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This is the page I am referring to. It seems counterintuitive to me that the Sun should be on the opposite side of the barycenter's wobble. I realize I am wrong, but I cannot see why I am wrong. Can someone explain why the wobble is away from Jupiter, not towards Jupiter?

Here is an unedited screenshot of the NASA animation - it shows the sun on the opposite side of the green line (barycenter) as Jupiter.

NASA's Image

My logic says, since gravity is in play here, the Sun and Jupiter should be on the same side of the green line. I have edited NASA's image in MS Paint to show what I think should be happening:

What my logic says should be happening

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    $\begingroup$ The green circle is the Sun's orbit around the barycenter. $\endgroup$
    – PM 2Ring
    Jun 27 at 0:38
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    $\begingroup$ Think centrifugal force. When you swing a bucket full of water around you on a rope, why do you lean AWAY from the bucket not towards it? ((i know this is inaccurate or incomplete on many levels, but it should point the OP's thoughts in the right direction)) $\endgroup$
    – PcMan
    Jun 27 at 9:35
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    $\begingroup$ The common barycenter has to be between the barycenters of the two bodies in the first place. $\endgroup$
    – fraxinus
    Jun 27 at 10:41
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    $\begingroup$ @pcman I wish you had posted that as an answer so I could green check it. $\endgroup$
    – Bookaholic
    Jun 27 at 14:55
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    $\begingroup$ Stare at these until they make sense. $\endgroup$
    – J...
    Jun 28 at 15:48
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The part of your intuition that is correct is that Jupiter pulls the Sun towards it. The problem is that "pulls towards" does not mean "brings closer"! The gravitational force results in an acceleration towards an attracting body, which is not a displacement or even the derivative of displacement, but the second derivative of displacement. Oscillatory or circular motion has the property that the second derivative carries a minus sign. For example, when the Sun is on the right side of the green circle, its acceleration is to the left, because it is changing from rightward motion to leftward motion. Thus, by being on the opposite side from Jupiter, the Sun is continually accelerating towards it.

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    $\begingroup$ I dont think that he means that the two objects gravitating with one another means the objects will always move towards each other. He even thinks they move side by side around a common center. They dont move towards each other in this view. This view can actually be right if there is mass in the bary center. I think this is the cause of the trouble. The very word "barycenter" implies a center with mass. $\endgroup$ Jun 27 at 12:09
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    $\begingroup$ @DescheleSchilder I interpreted the confusion as "Jupiter's attractive force on the Sun should bring the Sun closer than it would be in the absence of such attraction". And if these bodies were, say, quasistatic masses on springs, that would be correct (e.g., Cavendish experiment). But in fact for orbital dynamics, acceleration is 180° out of phase with displacement. $\endgroup$
    – nanoman
    Jun 27 at 20:54
  • $\begingroup$ Im not sure I understand why you consider the acceleration having a minus sign. The acceleration is directed towards the center. Why do you consifer it negative? Because beiñg on a side means the acceleration is directed to the other side? Why not consider it positive? The change in velocity is directed to the center too.is the velocity negative then? Or do you consider the line from the center to the sun as a positive axis with the center as origin? In which case the cöordinate system would be rotating. $\endgroup$ Jun 29 at 0:07
  • $\begingroup$ @DescheleSchilder The minus sign I refer to is simply that for circular motion with angular velocity $\omega$, the acceleration is $\ddot{\mathbf{r}} = -\omega^2 \mathbf{r}$, i.e., opposite to the displacement. I thought this explained why the Sun is on the opposite side from where OP initially thought it should be. $\endgroup$
    – nanoman
    Jun 29 at 1:04
  • $\begingroup$ Thats clear! The best way to put it.:) $\endgroup$ Jun 29 at 1:20
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Two massive bodies always rotate around a stationary point between them. They rotate on opposite sides of this point (the center of the green and red circles). So they can't both be on the same side of this point rotating in tandem. Their momenta are always opposite.

To be a bit more specific: from where would the force come to make both rotate around the barycenter (center of mass) on the same side? The barycenter is not a point with mass in it (though it might sound so). It's a point between two masses. It's always the closest to the highest mass (or even inside it). Only when the two masses are equal then this point lies exactly between the two masses (that is, between their own centers of mass).

If the distance of one mass $M_1$, to the barycenter (on the line connecting the masses) is $D_1$ and the distance of the second mass, $M_2$, is $D_2$, then:

$$\frac{M_2}{M_1}=\frac{D_1}{D_2}$$

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    $\begingroup$ I think I see that now! Thank you. Now opposites make sense. $\endgroup$
    – Bookaholic
    Jun 26 at 22:23
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    $\begingroup$ Alright then. Now you are on the opposite side too (the understanding side)... $\endgroup$ Jun 26 at 22:36
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The barycenter, by definition, is located between the centers of two bodies.

Your second picture would have the barycenter on the far side of the sun's center opposite Jupiter's.

Both objects will orbit the barycenter, which will remain directly between the objects, which requires (again, by definition) that both objects will be on the opposing "sides" in their orbit (or wobble) around the barycenter.

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The short answer is angular momentum, which for whatever reason was created at the very beginning of the solar system, would not change without the existence of external force. Therefore Jupiter will rotate with the Sun around their common center of mass forever (ideal case) as long as no other objects collides with Jupiter to annihilate its angular momentum around the Sun.

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    $\begingroup$ If Jupiter and the sun would rotate rotate both on the same side of the barycenter angular momentum would still be conserved. Its the llinear momentum that isnt conserved in that case. $\endgroup$ Jun 28 at 23:43
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    $\begingroup$ @DescheleSchilder Agreed. The key thing that never seems to get mentioned in these questions and answers is that, per Newton, it is the combination of acceleration (gravity) and movement (momentum) that causes orbits. $\endgroup$ Jun 29 at 12:49
  • $\begingroup$ @DescheleSchilder Yes I didn't say the conservation of angular momentum explicitly in my previous answer but yes it is the conservation of angular momentum that guarantees the circular motion of the Sun and Jupiter around their mutual center of mass. Liner momentum is not conserved as there's an exchange between the kinetic and potential energy between the two objects (total energy is still conserved), hence making their linear velocity faster and slower at different points. $\endgroup$
    – fdc
    Jul 1 at 6:39

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