Extended Data Fig. 6: Topology of the Quaoar 1/3 Spin-Orbit Resonance (SOR).
From: A dense ring of the trans-Neptunian object Quaoar outside its Roche limit
The bottom graph shows the maximum eccentricity \({e}_{\max }\) reached by a particle starting on an initially circular orbit of semi-major axis a perturbed by the Quaoar 1/3 SOR resonance. This resonance is driven by a mass anomaly whose amplitude is quantified by the dimensionless parameter ϵ1/3. The exact resonance radius a1/3 is marked by the dashed vertical tick mark. The top plots show the phase portraits in the eccentricity vector space (X, Y) corresponding to particular values of a, with \(X=e\cos ({\phi }_{1/3})\), \(Y=e\sin ({\phi }_{1/3})\), where e is the orbital eccentricity and \({\phi }_{1/3}=(-{\lambda }^{{\prime} }+3\lambda -2\varpi )/2\) is the resonant critical angle, see Methods for details. In an interval of width W = (16∣ϵ1/3∣/3)a1/3 in semi-major axis centerd on the resonance, the origin of the phase portrait is an unstable hyperbolic point. Particles are then forced to reach a maximum eccentricity \({e}_{\max }=\sqrt{4/3}\sqrt{(W/2-\Delta a)/{a}_{1/3}}\), where Δa = a − a1/3 is the initial distance of the particle to the resonance. The value of \({e}_{\max }\) peaks at \({e}_{{\rm{peak}},1/3}=(8/3)\sqrt{| \,{{\epsilon }}_{1/3}\,| /3}\) for Δa = − (8/9)∣ϵ1/3∣a1/3. Outside this interval, \({e}_{\max }=0\). Units are arbitrary in all the plots.
