A quick note before I drop the next section of the draft: I caught myself making several errors in the mathematics while developing this step. I think I’ve weeded all of those out, but if anyone is experimenting with this material as it appears, let me know if you come across any odd results.

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Step Sixteen: Determine Rotation Period

The next three steps in the sequence all have to do with planetary rotation. Every object in the cosmos appears to rotate around at least one axis, and in fact some objects appear to “tumble” by rotating around more than one.

Planets and their major satellites usually have simple rotation, spinning in the same direction as their orbital motion, around a single axis that is more or less perpendicular to the plane of their orbital motion. There are, of course, a variety of exceptions to this general rule.

In this step, we will determine the rotation period of a given world. In this case, we will be dealing with what’s called the sidereal period of rotation – the time it takes for a world to rotate once with respect to the distant stars.

Worlds appear to form with wildly varying rotation periods, the legacy of the chaotic processes of planetary formation. However, many worlds will have been affected by tidal deceleration applied by the gravitational influence of nearby objects. Tidal deceleration may cause a world to be captured into a special status called a spin-orbital resonance, in which the world’s orbital period and its rotational period form a small-integer ratio.

Procedure

Begin by noting the situation the world being developed is in: is it a major satellite of a planet, a planet with its own major satellite, or a planet affected primarily by its primary star?

First Case: Major Satellites of Planets

Major satellites of planets, as placed in Step Fourteen, will almost invariably be in a spin-orbit resonance state. Most models of the formation of such satellites suggest that they are captured into such a state almost immediately after their formation.

Since a major satellite’s orbit normally has very small eccentricity, the spin-orbit resonance will be 1:1. The satellite’s rotation period will be exactly equal to its orbital period.

Second Case: Planets with Major Satellites

A Leftover Oligarch, Terrestrial Planet, or Failed Core which has a major satellite may be captured into a spin resonance with the satellite’s orbit. This is actually somewhat unlikely; for example, Earth is not likely to become tide-locked to its own moon within the lifetime of the sun. However, a satellite’s tidal effects on the primary planet will tend to slow its rotation rate.

To estimate the probability that a planet has become tide-locked to its satellite, and to estimate its rotation rate if this is not the case, begin by evaluating the following:

Here, A is the age of the star system in billions of years. M_S and M_P are the mass of the satellite and the planet, respectively, in Earth-masses. R is the radius of the satellite, and D is the radius of the satellite’s orbit, both in kilometers.

If T is equal to or greater than 2, the planet is almost certainly tide-locked to its satellite. Its rotation period will be exactly equal to the orbital period of the satellite.

Otherwise, to generate a rotation period for the planet at random, multiply T by 12, round the result to the nearest integer, add the result to a roll of 3d6, and refer to the Rotation Period Table.

Rotation Period Table
Modified Roll (3d6)	Rotation Rate
3	4 hours
4	5 hours
5	6 hours
6	8 hours
7	10 hours
8	12 hours
9	16 hours
10	20 hours
11	24 hours
12	32 hours
13	40 hours
14	48 hours
15	64 hours
16	80 hours
17	96 hours
18	128 hours
19	160 hours
20	192 hours
21	256 hours
22	320 hours
23	384 hours
24 or higher	Resonance Established

Feel free to adjust a result from this procedure to any value between the next lower and next higher rows on the table.

The planet will be tide-locked to its satellite on a result of 24 or higher, or in any case where the randomly generated rotation rate is actually longer than the satellite’s orbital period. In these cases, again, its rotation period will be exactly equal to the orbital period of the satellite.

Third Case: Planets Without Major Satellites

A Leftover Oligarch, Terrestrial Planet, or Failed Core which has no major satellite may be captured into a spin-orbit resonance with respect to its primary star. Even if this does not occur, solar tides will tend to slow the planet’s rotation rate.

To estimate the probability that such a planet has been captured into a spin-orbit resonance, and to estimate its rotation rate if this is not the case, begin by evaluating the following:

Here, A is the age of the star system in billions of years, M_S is the mass of the primary star in solar masses, M_P is the mass of the planet in Earth-masses, R is the radius of the planet in kilometers, and D is the planet’s orbital radius in AU.

Again, if T is equal to or greater than 2, the planet has almost certainly been captured in a spin-orbit resonance. Otherwise, to generate a rotation period for the planet at random, multiply T by 12, round the result to the nearest integer, add the result to a roll of 3d6, and refer to the Rotation Period Table. The planet will be in a spin-orbit resonance on a result of 24 or higher, or in any case where the randomly generated rotation rate is actually longer than the planet’s orbital period.

Planets captured into a spin-orbit resonance are not necessarily tide-locked to their primary star (or, in other words, the resonance is not necessarily 1:1). Tidal locking tends to match a planet’s rotation rate to its rate of revolution during its periastron passage. If the planet’s orbit is eccentric, this match may be approximated more closely by a different resonance. To determine the most likely resonance, refer to the Planetary Spin-Orbit Resonance Table:

Planetary Spin-Orbit Resonance Table
Planetary Orbit Eccentricity	Most Probable Resonance	Rotation Period
Less than 0.12	1:1	Equal to orbital period
Between 0.12 and 0.25	3:2	Exactly 2/3 of orbital period
Between 0.25 and 0.35	2:1	Exactly 1/2 of orbital period
Between 0.35 and 0.45	5:2	Exactly 2/5 of orbital period
Greater than 0.45	3:1	Exactly 1/3 of orbital period

On this table, the “most probable resonance” is the status that the planet is most likely to be captured into over a long period of time. It’s possible for a planet to be captured into a higher resonance (that is, a resonance from a lower line on the table) but this situation is unlikely to be stable over billions of years.

Examples

Both Arcadia IV and Arcadia V are planets without major satellites, so they both fall under the third case in this section. The most significant force modifying their rotation period will be tidal deceleration caused by the primary star.

The age of the Arcadia star system is about 5.6 billion years. Arcadia IV has mass of 1.08 Earth-masses and a radius of 6450 kilometers. Alice computes T for the planet and ends up with a value of about 0.083. Arcadia IV is probably not in a spin-orbit resonance, but tidal deceleration has had a noticeable effect on the planet’s rotation. Alice rolls 3d6+1 for a result of 11 and selects a value slightly lower than the one from that line of the Rotation Period Table. She decides that Arcadia IV rotates in about 22.5 hours.

Meanwhile, Arcadia V has mass of 0.65 Earth-masses and a radius of 5670 kilometers. Alice computes T again and finds a value of about 0.007. (Notice that the amount of tidal deceleration is very strongly dependent on the distance from the primary.) Alice rolls an unmodified 3d6 for a value of 12, this time selecting a value slightly higher than the one from the table. She decides that Arcadia V rotates in about 34.0 hours.

Citations

Gladman, Brett et al. (1996). “Synchronous Locking of Tidally Evolving Satellites.” Icarus, volume 122, pp. 166–192.

Makarov, Valeri V. (2011). “Conditions of Passage and Entrapment of Terrestrial Planets in Spin-orbit Resonances.” The Astrophysical Journal, volume 752 (1), article no. 73.

Peale, S. J. (1977). “Rotation Histories of the Natural Satellites.” Published in Planetary Satellites (J. A. Burns, ed.), pp. 87–112, University of Arizona Press.