The moon's new faces

The moon shows a new face every night, but the sun always lights the same half of it. What changes is how much of that lit half we can see from the earth. One whole cycle, from one new moon to the next, takes about twenty-nine and a half days.

The sun always lights half of the moon; what we see of it changes. One cycle is $P\approx 29.5$ days.

Lit half fixed; cycle $P\approx 29.5$ days.

In that time the moon travels once round the earth, three hundred and sixty degrees. So each day it moves about twelve degrees. The sky turns fifteen degrees every hour, so the moon rises about forty-nine minutes later each day.

In $P$ the moon goes round once: $360^{\circ }/29.5\approx 12.2^{\circ }$ per day. The sky turns $15^{\circ }$ per hour, so it is $12.2/15\approx 0.81$ h, about $49$ min, later each day.

Speed $\omega =360^{\circ }/29.5\approx 12.2^{\circ }$ per day; sky $15^{\circ }$ per hour; delay $\Delta t=12.2/15\approx 0.81$ h.

From a full moon to the next new moon is about fifteen days. A quarter of the cycle, about seven days, takes us from a new moon to a half moon.

Full moon to new moon: $P/2=14.75$, about $15$ days. New moon to half moon: $P/4\approx 7.4$ days.

$P/2=14.75\approx 15$ days; $P/4\approx 7.4$ days.

Twelve cycles last twelve times twenty-nine and a half days, which is three hundred and fifty-four days: eleven days short of a year. So each year the full moon comes about eleven days earlier.

$12\times 29.5=354$ days and $365-354=11$: the full moon comes about $11$ days earlier each year.

$12P=354$ days; $365-354=11$ days earlier each year.