Planetary Alignment

If several planets are orbiting the sun at known speeds and current
positions, how many years will it be before those planets are
perfectly aligned?  This is a frequently asked question, but it's
a bit tricky, because it gets into the issue of whether irrational 
numbers actually exist.  Also, there is the issue of gravitational 
harmonics that tend to force the periods of planets to be rational 
multiples of each other.

However, setting all that stuff aside, and assuming arbitrary real 
constants for the periods of the planets, the answer to your question 
is that, except in very special cases, the planets will *never* be 
perfectly aligned (assuming you have more than 2 planets). 

Consider the hands of a clock-like mechanism.  Suppose the hands are
initially aligned at "12 o'clock", and the angular velocities of the 
first two hands are w1 and w2 radians/sec, respectively.  How often 
will those two hands be perfectly aligned?  The positions of the two 
hands at any time t are w1*t and w2*t, and they are aligned if and 
only if t*(w1-w2) is an integer multiple of 2*pi.  Therefore, the 
two hands are aligned at the time values given by

                         2 k pi
                    t = ---------     k = 0,1,2,....
                         (w1-w2)

However, if you include a third hand on the clock with angular 
velocity w3, and assume it was also aligned initially at t=0, then 
it will be aligned with the first hand at the times

                         2 j pi
                    t = ---------     j = 0,1,2,....
                         (w1-w3)

All three hands will be aligned simultaneously only when both of 
these equations are satisfied, which implies

                  k (w1-w3)  =  j (w1-w2)

and so
                     w1-w3          j
                    -------   =    ---
                     w1-w2          k

Since j and k are integers, this proves that (w1-w3)/(w1-w2) must be 
a rational number.  But suppose 

                        w1 = 2 radians/sec 
                        w2 = 1 radian/sec
                        w3 = 2-sqrt(2) radians/sec

In this case the ratio (w1-w3)/(w1-w2) is equal to sqrt(2), which is
irrational.  Thus, the three hands will *never* come into perfect 
alignment all at once.  

Only in the special case when all three speeds are rational multiples 
of each other (as in the case of a real clock mechanism) will they 
all become aligned periodically.