Is there a planet or planetary moon flyby path that could send a solar prograde orbiting asteroid into a retrograde solar orbit?

If an asteroid on an inner orbit catches up with a planet on an outer orbit, and passes close enough to the planet to be strongly influenced by the planet's gravity, it can whip around the planet ahead of the planet and then move in and orbit backwards (retrograde) farther from the sun than the planet.

So the asteroid might be captured by the planet if its retrograde speed is slow enough, and then move in a backwards or retrograde orbit around the planet. A lot of small asteroid sized moons of the giant planets have retrograde orbits indicating they were captured that way - also Triton, the large moon of Neptune. I just counted abut 119 natural satelelites with retrograde orbits in our solar system.

https://en.wikipedia.org/wiki/List_of_natural_satellites[1]

And the the asteroid's retrograde speed after passing around the planet is fast enough, it won't be captured by the planet but will continue to orbit the Sun, but in a retrograde orbit.

In fact, a very tiny percentage of all known asteroids have retrograde orbits, only 82 out of more than 726,000.

• "...is fast enough, it won't be captured by the planet..." An asteroid passing a planet at any speed can't be captured into orbit in two-body dynamics unless it enters the planet's atmosphere, at which point even if it is captured it's orbit will quickly decay. Neither slow nor fast objects can be captured into long-lived orbits unless there's at least a third body involved.
– uhoh
Mar 12, 2021 at 1:03
• Can you cite an example of an asteroid trajectory and a planet such that the asteroid's heliocentric orbit begins as prograde and ends as retrograde? I think that without numbers and math or at least supporting sources, the OP's question as asked hasn't been answered yet: "Is there a planet or planetary moon flyby path that could send a solar prograde orbiting asteroid into a retrograde solar orbit?"
– uhoh
Mar 12, 2021 at 1:03
• @uhoh YES! This is a disgustingly difficult little problem. Easily stated, but hard to solve. M.A. Golding has a good start, but is a long way from a solution! Mar 12, 2021 at 5:18