All Figures

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f1.ps}\end{figure}$ Figure 1: The Vesta region projected on the (proper a-proper e) plane (above) and on the (proper a-proper I) plane (below). On the left, all the numbered asteroids. On the right, only those larger than 7 km (assuming the same albedo as Vesta).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f2.ps}\end{figure}$ Figure 2: Above: size distribution of the asteroids in the Vesta region (within the box given in the text); only the asteroids with diameter <7 km have been included, since the larger ones are mostly interlopers, as can be seen from Fig. 1. Below: for the same asteroids, distribution of the escape velocity, computed from the difference in proper elements, with respect to the family parent body, that is Vesta. Given the distribution, it is unlikely that it corresponds to the original escape velocity: it is very likely that the relative velocities have been increased by subsequent dynamical evolution.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f3.ps}\end{figure}$ Figure 3: The Phocaea group in proper elements space. Upper left: proper a and proper e; upper right: proper a and proper $\sin I$ ; lower left: proper e and proper $\sin I$ . Lower right: histogram of accuracies of the proper elements, measured as the RMS value of changes of both proper e and $\sin I$ in the running box test.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f4.ps}\end{figure}$ Figure 4: Proper frequency g of the longitude of perihelion and s of the longitude of the node, in arcsec/yr. Note the alignment at the libration center due to averaging.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f5.ps}\end{figure}$ Figure 5: Trans-Neptunians (stars) and main belt asteroids brighter than absolute magnitude 8.5 (dots) plotted together on the proper a-proper e plane. For the main belt, the dual semimajor axis has been used (see formula in the text). This representation shows that the known trans-Neptunians correspond only to the outer edge of the main belt, but it is not clear whether a "main belt'' beyond the 1/2 resonance with Neptune does exist.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f6.ps}\end{figure}$ Figure 6: Locations of main mean motion and secular resonances in the trans-Neptunian region in the a-e plane; the frequencies have been computed with the analytical theory for $I=3^\circ$ . The stars show the proper elements of numbered and multi-opposition trans-Neptunians.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f7.ps}\end{figure}$ Figure 7: Locations of main mean motion and secular resonances in the trans-Neptunian region in the a-i plane; the frequencies have been computed for e=0.04.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f8.ps}\end{figure}$ Figure 8: For the multi-opposition trans-Neptunian 1999 CL₁₅₈ the numerical propagation of the orbit for 10 Myr resulted in macroscopic instability. The semimajor axis (upper left) shows large changes, in particular after 6 million yr, when the eccentricity (upper right) begins to increase to Neptune-crossing values. At that time the inclination (lower left) begins irregular oscillations, and the logarithm of the divergence of infinitesimally nearby orbits (lower right) grows to very large values, indicating positive Lyapounov exponents.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f9.ps}\end{figure}$ Figure 9: For the numbered trans-Neptunian (15 760), that is 1992 QB₁, the filtered semimajor axis (upper left) undergoes only small oscillations. The eccentricity (upper right) and inclination (lower left) show the effect of the oscillations with frequencies g-g₈, s-s₈ superimposed on a longer period oscillation, which can be attributed to the small divisor g+s-g₈-s₈: the corresponding critical argument (lower right) circulates, but it performs only 1.5 revolutions in 100 million yr.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f1.ps}\end{figure}$	Figure 1: The Vesta region projected on the (proper a-proper e) plane (above) and on the (proper a-proper I) plane (below). On the left, all the numbered asteroids. On the right, only those larger than 7 km (assuming the same albedo as Vesta).
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f3.ps}\end{figure}$	Figure 3: The Phocaea group in proper elements space. Upper left: proper a and proper e; upper right: proper a and proper $\sin I$ ; lower left: proper e and proper $\sin I$ . Lower right: histogram of accuracies of the proper elements, measured as the RMS value of changes of both proper e and $\sin I$ in the running box test.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f4.ps}\end{figure}$	Figure 4: Proper frequency g of the longitude of perihelion and s of the longitude of the node, in arcsec/yr. Note the alignment at the libration center due to averaging.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f6.ps}\end{figure}$	Figure 6: Locations of main mean motion and secular resonances in the trans-Neptunian region in the a-e plane; the frequencies have been computed with the analytical theory for $I=3^\circ$ . The stars show the proper elements of numbered and multi-opposition trans-Neptunians.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{MS3507f7.ps}\end{figure}$	Figure 7: Locations of main mean motion and secular resonances in the trans-Neptunian region in the a-i plane; the frequencies have been computed for e=0.04.
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