-off as the average minimum element domain in the domain of frequencies, I still do not observe distance between all the neighbouring asteroids in the same region a much larger value of max Ni with respect to Nmin.Again,thismay suggest that the local background of the Phocaea family is indeed of the asteroid belt. The value of Nmin is defined by Zappalaetal.` (1995) as dominated by the Phocaea family itself (see Section 6). Since the Phocaea family is by far the most prominent family = + Nmin N0 2 N0, (1) in the region, I started by using the CHCM for this family. Fig. 2 (panel A) displays the number of family members (left ordinate) where N0 is the average number of orbits within a sphere of radius and the differential number dn/dv (right ordinate) as a function d0 at every point of the proper element space. A cluster with a number of objects larger than this critical value is called a clump, of the velocity cut-off. The red vertical line displays the nominal while a family is a cluster with a number of members larger than cut-off according to Beauge´ & Roig (2001). The numbers on the top of the peaks of differential asteroid numbers are related to the 2.5 × N min. As we shall see in more detail in Section 8, the region cluster of objects englobed by the Phocaea family.