Gravitational Slingshot Calculator

Calculate spacecraft velocity gained from planetary gravitational slingshot maneuvers.

Jupiter: 13.07, Saturn: 9.69, Mars: 24.1
Velocity at infinity relative to planet
Depends on flyby distance
Jupiter=317.8, Saturn=95.2
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A Gravity Assist Exchanges Momentum with a Moving Planet

A gravitational slingshot changes a spacecraft’s velocity in the Sun-centered frame by bending its trajectory through the gravitational field of a moving planet. In the planet’s rest frame, an ideal unpowered flyby leaves the spacecraft’s far-away speed relative to the planet approximately unchanged; gravity mainly rotates the relative-velocity vector. Transforming back to the heliocentric frame can produce a large speed increase or decrease.

The energy comes from an unimaginably tiny change in the planet’s orbital energy and momentum, not from gravity creating energy. Flyby effectiveness depends on planet speed, closest approach, gravitational parameter, and incoming hyperbolic excess speed.

vout,Sun=vplanet+v∞,out, with |v∞,out|=|v∞,in| ideally
SymbolMeaningWhy it appears / units
v∞Hyperbolic excess velocitySpacecraft velocity relative to planet far from encounter.
vplanetPlanet heliocentric velocityVector added during frame transformation.
δTurning angleAngle through which planet gravity rotates v∞.
rpPeriapsis radiusClosest center-to-center flyby distance.

A gravity assist can accelerate, decelerate, or redirect a spacecraft depending on encounter geometry. A flyby behind a planet in its orbital motion can add heliocentric energy; passing in front can remove it.

Always identify the reference frame in a gravity-assist calculation. A velocity change quoted relative to the Sun is not the same as the spacecraft’s speed change relative to the planet. The planet-centered incoming and outgoing asymptotic speed magnitudes should match in an ideal unpowered flyby.

Worked Examples

Example 1: Voyager 1 Jupiter flyby: V_J=13.07km/s, v∞=10km/s, δ=90°
Gain = 2×13.07×sin45°
Result: +18.5 km/s gain — massive!
Launched at ~11 km/s, left solar system at ~17 km/s
Example 2: Mars gravity assist: V_M=24.1km/s, δ=60°
Gain = 2×24.1×sin30°
Result: +24.1 km/s max gain
Mars can provide significant assist due to fast orbit
Example 3: Vector rotation idea
|v∞|=5km/s before and after encounter in planet frame
Result: relative speed stays 5km/s ideally
The key change is direction, not magnitude, in the planet-centered frame.
Example 4: Closer safe flyby
smaller periapsis with same v∞
Result: larger turning angle
Stronger deflection can produce a larger heliocentric velocity change, subject to atmosphere and safety limits.

Common Mistakes

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Adding a fixed “planet speed bonus” to every flyby

The result is a vector transformation and depends strongly on encounter geometry.

⚠️
Claiming spacecraft gains energy in the planet frame

In an ideal gravity-only flyby, far-away planet-relative speed is unchanged; the heliocentric energy changes because the planet itself is moving.

⚠️
Ignoring minimum safe periapsis

Atmospheres, rings, terrain, radiation, and tidal or navigation constraints limit how close a real spacecraft can pass.

Frequently Asked Questions

How does slingshot conserve energy?
In the planet's reference frame, spacecraft bounces elastically off the planet's gravity — speed unchanged. In the Sun's frame, the planet is moving, so spacecraft gains or loses energy. The planet loses an imperceptibly tiny amount of orbital energy.
Real missions using slingshots?
Voyager 1&2 (Jupiter, Saturn, Uranus, Neptune). Cassini (Venus×2, Earth, Jupiter). New Horizons (Jupiter slingshot to Pluto). Parker Solar Probe (Venus×7). Without slingshots, these missions would be impossible with current propulsion.
Where does the spacecraft’s added energy come from?
It comes from the planet’s orbital motion. Conservation of energy and momentum gives the planet an immeasurably small opposite change because its mass is enormous compared with the spacecraft.
Can a gravity assist slow a spacecraft down?
Yes. Encountering the leading side of a moving planet can rotate the velocity vector so the spacecraft loses heliocentric energy, useful for missions heading inward.
Why is Jupiter effective for gravity assists?
Jupiter has a very large gravitational parameter and significant orbital speed, allowing substantial trajectory bending and heliocentric energy exchange.
Does the spacecraft need to fire its engine during a slingshot?
Not for the basic gravity assist. Small powered maneuvers may be used for targeting, and a burn near periapsis can combine with the Oberth effect for additional benefit.
Can a gravity assist create energy from nothing?
No. In the planet-centered frame, an ideal flyby mainly redirects the spacecraft velocity and approximately preserves its hyperbolic excess speed. In the Sun-centered frame, the spacecraft can gain or lose orbital energy by exchanging a tiny amount of momentum and energy with the moving planet.

Formula Explorer connections

Interpretation: This relationship connects mass, distance, orbit or spacetime behavior through gravitation and astrophysical scaling. Assumption: Many calculations assume spherical bodies, point masses, circular orbits, weak fields or Newtonian gravity; relativistic regimes require the stated correction.

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