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Do all stars in orbit around Sagittarius A have approximately circular orbits or are there stars that orbit the Milky Way center very elliptically, like being in the Milky Way's outer rim at apoapsis and in the core worlds close to Sagittarius A at periapsis? Do we know what orbital eccentricity the Sun (respectively the Sun-Jupiter barycenter) has?

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I'm not sure what the focus is on Sgr A*? Only the stars that are very close to the Galactic center can be said to be "orbiting Sgr A*", the rest of the stars in the Galaxy orbit in the the entire Galactic potential, to which Sgr A* is a minor contributor.

The stars near Sgr A* have a wide variety of orbital eccentricities. This is shown clearly in the picture below (credit: Cmglee, Wikimedia commons).

enter image description here

Stars in the Galaxy at large also show a variety of orbital eccentricities in their non-Keplerian orbits. The orbits are non-Keplerian because they are not orbiting a point-like source of gravity and the orbits do not form closed shapes.

Many, in fact most, of the stars near the Sun are in the Galactic disc population and have roughly circular orbits, confined close to the plane of the Galaxy. About 1% of nearby stars are part of the old halo population which have a wide dispersion of eccentricities and orbital inclinations.

The Sun is currently in a mildly "elliptical orbit" (it is not a closed ellipse as discussed above and it wobbles above and below the Galactic plane), since its motion is not entirely tangential to the Galactic centre - it has components of 5-10 km/s towards the Galactic centre and upwards out of the Galactic plane, compared with a tangential orbital velocity of around 240 km/s. See https://astronomy.stackexchange.com/a/12311/2531 and How far is the Earth/Sun above/below the galactic plane, and is it heading toward/away from it?

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    $\begingroup$ What's a non-Keplerian orbit? I thought every orbit is "closed", what would such non-Keplerian orbits look like? Also, do you know the scale for your above diagram, how long e.g. is a light-year or parsec there? $\endgroup$
    – John
    Oct 31 '20 at 11:32
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    $\begingroup$ @John the diagram has a scale. The orbit of Pluto is a little bit less than one small graph paper square across, so I guess one small graph paper square is about 120 au. Non-Keplerian means it isn't an ellipse. Not a closed orbit means it doesn't return to the same point. $\endgroup$
    – ProfRob
    Oct 31 '20 at 13:34
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    $\begingroup$ @John stable ellipses are the solution to orbits around a spherically-symmetric mass. The galactic distribution of mass is not spherically-symmetric. $\endgroup$
    – BowlOfRed
    Oct 31 '20 at 18:32
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    $\begingroup$ @John "What's a 'non-Keplerian' orbit? What are some examples, and can some still be closed?" is an excellent question and can have several interesting answers that belong in a new post rather than just in some comments. Please consider asking something like that separately! $\endgroup$
    – uhoh
    Nov 1 '20 at 1:14
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    $\begingroup$ @uhoh But my question is kind of answered already, just like the one about precipitation on Mars. You can still ask these questions as a post if you wanna. $\endgroup$
    – John
    Nov 1 '20 at 6:54

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