Sun Orbits Center of Mass w/ Jupiter & Earth: How To Calculate?

[whatisreality]

I know that if I have a two-body system, the sun and the earth, then the common centre of mass is orbited every 365 days. What about if I add another planet in, say jupiter? Does the sun still orbit the centre of mass every 365 days? With two bodies, their orbital period about the centre of mass is the same. But Jupiter and Earth have different orbital periods, so how do you know how often the sun orbits now, when there are two periods to choose from?

 

[Bandersnatch]

The motion around common centre of mass of the Earth-Sun system can be seen as a 'wobble' in the motion of the Sun-Jupiter system. Or vice versa, depending on chosen reference frame, but considering how the latter is much more pronounced, it's easier to treat as perturbations the influence of the smaller planet.
The period of those wobbles are equal to the period of the perturbing body.

 

[whatisreality]

The motion around common centre of mass of the Earth-Sun system can be seen as a 'wobble' in the motion of the Sun-Jupiter system. Or vice versa, depending on chosen reference frame, but considering how the latter is much more pronounced, it's easier to treat as perturbations the influence of the smaller planet.
The period of those wobbles are equal to the period of the perturbing body.

I want to do everything in the centre of mass frame. So I want the orbital period of the sun, Earth and Jupiter about the centre of mass, so I can work out their velocities around the centre of mass and therefore also the velocity of the centre of mass itself.

But I really don't know how to work out the velocity (particularly of the sun) around the centre of mass. Because I don't know its orbital period about the centre of mass.

 

[tfr000]

I want to do everything in the centre of mass frame. So I want the orbital period of the sun, Earth and Jupiter about the centre of mass, so I can work out their velocities around the centre of mass and therefore also the velocity of the centre of mass itself.

But I really don't know how to work out the velocity (particularly of the sun) around the centre of mass. Because I don't know its orbital period about the centre of mass.

You have to figure the positions of all the planets in their orbits. The Sun is on the opposite side of the center of mass for each. Then you add them all up. Not terribly difficult, but non-trivial. That's the numerical way to do it. Analytically, I don't know. You could assume circular orbits as a first cut.
The center of mass itself has no velocity relative to all of that motion.