<p>Frankly, four bodies do not periodically line up, even if the orbits are planar, unless there exist a common multiplum. Because all your periods are <strong>integers</strong>, that holds, but be aware that your double alignments, like the double solar eclipse or double lunar eclipse is not periodic if one of the periods does not <strong>exactly</strong> follow that integer system. (so it does not hold for 31.01 days, for example. How you handle quazi-periods is talked about <a href="https://physics.stackexchange.com/a/232918/102747">here</a>).</p>

<blockquote>
  <ol>
  <li>How often would both moons be full at the same time, overlapping each other like a bullseye?  </li>
  </ol>
</blockquote>

<p>Generally non-periodic, but can only happen at the <a href="https://en.wikipedia.org/w/index.php?title=Orbital_period&amp;redirect=no" rel="nofollow noreferrer">synodic period</a> of the two moons, that is:</p>

<p><a href="https://i.sstatic.net/Iu5t9.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Iu5t9.png" alt="synodic"></a>.</p>

<p>$\frac{2418}{47}$ for the two moons. However, that must also happen when both of them is on the same line as the Sun, in other words when The relative synodic period between one of the moons and the sun, and the moons' synodic period both divides the requested period. For the inner moon and the Sun, the synodic period is $\frac{10385}{304}$. Since 47 does not have any factors in common with 304, the first common multiplum appears at.</p>

<p>$$\frac{10385}{304} \cdot \frac{2418}{47} = \frac{12555465}{7144}$$, or 1757.484 days. (@Jonathan gets 2418, because his model consider the Sun to stay static in the sky, but it does in fact revolve around one time a year.) The exact same period is also valid for your <strong>question 4</strong>.</p>

<blockquote>
  <ol start="2">
  <li>How often would lunar eclipses occur for Moon A?</li>
  </ol>
</blockquote>

<p>Here, the answer is the synodic period of the Sun and the moon, already stated to be $\frac{10385}{304}$, or 34.161 days. That is a little bit longer than the Moon's orbital period, because the planet has moved a little in its orbit, slightly changing the Sun's position in the sky.</p>

<blockquote>
  <ol start="3">
  <li>How often would lunar eclipses occur for Moon B?</li>
  </ol>
</blockquote>

<p>Synodic period here too, $\frac{26130}{257}$ = 101.673 days.</p>

CC BY-SA 3.0
2016-03-03T07:49:20.323Z
2016-03-07T08:18:49.200Z
2017-04-13T12:40:15.377Z
SE - stop firing the good guys
<p>Here is my attempt, <strong>assuming the orbits are circular, and along the same plane</strong>.  Otherwise, calculations become more complex, and those variables would have to be specified.  Size of the planet and size of the moons would also impact the calculation (e.g. they might never fully eclipse if the planet is fairly small / dense).  Also note that Moon A may eclipse Moon B and Moon B may eclipse Moon A depending on their size.  I do not consider this in my calculations (I only consider the planet eclipsing them).</p>

<blockquote>
  <p>1.How often would both moons be full at the same time, overlapping each other like a bullseye?</p>
</blockquote>

<p>Not including the planets orbit, 31 is a prime number, and 78 breaks down to the factors of 2, 7, 7.  Since there is no shared common denominator, it does not appear these will synch up more frequently than 78 * 31 =
<strong>2418 days</strong></p>

<blockquote>
  <p>2.How often would lunar eclipses occur for Moon A?</p>
</blockquote>

<p><strong>About 78 days</strong> (+- about 78/335 days due to the planet orbiting the sun)</p>

<blockquote>
  <p>3.How often would lunar eclipses occur for Moon b?</p>
</blockquote>

<p><strong>About 31 days</strong> (+- about 31/335 days due to the planet orbiting the sun)</p>

<blockquote>
  <p>4.How often would both eclipse at the same time?</p>
</blockquote>

<p>Same answer as #1, which I calculate at <strong>2418 days</strong></p>

<p>I welcome refinement / correction of my calculations, this is a "rough attempt".</p>

CC BY-SA 3.0
2016-03-03T07:58:41.063Z
So I had a lunar eclipse of the outer moon. 78 days later, the Moon must be at the same position again, but the planet has moved almost a quarter around in its orbit. The moon can not move almost 90 degrees in just "+- about 78/335 days".
SE - stop firing the good guys
CC BY-SA 3.0
2016-03-02T18:05:54.343Z
2016-03-02T19:54:43.417Z
2016-03-02T19:54:43.417Z
Jonathan
<p>I am writing a work of fiction about a world where:</p>

<p>The length of a year is 335 days.
The world has two Moons.
Moon A is Larger and further away with a 78 day lunar orbit.
Moon B is smaller and closer with a 31 day orbit.</p>

<ol>
<li>How often would both moons be full at the same time, overlapping each other like a bullseye?</li>
<li>How often would lunar eclipses occur for Moon A?</li>
<li>How often would lunar eclipses occur for Moon b?</li>
<li>How often would both eclipse at the same time?</li>
</ol>

<p>This is a work of fiction, so liberties can be taken.  I just want it to be close.</p>

CC BY-SA 3.0
2016-03-02T14:41:57.007Z
As @barrycarter correctly implies, the angle of respective orbits matters. For example if everything was perfectly in plane in our Solar system, we would see lunar and solar eclipses every month (you'd need to be in the right place to see the Solar ones though.)
Andy
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2016-03-02T15:02:42.967Z
By the way you might find this interesting: (not on topic as it's not an eclipse, merely an occultation) http://www.planetary.org/blogs/emily-lakdawalla/2013/movie-of-phobos-and-deimos.html
Andy
CC BY-SA 3.0
2016-03-02T18:41:54.257Z
Would the question be a better fit for "Worldbuilding"?
James K
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2016-03-03T08:01:01.187Z
@JamesKilfiger It is on-topic both places.
SE - stop firing the good guys
CC BY-SA 3.0
2016-03-07T07:54:09.153Z
I'm voting to close this question as off-topic because questions about purely hypothetical situations are off topic, however there is worldbuilding
James K
<solar-eclipse><lunar-eclipse>
Eclipse Cycle Calculations Needed
Urlord
# Eclipse Cycle Calculations Needed

(Asked by: username_0 on 2016-03-02)

I am writing a work of fiction about a world where:

The length of a year is 335 days.
The world has two Moons.
Moon A is Larger and further away with a 78 day lunar orbit.
Moon B is smaller and closer with a 31 day orbit.

1. How often would both moons be full at the same time, overlapping each other like a bullseye?
2. How often would lunar eclipses occur for Moon A?
3. How often would lunar eclipses occur for Moon b?
4. How often would both eclipse at the same time?

This is a work of fiction, so liberties can be taken. I just want it to be close.

**Tags:** solar-eclipse, lunar-eclipse
> **Comments:**
> **username_1**: As @barrycarter correctly implies, the angle of respective orbits matters. For example if everything was perfectly in plane in our Solar system, we would see lunar and solar eclipses every month (you'd need to be in the right place to see the Solar ones though.)  
> **username_1**: By the way you might find this interesting: (not on topic as it's not an eclipse, merely an occultation) http://www.planetary.org/blogs/emily-lakdawalla/2013/movie-of-phobos-and-deimos.html  
> **username_2**: Would the question be a better fit for "Worldbuilding"?  
> **username_3**: @JamesKilfiger It is on-topic both places.  
> **username_2**: I'm voting to close this question as off-topic because questions about purely hypothetical situations are off topic, however there is worldbuilding  

---

## 2 answers

### Answer by username_3 on 2016-03-03. Score: 3

Frankly, four bodies do not periodically line up, even if the orbits are planar, unless there exist a common multiplum. Because all your periods are **integers**, that holds, but be aware that your double alignments, like the double solar eclipse or double lunar eclipse is not periodic if one of the periods does not **exactly** follow that integer system. (so it does not hold for 31.01 days, for example. How you handle quazi-periods is talked about [here](https://physics.stackexchange.com/a/232918/102747)).

> 1. How often would both moons be full at the same time, overlapping each other like a bullseye?

Generally non-periodic, but can only happen at the [synodic period](https://en.wikipedia.org/w/index.php?title=Orbital_period&redirect=no) of the two moons, that is:

[![synodic](https://i.sstatic.net/Iu5t9.png)](https://i.sstatic.net/Iu5t9.png).

$\frac{2418}{47}$ for the two moons. However, that must also happen when both of them is on the same line as the Sun, in other words when The relative synodic period between one of the moons and the sun, and the moons' synodic period both divides the requested period. For the inner moon and the Sun, the synodic period is $\frac{10385}{304}$. Since 47 does not have any factors in common with 304, the first common multiplum appears at.

$$\frac{10385}{304} \cdot \frac{2418}{47} = \frac{12555465}{7144}$$, or 1757.484 days. (@Jonathan gets 2418, because his model consider the Sun to stay static in the sky, but it does in fact revolve around one time a year.) The exact same period is also valid for your **question 4**.

> 2. How often would lunar eclipses occur for Moon A?

Here, the answer is the synodic period of the Sun and the moon, already stated to be $\frac{10385}{304}$, or 34.161 days. That is a little bit longer than the Moon's orbital period, because the planet has moved a little in its orbit, slightly changing the Sun's position in the sky.

> 3. How often would lunar eclipses occur for Moon B?

Synodic period here too, $\frac{26130}{257}$ = 101.673 days.


---

### Answer by username_4 on 2016-03-02. Score: 0

Here is my attempt, **assuming the orbits are circular, and along the same plane**. Otherwise, calculations become more complex, and those variables would have to be specified. Size of the planet and size of the moons would also impact the calculation (e.g. they might never fully eclipse if the planet is fairly small / dense). Also note that Moon A may eclipse Moon B and Moon B may eclipse Moon A depending on their size. I do not consider this in my calculations (I only consider the planet eclipsing them).

> 1.How often would both moons be full at the same time, overlapping each other like a bullseye?

Not including the planets orbit, 31 is a prime number, and 78 breaks down to the factors of 2, 7, 7. Since there is no shared common denominator, it does not appear these will synch up more frequently than 78 \* 31 =
**2418 days**

> 2.How often would lunar eclipses occur for Moon A?

**About 78 days** (+- about 78/335 days due to the planet orbiting the sun)

> 3.How often would lunar eclipses occur for Moon b?

**About 31 days** (+- about 31/335 days due to the planet orbiting the sun)

> 4.How often would both eclipse at the same time?

Same answer as #1, which I calculate at **2418 days**

I welcome refinement / correction of my calculations, this is a "rough attempt".

> **Comments:**
> **username_3**: So I had a lunar eclipse of the outer moon. 78 days later, the Moon must be at the same position again, but the planet has moved almost a quarter around in its orbit. The moon can not move almost 90 degrees in just "+- about 78/335 days".  

---

