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I know that they will eventually get close (within 10,000 light years) of our solar system, but will the orbit be circular or elliptical, around how much eccentricity, what would be a reasonable apoapsis and periapsis estimate... I know that these are kind of hard to predict, but could we make reasonable estimates?

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    $\begingroup$ Since the galaxy is an extended, lumpy distribution of mass (both light and dark) and not a single point, orbital motion of Voyager or of any of the other objects of the galaxy won't be Keplerian, so the shape of the motion won't be circular or elliptical and things like eccentricity won't apply. But it will still "orbit" the center of galaxy in the sense that like the Sun it will slowly go round and round. But I don't know how much different Voyagers trajectory will differ from that of the Sun. $\endgroup$
    – uhoh
    Commented Feb 9, 2022 at 21:30

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Wikipedia's Galactic year (yes I'm not making it up) says:

The galactic year, also known as a cosmic year, is the duration of time required for the Sun to orbit once around the center of the Milky Way Galaxy. One galactic year is 230 million Earth years. The Solar System is traveling at an average speed of 230 km/s (828,000 km/h) or 143 mi/s (514,000 mph) within its trajectory around the galactic center, a speed at which an object could circumnavigate the Earth's equator in 2 minutes and 54 seconds; that speed corresponds to approximately 1/1300 of the speed of light.

Voyagers 1 and 2 have escape velocity and are currently traveling at only 17.5 and 15.5 km/s and in somewhat different directions, but since it's a small fraction of the Solar System's velocity we can say that they're roughly traveling on the same general path as the Sun.

It's hard to say anything more because the Milky Way has a lumpy, irregular mass distribution, so one can't model where either of them will end up exactly in another galactic year; their orbits won't be circles nor ellipses.

From Wikipedia's The Sun; Orbit in the Milky Way:

The Sun's orbit around the Milky Way is perturbed due to the non-uniform mass distribution in Milky Way, such as that in and between the galactic spiral arms. It has been argued that the Sun's passage through the higher density spiral arms often coincides with mass extinctions on Earth, perhaps due to increased impact events. It takes the Solar System about 225–250 million years to complete one orbit through the Milky Way (a galactic year),[146] so it is thought to have completed 20–25 orbits during the lifetime of the Sun.

From The galactic cycle of extinction (also archived)

Fig. 1. Cycle of high extinction events and superchrons in a four spiral arm model of the Milky Way Galaxy.

Fig. 1. Cycle of high extinction events and superchrons in a four spiral arm model of the Milky Way Galaxy. The current position of the Sun is indicated with an approximately circular orbit shown by the light blue circle (7.9 kpc radius). The positions of the six extinctions (jagged symbol) and three superchrons (purple lines) are determined from 703.8 Myr (175.96r4) as the time to pass through all four arms. Original figure of spiral arms from Valle´e (2005).

What's that weird black chevron at the bottom?

The reason will likely be related to Why the blank wedges in this very early 21 cm map of the Milky Way? (Oort et al. 1958) See also this map.

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    $\begingroup$ The weird black chevron poorly represents the portion of the milky way blocked from 21cm radio astronomy observation by the galactic core and bar (represented by the black bar-like structure). Better graphic: upload.wikimedia.org/wikipedia/commons/thumb/1/1d/… $\endgroup$
    – Woody
    Commented Aug 5, 2023 at 15:57

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