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The science
What a gravity assist really does
Seen from the planet, a flyby is an ordinary two-body hyperbola: the probe arrives with speed v∞ and leaves with the same speed, but its velocity vector is rotated by the deflection angle δ = 2 arcsin(1/e), where the eccentricity is e = 1 + rp v∞2/(G mp) and rp is the closest approach. Seen from the Sun, the planet is moving with velocity u, so the probe's velocity changes from vin = u + v∞,in to vout = u + v∞,out. Rotating v∞ toward the planet's direction of motion adds energy; rotating it the other way removes energy. The maximum speed change is Δv = 2 v∞ sin(δ/2). Energy is taken from the planet's orbital motion, which changes by an immeasurably small amount because mp ≪ Msun.
The model: circular restricted three-body problem
Units: the Sun-planet distance is 1, the total mass is 1 and G = 1, so the planet circles with angular rate 1 (period 2π). The Sun (mass 1-μ) and planet (mass μ) move on exact circles about their barycentre; the probe is a massless test body integrated with an adaptive fourth-order Runge-Kutta step that shrinks near the planet. In the rotating frame the quantity C = x2 + y2 + 2(1-μ)/r1 + 2μ/r2 - v2 (the Jacobi constant) is exactly conserved, so its drift measures the integration error. The "energy change" shown is v2/2 - GM/r about the Sun, which is not conserved: that change is the assist.
Honest simplifications
The planet mass ratio is exaggerated by default (μ = 0.01, about ten times Jupiter) so the effect is visible in one orbit; select the Jupiter preset for the real ratio. The motion is two-dimensional (the orbital plane), the planet is on a circular orbit, there is no atmosphere or planet rotation, and the probe never fires its engine (an Oberth-effect burn at closest approach would add more). The planet radius is drawn at a visual scale, not the true one.
Model checks: Jacobi-constant conservation over the live flyby; an isolated hyperbolic encounter against 2 arcsin(1/e); and a circular two-body orbit that must return to its start after 2π √(r3/GM).