3.2 Resonant and ``pathological'' cases

Another group of troublesome cases is more heterogeneous. It again consists mostly of resonant asteroids, but there are also a few asteroids for which the determination of some proper value(s) was seriously deteriorated due to other reasons. The approach we adopted for these bodies is based upon the idea that all the data which could have been computed, even with a low accuracy, might be useful at least for statistical purposes; therefore, we decided to keep all these bodies in the files, correcting very few of them in a special way, whenever this was necessary. The users are warned that the consultation of the files containing the RMS and the excursion in the runnig box test is mandatory, at least before drawing any conclusion from the values of the proper elements. It is to be understood that we could not analyze in detail every troublesome case found by our automated procedure. Thus, for most of the bodies we only report the deteriorated quantities. We defined as ``pathological'' all the cases for which one or more of the following conditions were met:

$$\begin{array}{rrrrr} \sigma a_p>0.01 & \sigma e_p>0.1 & \sig... ...Delta\sin I_p>0.03 & \Delta g>30 & \Delta s >30 \end{array}$$

Here $\sigma$ and $\Delta$ are standard deviation and maximum excursion; the statistics for frequencies of perihelion g and node s are given in arcsec/yr, these for the proper semimajor axis a_p in AU. In Table 2 we give a list of objects for which one or more of the proper values, as derived from the basic 2 Myr integration, exceeded the above thresholds.

 

Table 2: ``Pathological'' cases for which some proper value(s) had very large errors in 2 Myr run. The columns contain asteroid identification, statistical parameters found to be excessively large, and a remark on the specific dynamical features of the bodies. See the text for notation and units.

Asteroid	Erroneous values	Remark
531	$\Delta g$
759	$\Delta g$
1263	$\sigma g$; $\Delta g$
3480	$\sigma a$
3579	$\Delta e$; $\sigma g$; $\Delta g$
4112	$\sigma g$
4915	$\Delta a$
5222	$\Delta g$
5330	$\Delta \sin I$; $\sigma g$; $\Delta g$
5765	$\sigma g$
6626	$\sigma a$
7306	$\sigma g$; $\Delta g$
7598	$\Delta a$
8009	$\Delta g$
1977RD	$\sigma g$
1979OA	$\Delta \sin I$
1981EJ40	$\sigma g$
1981QN3	$\sigma g$	2/1
1984LK	$\sigma a$; $\sigma s$; $\Delta s$
1987UW	$\Delta g$
1991QG	$\Delta a$
1991UC	$\sigma g$	e=0.0031
1993FB35	$\sigma a$; $\sigma s$; $\Delta s$
1993QN	$\Delta g$
1993UT	$\Delta \sin I$; $\sigma g$; $\Delta g$
1993VV7	$\Delta e$; $\sigma g$; $\Delta g$
1994TA1	$\sigma g$; $\Delta g$
1995SU32	$\sigma g$; $\Delta g$
1998QX	$\Delta e$; $\sigma g$
2705P-L	$\Delta a$
3230T-2	$\sigma g$	2/1

For a comparison we show a similar table (Table 3) with the ``pathological'' data detected in 10 Myr runs. Here we have chosen to include only asteroids for which one or more of the proper elements had standard deviation in excess of the adopted threshold. In the final output there are additional 20 objects with ``pathological'' standard deviations in frequencies, 9 objects with maximum excursion in frequency above the critical value, and 73 objects for which the maximum excursions of elements were found to be above the adopted threshold.

 

Table 3: The same as Table 2, but for the 10 Myr integration. Included are only objects with excessive standard deviations of proper elements.

Asteroid	Erroneous values	Remark
1301	$\sigma e$; $\sigma g$	Pallas
4177	$\sigma \sin I$	2:1
4997	$\sigma e$; $\sigma g$	g=s; Pallas ?
1979OA	$\sigma e$; $\sigma g$; $\sigma s$	$g=g_6; T_L\approx 4500$
1981EJ40	$\sigma e$; $\sigma g$	Pallas
1991QG	$\sigma a$; $\sigma g$; $\sigma s$	$T_L\approx 6000$
1993SQ3	$\sigma a$; $\sigma \sin I$; $\sigma g$	2:1
1993TC23	$\sigma a$; $\sigma g$	$T_L\approx 6300$
1993UT	$\sigma e$; $\sigma g$	Pallas
1995CT1	$\sigma e$; $\sigma g$	Pallas
1996KL1	$\sigma a$	$T_L\approx 2500$
2705P-L	$\sigma a$	$T_L\approx 11000$

Not all of the asteroids with large standard deviations of proper elements listed in Table 2 appear again in Table 3. For some cases, namely 3480, 6626, and 1984 LK, this is because they went hyperbolic, as discussed in Section 3.1. Many other bodies from Table 2 do not qualify for the ``pathological'' category in the extended run because of the tighter definition (based on the RMS only): e.g., the numbered asteroids 531, 3579, 4915, 5330, 7598. However, there are indeed cases in which the standard deviation of the proepr elements was improved in the longer integration: e.g. for 1993 FB35 $\sigma a_p$ was improved. The running boxes of the 10 Myr integration being 2 Myr, twice as long as the ones of the shorter integration, the decrease of the instability of some proper element indicates that averaging over an extended time span better removes some long periodic perturbation. Overall, however, we found more ``pathological'' cases in the long run than in the short run. This could be expected since the deterioration occurred for some chaotic asteroids and for a number of asteroids near secular resonances. Thus in the long run the perturbation builds up more effectively; this happens more often for large eccentricities and/or inclination. As an example, a number of members of the high inclination Pallas family originally discovered by Lemaitre and Morbidelli ([1994]) appear in our Table 3. As a by-product of our analysis, we have identified 12 new potential members of this family: 5222, 5234, 8009, 1977 RD, 1981 EJ40, 1993 UT, 1993 VV7, 1994 PP, 1994 TA1, 1995 CT1, 1998 QX, 1998 QZ85. However, this is possibly not a complete list, also because a number of high eccentricity objects have been a-priori eliminated from our integrations due to the osculating perihelion distance criterion (Section 1). It is obvious that some dynamical features (e.g. mean motion and secular resonances) limiting the accuracy of the analytical theories correspond to real effects in the exact dynamics, and therefore have to appear also in the numerical integration output. Thus the derived elements are as good as they can be, given that the definition of proper elements must be uniform to satisfy the basic requirements of the family identification procedures. Our synthetic theory makes use of the averaging principle, which is well known not to work for resonant asteroids. As a typical example of the outcome of an inappropriate application of averaging, and of the ``erroneous'' definition of a proper element, let us mention the Griqua group of asteroids, located in the 2:1 mean motion resonance with Jupiter. As can be seen in Figures 7 and 8 (Section 3.4), all these bodies appear at the libration center, that is, as if they all had the same proper semimajor axis of $\simeq 3.28$ AU. Thus, even if these bodies are kept in the files, one should not use these averaged semimajor axes for family identification purposes. A well-known solution to the problem of resonant asteroids is to adopt some other proper value, taking into account the resonance in its very definition, instead of the usual one. For example, the libration amplitude of the semimajor axis could replace the averaged semimajor axis (see, e.g., Milani [1993]), but then the question is how to use different proper elements (of presumably different accuracy) in the family identification procedure. Even if the bodies are only close to strong resonances the accuracy of the determination of proper elements is degraded and the same problem applies (see Milani and Knezevic [1999], for an attempt to improve upon this situation). Another approach is to compute special ``resonant proper elements'' (Morbidelli et al. [1995]), but these do not belong to the same 3D-space; such proper elements can be very useful locally (to study dynamics), but cannot be used for large scale structures (e.g., to identify families) because of the discontinuities and/or topological changes. Let us just mention here that we shall later (Section 3.5) revisit the problem of resonant asteroids, to discuss the leakage of asteroid families members due to chaotic diffusion along the resonances. There are other reasons why synthetic proper elements and frequencies have been poorly determined in some special cases. For asteroids with very low eccentricities and high inclinations ( $\sin I \geq 0.3$) two problems simultaneously affect the computation of proper elements. In the real dynamics, strong secular resonances (with frequencies either g-g₅, g-g₆, g-g₇ or 2g-2s near zero) affect the determination of the proper values. At the same time the argument of the proper mode may be difficult to compute from the time series (k(t),h(t)) because the point (k,h) may sometimes be too close to zero. Both effects combine in an inaccurate computation of the proper frequency g; the proper value e_p is nevertheless accurate. For example, this appears to be the case with the asteroid 1991 UC, which has so low proper eccentricity ( e_p=0.0031) that an automatic computation of the proper longitude of perihelion was not possible. However, we have not performed any ``manual intervention'' in such cases, both because it would necessarily introduce subjective choices in the very definition of proper elements, and because it would make our procedure impossible to use in the context of an automated information services such as AstDyS (and the companion information system NEODyS http://newton.dm.unipi.it/neodys/). Some other cases of inaccurate computations of proper elements and/or Lyapunov exponents are the result of very chaotic motion and macroscopic instabilities. In few such cases this forced us to manually correct some wild results. As an example, for the asteroid 1991 QG an output format overflow occurred in the 10 Myr integration for the maximum excursions of the nodal frequency and of the mean longitude residuals with respect to the linear fit (we simply set these two values to zero keeping the rest of the data). Note that the semimajor axis of this asteroid was very poorly determined even in the short run (Table 2), and that it became even worse in the extended one: we are dealing with a strongly chaotic object (Lyapunov time on the order of few thousand years), and such wild oscillations are not surprising.