Please make whatever assumptions are needed. Assume sun approaches black hole on some sort of hyperbolic trajectory. Assume size of supermassive black hole is same as the one at center of our galaxy. How close would sun have to get in order for Jupiter to be “slingshot” into interstellar space.
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2$\begingroup$ I think "An orbiting pair of objects (m2, m3) passes a much more massive object m1 on a hyperbolic trajectory, at what distance of closest approach will it be possible for the orbiting pair to separate after the encounter?" is on-topic. But I think you should ask your unrelated follow-up question as a new question since the answers are totally different. $\endgroup$– uhohCommented Mar 22, 2020 at 2:43
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$\begingroup$ One question per post is the rule. $\endgroup$– StephenG - Help UkraineCommented Mar 22, 2020 at 3:18
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$\begingroup$ I've removed the second question, as per the "one question per post" guideline. $\endgroup$– James KCommented Mar 22, 2020 at 8:14
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Closer than about 10000AU, 1.5 trillion km or about 2000 times the orbital radius of Jupiter.
Three body systems are peculiar beasts, and but I'm going to change the question to "Would Jupiter be in a stable solar orbit?" This makes it a question of Hall radius.
The Hall radius is $r_H = a\sqrt[3]{m/3M}$ where $m/M=1/4000000$ is the ratio of solar mass to black hole mass (the mass of Jupiter is treated as negligible!) and $a$ is the orbital distance of the sun about the black hole
If we set $r_H$ to 5 (AU) and solve for $a$ we get a value of about 10000 (I'm only working at 1 s.f. accuracy here. In other words, if the sun is orbiting the black hole at 10000AU then Jupiter would be in a marginally stable orbit. If the sun got closer than 10000AU then Jupiter would tend to orbit the black hole and not the sun.
However this is only an upper bound. On a hyperbolic orbit the sun would spend little time near the black hole and so even if Jupiter had drifted in the black hole's gravity as the sun passed by, as the sun and jupiter moved away, it would tend to fall back into orbit around the sun. To model whether Jupiter would actually be lost would require knowledge of the initial conditions and numerically solving the differential equations of motion.
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$\begingroup$ Ok I hadn’t considered that Jupiter being captured into orbit around the black hole would be very likely. It sounds like (from your answer) that there may be a somewhat narrow window of initial conditions whereby Jupiter would be flung into interstellar space. $\endgroup$ Commented Mar 22, 2020 at 16:31
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$\begingroup$ jupiter being captured is the alternative, if the Earth is "slingshotted" away, that momentum comes from somewhere, and it can come from Jupiter, which by losing momentum is captured into an orbit. In 3 body dynamics many things are possible. $\endgroup$– James KCommented Mar 22, 2020 at 16:49
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$\begingroup$ But this is sort of an anti-answer: If you are more than 10000au away, everything should be okay. If you are less than this, then watch out. but initial conditions matter and $\endgroup$– James KCommented Mar 22, 2020 at 16:50
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$\begingroup$ I just read this from another website “Near the galactic center, the average distance between neighboring stars is expected to be about 1000 AU”. Intuitively, it seems unlikely for a planet, at Jupiter’s distance from the sun, to survive a trip through the galactic center without being captured by a more massive object. $\endgroup$ Commented Mar 22, 2020 at 18:45
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$\begingroup$ Ooh and then there’s this google.com/amp/s/www.sciencealert.com/… $\endgroup$ Commented Mar 22, 2020 at 18:54