Three bodies cannot be solved, and yet spacecraft fly. How are their paths worked out?
The idea is simple. Near a planet, deal with the planet alone; far from it, with the Sun alone. Each is a two-body problem and each has a closed solution.
Where, then, to switch? The naive answer is where the two pulls are equal: for the Earth and the Sun, km out, which is inside the orbit of the Moon.
The boundary actually used sits at km. It is defined differently: not by the pulls themselves, but by which choice of primary leaves the smaller disturbance behind.
That boundary is the sphere of influence. Its size is the distance from the Sun times the mass ratio raised to the power .
, in the form Laplace derived. The exponent being neither nor is a sign that this is not a simple comparison of forces.
For the Earth it is km. The Moon orbits at , so the Moon is safely inside.
Jupiter, heavy and far, has million km, more than fifty times the Earth. Even so, that is under one percent of its orbital radius. Almost all of the solar system belongs to the Sun.
The procedure follows. Outside the sphere, solve about the Sun; inside it, solve about the planet.
Outside gives an ellipse or a hyperbola; inside is almost always a hyperbola, since anything passing through without being captured has speed to spare there.
Leave the Earth, curve round the Sun, enter the sphere of Mars: the journey is three conic sections stitched end to end. This is the patched conic method.
The three-body problem has been avoided. An unsolvable problem has been replaced by a splicing of solvable ones.
One thing happens at a seam: position and velocity carry across, and the centre changes.
Subtract the velocity of the planet from the velocity seen by the Sun and you have the velocity seen by the planet. Nothing jumps. It is the same change of frame as in the gravity assist of the previous chapter.
A mismatch is left, however. Inside the sphere the Sun still pulls, and outside it the planet still pulls, and the splicing drops both.
The mismatch is small but not zero. Flown as it stands, the craft misses its target.
Why use something that misses? Because it is fast.
The splicing gives a trajectory by solving formulas. Which planets in which order, when to leave, where and how hard to burn: tens of thousands of options can be tried and the cheapest chosen.
The rough trajectory that comes out is then used as a starting guess, and from there every body is included and the whole thing stepped forward numerically. Only the final refinement is done that way.
Voyager, Hayabusa and the James Webb telescope all flew on this two-stage method. It is the most practical accommodation anyone has made with an unsolvable problem.