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How common are asteroids that orbit retrograde with respect to the motion of all other bodies? How stable are said orbits in the Solar System, and where can we find most of them?

I presume that there are a large amount of them in the scattered disk, as they probably were ejected by the giant planets.

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In my opinion, most asteroids in retrograde orbits should continue to orbit in retrograde orbits for billions of years. Experts in orbital mechanics would have more expert opinions, of course.

Asteroids usually have a prograde orbit around the Sun. Only a few dozen asteroids in retrograde orbits are known.

Some asteroids with retrograde orbits may be burnt-out comets,[16] but some may acquire their retrograde orbit due to gravitational interactions with Jupiter.[17]

Comets from the Oort cloud are much more likely than asteroids to be retrograde.[16] Halley's Comet has a retrograde orbit around the Sun.[24]

Most Kuiper belt objects have prograde orbits around the Sun. The first Kuiper belt object discovered to have a retrograde orbit was 2008 KV42.[25] Other Kuiper belt objects with retrograde orbits are (471325) 2011 KT19,[26] (342842) 2008 YB3, (468861) 2013 LU28 and 2011 MM4.[27] All of these orbits are highly tilted, with inclinations in the 100°–125° range.

Meteoroids in a retrograde orbit around the Sun hit the Earth with a faster relative speed than prograde meteoroids and tend to burn up in the atmosphere and are more likely to hit the side of the Earth facing away from the Sun (i.e. at night) whereas the prograde meteoroids have slower closing speeds and more often land as meteorites and tend to hit the Sun-facing side of the Earth. Most meteoroids are prograde.[28]

https://en.wikipedia.org/wiki/Retrograde_and_prograde_motion#Solar_System_bodies

Stars and planetary systems tend to be born in star clusters rather than forming in isolation. Protoplanetary disks can collide with or steal material from molecular clouds within the cluster and this can lead to disks and their resulting planets having inclined or retrograde orbits around their stars.3 Retrograde motion may also result from gravitational interactions with other celestial bodies in the same system (See Kozai mechanism) or a near-collision with another planet,1 or it may be that the star itself flipped over early in their system's formation due to interactions between the star's magnetic field and the planet-forming disk.[31][32]

The accretion disk of the protostar IRAS 16293-2422 has parts rotating in opposite directions. This is the first known example of a counterrotating accretion disk. If this system forms planets, the inner planets will likely orbit in the opposite direction to the outer planets.[33]

WASP-17b was the first exoplanet that was discovered to be orbiting its star opposite to the direction the star is rotating.[34] A second such planet was announced just a day later: HAT-P-7b.[35]

In one study more than half of all the known hot Jupiters had orbits that were misaligned with the rotation axis of their parent stars, with six having backwards orbits.2

https://en.wikipedia.org/wiki/Retrograde_and_prograde_motion#Exoplanets

How long can an object last in a retrograde orbit?

All retrograde satellites experience tidal deceleration to some degree. The only satellite in the Solar System for which this effect is non-negligible is Neptune's moon Triton. All the other retrograde satellites are on distant orbits and tidal forces between them and the planet are negligible.

https://en.wikipedia.org/wiki/Retrograde_and_prograde_motion#Natural_satellites_and_rings

Tidal interactions also cause Triton's orbit, which is already closer to Neptune than the Moon's is to Earth, to gradually decay further; predictions are that 3.6 billion years from now, Triton will pass within Neptune's Roche limit.[25] This will result in either a collision with Neptune's atmosphere or the breakup of Triton, forming a new ring system similar to that found around Saturn.[25]

https://en.wikipedia.org/wiki/Triton_(moon)#Orbit_and_rotation

So Triton has been orbiting Neptune for billions of years, and still has billions of years left before crashing into neptune, and other retrograde natural satellites areso far from their planets that the tidal forces decaying their orbits are insignificent compared tothose acting on Triton. So it is my guess that retrograde asteroids can orbit the Sun for many billions of years without their orbits decaying much.

But possibly an expert on orbital mechanics would know reason why retrograde asteroid orbits might not last that long.

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