I have now learned that the Sun is not fixed in place, so I asked myself this question: It is possible, that the Sun and our Solar System will one day leave our galaxy? If yes, how, and if no, why not?
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It's certainly possible for a star system to be stripped away if its galaxy interacts gravitationally with a second galaxy. Tidal forces between the two can disrupt stellar orbits and form tidal tails and other possibly transient structures. This won't happen on human timescales - no satellite galaxies are anywhere close enough to us to pose a threat - but in several billion years, the Milky Way will collide with Andromeda in a process that will completely disrupt the structures of both galaxies.
It's been suggested that the Solar System may be stripped from its present galactic orbit during the collision. Cox & Loeb 2008 suggested that there is a 12% chance that we will be swept into a tidal tail >20 kpc from the center of the merger remnant, and a >50% chance that we will end up >30 kpc from the center, will in the outer halo - compare both of these with our current distance of ~8 kpc from the galactic center. Andromeda could even strip us away on its first pass at the Milky Way, though the authors regard this as unlikely.
All of these outcomes, of course, do still end with the Solar System being bound to the remnant. It's certainly true that we could be ejected from the system entirely, but that probability appears low compared the odds that we will simply move into a comparatively wide halo orbit.
If you regard the scenario where we're temporary bound to Andromeda as leaving the galaxy, then yes, that's indeed a possibility. Unfortunately (or fortunately?), we will soon coalesce once again with the Milky Way on timescales of ~1-2 billion years.
As an aside: There are other ways for a star to be ejected from a galaxy. One of the more common involves an encounter with a supermassive black hole. If a binary star approaches the black hole, the resulting three-body encounter will disrupt the system; one star may be captured by the black hole, while the other may be ejected at speeds greater than the galactic escape velocity. We know of several dozen of these so-called hypervelocity stars, including US 708 and S5-HVS1.
answered Jul 10, 2020 at 14:25
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2$\begingroup$ And in a quadrillion years or so, many of the remaining stars will "boil" out of the galaxies. math.ucr.edu/home/baez/end.html Of course, the Sun will be a black dwarf long before that happens. $\endgroup$– PM 2RingJul 10, 2020 at 16:28
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$\begingroup$ While being ejected, will the Solar system as a whole (i.e. the orbits of its planets relative to the Sun) remain mostly unaffected? $\endgroup$– RuslanJul 11, 2020 at 20:39
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1$\begingroup$ @Ruslan The individual planets will almost certainly be unaffected. They could only be disrupted by very close encounters with nearby stars, which could only happen on timescales of $\sim10^{12}$ to $\sim10^{15}$ years (Laughlin & Adams 1997). $\endgroup$– HDE 226868 ♦Jul 11, 2020 at 20:52