Worldbuilding Guide Part 5: Moons and Rings

Links to previous parts: Part 1, Part 2, Part 3, and Part 4
Previously, we created stars, planets and their orbits around the aforementioned stars. However. All of these planets have something missing: Moons. Moons are very important to a vivid depiction of a world, and are also rather fun to make. They can also often be better at having life on them than their star-orbiting counterparts, as they can be protected by their parent planet's (possibly very strong) magnetic field.
== Terrestrial Moons ==
=== Intro ===
For the purposes of this tutorial, we will classify moons by size and composition. 
==== Size ====
All natural satellites can be either Major or Minor moons. Minor moons are small satellites that lack the gravity to pull themselves into a sphere, and as such they usually have funny shapes. These bodies, like the moons of mars, can be no larger than 200-300 km in radius. Major moons on the other hand, like our moon Luna, are gravitationally rounded, and they must have radii larger than 200-300 km.
==== Composition ====
All moons can be either predominantly rocky, like our moon, or predominantly icy, like Mimas. A grid such as this, although very simplistic, is a very useful tool for making worldbuilding easier. 
=== Guidelines ===
Terrestrial planet moons generally follow a few simple guidelines.
#Amount - For the most part terrestrial planets will have very few satellites if any. They may occasionally capture one or two asteroid minor moons, but rarely if ever have any major moons.
#Habitability - if your planet is intended to be habitable, it should probably have a major moon. There are exceptions of course, as with tidally locked planets around red dwarves, but this is a general rule you should follow.
#Distance - Close in planets will have less moons than that of more distant planets, all things being equal, as they have smaller hill spheres where a moon can orbit.
#Composition - Inner system moons (planet SMA<Frost Line) will be predominantly rocky (with exceptions), and outer system moons (planet SMA>Frost Line) will be predominantly icy.
=== Major Moon construction ===
Major moons will not often occur around terrestrial planets, but they obviously can (example: Luna). Major moons surrounding terrestrial planets will generally only occur if the parent planet got whacked by another, preferably smaller, planet in its early history. This mostly disintegrates the parent planet, and the debris created by the collision forms a Synestia, a cloud of debris which will eventually recoalesce into the planet and its new moon. 
To begin, choose a radius for your moon larger than 200-300 km, then choose the percentages of materials which form your moon. The main materials which form planets and moons are iron, silicate materials, and water ice. After you choose your percentages (ensuring they sum to 1), simply multiply your percentage by the corresponding density of the material, and add the three values together:
Densities of planet and moon-forming materials
Iron 	Silicates 	Water Ice
7.87 g/cm2	3.25 g/cm2	0.93 g/cm2
Next we can use the following equation to find the mass of the moon from the radius, which we already have, and the density, which we already have. M is the mass in earth masses, R is the radius in kilometers, and ρ is the density in g/cm2.
M=(R/6371)^3 * ρ/5.51
We can then plug our mass and radius into the following equation to get our moon's surface gravity.
g=M/R^2
=== Minor Moon construction ===
Minor moons aren't spheres, so there's less we can really say about them without some 3d modeling program or something. The stabilization effect of minor moons on their parent planet's axial tilt is nonexistent, which means they don't affect planetary habitability in any way, which means we can do WHATEVER WE WANT WITH THEIR ORBITS!
Instead of going through the densities of different materials, just pick a reasonable-sounding number to avoid endless tedium. If it's rocky, go for something near 2 or 3 g/cm2. If it's icy, go for something a little above 1 g/cm2. If it has a lot of iron (which is uncommon), go for something a little less than 5 g/cm2.
The moon is non-spherical, so we can have it be any shape we want. However, for ease of construction, I would advocate either copypasting the statistics of an unimportant asteroid (preferably one that's been imaged up close) or modeling your minor moon as an Ellipsoid. This means that we need to specify 3 axes, a, b, and c.
Sidenote: ellipsoids come in 3 distinct flavors. Choose one for your modeling at will.
Ellipsoid classes
Name 	Dimension ratios
Tri-axial Ellipsoid	A>B>C
Oblate Spheroid 	A=B>C
Prolate Spheroid 	A=B<C
Now that we have density, size, and shape, we can find the mass in kilograms of our minor moon with the following equation, where ρ is the density, π is well, pi, and a, b, and c are the three axes.
M = ρ(4/3)πabc
Your number might seem like a lot but really it's absolutely tiny. We could try and find the surface gravity, but given the irregularity and tininess of minor moons, that's REALLY overkill. Even finding the mass is a little overkill, it's not entirely necessary. Knowing its composition, size, and shape is more than enough.
Once this is done, we can put our moon into orbit as below: 
=== Orbits ===
To put our moon into orbit, we are going to know the physical characteristics of our planet, and its parent star, both of which have been covered in previous parts. Planets, like all astronomical objects, are encased in a dynamically defined sphere surrounding the object known as a Hill Sphere. Anything inside this zone is more gravitationally affected by this object than whatever its parent object is, allowing it to orbit the secondary object. Therefore, satellites must be placed within this zone, the outer limit of which in earth radii is given by: 
Hs = a * ∛(m/M) * 235 
However, this is not the only limit involved here. Many of you will know that moons cannot orbit arbitrarily close to their parent planet. The inner limit here is given by the Roche Limit, depending on the density of the moon as well as the planet's properties, and given by:
2.44 * R * (ρp/ρs)^(1/3)
Sidenote: Moons have hill spheres and roche limits too, calculable with the same formulae, so in special circumstances, you can place moons of moons.
Anyway, between the limits given with those two equations, you can place your moon! You will have to separately calculate roche limits for moons in some cases, but this will not usually be necessary unless you wish to place a moon unreasonably close to a planet.
You can place moons anywhere between these two limits, but its generally best to place most moons less than half the distance to the edge of the hill sphere, to ensure long-term stability.
The orbital period of our moon in earth days can be found by the following equation, where R is the radius of the orbit in earth radii, M is the mass of the planet and m is the mass of the moon.
P = 0.0588 * √(R^3/(M+m))
Now that we have semi-major axis and orbital period, we must decide whether our moon is regular or irregular. Major moons are almost invariably regular satellites, which means that the have low-eccentricity (>0.1), low-inclination (>10º), prograde orbits. Minor moons, on the other hand, with a few exceptions, are almost always irregular satellites, which means that they can have pretty much any sort of orbit, and are often captured. Irregular satellites can also result from a regular satellite getting too close to a major moon and getting thrown onto a weird orbit.
Knowing the general type of orbit our moon has, we can then decide precise values for inclination and eccentricity, then find the periapsis and apoapsis in the same manner as we do for planets. Next up: Gas Giant Moons!
== Gas Giant Moon Systems ==
=== Rings! ===
Gas Giants very often have rings. They are usually placed from 1.34-2.44x the radius of the giant planet (plus or minus 0.2 in either direction if you want for both of those), or if you want to go overkill, calculate the roche limit of whatever moon you're breaking up and use that as the outer radius. 
=== Moons ===
Most gas giants will have up to three wildly different groups of satellites, conveniently named A, B, and C, each with different properties.
==== Group A ====
Group A moons are irregular minor moons that orbit in the outer half of the ring system and just outside it, orbit 0.25-1.5 Earth Radii apart from each other, are from 10s to early 100s of kilometers in diameter, and, most notably, are held together by tensile strength rather than gravity. This has to be so because if they were held together by gravity, they would be ripped apart by the planet's Roche Limit.
The Reason why these moons will be so tightly packed together is because Mean Motion Resonances (discussed in part 3) stabilize their orbits. However, you dont have to lay out an absurd web of resonances. The 0.25-1.5 Earth Radii rule is enough because no one really looks closely at these anyway.
Another thing - Most of your Group A moons will be embedded within the rings. Because of this, they will leave some small gaps in the rings at their orbits, their width usually a little over ten times the diameter of the moon itself. Plot these.
==== Group B ====
The B group will contain a satellite system's most well-known and largest satellites by far. It consists of a handful of large, regular major moons. They can go from roughly 3-15 planetary radii from the gas giant's center, and no two major moons should go closer than 1 planetary radius from each other. 
We can also include minor moons in our major moon layout, as either lagrangian companions or orbiting in resonance and in between the major moons. Real world examples include Methone and a few other obscure saturnian moons. There's no clear group that these fall into so we'll just call them group B2, and you should definitely throw them in!
It's a good idea to check whether any of your moons have a 2:1 Mean Motion Resonance which falls within the ring system. If so, make a nice big gap there of roughly 0.5 earth radii. Just remember when filling in the characteristics of your moons, give the moon causing the resonance a relatively high mass. 
==== Group C ====
The final group, the C group, consists of a collection of distant, irregular minor moons orbiting on random orbits far from the main moon system. This means they orbit from ~20 planetary radii all the way out to the edge of the Hill Sphere, and orbit on highly inclined and eccentric orbits, some of which might even be retrograde. With few exceptions, they will be roughly 10s of kilometers in size.
Because most gas giants orbit relatively far from their star and are absolutely gigantic, you have a huge amount of room for your captured moons. You can set up several different classes, where members of the same class have relatively similar orbits. If you wish, you can go through each satellite and meticulously calculate them, but that is overkill, and you can just give a general idea with your classes.
=== Habitable Moons? ===
If we wish to make a habitable moon, we must move our gas giant into the habitable zone, and give it a large, earthlike moon. There are two ways to do this, either the gas giant migrates inward and holds onto an in-situ earth-mass-ish moon, or it migrates and captures a planet from the habitable zone on the way in. They both work, but neither is perfect. 
To ensure atmospheric retention, a habitable moon must be at least 0.25 earth masses. Since most moons are less than 1/5 that, a moon this size is very unlikely to form in-situ. However, captured major moons are very rare and would require something to slow them down into orbit which would be very likely to be destructive. 
But the in-situ problem isn't the biggest problem: The bigger problem is that such a moon, having been formed beyond the frost line, will be icy. This means that when temperatures increase, the moon would melt, creating an ocean planet rather than an earthlike planet. Because of this, the capture-after-formation method is probably the most likely.
Great! However, there are still more hoops to jump through: Habitable moons should be placed around stars greater than 0.5 solar masses. Below this, the hill sphere of your gas giant will be too small to allow long-term stability in the habitable zone.
Once ensnared, ensure the moon orbits between 10 and 20 planetary radii, and ensure that its orbital eccentricity is less than 0.01. Just in case, make sure that the ratio between the moon's orbital period and the planet's orbital period is less than 1/9. However, all known captured objects are irregular satellites. That is, they have highly inclined and eccentric orbits, which means in theory you could have your habitable moon have a very wacky orbit. But this is unadvisable. 
So yeah! We've done our habitable moon!
Oh. 
You want something cooler.
Well you asked for it! Saturn's moons Janus and Epimetheus have a type of orbit known as a Horseshoe Orbit. Here's how it works: They both share near-identical orbits. Epimetheus, orbiting slightly closer, will be traveling farther and will approach Janus from behind. The closer they get, the more they gravitationally effect each other. Epimetheus slows Janus down, which makes it fall towards Saturn. Epimetheus likewise speeds up, increasing the size of its orbit. In other words, they switch places. This all repeats four years later.
You could set up something like this with two moons or even two habitable moons. It'd be unlikely as hell, but hey! It's possible, and would look awesome, with the other moon getting really close, then turning around and receding. That is the definition of awesome. If one of you doesn't do this I'm suing.
Right! We have finished the moon systems of your planets, and now we can get down to business with the most difficult part in Worldbuilding Guide Part 6: Planetary Mapping!
Category:Helpful
Category:Worldbuilding Guide