References and Notes

1. For comet Halley, the computation by Kiang and Yeomans was terminated in 1404 B.C., because Halley passed close to Earth (0.04 AU). Earth’s gravitational perturbation would have injected a large error that could not be rectified by a human observation even further back in time. With comet Swift-Tuttle, the backward computation by Yeomans et al. was terminated in 702 B.C for a similar reason.

Ending the backward projection was proper, because too much precision would have been lost for all earlier dates. An analogous situation occurs if someone is adding a thousand numbers, one of which has great uncertainty compared to all others. It makes no sense to claim high precision for the sum or to strive for further precision for the 999 good numbers when one number has little accuracy. That low precision number becomes the weak link in the chain.

However, with the statistical method used here, we are not forcing Halley and Swift-Tuttle to pass through (or near) points in space at specific times long ago.We are only looking for the tightest clustering of three bodies (Halley, Swift-Tuttle, and Earth) within a 2,000-year window: 4,000–6,000 B.C. Then we compare that tightness with the tightest clustering of those three bodies in that same 2,000-year window for each of a million **random** orbits of the two comets back to 6,000 B.C. Using the same step-back procedure, each of the million “step backs” begins not at their earliest known perihelion point, but at a random point on its earliest known orbit.

It turns out that less than 1% of the random orbits can produce a tighter clustering. Therefore, even though each comet experienced a large perturbation error, less than 1% of the random orbits could beat our actual orbit—a highly significant result.