Do the planets really orbit the Sun?

Question

We often say that the planets orbit the Sun, which is usually a reasonable approximation. But in reality both Sun and the planets orbit the center-of-mass/center-of-gravity of the whole solar system, not the center of the sun.

The Sun is by far the most massive body in the solar system, so it's plausible that the center-of-gravity lies within it. On the other hand Jupiter is rather massive as well and pretty far from the Sun. Thus the center-of-mass might not actually be inside the Sun, only near it. The influence of Saturn might not be negligible either.

Answer 1

You are correct, the centre of the Sun is not the Solar System's centre of gravity.

^{A diagram (courtesy Wikimedia Commons), showing how the barycentre of the Solar System has changed over time.}

The Sun is affected by the gravity of all planets in the Solar System, but you are right, it is most affected by the two most massive ones; Jupiter and Saturn. You can see in this animation (a representation, not a simulation), how two bodies affect each other in a normal orbit like that between the Earth and the Sun:

^{Courtesy Wikimedia}

Apply this relation to every body in the Solar System (adjusting for mass and distance of course) and you get a rough idea of how the Sun is affected by the rest of the Solar System.

Due to this somewhat chaotic dance the centre of mass of the Solar System continuously moves around, sometimes under the Sun's surface and sometimes outside it. The further away from the centre of the Sun this barycentre is, the more the Sun appears to oscillate.

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Another example of this phenomenon is the Pluto-Charon system:

^{Courtesy Wikimedia.}

Charon is about one-tenth the mass of Pluto (thank you for the correction @Hobbes), and yet exerts a significant gravitational pull on Pluto. Hence they both orbit around the centre-of-gravity of their system, well outside the surface of Pluto.

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Bonus: We use this phenomenon to find planets outside the Solar System! If a distant star is observed to 'wobble' or oscillate about it's mean position, we can use that data to infer the presence of one or more exoplanets, and calculate their mass.

Further Reading:

http://homepages.wmich.edu/~korista/solarsystem_barycenter.pdf http://spaceplace.nasa.gov/barycenter/en/

Answer 2

Using the approximation

$$\Delta \mathrm{center}=\frac{m_p}{m_\odot}\cdot\frac{\mathrm{dist}_p}{r_\odot}$$

and data from List of gravitationally rounded objects of the Solar System - Wikipedia:

Name       Distance   Mass/[kg]  Mass       Distance   ΔCenter
           /km        /kg        /sun mass  /sun radius
--------------------------------------------------------------
Sun        0.00E+000  1.99E+030  1.00E+000  0.00E+000  0.00000
Mercury    5.79E+007  3.30E+023  1.66E-007  8.32E+001  0.00001
Venus      1.08E+008  4.87E+024  2.45E-006  1.55E+002  0.00038
Earth      1.50E+008  5.97E+024  3.00E-006  2.15E+002  0.00065
Mars       2.28E+008  6.42E+023  3.23E-007  3.27E+002  0.00011
Jupiter    7.78E+008  1.90E+027  9.55E-004  1.12E+003  1.06735
Saturn     1.43E+009  5.69E+026  2.86E-004  2.05E+003  0.58576
Uranus     2.87E+009  8.68E+025  4.37E-005  4.12E+003  0.18007
Neptune    4.50E+009  1.02E+026  5.15E-005  6.46E+003  0.33278

Which sums to a shift of 2.17 sun radii if all planets were aligned. Jupiter clearly dominates (about half of the total effect) and would be enough to move the center-of-gravity outside the sun. Only the heavy outer planets (Jupiter, Saturn, Uranus, Neptune) have a non negligible influence.

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C# source code for reproducibility:

void Main()
{
    // http://en.wikipedia.org/wiki/List_of_gravitationally_rounded_objects_of_the_Solar_System
    Body[] planets={
        new Body{Name="Mercury",Distance=57909175,Mass=3.302E23},
        new Body{Name="Venus",Distance=108208930,Mass=4.8690E24},
        new Body{Name="Earth",Distance=149597890,Mass=5.9742E24},
        new Body{Name="Mars",Distance=227936640,Mass=6.4191E23},
        new Body{Name="Jupiter",Distance=778412010,Mass=1.8987E27},
        new Body{Name="Saturn",Distance=1426725400,Mass=5.6851E26},
        new Body{Name="Uranus",Distance=2870972200,Mass=8.6849E25},
        new Body{Name="Neptune",Distance=4498252900,Mass=1.0244E26},
    };
    var bodies=new[]{Sun}.Concat(planets);
    bodies.Select(planet=>new{
        Name=planet.Name,
        Distance=planet.Distance.ToString("E2"),
        Mass=planet.Mass.ToString("E2"),
        DistanceInSunRadii=planet.DistanceInSunRadii.ToString("E2"),
        MassInSuns=planet.MassInSuns.ToString("E2"),
        CenterOfGravityShift=planet.CenterOfGravityShift.ToString("n5")
    }).Dump();
}

static Body Sun=new Body{Name="Sun",Distance=0,Mass=1.98855E30};
const double SunRadius=696342;

class Body
{
    public string Name{get;set;}
    public double Distance{get;set;}// in km
    public double Mass{get;set;}// in kg

    public double MassInSuns{get{return Mass/Sun.Mass;}}
    public double DistanceInSunRadii{get{return Distance/SunRadius;}}
    public double CenterOfGravityShift{get{return MassInSuns*DistanceInSunRadii;}}
}

Answer 3

We often say that the planets orbit the Sun, which is usually a reasonable approximation. But in reality both Sun and the planets orbit the center-of-mass/center-of-gravity of the whole solar system, not the center of the sun.

That depends on what you mean by "orbit". If you mean that the equations of motion take on their simplest form in a non-rotating frame centered at the solar system barycenter, that is correct. Ignoring perturbations from the galaxy and nearby stars, this barycentric frame is an Newtonian inertial frame of reference. (Taking those small perturbations into account means that a barycentric frame is only approximately an inertial frame.) A heliocentric (Sun-centered) frame is also very close to inertial, but less so than is a barycentric frame. You have two choices with a frame that is known to be rotating and/or accelerating: Account for that rotation/acceleration via fictitious forces, or ignore them. Accounting for them makes the equations of motion more complex.

If on the other hand you mean that each body in the solar system is attracted toward the solar system barycenter, that is incorrect.

Answer 4

Mach's principle (which is somewhat controversial, but has led to some inspired thinking - see Wikipedia) can be interpreted as saying there is no center really, it is the overall distribution of mass that determines motion. And that is where Einstein's General Relativity takes over and says things just follow their local geodesics which are determined by the overall mass distribution. In that view some (well, many) configurations look like one object is orbiting another, but really every object is just doing its thing oblivious to all other objects. For a great book about this with wonderful illustrations (but lots of math) see Gravitation by Misner, Thorn and Wheeler and here.

I don't think this is always the simplest and most useful way to look at things, but it certainly has its uses (more so in complex configurations) and I just wanted to put it out there because no one else has in this thread.