I was pondering a recent comment by Henning Makholm to this answer, he was addressing whether the distance from us of objects beyond Uranus are more affected by our orbit, or their orbits, depending on their orbital eccentricity:
Assuming that our hypothetical object orbits (very) roughly in the ecliptic plane, I don't think eccentricity matters... if the object moved fast enough to get away from Earth faster than Earth goes around half its orbit, it would have more than solar escape velocity!
I had to chew on that for a while.
What is the relationship between an object's distance from its primary, and how eccentric its orbit can be before it will escape the primary? Does this depend on how deep the gravity well is? Is there a term for this, and a formula?