Identifying the population of stable &#957;6 resonant asteroids using large databases
Authors: Carruba Valerio , Aljbaae Safwan , Domingos Rita C. , Huaman Mariela , Martins Bruno 
Published: September 2022

&lt;p&gt;&lt;strong&gt;1. Introduction&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;&lt;br /&gt;Linear secular resonances occur when there is a commensurability between the frequency of precession of an asteroid perihelion, g, or node, s, and that of a planet. The &amp;#957;&lt;sub&gt;6&lt;/sub&gt; involves a relationship between the frequency g of an asteroid and the g6 frequency of Saturn. Since it is a pericenter resonance, it can increase most asteroids&amp;#8217; eccentricities to planet-crossing levels. It can, therefore, destabilize most of the bodies that interact with it, and it is a major source of NEAs through Yarkovsky effect [2]. Despite this fact, stable orbital configurations exist inside the &amp;#957;&lt;sub&gt;6 &lt;/sub&gt;resonance. In such configurations, the resonant angle &amp;#963; = &amp;#982;&amp;#160; &amp;#8722;&amp;#982;&lt;sub&gt;6&lt;/sub&gt;, with &amp;#982;&amp;#160; the longitude of pericenter of the asteroid and &amp;#982;&lt;sub&gt;6&lt;/sub&gt; that of Saturn, can oscillate around either 0&amp;#9702; or 180&amp;#9702;. In the first case, we have an &amp;#8220;aligned libration&amp;#8221;, since the pericenters of both Saturn and the asteroid are pointing in the same direction. For the second case, we have an &amp;#8220;anti-aligned libration&amp;#8221;.&lt;/p&gt;
&lt;p&gt;[4] identified the first case of an asteroid family completely made of asteroids in anti-aligned libration states, that of 1222 Tina. [6] extended this analysis to the whole main-belt, and found the first seven asteroids in aligned states of the same resonance. Here, we extended the previous analysis using the largest database of asteroid proper yet obtained.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;2. Identification of the resonant population using large databases&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;872&quot; height=&quot;412&quot; /&gt;&lt;/p&gt;
&lt;p&gt;To identify the population of &amp;#957;&lt;sub&gt;6 &lt;/sub&gt;resonators, first, we need to select the asteroids most likely to be affected by this resonance. For this purpose, we use the criterion described in [6] and the approach outlined in [3].&amp;#160; We identified 15 asteroids on aligned orbital configurations, and 1713 on anti-aligned ones (see figure (1)). We then proceeded searching for asteroid groups among the newly identified anti-aligned population using the Hierarchical Clustering Method (HCM) [1]. Our results are shown in figures (2). We identify a new group around the asteroid 12988 Tiffanykapler.&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;904&quot; height=&quot;346&quot; /&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;3. 12988 Tiffanykapler: the first &lt;/strong&gt;&lt;strong&gt;young family in the &amp;#957;&lt;sub&gt;6 &lt;/sub&gt;secular &lt;/strong&gt;&lt;strong&gt;resonance&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;910&quot; height=&quot;383&quot; /&gt;&lt;br /&gt;Young asteroid families can be dated using methods based on numerical integrations of family members in the past, like the Backward Integration Method (BIM, [7]) and the Close Encounters Method (CEM, [5]). In the BIM method, from time-reversal numerical simulations, past discrepancies in the longitudes of node &amp;#937; of family members with regard to those of the claimed parent body are obtained. These differences should converge to values approaching zero at the time of family formation. The CEM approach works by integrating into the past multiple clones of the parent body and of the other family members. Close encounters that occur at low relative distances and speeds between two clones are recorded, and the median value of these times is used to estimate the asteroid pair&amp;#8217;s age.&amp;#160; The results of both methods are shown in figure (3). The BIM method suggests a possible solution for the 12988 group in the time range from 3.3 to 6.5 Myr. The weighted average of the age determination of all asteroid pairs in the family, one example of which is shown in the right panel of figure (3), provides an age estimate for the family of 3.05 &amp;#177; 1.15 Myr.&lt;/p&gt;
&lt;p&gt;As discussed in [4], at the simplest level of perturbation theory the quantity:&lt;br /&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;318&quot; height=&quot;45&quot; /&gt;&lt;/p&gt;
&lt;p&gt;is preserved for asteroids in the &amp;#957;&lt;sub&gt;6&lt;/sub&gt; resonance in the conservative case. The preservation of the original values of K&lt;sub&gt;2&lt;/sub&gt;' permits to estimate the initial ejection velocity field with a method not generally available for non-resonant families. Our analysis yields a value of the ejection velocity parameter V&lt;sub&gt;EJ&lt;/sub&gt; = 15&lt;sup&gt;+6&lt;/sup&gt;&lt;sub&gt;-3&lt;/sub&gt; m/s.&lt;/p&gt;
&lt;p&gt;&lt;br /&gt;&lt;strong&gt;4. Summary and Conclusions&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; width=&quot;894&quot; height=&quot;389&quot; /&gt;&lt;br /&gt;In this work, we were able to obtain a sample of 15 asteroids on aligned orbits and 1669 asteroids on anti-aligned orbits, the largest database so far for this asteroidal population. Two new asteroid groups were identified for the first time in this work: those of Tiffanykapler in the inner main-belt and 138605 QW177 in the outer belt. Tiffanykapler is the first young asteroid family in a linear resonant configuration ever to be found. Combining methods based on backward numerical simulations and constraints from resonant dynamics, we found that the family has an age of 3.05 &amp;#177; 1.15 Myr and an ejection velocity field parameter of V&lt;sub&gt;EJ&lt;/sub&gt; = 15&lt;sup&gt;+6&lt;/sup&gt;&lt;sub&gt;-3&lt;/sub&gt; m/s. Finally, using resonant proper elements ([4]), we also identified a highly eccentric population of asteroids near the Tina family, that are likely to be part of its halo. It is the first example of a &amp;#8220;resonant halo&amp;#8221;.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Acknowledgements&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;We thank the Brazilian National Research Council (CNPq, grant 301577/2017-0), and the Foundation for Research Support of S&amp;#227;o Paulo state&lt;br /&gt;(FAPESP, grant 2016/024561-0). More information on this work is available on Carruba et al. 2022, MNRAS, under review, available on ArXiv: 2203.15763.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;
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