We have to distinguish “the universe” with local space. The *universe* is, as far as we know, expanding, but it’s not expanding at a rate that makes an appreciable difference to local systems (yet. One theory posits that the speed of expansion will continue to get faster and faster until eventually not even individual atoms will maintain their composition, the theory is sometimes called the Big Rip)
Comets like Halley’s Comet are still gravity bound to our sun and orbit it just like Earth does.
The fact that *space* is expanding doesn’t mean it’s expanding *fast enough* to impact orbits of objects in a local system. It might. Eventually. But not yet.
Those asteroids and comets reappear because they’re orbiting the sun, just like Earth and everything else in the solar system. Their orbits are often eccentric (highly elliptical, and/or tilted with respect to the planets), but they’re still gravitationally bound to the sun. The effect of expansion is only seen in the distances between galaxies.
Same reason the planets do.
They’re basically all going around the Sun with us, just at different speeds and distances.
Asteroids are basically mini-planets; small lumps of rocks (anywhere up to a thousand kilometres across) orbiting the Sun, minding their own business, but not big enough to collapse into a full planet and “clear” the space around them.
Comets tend to have fairly squished/elliptical orbits, which means they can kind of [swing by](https://en.wikipedia.org/wiki/File:Comet_Kohoutek_orbit_p391.svg) and then zoom off into the outer Solar System for a long time before coming back again.
It’s actually really hard for things to “escape” the Solar System; they have to be going very fast – if not they’ll end up looping back in again eventually. Kind of like throwing something up in the air – unless you throw it very hard it will fall back down again.
There are also an awful lot of these things. [This little animation](https://en.wikipedia.org/wiki/File:PIA22419-Neowise-1stFourYearsDataFromDec2013-20180420.gif) shows the things detected by Nasa’s “Near-Earth Object Wide-field Survey Explorer” programme, from 2014-2018. The blue rings are the orbits of the inner planets (the paler blue one is us, Earth). Each dot represents an asteroid or comet – the green ones are asteroids that pass near the Earth at some point in their orbit, and the yellow dots are comets that tend to be more far-flung and just zoom by (but will come back eventually).
While space is ever expanding, it is only expanding on truly huge scales – at scales between clusters of galaxies. Asteroids and comets are things within the Solar system, so much, much more local.
Interestingly the comets don’t fly that far away from sun. Orbital period of Halley’s comet is like 75 years and the furthest it goes is just around Neptune.
Or the Hale-Bop comet (seen in 1995) will return in around year 4400 and max distance will be just 10x the distance of Neptune (2 light days). Voyager 1 is already half that much away from us.
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