Understanding the moon’s cycle:

A. The moons orbit and its dark/light periods:
We all witness throughout the month the cycle of the moon going from being invisible [i.e. New Moon] to being a full moon and then once again to being invisible [i.e. New Moon]. What causes this change to occur? In this paragraph we will investigate the moon’s orbit which affects its level of visibility from earth.
Each month, the moon orbits around the earth. That which effects earths’ visibility of the moon is the angle from which the moon is found opposite the sun and the earth. It is the sun that is responsible for the moon’s light, the moon having no light of its own. Depending on the angle in which the moon is being seen from the earth will affect our visibility of the moon. Half of the moon constantly faces earth throughout its monthly orbit. There is never a time of the month that we have more or less than half of the moon facing us. Nonetheless, the half that we see is constantly changing due to the orbit of the moon, and this is what effects our ability to see the moon’s light. The moon itself remains alight in half of its body size throughout the month, as the sunlight constantly hits the half that faces the sun. This half goes through a change during orbit, on a two week basis. Meaning, every part of the moon goes through a two week cycle of light and darkness, two weeks of light and two weeks of darkness. As the moon orbits earth and rotates, its angle from the sun and earth changes, causing the people on earth to see either more of its dark side or more of its light side. The following is its cycle: In the beginning of the moons cycle the moon is directly opposite the sun, between the sun and the earth. We will consider this the 0 degree angle from the earth. At this point the moon’s half that faces earth is completely dark, as the sun only shines the half that is directly facing it, leaving us to see only its unlit side. This stage is called New Moon. As the moon begins its orbit cycle it slowly begins to reveal part of its shining half to the earth, until it reveals approximately 1/4 of the moon’s shining half. This stage is called the Waxing Crescent, in which the moon is at a 45 degree angle from the earth. It takes a few days for the moon to reach this position during orbit. After a full week, the moon’s orbit reaches a 90 degree angle from the earth, in which case we have Half Moon. In this position, we see exactly one half of the lit side of the moon. After two weeks into orbit the moon is now at a 180 degree angle from the earth and its entire shining half is visible to earth. This is called Full Moon. In this stage the moon is behind the earth, and the earth is between the moon and the sun. This allows earth to see the full shining half of the moon, which is opposite the sun. After three weeks into orbit the moon is now 270 degrees from the earth in which case we once again see Half Moon. After four weeks the moon has made a complete 360 degree turn and reaches back to point 0 in which the entire dark half of the moon is facing the earth and we return to have zero visibility of the moon until it restarts its cycle.    

B. How long does it take the moon to make a full orbit around the earth?[1]
It takes the moon exactly 29 days, 12 hours and 793 Chalakim to make a full orbit [360 degree circle] around the earth.[2] That amounts to four full weeks, one day, 12 hours and 793 Chalakim.[3] Each 60 minute hour contains 1080 Chalakim[4]; each minute contains 18 Chalakim [18×60=1080]; and each Cheleik contains 3.333 seconds [60/18=3.3]. Accordingly, 793 Chalakim of an hour is 44 minutes [44×18=792 Chalakim] and 3^1/3 seconds [1 Chelek for a total of 793 Chalakim]. Thus, between each Molad [new moon] there is always exactly 29 days 12 hours 44 minutes and 3.3 seconds.[5]

C. Calculating the Molad:
Once one knows the Molad of any month he can calculate the Molad of the next month by simply skipping four weeks and adding one day, 12 hours and 793 Chalakim [44 minutes and 3.3 seconds].

D. When is the moon not visible?[6]
The moon becomes invisible from earth during the time of the Molad, and also a day prior and post the Molad. Upon approaching the Molad the moon reaches completion of its orbit and becomes aligned between the sun and the earth, thus blocking earth from seeing any of its lit side which is now lying opposite the sun. The period of time that the moon remains in the dark is approximately two days, give or take some. It is in the dark for approximately one day before the orbit is complete and one day after the orbit is complete.

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[1] Rambam Kiddush Hachodesh 6/3-5

[2] Rambam ibid 3

[3] Rambam ibid 5

[4] Rambam ibid 2

[5] Rambam ibid 3

[6] Rambam Kiddush Hachodesh 1/3