She detached the spring-shot, then reattached it pointing in the opposite direction: sharq along the Null Line. Now the stone would be moving backward from the start, making it slower than the Splinter at the point where their orbits touched.
Its path followed exactly the same pattern as before, except that rarb became sharq, and sard became garm. After leaving the spring-shot the stone veered garmwards; halted its leisurely sharqwards progress and went rapidly into reverse; reached a maximum distance garmwards from the Null Line and started back toward it; then, close to the Null Line, performed a small loop that took it back into the same cycle, albeit many spans rarb of where it had begun.
Roi said, “Its orbit was smaller than ours, so it was racing ahead of us?”
“Yes.”
“And the way it moved away from the Null Line and then back again, that’s because the orbit wasn’t a perfect circle?”
“Right,” Zak said. “We remain a constant distance from the Hub, but there are orbits like this that draw closer to the Hub and then move away again.”
Roi contemplated this. “What if we could put a stone into an orbit that wasn’t a perfect circle, but was still the same size as ours, overall? With the same period?”
Zak didn’t reply immediately, but his posture made it clear that he was intrigued by her suggestion. “That could be very useful,” he said eventually. “We ought to see it execute a fixed, cyclic motion instead of running away across the chamber.”
Roi detached the spring-shot from the Null Line again, and attached it pointing sardwards: perpendicular to the Null Line, “halfway between” the two directions she’d already tried. She was acting purely on instinct, and even as she tightened the clips she wondered if launching the stone away from the Hub meant she’d be putting it into an orbit that would keep it perpetually on one side of the Null Line. But then, shooting the stone along the Null Line itself, which seemed more symmetrical in that respect, certainly didn’t work, so what she was trying made as much sense as anything.
She squeezed the plunger one notch, then released it.
The stone veered sideways as it emerged, but not as sharply as it had in the previous two experiments. As it moved, it picked up pace, but nowhere near as rapidly as before. Roi was surprised; she’d half expected the sardwards weight to take over and drag the stone into a frenzied spiral as the weight of motion twisted its ever-quickening flight. Instead, the stone continued to turn in a smooth, shallow arc, still progressing sardwards while swinging around ever more to the sharq.
Eventually, its sardwards motion leveled off, about two spans from the Null Line. It was moving perhaps three times faster now than it had when she’d launched it. It continued to swing around gently, coming back toward the Null Line, while its sharqwards speed lessened.
As it approached the Null Line, Roi tensed. It was no longer traveling sharqwards, but it would probably perform the same annoying little loop as the others, and then it would be lost to them, drifting away across the chamber.
It didn’t. It crossed straight over the Null Line, at about the same speed as it had left the spring-shot, and veered to the rarb. The symmetry was unmistakable: it was performing exactly the same kind of motion as it had when she’d fired it, only with garm in place of sard and rarb in place of sharq. If that symmetry held true, there was only one place where it could cross the Null Line again.
When the stone finally approached the spring-shot, Roi thought it might collide with it, but her aim hadn’t been that perfect. It was close, though. The stone passed less than half its width from the tube before continuing on around the same closed loop as before.
“I can’t believe I missed this,” Zak said. “A new periodic motion! Congratulations!”
Roi said, “What are we seeing, exactly? Is this showing us the Splinter rotating?”
“What we’re seeing is a stone in orbit, moving back and forth between its nearest and furthest points from the Hub,” Zak replied. “If we try to explain that from a Splinter’s eye-view, the motion will depend on the strength of the garm-sard weight, as well as the speed of the Splinter’s rotation. Once I would have said that those two things should combine in a very simple way, but now I’m not so sure. The two and a quarter has taught me to be more cautious.”
Roi launched another stone directly sardwards, this one faster than the first. The loop it followed was larger, but the shape was the same—about three times as long as it was wide—and the faster stone completed each circuit in the same time as its slower companion. These stones were spending half their time sard of the Null Line, moving more slowly than the Splinter, and the other half garm of the Null Line, moving faster, so over time they were keeping pace both with the Splinter and with each other. Surely that meant that each cycle they completed marked the time for all three to complete an orbit around the Hub? And surely the Splinter’s rotation around its axis shared the same period, too, as a matter of simple geometry?
Zak said, “I know what we should do.” He found an empty tube, attached it to the Null Line aligned shomal-junub, then placed a stone in its mouth and let it begin its slow fall. Now they could compare the two kinds of motion directly, without having to worry about the accuracy of their counts to time the periods.
It soon became obvious that the periods were not the same: the looping stones were taking far longer to complete each cycle than the stone falling shomal and then junub. For a while, Roi wondered if the slower cycle might be exactly twice as long as the faster one—and if some simple aspect of the geometry that she’d neglected could make sense of this—but that hope proved misplaced. The shomal-junub stone completed seventeen cycles while the looping stones completed nine. There was nothing simple about that.
Zak seemed forlorn at first, but then he proclaimed, “There’s something encouraging about the way these numbers demolish half of my assumptions, yet the whole notion of orbits seems to survive. Watching these stones, can you honestly tell me that you don’t believe they’re going around the Hub?”
Roi said, “The idea still makes sense, but we’re missing something.”
Zak regarded the shomal-junub stone. “If orbits still make sense at all, then this stone tells us how much time passes for something orbiting at a small angle to the Splinter’s orbit to come back to the same height above us each time. The stone doesn’t go wandering off along the Null Line, so the periods of the two orbits must be the same. But what if the place where the orbits are farthest apart isn’t fixed? What if that point moves around? Then this need not be telling us how long an orbit actually takes.”
He moved to the looping stones. “And when you deform an orbit so it’s no longer circular, what if the point of closest approach to the Hub isn’t fixed either? That point, too, might wander around.”
Roi struggled to picture what he was describing. “So these other orbits wouldn’t close up? The Splinter would follow a perfect circle, but these stones would be weaving up and down, or back and forth around that circle, never quite repeating their paths?”
“Yes.”
Roi was dismayed. “If the things we thought were landmarks can’t be trusted to stay still, how can we ever decide how long it takes for the Splinter to complete an orbit?”
Zak said, “Good question.”
Neither of them had the answer to it, so they set about calculating what Roi’s looping stones actually did tell them. They worked side by side until the end of the shift, slept, then worked through two more shifts.
Finally, they had templates describing the relationship between three things: the strength of the garm-sard weight, the period of the Splinter’s rotation, and the period of the looping stones. These calculations made no assumptions at all about the existence of “orbits around the Hub”; they just followed the effects of the weights directly—though they did rely on a correct understanding of how spin contributed to weight.