Eclipse Cycle Calculations Needed - Astronomy

NOTE: I am using a "geocentric" frame of reference, where both the
moons and the sun orbit the planet, and am creating an arbitrary xy
coordinate system.

We note from @Hohmannfan's answer that (answering your questions out
of order for simplicity):

• Moon B will eclipse the sun every $frac{10385}{304}$ (~ 34.16)
  days. In this time period, the sun completes $frac{31}{304}$th of
  an orbit and Moon B completes $1frac{31}{304}$ orbits, lapping the
  sun once.




• Moon A will eclipse the sun every $frac{26130}{257}$ (~ 101.67)
  days. The sun will complete $frac{78}{257}$ of an orbit, and Moon A
  will lap it by completing $1frac{78}{257}$ orbits.




• Moon B will overlap Moon A once every $frac{2418}{47}$ (~ 51.44)
  days, in which Moon A will complete $frac{31}{47}$ of an orbit and
  Moon B will lap it by completing $1frac{31}{47}$ orbits.

However, as @Hohmannfan notes, there's no guarantee that the moons
will be full when they overlap.

There's also no guarantee that the two moons will ever both eclipse
the sun at the exact same time, although they will get arbitrarily
close to doing so:

In the $frac{2418}{47}$ days between two successive lunar overlaps, the sun
moves $frac{2418}{47} times frac{1}{335}$ of an orbit.

As above, the moons have advanced $frac{31}{47}$ of an orbit.

Thus, compared to the sun, the moons have advanced $frac{31}{47} - frac{2418}{47} times frac{1}{335}$ or $frac{7967}{15745}$ of an
orbit (this number is surprisingly close to $frac{1}{2}$ but that's
just a coincidence).

This happens between every pair of overlaps, so the sun's angular
distance (in orbits) from the overlapping moons is $frac{7967 n}{15745}+r$ where $r$ is the angular distance at a specific overlap
and $n$ is any integer.

For the overlapping moons to eclipse the sun $frac{7967 n}{15745}+r$
must be an integer. If $r$ is irrational, this can never happen.

However, the angular distance can get arbitrarily small, even to the
point where an observer wouldn't realize the double moon eclipse isn't
100% perfect.

By a similar argument, you can show the two full moons will get
arbitrarily close to overlapping.

NOW, if we make the simplifying assumption that both moons are
eclipsing the sun at year 0 (perhaps your astronomer-priests have
decided this unusual occurence is a good time to start numbering the
years, and believe zero (not one) is a good first year), we can make
some other calculations.

Since the moons line up every $frac{2418}{47}$ days and the sun and
Moon B line up every $frac{10385}{304}$ days, all three will line up
(to form a double moon eclipse of the sun) on the least common
multiple of these numbers, or 810,030 days (which would be exactly
2418 of your years, and note that 2418 is the product of the two lunar
orbits in days). In this time:

• Moon A will have completed exactly 10,385 orbits.




• Moon B will have completed exactly 26,130 orbits.




• As above, the sun will have completed exactly 2,418 orbits.

As it turns out, there can never be a perfect double full moon bullseye:

• Moon B will be full on day $frac{10385}{608}$ (~ 17.08), at which
  point it will have completed $frac{335}{608}$ of an orbit and the
  sun will have completed $frac{31}{608}$ of an orbit, so Moon B will
  have gained half an orbit on the sun, which is required for a full
  moon. After that, the moon will be full every $frac{10385}{304}$
  days, the period of time it takes the sun to complete
  $frac{31}{304}$ orbits, and Moon B to complete $1frac{31}{304}$
  orbits.




• By similar calculation, Moon A will be full on day
  $frac{13065}{257}$ (~ 50.84) and every $frac{26130}{257}$ days
  thereafter.




• To find when they're both full at the same time, we solve this
  linear Diophantine equation:

$frac{10385 n}{304}+frac{10385}{608}=frac{26130 m}{257}+frac{13065}{257}$

where n and m are integers. This reduces to:

$nto frac{47424 m+15745}{15934}$

Unfortunately, $47424 m$ is always even, so $47424 m+15745$ is always
odd. Since the denominator ($15934$) is even, you are dividing an odd
number by an even number, and the result can never be an integer.

However, this doesn't tell the full story. For example, if we compute
the positions on day $frac{34987576465283}{92766720}$ (~ 377156.55),
we find:

• Moon B is at 122.5656 degrees.




• Moon A is at 122.5581 degrees, only ~ 27 arcseconds away.




• The sun is at 302.5658 degrees, 179.9998 degrees from moon B, and
  179.9924 degrees from moon A (~ 28 arcseconds from opposition).

In other words, this is pretty close to a double full moon, even
though it's not exact.

In a similar vein, even though double solar eclipses only occur once
every 810,030 days, there are several close calls:

$begin{array}{cc} text{Day} & text{Sep (')} \ -810030.00000 & 0.00 \ -754313.10860 & 0.91 \ -698596.21710 & 1.82 \ -642879.32570 & 2.73 \ -587162.43420 & 3.64 \ -531445.54280 & 4.55 \ -475728.65130 & 5.47 \ -445735.13160 & 7.29 \ -420011.75990 & 6.38 \ -390018.24010 & 6.38 \ -364294.86840 & 7.29 \ -334301.34870 & 5.47 \ -278584.45720 & 4.55 \ -222867.56580 & 3.64 \ -167150.67430 & 2.73 \ -111433.78290 & 1.82 \ -55716.89145 & 0.91 \ 0.00000 & 0.00 \ 55716.89145 & 0.91 \ 111433.78290 & 1.82 \ 167150.67430 & 2.73 \ 222867.56580 & 3.64 \ 278584.45720 & 4.55 \ 334301.34870 & 5.47 \ 364294.86840 & 7.29 \ 390018.24010 & 6.38 \ 420011.75990 & 6.38 \ 445735.13160 & 7.29 \ 475728.65130 & 5.47 \ 531445.54280 & 4.55 \ 587162.43420 & 3.64 \ 642879.32570 & 2.73 \ 698596.21710 & 1.82 \ 754313.10860 & 0.91 \ 810030.00000 & 0.00 \ end{array}$

The table above lists all near-eclipses within 7.5 minutes of arc,
where day is the number of days from year 0 (including days before
year 0), and sep is the maximum separation (in minutes of arc) of any
two of Moon A, Moon B, and the sun. Note that days $0$ and $pm 810030$ are perfect eclipses, as expected.

Similarly, the closest we get to double full moons is below. In this
case, sep is (in minutes of arc) the maximum of:

• the angular distance of Moon A from opposition




• the angular distance of Moon B from opposition




• the angular distance between Moon A and Moon B

$begin{array}{cc} text{Day} & text{Sep (')} \ -797168.29790 & 10.29 \ -767174.80850 & 8.92 \ -711457.91490 & 7.55 \ -655741.02130 & 6.17 \ -600024.12770 & 4.80 \ -544307.23400 & 3.43 \ -488590.34040 & 2.06 \ -432873.44680 & 0.69 \ -377156.55320 & 0.69 \ -321439.65960 & 2.06 \ -265722.76600 & 3.43 \ -210005.87230 & 4.80 \ -154288.97870 & 6.17 \ -98572.08511 & 7.55 \ -42855.19149 & 8.92 \ -12861.70213 & 10.29 \ 12861.70213 & 10.29 \ 42855.19149 & 8.92 \ 98572.08511 & 7.55 \ 154288.97870 & 6.17 \ 210005.87230 & 4.80 \ 265722.76600 & 3.43 \ 321439.65960 & 2.06 \ 377156.55320 & 0.69 \ 432873.44680 & 0.69 \ 488590.34040 & 2.06 \ 544307.23400 & 3.43 \ 600024.12770 & 4.80 \ 655741.02130 & 6.17 \ 711457.91490 & 7.55 \ 767174.80850 & 8.92 \ 797168.29790 & 10.29 \ end{array}$

Other notes:

• Even though you said this was fiction, note that it's highly
  unlikely that the moons' orbital period will be an exact multiple of
  the planets day. The only exception to this is if the moon(s) are
  tidally locked, in which case the orbital period will equal exactly
  one day.




• Similarly, it's unlikely the planet's orbital period would be an
  exact multiple of its rotation period (ours certainly isn't).

This is an interesting problem in general, and I am writing
https://github.com/barrycarter/bcapps/blob/master/MATHEMATICA/bc-orrery.m
to solve a similar problem:
http://physics.stackexchange.com/questions/197481/

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