A study of the high-inclination population in the Kuiper belt - II. The Twotinos

As the second part of our study, in this paper we proceed to explore the dynamics of the high-inclination Twotinos in the 1:2 Neptune mean motion resonance (NMMR). Depending on the inclination $i$, we show the existence of two critical eccentricities $e_a(i)$ and $e_c(i)$, which are lower limits of the eccentricity $e$ for the resonant angle $\sigma$ to exhibit libration and asymmetric libration, respectively. Accordingly, we have determined the libration centres ${\sigma }_0$ for inclined orbits, which are strongly dependent on $i$. With initial $\sigma ={\sigma }_0$ on a fine grid of $(e,i)$, the stability of orbits in the 1:2 NMMR is probed by 4-Gyr integrations. It is shown that symmetric librators are totally unstable for $i\ge {30}^{\circ }$; while stable asymmetric librators exist for $i$ up to ${90}^{\circ }$. We further investigate the 1:2 NMMR capture and retention of planetesimals with initial inclinations $i_0\le {90}^{\circ }$ in the planet migration model using a time-scale of $2\times {10}^7$ yr. We find that: (1) the capture efficiency of the 1:2 NMMR decreases drastically with the increase of $i_0$, and it goes to 0 when $i_0\ge {60}^{\circ }$; (2) the probability of discovering Twotinos with $i>{25}^{\circ }$, beyond observed values, is roughly estimated to be $\le 0.1$ per cent; (3) more particles are captured into the leading rather than the trailing asymmetric resonance for $i_0\le {10}^{\circ }$, but this number difference appears to be the opposite at $i_0={20}^{\circ }$ and is continuously varying for even larger $i_0$; (4) captured Twotinos residing in the trailing resonance or having $i>{15}^{\circ }$ are practically outside the Kozai mechanism, like currently observed samples.