Past Blog Tie-ins:
• The Hill Sphere and The Roche Limit
• Planets and Worlds, Part 2: Planetary Pairs
Etymology of "Moon"
As a proper noun, the name Moon comes from Old English mōna, stemming from Proto-Germanic *mēnô, itself from Proto-Indo-European (PIE) *mḗh₁n̥s, which meant "month", ultimately derived from PIE *meh₁-, "to measure", because the Moon's phases were used to measure that unit of time.
* For a discussion on whether the Moon is a satellite of Earth or whether the two comprise a double-planet system, see my blog Planets and Worlds, Part 2: Planetary Pairs.
Types of Moons: Regular and Irregular
Regular moons are those which:
Irregular moons are those which:
* "Prograde" means that they orbit in the same direction as the planet's axial rotation.
** "Retrograde" means that they orbit in the opposite direction from the planet's axial rotation.
Classes of Moons: Major and Minor
This is not an unbreakable rule, however; Neptune’s moon, Triton, appears to have been a dwarf planet that passed within the planet’s Hill Sphere at a slow enough angular momentum to be captured. Its orbit is very nearly circular and it orbits close to its primary, like a regular moon, but its orbit is also retrograde and highly inclined, like an irregular moon. Thus, it would be defined as a major-irregular moon.
Examples of minor-irregular moons would be the aforementioned Phobos and Deimos; they are irregular despite the fact that they both orbit close to Mars on nearly circular, prograde orbits with less than 2° inclinations. Interestingly, Phobos orbits Mars in less time than Mars takes to rotate on its axis, so as seen from Mars, it rises in the west and sets in the east, on a cycle of about 11 hours.
(See this article at ScienceAdvances.com for more details).
- The Galilean Moons of Jupiter, Saturn's moon, Titan, and the Moon (Luna) are the largest major moons in the Outer Solar System;
- The ice giant moons are significantly smaller in mass and radius than the gas giant moons;
- Excepting Earth's Moon, the mass and radius ranges of major moons in the Solar System overall decreases with distance from the Sun.
- The most massive (Ganymede) is 2272.7 times more massive than the least massive (Miranda).
- The largest (Ganymede) has 11.17 times the radius of the smallest (Miranda).
Among the Galilean moons, Europa is the least massive and Ganymede is the most massive. The range of their masses of the is 0.017, and their average mass is 0.0165 (which also happens to be the median mass for this data).
Among the Uranian moons, Miranda is the least massive and Titania is the most massive. The range of their masses is 0.00058; their average mass is 0.0003066; their median mass is 0.000226.
Technically, Pluto-Charon together constitute a dwarf binary planet—by IAU standards.
By DeBenedictis' standards, Pluto would be a major planet and Charon, then, would be a major moon.
I leave it up to the readers to decide for themselves as to which solution better adheres to Occam's Razor.
Major Moons of "Terrestrial" Planets
Note: Major, regular satellites need to have orbits spaced at greater than 10 planetary radii from one another, or they will interfere gravitationally and the system will be unstable.
Gas- and Ice-Giant Moons
It would seem to make sense to divide giant planet major moons into three broad mass classes:
- Class A major moons include the moons of Earth, Jupiter, and Saturn
- Class B major moons would includes those in mass ranges centered more-or-less around Neptune's moon, Triton
- Class C major moons include Uranus' entire complement of companions.
This provides us with the following table:
Dermott's Law and The Major Moons of Giant Planets
T(n) is the orbital period of the nᵗʰ ordinal major moon of the planet;
T0 is a constant Dermott worked out for each planet;
Cⁿ is a constant for each planet, raised to the power of the moon's ordinal.
... with the constants:
Jupiter: T0 = 0.444; C = 2.03
Saturn: T0 = 0.462; C = 1.59
Uranus: T0 = 0.760; C = 1.80
For Jupiter, with values of n in the range [1, 6], the predicted orbital periods compared to the actual measured orbital periods are:
This relation can be modified to provide possible major moon orbits for gas- and ice-giant planets by applying a multiplier based on the ratio of the mass of the fictional planet in question, in relation to that of Jupiter, Saturn, or Uranus/Neptune.
Using the expanded mass classification table as a guide, we can (somewhat arbitrarily) set mass limits for each type:
Ice Giant I: 10 - 59 Earth masses (0.688 - 4.059 Uranus masses)
Ice Giant II: 60 - 79 Earth masses (4.06 - 5.435 Uranus masses)
Gas Giant I: 80 - 149 Earth masses (0.841 - 1.566 Saturn masses)*
Gas Giant II: 150 - 449 Earth masses (0.472 - 1.416 Jupiter masses)
Gas Giant III: 450 - 1249 Earth masses (1.416 - 3.930 Jupiter masses)**
Gas Giant IV: 1250 - 3500 Earth masses (3.931 - 11.013 Jupiter masses)
* The gas giants have to be divided into four classes because treating them all as a single class results in moons orbiting faster the farther they are from the planet, which breaks physics.
* Above about 500 Earth masses, gas giant planets are actually somewhat smaller in diameter than Jupiter due to electron degeneracy. Gas giants can actually reach 13 Jupiter masses (4131 Earth masses), but at those masses they become brown dwarfs, and thus should be treated as stars.
Stephen L. Gillet, in his book, World-building: A Writer's Guide To Constructing Star Systems and Life-Supporting Planets, hints on page 31 that no planet should exceed 4.0% of the mass of the star(s) it orbits.
The brown-dwarf pair DENIS-P J082303.1-491201 and DENIS-P J082303.1-491201 b have a mass ratio of 36.26143%, but among actual, confirmed exoplanet-star pairs, one of the highest planet-to-mass ratios is that of HD 180314 b and HD 180314, which is 0.87944%.
In comparison, Jupiter weighs in at 0.09542% of the mass of the Sun.
Ice Giant I: 34.500 Earth masses (2.373 Uranus masses)
Ice Giant II: 69.500 Earth masses (4.781 Uranus masses)
Gas Giant I: 114.5 Earth masses (1.203 Saturn masses)
Gas Giant II: 299.5 Earth masses (0.944 Jupiter masses)
Gas Giant III: 849.5 Earth masses (2.675 Jupiter masses)
Gas Giant IV: 2375.0 Earth masses (7.473 Jupiter masses)
... and round the first five to 35.0, 70.0, 115.0, 300.0, and 850.0, respectively, for simplicity.
We can then use Dermott's values for Uranus as the standard for both classes of ice giants, Saturn's values for Class I gas giants, and Jupiter's values gas giant Classes II, III, and IV.
Ice Giant I: T0 = 0.760; C = 1 80; M = 35.0
Ice Giant II: T0 = 0.760; C = 1.80; M = 70.0
Gas Giant I: T0 = 0.462; C = 1.59; M = 115.0
Gas Giant II: T0 = 0.444; C = 2.03; M = 300.0
Gas Giant III: T0 = 0.444; C = 2.03; M = 850.0
Gas Giant IV: T0 = 0.444; C = 2.03; M = 2375.0
And a complete table becomes:
Ice Giant I: 10 - 59 Earth masses; T0 = 0.760; C = 1.80; M = 35.0
Ice Giant II: 60 - 79 Earth masses; T0 = 0.760; C = 1.80; M = 70.0
Gas Giant I: 80 - 149 Earth masses; T0 = 0.462; C = 1.59; M = 115.0
Gas Giant II: 150 - 449 Earth masses; T0 = 0.444; C = 2.03; M = 300.0
Gas Giant III: 450 - 1249 Earth masses; T0 = 0.444; C = 2.03; M = 850.0
Gas Giant IV: 1250 - 3500 Earth masses; T0 = 0.444; C = 2.03; M = 2375.0
When developing a giant planet, first decide whether the planet will be an ice giant (of Class I or II) or a gas giant (of Class I, II, III, or IV), then calculate its mass relative to the "standard" for the class, and use the result as a multiplier for T0 and C.
Thus, if our Class I ice giant planet (let's call her Ulthia) has a mass of 22 Earth masses, she is 22.0 ÷ 35.0 = 0.629 times the mass of the average Class I ice giant, and we'd use the following values for T0 and C in calculating potential orbital periods for her moons:
T0Ulthia = 0.760 ⨉ 0.629 = 0.478
CUlthia = 1.800 ⨉ 0.629 = 1.131
... and the potential orbital periods for a set of six major moons would be:
For a Class II gas giant (Arada) of 272.16 Earth masses (0.72 Jupiter masses), the multiplier would be 272.16 ÷ 300.0 = 0.907:
T0Arada = 0.444 ⨉ 0.907 = 0.403
CArada = 2.03 ⨉ 0.907 = 1.842
... and the potential orbital periods for a set of six major moons would be:
Variety is The Spice Of Life
- Determine the kind of giant planet you have (Ice- or Gas-Giant);
- Determine the planet's mass;
- Find the range on the table into which the planet's mass falls;
- Divide the mass of the planet by the M value from the table;
- Multiply T0 by the value calculated in Step 4.
- Use the value calculated in Step 5 in your iteration for as many major moons as you wish to generate for your planet.
T0, C, and M are the values from the table for the "Dermott Class" of the planet, and n is the value of the moon ordinal under consideration.
Remember that just because there is an orbit for a moon, that doesn't require that a moon be found on that orbit; you are always free to leave orbits vacant.
Moons: Orbital Period
Mass of the Moon in Earth masses: 0.0123
Radius of Moon's orbit in Earth radii: 60.336
Mass of Mars: 6.4171E+26 grams / Mass of Earth in grams: 5.972E+27 = 0.10745M⨁
Mass of Deimos in Earth masses:
Mass of Deimos: 1.4762E+18 grams / Mass of Earth in grams: 5.972E+27 = 2.472E-10 (for all practical purposes, this is zero.
Radius of Deimos' orbit around Mars in Earth radii:
Deimos' orbit in meters: 2.3463E+7 / Radius of Earth in meters: 6.371E+6 = 3.6828R⨁
Satellite Orbital Speed
Putting It All To Work
Rs = orbital radius of the planet (in Earth-radii);
Ps = the calculated orbital period of the moon (in Earth-days);
Mp = the mass of the planet (expressed in Earth-masses);
Ms = the mass of the moon (expressed in Earth-masses)*.
* Note: The mass of the moon is likely to be negligible. Saturn's mass in Earth masses is 95.159 and Titan's mass in Earth masses is 0.0225, which gives a moon-to-planet mass ratio of 0.000236; in other words, Saturn is 4229.3 times more massive than Titan. The sum of the two masses (95.159 + 0.0225 = 95.182) is only 1.000236 times more than the mass of Saturn, alone. Thus, ignoring the mass of the moon in this equation is not likely to make a significant difference in the final calculation.
Ms = the mass of the moon (expressed in Earth-masses);
Rs = the orbital radius of the planet (in Earth-radii);
Ps = the calculated orbital period of the moon (in Earth-days);
Mp = the mass of the planet (expressed in Earth-masses).
Let's specify that Ulthia has a moon (Aras; of negligible mass) with an orbital period of 0.692 Earth days. What is its orbital distance from Ulthia (in Earth-radii)?
We specified earlier that Ulthia's mass is 22 Earth masses, so our equation is (again, ignoring the mass of the moon Aras):
If Ulthia is 1.137 times the radius of Uranus, it is 4.556 Earth-radii in size, so Aras' orbit is 14.498 ÷ 4.556 = 3.182 times Ulthia's radius.
Affects of Moons On The Potential for Life on the Primary
But, Mike Wall, in an article at Space.com , says that this may not necessarily be true. Quoting Jack Lissauer of NASA’s Ames Research Center, Wall reports that while the Moon does stabilize Earth’s wobble, the extent of the wobble without the Moon might not have been as drastic as first assumed. In the simulations that Lissauer’s team ran on a Moon-less Earth, the axial tilt over periods of 100 million years never exceeded 40 degrees, nor fell below 10 degrees . While these variations are certainly larger than the range of 22-to-24 degrees that the Earth currently experiences, they are not enough to preclude environments suitable to the advent and success of life.
National Public Radio’s Ira Flatow, in the transcript of a broadcast called “Is A Moon Necessary For A Planet To Support Life” , interviewed Jason Barnes of the University of Idaho, who corroborates Lissauer’s contention that “…you really don't need a moon to stabilize the Earth—or to stabilize a typical exosolar planet, I should say—and that probably 80 percent or so of exosolar planets will have … climate stability….” 
Moons With Moons
M = 0.0123 (mass of the Moon in terms of Earth's mass)
a = 221.301 (distance to the Moon in terms of the Moon's radius)
... and solving:
Note that if you do the equation without any value for a, the result always returns the percentage of the distance between the two objects at which the Hill radius falls; in Earth's case this is 1.0002% of the distance from the Earth to the Sun.
When Is A Moon Not A Moon?
(frantically waving hands in the air)
Hold on! Hold on, let me explain!
Of the major moons in the Solar System, the most massive, Ganymede, is about 45% of the mass of Mercury, and Ganymede and Titan each are actually larger than Mercury in diameter.
(Earth's Moon, though only about a quarter of the mass of Mercury, nevertheless has a larger diameter than seven of the other eleven major moons of the Solar System. In fact, the Moon is 5.64 times more massive than Pluto and 1.46 times Pluto's diameter).
It is safe to say that, were any of these bodies to be in orbit around the Sun independently, they would be called planets, even by the (contentious) 2006 IAU definition.
Looking at the following table, we can readily see the problem:
Glancing briefly at the issue from the other direction, even Jupiter—a gas giant of somewhat modest mass—could potentially have a "moon" with the mass of the Earth.
I think that this can't be resolved by looking at masses alone. We'll have to fall back (albeit reluctantly) on the 2006 IAU definition, by specifying that any object smaller in mass than a giant planet, and which is in orbit around a giant planet or a gas-giant planet, is a "moon" of that giant or gas giant planet—even if said object would be a planet in its own right it were orbiting the central star(s) independently.
Thus, an Earth-mass body orbiting a gas giant is, by definition, a moon of the gas giant planet (though, personally, I would be inclined to say that any object of the mass of Mercury (0.055 Earth-masses) or greater, which is in orbit around a gas giant planet—and certainly around a giant planet—constitutes a double-planet system ... but I'd probably fall back on Asimov's "Tug-of-War" calculation, just to be safe).