Derivation of Solar Position Formulae

TL;DR

This paper derives simplified formulae for calculating solar positions from orbiting planets, including sunrise, declination, and azimuth, and validates the models with Earth data achieving high accuracy.

Contribution

It introduces a unified derivation of solar position equations using effective axial tilt and Rodrigues Rotation, simplifying calculations for planetary observations.

Findings

Average sunrise direction accuracy of 1.25 degrees

Estimated Earth's axial tilt of 23.52 degrees

Validated formulas with actual Earth sunrise data

Abstract

Derivation of the following formulae for solar position as seen from orbiting planet based on a simplified model: sunrise direction formula, solar declination formula, sunrise equation, daylight duration formula, solar altitude formula, solar azimuth formula. Use of notion of effective axial tilt, and Rodrigues Rotation Formula, and reflections of the orbital quadrants to reduce the general case to the simpler case of the day of the winter solstice. Derivation of equations for solar time to clock time conversion, and sunrise, sunset, and solar noon times. Implementation of an analemma calculator. Comparison of the sunrise direction formula with 304 point dataset of actual sunrise data from Earth, obtaining average accuracy of 1.25 degrees, and estimate of Earth's axial tilt of 23.52 degrees, within 0.1 degrees of the currently accepted value 23.44 degrees.