Do the planets really orbit the Sun?

I suggest that it doesn't make much sense to say that the planets orbit the barycenter of the solar system.

Beware that you are going very much against the grain of the best models of the solar system in writing that. All three of the leading ephemeris models (JPL's Development Ephemeris, the Russian Institute for Applied Astronomy's Ephemerides of the Planets and Moon, and the IMCCE's INPOP) use a barycentric rather than heliocentric coordinate system to model the motions of the bodies that comprise the solar system.

That said, the following plot shows the distances between Venus and the Sun (red) and Venus and the solar system barycenter (black) from January 1970 to December 2014. The horizontal (time) axis is in days from 12 Noon TT, 1 January 2000.

I used Venus rather than the Earth for two reasons: The Earth's orbit is a bit more complex than is Venus' orbit because of the Moon, and because Venus' orbit is the closest to circular of all the planet,s so variations should be small. The red curve, the distance between Venus and the Sun, is very close to a small amplitude sinusoidal. The black curve exhibits beats and other nastiness. A much simpler picture results when Venus is modeled as orbiting the Sun as opposed to the solar system barycenter.

So why do solar system modelers inevitably use a barycentric rather than heliocentric coordinate system? First, I'll digress a bit and look at the question "Do the planets orbit the Sun or the solar system barycenter?" This is posed as an either-or question, implying that only one point of view is valid. Both points of view are equally valid. All frames of reference are equally valid. Do the math right, and even a Nix-centered, Nix-fixed coordinate system would work just fine, at least for a little while. One problem with a Nix-centered, Nix-fixed frame is that Nix's rotational motion is chaotic (see https://www.youtube.com/watch?v=zwSFC-aPEG0). Another is that it just doesn't make much sense, physically. But it can be done!

Solar system modelers have moved beyond using perturbed ellipses to describe the behaviors of the planets. Numerical integration provides more precise and faster predictions, so long as a frame with simple dynamics is used. A barycentric frame is preferred because that is the frame in which the equations of motion take on their simplest form.

You seem to be groping towards the fact that the gravitational three-body problem is, in general, not solvable.

We can get away with saying "the sun is at one focus of an elliptical orbit" in the solar system because the sun is so much larger than anything else around. The sun is 1000 times more massive than Jupiter, so the sun-Jupiter barycenter is about 1/1000th of the way from the sun's center of mass to Jupiter's center of mass. That's actually closer to the sun's surface than to its center: Jupiter's orbital radius of 5 AU is about 1000 solar radii.

Generally, as the mass hierarchy becomes more equitable, Kepler's approximation of "sun at one focus" becomes poorer, and the long-term behavior of the system becomes more chaotic. We concentrate on the perturbation approach to $n$-body systems because it's possible to get answers, and those answers happen to be useful for our particular solar system.

If you wanted to have a look at whether Earth's orbit is better described as an ellipse about the sun's center of mass or as an ellipse about the sun-Jupiter barycenter, you could try and dig up a table of perihelion distances over the past few decades. Because Earth's orbit is eccentric, Earth is about seven solar radii nearer to the sun in January than in July.

• In years when Jupiter is at or near opposition in January (i.e. in Gemini or Taurus, as in 2013–2014) then the Sun will be on the July side of the sun-Jupiter barycenter: a little more distant at perihelion and a little nearer at aphelion. Perhaps the difference between Earth's perihelion and aphelion distances will only be about six solar radii in those years.

• By contrast, when Jupiter is near opposition in July (as, I suppose in 2008-ish and 2020-ish) then the sun will swing to the January side of the barycenter and the contrast between Earth's perihelion and aphelion distances will be more pronounced: maybe eight solar radii difference.

• On the third hand, if it's a better approximation to say (as I think you do) that the Earth's orbit is about the sun's center of mass and Jupiter be damned, then you should see comparable differences between perihelion and aphelion distance regardless of where Jupiter is.

• On the fourth hand, it may be the least unreasonable thing to say is that the sun-Earth barycenter, esssentially indistinguishable from the center of the sun, orbits the sun-Jupiter barycenter … in which case the differences I was thinking of would become quite challenging to measure.

This is a small effect you're asking to suss out: you want to take the differences between two pairs of large numbers (sun-Earth distances at different times of the year) and compare those differences to a few percent. You would have to hunt down sun-earth distance data that you believe to four or five significant figures. You might be able to get this information out of free or commercial planetarium software, like Stellarium or Redshift, but you'll want to compare a few of them for consistency. Interesting question.

(I used Stellarium to look briefly at the difference between perihelion and aphelion differences for 2005—2015, odd years only because it was kind of labor-intensive, and saw year-on-year differences of order $0.01R_\text{sun}$, much less than my swinging-barycenter predictions above. I consider this interesting but not conclusive.)

Some very good answers here. Basically nothing orbits a barycentre since barycentres do not attract. Gravitation works with inv. sqr. law and vector addition of individual forces, this is not the way a barycentre is calculated.

Much confusion arises from a use of the term "centre of gravity" to represent the centre of mass ( barycentre ). This is used on a human scale where the gravitational field surrounding us can be considered as constant and non-divergent. If you are considering planetary motions, the field is highly variable in both space and time and this identification of the two terms is not applicable.

The barycentre lies along the line of mutual attraction for a two body system. Yet it is a direction of attraction, not a spot. Many people spuriously extend this by thinking that there is some central spot of gravitational attraction that applies to multi-body systems.

While multiple gravitational vectors can be reduced one instantaneous resultant force, there is no spot in space where this originates.

The Venus plot is a very good illustration. Only once in about 8000y can Venus be said to orbit the barycentre !