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This answer to A moon in eccentric orbit dipping below Roche limit includes the following about Comet Shoemaker–Levy 9

Here is a nice figure of its last passes:

last pass of Comet Shoemaker–Levy 9

I looked up "fragment A" in JPL's Horizons and it provides a modeled trajectory only between 1992-07-15 and 1994-07-17 but the link cited above says

Computations by Paul Chodas, Zdenek Sekanina, and Don Yeomans, suggest that the comet has been orbiting Jupiter for 20 years or more, but these backward extrapolations of motion are highly uncertain. See "elements." and "ephemeris." at SEDS.LPL.Arizona.EDU in /pub/astro/SL9/info for more information.

If I paste some of that into a search engine I get http://spider.seds.org/sl9-list.html with lots and lots of links, but now I'm in way over my head.

Question: What calculations show that Comet Shoemaker–Levy 9 was orbiting Jupiter for several decades before its spectacular impact? (Chodas, Sekanina & Yeomans) Where can I read about these calculations? Have there been further refinements since, based on any subsequently discovered historical precovery data?

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Chodas and Yeomans published a paper called The orbital motion and impact circumstances of Comet Shoemaker-Levy 9. In it they describe the algorithms and techniques they used to reconstruct the orbit backwards from the discovery of the comet. SL9 had an incredibly chaotic orbit in the mathematical sense in that small perturbations in the position would result in large effects in reverse-time orbital simulations. In the words of the paper authors:

Unfortunately, SL9'sorbit about Jupiter was among the most chaotic of any known solar system body, with an effective Lyapunov time on the order of 10 years (Benner and McKinnon 1995). As a result, a single backward numerical integration does not provide definitive answers on the orbital history of this object. A better approach is to account for the uncertainties in the initial conditions of the backward integrations, and to investigate the motion in a statistical manner using a Monte Carlo analysis (Chodas and Yeomans 1995).

For comparison, the Lyapunov time for the Solar System is on the order of millions of years. An orbit trajectory plot for SL9 is the most complicated I've ever seen (excluding human-built craft)!

enter image description here

I don't know if precovery data was ever used to refine the orbit, though papers are still being published on SL9's orbit. Unfortunately, Invariant Manifolds and the Capture of Comet Shoemaker-Levy 9, published in 2019, is behind a paywall.

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    $\begingroup$ I see, thanks! You've sent me off on a reading expedition now ;-) $\endgroup$
    – uhoh
    Mar 11 at 2:46
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    $\begingroup$ @uhoh I hope you enjoy your trip! I encourage you to drop by the Quasi-Hilda Comets on your way! $\endgroup$
    – Connor Garcia
    Mar 11 at 5:53

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