Astronomy & Celestial Mechanics
the sky, computed
Generator

The gravity-assist generator

A gravity assist with a 70° turn
A gravity assist with a 70° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.

The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.

4 essays call gravity-assist. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

The aim point and the miss distance are the same number far out and nothing like it close in. Periapsis distance against aim point for a hyperbolic approach to Jupiter at v∞ = 5.6 km/s, both in planet radii. The diagonal is where the two would be equal — where gravity did nothing — and the curve falls below it everywhere, by more the closer in the aim is. A trajectory aimed at 10.68 radii grazes the surface, because gravitational focusing means the planet's effective size is √(1 + v_esc²/v∞²) times its radius. The slope of this curve is what a navigation team cares about: it is 0.208 at an aim of 12 radii and 0.936 at 150, so the same correction manoeuvre changes the periapsis distance by 4.5 times as much at one end of the range as at the other, while changing B by exactly the same amount at both; far outside this plot, where focusing has run out, it reaches 1.000 and the two numbers become the same one. That is why the aim point is the coordinate a manoeuvre is quoted in, why an error ellipse is published in the B-plane, and why the turn angle — 2 arctan(μ/Bv∞²), 156.0° at 12 radii and 77.8° at 70 — is thought of as a function of B rather than of anything the spacecraft does. Orbits

Aiming at a plane instead of at a planet

A spacecraft arriving at a planet is not aimed at a periapsis distance. It is aimed at a point in a plane perpendicular to its own incoming asymptote, because that is the one coordinate in which the miss distance responds linearly to a correction — and every navigation product ever published for a flyby is written in it.

 A gravity assist with a 70° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun. Spaceflight

Stealing speed from a planet, which does not notice

A flyby cannot change a spacecraft's speed relative to the planet. It changes its direction — and adding the planet's own motion back turns that into free velocity.

 What a 1 km/s burn is worth, against where it is spent. A vehicle arriving at Jupiter with an excess speed of 5.6 km/s, burning 1 km/s along its velocity at one point of the hyperbola. The vertical axis is the excess speed it leaves with. Spent at the surface the burn is worth 12.33 km/s of departure speed; spent far away it is worth 7.20. The energy bought is v·Δv, so the same propellant is worth 5.9 times as much at the bottom of the well — and nothing about the rocket has changed. Spaceflight

The same burn is worth more when moving fast

A rocket firing for ten seconds delivers the same change of speed wherever it is. It does not deliver the same change of energy, because energy is quadratic in speed — so the identical burn buys six times as much at the bottom of a gravity well as at the top, and every escape manoeuvre ever flown is arranged around that fact.

 A gravity assist with a 68° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun. Spaceflight

The planet pays, and it shows

A gravity assist takes energy from a planet and gives it to a spacecraft, and the planet's loss is exactly the spacecraft's gain. For a two-tonne probe past Jupiter that loss is unmeasurable. Do it with a hundred Earth masses of icy debris and the same bookkeeping moves Neptune outward by several astronomical units.

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