Barycentre, barycentre. It’s the kind of word that sounds like it should be easy to pun, but really isn’t.

In preparation for the post on Lagrange points coming later on I should start by defining the concept of a gravitational barycentre (or centre of mass). For a single roughly-spherical object like a planet we can treat this as a single point mass located at the very centre of the sphere – that is, if we took the planet away and replaced it with a single infinitesimally small object with the same mass located at the planet’s centre of mass, this tiny object would (broadly) be gravitationally indistinguishable from the planet. If we add in another spherical object – say a moon – and attempt to calculate the centre of mass of the planet-moon system, things get a little trickier. The planet exerts a gravitational force on the moon and the moon exerts a gravitational force on the planet, meaning that the gravitational centre of mass of the combined system will be located somewhere in between.

This is actually kind of a bad picture for illustrating the concept because m₁ is the heavier mass, and yet it’s rendered as smaller than m₂. Never mind. Pretend m₁ is made of something really dense, like neutronium. The centre of mass of a two body system is defined by factoring each body’s mass by its distance from some arbitrary reference point – in this case x₁ and x₂ – and dividing by the system’s total mass. Since the reference point is entirely arbitrary we can cheat and say it’s located at the centre of mass of m₁; this means that x₁ is zero, x₂ is now the distance d between the two centres of mass and the equation reduces to

 In other words the centre of mass of the two body system is located a distance X_cm from the centre of mass of m₁along a line connecting the centres of mass of m₁ and m₂. This is the gravitational barycentre of the system, and it’s very important when considering orbital mechanics because in a two-body system like a moon and a planet, or a planet and the sun, both bodies will orbit the gravitational barycentre. If you have two masses of comparable size like a binary star system the barycentre will lie roughly in the middle of the two bodies, leading to very obvious mutual orbital behaviour like this:

(This is actually wikipedia’s example for Pluto + Charon, but I don’t like the one for the binary star system because it implies the stars have exactly the same mass, which would be unlikely.)

If one body is very much larger than the other, however, then X_cm will be smaller than the radius of the larger body, which is another way of saying that the gravitational barycentre will be located inside the larger mass. The existence of the barycentre is much less obvious in this case since the centre of mass of the combined system is so close to the centre of mass of one of the bodies. For example, in the case of the Earth-Sun system the barycentre is located

or about 450km away from the Sun’s centre of mass. Given that the Sun has a radius of 696,000km this does not produce much appreciable movement on the part of the Sun! I mention this so that the following gif doesn’t give you the wrong idea.

This is a very exaggerated portrayal of the sort of motion we’re talking about here; in actuality for most planets the motion on the part of the Sun will be tiny, and even a really big planet like Jupiter can only shift the barycentre of their mutual orbits to roughly the surface of the Sun, meaning that it’s less of a mutual orbit than it is a slight “wobble” on the part of the Sun. Nevertheless this motion can be detected, and as previously mentioned it’s one of the methods we use to detect exoplanets orbiting other stars. Hopefully by this point it should be obvious why we’ve mostly only found really big exoplanets, since they’re the ones which produce the biggest wobble in the motion of their parent star.

Now, remember, this applies for all two-body systems. The Earth and the Moon orbit a mutual gravitational barycentre. The Sun and Pluto orbit a mutual gravitational barycentre. Even tiny satellites like the Martian moons Phobos and Deimos will cause Mars to shift slightly in position, orbiting a mutual gravitational barycentre. This is an important factor to take into consideration when plotting celestial trajectories, and it also gives rise to some interesting side-effects which I’ll tackle on Thursday.

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