I'm curious to know, does the sun really revolve around the black hole at the center of the galaxy?
Sun’s Movement – Does the Sun Revolve Around the Supermassive Black Hole at the Center of the Milky Way?
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As it takes the sun about 250 million years (250 My) to orbit the galaxy, the proper motion of stars relative to the Sun will be the dominant effect of changes in the sky. The visible effects of the rotation will be far slower.
All stars move in the sky, some faster some slower, and in more or less random directions, not just moving around the galaxy. For example, Vega moves about 1 degree every 11,000 years. Around 12,000 BCE it was the pole star, and will be so again around 14,000 CE. Between now and then, other stars like Gamma Cephei and Iota Cephei will temporarily take the role of Polaris.
By 250 million years most stars will be far from their current position in the sky, but because of uncertainties it's impossible to say just where they'll be. For example, if the estimate of 250 My is wrong by just 1% (or 2.5 My), that means about 100 periods of Vega. Hence by that time Vega could be anywhere at all even if it stays in our general neighbourhood - which is certainly not guaranteed.
Galaxies move as well, but because they are much further away, their apparent position changes much slower than that of stars. It will be mainly our rotation around the galaxy that moves them in the sky.
Using telescopes we have already seen differences in the positions of the closer stars. In 10,000 years many changes will be visible to the naked eye. By the year 250 My, the sky won't look even remotely like the present.
Short answer, no.
The Sun's orbit is non-Keplarian; there are many perturbations and a general unevenness in the motion of the Sun around the Galactic centre. This is a result of non-uniform mass distributions, the galaxy not being a point mass, and the impact the relative motions of neighbour stars has on measuring. Thus, giving a particular eccentricity for the Sun is almost meaningless. For instance, it fluctuates up and down roughly $2.7$ times per orbit and it passes through high density regions which cause major perturbations. This creates instability in any average eccentricity.
Long answer, it is not impossible.
In theory, we could measure it. However, we have two rest frames; local and standard rest. The local rest frame refers to how we can take the average motion of stars within (say) $100~pc$ and use this average to compute our approximate orbital properties. The standard rest frame refers to us using Oort constants/properties and similar things in order to determine our more specific motion around the galaxy based on accelerational perturbations, etc. Both frames have their own advantages and both give slightly different values for our currently computed orbital characteristics. The problem lies with determining the relative weights each might contribute to an eccentricity value.
While the motion of the Sun may be non-Keplarian, we do know that the circular velocity is around $230{km\over s}$ and the peculiar velocity is on order $15{km\over s}$. This leads many astronomers to say that while measuring the eccentricity would be very hard and calculating it would be near impossible, they can say that it is most likely on the order of a few percent. Definitely less than $10\%$, but a value in the range of $e=0.02-0.08$ would be the most likely.
Best Answer
Sun revolves around the center of our galaxy. It takes about 225-250 million years (the timespan known as galactic year) to complete full circle. Taking this image from the wiki
The center of the galaxy indeed contains the supermassive black hole. However the situation is significantly different from say the solar system. In the solar system almost all the matter is contained in Sun. The rest of the planets, moons, asteroids and comets contributes very little. So planetary orbits are determined mostly by Sun's gravity. In contrast the supermassive black hole in the center of our galaxy constitutes a miniscule proportion of the total galaxy mass. Also the significant part of the galaxy mass is in the dark matter halo which is spread even more than a visible matter. So saying that Sun revolves around supermassive black hole in the center is probably misguiding - its motion is determined mostly by other matter gravity.