Tadpole type motion of charged dust in the Lagrange problem with planet Jupiter

The paper’s claims are supported by numerical experiments and a simplified Gauss-averaged model: (i) the nominal 1:1 location a ≃ aJ(1−β)^{1/3} is explicitly derived and used in the minimization procedure for ‘minimum-libration amplitude’ solutions, with observed β-dependent shifts and an L4–L5 asymmetry consistent with earlier work ; (ii) the resonance width δa decreases with β and differs between L4 and L5 (uncharged case), as shown in Fig. 11 and discussed in the text ; (iii) with charge, the paper documents secular i–Ω oscillations and identifies turning longitudes at Ω=Ω0 and Ω=Ω0+180°, matching a simplified analytical argument ; (iv) ‘spikes’ in σ-libration amplitude occur when the nodal period commensurates with the σ-libration period, and the nodal period scales inversely with γ as observed in the simulations and noted by the authors ; (v) the time of temporary capture is shorter for charged dust than neutral across the surveyed regimes . In contrast, the candidate model asserts a nonzero secular along-track Lorentz forcing ⟨TL⟩ ∼ O(γ i) in the Parker-field case with BN=0, which the paper explicitly neglects (and prior work shows to average out over an orbital period), relying instead on the normal component’s role for i,Ω dynamics . The candidate also derives a commensurability criterion that effectively makes the nodal precession rate sΩ independent of γ (via i_rms ∝ γ), contradicting the paper’s (and prior) finding that the nodal period is ∝ 1/γ, hence sΩ ∝ γ . These are central to items (2), (4)–(6) in the candidate’s outline, so we judge the paper’s account correct and the model’s key dynamical claims flawed.