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Gravity assist: gaining speed at a planet (3D)

About this app

A probe flies past Venus, Earth or Jupiter. Seen from the planet it follows a hyperbola and keeps its speed; only its direction turns by δ. Seen from the Sun, v = v_P + v∞: passing behind the planet speeds it up, passing in front slows it down. Vector diagram, orbits around the Sun before and after, route Earth → Venus → Mercury as flown by Mariner 10.

With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.

The physics

In a gravity assist (swing-by) a spacecraft flies close past a planet and lets the planet’s gravity bend its path. Seen from the planet it travels on a hyperbola: it arrives at speed v∞ and leaves with the same magnitude, only in a different direction. The deflection angle δ follows from sin(δ/2) = 1 / (1 + r_p·v∞²/μ), with r_p the closest distance and μ the planet’s gravitational parameter. Seen from the Sun, the planet itself is moving, and v = v_P + v∞. Because v∞ is turned, the magnitude of the sum changes: if the probe passes behind the planet it speeds up, if it passes in front of it it slows down.

What you see

A probe flies past Venus, Earth or Jupiter in 3D. A switch changes the reference frame: with the planet at rest, the arrows at the start and end of the path are equally long, together with asymptotes, deflection angle and impact parameter. With the Sun at rest, the planet moves across the view at its orbital speed and the arrows differ in length. A vector diagram shows v_out = v_P + v∞,out with the circle of radius v∞. The second view shows the solar system from above with the probe’s orbit before and after the flyby, including, as a worked example, the route Earth → Venus → Mercury that Mariner 10 flew from 3 November 1973.

Things to try

Let the probe fly past Venus and read off the speed at closest approach: it rises from 8.5 to 13.0 km/s and then falls back to 8.5 km/s. Switch to “Sun at rest”: passing in front of Venus, the speed drops from 36.8 to 29.9 km/s; passing behind it, it rises to 42.1 km/s. Change the closest distance and the approach speed and watch the deflection angle. Switch to the solar system view: after braking at Venus the probe falls to inside Mercury’s orbit. At Jupiter, a pass behind the planet is enough to leave the solar system.

Frequently asked questions

How does a gravity assist work?
The probe flies close past a planet and is deflected by its gravity. Relative to the planet it keeps its speed; only the direction changes. Because the planet orbits the Sun, its velocity adds to the probe’s. With the new direction the sum becomes larger or smaller, depending on whether the probe passes behind or in front of the planet.
Where does the energy in a gravity assist come from?
From the planet’s orbital motion. If the probe gains energy, the planet slows down by the same amount; if the probe loses energy, the planet speeds up. Because of the planet’s enormous mass the change is immeasurably small. Overall, energy and momentum are conserved.
Why did Mariner 10 fly past Venus to reach Mercury?
To fall from Earth’s orbit to Mercury, a probe has to lose speed relative to the Sun. Mariner 10 launched on 3 November 1973 and flew past Venus on 5 February 1974 so that it was slowed down. Its new orbit then reached Mercury, which it met on 29 March 1974 and passed three times in total.
What can I set in the app “Gravity assist: gaining speed at a planet”?
You choose Venus, Earth or Jupiter, a pass behind or in front of the planet, the closest distance, the approach speed and the approach direction. A switch toggles between “planet at rest” and “Sun at rest”, and a second view shows the orbits around the Sun before and after. The values and the vector diagram update live.

Subject: Astronomy

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