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Jupiter's magnetosphere extends "seven million kilometers in the Sun's direction and almost to the orbit of Saturn in the opposite direction" and Saturn's magnetosphere extends towards the Sun on average 22 Rs (Rs=60,330 km is the equatorial radius of Saturn). My question is: do these magnetospheres actually touch each other when the planets are on the same line?

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The simple answer is yes, because magnetic fields extend to infinity.

Your question should ask if a significant interaction is present. My question would be, what is a significant interaction?

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my question was about magnetospheres, I corrected the title to better reflect this intent –  symbiotech Apr 27 at 2:58
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@symbiotech The magnetospheres are created by the magnetic fields from the planets. Like I said, how weak does the magnetosphere need to be before you consider it is too weak for an interaction? –  LDC3 Apr 27 at 3:14

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