Does the Barycenter shift/combine when 2 planets line up with the sun?

Question

On August 1st-2nd, Earth & Saturn lined up together in relation to the sun. Here is a website that discusses it: https://www.healththoroughfare.com/science/saturn-will-line-up-with-earth-and-the-sun-in-beautiful-celestial-show/34561

While the two planets were in alignment, did the Barycenter between the sun & both planets combine into 1 Barycenter? Or did both planets retain their usual Barycenters?

Answer 1

The barycenter of two point masses is a mathematical concept. It's just the mass "weighted" average of their positions:

$$\mathbf{r_{B,12}} = \frac{m_1 \mathbf{r_1} + m_2 \mathbf{r_2}}{m_1+m_2}$$

If you have three bodies you can also write

$$\mathbf{r_{B,123}} = \frac{m_1 \mathbf{r_1} + m_2 \mathbf{r_2} + m_3 \mathbf{r_3}}{m_1+m_2+m_3}$$

Barycenters are just math tools that are useful when simplifying a real world problem to something, well... simpler. In a complicated situation like the solar system, the barycenter of any on planet with the Sun isn't a very useful concept at all, unless that planet happens to be Jupipter which is a real solar system bully.

Does the Barycenter shift/combine when 2 planets line up with the sun?

The Sun-Earth barycenter and Sun-Saturn barycenter just keep moving along pretty much independently of each other, to the same extent that the two planets just keep orbiting the much, much more massive Sun pretty much independently of each other.

And of course the Sun also moves around the barycenter for the whole solar system in a squiggly sort of way as shown in the question Is the barycenter of the solar system usually outside of the sun?

If you like, you can think of that residual motion as a bit like the average of the motions of all the individual planet's Sun-planet barycenters.