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The science
Why the speed changes
In the planet's frame the flyby is elastic: the probe arrives and leaves with the same speed v∞ but its velocity is rotated by the deflection angle δ = 2 asin(1/e), where the hyperbola eccentricity is e = 1 + rp v∞2 / μ (μ = GM of the planet). The Sun-frame velocity is v = U + v∞, so rotating v∞ changes |v| even though |v∞| is unchanged: Δ(v2/2) = U · (v∞,out - v∞,in). The energy comes from, or goes to, the planet's orbital motion, which changes by an immeasurably small amount.
Numerical solution
The page integrates the planet-frame equation of motion r'' = -μ r / r3 with a fourth-order Runge-Kutta scheme from 105 periapsis radii away, and compares the outgoing direction with the analytic deflection.
Honest simplifications: a single planet in a circular orbit with the Sun's pull ignored during the short encounter (patched conics), no atmosphere, no J2 or moons, and the probe is a point mass.
What to try
- Choose Jupiter with approach direction 180 deg: the probe leaves faster in the Sun frame. Try 0 deg to see it slow down.
- Lower the periapsis to 1.05 radii: the deflection grows but never exceeds the limit set by the eccentricity.
- Flip the flyby side and watch the outgoing vector swing to the other side.
- Raise v-infinity: the encounter gets gentler and the speed gain shrinks.