tokens: found that Centaurs can produce HTCs, in particular they are trapped very often in the 1:6 MMR with Jupiter (energy x ≃ 0.058 au −1 ) which coincides with the clustering in the observed energy distribution of HTCs (Fig. 3 ). It is worth noting that our sample of observed Centaurs contains a few objects in retrograde orbits. To produce both the observed HTCs and JFCs, a flattened source is required but it should contain a small fraction of high-inclination and retrograde comets. A near-ecliptic, prograde population of Centaurs ( i ≲ 25°) will essentially lead to a near-Earth population of bodies with Tisserand parameters T J > 2, typical of JFCs (Levison Duncan 1997 ). It is not the purpose of this paper to discuss the ultimate origin of Centaurs, though we can say a few words. A flat source in the outer planetary region such as the Scattered Disk will essentially produce a population of Centaurs in prograde, low-inlination orbits (Di Sisto Brunini 2007 ), though it may be possible that some SDOs scattered outwards attain Oort cloud distances where galactic tides switch their orbits from prograde to retrograde (Levison et al. 2006 ). Other possibility is that some Centaurs come from the capture of Oort cloud comets in the outer planetary region as discussed by Emel'yanenko Bailey ( 1998 ) and Emel'yanenko, Asher Bailey ( 2005 ). In the end, some HTCs may come from the Oort cloud but through a different dynamical route that direct insertion of Oort cloud comets in the near-Earth or the Jupiter region. In order to check the results found for the simulated HTCs produced from the evolution of the Centaur sample, we also studied the dynamical evolution for 1 Myr to the past and to the future of 15 observed HTCs with q 1.3 au and energies 0.04 x 0.08 au −1 (orbital periods 44.2 P 125 yr). We confirm that HTCs do not diffuse randomly in a but tend to be captured in the 1:6 exterior resonance with Jupiter generating a concentration at x ∼ 0.058 au −1 . This peak is clearly shown in Fig. 17 which is a histogram of frequencies of the average 〈 a 〉 over periods of 500 yr. Figure 17. Frequency distribution of the time-average 〈 a 〉 (over 500 yr) for the observed sample of 15 HTCs with q 1.3 au and energies 0.04 x 0.08 au −1 (periods 44.2 P 125 yr) integrated over 10 6 yr in the past and in the future. Other remarkable feature is that even though we obtain about 4.7 per cent of LPCs that end up colliding with the Sun for a purely dynamical evolution, this percentage is practically null when we consider appropriate physical lifetimes ( N phys ( q = 1) ≲ 500 revolutions). The reason is that LPCs approach the Sun in a smooth way through a multi-step process, that takes hundreds to thousands revolutions, so it is very likely that such comets will disintegrate well before reaching sungrazing states. The inclination distribution of the old LPCs shows clear departures from randomness that can be described as a slight predominance of prograde orbits that is
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