Dermott's Law <1> is an equation developed by Stanley Dermott, based on observations of the major moons of Jupiter, Saturn, and Uranus. The equation takes the general form:

Where:

D

… using the constants:

Jupiter: 𝜏

Saturn: 𝜏

Uranus: 𝜏

For Jupiter, with values of

*m*= the maximum value of orbital periods desired <2>;D

*n*= the orbital period (in days)of the*n*th ordinal major moon of the planet;*τ*= a constant for each type of planet;*η*= a constant for the orbital system… using the constants:

Jupiter: 𝜏

*= 0.444*;*η**= 2.03*Saturn: 𝜏

*= 0.462*;*η**= 1.59*Uranus: 𝜏

*= 0.760*;*η**= 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:The numbers produced by the equation do not return exactly the observed orbital periods; this is probably due to complex gravitational interactions between Jupiter, its 67 moons, and the rest of the Solar system bodies. Also —importantly —

This relation can be modified to provide a general formula for 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.

We can interpolate in each mass range and define the average masses for each class as:

*a predicted orbital period does not necessarily correspond to an actual moon*; the first and sixth orbital periods in the table above are not relatable to any of Jupiter's major moons. The predicted orbital period only indicates the possibility of the presence of a moon, it does not dictate that a moon must occupy an orbit with that orbital period.This relation can be modified to provide a general formula for 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.

We can interpolate in each mass range and define the average masses for each class as:

We can then use Dermott's values for Uranus as the standard for both classes of ice giants, Saturn's values for Kronean gas giants, and Jupiter's values for Joveans and Kroneans.

And a complete table becomes:

When developing a giant planet, first decide whether the planet will be a Titanean (Promethean or Atlasean) or an Olympean (Kronean, Jovean, or Empyrean), then calculate its mass relative to the "standard" for the class, and use the result as a multiplier for

*τʼ*and*ηʼ*.**Example 1: Ulthia**

Let’s specify a planet called Ulthia which has a mass of 22.0 terran. This puts Ulthia in the Argesean mass range. We calculate a modifier value for Ulthia by dividing its mass by the standard mass (μ) for Argesean class (40), and using the resulting figure to calculate a

*τ*value and a*η*value for Ulthia:… and use these values in the Dermott equation to produce a potential orbits table:

… all expressed in Earth days, of course.

**Example 1: Ulthia**

What about a planet called Arada with a mass of 272.16 terran?

This mass is in the Jovean range, so the standard values are:

𝜏ʼ

𝛾ʼ

This mass is in the Jovean range, so the standard values are:

𝜏ʼ

*= 0.444*𝛾ʼ

*= 2.03**μ**= 320.0*… again, all expressed in Earth days.

**Variety Is The Spice Of Life**

One side-effect of using Dermott's Law—even as I've modified it—is that the interval between any two moon orbits will always be the value of

*η*used in the equation. A way to juggle the intervals up a bit is to add the value of the common logarithm of the orbit ordinal to the result of the equation, so that the equation becomes:… an algorithm might be:

- Determine the planet's mass;
- Find the range into which the planet’s mass falls.
- Determine the 𝜅-value by dividing the planet’s mass by the standard mass for the class determined in Step 2;
- Calculate the 𝜏 and
*η*values by multiplying the standard values by the value obtained in Step 3. - Use the value calculated in Step 4 in the extended Dermott’s Law equation.

Where:

D

𝜏ʼ= the Dermott constant for the giant planet class;

So, revisiting Arada, the first potential orbital period calculates as:

*n*= the ordinal of the given orbit;D

*n*= the orbital period of the*n*th ordinal major moon of the planet;𝜏ʼ= the Dermott constant for the giant planet class;

*ηʼ*= the Dermott orbital system constant for the giant planet class;*M*= the actual mass of the planet in question, in terran units;*μ*= the standard mass for the giant planet class

So, revisiting Arada, the first potential orbital period calculates as:

Remember that just because there is a potential orbital period for a moon, that doesn't require that a moon be found on the corresponding orbit; you are always free to leave orbits vacant.

## Notes

- "Dermott's Law," Wikipedia, July 24, 2018, , accessed July 28, 2018, https://en.wikipedia.org/wiki/Dermott's_law.
- Note that the values of
*n*need not be integers.