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solar year range

by Irv Bromberg :: Rate this Message:

Dear Calendar People:

I have added "Longest Solar Year" (grey) and "Shortest Solar Year" (black) curves to my equinoctial and solstitial year length plot, attached as a PDF (36 KB).

These curves are almost tangent to the helical curves that they contain, a surprisingly good fit despite the fact that the helical curves are based on numerical integration (SOLEX 9.1) whereas the solar year range curves are based on astronomical algorithms (Meeus' mean aphelion polynomial to find the target solar longitude of aphelion and perihelion, and the Reingold/Dershowitz/Meeus truncated VSOP87 for the solar longitude). I am especially surprised to see no evidence that the polynomial goes "wild" within this relatively wide range of years.

The longest solar year is calculated as follows:

For each 200-year interval, find the ecliptic longitude of mean perihelion.

Find the moment when Sun was/will be at same ecliptic longitude (not the longitude of perihelion) 100 years before and after, and calculate the average year length by subtracting them and dividing by 200 years.

Express the year length as the time in excess of 365 days 5 hours by subtracting 365 days 5 hours.

This generates an array of points follow along the top edge of the helical curves with some scatter, so I fit a 3rd order polynomial (least squares regression) to that, which is the grey plotted trendline, then I hid the scattered points to get them out of the way. I found that higher order polynomials didn't improve the fit.

The shortest solar year is calculated the same way, but substitute aphelion for perihelion, and fit a 3rd order polynomial to the array of points following along the bottom edge of the helical curves, and assigned it a black color.

Stable calendar seasons can only exist between the longest and shortest solar year curves as appropriate to the target era.

Note that these year lengths are very different from the mean anomalistic year, the time required for Earth to revolve from perihelion to perihelion or from aphelion to aphelion, which is presently about 365 days 6 hours 13 minutes and 52 seconds, slightly longer than the mean sidereal year because of the advance of perihelion and aphelion.

It would be nice at some point (when I figure out how) to redo this chart using SOLEX 10.1, in "planets only" numerical integration mode to generate the data for plotting all of the curves according to the Earth-Moon barycenter, taking advantage of several of the new SOLEX features.

(Note that SOLEX doesn't numerically integrate the Earth axial tilt = obliquity of the ecliptic, but simply uses Williams' polynomial for ±10000 years from the present era.)

-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/leap/>

<http://www.sym454.org/seasons/>

Mean_Solar_Years_15K_Solar_Year_Range.pdf (47K) Download Attachment

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