The fact that all planets orbit the sun and the fact that mass of the sun is equivalent to 99% mass of our solar system doesn’t automatically means that the sun is stationary relative to the barycenter of our solar system.
Due to the lack of time to explain heliocentric principle in detail in most classes, some students get the impression that Sun is stationary. In reality, the Sun is only stationary relative to its own reference frame. But anything in this universe is stationary relative to its own reference frame. Our Earth is stationary relative to Earth reference frame, the Moon is stationary relative to Moon reference frame and even a chicken crossing the road is also stationary relative to its own reference frame.
Of course it is legitimate to argue that Earth reference frame, Moon reference frame and chicken reference frame are not inertial reference frame. But the same is also true for Sun reference frame. Sun reference frame is not an inertial reference frame.
Sun orbit the barycenter of our solar system
Unlike most people believe, the Sun is not even stationary relative to our solar system. Because of the gravitational acceleration exerted by planets orbiting it, the Sun actually also orbit a center of gravity known as barycenter. This barycenter is located inside the Sun itself, but is not exactly in the very center of the Sun itself. This is why we say that the Sun is the center of our solar system. Every planets in the solar system and the Sun itself actually orbit a point located inside the Sun. This means that even relative to our solar system, the Sun is not stationary.
Sun orbit the core of our galaxy
Sun orbit the core of our galaxy with orbital speed of 220 kilometers per second. This is because the galaxy have far more mass than our sun. It is estimated that our galaxy is about 100 billion times more massive that our sun. All massive objects always pull any other objects toward itself due to gravity. This means that if two objects don’t have enough relative velocity element perpendicular to a straight line between them, those two objects are going to collide.
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So for any two objects with different mass, the smaller one only have three possible fate. The first possible fate is to smaller object is going to fall into the bigger object. The second possible fate is the smaller object is going to orbit the bigger object. The third possible fate is the smaller object is going to escape the bigger object. Being stationary is not an option for smaller objects when there are bigger objects around.
Sun is not stationary relative to other stars
The fact that Sun is not stationary relative to other stars is known from the observed proper motion of stars and the Doppler shift of radiation we received from other stars. This is because all stars orbit the galaxy at slightly different orbital velocity. The proper motion means that nearby stars always change their positions relative to very distant stars in the background. Barnard star has the largest proper motion of all stars. It move about 10.3 arc seconds every year.
The Doppler shift on the other hand measure the vectoral component of a star relative velocity to our solar system perpendicular to its proper motion. We know from high school physics lessons that the frequency of sound from an incoming car is higher than the same sound coming from a car at rest relative to us. This is effect is called Doppler shift. By measuring the Doppler shift of radiation we received from each star, we can calculate their velocity relative to us. Some stars are blue-shifted which means that they are closing their distance relative to us, while the other are red-shifted which means that they are going away from us.
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