Stealing speed from Jupiter

Send a spacecraft past a planet and it can come out the far side moving faster than it went in, having burned nothing. The planet's gravity did it — provided you do the arithmetic in the right point of view.

An elastic encounter that changes nothing

Do the flyby from Jupiter. The spacecraft falls in, swings around and climbs back out. Far away on either side gravity is negligible, and the planet is far heavier, so it barely moves. It arrives at one speed relative to the planet and leaves at the same speed, merely pointed somewhere new. In the planet's frame this is an elastic collision, which cannot change speed. Take that frame for the whole story and a flyby does nothing.

The same encounter, seen from the Sun

Now watch from the Sun, where the planet is not standing still. A velocity is a vector: the spacecraft's motion through the solar system is its motion past the planet plus the planet's own motion — a Galilean addition, no relativity needed. Jupiter carries the encounter along with it at 13.1 km/s. Add that one vector to the arriving velocity and to the departing one and the two sums come out different lengths, though the vectors added were identical in length and differed only in direction. The speed past the planet is untouched; what changed is the frame.

Paid for, not free

The energy the spacecraft gains is energy the planet loses — same quantity, opposite sign. The transfer is far too small a share of Jupiter's orbital energy to alter the planet's motion: a withdrawal no bank would notice. A spacecraft is light enough to be handed a useful shove; a planet is too heavy to feel it.

Behind or in front

Which side you pass sets the sign. This is our reading of the paper's geometry, and the equations support it: pass behind the planet, crossing its track where the planet has already been, and on the way out the planet's motion is partly lined up with yours, so you leave faster than you arrived. Pass in front, where the planet has not yet reached, and the same addition works against you: slowed and turned back. Aim closest to the planet and the turn is hardest — the slider stops at one Jupiter radius, and it is how close you pass that sets the size of the turn.

Ulysses

A flyby is how a mission leaves the flat plane the planets share without spending fuel it does not have. Ulysses massed 366.7 kg at launch and arrived at 13.896 km/s relative to Jupiter — 16.184 km/s relative to the Sun — passing 6.3 Jupiter radii from its centre. Jupiter circles the Sun at 13.1 km/s; the speed past the planet and those 6.3 Jupiter radii, with Jupiter's mass, give a scattering angle of 74° in Jupiter's frame. Run them through the paper's three-dimensional model at the elevation angle Ulysses reached, about 80°, and it gives a final speed of 7.4 km/s, slower than the 16.184 km/s it arrived with, and a semi-major axis of 3.10 AU, against the mission's real 3.37 AU. The same model sets a price: below roughly 48° of elevation the spacecraft leaves the solar system instead of staying bound to the Sun, and a final speed above 18.5 km/s at Jupiter's distance does too.

In short

The encounter is elastic in the planet's frame and anything but in the Sun's; the difference is one Galilean addition. The energy comes out of the planet's orbital energy: too small to alter the planet, large enough to matter to a spacecraft. Which side you pass decides whether the addition speeds you up or slows you down. So Ulysses left the plane of the planets: past Jupiter, into an orbit almost at right angles to the ecliptic, bought with speed rather than fuel.