A sidereal month is the time it takes for the moon to change phases once. While the Moon revolves around the Earth, the Earth also revolves around the Sun, so after one sidereal month, the Earth is about 27° more orbited around the Sun.
Therefore, for the moon to be in the same phase, the moon would need to orbit the earth about 27° more, which is about 2.2 days longer than a sidereal month, which is the time it takes for the earth to complete one revolution around a star.
In other words, one sidereal month is about 27.32 days, while one capricorn month is about 29.53 days.
Strictly speaking, Earth’s orbit is elliptical, which means that its angular velocity is not constant. It rotates faster when it’s closer to the Sun and slower when it’s farther away.
The faster the Earth rotates, the longer the lunar month is because the Moon has to travel an additional distance, and the shorter the lunar month is because the Moon has to travel a shorter distance. Therefore, one lunar month is approximately between 29.2 and 29.8 days.
Sidereal months and the tooth method
What we commonly refer to as the lunar calendar is a lunar calendar that defines months based on the phases of the moon. The lunisolar calendar is a calendar that reflects the changes in the seasons so that the months and seasons almost coincide.
Technically, the lunar calendar used in Asia, such as Korea and Japan, is the lunisolar calendar, while the Islamic calendar is the solar calendar.
In the lunisolar calendar, a month is 29.5 days on average, so a small month is 29 days and a large month is 30 days. Twelve months in a year equals 354 days in a year, so the cumulative effect of these errors would be a difference of more than a month in less than three years, and a summer month could become a winter month in about 16 years.
For this reason, the T’ai Chi solar calendar has seven leap months every 19 years to align the seasons with the calendar. A leap month rule called the unweighted leap month rule is used, in which the month that does not fall in the middle of the 24 seasons is the leap month, and the month is named after the passage.
If there are two non-midwinter months in a year, only the first month is a leap month. A year in the lunar solar calendar has 354 days in a normal year and 383 to 384 days with leap months. The period when the lunar and solar calendars coincide is called the metonic cycle.
Other rotational periods of the Moon
In addition to sidereal and lunar months, there are other ways to define when the moon has completed a full revolution. There’s the tropical month, which is the time it takes to go from being in the direction of the vernal equinox to being in the same direction again.
At 27.3216 days, it’s a few seconds shorter than the sidereal month due to precession. There is an anomaistic month, measured relative to the perigee, with a period of 27.5546 days. The 15 perigee months have a length of 413.32 days and the 14 apogee months have a similar length of 413.43 days, which is the time it takes for a supermoon (a full moon that appears close to the perigee) to turn back into a supermoon. The length of a Draconic month, a period measured by the intersection of the ecliptic and the celestial equator (either ascending or descending), is 27.2122 days.