Mean tropical year length at arbitrary ecliptic longitude

Astrophysics > Earth and Planetary Astrophysics [Submitted on 4 May 2026 (v1), last revised 31 May 2026 (this version, v2)] Title:Mean tropical year length at arbitrary ecliptic longitude View PDFAbstract:We compute the mean interval between successive returns of the apparent geocentric solar longitude $\lambda$ 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$5{,}700 years for any fixed intercalation rule. Submission history From: Daniel Quigley [view email][v1] Mon, 4 May 2026 05:30:53 UTC (318 KB) [v2] Sun, 31 May 2026 03:48:34 UTC (318 KB) Current browse context: astro-ph.EP Change to browse by: References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) IArxiv Recommender (What is IArxiv?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.