Measuring the Severity of Close Encounters Between Ringed Small Bodies and Planets

Rings have recently been discovered around the trans-Neptunian object (TNO) 136108 Haumea and the Centaur 10199 Chariklo. Rings are also suspected around the Centaur 2060 Chiron. As planetary close encounters with ringed small bodies can affect ring longevity, we previously measured the severity of such encounters of Chariklo and Chiron using the minimum encounter distance, d(min). The value of d(min) that separates noticeable encounters from non-noticeable encounters we called the 'ring limit', R. R was then approximated as 10 tidal disruption distances, 10R(td). In this work, we seek to find analytical expressions for R that fully account for the effects of the planet mass, m(s), small body mass, ms, ring orbital radius, r, and velocity at infinity, v(infinity), for fictitious ringed Centaurs using ranges 2 x 10(20) kg <= m(s)<= 1 Pluto mass and 25 000 <= r <= 100 000 km. To accomplish this, we use numerical integration to simulate close encounters between each giant planet and ringed Centaurs in the three-body planar problem. The results show that R has a lower bound of approximately 1.8R(td). We compare analytical and experimental R values for a fictitious Haumea, Chariklo, and Chiron with r= 50 000 km. The agreement is excellent for Haumea, but weaker for Chariklo and Chiron. The agreement is best for Jupiter and Saturn. The ring limits of the real Haumea, Chariklo, and Chiron are <4R(td). Experimental R values for the fictitious bodies make better approximations for the R values of the real bodies than does 10R(td). Analytical values make good first approximations.