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Knowing it has exited the solar system, is it traveling in the direction of the center of the galaxy, away from it, orbiting it, or maybe even going perpendicular from the disk-shaped galaxy?

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"Stars and gases at a wide range of distances from the Galactic center orbit at approximately 220 kilometers per second. " from Wikipedia: http://en.wikipedia.org/wiki/Milky_Way

That's much faster than the Voyager probes relative to the sun (Voyager 1: 17 km/s). Hence the probes will orbit the galactic center roughly the same way as our solar system, even after occasional hyperbolic encounters with other stars.

Voyager 1 is travalling in the direction of Ophiuchus, seen from Earth. She's at or beyond the border of the heliosphere.

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Are they then on the same orbit as our solar system, but slightly ahead of us? Will the distance be increasing between the voyagers and solar system on this orbit? –  Ska Jan 7 at 1:40
    
The distance will be increasing for the next few million years; they are faster than the escape velocity from the sun. But they won't escape from the Milky Way, unless they get accelerated by an unpredictable close encounter to a star in a very distant future. –  Gerald Jan 7 at 2:39
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...Eventually the probes may again come closer to our solar system, but only in timescales of about a galactic year (en.wikipedia.org/wiki/Galactic_year), hundreds of millions of years, if they follow Kepler ellipses around the galactic center. But that's hard to predict over such long time scales. –  Gerald Jan 7 at 2:45
    
Why Ophiuchus? Some specific reason? –  Magno C Feb 4 at 11:49
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That's probably just a consequence of the Saturn flyby, coming from Jupiter. –  Gerald Feb 4 at 14:50

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