Re: Find Solar Calendar Seasons spreadsheet posted

Although the mean year of any particular leap cycle may not exactly match the mean year of an equinox or solstice, each reasonably accurate leap cycle has a mean year that is a stable match to one or two points in the annual solar cycle, which we can call "calendar season(s)" for that leap cycle. In the present era, such calendar seasons are stable for calendar mean years that are as short as about 365 days 5 hours 47 minutes 52 seconds or about 365+43/178 days to as long as about 365 days 5 hours 49 minutes 31 seconds or 365+25/103 days, a range of only about 1 minute 39 seconds, and that stability endures for about 10 millennia.

In any given era the longest cycle mean year having a stable calendar season will match the mean year (in terms of mean solar days) at the ecliptic longitude of the Earth orbital perihelion, and the shortest cycle mean year having a stable calendar season will match the mean year at the ecliptic longitude of the Earth orbital aphelion. The advance of perihelion (and aphelion, always 180° away) together with the tidal slowing of the Earth rotation rate cause calendar seasons to evolve and migrate as the millennia pass.

A leap cycle will not have any calendar seasons in an era in which its mean year is a few seconds shorter than the mean year at the ecliptic longitude of aphelion. With tidal slowing of the Earth rotation rate, however, eventually the mean year at aphelion will equal the cycle mean year, so a calendar season will appear at the ecliptic longitude of aphelion. With progressive tidal rotation slowing that calendar season will split into a more stable season that will migrate ahead of aphelion (to earlier solar longitudes) as well as a less stable season that will migrate after aphelion (to later solar longitudes). As tidal rotation slowing continues, eventually the average length of the solar cycle will approximately equal the cycle mean year, and then both calendar seasons will be optimally stable and perihelion and aphelion will be situated approximately midway between them, with perihelion having the less stable calendar season behind it (prior solar longitude) and the more stable calendar season ahead of it (later solar longitude). Further tidal rotation slowing will cause the calendar seasons to converge towards perihelion, disappearing in later years when the mean year near the ecliptic longitude region of perihelion becomes longer than the cycle mean year. The region near perihelion having the longest mean year spans about 45° of ecliptic longitude, ranging from about 15° before to about 30° after the ecliptic longitude of perihelion, so both calendar seasons disappear before ever reaching perihelion.

With that description, the reader might think that aphelion should eventually "catch up" to perihelion, but that can never happen because they both advance in unison around Sun, always 180° apart. As aphelion passes through each point of the solar cycle, however, that solar longitude has the shortest mean year, and the opposite solar longitude 180° away at perihelion has the longest mean year.

(I also have just posted another new version of the spreadsheet, extending the continued fraction convergent list from 10-->12, making it easier to get exact fractions.  For experimenting with characterizing features such as those outlined above, it is helpful to use a small step size such as 100 years.)