Borrow some motion.
Try a trailing-side pass. Watch the white velocity arrow turn toward the planet’s motion. Compare the far-away arrival and departure speeds in the Sun frame.
FIELD NOTE 11 / GRAVITY ASSIST
A little help from a world.
No engine burn. Just a carefully chosen encounter.
Watch from
Path travelledPath aheadVelocity arrowGravity arrowPlanet motion
ApproachingClosest passDeparting
The planet stays at the origin in this view. Earth and distances are to scale; the spacecraft marker is enlarged.
- Speed · Sun frame
- — km/s
- Speed · planet frame
- — km/s
- Current altitude
- — km
Try an encounter
THE LASTING CHANGE · FAR FROM THE PLANET
Borrowing a little orbital energy.
Sun-frame speed — → — km/s
These are the calculated far-away limits, beyond the displayed one-hour encounter.
—km/s speed change
— MJ/kg kinetic energy
The arrival and departure speeds match in the planet frame.
Follow the exchange.
Bright traces show elapsed time. Faint traces show the rest of the calculated encounter.
One spacecraft. Two speeds.
Sun framePlanet frame
Both speeds can rise near the planet. The far-away comparison above isolates the lasting change.
Energy in the planet frame.
KineticPotentialTotal
Falling inward trades potential energy for kinetic energy. The total stays at — MJ/kg.
The encounter is playing. Change the settings or viewpoint as it runs.
Make a prediction. Change one thing. See what happens.
Model notes & references ↗FOLLOW THE SCIENCE
A bend in the path.
A change in possibility.
A moving planet can change a passing spacecraft’s speed relative to the Sun. For the geometry here, passing behind the planet gains speed; passing ahead loses speed. Mission planners use these encounters to reshape trajectories while saving propellant. ESA: what are gravity assists? ↗
The planet is moving too. The same flyby looks different from a viewpoint travelling alongside it and from a viewpoint at rest relative to the Sun. Switch between the two while the encounter runs.
A FEW QUESTIONS TO TRY
Where does the extra speed come from?
Give some back.
Switch to the leading side. The planet now pulls the spacecraft against its own direction of travel. Can you lose Sun-frame speed without firing an engine?
Stop the planet.
Set the planet’s speed to zero. The path still bends and the spacecraft still speeds up near the planet. What happens to its far-away speed change?
Speed depends on your point of view.
TRAVELLING WITH THE PLANET
A turn, without a lasting speed gain.
The spacecraft accelerates on the way in and slows on the way out. At equal distances, the speeds match. Far away, its arrival and departure speeds both approach v∞, but its direction has changed.
WATCHING FROM THE SUN
The same turn can change the speed.
Add the planet’s velocity to the spacecraft’s relative velocity. Turning the relative vector toward the planet’s motion makes the sum longer; turning it away can make the sum shorter. Velocity is a vector: direction matters.
v⃗Sun = V⃗planet + u⃗relative
In a real gravity assist, the spacecraft and planet exchange energy and momentum. A spacecraft gaining orbital energy takes a tiny amount from the planet. Its immense mass makes the planet’s response negligible for ordinary trajectory calculations. There is no free creation of energy. NASA: trajectories and gravity assists ↗
A closer pass bends more.
At a fixed arrival speed, reducing the closest distance produces a larger deflection. A faster arrival is harder to turn. Try both controls while watching the trajectory and deflection angle. The greatest turn does not always give the greatest speed boost: the direction of the outgoing velocity also matters.
Speeding up near a planet is not the whole assist.
Even a stationary planet accelerates an incoming spacecraft. It gives that local speed increase back as the spacecraft climbs away. The lasting Sun-frame change comes from encountering a moving planet.
The mathematics, assumptions & references
An exact local flyby
The planet is spherical, with gravitational parameter μ = 398,600.4418 km³/s² and radius R = 6,371 km. Its centre moves at constant velocity V along +x. In the planet frame, the incoming asymptotic velocity is (0, v∞). The spacecraft is a test particle following an unpowered, two-body hyperbola. Closest approach occurs at t = 0. The displayed window is −30 to +30 minutes.
rp = R + altitude
e = 1 + rpv∞²/μ
δ = 2 asin(1/e)
b = rp√(1 + 2μ/(rpv∞²))
e = 1 + rpv∞²/μ
δ = 2 asin(1/e)
b = rp√(1 + 2μ/(rpv∞²))
Here rp is distance from the planet’s centre at closest approach, e is eccentricity, δ is the magnitude of the asymptotic turning angle, and b is the impact parameter of the incoming asymptote. The closest-altitude control sets rp, not b. The allowed altitude range stays above the surface.
Following time along the curve
a = μ/v∞² > 0
nt = e sinh H − H · n = v∞³/μ
x′ = a(e − cosh H)
y′ = ±a√(e² − 1) sinh H
nt = e sinh H − H · n = v∞³/μ
x′ = a(e − cosh H)
y′ = ±a√(e² − 1) sinh H
a is the positive magnitude of the hyperbola’s semimajor axis. The hyperbolic Kepler equation is solved for H, then the curve and its analytic velocity are rotated so the incoming asymptote points upward. The sign selects the leading or trailing side. Adding (Vt, 0) to each position and (V, 0) to each velocity produces the Sun-frame view. Both coordinate axes use the same distance scale. Each view fits the complete encounter; switching frames also changes the view’s scale. Arrows indicate direction, with lengths scaled separately for readability.
Energy and far-away limits
εplanet = ½|u⃗|² − μ/r = ½v∞²
|u⃗in| = |u⃗out| = v∞
ΔKSun/m = V⃗ · (u⃗out − u⃗in)
|u⃗in| = |u⃗out| = v∞
ΔKSun/m = V⃗ · (u⃗out − u⃗in)
The planet-frame graph shows kinetic, potential and total energy per unit spacecraft mass. Zero potential energy is at infinity. Values in km²/s² equal values in MJ/kg. The outcome panel uses the incoming and outgoing asymptotic velocity vectors, not the finite endpoints of the animation. It reports the change in speed magnitude and in Sun-frame kinetic energy per kilogram. Neither is an engine’s Δv.
The Sun’s gravitational field, the curvature of the planet’s solar orbit, other bodies, atmosphere, rotation, oblateness, tides and the planet’s tiny recoil are omitted. V is adjustable independently of Earth’s actual orbit. This is the local encounter part of a patched-conic picture, not a complete Earth flyby plan, capture model or mission predictor. The asymptotic comparison assumes essentially the same solar potential before and after the local encounter. A real mission must account for the Sun and the planet’s sphere of influence.
Changing settings evaluates a new complete trajectory at the same model time; it does not add a physical force or conserve energy across your edits. Play resumes exactly from Pause. At the end of the displayed window, the animation stops; use Replay or move the time slider to revisit the encounter. Reduced-motion preferences start it paused. Optional audio is a synthesized speed indicator, muted during pause or while the page is hidden.