The science
The spacecraft is a massless test particle moving in the Sun's field plus the pull of Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune: a = −μ☉ r/|r|³ + Σ μi (ri−r)/|ri−r|³, integrated with adaptive RK4 (step shrinks near planets). The Sun is fixed and the planets follow circular, coplanar orbits at their mean distances: a simplification (real orbits are slightly eccentric and inclined, the Sun wobbles around the barycentre).
Seen from the planet the flyby is a hyperbola: e = 1 + rp v∞²/μ, turn angle δ = 2 asin(1/e). The speed relative to the planet is unchanged (|v∞,in| = |v∞,out|) but its direction rotates by δ, so in the Sun's frame vout = vplanet + R(δ) v∞,in. Passing behind the planet adds speed, passing in front removes it. Energy comes from the planet's orbital motion (its change is immeasurably small).
The launch date is found automatically: a launch-window scan plus bisection makes the closest approach equal to the altitude you set. Initial speed is the hyperbolic excess speed, not the rocket's burn speed (the burn from a 300 km orbit is shown in the Mission Planner). Planet sizes and path clearance in the 3D view are exaggerated (path near the target is pushed radially outward so it clears the enlarged globe); physics always uses true sizes. The comet, asteroid belt and real-mission figures are illustrative; flagged typical values should be spot-checked. Numbers in the reference screenshot are illustrative; here every number comes from the integration.
Models and textures built in Blender; rendering with Three.js. Not affiliated with any space agency.