In a spectroscopic binary, if a pair shows a combined set of lines tonight, but a maximum split two nights later, what must its orbital period be?
Think of the splitting of the lines as two sine waves. They are 180 degrees out of phase because when Star A moves toward Earth making a blue shift, the Star B is moving away from Earth making a red shift.
Intial state: spectral lines together.
1/4 period: Star A reached its maximum blue shift (think of it as a peak in the sine wave), Star B reaches its maximum red shift (trough).
1/2 period: Spectral lines are back together.
3/4 period: Star A has a maximum red shift (trough), Star B a maximum blue shift (crest). Reverse of 1/4 period.
Full period: Back to the initial state for another cycle.