The time it takes the Moon to return to a given position as seen against the background of fixed stars, $27.3$ days, is called sidereal month. The time interval between identical phases of the Moon is called a lunar month. A lunar month is longer than a sidereal month. Why and by how much?
I do understand the concept that in $27.3$ days, the moon may have orbited $360^\circ$ but since the Earth moved through $(27.3/365) \times 360^\circ = 27^\circ$. So, the Moon needs to move $27^\circ$ to catch up. That should take $(27^\circ / 360^\circ) \times 27.3$ days $= 2.05$ days , but in that time Earth would have been moved on yet much farther.
Is there any method to calculate the exact time taken i.e. (the taken by the Moon to be in the same phase and the same background positioning among the fixed set of stars) rather than doing this cumbersome process over and over again?
Are there any new methods to solve this question?