Binary planets and dual moon alignment occurring once every 200 years

Question

I am trying to figure out a 200 year cycle where Binary planets share 1 moon Planet 1 has its own moon that maintains orbit around it. The cycle would be as illustrated below. Moon 1 would belong to planet 1. Moon 2 would be shared. Moon 1 has a 24 day orbit around Planet 1. What I need to know is what would be the moon orbit cycle in days to complete the dual orbit around both planets so that once every 200 years the moons would align with Moon 2 in the nexus and moon 1 would be on the exact opposite side of planet 1.

1:

Answer 1

1. You did not say how many hours are in a day and how many days are in a year.

   Note that since the duration of MoonOne's orbit is 24 hours, and after 200 years it must have completed an integer number of orbits, the duration of a year is constrained to be of the form $k \times 3/25$ hours, with $k$ being an integer.

   Justification: 200 years is then $k \times 3/25 \times 200 = k \times 24$.

2. It is plain obvious that from a purely arithmetical point of view there are infinitely may solutions.

3. Let's assume that one day is 24 hours, and there are $N$ days in a year. You can easily see that any duration of MoonTwo's orbit of the form $n + 1 / (200 \times N)$, with $n$ an integer fraction of $N$, will satisfy the requirement.

   Justification: After one year of $N$ days, the position of MoonTwo would have advanced, on the average, by $1/200$ of its orbit; the $n$ vanishes because it is an integer fraction of $N$. After 200 years it would have advanced an entire orbit.