Solar Longitude Clock

While solar longitude can be treated as a useful calendar, it is actually defined as the position of the Sun from the perspective of a given planet, expressed as the angle of the Sun from the vernal equinox, measured in the planet's orbital plane, which on Earth is simply the ecliptic longitude of the Sun. For a planet with a circular orbit and no moons, the solar longitude would always increase at a constant rate. In an eccentric orbit, the solar longitude increases slower when the planet is farther from the sun and faster when it is close to the Sun. This is relatively simple to predict since the planet still follows a simple elliptical path. Moons orbiting the planet make the situation far more complicated. Gravity from an orbiting moon pulls on the planet, making the planet speed up and slow down depending on the position of the moon in its orbit. These perturbations in the planet's orbital velocity cause small changes in the derivative of solar longitude, and if the moons are large enough compared to the planet, they can even make the derivative of solar longitude temporarily flip signs.

Why is Pluto's L_s changing so fast compared to Uranus and Neptune? Charon, Pluto's largest moon, is so large compared with Pluto itself that the two bodies form a binary system centered on a point in space between them. Whenever Charon is ahead of Pluto in its orbit, the moon pulls Pluto along and makes it speed up, and when Charon is trailing behind, it slows Pluto so much that the planet fully changes direction and orbits backwards about 49.6% of the time. So while it may appear from the rapidly changing solar longitude that Pluto is orbiting too fast, as much as 0.13° per Earth day, its frequent backward motion negates the rapid changes so that on average Pluto only orbits about 1.45° every Earth year, less than 0.004° per Earth day.

A mean solar longitude

The potential for solar longitude to temporarily change direction makes it significantly less useful as a calendar, since a calendar should be able to uniquely specify a given point in time. Instead, we can take a cue from historical orbital mechanics, which often defines true and mean quantities, such that the mean variable is equivalent to the true variable if the orbit was idealized in some way. For example, the true anomaly is the actual angular position of a body in its orbit, while the mean anomaly is the angular position it would be at if the orbit was circular. This is convenient because the mean anomaly angle increases at a nearly constant rate, so it can be calculated by simply multiplying the mean orbital speed by time. On the other hand, calculating the true anomaly involves accounting for the eccentricity, which can be difficult.

Neither the mean nor the true anomaly make good calendars, since they are both based on periapsis rather than the solstice, but we can image a similar "mean solar longitude" that preserves the seasonal information but not perturbations due to moons. I define "mean solar longitude" as the solar longitude as observed from the barycenter of the planetary system, but still measured from the longitude of the vernal equinox of the planet primary. The values calculated and listed below are the standard solar longitude and not the "mean solar longitude."