1. Figure 3. Inclination instability in the outer Solar system
Figure 3. Inclination instability in the outer Solar system. (Left): orientation angles measured for minor planets in the Solar system. We show all known minor planets with pericentre distances greater than Neptune's orbit (30 au) and with semi-major axes greater than 90 au (small circles) and 150 au [large circles; cf. Trujillo & Sheppard (2014)]. We plot the longitude of ascending node Ω and the angle $i_{{\rm e}} \equiv \arctan ( a_{{\rm y}}, {a_x})$, which represents the orientation of the semi-major axis (or eccentricity vector) within the reference plane. Orbits randomly distributed within a disc would fill out this space; however, the minor planes only occupy a narrow band. Error bars are smaller than symbols. (Right): N-body simulation of the inclination instability in a disc with 100 particles and a total mass M_disc = 10⁻⁴ M. The top panel shows the initial condition for the simulation; orbits randomly distributed within the disc populate the entire Ω–i_e space. During the inclination instability, this distribution collapses to a narrow band in Ω–i_e space, as in seen in the outer Solar system. In all plots, colour indicates the angle of pericentre ω, banding in Ω–i_e space corresponds to clustering in ω.
From: A new inclination instability reshapes Keplerian discs into cones: application to the outer Solar system
Mon Not R Astron Soc Lett. 2016;457(1):L89-L93. doi: /mnrasl/slv203
Mon Not R Astron Soc Lett | © 2016 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

2. Figure 4. Geometric meaning of ‘clustering in ω
Figure 4. Geometric meaning of ‘clustering in ω.’ (Middle): a disc of 20 orbits, all with identical semi-major axis ${\boldsymbol a}$, eccentricity ${\boldsymbol e}$ = 0.8, inclination i = 0.5, and angle of pericentre ω = −0.75. Longitude of ascending node Ω uniformly fills [0, 2π) and is indicated with colour. Discs which cluster in ω represent ‘cones’ with the centre of mass above or below the disc plane, and with each orbit tilting in the same fashion. (Bottom): the same disc, but each orbit is given a random value of ω ∈ [0, 2π); in this (much more generic) case, the disc becomes a thick torus. (Top): orbits for minor planets with semi-major axis between 90 and 110 au; the disc clusters in ω and has a cone shape as seen in the middle panel. (We restrict the range in semi-major axis for clarity, the results look similar for other ranges.)
From: A new inclination instability reshapes Keplerian discs into cones: application to the outer Solar system
Mon Not R Astron Soc Lett. 2016;457(1):L89-L93. doi: /mnrasl/slv203
Mon Not R Astron Soc Lett | © 2016 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

3. Figure 2. Quantitative evolution of the inclination instability
Figure 2. Quantitative evolution of the inclination instability. (Top): exponential growth of the inclination angles i_a and i_b. Lines show the median values of all the orbits and the coloured bands show the width of the distribution enclosing 50 per cent of the orbits. Solid lines indicate positive values, and dashed lines indicate negative values. At early times, the individual angular momentum vectors precess incoherently; this precession implies oscillations in i_a and i_b with |i_a| > |i_b|. The precession averages to zero upon taking the median, revealing exponential growth with |i_b| > |i_a|. The instability is present even at low inclinations (<1°) though this may not be observable in systems of small numbers of bodies. (Middle): evolution of the ratio 〈i_b〉/〈i_a〉. Though 〈i_b〉 and 〈i_a〉 individually grow by more than two orders of magnitude from t ∼ 0–1.5, their ratio remains constant to within a factor of 2. (Bottom): evolution in mean angle of pericentre. We overplot a theoretical prediction. The coloured bands show the width of the distribution enclosing 50 per cent of the orbits. The distribution in ω remains tight even after the exponential growth phase ends.
From: A new inclination instability reshapes Keplerian discs into cones: application to the outer Solar system
Mon Not R Astron Soc Lett. 2016;457(1):L89-L93. doi: /mnrasl/slv203
Mon Not R Astron Soc Lett | © 2016 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

4. Figure 1. (Top): evolution of the inclination angles i_a and i_b (defined in equation 1) for particles initially forming a thin, near-Keplerian disc. Lines show median/mean angles and the shaded region encloses the twenty-fifth and seventy-fifth quantiles, i.e. enclosing 50 per cent of the particles. Orbital inclinations initially grow rapidly due to the inclination instability. Each orbit has approximately the same inclination, with i_a and i_b initially growing together in a fixed ratio <0. After the instability saturates around a time t ∼ 3, the orbits precess coherently about the mean angular momentum axis of the disc. (Bottom): evolution of the angle of pericentre ω. The inclination instability causes the orbits to collapse into a narrow distribution of ω around a time t ∼ 2–3. This clustering in ω is not actively maintained after the instability saturates, and spreads due to differential precession by a time t ∼ 6.5. As in the top panel, the coloured region shows the width of the distribution enclosing half of the orbits. We mark this as green where the width is less than ±55°, the observed scatter for minor planets beyond Neptune (see Section 3). (Inset): angular distribution of ω during the linear, exponentially growing phase from t ∼ 2.5–3 (blue) and in the saturated state (t ∼ 5.5–6, yellow). The asymmetry in ω persists even at late times (t ∼ 11.5–12, grey).
From: A new inclination instability reshapes Keplerian discs into cones: application to the outer Solar system
Mon Not R Astron Soc Lett. 2016;457(1):L89-L93. doi: /mnrasl/slv203
Mon Not R Astron Soc Lett | © 2016 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society