## Mean motion resonances in trans-neptunian region

The following figure shows the two-body mean motion resonances with Jupiter (white), Saturn (red), Uranus (green), and Neptune (blue). Three lines per resonance are plotted: one line showing the resonant center and two lines denoting the resonant width. Inclinations assumed zero, planetary eccentricities are 0.048, 0.044, 0.038, and 0.009 (equal to mean values) andvarpi-varpi_Pl = PI ,wherevarpiare perihelion longitudes. This value of perihelion longitudes corresponds to maximal resonant widths. All resonances up to order 50 are shown. The resonances become large near planet crossing limit.

Using the Chirikov criterion of resonace overlap, we compute the onset of chaos due to mean motion resonances overlap. This is done by an algorithm which searches for separatrix intersections in the above figure. The following figure shows the limiting excentricity (blue line - in fact we repeated the computation in various models - green and red color - but these nearly overlap with the blue curve showing an insensivity of the limiting eccentricity to planet's eccentricity and perihelion longitudes configuration). At given semimajor axis, all excentricities above this line are strongly chaotic. Note that for small semimajor axis the limiting perihelion distance is 35 AU (in good corespondence with numerical works, e.g. Duncan at al. 1995) while at large semimajor axis this limit is near

q=30 AU. The later may happen due to the fact that only the mean motion resonances with order<50 are taken into account (see later). For comparison, the white line shows a curve, empirically obtained by Gladman et al. 2000 for 17deg inclinations (similar to 2000 CR105). This curve shows the lower boundary of a region where most trajectories have the Lyapunov time shorter than 20 orbital periods.

Now, we zoom out the region near 2000CR105 accounting for even higher order resonances (we take into account all resonances up to order 100). Blue resonances are resonances with Neptune, green ones are with Uranus. The mean motion resonances with Jupiter and Saturn (red) are negligible in this region. There are two panels corresponding to two different phases of secular angles. The difference is negligible (unlike what happens with less remote mean motion resonances). The prominent resonances are 1:18 (206.8 AU), 1:19 (214.4 AU), and 1:20 (221.8 AU) with Neptune. The pattern of resonances has the following rule: the one at between 1N:18 and 1N:19 is 2N:35, i.e. (1+1)N:(18+19). The orbital elements of 2000 CR105 and their error bars are shown in white. The white line corresponds to Neptune crossing limit.

Note the red horizontal curve which is q=30+0.09(A-30) , i.e. the lower boundary of orbits having Lyapunov times shorter than 20 orbital periods. There is an amazingly good correspondence between this corve obtained empirically by Gladman et al. (2001) and the intersection of different resonances!!! This shows that chaos observed in the numerical simulation can be interpreted in terms of the overlap criterion of mean motion resonances of very high orders.But even under the red line, the mean motion resonance are likely to be chaotic. One can roughly estimate their Lyapunov time from the magic formula: T_lyap = sqrt(32/3)*a_res / (da * |k|) (in time units of an orbital period), where a_res is the resonant semimajor axis, da is the resonant width and k is integer in front of small bodies mean longitude in the resonant angle (i.e. for 1:18 resonance, |k|=18). If

T_lyap < 20orbital periods, then appr.da > 35 / |k|(assuminga_res=215 AU), which means that for resonances where |k|~20, da must be larger than about 1.75 AU. This does not happen. For this reason, it is expected that the transition from regular to chaotic motion at the red line is abrupt.2000 CR105 can be anywhere in the shown interval of the semimajor axis: in addition to a uncertainity of orbital elements, there is a large difference between an instantaneous position of 2000 CR105 shown here (at J2000) and its mean value, mainly because of the perturbations by Jupiter.

## Another interesting object is 1995 TL8

As Morby pointed to me, this object is `deeper' in `eccentric' classical Kuiper belt than 2000 CR105. This can be see in the following figure:

1995 TL8 is located in eccentricities where MMRs do not overlap (coversely to 2000 CR105 which is in the chaotic overlap zone). Its instantaneous orbital elements are close to the 3:7 resonance with Neptune at 52.97 AU. Is 1995 TL8 a resonant body? A short integration should answer this question by computation of the mean semimajor axis. The osculating semimajor axis places this object near separatrices of the 3:7 MMR with Neptune:

The above figure shows trajectories for

e=0.24near the 3:7 MMR with Neptune (the eccentricity of 1995 TL8 is ~0.238). The resonant angle defined as3/4 lambda_N - 7/4 lambda + varpiis on X-axis (in radians) whilea - 52.969AU (which is the resonant semimajor axis of the 3:7 MMR) is on Y-axis. Position of 1995 TL8 is marked by asterisk. If there is no large difference between osculating and mean semimajor axes, the nominal orbit of 1995 TL8 seems to indicate that this is a scattered disk object which sticked to 3N:7 and decreased eccentricity. What is the error of the nominalafitted to observations?

Last revised: May 1, 2001

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