I have seen it said on Quora and Reddit that it's about every 220-250 million years to go around the Milky Way from Earth. Wondering what the rough equation is for calculating it. That is, I guess, the angular momentum and distance from the center, so I could visualize it on a sped up animation and otherwise derive something like those numbers 220 to 250.
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I'm not 100% sure what equation you would like, but I'll try at an answer to see how it goes.
The observed speed of the Sun orbiting around the center of the Milky way is about 220 km/sec. The distance to the center is about 8.1 kiloparsec, which is 2.5E+17 kilometers.
The time it takes to travel around a circle is the circumference divided by the speed,
$$\frac{2 \pi R}{v} \approx \frac{6.28 \times 2.5\times 10^{17}}{220} \approx 7.17 \times 10^{15} \text{seconds} \approx 227 \text{ million years}$$
The speed is a measured quantity. There isn't a simple way to calculate the speed with an equation, because the mass distribution in the galaxy is complicated and dominated by dark matter, whose presence we can only infer from the orbital motion of our own star as well as a large number of other stars in our galaxy.
From Wikipedia's Milky Way:
Galaxy rotation curve for the Milky Way. Vertical axis is speed of rotation about the Galactic Center. Horizontal axis is distance from the Galactic Center in kpcs. The Sun is marked with a yellow ball. The observed curve of speed of rotation is blue. The predicted curve based upon stellar mass and gas in the Milky Way is red. Scatter in observations roughly indicated by gray bars. The difference is due to dark matter
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