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We know the speed that the sun travels around the galaxy is $220\, {\rm km/s} = 0.00021 \, {\rm parsec/year}$. I'm unsure of how to find the period?

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    $\begingroup$ Find the distance of the Sun from the Galactic Center and then, assuming a circular orbit, find the circumference... $\endgroup$ Commented Mar 5, 2020 at 2:17
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    $\begingroup$ or the value is easily found by searching for google.com/… $\endgroup$
    – James K
    Commented Mar 5, 2020 at 7:14

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The time it takes for the Sun to complete one orbit around the center of the Galaxy is called a Galactic year. It is estimated to last 225 to 250 million terrestrial years.

Now if you know the speed of the Sun in the Galaxy and you know how far the Sun is from the center, you can estimate this period. Let's assume a circular orbit and ignore the (large) error bars, for the sake of simplicity. The Sun is roughly $8\,200$ parsecs away from the Galactic center. So the circumference of the Sun's orbit is $8\,200 \cdot 2 \cdot \pi = 52\,000 {\rm \,parsecs}$. You mention the speed of the Sun is $2.1\cdot 10^{-4} {\rm \,parsec/year}$. So to complete one orbit, it takes circumference/speed = $52\,000 / 2.1\cdot 10^{-4} = 245 \cdot 10^6 {\rm years}$, that is 245 million years. Pretty close to the actual value!

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    $\begingroup$ The 'actual value' is calculated not more complex than you show. Just the uncertainty in radial distance speed gives the uncertainty $\endgroup$ Commented Mar 5, 2020 at 13:45
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    $\begingroup$ This is fine for a ballpark figure, but the Sun's orbit is not a circle. Perhaps you could address how circular it is and how inaccurate that might make your estimate? $\endgroup$
    – ProfRob
    Commented Apr 7, 2021 at 12:04

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