Earthv∞ 5 km/sPeriapsis alt. 150 kme 1.41Turn δ 90°Sun-frame speed 0 → 0 km/sGain 0 km/sFPS 0

Drag to orbit, wheel or pinch to zoom. Orange = incoming leg, cyan = outgoing leg (planet frame). Thin rings are gravity-potential levels (schematic). The planet turns the spacecraft's velocity; in the Sun's frame that turn becomes a speed change.

Planetary Flyby: Hyperbola Turn Angle and Heliocentric Speed Gain

Fly a spacecraft past a planet with no fuel. Choose the planet, approach speed, how close to pass and the approach angle, and read how the planet's gravity turns your velocity and changes your speed in the Sun's frame.

Approach

Animation

Live readouts

Velocity triangle in the Sun's frame

Speed along the flyby (planet frame)

Missions

Independent numerical checks

Controls

Drag / one finger: orbit; wheel / pinch: zoom; buttons jump to the hero shot, a periapsis close-up or the whole path.
Approach angle φ is the angle between the planet's orbital velocity and the spacecraft's incoming velocity relative to the planet (180° = head-on, 0° = overtaking from behind).
Turn left or right chooses on which side of the planet you pass; that decides whether you gain or lose Sun-frame speed.
The periapsis slider cannot go below the planet's radius plus a safe altitude for its atmosphere.

The science

Patched conics. In the planet's frame the spacecraft follows a hyperbola with excess speed v∞ (the speed far from the planet). Eccentricity e = 1 + r_p v∞² / μ, turn angle δ = 2 asin(1/e), periapsis speed v_p = √(v∞² + 2μ/r_p). Far from the planet |v∞| is the same before and after: gravity only rotates the vector.

Sun frame. Add the planet's orbital velocity V: v_in = V + v∞,in, v_out = V + v∞,out. Because |v∞| is unchanged, |v_out|² − |v_in|² = 2 V·(v∞,out − v∞,in): the speed change is the planet's momentum exchange. The planet's own speed changes by a negligible amount (its mass is vastly larger).

Numerical trajectory. The drawn path is an RK4 integration of r̈ = −μ r / r³ with an adaptive step proportional to r/v, started at 40 r_p on the incoming leg. The check table compares the integrated turn angle, periapsis and energy with the closed form.

Simplifications. Planar encounter, point-mass planet (no oblateness, atmosphere or moons), circular coplanar planetary orbit with the speeds shown, the Sun's gravity ignored during the (hours long) encounter, instantaneous patching. Planet constants are rounded fact-sheet values. The thin rings are a schematic gravity-potential picture, not magnetic field lines. Models built with Blender, scene by Three.js. Orbital physics only.