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When I was small, I read that Sun is fixed at the center of the solar system and that all the other planets rotate around it.
But later I heard that even the Sun is not fixed; it moves. Is this true?
Why had people previously thought that the Sun is fixed?
Is this correct planets in helix path or spring like path?
The Sun moves, even in the context of the solar system. Gravity of the planets (mostly Jupiter) pulls the Sun out of position with respect to the centre of gravity of the solar system. This wikipedia entry explains it in a lot more detail, and also explains that their common centre of gravity lies outside of the sun.
This wobbling of a star due to planets orbiting them is also one method of how we detect planets around other stars.
Assuming that the Sun is fixed within the context of the solar system is a fairly good approximation. You need accurate and long term observations to detect the wobble.
The Sun orbits the Galactic center once approximately every 226 million years as the galaxy spins, which is also travelling through space towards an unknown source of gravity (that is currently blocked from view by the luminosity of the galaxy itself) known as "The Great Attractor".
The Sun moves in many ways. For one the sun does wobble, as stated above, due to Jupiter's pull, this is actually how astronomers are able to find new planets! They can use physics and Mathematics to figure out a planets size and number of planets orbiting a star just by studying the stars wobble.
Secondly the Sun along with the Solar System orbits the center of the Milky Way Galaxy. The length of this process is called a Galactic Year. The Solar System's Galactic year ranges somewhere from 225 to 250 million years.
Lastly our Galaxy and the Sun move as a whole through space, which is what will eventually cause the Milky Way Galaxy to collide with the Andromeda Galaxy.
The answer depends on your frame of reference. In a heliocentric frame, the Sun is fixed at the center by definition. In a barycentric frame, defined relative to the Solar system's combined center of mass, the Sun moves in a complex multi-year loop opposite the planets' orbits, never wandering more than its own diameter away from the barycenter as it is the most massive body by far. The distinction between these frames may not have been obvious until Newton published his laws of motion and gravitation. In a galactic or supergalactic context, the difference in motion between the Sun and planets is negligible.
One frame is not inherently more correct than another, but one may be more useful for a given purpose. When astronomers discover an asteroid, they measure its geocentric spherical coordinates. Having several such measurements over a span of time, they approximate its current trajectory with an ideal heliocentric ellipse. To estimate the asteroid's trajectory 100 years in the future, their N-body gravitational simulation would use barycentric coordinates.
Two questions that don't seem to have been answered yet:
Why had people previously thought that the Sun is fixed?
In fact, since the time of Newton in the late 1600's people have known that the Sun is in motion, wobbling around the center of mass of the solar system. And, before Copernicus, people thought the Earth was fixed, and the Sun moved around it. So the idea that the Sun was perfectly fixed really only lasted about 130 years.
And the reason Copernicus assumed it was fixed is that, at the time, that was the simplest model that would match the motions of the planets that he could observe.
Is this correct planets in helix path or spring like path?
Well, sort of. That picture looks like a frame from a video that's been circulating on the internet, and the video overall is hugely wrong in a lot of ways. But it is true that the whole solar system is moving around the center of the galaxy, and the plane of the solar system is tilted about 60$^\circ$ compared to the plane of the galaxy. So, if you combine the motion of the planets around the sun with the motion of the whole solar system around the center of the galaxy, you do get a kind of corkscrew pattern. But if you hear the term "vortex" for the solar system, that's a sign that you're in nonsense territory.
