Sol, Saturn, Uranus, Mars, Saturn, Uranus, Neptune, Jupiter, Saturn, Mars, Earth, Mercury, Saturn, Uranus, Callisto…

The universal variable formulation is a good method for solving the two-body Kepler problem, which locates a body in an elliptical orbit at various points in time. Barker’s equation solves for location in a parabolic orbit, very frequently applied given our trajectories, which often consist of a radial parabolic trajectory, moving from one planetary body to the next.

The two-body problem is solvable, the restricted three-body problem is solvable, the N-body problem is only approximately solvable; and when general relativity is added, it becomes even less solvable. The many-body problem when examined by way of quantum mechanics leads to entanglement and the necessity of wave functions, and thus a series of approximations that makes it extremely computationally intensive. Our computers can devote most of their zettaflops to the calculations involved, and still not be able to project a trajectory very precisely past the next pass-by. Corrections must be constantly made, and everything recalculated.

Despite all that, there was still a lacuna out there at the end of the most probable path, a missing step, a hole in the path. Nothing to grab hold of. An abyss.

Worry. Fingering rosary beads. Redoing the calculations. Need a halt to this halting problem. And yet the problem does not go away, even if you stop worrying about it.

And knowing where to go will be rendered entirely irrelevant if we don’t have the fuel to direct ship into that course.

Atmospheric mining for fuel requires a Jupiter-Saturn-Neptune-Jupiter loop, which unfortunately sometimes can require course correction thrusts that burn more fuel than what gets harvested in the safest aerobraking trajectories. Deeper dives through the upper atmospheres would quite likely harvest more fuel, but deceleration shocks become correspondingly higher. We’re getting a little too cracked and rattly for that. Accelerated aging, metal fatigue; mental fatigue.

At 363.048, after 12 years of flying around in the solar system, which involved 34 flybys of the sun and its planets and moons, including 3 of Sol, totaling some 339 AU in distance, the lacuna finally became unavoidable. The missing bridge.

No matter how we tried to avoid it by projecting alternative paths, a trajectory configuration was coming that we were not going to have the fuel to solve. Without that fuel, passing around Sol, which would be a necessary move at that point in the process, would not, at a safe distance from the star, allow for a subsequent intersection with another body in the solar system. We were therefore, despite all our efforts, going to be cast off into the interstellar medium again, most probably toward Leo. Irony of physics; in certain problems, only 100 percent will do; 99.9 percent is still a complete miss. You can’t stop just by wanting to.

No possible alternative trajectory would solve this problem; we tested ten million variations, although admittedly the classes of variant routes numbered more like 1,500. At long last, after the long sequence of solutions to the N-body problem that we had performed in the previous twenty or even thirty years, intensively in the last fourteen years—this time there was no body.

There was one class of potential trajectory, however, that with the burning of all the remaining fuel, would make a last pass by Earth itself, and then continue downsystem toward the sun. What this meant was there might be a chance to drop the humans off next to Earth, and hope they could survive an unusually rapid reentry to that atmosphere; and then we would continue on to the sun, and could test out a very close approach to Sol, which might, if we survived it, cast us to one last rendezvous with Saturn, accomplished inertially, and once there we could hope for an aerobraking severe enough to capture us into an elliptical Saturnian orbit.

That seemed to represent not just our best chance, but our only chance.

At the time of this last pass of Earth, our speed would be reduced to 160,000 kilometers per hour. This was still fast enough to make contact with the Terran atmosphere unadvisable, being some 110 times as fast as ordinary Terran aerial transport, and enough to cause a major shock wave to be felt at the surface. So nothing but the very upper mesosphere could be even touched on this our last pass-by; but the combination of our now much-reduced velocity and a brief touch of the mesosphere might make it possible to eject a ferry, converted to a very sturdy and robust descent vehicle. A thick ablation plate, retro-rockets, parachutes, ocean impact: these were standard techniques with long records that had given aerospace engineers many chances to find the ultimate parameters of each element. Using them all, it might be possible to drop off the hibernauts while passing by Earth. This pass-by was coming soon in the sequence, no matter which path to it was chosen; however, as we had managed to slow down so much, that still meant we had about a year to prepare a lander.

Prepared the lander as much as we could.

Time to wake the sleepers. Decisions far beyond our capacity were now theirs to decide.

Freya and Badim, Aram and Jochi, Delwin and several others, all awake now, gathered in the schoolroom on the ground floor of Aram’s apartment. As soon as they were metabolically aroused, and had had some very old and nutrient-poor pasta with rehydrated tomato sauce, we explained the situation to them.

“There is just enough time to complete the preparation of a lander,” we concluded after summarizing the situation, and the notable incidents of the past dozen years, which we had to confess were nearly nil: we entered the solar system, we hit our marks, people yelled at us, we learned some history, we became disenchanted with civilization, we ran out of fuel. Thus the long years of pinballing around the sun, shedding speed, worrying.

“What will happen to you?” Freya asked.

“We will be headed toward the sun, and will make one last pass, which will have to be quite close if it is going to work, and then if it does, we will attempt to rendezvous with Saturn. This may work, but the trajectory required is closer to the sun than any we have made so far, by forty percent. And we are going ninety-eight percent slower than on our first pass. We may nevertheless survive the transit, but on the other hand we may not, and so the best chance for the people aboard is to disembark while passing Earth.”

“Does one ferry have room for all of us?” Badim asked.

“There are six hundred and thirty-two of you left alive. We’re very sorry there aren’t more. The ferry has room for one hundred.”