Direct low-energy trajectories to Near-Earth Objects

The paper proposes a patched Sun–Earth CR3BP/heliocentric-2BP method and explicitly states that, for a fixed three‑body state at the circle of influence (CI), the heliocentric osculating semimajor axis, eccentricity, and true anomaly depend only on the CI state, while the longitude of perihelion rotates linearly with the Sun–Earth mean motion; it also gives the relation ω_S/C(t2) = ω_S/C(t1) + n_ES (t2 − t1) and reduces the planar rendezvous search to finding intersections of confocal ellipses, using sampling Δω = 0.4°, a phasing tolerance ω_e = 0.02°, and a single‑burn limit ΔV# ≤ 1500 m/s, with an optional two‑impulse Lambert refinement and the practical observation that TO‑based departures often reach the CI faster than MT‑based ones. These claims appear verbatim in the manuscript and figures and are used in large‑scale enumerations of opportunities. The model’s solution reproduces the same method but adds compact derivations of the invariance of (a, e, ν_CI) under epoch shift and a clean radial‑interval existence proof for ellipse intersections; it also carries over the same sampling and acceptance thresholds and the Lambert refinement. No conflicts were found; the model simply fills in proofs the paper states or sketches. See the paper’s statements on patched CR3BP/2BP and ellipse intersections, the epoch–orientation relation and formula, the CI radius, the neglect of the Sun–barycenter offset in the transformation, the sampling tolerances and ΔV cap, and the TO vs MT time‑to‑CI observation .