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What are the orbital parameters of the Sun such as orbit velocity etc in it's orbit around the Solar System's center of mass? Consider the Sun pointlike or alternatively when talkin about the Sun's movement I mean it's center of mass.
Do not tell me that the Sun is stationary because the planets' masses can be neglected. I do no want such oversimplifications.
Since the Solar system is a multi-body system (with $N>2$ bodies of significant mass), the orbits of its constituents are not exact Keplerian orbits.
To lowest order, each planet orbits the Sun (or rather the centre of mass of all interior planets) on a Keplerian orbit, but the interactions with the planets as well as the fact that the centre of mass of the interior is not fixed lead to deviations of the true orbit from this simplification. These deviations can be treated either numerically or via perturbation theory, but are non-trivial functions of time.
The same holds for the Sun: to lowest order one can neglect all planets but Jupiter (which is more then twice as massive as all the remaining planets combined), when the Sun follows an elliptic orbit with semi-major axis of about 0.005AU (smaller than that of Jupiter by their mass ratio). This is of the same order as the radius of the Sun, i.e. the barycentre of the Solar system hardly leaves the Sun. However, as above, the pull by all other planets lead to a deviation from this simple model. Again, these deviations are non-trivial.
