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In every derivation of Kepler's Laws that I have seen, we assume that the sun is stationary. However, in other places I have read that celestial bodies move about their barycentre (center of mass). So are planets actually moving in elliptical orbits around the Sun or do they move in circular orbits around their center of mass?

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  • $\begingroup$ Related: physics.stackexchange.com/q/25110/2451 and links therein. $\endgroup$ – Qmechanic Apr 3 '17 at 7:21
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    $\begingroup$ Also related: physics.stackexchange.com/q/188650 $\endgroup$ – David Hammen Apr 3 '17 at 15:34
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    $\begingroup$ Yep, they move in an ellipse. Kepler figured that out after many years of research and published his finding in the book "Astronomia Nova". Read that book and find out all about it. (you might need to learn Latin first) $\endgroup$ – Ambrose Swasey Apr 3 '17 at 17:04
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    $\begingroup$ "In every derivation of Kepler's Laws that I have seen, we assume that the sun is stationary. " Starting at the upper-division level every derivation of the 2-body orbit behavior you see will include the canonical transformation to center of mass coordinates. Which won't change the math a whit besides replacing the mass of the small body by the reduced mass. $\endgroup$ – dmckee Apr 3 '17 at 17:16
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    $\begingroup$ -1. You missed other options, including that they move in elliptical orbits about the centre of mass. $\endgroup$ – sammy gerbil Apr 3 '17 at 21:33
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In an ideal two body system (say a sun and a planet), both bodies would move around their barycenter. An ideal periodic orbit would be an ellipse or a circle.

EDIT : See comment by @user11153 regarding the barycenter of the solar system and related links.

In a more complex system like our solar system, to a good approximation the planets can be modeled by a two body system (i.e. the Sun being so massive it is the dominant effect) and for many practical purposes the motion of the Sun around the barycenter is not significant, as the barycenter is actually inside the Sun.

More precise calculations the motion of a planet requires allowing for the gravitational perturbation of other planets as well as allowing for the center of mass and relativistic effects. The net effect is that no planets actually orbit in ideal elliptical orbits.

So are they actually moving in elliptical orbits around the sun or do they move in circular orbits around their center of mass?

I have the impression from this question that you think the elliptical orbits are a result of using the barycenter as a center of motion and that otherwise a circle would be the orbit's shape.

This is not the case. The general shape for an orbit in an ideal two body system with a Newtonian gravitational force is an ellipse. A circle is a special case of an ellipse.

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    $\begingroup$ "motion of the Sun around the barycenter is not significant, as the barycenter is actually inside the Sun." - not true $\endgroup$ – user11153 Apr 3 '17 at 10:34
  • $\begingroup$ @user11153 : I was thinking of this simulation showing the barycenter. However I won't argue the point. $\endgroup$ – StephenG Apr 3 '17 at 11:15
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    $\begingroup$ "The general shape for an orbit in an ideal two body system with a Newtonian gravitational force is an ellipse. A circle is a special case of an ellipse": false. It is a standard exercise to show that according to the initial energy you could have whichever conic section you want (usually NOT an ellipse). $\endgroup$ – gented Apr 4 '17 at 8:56
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    $\begingroup$ @GennaroTedesco True, but most laypeople understand "orbit" to mean "closed orbit", which is elliptical. $\endgroup$ – Will Vousden Apr 4 '17 at 12:47
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    $\begingroup$ @gennaro-tedesco : I was taking the context of the question (planets) to imply quasi-periodic orbits, but of course you're correct. Regarding "being precise", again I'd suggest that precision within the question's context is what matters. $\endgroup$ – StephenG Apr 4 '17 at 13:00
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Consider a 2-body system, a mass-$m$ planet of position $\mathbf{r}$ orbiting a mass-$M\gg m$ star of position $\mathbf{R}$. (I'm referring to the bodies' centres of mass.) Newton's third law implies the system's barycentre $\mathbf{b}:=\frac{m\mathbf{r}+M\mathbf{R}}{M+m}$ is conserved. In Newtonian mechanics, each body moves in an ellipse of which one focus is $\mathbf{b}$, but of course the planet's orbit is larger than the star's. (Indeed, since neither body is a point mass, the barycentre may well be inside the star's volume.) Real-life planetary systems are more complicated, not only because of more bodies but also because even a 2-planet system is predicted, in special relativity, to suffer orbital precession so that an unchanging closed ellipse is not repeatedly followed. However, these are small corrections that don't change the approximate elliptical behaviour of the orbits.

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Assuming an ideal two body case such as Sun and Earth, both are orbiting around the barycenter in two elliptical orbits. These two orbits are similar in geometry and scaled proportional to their relative distance from the barycenter.

When the earth is at the farthest point in its orbit, sun is at the farthest point in its orbit; and both are moving at thier slowest speed. When they are at the closest point in their relative elliptical orbit they move fastest. They both have the greatest acceleratio to and from each other at about January when the Earth is closest to the sun.

Then if we add the effect of other planets the system's​ barycenter wobbles in a rotating complex path, however the path of the planets is not a perfect elliptical orbit anymore.

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At the beginning of the answer, I have to mention all those interested in the movements of celestial bodies and all scientific institutions that Kepler's law is incorrect as far as the actual paths of the heavenly bodies. Each system has its center of mass (peričenter), around which rotate all the participants of the system, including the main body around which rotate other companions. The center of mass of the solar system is the sun and around it rotate all centers of mass of other systems (planetary), and the sun and the variable sinusoidal radius, except that the sun has its spin (rotation around its axis). The vectors of the spin and the rotation of the sun around the center of mass of the solar system, have the same value but opposite directions. This is true for planets when rotating around the center of mass of the planetary system. Thus the center of mass of the planet (Earth and Moon) travels the Kepler ellipse, or the Earth and the moon have their spin, which is equal to the rotation around the center of mass of the planets and the moon, around the center of mass mjeseca.Ta two spin are the same size, or opposite directions (coupling of moments). So the earth rotates around the center of mass of the planet (located about 1,500 km below the Earth's surface) and around the ellipse, at which moves the center of mass, form a sinusoid whose true anomaly angle. The Moon rotates around the center of mass of the planet (and its center of mass) varies according to Kepler's ellipses, while I have a month to spin and rotation around its own center of gravity, the same size, but opposite directions. This evidence I have made about 15 pages of formulas and diagrams. With him is refuted Einstein's "proof" precession of the perihelion of the planet, and also to see why our moon has always one and the same with his side to the Earth, and something

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