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According to Centaur minor planet

Centaurs are small Solar System bodies with a semi-major axis between those of the outer planets. They generally have unstable orbits because they cross or have crossed the orbits of one or more of the giant planets

Of all the known ones which I've found and I've seen data, all of them overlaps the orbit of Saturn, with the exception of (514107) 2015 BZ509 which doesnt overlap the orbit of Saturn but the orbit of Jupiter.

My question is, are there known centaurs which dont overlap the orbits of Jupiter and Saturn but which orbit between those 2 orbits? Or does the definition of Centaur mean they have to cross orbits and an asteroid orbiting between those 2 orbits would be called another name if exists?

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The JPL Small Body Database lists 5 asteroids and 5 comets with q > 4.56 au and Q < 9.04 au. Two of those asteroid orbits have large uncertainties due to few observations and may not actually meet your criteria. These are fairly well determined:

  • Asteroid 494219 has a highly inclined 14-year orbit which occasionally brings it within 2 au of Jupiter. It was discovered in 2012.
  • Asteroid 2000 GM137 has a 22-year orbit which occasionally passes within 2 au of Saturn. It was discovered in 2000, unobserved for a few years, and recovered in 2014.
  • Comet 29P/Schwassmann-Wachmann 1 was discovered in 1927, has a nearly circular 15-year orbit, and is known for frequent outbursts.
  • Comet 39P/Oterma was discovered in 1943 on an 8-year orbit. A 1963 encounter with Jupiter switched it to its present 20-year orbit, where it became dormant and was lost until 2001. Numerical simulations indicate a prior Jupiter encounter in 1937; it was likely also a Centaur before that.
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