In this step, we will determine elements of the local calendar on the world being developed: the length of the local day, the length of any “month” determined by a major satellite, and so on.

Length of Local Day for a Planet

To determine the length of a planet’s day – the planet’s rotation period with respect to its primary star rather than with respect to the distant stars – compute the following:

Here, P is the planet’s orbital period as determined in Step Fifteen, while R is the planet’s rotational period as determined in Step Sixteen, both in hours. T is the apparent length of the planet’s day, also in hours.

Note that this equation is undefined in cases when the orbital period and rotational period are equal (that is, the planet is in a spin-orbital resonance of 1:1 and is “tide locked”). In this case, the length of the local day is effectively infinite – the sun never moves in the sky!

At the other extreme, if the orbital period is much longer than the rotational period, then the day length and the rotational period will be very close together.

To determine the length of the local year in local days, simply divide the planet’s orbital period by the length of the local day as computed above.

Length of Apparent Orbital Period for a Satellite

To determine the length of a satellite’s orbital period, from any position on the planet’s surface, use the same equation:

Here, P is the satellite’s orbital period as determined in Step Fifteen, while R is the planet’s rotational period as determined in Step Sixteen, both in hours. T is the apparent length of the satellite’s orbital period, also in hours.

Again, this equation is undefined in cases when the satellite’s orbital period and the planet’s rotational period are equal (that is, the planet is tide-locked to its satellite, or the satellite happens to orbit at a geosynchronous distance). In this case, the length of the satellite’s apparent orbital period is effectively infinite – the satellite never moves in the sky.

At the other extreme, if the satellite’s orbital period is much longer than the planet’s rotational period, then the apparent orbital period and the rotational period will be very close together. Earth’s moon is a familiar example – its apparent motion in the sky is dominated by Earth’s rotation.

It’s possible for a satellite’s orbital period to be shorter than the planet’s rotational period. For example, a moonlet that orbits very close is likely to fall into this case. The apparent orbital period will therefore be negative, indicating that the satellite appears to move backward over time. The satellite will rise in the west and set in the east.

Length of Synodic Month for a Satellite

To determine the length of a satellite’s synodic month, use the same equation once more:

Here, P is the planet’s orbital period as determined in Step Fifteen, while R is the satellite’s orbital period as determined in Step Fifteen, both in hours. T is the length of the satellite’s synodic month.

It’s very unlikely for a satellite to have the same orbital period around its planet as the planet does around its primary star, so the undefined or negative cases almost certainly will not occur. T will indicate the period between (e.g.) one “full moon” and the next, as observed from the planet’s surface.

Examples

Arcadia IV has no satellite, so the only item of interest will be the length of its local day. Alice computes:

The local day on Arcadia IV is only slightly longer than its rotation period. Alice can also determine the length of the local year in local days, by dividing the orbital period by this day length. Arcadia IV has a local year of about 184.35 local days.

Arcadia V has a satellite, so that satellite’s apparent orbital period and synodic month might be of interest. For the apparent orbital period, Alice computes:

Arcadia V’s moonlet appears to move retrograde or “backwards” in the sky, rising in the west and setting in the east, with an apparent period of about 31.7 hours.

Meanwhile, for the synodic month, Alice computes:

The satellite’s synodic month – the period between “full moon” phases – is much shorter than its apparent orbital period. From the surface of Arcadia V, the moonlet will appear to move slowly through the sky, its phase visibly changing as it moves, passing through almost a complete cycle of phases before setting once more in the east. Very strange, for human observers accustomed to the more sedate behavior of Earth’s moon!