Final answer:
The Moon takes 27.3217 days for its sidereal period and 29.5306 days for its solar month or synodic period, but it will take one year for the Moon to go from one vernal equinox to the next.
Step-by-step explanation:
The question refers to the motion of the Moon around the Earth and its relation to the Sun's position in the celestial sphere, specifically considering the vernal equinox.
The Moon's sidereal period, which is the time it takes to return to approximately the same position among the stars, is 27.3217 days. However, due to Earth's motion around the Sun, there's a difference between the sidereal period and the time it takes the Moon to complete its cycle as seen from Earth in terms of phases, known as the solar month, or synodic month, which is 29.5306 days.
The Moon must complete more than one full orbit around the Earth to catch up with the Sun and reach the same phase again because the Sun has moved as well during this time. For instance, from new moon to new moon, the cycle is 29.5 days, but this doesn't necessarily align with the vernal equinox specifically.
Due to the Earth's orbit around the Sun, the vernal equinox occurs once every year, so the time it takes for the Moon to move from one vernal equinox to the next is simply one year. The Moon will go through many of its synodic and sidereal cycles within that year.