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Yeah, obviously one is an asteroid and one a comet, duh; but what does really differentiate them? I've read that active asteroids have an asteroid-like orbit, but what does that mean? Thanks in advance

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    $\begingroup$ A comet-like orbit is a very eccentric one, typically going beyond Neptune at one extreme, and into the inner-solar system on the other. They can also be highly inclined. An asteroid-like orbit is, by contrast, one that isn't like that. $\endgroup$ Commented May 20 at 16:23

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Active asteroids are (as far as we can tell) comets in every respect except that they're inside the asteroid belt, and Jupiter-family comets are comets that stay inside Jupiter's orbit but aren't in the asteroid belt. They're the same kind of object, just with slightly different orbital characteristics. This distinction is complicated by the fact that the asteroid belt is not a clearly visible band of objects -- it's fuzzy at the edges and any distinctions we draw will necessarily be splitting hairs. (This happens a lot whenever we humans start trying to categorize natural systems.)

The only real distinction between active asteroids and Jupiter-family comets is basically the Tisserand parameter with respect to Jupiter. It's common, though not universal, to define asteroids (whether active or not) as having a $T_J$ parameter greater than 3, while Jupiter-family comets have a $T_J$ between 2 and 3 and long-period comets have a $T_J$ of less than 2.

Tisserand's parameter is determined by a formula that takes into account a small body's eccentricity and inclination, and the orbital distance of the larger body you want to compare against (Jupiter in this case), using the Sun as the center of all the calculations. The main thing of note is that $T$ is conserved when the three bodies interact. Like, if a rock has $T_J$ = 2.454 and then gets close to Jupiter so that its orbit gets wildly changed, it'll still have $T_J$ = 2.454 afterward (give or a take a little, since $T$ only considers the influence of three bodies rather than the entire solar system). This is useful for tracking small bodies after major orbital changes; it acts like a fingerprint we can use to identify a particular object over time.

That said, I don't really understand how to interpret a Tisserand parameter in isolation. I'm not sure $T_J$ actually tells us anything all by itself, other than serving to semi-arbitrarily define these families of objects.

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    $\begingroup$ Could you provide some explanation (even if only heuristic) as to what the Tisserand parameter is, what it means, and how we should interpret it? $\endgroup$ Commented May 22 at 20:20
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    $\begingroup$ I did what I could. $\endgroup$ Commented May 23 at 15:05
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As per Wikipedia:

Active asteroids are small Solar System bodies that have asteroid-like orbits but show comet-like visual characteristics i.e. they show a coma, tail, or other visual evidence of mass-loss (like a comet), but their orbits remain within Jupiter's orbit (like an asteroid).

So, active asteroid do shows comet like behavior but their orbit is more like that of asteroids in asteroid belt (but their orbit is not within the main asteroid belt)

Jupiter-family comets on the other hand are also comets but they do not lie in the asteroid belt (not even outskirts) and the majority of the orbit lies in the vicinity of Jupiter (their orbit is heavily influenced by the gravity of Jupiter).

The distinction is quite fuzzy as such bodies are quite rare (as of now 60 active asteroids and 400 Jupiter-family comets were discovered). The only thing that can be used here Tisserand parameter (which is well explained in @Darth's answer).

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