Celestial mechanics in a nutshell

(CMiaNS)

Marc van der SluysRadboud University,Nijmegen, the NetherlandsSeptember 30, 2021

Copyright © 2013–2021 by Marc van der Sluys

All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any formor by any means, including photocopying, recording, or other electronic or mechanical methods, withoutthe prior written permission of the author, except in the case of brief quotations embodied in criticalreviews and certain other noncommercial uses permitted by copyright law. For permission requests, contactthe author at the address below.

http://cmians.sf.nethttp://astro.ru.nl/∼sluys/

Manuscript rev.74, git hash 1a9e41c (2021-09-30 16:23), compiled on Thu 30 Sep 2021, 16:49 CEST.

This document was typeset in LATEX by the author.

Cover image:Ptolemaic diagram of a geocentric system, from the star atlas Harmonia Macrocosmica by the cartog-rapher Andreas Cellarius, 1660. Photos.com/Thinkstock.

Contents

1 Introduction 4

2 Variables 52.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Astronomical time systems 63.1 Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1.1 Main timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.2 Offsets between timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Julian day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.1 Computing the Julian day from the UNIX time . . . . . . . . . . . . . . . . . . 7

3.3 Time since J2000.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Sidereal time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 The Sun 94.1 Position of the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Conversion to local coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Insolation on an inclined surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 Air mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.5 Atmospheric extinction of sunlight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Coordinate systems 135.1 Equatorial coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.1.1 Right ascension and declination . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.1.2 Hour angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.2 Horizontal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2.1 Atmospheric refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2.2 Parallax and topocentric coordinates . . . . . . . . . . . . . . . . . . . . . . . . 14

5.3 Angular separation and position angle between two celestial objects . . . . . . . . . . 15

6 Transit, rise and set of objects 166.1 Transit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Rise and set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

A Rotations 18A.1 Definition of rotations and coordinate systems . . . . . . . . . . . . . . . . . . . . . . . 18A.2 2D rotations of points and coordinate systems . . . . . . . . . . . . . . . . . . . . . . . 18A.3 Derivation of the 2D rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19A.4 3D rotations of coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

B Derivation of common coordinate transformations 21B.1 Conversion between ecliptical and equatorial coordinates . . . . . . . . . . . . . . . . . 21B.2 The rotation over a phase angle: right ascension to hour angle . . . . . . . . . . . . . . 22B.3 Conversion between equatorial and horizontal coordinates . . . . . . . . . . . . . . . . 22B.4 Angular separation and position angle between two celestial objects . . . . . . . . . . 23

B.4.1 Projection of sunlight on a surface . . . . . . . . . . . . . . . . . . . . . . . . . 24

3

1 Introduction

This document is work in progress. I intend it to evolve into a rich source of information for thosewho want to compute positions of celestial objects or find events in the night sky. Until that time,the bits of text I have gathered together may be useful for some people already.

The purpose of this document, with the working title Celestial mechanics in a nutshell or CMiaNS,is to provide the interested reader with the equations, explanations and background that are neededto compute celestial events, understand why these equations are used and allow the reader to trackwhere (in the literature) they came from. For example, in the appendix we derive the default rotationsfor 2D and 3D rotations, and from there the coordinate transformations between the most commoncoordinate systems.

This is a living document. The current version was compiled on September 30, 2021. You may finda newer version on http://CMiaNS.sf.net. This document may well contain errors. Please reportthem to the same website.

4

2 Variables

2.1 Functions

floor() Round variable down to nearest integer: floor(0.8) = 0; floor(−0.3) = −1

2.2 Time

JD Julian day

JD2000 Julian day of 2000.0 = 2451545.0

θ Local mean sidereal time

θ′ Local apparent sidereal time

θ0 Greenwich mean sidereal time

θ′0 Greenwich apparent sidereal time

2.3 Position

ϕobs Geographic latitude of observer, > 0 if on northern hemisphere

λobs Geographic longitude of observer, > 0 if east of Greenwich

l Ecliptical longitude

b Ecliptical latitude

αobj Right ascension of an object

δobj Declination of an object

A Azimuth

h Altitude

ε0 Mean obliquity of the ecliptic, without nutation

∆ε Nutation in obliquity

ε True obliquity of the ecliptic, corrected for nutation ε = ε0 + ∆ε

∆ψ Nutation in longitude

5

3 Astronomical time systems

3.1 Timescales

In order to perform ephemeris calculations, the “civil time” on our clocks (e.g. UTC), which is basedon the solar day, often needs to be converted to an “ephemeris time” that approximates “the time inthe universe” (usually TT). Since the Earth’s rotation is not constant, the difference between thesetwo timescales, called ∆T , changes unpredictably and must be derived from observations. Below, Ibriefly describe the main timescales of importance and their offsets.

3.1.1 Main timescales

UTC = Coordinated Universal Time; “civil time” at Greenwich, differs from local time by the timezone (usually an (semi)integer number of hours), updated only by leap seconds to stay approxi-mately synchronised with the Earth’s rotation.

UT1 = Universal Time 1 ; changes continuously with respect to UTC, since the rotation of the Earthis variable. When UT1−UTC reaches ∼ −0.6 s, a leap second is introduced, which changes UTCsuch that UT1−UTC ∼+0.4 s.

TAI = Temps Atomique International ; atomic time based on continuous counting of the SI second;independent of the rotation of the Earth. Approximation of TT, but affected by relativisticeffects. The GPS time, used by the Global Positioning System, is defined as GPS ≡ TAI−19 s.

TT = Terrestrial Time; replaces “Terrestrial Dynamical Time” (TDT) and “Ephemeris Time” (ET);TT ≈ TAI + 32.184s ≈ GPS + 51.184s, independent of the Earth’s rotation. Idealised conceptof “the time in the universe”, and hence used as input for ephemeris calculations. Approximatedby TAI, but without relativistic effects.

3.1.2 Offsets between timescales

∆ T = 32.184 s + (TAI−UTC) − (UT1−UTC) = 32.184 s + TAI − UT1 = TT − UT1. Cumulativedifference between “idealised time in the universe” or “dynamical time”, and “civil time”. ∆ Twas negative between ∼ 1871 and 1902, and positive at all other times (barring giant impacts).The library van der Sluys (2018a) contains inter- and extrapolation routines to approximate∆T . See also the IERS (2015) website and Extrapolation of Delta T (2015) for a compilation ofhistorical values, future extrapolations, and more references.

TAI−UTC = cumulative number of leap seconds = 35 s on 2012-07-01; changes every now and thenby one second.

UT1−UTC = ∼ −0.6 – +0.4 s; measured by e.g. the IERS (2015), and reset when a leap second isadded.

TAI−GPS = 19 s, and fixed.

GPS−UTC = (TAI−UTC) − (TAI−GPS) = 35 s − 19 s = 16 s on 2012-07-01; changes whenever aleap second is introduced.

3.2 Julian day

The Julian day (JD) is the default variable used in astronomy to express the date and time of aninstant. The every-day variables of month, day, hour minute and second are not monotonous, but areregularly reset (from 59 or 23 to 0, or from 28–31 or 12 to 1), which makes calculations more difficult.Instead, the Julian day is a continuous time variable, expressed in decimal days, so that the time ofday can be contained in the fraction. JD=0 has been defined as January 1 of the year -4712 CE, atnoon UT. Hence, a new Julian day starts at noon UT, not at midnight. January 1, 2000 at noon

6

corresponds to JD=2451545.0 and is expressed as JD2000.0 or JD2000.1

The calculation of the position of a celestial object for any instant always starts by converting thedate and time of that instant to JD. Algorithms to compute the Julian day are provided in the libraryby van der Sluys (2018b); an algorithm in pseudocode can be found in Box 3.1.

Box 3.1 Julian day

Input:

y: year (integer), m: month number (integer)

d: decimal day of month in UT - e.g. d + h/24 (float)

// Months 1 and 2 are considered as 13 and 14 of the previous year:

if(m <= 2) y = y-1

m = m+12

b = 0

// Always use the Gregorian calendar for modern dates:

if(Gregorian calendar) a = floor(y/100.)

b = 2 - a + floor(a/4.)

JD = floor(365.25*(y+4716)) + floor(30.6001*(m+1)) + d + b - 1524.5

3.2.1 Computing the Julian day from the UNIX time

Another continuous time variable that is available on many computing systems is the UNIX time,which is the time in seconds since 1970-01-01 at midnight UT. This moment corresponds to JD =2440587.5. Hence, the Julian day can be computed from the UNIX time using

JD =UNIX time

86400+ 2 440 587.5, (3.2.1)

where the denominator is the number of seconds in a Julian day.

3.3 Time since J2000.0

For fits to parameters that vary over long timescales, the time since J2000.0 is often used, expressedin Julian days, years, centuries or millennia:

tJd = JD(E)− 2 451 545; (3.3.1)

tJy =JD(E)− 2 451 545

365.25; (3.3.2)

tJc =JD(E)− 2 451 545

36525; (3.3.3)

1Note that the Julian day number is an integer that counts the days since JD=0. Also, the Julian day is often calledJulian date. However, the latter expression is ambiguous, since it also has the meaning of a date in the Julian calendar(as opposed to a date in the Gregorian calendar). Hence, I will use the term Julian day here.

7

tJm =JD(E)− 2 451 545

365250. (3.3.4)

3.4 Sidereal time

Where the civil time we use every day is (loosely) based on the position of the Sun in the local sky(the Sun transits around noon, barring time zones and daylight savings), the sidereal time representsthe orientation of the night sky (a given star always transits at the same sidereal time). The twodiffer in day length by about four minutes, since by the time the Earth has completed a full revolutionabout its axis with respect to the distant stars, it has also moved in its orbit around the Sun andneeds another four minutes before it assumes the same orientation with respect to our star. Hence,the sidereal time quantifies the mismatch between the night sky and our clock, and is important tocompute positions and events like rise, transit and setting of objects in our local sky. This is usuallydone using the hour angle, as explained in Section 5.1.2.

In order to compute the local sidereal time, one starts with an expression for the sidereal time forthe meridian of Greenwich and compensates for the observer’s geographical longitude. To take intoaccount the effect of precession of the equinoxes, the mean sidereal time can be converted to theapparent sidereal time.

The Greenwich mean sidereal time, in radians, is given by Urban & Seidelmann (2012, Eq. 6.66):

θ0 = 4.89496121042905 + 6.30038809894828323 · tJd + 5.05711849× 10−15 · t2Jd− 4.378× 10−28 · t3Jd − 8.1601415× 10−29 · t4Jd − 2.7445× 10−36 · t5Jd+ 7.0855723730× 10−12 ·∆T,

(3.4.1)

where ∆T = TT −UT1 (see Sect. 3.1.2) is expressed in seconds of time. If ∆T is not available, removethe last term and replace the first with 4.89496121088131 radians. Note that you should use the JDto compute tJd, not the JDE.

The Greenwich apparent sidereal time can be computed using Urban & Seidelmann (2012, Eqs. 6.7,6.67):

θ′0 = θ0 + ∆ψ cos ε (3.4.2)

The local mean sidereal is then given by

θ = θ0 + λobs, (3.4.3)

where λobs > 0 indicates a geographical location east of Greenwich, and the local apparent siderealtime by

θ′ = θ′0 + λobs = θ + ∆ψ cos ε. (3.4.4)

Before printing the variables θ0, θ′0, θ and θ′, you should ensure that their values lie between 0 and

2π.

8

4 The Sun

4.1 Position of the Sun

A simple, fast and accurate method to compute the ecliptical coordinates of the Sun at any instantis given below. This method is used in the routine sunpos la() in van der Sluys (2018a), and has anaccuracy of 11.9 ± 5.7” for the years 1900–2100 when compared to VSOP87 (Bretagnon & Francou,1988) for 105 random moments. All angles are expressed in radians. The time is expressed in Juliancenturies since J2000.0 (tJc, see Eq. 3.3.3).

Semi-major axis of the Earth in AU (Simon et al., 1994, Sect. 5.9.3):

a ≈ 1.0000010178 (4.1.1)

Mean longitude of the Sun (λ+ π in Simon et al., 1994, Sect. 5.9.3):

λ ≈ 4.895063113086 + 628.331965406500 · tJc + 5.2921335× 10−6 · t2Jc + 3.4940522× 10−10 · t3Jc (4.1.2)

Mean anomaly of the Sun (l′ in Chapront-Touze & Chapront, 1988, Table 6):

m ≈ 6.24006012697 + 628.3019551680 · tJc − 2.680535× 10−6 · t2Jc + 7.12676× 10−10 · t3Jc (4.1.3)

Eccentricity of the Earth’s orbit (Simon et al., 1994):

e ≈ 0.0167086342− 4.203654× 10−5 · tJc − 1.26734× 10−7 · t2Jc + 1.444× 10−10 · t3Jc (4.1.4)

Sun’s equation of the centre (Brown, 1896, Chap. III, Eq.7, up to e3):

C ≡ ν −m ≈(

2e− 1

4e3)

sin m+5

4e2 sin(2m) +

13

12e3 sin(3m) (4.1.5)

Alternatively, combining Eqs. 4.1.4 and 4.1.5, we find (using the leading terms in tJc only):

C ≈(3.34161022× 10−2 − 8.40643× 10−5 · tJc − 2.536× 10−7 · t2Jc

)· sin m

+(3.489731× 10−4 − 1.75593× 10−6 · tJc

)· sin(2m) + 5.053× 10−6 · sin(3m) (4.1.6)

The Sun’s true geometric longitude, for the mean equinox:

= λ+ C (4.1.7)

The Sun’s true anomaly:ν = m+ C (4.1.8)

Heliocentric distance of the Earth, or geocentric distance of the Sun:

r =a(1− e2)1 + e cos ν

(4.1.9)

Longitude of the Moon’s mean ascending node (Chapront-Touze & Chapront, 1988, Table 6):

Ωm ≈ 2.18243919722− 33.7570446083 · tJc + 3.623594× 10−5 · t2Jc + 3.734035× 10−8 · t3Jc (4.1.10)

Mean longitude of the Moon (L in Chapront-Touze & Chapront, 1988, Table 6):

Lm ≈ 3.810344430588+8399.709113522267·tJc−2.315615585×10−5 ·t2Jc+3.23904×10−8 ·t3Jc (4.1.11)

9

Nutation in longitude (Seidelmann, 1982, Table I, lines 1,9,31,2,10):

∆ψ ≈ (−8.338601× 10−5 − 7.1365× 10−8 tJc) · sin Ωm − 6.39324× 10−6 · sin(2λ)

− 1.1025× 10−6 · sin(2Lm) + 9.9969× 10−7 · sin(2 Ωm) + 6.9134× 10−7 · sin m

(4.1.12)

Annual aberration (Kovalevsky & Seidelmann, 2004):

θab ≈ −9.9365085× 10−5a(1− e2)

r(4.1.13)

Apparent geocentric longitude, for the true equinox:

l = + θab + ∆ψ (4.1.14)

Apparent geocentric latitude:b ≈ 0 (4.1.15)

Before printing the angles , ν or l, you may want to ensure they fall between 0 and 2π.

4.2 Conversion to local coordinates

In order to convert the ecliptic coordinates of the Sun to a local coordinate system, we can useSection 5.1 to compute the equatorial and thence the parallactic coordinates of the Sun, and Sect. 5.2to convert these to horizontal coordinates.

Note that if we assume that for the Sun b ≈ 0, then consequently sin b ≈ tan b ≈ 0 and cos b ≈ 1.Equations 5.1.1 and 5.1.2 then reduce to:

tanα =sin l cos ε

cos l= tan l cos ε; (4.2.1)

sin δ = sin l sin ε. (4.2.2)

If accurate results are desired, atmospheric refraction (Sect. 5.2.1) and even parallax (Sect. 5.2.2) canbe taken into account.

4.3 Insolation on an inclined surface

To compute the insolation of direct solar radiation (beam radiation) on an inclined surface, for examplesolar PV panels or a solar collector, we need to compute the angle θ between the local position vectorof the Sun (A, h) and the normal vector of the surface (β, γ):

cos θ = sinh cosβ + cosh sinβ cos(A− γ). (4.3.1)

Here, β is the inclination angle of the surface w.r.t. the horizontal (i.e. ‘zenith angle’) and γ theazimuth angle of the surface’s normal vector w.r.t. the south (in the northern hemisphere).

Note that if the sunlight falls in perpendicular to the surface, h = 90- β and A = γ, so that Eq. 4.3.1reduces to cos θ = cosβ cosβ + sinβ sinβ cos(0) = 1, so that indeed θ = 0. A full derivation ofEq. 4.3.1 can be found in Appendix B.4.1.

If the beam normal radiation (or DNI) is provided, it can directly be multiplied with cos θ. If thebeam/direct horizontal radiation is known, it must be multiplied with cos θ

sinh instead, where h is theSun’s altitude. If both are available, using the DNI is preferred, since dividing by sinh can produceunexpected results for low Sun (h ≈ 0).

10

4.4 Air mass

The air mass (AM) quantifies the column density of atmospheric gasses in the path of the light of theSun (or another celestial object). The air mass at the top of the Earth’s atmosphere equals zero, foran object in the zenith AM = 1 (under standard conditions, at sea level), and the air mass increaseswhen an object is closer to the horizon. The air mass can be computed from

AM = max

[1,

1.002432 sin2 h+ 0.148386 sinh+ 0.0096467

sin3 h+ 0.149864 sin2 h+ 0.0102963 sinh+ 0.000303978

], (4.4.1)

where h is the true altitude of the object (uncorrected for atmospheric refraction) and which has amaximum supposed error (at the horizon) of 0.0037 air mass (Young, 1994).

4.5 Atmospheric extinction of sunlight

The bolometric2 atmospheric extinction of sunlight is a function of the air mass. My own fit todetailed spectra of direct sunlight computed using the SMARTS code (Gueymard, 1995, 2001) yieldsthe following polynomial:

E ≈ exp

[11∑i=1

Ci · (AM)i−1], (4.5.1)

with

C =(0.091619283, 0.26098406,−0.036487512, 0.0064036283,−8.1993861 · 10−4, 6.9994043 · 10−5,

−3.8980993 · 10−6, 1.3929599 · 10−7,−3.0685834 · 10−9, 3.7844273 · 10−11,−1.9955057 · 10−13).

(4.5.2)

I find a maximum deviation in the extinction factor of 0.0914 for AM ≈ 36.9 (see Fig. 4.5.1). Note thatthis expression holds for the irradiation of sunlight, not for the visibility of faint objects like stars.3

The solar constant4 can be divided by the extinction factor E the Sun distance in AU r squared toobtain the direct normal insolation (DNI) of sunlight:

DNI ≈ 1361.5 W m−2

E · r2. (4.5.3)

2i.e. over the whole spectrum3For the visibility of stars, night vision provided by the rods in the human retina plays a role.4∼ 1361.5 W/m2 insolation at the top of the Earth’s atmosphere

11

01

02

03

04

0

Exti

nct

ion

0 10 20 30 40

-6-4

-2

log |Δ

|

Air mass

Figure 4.5.1: Top panel (a): polynomial fit (dashed blue line) to the atmospheric extinction ascomputed by the SMARTS code (solid red line), as a function of air mass. Bottom panel (b): logarithmof the absolute residual of the fit.

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5 Coordinate systems

5.1 Equatorial coordinates

5.1.1 Right ascension and declination

The equatorial coordinate system is a spherical system with the same orientation as the Earth’sgeographical coordinate system, hence with the Earth’s equator as the base plane. The two coordinatesare right ascension (RA, α; comparable to geographical longitude) and declination (dec., δ; “latitude”).Right ascension is usually expressed in hours (24h for a full circle), increasing eastward (opposite ofwhat you might expect). Declination is expressed in degrees (-90–+90 for south pole to north pole).For calculations these are often converted to radians. The positions of “fixed objects” (stars and deep-sky objects) are usually listed in RA and dec., and the positions of solar-system objects are convertedfrom ecliptical to horizontal coordinates through the equatorial system.

The coordinate transformation needed for the conversion from ecliptical coordinates (l, b) to equatorialcoordinates (see e.g. Sect. 4.2) is derived in Appendix B.1 and is given by

tanα =sin l cos ε− tan b sin ε

cos l; (5.1.1)

sin δ = sin b cos ε+ sin l cos b sin ε, (5.1.2)

where ε is the obliquity of the ecliptic. Make sure you use the atan2() function to compute the rightascension from Eq. 5.1.1.

The inverse transformation, from equatorial to ecliptical coordinates, is also derived in Appendix B.1:

tan l =sinα cos ε+ tan δ sin ε

cosα; (5.1.3)

sin b = sin δ cos ε− sinα cos δ sin ε. (5.1.4)

5.1.2 Hour angle

When using an equatorial telescope mount (one for which the main axis is aligned with the Earth’srotation axis), the right ascension is often converted to the hour angle (HA, H), which is the localcounterpart of the RA. The hour angle is measured in hours and usually defined as 0h (sometimes12h) when an object is transiting, and negative before and positive after transit. Hence, the hourangle essentially measures the time since the transit of that object. The HA can be obtained fromthe right ascension by

H = θ − α, (5.1.5)

where θ is the local sidereal time, which depends on the geographic longitude of the observer (seeEq. 3.4.3 in Sect. 3.4). Note that the HA increases in the opposite direction from the RA. A non-tracking telescope points at a constant sky position when expressed in hour angle and declination.

5.2 Horizontal coordinates

While equatorial coordinates are based on the plane of the Earth’s equator, an observer will wantto know where an object is with respect to the local horizon: above it or below, by how much (thealtitude h of the object) and the wind direction where the object is above or below the horizon, calledthe azimuth (A).

The conversion from equatorial coordinates hour angle H and declination δ to horizontal coordinatesis derived in Appendix B.3 and results in:

tanA =sinH

cosH sinϕobs − tan δ cosϕobs; (5.2.1)

sinh = sin δ sinϕobs + cosH cos δ cosϕobs, (5.2.2)

13

where ϕobs is the geographic latitude of the observer. Note that the hour angle contains the informationof the observer’s longitude (see Eq. 5.1.5). Make sure you use the atan2() function to compute theazimuth from Eq. 5.2.1.

The reverse transformation, from horizontal to equatorial coordinates, is also found in Appendix B.3:

tanH =sinA

cosA sinϕobs + tanh cosϕobs; (5.2.3)

sin δ = sinh sinϕobs − cosA cosh cosϕobs. (5.2.4)

Sometimes, the zenith angle (z) is used instead of the altitude:

z = 90 − h. (5.2.5)

5.2.1 Atmospheric refraction

For precise calculations, one may want to take into account atmospheric refraction, which can causeobject to appear higher in the sky when compared to the case where the Earth had no atmosphere.For an object in the zenith, the effect is zero, but it amounts to about half a degree close to thehorizon.

The following approximation is taken from Saemundsson (1986), but converted to radians:

∆h =2.967× 10−4

tan

(h+

3.138× 10−3

h+ 8.919× 10−2

) (P

1010

) (283

273 + T

). (5.2.6)

Here, h is the uncorrected altitude of the Sun, and the correction ∆h must be added to that. The lasttwo factors take into account the air pressure (P in millibars) and temperature (T in C). If these areunknown, or high precision is not needed, they can be left out (implicitly assuming P = 1010 mb andT = 10C).

5.2.2 Parallax and topocentric coordinates

If precise results for nearby objects (especially the Moon, and to a lesser extent other solar-systembodies) are needed, the diurnal parallax should be taken into account. The parallax arises from the factthat a nearby body like the Moon appears at a slightly different position with respect to the distantstars, when observed from different locations on Earth. This is similar to the different positions ofa nearby finger against a distant background, when alternatingly seen from the left and right eye.The result of the diurnal parallax is that objects appear lower in the sky than when observed fromthe centre of the Earth. Hence, the parallax is used to convert the geocentric coordinates we haveconsidered so far to topocentric coordinates.

The equatorial (i.e. using the equatorial radius of the Earth) horizontal parallax of the Sun equals

π = arcsin

(Req

AU

)≈ 8′′.794. (5.2.7)

For an object with distance r, the parallax yields

sinπobj =sinπr

, (5.2.8)

where in many cases sufficient accuracy is obtained using the approximation

πobj ≈πr. (5.2.9)

14

The correction to the altitude h of an object is then given by

sin ∆h = −ρ sinπ cosh, (5.2.10)

where ρ is the distance of the observer to the centre of the Earth, expressed in Earth radii. Note thatthe parallax is largest for an object near the horizon (h ∼ 0) and vanishes for an object in the zenith(h ∼ 90).

For low-accuracy results, a spherical Earth can be assumed (ρ = 1), and the sines can be assumedequal to their angles:

∆h = −π cosh. (5.2.11)

The topocentric altitude of the object is then computed from the geocentric altitude:

htopo = hgeo + ∆h. (5.2.12)

5.3 Angular separation and position angle between two celestial objects

Given two (nearby) objects in the sky, we may be interested in their relative positions, which we canexpress as the angular separation or angular distance (i.e., the distance in the sky expressed inradians or degrees) and the position angle, which indicates e.g. whether object 2 is more to thetop/north or right/west, etc. when compared to object 1. To do this, we will express the position ofobject 2 in a polar coordinate system (θ, ϕ), where 0 ≤ θ ≤ π is the polar angle and 0 ≤ ϕ ≤ 2π aphase angle, and where object 1 will be at the origin. The positions of the two objects are given in ina generic longitude-latitude coordinate system: (l1, b1) and (l2, b2), where l could be e.g. the eclipticlongitude, the right ascension or the azimuth, and b could be the corresponding ecliptic latitude,declination or altitude.

The angular separation between the two objects is given by

cos θ = cos b1 cos b2 cos(l2 − l1) + sin b1 sin b2, (5.3.1)

while the position angle of object 2 with respect to object 1 can be computed with

tanϕ =sin(l2 − l1)

cos b1 tan b2 − sin b1 cos(l2 − l1). (5.3.2)

Note that you need the atan2() function to compute ϕ from Equation 5.3.2. This uses ϕ = 0 indicatingthat object 2 is above/N of object 1, ϕ = π/2 = 90 for to the right, etc.

Note that increasing right ascension is defined opposite to increasing ecliptic longitude or azimuth, sothat a negative sign must be added to the numerator at right-hand side of Equation 5.3.2 (effectivelyswapping the l1’s and l2’s), which comes down to

tanϕ =sin(α1 − α2)

cos δ1 tan δ2 − sin δ1 cos(α1 − α2). (5.3.3)

See Appendix B.4 for the derivation of these equations.

15

6 Transit, rise and set of objects

6.1 Transit

The hour angle H of an object represents the time since its transit. Hence, the negative of the hourangle of that object computed for midnight local time −H0 gives the time of transit. Using this, andEqs. 5.1.5 and 3.4.3 for an object with right ascension αobj, the time of transit can be computed by

ttr = −H0 = αobj − θ′ = αobj − θ′0 − λobs, (6.1.1)

If the object is in the solar system, its position αobj may have changed between the instance for whichits position was computed and the transit time. Since we computed θ′0 for midnight, we will use thefollowing equations:

Hi = θ′0 + 1.002737909350795 · ttr,i − αobj,i + λobs; (6.1.2)

ttr,i = ttr,i−1 −Hi, (6.1.3)

where the constant is the number of siderial days in a Julian day for J2000.0 (Eq.3.17 in Urban &Seidelmann, 2012, with corrected typo). This exercise should be iterated using the newly computedtime until the desired accuracy has been achieved (i.e., the new transit time differs less than e.g. 1 minor 1 s from the previous value). Once converged, the angle should be converted to a time (e.g. bydividing by π and multiplying with 12 if radians and a 24h clock are used). The final value should liebetween 0 and 24 hours. If it does not, add or subtract 24 until it does (or use the modulo function).

The local transit altitude for an object can be found from Eq. 5.2.2 with H = 0, and depends only onits declination δobj and the observer’s latitude ϕobs:

htr = arcsin (sinϕobs sin δobj + cosϕobs cos δobj) . (6.1.4)

6.2 Rise and set

In order to quickly estimate the time of rising or setting of an object with given equatorial coordinates,one can find the hour angle at which the altitude equals zero in Eq. 5.2.2. From the hour angle andright ascension of the object, one can compute the sidereal time through Eq. 5.1.5, which can then bereduced to a local time.

However, there are some complications in case some accuracy is required. Firstly, due to atmosphericrefraction, an object seems to have a higher altitude than in the case where the Earth would have noatmosphere, especially near the horizon. At the horizon, and for default temperature and pressure,this effect amounts to about 0.567 and should be taken into account. In addition, for extendedobjects like the Sun and the Moon, rise and set times are usually defined at the moment their upperlimb, rather than centre, crosses the horizon. This effect amounts to their apparent radius, about0.27, and combined with the effect of refraction we will want to determine when the Sun or Moonhave an altitude of -0.833. Thirdly, the Moon is relatively close to the Earth, which causes a parallaxbetween the geocentric and topocentric positions of the Moon (see Sect. 5.2.2). This horizontal parallaxis largest at rise and set time, and will depend on the observer’s location. On average, this makesthe Moon appear about 0.958 lower in the sky than indicated by the geocentric position, so that weshould add this value to the desired altitude, resulting in +0.125. If a higher accuracy is desired, onecan compute the refraction at the horizon from the current weather conditions, or compute the actualapparent radius of the Sun, the Moon and the planets and use these values instead.

Hence, the altitude at which rising and setting are defined are

Sun: h0 ≈ −0.833;

Moon: h0 ≈ +0.125;

Other objects: h0 ≈ −0.567.

16

Substituting h0 in Eq. 5.2.2, we can compute the hour angle H0 for which an object rises or sets:

cosH0 =sinh0 − sin δ sinϕobs

cos δ cosϕobs, (6.2.1)

where δ is the declination of the object and ϕobs is the latitude of the observer. Note that if theright-hand side of Eq. 6.2.1 does not fall between -1 and +1, the object never rises or sets, but isalways below or above the horizon. If it does fall in that range, we can compute H0 by taking thearccosine and converting from radians (or degrees) to hours. Two solutions are possible — we arecurrently only interested in the solution that lies between 0 and 12h.

Because the hour angle expresses the time since transit (see Sect. 5.1.2), the rise and set times of theobject can be computed from the transit time in Eq. 6.1.1:

trise = ttr −H0; (6.2.2)

tset = ttr +H0. (6.2.3)

As with the transit time, we will need to iterate this process in the case of solar-system objects (useseparate iteration loops for the transit, rise and set events). Again, the final values should lie between0 and 24 hours. If they do not, add or subtract 24 until they do (or use the modulo function).

17

A Rotations

Because we are living on a planet that rotates about its axis, whilst revolving around the Sun, rotationsform the basis of many operations in celestial mechanics. In addition, coordinate transformationstypically involve one or more rotations in 3D space. We will discuss the basic rotations in 2D and 3Din this section.

A.1 Definition of rotations and coordinate systems

The default convention for a rotation over a positive angle in the x–y-plane is as follows (Wikipedia,2017):

1. We rotate a point . . .

2. counterclockwise (looking from positive x× y ≡ z to the origin). . .

3. in a right-handed coordinate system. . .

4. using pre-multiplication.

The first two steps define an active transformation or alibi, which is used in the case where an objecthas moved in space (in a rotation about the origin) and the new position must be computed in thesame, fixed coordinate system. The opposite case, where in step 1 the coordinate system is rotated(CCW) rather than a point, is called a passive transformation or alias (the same would happen if wewould rotate a point in a clockwise direction in step 2). In this case, the object does not move, butthe coordinate system is rotated, and we wish to express the position of the same point in the newsystem. Hence, the passive transformation is used in coordinate transformations, where the position ofan object is first known or computed in e.g. equatorial coordinates (right ascension and declination),and then transformed to the horizontal system (azimuth and altitude) for a local observer.

A right-handed coordinate system consisting of axes x, y and z (in the three-dimensional case) isdefined as follows:

1. Choose orthogonal x and y axes; they define the x–y-plane;

2. Position your open right hand with the stretched fingers pointing in the positive x-direction;

3. Bend your four fingers by 90, so that they point in the positive y-direction.

4. When you stick out your thumb, it points in the positive z-direction, perpendicular to thex–y-plane (z ≡ x× y).

The pre-multiplication rule specifies that the rotation matrix R is multiplied with the column vector~v. In order to achieve the same result with row vector ~w one should use post-multiplication with thetranspose of R:

R~v = ~wRT . (A.1.1)

A.2 2D rotations of points and coordinate systems

The default two-dimensional rotation using the convention from the previous section, i.e. rotating apoint represented by the vector ~v counterclockwise about an angle θ in a right-handed coordinatesystem using pre-multiplication, is given by

~v′ = R(θ)~v; R(θ) =

(cos θ − sin θ

sin θ cos θ

). (A.2.1)

Instead, we are interested in rotating the coordinate system rather than a point, in which case we willuse the passive transformation

~v′ = S(θ)~v; S(θ) = R(θ)T =

(cos θ sin θ

− sin θ cos θ

). (A.2.2)

18

A default 2D coordinate transformation is derived below.

A.3 Derivation of the 2D rotation

The passive rotation matrix for a default coordinate transformation in two dimensions can be derivedas follows. We start with the coordinate system in the x− y plane in which the coordinates for pointP are (x, y). We want to express this position in the u − v plane, which has been rotated over anangle θ (see Figure A.3.1). This is equivalent to computing the values (u, v) as a function of (x, y).To facilitate this, we have split the coordinate variables into x = x1 − x2, y = y1 + y2, u = u1 + u2and v = v1 − v2.

P

x

y

u

v

x

x1

x2

y y1

y2

u1

u2

v

v1

v2

x →

y →

u →

v →

θ

Figure A.3.1: Derivation of the rotation of the coordinate system about an angle θ around the origin.

u: v:u1 = x

cos θ v1 = ycos θ

u2: y2 = x tan θ v2: x2 = y tan θu2 = y1 sin θ = (y − y2) sin θ v2 = x1 sin θ = (x+ x2) sin θ

= (y − x tan θ) sin θ = (x+ y tan θ) sin θu = u1 + u2 = x

cos θ + (y − x tan θ) sin θ v = v1 − v2 = ycos θ − (x+ y tan θ) sin θ

= x(

1cos θ −

sin2 θcos θ

)+ y sin θ = y

(1

cos θ −sin2 θcos θ

)− x sin θ

u = x cos θ + y sin θ v = −x sin θ + y cos θ

This gives the rotation (uv

)=

(cos θ sin θ− sin θ cos θ

)(xy

), (A.3.1)

which contains the matrix S(θ) as given in Eq. A.2.2.

19

A.4 3D rotations of coordinate systems

A rotation about one of the main axes (x, y or z) over a positive angle θ in three dimensions is definedas a counterclockwise rotation in a right-handed coordinate system, where the positive direction of therotation axis points at the observer (who is looking back at the origin to observe the rotation). Assuch, a 3D rotation about a main axis is basically a passive 2D rotation in the plane spanned by thetwo remaining axes, as given by Equation A.2.2:

Rx(θ)=

1 0 0

0 cos θ sin θ

0 − sin θ cos θ

; Ry(θ)=

cos θ 0 − sin θ

0 1 0

sin θ 0 cos θ

; Rz(θ)=

cos θ sin θ 0

− sin θ cos θ 0

0 0 1

,

(A.4.1)where Ry may look like the transpose of what you expected, since the x − z plane is a left-handedsystem. Hence, the counterclockwise rotation when viewed from the positive y-direction, as we use ithere, is actually a clockwise rotation when viewed from the negative y-direction, i.e. −θ was used toobtain Ry in Eq. A.4.1.

In three or more dimensions, the multiplication of rotation matrices is not commutative. Hence, in asequence of rotations, the order is important; the first rotation is indicated by the last matrix. Forexample, if we want to rotate about the z-axis over an angle ϕ first, followed by a rotation over θabout the (new!) x-axis, we should use

~v′ = Rx(θ) Rz(ϕ) ~v. (A.4.2)

If a 3D rotation is required about an axis that is not one of the main axes, one can perform one ortwo rotations in order to align one of the main axes with the desired rotation axis, and then performthe intended rotation.

20

B Derivation of common coordinate transformations

B.1 Conversion between ecliptical and equatorial coordinates

When expressing ecliptical and equatorial coordinates in rectangular format, we can use the fact thatthe two systems differ be a rotation about the location of the vernal (or spring) equinox, known asthe (first) point of Aries. We choose our rectangular coordinates such that the x-axis points to thatlocation in both systems, while the z-axis points to the ecliptical or celestial north pole:xy

z

=

cos l cos b

sin l cos b

sin b

;

x′

y′

z′

=

cosα cos δ

sinα cos δ

sin δ

. (B.1.1)

The transformation from the ecliptical to the equatorial system is then given by a reverse rotationover the obliquity of the ecliptic (−ε) about the x-axis:cosα cos δ

sinα cos δ

sin δ

= Rx(−ε)

xyz

=

1 0 0

0 cos ε − sin ε

0 sin ε cos ε

cos l cos b

sin l cos b

sin b

=

cos l cos b

sin l cos b cos ε− sin b sin ε

sin l cos b sin ε+ sin b cos ε

, (B.1.2)

so that

tanα =sinα cos δ

cosα cos δ=

sin l cos b cos ε− sin b sin ε

cos l cos b=

sin l cos ε− tan b sin ε

cos l, (B.1.3)

andsin δ = sin l cos b sin ε+ sin b cos ε. (B.1.4)

For the inverse transformation, from equatorial to ecliptical coordinates, we need to rotate in theopposite direction:

R−1x (−ε) = Rx(ε) =

1 0 0

0 cos ε sin ε

0 − sin ε cos ε

. (B.1.5)

Hence, cos l cos b

sin l cos b

sin b

= Rx(ε)

cosα cos δ

sinα cos δ

sin δ

=

cosα cos δ

sinα cos δ cos ε+ sin δ sin ε

− sinα cos δ sin ε+ sin δ cos ε

, (B.1.6)

from which we find

tan l =sin l cos b

cos l cos b=

sinα cos δ cos ε+ sin δ sin ε

cosα cos δ=

sinα cos ε+ tan δ sin ε

cosα, (B.1.7)

andsin b = sin δ cos ε− sinα cos δ sin ε. (B.1.8)

See also Urban & Seidelmann (2012) Sect. 14.4.4.1 and their Equations 14.42 and 14.43.

21

B.2 The rotation over a phase angle: right ascension to hour angle

When performing a 3D rotation over the phase angle (longitude, right ascension, etc.) only, keepingthe polar angle (latitude, declination, etc.) constant, we would intuitively expect that we can simplysubtract the rotation angle from the phase angle. Indeed, this follows from the general 3D rotationmatrices derived in Sect. A.4.

An example is the conversion from right ascension to hour angle, which is given by

H = −α+ θ, (B.2.1)

where H is the hour angle, α the right ascension, and θ the local sidereal time of the observer.The negative sign indicates that the right ascension is measured counterclockwise, and the rotationangle θ is simply added to −α. We expect the declination δ to be unchanged, but we will call thenew corresponding coordinate d to see whether this is indeed the case. Hence we will transform thecoordinates (α, δ) to (H, d).

Expressing our input and output positions in Cartesian coordinates, and using the rotation matrix fora rotation about the z axis from Eq. A.4.1, we getx

′

y′

z′

= Rz(−θ)

xyz

=

cos(−θ) sin(−θ) 0

− sin(−θ) cos(−θ) 0

0 0 1

xyz

,

cosH cos d

sinH cos d

sin d

=

cos θ − sin θ 0

sin θ cos θ 0

0 0 1

cos(−α) cos δ

sin(−α) cos δ

sin δ

=

cosα cos δ cos θ + sinα cos δ sin θ

cosα cos δ sin θ − sinα cos δ cos θ

sin δ

. (B.2.2)

Thus,

tanH =y′

x′=

cosα sin θ − sinα cos θ

cosα cos θ + sinα sin θ=

sin(θ − α)

cos(θ − α)= tan(θ − α)

andsin d = z′ = sin δ,

so that we indeed find that H = −α+ θ and d = δ, as expected.

B.3 Conversion between equatorial and horizontal coordinates

In many cases, one wants to know where in the local sky a celestial object can be found, usually withrespect to the local horizon: in which wind direction (azimuth) and how far above of below (altitude)the horizon is the object at a given time. The horizontal coordinates of an object depend on thegeographical location of the observer (λobs, ϕobs) and, even for ‘fixed’ objects like stars, the time.

One can imagine that for an observer standing on the Earth’s North Pole (ϕobs = π/2 = 90), theequatorial plane of the Earth or the sky and the local horizontal plane coincide. Hence, the calculationof horizontal coordinates can be most easily be done from equatorial coordinates by a single rotationthat takes into account the observer’s latitude and perhaps a second rotation about the local verticalaxis if desired. If we define that the origin of the hour angle (H = 0) and that of the azimuth(A = 0) coincide (i.e., A = 0 when the object is transiting in the south for an observer in the northernhemisphere) and the positive x-axis points there, the rotation simplifies to a single rotation aroundthe y-axis over an angle π/2− ϕobs. We can find the corresponding rotation matrix from Eq. A.4.1:

Ry

(π2− ϕobs

)=

cos(π2 − ϕobs

)0 − sin

(π2 − ϕobs

)0 1 0

sin(π2 − ϕobs

)0 cos

(π2 − ϕobs

) =

sinϕobs 0 − cosϕobs

0 1 0

cosϕobs 0 sinϕobs

. (B.3.1)

22

Hence, the coordinate transformation becomes:cosA cosh

sinA cosh

sinh

=

sinϕobs 0 − cosϕobs

0 1 0

cosϕobs 0 sinϕobs

cosH cos δ

sinH cos δ

sin δ

=

cosH cos δ sinϕobs − sin δ cosϕobs

sinH cos δ

cosH cos δ cosϕobs + sin δ sinϕobs

, (B.3.2)

so that

tanA =sinA cosh

cosA cosh=

sinH cos δ

cosH cos δ sinϕobs − sin δ cosϕobs=

sinH

cosH sinϕobs − tan δ cosϕobs, (B.3.3)

andsinh = cosH cos δ cosϕobs + sin δ sinϕobs. (B.3.4)

For the inverse transformation, from horizontal to equatorial coordinates, we need the inverse matrixof Eq. B.3.1 (the same rotation, but in the other direction):cosH cos δ

sinH cos δ

sin δ

=

sinϕobs 0 cosϕobs

0 1 0

− cosϕobs 0 sinϕobs

cosA cosh

sinA cosh

sinh

=

cosA cosh sinϕobs + sinh cosϕobs

sinA cosh

− cosA cosh cosϕobs + sinh sinϕobs

, (B.3.5)

so that

tanH =sinH cos δ

cosH cos δ=

sinA cosh

cosA cosh sinϕobs + sinh cosϕobs=

sinA

cosA sinϕobs + tanh cosϕobs, (B.3.6)

andsin δ = sinh sinϕobs − cosA cosh cosϕobs. (B.3.7)

B.4 Angular separation and position angle between two celestial objects

Given two (nearby) objects in the sky, we may be interested in their relative positions, which we canexpress as the angular separation (i.e., the distance in the sky expressed in radians or degrees) andthe position angle, which indicates e.g. whether object 2 is more to the top/north or right/westwhen compared to object 1. To do this, we will express the position of object 2 in a polar coordinatesystem (θ, ϕ), where 0 ≤ θ ≤ π is a polar angle and 0 ≤ ϕ ≤ 2π a phase angle, and where object 1will be at the origin. We start with the coordinates of the two objects in a generic longitude-latitudecoordinate system: (l1, b1) and (l2, b2), where l could be e.g. the ecliptic longitude, the right ascensionor the azimuth, and b could be the corresponding ecliptic latitude, declination or altitude.

xyz

=

cosϕ sin θ

sinϕ sin θ

cos θ

;

x′

y′

z′

=

cos l2 cos b2

sin l2 cos b2

sin b2

.

To make the coordinate transformation from and origin at (l, b) = (0, 0) to an origin at (l, b) = (l1, l2),we need to make two rotations. First, we rotate about the axis connecting the two poles of theoriginal (l, b) coordinate system (i.e. a rotation in the base plane of the system) over the angle l1.This corresponds to the rotation Rz(l1) from Equation A.4.1. Next, we rotate away from the original

23

base plane, about the (new) y axis over the angle b1, i.e. Ry(b1) from Eq. A.4.1. The total rotationthen becomes:cosϕ sin θ

sinϕ sin θ

cos θ

= Ry(b1)Rz(l1)

cos l2 cos b2

sin l2 cos b2

sin b2

=

sin b1 cos l1 sin b1 sin l1 − cos b1

− sin l1 cos l1 0

cos b1 cos l1 cos b1 sin l1 sin b1

cos l2 cos b2

sin l2 cos b2

sin b2

=

sin b1 cos l1 cos l2 cos b2 + sin b1 sin l1 sin l2 cos b2 − cos b1 sin b2

− sin l1 cos l2 cos b2 + cos l1 sin l2 cos b2

cos b1 cos l1 cos l2 cos b2 + cos b1 sin l1 sin l2 cos b2 + sin b1 sin b2

,(B.4.1)

so that, from comparing the third terms in Equation B.4.1, we find

cos θ = cos b1 cos l1 cos l2 cos b2 + cos b1 sin l1 sin l2 cos b2 + sin b1 sin b2

= cos b1 cos b2 cos(l2 − l1) + sin b1 sin b2, (B.4.2)

where I applied the identity cos l1 cos l2 + sin l1 sin l2 = cos(l2− l1) in the last step. This result is usedin Section 4.3.

We can find ϕ by dividing the second and first terms of Eq. B.4.1, to yield

tanϕ =sinϕ sin θ

cosϕ sin θ

=− sin l1 cos l2 cos b2 + cos l1 sin l2 cos b2

sin b1 cos l1 cos l2 cos b2 + sin b1 sin l1 sin l2 cos b2 − cos b1 sin b2

=sin(l2 − l1)

sin b1 cos(l2 − l1) − cos b1 tan b2, (B.4.3)

where I applied the aforementioned identity, as well as cos l1 sin l2−sin l1 cos l2 = sin(l2−l1) and dividedboth numerator and denominator by cosh in the last step. Note that this gives ϕ = π = 180 if object 2is above or north of object 1, whereas the convention ϕ = 0 indicating above/N, ϕ = π/2 = 90 for tothe right, etc. is more common and is achieved by changing the sign in the denominator:

tanϕ =sin(l2 − l1)

cos b1 tan b2 − sin b1 cos(l2 − l1). (B.4.4)

Note that increasing right ascension is defined opposite to increasing ecliptic longitude or azimuth, sothat a negative sign must be added to the numerator at right-hand side of Equation B.4.4 (effectivelyadding a negative sign to all l’s).

B.4.1 Projection of sunlight on a surface

Computing the projected insolation of direct solar radiation on an inclined surface, for example a solarpanel, is similar to computing the angular separation between two celestial objects, since the anglebetween the normal vector of the surface and the solar vector (i.e., the angular separation betweenthe Sun and the sky position the normal vector is pointing at) must be computed.

The local position of the Sun is usually expressed in horizontal coordinates (A, h) and the normalvector of a surface as (γ, β), where γ is the ‘azimuth’ of the panel orientation, defined similarly to Aand β is defined as the inclination of the surface with respect to the horizontal. Note that while γ takesthe role of a longitude analogous to l1 in Section B.4, β is a polar angle rather than a latitude-likecoordinate, so that β = π/2− b and sines and cosines are swapped when replacing b1 and β.

We start with a horizontal panel, aligned north-south, i.e. β = γ = 0. We first need to rotate it abouta vertical axis to give it the proper angle γ, i.e. the rotation Rz(γ) (see Eq. A.4.1). Next, we give the

24

panel its inclination β, in such way that if γ = 0 it will be inclined to the south; in other words, werotate it about the (new) horizontal y axis: Ry(β) from Eq. A.4.1. The total rotation then becomes(compare Equation B.4.1)cosϕ sin θ

sinϕ sin θ

cos θ

= Ry(β)Rz(γ)

cosA cosh

sinA cosh

sinh

=

cosβ cos γ cosA cosh + cosβ sin γ sinA cosh − sinβ sinh

− sin γ cosA cosh + cos γ sinA cosh

sinβ cos γ cosA cosh + sinβ sin γ sinA cosh + cosβ sinh

, (B.4.5)

so that, from comparing the third terms in Equation B.4.5, we find

cos θ = sinβ cosh cos(A− γ) + cosβ sinh. (B.4.6)

This result is comparable to Eq. B.4.2 and used in Section 4.3.

We can find ϕ similarly to Eq. B.4.4 by dividing the second and first terms of Eq. B.4.5, to yield

tanϕ =sin(A− γ)

sinβ tanh − cosβ cos(A− γ). (B.4.7)

25

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Bretagnon, P. & Francou, G. 1988, A&A, 202, 309 [ADS]

Brown, E. W. 1896, An introductory treatise on the lunar theory

Chapront-Touze, M. & Chapront, J. 1988, A&A, 190, 342

Extrapolation of Delta T. 2015, Extrapolation of ∆T http://hemel.waarnemen.com/Computing/deltat.html

Gueymard, C. 1995, Professional Paper FSEC-PF-270-95 https://www.nrel.gov/grid/solar-resource/smarts.html

Gueymard, C. A. 2001, Solar Energy, 71, 325

IERS. 2015, International Earth Rotation and Reference Systems Service http://maia.usno.navy.mil

Kovalevsky, J. & Seidelmann, P. K. 2004, Fundamentals of Astrometry

Saemundsson, T. 1986, S&T, 72, 70

Seidelmann, P. K. 1982, Celestial Mechanics, 27, 79

Simon, J. L., Bretagnon, P., Chapront, J., Chapront-Touze, M., Francou, G., & Laskar, J. 1994, A&A,282, 663

Urban, S. E. & Seidelmann, P. K. 2012, Explanatory Supplement to the Astronomical Almanac (3rdEdition) (University Science Books)

van der Sluys, M. 2018a, A Fortran library to compute the positions of and events for celestial bodieswith great accuracy http://libthesky.sf.net

—. 2018b, A library containing Some Useful Fortran Routines http://libsufr.sf.net

Wikipedia. 2017, Rotation matrix — Wikipedia, The Free Encyclopedia, accessed 2017-02-18https://en.wikipedia.org/wiki/Rotation matrix

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