Observers in a distant solar system are watching our Sun and using the radial velocity method to try to determine whether the Sun has planets. Because of the gravitational pull of Jupiter, the observers see the Sun's spectrum slowly redshift and then blueshift by a tiny amount. How long must they observe to see a full cycle?

almost 12 years

Explanation:

Using the radial velocity method, planet hunters can track a sun's spectrum, to determine whether the sun has planets. The spectrum appears first slightly blue-shifted, and then slightly red-shifted. If the shifts are regular, repeating themselves at fixed intervals of days, months, or even years, it  is almost certainly caused by a body orbiting the sun, tugging it back and forth over the course of its orbit. Because of the gravitational pull of Jupiter (about 3,000 times the mass of Earth), In which its average distance from the sun is 480 millions miles and takes nearly 12 years to make one revolution.