The number of days in a lunar cycle
2 answers
1 lunar cycle = 29.53059 days
The number of days in a lunar cycle
The mean lunar month, 29.53 days, runs from New Moon to New Moon, the points where the sun, moon, and earth, in that order, are all on a straight line, looking down on the ecliptic plane. It takes 27.32 days for the moon to complete one "360 degree" revolution about the earth. But, since the earth has also revolved ~27 degrees around the sun during that time period, the moon must therefore revolve the same 27 additional degrees about the earth in order to return to the same New Moon position. Lets explore this in some more detail.
The moon revolves about the earth at the mean angular rate of 360/27.32 = 13.1771 degrees per day. Starting from a New Moon, in 27.32 days the moon has completed a 360 degree revolution about the earth. But, as I said earlier, the earth, revolving about the sun at a mean rate of .985626 degrees per day, has also moved 27.32(.985626) = 26.9273 degrees in its orbit and is now behind the line joining the moon and the sun, or the moon is ahead of the line joining the sun and the earth. In order for the moon to return to the earth-sun line for the next New Moon phase, it has to revolve the additional 26.9273 degrees to catch up, in roughly 2.0434 more days. Thus 27.32 + 2.0434 = 29.36 days. But, wait a minute. The lunar month is 29.53 days long. We have another iteration to perform here. In the 2.0434 days it took for the moon to catch up with where the earth was after 27.32 days, the earth moved another 2.1043(.985626) = 2.0140 degrees in its orbit and the moon must now also move an additional 2.0140/13.1771 = .1528 degrees to catch up. So now we are up to 27.32 + 2.0434 + .1528 = 29.516 days. As you can readily see, the moon ultimately reaches the earth-sun line in 29.53 days and we again have a New Moon.
The mean lunar month, 29.53 days, runs from New Moon to New Moon, the points where the sun, moon, and earth, in that order, are all on a straight line, looking down on the ecliptic plane. It takes 27.32 days for the moon to complete one "360 degree" revolution about the earth. But, since the earth has also revolved ~27 degrees around the sun during that time period, the moon must therefore revolve the same 27 additional degrees about the earth in order to return to the same New Moon position. Lets explore this in some more detail.
The moon revolves about the earth at the mean angular rate of 360/27.32 = 13.1771 degrees per day. Starting from a New Moon, in 27.32 days the moon has completed a 360 degree revolution about the earth. But, as I said earlier, the earth, revolving about the sun at a mean rate of .985626 degrees per day, has also moved 27.32(.985626) = 26.9273 degrees in its orbit and is now behind the line joining the moon and the sun, or the moon is ahead of the line joining the sun and the earth. In order for the moon to return to the earth-sun line for the next New Moon phase, it has to revolve the additional 26.9273 degrees to catch up, in roughly 2.0434 more days. Thus 27.32 + 2.0434 = 29.36 days. But, wait a minute. The lunar month is 29.53 days long. We have another iteration to perform here. In the 2.0434 days it took for the moon to catch up with where the earth was after 27.32 days, the earth moved another 2.1043(.985626) = 2.0140 degrees in its orbit and the moon must now also move an additional 2.0140/13.1771 = .1528 degrees to catch up. So now we are up to 27.32 + 2.0434 + .1528 = 29.516 days. As you can readily see, the moon ultimately reaches the earth-sun line in 29.53 days and we again have a New Moon.