Mean tropical year length at arbitrary ecliptic longitude

Original Abstract

We compute the mean interval between successive returns of the apparent geocentric solar longitude $λ$ to a fixed value $L \in \{0^\circ, 45^\circ, 90^\circ, \ldots, 315^\circ\}$, averaged over a multi-millennium window; this gives eight ``mean years'' against which calendar leap rules can be tuned: four cardinal-point years (equinoxes and solstices); four cross-quarter years. The construction is built on Meeus's low-precision solar theory (Astronomical Algorithms, 2nd ed., 1998), itself a low-order truncation of Newcomb's Tables of the Sun, re-expanded around J2000.0. Where Meeus presents polynomial coefficients without justification, we draw on Smart's Textbook on Spherical Astronomy (6th ed., revised by Green, 1977) for the underlying derivations. Numerical accuracy is validated against the cardinal-point intervals tabulated in Meeus, More Mathematical Morsels, 2002. We close with a derivation of the secular drift equation, showing that, regardless of how well a leap rule is tuned, the slow shrinkage of the tropical year produces a quadratic cumulative error that reaches one day in $\sim$57,000 years for any fixed intercalation rule.