Design of a low-thrust gravity-assisted rendezvous trajectory to Halley’s comet

The paper derives ΔV3^2 = K + 2 v∞(d1 cos δ + d2 sin δ cos α + d3 sin δ sin α), then shows the first-order conditions tan α = d3/d2 and tan δ = (d2 cos α + d3 sin α)/d1, and uses second-derivative checks to confirm a minimum; if δopt exceeds the pericenter cap, the boundary δ = δmax is optimal; finally, given t2, the GA geometry is determined by {t2, t4}, reducing the outer search to 1D in t4 (Brent) over the June 2053–May 2061 window. These are precisely the steps and conclusions in the candidate solution, including the α-minimization via T(α)=d2 cos α + d3 sin α, the reduction to a single harmonic in δ, the sign conditions for the Hessian, and the boundary-case logic. The algebra, conditions, and dimensionality reduction match the paper’s Eq. (17)–(26) and the surrounding discussion, as well as the mission-design workflow.