The shortest period of time after which we see the Moon in the same position relative to the stars, or the time it takes for the Moon to rotate exactly $360^{\circ}$ relative to the center of the Earth, is on average 27 days 7 hours 43 minutes. The same period of time relative to the Sun is 365 days 6 hours 9 minutes, which is the time it takes for the Earth to complete one orbit around the Sun (this is the so-called sidereal, or star-related, month or year). How often would the phases of the Moon repeat if the Earth's orbit around the Sun and the Moon's orbit around the Earth were circular, their motion were uniform, and the planes of the two orbits coincided, that is, how often would the centers of the Sun, Earth, and Moon fall in a straight line in this order (i.e., what would be the so-called synodic, or conjunction, month)? Would new moons, first and last quarters also repeat at the same intervals? (The Moon orbits the Earth in the same sense as the Earth orbits the Sun.)
In our diagram, the Earth and the Moon are in positions $F_{1}, H_{1}$, where there is a full moon, and the Sun, Earth, Moon, and a (distant) star C are in a straight line. (At such times, a lunar eclipse also occurs, but we will not consider this here.)

![](https://cdn.mathpix.com/cropped/2024_05_02_14a1e1c6066767e19d17g-1.jpg?height=621&width=487&top_left_y=264&top_left_x=800)

The Moon completes a $360^{\circ}$ rotation, or one sidereal month, during which the Earth travels approximately one-thirteenth of its orbit, so in the $F_{2}, H_{2}$ position, the next full moon has not yet occurred, only in the slightly later $F_{3}, H_{3}$ position. This position is characterized by the fact that the directions $N F_{3}$ and $N H_{3}$ form the same angle $\alpha$ with the direction $N C s$. So far, the Earth has rotated by $\alpha$, and the Moon by $360^{\circ} + \alpha$, giving the Moon a $360^{\circ}$ lead. The time $T$ required for this lead must be calculated from the orbital periods of the Moon and the Earth (measured in days) $T_{H}, T_{F}$.

The Earth rotates by $360^{\circ} / T_{F}$ degrees per day, and the Moon by $360^{\circ} / T_{H}$ degrees per day (which is larger due to $T_{H} < T_{F}$), so the Moon gains $\beta = 360^{\circ} \left( \frac{1}{T_{H}} - \frac{1}{T_{F}} \right)$ degrees per day. The time required to gain a $360^{\circ}$ lead is:

$$
T = \frac{360^{\circ}}{\beta} = \frac{1}{\frac{1}{T_{H}} - \frac{1}{T_{F}}} = \frac{T_{H} T_{F}}{T_{F} - T_{H}}
$$

With the given orbital periods, $T = 29$ days 12 hours 44 minutes. (It is unnecessary to calculate seconds, as the data is rounded to minutes.)

This reasoning can be directly applied to the calculation of the recurrence time of new moons and quarter phases. The phases of the Moon (under the given assumptions) depend only on the angle $N F H$, and any value of this angle repeats every $T$ time interval.

Denes Nagy Lajos (Budapest, Rákóczi F. g. I. o. t.)

Remarks. 1. Our reasoning would also be valid if the Moon orbited the Sun, that is, if we were calculating the recurrence time of planetary conjunctions, oppositions, and quarter phases.

2. One solution pointed out an inconsistency with the data for the sidereal year. It is surprising that only one did, as it is often stated that the sidereal year is 365 days 5 hours 48 minutes. However, this is the year in terms of the changing seasons, the time it takes for the Earth to return to a position where the equatorial plane passes through the center of the Sun. This issue cannot be easily simplified.