Eccentricity is a way of characterizing the curve the object makes around the sun. <1 is an ellipse, and so the object would be in orbit around the sun, and hence probably not interstellar. >1 is hyperbolic, and thus not gravitationally bound to the sun, and so probably interstellar.The closer the eccentricity is to 1 without being equal or less, the more "curved" the trajectory is and the nearer its closes approach to the sun will be.

 > The closer the eccentricity is to 1 without being equal or less, the more "curved" the trajectory is and the nearer its closes approach to the sun will be.Is this true? I thought it depends on the objectâ€™s speed. You can have an object with e=3 have a closer approach to the sun than an object with e=2 if the first object is traveling sufficiently faster.In other words, at a given perihelion, you can change an objectâ€™s e by accelerating or decelerating. Not a orbital mechanics major, just played too much KSP.
 ... for a given semimajor axis, is what the OP forgot to add.
 The absolute nearness of approach depends on the energy of the orbit as well as it's eccentricity.Talking about the "speed" is potentially unclear as an object's speed can vary greatly within its orbit.

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