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From the simple perspective of someone like myself, it appears that our sun is "fixed in place" (from the perspective of the solar system itself) and that everything else of lower mass orbits around it. I suspect that the sun wobbles due to the other objects (that the barycenter is either near it or within its radius).

It appears from reading about binary stars of more-or-less equal mass that they cannot have one "fixed in place" and the other orbiting it. The barycenter is basically between them.

Is that perspective correct? Is it ever possible for a low mass star to be the "fixed in place" center with a large mass star orbiting it? Or does the barycenter always shift toward the more massive object?

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The barycenter of a system consisting of two objects of mass m1 and m2 with their centers separated by a distance d is always along the line connecting the two centers and m1/(m1+m2) of the way along the line from the center of m1 to the center of m2.

So for the Sun and the Earth, where the Sun weighs in at about 333,000 Earth masses, the barycenter will be about 1/333001 of the way from the center of the Sun to the center of the Earth, or about 280 miles from the center of the Sun in the direction of Earth.

(So it might as well be fixed in place for all you or I could tell without high-quality instruments, but it does wobble in a 270-mile orbit once each year.)

The presence of the rest of the planets makes this all a lot more complicated, of course and the Sun actually traverses a much more complicated path than it would if the Solar System was a simple two-body system. In this case, the barycenter is possibly better thought of as the center of mass.

OTOH, since Jupiter is so much heavier than everything else, to a decent first approximation the Solar System is a binary system. The barycenter for the Sun-Jupiter pair is actually outside the Sun, at about 1.09 Solar radii.

See my answer at What point does Earth actually orbit? for more.

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"Barycenter" is really just a fancy term for the center of mass of a system. You can compute it quite easily if you know the positions of all the objects in a system at a given time. This is usually quite handy, at least when dealing with either a pair of similar-mass objects, such as a binary star, or in systems where one object is gravitationally dominant and most other objects orbit it, like in a planetary system.

A barycenter isn't really "fixed", because no reference frame is absolute; we know this from the principle of relativity, which discards the idea of privileged inertial reference frames. However, when discussing orbital mechanics, it's often quite convenient to use barycentric coordinates. In a binary star system, for instance, the barycenter is a focus of the orbits of both bodies, and so it is an obvious choice for designating as a reference point, as we can then calculate the stars' positions.

Now, things get complicated if we look at more complex systems. For instance, Earth doesn't really orbit the Solar System's barycenter; it orbits the Sun-Earth barycenter - which is, for most purposes, the center of the Sun. Finding the barycenter of the Solar System isn't too important if you're computing Earth's orbit approximately (for more exact calculations, you might want to take other bodies into account, namely, the giant planets).

Note that the equation for the center of mass of two bodies of masses $m_1$ and $m_2$ and positions $\mathbf{x}_1$ and $\mathbf{x}_2$ is $$\mathbf{x}_{\text{cm}}=\frac{m_1\mathbf{x}_1+m_2\mathbf{x}_2}{m_1+m_2}$$ and if $m_1\ll m_2$, $$\mathbf{x}_{\text{cm}}=\frac{m_1\mathbf{x}_1}{m_1+m_2}+\frac{m_2\mathbf{x}_2}{m_1+m_2}\approx\frac{m_1}{m_2}\mathbf{x}_1+\frac{m_2}{m_2}\mathbf{x}_2\approx\mathbf{x}_2$$

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