Is there an astrophysicist in the room?

moved to exploration subforum. better chance of meeting an astrophycist :)

btw. - there is also an astronomy subforum.
 
Not an astrophysicist, but I suspect you're right that the rings wouldn't be uniform like that with the two worlds as close together as they seem to be, especially at the tilted axes like that. I'm not even sure that you could get a stable ring system in that kind of configuration at all. Maybe one further out orbiting both worlds, or if one of the worlds were a smaller moon orbiting with the rings (see Saturn).

Saturn's_Rings_PIA03550.jpg
 
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OP: fire up Universe Sandbox, put two ringed planets close to each other as in your example, and enjoy the show :)

edit: and here's what's going on... in short, chaos :)

3JohxnR.jpg
 
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