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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