# How is this possible: a moon with A) large apparent size B) that spins C) spins in a 5:1 spin-orbit resonance?

I wanted to give my planet a moon that spins every SIX days and waxes-wanes every THIRTY days.

The bad news: Luna always shows the same face to Terra because of something called tidal locking explained here. And not just Luna: "All twenty known moons in the Solar System that are large enough to be round are tidally locked with their primaries, because they orbit very closely and tidal force increases rapidly (as a cubic function) with decreasing distance" A small and irregularly-shaped moon like Phobos and Deimos could spin, but that's not pretty and the world should be pretty.

The workaround: I'm not sure that 'tidal locking' means 'no rotation'. I thiiink (and here I am asking clarification from someone who knows the science better than I) that the forces that create tidal locking are a strong pull between the orbiter and the orbitee, and that doesn't necessarily mean stillness, it can mean a steady rhythm. I got my current feeble understanding mostly from this page which talks about how Mercury is locked to Sol in spin-orbit resonance ("This was the first calculation that showed that locking in fractional ratio is actually possible.")

Why I wanted to do this: Terra has months (lunar), years (solar), and weeks, which are based on nothing celestial. Many Terran cultures have had a time-division of 7±2 days. I thought wouldn't it be cool if we have celestial weeks. So then we need three celestial events, to track months, years and weeks. I was going to add two moons (one with a weeklong cycle and one with a monthlong cycle), but "looking up at the two alien moons" feels too much like Astounding Science Fiction, feels too alien. Another issue is two moons + Sun has religious implications, rather than a Moon-Goddess + Sun-God you'd need a triad. Solution: if the moon changes faces once a week and waxes once a month, you track weeks and months with one celestial body. Then I thought it was an exciting bonus that the 'fractional locking' thing is a real scientific justification for a nice tidy perennial calendar; there are scientific reasons to expect nice ratios like 5:1 (rather than something messy like 4.418941:1)

So the question in a nutshell is: would this work? The moon is big and close, like our Luna is. That implies that it is tidally locked (according to my understanding). But is it possible that it's tidally locked in a 5-spins-per-month rhythm rather than a no-spin rhythm?

Possibly relevant: The world in question was engineered by alien scientists far more advanced than the people who live on it. Extraterrestrial creationism. Like Slartibartfast. If the orbits and periods seem too good to be true, suspiciously tidy, that is actually good; the people in the story will notice their world is suspiciously well-designed.

Background research:

• The 7 days are associated to the 7 shiny and oddly moving things one can see in the sky: Sun, Moon, Mercury, Mars, Venus, Jupiter, Saturn. I would say they are celestial as well.
– L.Dutch
Commented Apr 1 at 17:34
• @L.Dutch: In Romance languages five of them are, indeed, named for lights in the sky. (Only five because Saturday is "Sabbath Day" and Sunday is "Lord's Day" in Romance.) But in English only three of them are (Saturday, Sunday and Monday) with the other four named for old pagan gods and goddesses, in German only two of them are (Sonntag and Montag), and in Russian none of them are (the Russian week goes "Day after Holy Day", "Second Day", "Mid Week", "Fourth Day", "Fifth Day", "Sabbath Day", "Holy Day"). Commented Apr 1 at 18:08
• A tidally locked object rotates once per orbit. That's how the locked object keeps the same face toward the locking object.
– JBH
Commented Apr 1 at 18:39
• "Our use of the seven-day week can be traced back to the astronomically gifted Babylonians and the decree of King Sargon I of Akkad around 2300 BCE. They venerated the number seven, and before telescopes the key celestial bodies numbered seven (the Sun, the Moon and the five planets visible to the naked eye)." (Source) The article also credits the Jewish Genisis account, but it's unlikely a lot of other cultures would have picked that up. The Babylonians, on the other hand...
– JBH
Commented Apr 1 at 18:43
• All that aside, it appears that if you delete everything that's causing us to comment... all of the Earth-centric stuff that isn't relevant to your question at all, what's left is, "can I have a moon the size of Earth's moon that's still rotating more than once per orbit?" I believe it's believed that all moons (celestial objects) rotated and slowly became tidally locked. I believe it's also believed that this takes long enough that life didn't begin until after tidal locking. But is it really necessary to duplicate what happened on Earth? Why not spin the moon and move forward?
– JBH
Commented Apr 1 at 18:50

## TL;DR

While a 5:1 resonance between the rotation of the Moon and its orbital period is possible (albeit quite unlikely), it won't help.

## Long explanation

A lunar month, or a lunation, is the time between two consecutive new Moons, that is, the time it takes the Moon to come back into the same position with respect to the Sun as seen from Earth. This time is not equal with the time it takes the Moon to complete one revolution around the Earth.

• The time it takes the Moon to complete one revolution around the Earth is called the sidereal month, because after one revolution around the Earth the Moon comes into the same position with respect to the fixed stars. (The Latin word for "stars" is sidera. One star is sidus.)

A sidereal month is 27.321661 average Earth solar days, or 27 d 7 h 43 min 11.6 s.

• The time between two consecutive new Moons, that is, the time it takes the Moon to come back into the same position with respect to the Sun, is of course longer than a sidereal month; that's because after the Moon has completed a full revolution around the Earth, the Earth itself has advanced about 26° 56′ on its orbit around the Sun, so that the Moon still has some catching up to do in order to get back into position.

This period is called the synodic month; on the average, a synodic month is 29 Earth solar days, 12 hours, 44 minutes and 2.9 seconds.

The word synodic is Greek for "same path", syn- + -hodos "road, path". From the same Greek word meaning "road", the English language has "odometer", a device for measuring the distance travelled.

• From simple geometrical considerations, the relationship between Earth sidereal years $$T_\text{Earth}$$, sidereal months $$T_\text{Moon}$$, and synodic months $$S_\text{Moon}$$ is as follows:

$$S_\text{Moon} = T_\text{Moon} \times \frac{T_\text{Earth}}{T_\text{Earth} - T_\text{Moon}}$$

Note that what counts is Earth sidereal year, not the practically important tropical year.

Moreover, the duration of a synodic month is not constant, because of course Earth's orbit is not perfectly circular, but a little flattened into an ellipse, so that the angular speed of Earth as it moves on its path around the sun is variable. Practically, the duration of a synodic month varies from about 29 days 6 hours to about 29 days 20 hours in the course of a year.

The point of all this arithmetic is to show that a 5:1 rotation-to-orbital period resonance, while not absolutely impossible, won't help. What would be needed would be a to have the rotation of the Moon equal an integer fraction of the synodic month, and that is extremely unlikely, because the two durations do not have any particular relationship.

Your question is basically whether a 5:1 spin-orbit resonance is possible, and the answer seems to be 'yes', even if we don't have any confirmed existence proof in our own naturally created solar system, and such a configuration arising naturally is improbable.

First, a few corrections: in any reasonable orbital frame, Earth's Moon does spin—specifically, it rotates once for every orbit. That is, tidal-locking does not generally mean "no rotation".

Moreover, it also does not necessarily mean 1:1 rotation, as you suspected it might not:

A widely spread misapprehension is that a tidally locked body permanently turns one side to its host.

(astronomer quoted in paper, via Wikipedia article on tidal locking)
That is, a tidally locked body can rotate at other rates than 1:1; the 'locking' just means that the spin and the orbit are coupled. These are called 'spin-orbit resonances', and the 1:1 special (but also commonest) case is 'synchronous resonance'. Admittedly, the usage does vary somewhat, with tidal locking only usually indicating 1:1 resonance—no doubt this practice, combined with our solar system's predominantly 1:1 spin-resonances being the usual subject, exacerbates attendant confusion.

Intuitively, as stated in this paper:

Because of tides in their interiors, mostly solid exoplanets are expected eventually to despin to a state of spin-orbit resonance, where the orbital period is some integer or half-integer times the rotation period.

Mathematically, the Fourier transform of the forcing function has islands of stability around half-integer multiples of $$\nu_k/n$$ (frequency per mean motion). See Spin-orbit coupling for close-in planets Eqn. 10 and Fig. 1, reproduced below:

As you can see, higher resonances are less stable (i.e. require more 'fine-tuning'), but there's no particular reason certain ratios are allowed and other aren't. (A related analysis with phase-space plots.)

This paper goes over some of the details for hypothetical versions of Mercury, concurring with the plot above. 1:1 synchronous resonance is the most likely and higher-order resonances are possible but less probable. For what it's worth, this makes our solar system's observed preponderance of 1:1 resonances not particularly surprising.

An interesting additional result of that last paper is that the higher the eccentricity, the more likely the system is to fall into a more exotic resonance (the paper phrases this as that higher resonances require very lucky initial conditions). Happily, it has been widely noted that exoplanets have all kinds of eccentricities (though that's somewhat due to an observation bias which favors discovering larger eccentricities).

Putting it all together, a 5:1 resonance seems very possible, with a high eccentricity making it more probable (though still improbable in an absolute sense). If, as you mention, the scenario in-setting arises due to alien intervention, so much the better for explaining it.

I haven't addressed the aspect of large apparent size. It does not seem that this matters to the outcome beyond how it affects the orbit itself: a closer body will experience stronger forcing, but the 'shape' of the forcing doesn't change from that per-se.

• Thanks. To address your last paragraph: I was talking about big beautiful moons, not little lumps like Mars's moons that don't appear like the Moon we're used to. I know little ones don't get tidally locked, so they're different. Commented Apr 2 at 16:16

After our moon's formation, it took on the order of 100 million years for the moon to become tidally locked to Earth. That is a long time. If celestial motion is literally orchestrated to seem perfect, it will take a long time for natural forces to undo the engineering.

Yes, given advanced engineering, what you're asking can be done. If you're interested in the futurism side of sci-fi (for your engineers), I highly recommend checking out Isaac Arthur's work. His YouTube channel is very approachable and full of goodies.