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After reading this answer I wondered if there have been document comets with orbits on non-return, hyperbolic trajectories as they approached the Sun.

This means that they would not be gravitationally bound to the sun, so either they came from outside the solar system, or they managed to gain energy through a collision, or close pass with a large planet like Jupiter.

The quickest way to find the comet with the "highest" orbit is to ask for the one with the largest specific orbital energy, the total energy (kinetic + potential) divided by the mass, (Joules/kg). For a bound orbit the value would be negative, for an unbound, non-returning orbit it would be positive.

Question: What documented comet has/had the highest specific orbital energy? It should be an identified object with a known or measured orbit, not just a generic, nameless "edge-of-the-Oort Cloud placeholder."

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That is a very good timing for this question.

There is a comet with the name C/2017 U1 (PANSTARRS), that was just discovered a few days ago (on 18th October) which has an orbital eccentricity of 1.18, which is by far highest eccentricity recorded so far.

The eccentricity describes how strongly "elliptical" an orbit is and refers to the variable e in the formula of Kepler's orbit. 1.18 makes the orbit a hyperbolic one, meaning it is not an actual orbit, but has higher velocity than the required escape velocity and is basically just passing by in our solar system. And it's the first recorded comet so far that has an hyperbolic orbit.

This comet is the first candidate for an interstellar comet ever in recorded history. It is already pretty certain that it originated from interstellar space, but not official yet.

Quote from minorplanetscenter.net:

Further observations of this object are very much desired. Unless there are serious problems with much of the astrometry listed below, strongly hyperbolic orbits are the only viable solutions. Although it is probably not too sensible to compute meaningful original and future barycentric orbits, given the very short arc of observations, the orbit below has e ~ 1.2 for both values. If further observations confirm the unusual nature of this orbit, this object may be the first clear case of an interstellar comet.

Sources:

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    $\begingroup$ Wow, that's great news, and yes, quite a coincidence. OK maybe in a week or two as more astrometry can be collected the uncertainty in the orbital elements will settle down and more certain values will be available. Thanks! $\endgroup$
    – uhoh
    Oct 25, 2017 at 15:57
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    $\begingroup$ "It is already pretty certain that it originated from outer space, but not official yet." I think you mean outside of the solar system, not just outer space? $\endgroup$
    – uhoh
    Oct 25, 2017 at 15:58
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    $\begingroup$ yeah sorry, typo correcting it :) $\endgroup$
    – RononDex
    Oct 25, 2017 at 16:03
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    $\begingroup$ Best fit from MPC has e = 1.2 twitter.com/cosmos4u/status/923697091322429441 minorplanetcenter.net/mpec/K17/K17UI5.html This confirms a strongly hyperbolic orbit $\endgroup$
    – James K
    Oct 29, 2017 at 11:13
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    $\begingroup$ The asteroid seems to have been given a name. I'm not sure if it's 100% official yet. bbc.com/news/science-environment-42019778 Linked there is this, apparently submitted to ApJ letters ($e=1.197$) $\endgroup$
    – uhoh
    Nov 17, 2017 at 14:33

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