Sharp dynamic points in Earth-Sun physics

TL;DR

This paper presents a unified dynamical framework linking the Sun’s vertical path ($\delta$), the Equation of Time ($\delta^*$), and Earth's rotation speed ($\mathrm{ER}_\omega$) through the subsolar-point, or Natural Beam Irradiance (NBI). By constructing analytic models of $\delta$ and $\delta^*$ and their derivatives up to the fourth order, and by deriving $\mathrm{NBI}$ and $\mathrm{ER}_\omega$ along with their higher-order derivatives, the authors show that the lemniscate structure of $\mathrm{NBI}_\alpha(\delta)$ mirrors the analemma and aligns with the obliquity component of the EoT. They identify eight sharp dynamic-stress periods tied to midseason boundaries, revealing coordinated or opposing net drives between the SMD and EoT that drive variations in LSD and $\mathrm{ER}_\omega$. A key finding is a consistent northward offset (~3°) in the analemma segmentation, suggesting a systematic geometric shift in how SMD couples to the EoT and rotation. Overall, the work argues that the Sun–Earth gravity axis and the subsolar-point dynamics provide a causal, physically grounded explanation for Earth’s rotation within the Sun–Earth system, with potential implications for understanding long-term rotational behavior.

Abstract

The subsolar point, the closest location on Earth's surface to the Sun, marks the Sun-Earth line of gravity that governs Earth's coupled orbital-rotational cycle. We examined the dynamic interactions among the Sun meridian declination (SMD), the Equation of Time (EoT), Earth's rotational speed (ER$_ω$) -- equatorial and with respect to the Sun -- and the path of the subsolar point (NBI) across longitude, including time derivatives up to the fourth order (snap). A central finding was that the function NBI$_α$(SMD) traces a lemniscate whose temporal structure mirrors the analemma, EoT(SMD), and whose symmetry converges to the obliquity component of the EoT. The EoT velocity ($ω^*$) peaks at solstices, troughs near the equinoxes, and crosses zero every mid-season. ER$_ω$ decreases monotonically along trans-equinoctial phases where the net drives of EoT and SMD coincide, and increases along trans-solstitial phases, where their net drives oppose. Eight sharp kinematic periods were identified for the cycle SMD-EoT-ER$_ω$: two equinoctial, two solstitial, and one within each season. The non-solstitial sharp terms, defined by ZCPs and troughs of $ω^*$, display a consistent 3$^\circ$ northward offset from the function NBI$_α$(SMD). These results reveal a direct dynamical link between SMD, EoT, and Earth's rotational speed, providing a novel framework for understanding Earth's rotation.

Figures (2)

• Figure 1: Association between the parameters of the Sun's vertical path (y-axis) and those of the Sun’s horizontal path (x-axis), including position (a), velocity (b), acceleration (c), jerk (d), and snap (e). EV:spring, LV: Late spring (V means vernal), ES: Early summer, LS: Late summer, EA: Early autumn, LA: Late autumn, EW: Early winter, and LW: Late winter. Every section denoted late- is colored grey to avoid overmarking.
• Figure 2: Dynamics of the subsolar point (here denoted as NBI) vs dynamics of Earth's rotation, involving speed (a),acceleration (b), jerk (c), and snap (d). Additionally, seasonal variation of $\mathrm{NBI}_\alpha$ (e), and $\mathrm{ER}_\alpha$ (f) are plotted against solar meridian declination.