Left Column: Orbital evolution of NGC 6569 in our barred MW model (integrated with GALA). Top: Galactocentric cylindrical radius R(t). Middle: Bar-frame azimuth ϕ_rot(t). Bottom: Instantaneous bar-frame angular rate dϕ/dt (solid), its time–average (dashed), and the inertial-frame circular frequency Ω (dotted). The listed characteristic periods (T_ϕ inertial, T_rel bar frame, T_R, T_Z) are measured from the same integration. Quasiperiodic beating reflects pericenter passages and the coupling with the rotating bar. Right Column: Time variation of the Jacobi (Roche) radius (r_J) and the critical Jacobi (E_J,crit) energy along the same orbit and in the same rotating bar frame. Top: r_J(t) in arcminutes at D = 10.53 kpc. Bottom: E_J,crit(t), computed from the effective potential Φ_eff evaluated at the L1/L2 saddle points. The “breathing” of r_J (smaller near pericenter, larger near apocenter) is mirrored by the modulation of E_J,crit. These curves are used to assess boundness by comparing stellar E_J to E_J,crit(t).