Detecting Multiple Planets Around a Star

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cepheid
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In the Doppler shift or radial velocity method for detecting exoplanets, it is assumed that the orbital period of the star's planetary companion is equal to the period of the star's detected wobble. I'm assuming that this fact comes straight from the solution to the two-body problem.

My question is, what if a star has multiple planets in orbit around it? Say, for example, we were viewing our solar system from afar. Would we be able to detect anything other than Jupiter? I guess this is an n-body problem, although it seems that the typical thing to do is to assume that the interaction between each planet and the parent star is much greater than the interactions among planets, which can be ignored. So, I guess that this is like 8 independent two-body problems, each of which gives a solution that is a good (zeroth-order? first-order?) solution for that planet's orbit.

What's confusing me is that the solution to each two-body problem would have you believe that the parent star wobbles with a period equal to the orbital period of the second object. So, how could the sun wobble "around" the barycentre on 8 different timescales? What does the wobble look like if there are multiple planets, and how are we supposed to know that they are there?
 

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IIRC, the way it is done is to look for 'residuals': The Doppler wobble from the first planetary candidate gives an approximate orbit. Further observations improve the estimate. Later observations show that the first planet is running a bit faster then a bit slower than its predicted motion. Back-calculating these 'residuals' give an estimate for a second planet. Further measurements improve the estimate until new, cyclic 'residuals' appear...
 
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cepheid
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Interesting! Thanks Nik_2213 and Nabesin for the verbal and visual help, respectively.
 

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