The Sun’s position in the sky Alejandro Jenkins * High Energy Physics, 505 Keen Building, Florida State University, Tallahassee, FL 32306-4350, USA (Dated: Aug. 2012, last revised Mar. 2013; published in Eur. J. Phys. 34 , 633 (2013)) We express the position of the Sun in the sky as a function of time and the observer’s geographic coordinates. Our method is based on applying rotation matrices to vectors describing points on the celestial sphere. We also derive direct expressions, as functions of date of the year and geographic latitude, for the duration of daylight, the maximum and minimum altitudes of the Sun, and the cardinal directions to sunrise and sunset. We discuss how to account for the eccentricity of the Earth’s orbit, the precessions of the equinoxes and the perihelion, the size of the solar disk, and atmospheric refraction. We illustrate these results by computing the dates of “Manhattanhenge” (when sunset aligns with the east-west streets on the main traffic grid for Manhattan, in New York City), by plotting the altitude of the Sun over representative cities as a function of time, and by showing plots (“analemmas”) for the position of the Sun in the sky at a given hour of the day. Keywords: celestial sphere, rotation matrices, calendar, equation of the center, equation of time, precession, Manhattanhenge PACS: 95.10.Km, 02.40.Dr Contents I. Introduction 2 II. Spherical coordinates for the Sun 3 A. Ecliptic frame 3 B. Equatorial frame 4 C. Terrestrial frame 5 III. Astronomical adjustments 6 A. Equation of the center 6 B. Precession of equinoxes and perihelion 7 IV. Duration of daylight 8 A. Maximum and minimum solar altitudes 10 B. Correcting for size of solar disk and atmospheric refraction 10 V. Solar alignments 12 A. Direction to sunrise and sunset 12 B. Manhattanhenge 13 VI. Geographic longitude 13 A. Reference time 13 B. Solar azimuth 14 VII. Altitudes 15 A. Buenos Aires, Argentina 15 B. Alert, Nunavut, Canada 15 C. Singapore 16 D. San Jos´ e, Costa Rica 16 VIII. Analemmas and equation of time 17 * Electronic address: [email protected] arXiv:1208.1043v3 [physics.pop-ph] 1 Apr 2013
2 IX. Discussion 17 Acknowledgments 18 References 18 I. INTRODUCTION This article will show how to compute the position of the Sun in the sky, for any given location on the surface of the Earth, at any given time. Our method is based on describing the position of the Sun on the celestial sphere (a concept that should be very familiar to amateur astronomers) and performing several coordinate rotations on that sphere. The idea is to begin with the ecliptic reference frame, in which the position of the Sun during the year is most easily and directly expressed, and to end with a terrestrial reference frame, defined with respect to an observer standing at a given point on the surface of the Earth, at a given time. The mathematical training needed to understand this derivation is that which a student should have after a first course in linear algebra, since rotations will be described by matrices acting on three-dimensional vectors. Familiarity with the transformation between rectangular (“Cartesian”)
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