Celestial mechanics in a nutshell

Transcription

1 Celestial mechanics in a nutshell Marc van der Sluys CMiaNS.sf.net October 13, 2018

2 Copyright c by Marc van der Sluys All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the author, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests, contact the author at the address below. sluys/ This document was typeset in L A TEX by the author.

3 Contents 1 Introduction 4 2 Variables Functions Time Position Time, location, transit Timescales Main timescales Offsets between timescales Julian day Computing the Julian day from the UNIX time Time since J Sidereal time The Sun Position of the Sun Conversion to local coordinates Insolation on an inclined surface Air mass Atmospheric extinction of sunlight Coordinate systems Equatorial coordinates Right ascension and declination Hour angle parallactic coordinates Horizontal coordinates Atmospheric refraction Parallax and topocentric coordinates Transit, rise and set of objects Transit Rise and set A Rotations and transformations 18 A.1 Definition of rotations and coordinate systems A.2 2D rotations of points and coordinate systems A.3 Derivation of a 2D coordinate transformation A.4 3D rotations of coordinate systems B Derivation of common coordinate transformations 21 B.1 Conversion between ecliptical and equatorial coordinates B.2 The rotation over a phase angle: right ascension to hour angle B.3 Conversion between equatorial and horizontal coordinates

4 1 Introduction This document is work in progress. I intend it to evolve into a rich source of information for those who 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 needed to compute celestial events, understand why these equations are used and allow the reader to track where (in the literature) they came from. For example, in the appendix we derive the default rotations for 2D and 3D rotations, and from there the coordinate transformations between the most common coordinate systems. This is a living document. The current version was compiled on October 13, You may find a newer version on This document may well contain errors. Please report them to the same website. 4

5 2 Variables 2.1 Functions floor() Round variable down to nearest integer: floor(0.8) = 0; floor( 0.3) = Time JD Julian day JD 2000 Julian day of = θ Local mean sidereal time θ Local apparent sidereal time θ 0 Greenwich mean sidereal time θ 0 Greenwich apparent sidereal time 2.3 Position ϕ obs Geographic latitude of observer, > 0 if on northern hemisphere λ obs Geographic longitude of observer, > 0 if east of Greenwich l Ecliptical longitude b Ecliptical latitude α obj Right ascension of an object δ obj Declination of an object A Azimuth h Altitude ε 0 Mean obliquity of the ecliptic, without nutation ε Nutation in obliquity ε True obliquity of the ecliptic, corrected for nutation ε = ε 0 + ε ψ Nutation in longitude 5

6 3 Time, location, transit 3.1 Timescales In order to perform ephemeris calculations, the civil time on our clocks (e.g. UTC), which is based on the solar day, often needs to be converted to an ephemeris time that approximates the time in the universe (usually TT). Since the Earth s rotation is not constant, the difference between these two timescales, called T, changes unpredictably and must be derived from observations. Below, I briefly describe the main timescales of importance and their offsets Main timescales UTC = Coordinated Universal Time; civil time at Greenwich, differs from local time by the time zone (usually an (semi)integer number of hours), updated only by leap seconds to stay approximately synchronised with the Earth s rotation. UT1 = Universal Time 1 ; changes continuously, since the rotation of the Earth is variable. When UT1-UTC reaches -0.6s, a leap second is introduced, which changes UTC such that UT1-UTC +0.4s. 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 relativistic effects. TT = Terrestrial Time; replaces Terrestrial Dynamical Time (TDT) and Ephemeris Time (ET); TT TAI s GPS s, independent of the Earth s rotation. Idealised concept of the time in the universe, and hence used as input for ephemeris calculations. Approximated by TAI, but without relativistic effects Offsets between timescales T = s + (TAI-UTC) - (UT1-UTC) = s + TAI - UT1 = TT - UT1. Cumulative difference between idealised time in the universe or dynamical time, and civil time. T was negative between 1871 and 1902, and positive at all other times (barring giant impacts). The library libthesky ( ) contains inter- and extrapolation routines to approximate T. See also the IERS (2015) website and Extrapolation of Delta T (2015) for a compilation of historical values, future extrapolations, and more references. TAI-UTC = cumulative number of leap seconds = 35s on ; changes every now and then by one second. UT1-UTC = s; measured by e.g. the IERS (2015), and reset when a leap second is added. TAI-GPS = 19s, and fixed. GPS-UTC = (TAI-UTC) - (TAI-GPS) = 35s - 19s = 16s on ; changes when a leap 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 an instant. The every-day variables of month, day, hour minute and second are not monotonous, but are regularly reset (from 59 or 23 to 0, or from or 12 to 1), which makes computation difficult. Instead, the Julian day is a continuous time variable, expressed in decimal days. JD=0 has been 6

7 defined as January 1 of the year CE, at noon UT. Hence, a new Julian day starts at noon UT, not at midnight. January 1, 2000 at noon corresponds to JD= and is expressed as JD or JD The calculation of the position of a celestial object for any instant always starts by converting the date and time of that instant to JD. Algorithms to compute the Julian day are provided in the library libsufr ( ); 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( *(m+1)) + d + b 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 at midnight UT. This moment corresponds to JD = Hence, the Julian day can be computed from the UNIX time using JD = UNIX time where the denominator is the number of seconds in a Julian day , (3.2.1) 3.3 Time since J For fits to parameters that vary over long timescales, the time since J is often used, expressed in Julian days, years, centuries or millennia: t Jd = JD(E) ; (3.3.1) t Jy = JD(E) ; (3.3.2) Note that the Julian day number is an integer that counts the days since JD=0. Also, the Julian day is often called Julian 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

8 t Jc = JD(E) ; (3.3.3) t Jm = JD(E) (3.3.4) 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 represents the orientation of the night sky (the same star transits at a given sidereal time). The two differ in day length by about four minutes, since by the time the Earth has completed a full revolution about its axis with respect to the distant stars, it has also moved in its orbit around the Sun and needs 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 to compute positions and events like rise, transit and setting of objects in our local sky. This is usually done using the hour angle, as explained in Section In order to compute the local sidereal time, one starts with an expression for the sidereal time in Greenwich and compensates for the observer s geographical longitude. To take into account the effect of precession of the equinoxes, the mean sidereal time can be converted to the apparent sidereal time. Greenwich mean sidereal time, in radians (Urban & Seidelmann, 2012, Eq. 6.66): θ 0 = t Jd t 2 Jd t 3 Jd t 4 Jd t 5 Jd (3.4.1) T, where T = T T UT 1 (see Sect ) is expressed in seconds of time. If T is not available, remove the last term and replace the first with radians. Greenwich apparent sidereal time (Urban & Seidelmann, 2012, Eqs. 6.7, 6.67): Local mean or apparent sidereal time, λ obs > 0 if east of Greenwich: θ 0 = θ 0 + ψ cos ε (3.4.2) θ = θ 0 + λ obs (3.4.3) θ = θ 0 + λ obs = θ + ψ cos ε (3.4.4) Before printing the variables θ 0, θ 0, θ, θ, you should ensure their values lie between 0 and 2π. 8

9 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 instant is given below. This method is used in the routine sunpos la() in libthesky ( ), and has an accuracy of 11.9 ± 5.7 for the years when compared to VSOP87 (Bretagnon & Francou, 1988) for 10 5 random moments. All angles are expressed in radians. The time is expressed in Julian centuries since J (t Jc, see Eq ). Semi-major axis of the Earth in AU (Simon et al., 1994, Sect ): Mean longitude of the Sun (λ + π in Simon et al., 1994, Sect ): a (4.1.1) λ t Jc t 2 Jc t 3 Jc (4.1.2) Mean anomaly of the Sun (l in Chapront-Touze & Chapront, 1988, Table 6): m t Jc t 2 Jc t 3 Jc (4.1.3) Eccentricity of the Earth s orbit (Simon et al., 1994): e t Jc t 2 Jc t 3 Jc (4.1.4) Sun s equation of the centre (Brown, 1896, Chap. III, Eq.7, up to e 3 ): C ν m (2e 14 ) e3 sin m e2 sin(2m) e3 sin(3m) (4.1.5) Alternatively, combining Eqs and 4.1.5, we find (using the leading terms in t Jc only): C ( t Jc t 2 Jc) sin m + ( t Jc ) sin(2m) 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): Ω m t Jc t 2 Jc t 3 Jc (4.1.10) Mean longitude of the Moon (L in Chapront-Touze & Chapront, 1988, Table 6): L m t Jc t 2 Jc t 3 Jc (4.1.11) 9

10 Nutation in longitude (Seidelmann, 1982, Table I, lines 1,9,31,2,10): ψ ( t Jc ) sin Ω m sin(2 λ) sin(2 L m ) sin(2 Ω m ) sin m (4.1.12) Annual aberration (Kovalevsky & Seidelmann, 2004): θ ab a(1 e2 ) r (4.1.13) Apparent geocentric longitude, for the true equinox: l = + θ ab + 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 use Section 5.1 to compute the equatorial and thence the parallactic coordinates of the Sun, and Sect. 5.2 to 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 and then reduce to: tan α = 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 ) and even parallax (Sect ) can be 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 example solar PV panels or a solar collector, we need to compute the angle θ between the local position vector of the Sun (A, h) and the normal vector of the surface (β, γ): cos θ = sin h cos β + cos h sin β 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 the azimuth 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 reduces to cos θ = cos β cos β + sin β sin β cos(0) = 1, so that indeed θ = 0. If the beam normal radiation (or DNI) is provided, it can directly be multiplied with cos θ. If the beam/direct horizontal radiation is known, it must be multiplied with cos θ sin h instead, where h is the Sun s altitude. If both are available, using the DNI is preferred, since dividing by sin h can produce unexpected results for low Sun (h 0). 10

11 4.4 Air mass The air mass (AM) quantifies the column density of atmospheric gasses in the path of the light of the Sun (or another celestial object). The air mass at the top of the Earth s atmosphere equals zero, for an object in the zenith AM = 1 (under standard conditions, at sea level), and the air mass increases when an object is closer to the horizon. The air mass can be computed from [ AM = max 1, sin 2 ] h sin h sin 3 h sin 2, (4.4.1) h sin h where h is the true altitude of the object (uncorrected for atmospheric refraction) and which has a maximum supposed error (at the horizon) of air mass (Young, 1994). 4.5 Atmospheric extinction of sunlight The bolometric 2 atmospheric extinction of sunlight is a function of the air mass. My own fit to detailed spectra of direct sunlight computed using the SMARTS code (Gueymard, 1995, 2001) yields the following polynomial: [ 11 ] E exp C i (AM) i 1, (4.5.1) with i=1 C = ( , , , , , , , , , , ). (4.5.2) I find a maximum deviation in the extinction factor of for AM 36.9 (see Fig ). Note that this expression holds for the irradiation of sunlight, not for the visibility of faint objects like stars. 3 The solar constant 4 can be divided by the extinction factor E the Sun distance in AU r squared to obtain the direct normal insolation (DNI) of sunlight: DNI W m 2 E r 2. (4.5.3) 2 i.e. over the whole spectrum 3 For the visibility of stars, night vision provided by the rods in the human retina plays a role W/m 2 insolation at the top of the Earth s atmosphere 11

12 log Δ Extinction Air mass Figure 4.5.1: Top panel (a): polynomial fit (dashed blue line) to the atmospheric extinction as computed by the SMARTS code (solid red line), as a function of air mass. Bottom panel (b): logarithm of the absolute residual of the fit. 12

13 5 Coordinate systems 5.1 Equatorial coordinates Right ascension and declination The equatorial coordinate system is a spherical system with the same orientation as the Earth s geographical coordinate system, hence with the Earth s equator as the base plane. The two coordinates are 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 of what you might expect). Declination is expressed in degrees ( for south pole to north pole). For calculations these are often converted to radians. The positions of fixed objects (stars and deepsky objects) are usually listed in RA and dec., and the positions of solar-system objects are converted from ecliptical to horizontal coordinates through the equatorial system. The coordinate transformation needed for the conversion from ecliptical coordinates (l, b) to equatorial coordinates (see e.g. Sect. 4.2) is derived in Appendix B.1 and is given by tan α = sin l cos ε tan b sin ε ; (5.1.1) cos l 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 right ascension from Eq The inverse transformation, from equatorial to ecliptical coordinates, is also derived in Appendix B.1: tan l = sin α cos ε + tan δ sin ε ; cos α (5.1.3) sin b = sin δ cos ε sin α cos δ sin ε. (5.1.4) Hour angle parallactic coordinates When using a parallactic telescope mount (one for which the main axis is aligned with the Earth s rotation axis), the right ascension is often converted to the hour angle (HA, H), which is the local counterpart of the RA. The hour angle is measured in hours and usually defined as 0h (sometimes 12h) when an object is transiting, and negative before and positive after transit. Hence, the hour angle essentially measures the time since the transit of that object. The HA can be obtained from the 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 (see Eq in Sect. 3.4). Note that the HA increases in the opposite direction from the RA. Hour angle and declination are also known as parallactic coordinates. 5.2 Horizontal coordinates While equatorial coordinates are based on the plane of the Earth s equator, an observer will want to know where an object is with respect to the local horizon: above it or below, by how much (the altitude h of the object) and the wind direction where the object is above or below the horizon, called the azimuth (A). 13

14 The conversion from equatorial/parallactic coordinates hour angle H and declination δ to horizontal coordinates is derived in Appendix B.3 and results in: tan A = sin H ; cos H sin ϕ obs tan δ cos ϕ obs (5.2.1) sin h = sin δ sin ϕ obs + cos H cos δ cos ϕ obs, (5.2.2) where ϕ obs is the geographic latitude of the observer. Note that the hour angle contains the information of the observer s longitude (see Eq ). Make sure you use the atan2() function to compute the azimuth from Eq The reverse transformation, from horizontal to equatorial coordinates, is also found in Appendix B.3: tan H = sin A ; cos A sin ϕ obs + tan h cos ϕ obs (5.2.3) sin δ = sin h sin ϕ obs cos A cos h cos ϕ obs. (5.2.4) Sometimes, the zenith angle (z) is used instead of the altitude: z = 90 h. (5.2.5) Atmospheric refraction For precise calculations, one may want to take into account atmospheric refraction, which can cause object 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 the horizon. The following approximation is taken from Saemundsson (1986), but converted to radians: h = ) tan (h + h ( ) ( ) P 283. (5.2.6) T Here, h is the uncorrected altitude of the Sun, and the correction h must be added to that. The last two factors take into account the air pressure (P in millibars) and temperature (T in C). If these are unknown, or high precision is not needed, they can be left out (implicitly assuming P = 1010 mb and T = 10 C) Parallax and topocentric coordinates If precise results for nearby objects (especially the Moon, and to a lesser extent other solar-system bodies) are needed, the diurnal parallax should be taken into account. The parallax arises from the fact that a nearby body like the Moon appears at a slightly different position with respect to the distant stars, when observed from different locations on Earth. This is similar to the different positions of a 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 from the centre of the Earth. Hence, the parallax is used to convert the geocentric coordinates we have considered so far to topocentric coordinates. The equatorial (i.e. using the equatorial radius of the Earth) horizontal parallax of the Sun equals ( ) Req π = arcsin (5.2.7) AU 14

15 For an object with distance r, the parallax yields sin π obj = sin π, (5.2.8) r where in many cases sufficient accuracy is obtained using the approximation The correction to the altitude h of an object is then given by π obj π r. (5.2.9) sin h = ρ sin π cos h, (5.2.10) where ρ is the distance of the observer to the centre of the Earth, expressed in Earth radii. Note that the 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 assumed equal to their angles: h = π cos h. (5.2.11) The topocentric altitude of the object is then computed from the geocentric altitude: h topo = h geo + h. (5.2.12) 15

16 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 hour angle of that object computed for midnight local time H 0 gives the time of transit. Using this, and Eqs and for an object with right ascension α obj, the time of transit can be computed by t tr = H 0 = α obj θ = α obj θ 0 λ obs, (6.1.1) If the object is in the solar system, its position α obj may have changed between the instance for which its position was computed and the transit time. Since we computed θ 0 for midnight, we will use the following equations: H i = θ t tr,i α obj,i + λ obs ; (6.1.2) t tr,i = t tr,i 1 H i, (6.1.3) where the constant is the number of siderial days in a Julian day for J (Eq.3.17 in Urban & Seidelmann, 2012, with corrected typo). This exercise should be iterated using the newly computed time until the desired accuracy has been achieved (i.e., the new transit time differs less than e.g. 1 min or 1 s from the previous value). Once converged, the angle should be converted to a time (e.g. by dividing by π and multiplying with 12 if radians and a 24h clock are used). The final value should lie between 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 with H = 0, and depends only on its declination δ obj and the observer s latitude ϕ obs : h tr = arcsin (sin ϕ obs sin δ obj + cos ϕ obs cos δ obj ). (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 From the hour angle and right ascension of the object, one can compute the sidereal time through Eq , which can then be reduced to a local time. However, there are some complications in case some accuracy is required. Firstly, due to atmospheric refraction, an object seems to have a higher altitude than in the case where the Earth would have no atmosphere, especially near the horizon. At the horizon, and for default temperature and pressure, this effect amounts to about and should be taken into account. In addition, for extended objects like the Sun and the Moon, rise and set times are usually defined at the moment their upper limb, rather than centre, crosses the horizon. This effect amounts to their apparent radius, about 0.27, and combined with the effect of refraction we will want to determine when the Sun or Moon have an altitude of Thirdly, the Moon is relatively close to the Earth, which causes a parallax between the geocentric and topocentric positions of the Moon (see Sect ). This horizontal parallax is largest at rise and set time, and will depend on the observer s location. On average, this makes the Moon appear about lower in the sky than indicated by the geocentric position, so that we should add this value to the desired altitude, resulting in If a higher accuracy is desired, one can compute the refraction at the horizon from the current weather conditions, or compute the actual apparent 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: h ; Moon: h ; 16

17 Other objects: h Substituting h 0 in Eq , we can compute the hour angle H 0 for which an object rises or sets: cos H 0 = sin h 0 sin δ sin ϕ obs cos δ cos ϕ obs, (6.2.1) where δ is the declination of the object and ϕ obs is the latitude of the observer. Note that if the right-hand side of Eq does not fall between -1 and +1, the object never rises or sets, but is always below or above the horizon. If it does fall in that range, we can compute H 0 by taking the arccosine and converting from radians (or degrees) to hours. Two solutions are possible we are interested in the solution that lies between 0 and 12h. Because the hour angle expresses the time since transit (see Sect ), the rise and set times of the object can be computed from the transit time in Eq : t rise = t tr H 0 ; (6.2.2) t set = t tr + H 0. (6.2.3) As with the transit time, we will need to iterate this process in the case of solar-system objects (use separate iteration loops for the transit, rise and set events). Again, the final values should lie between 0 and 24 hours. If they do not, add or subtract 24 until they do (or use the modulo function). 17

18 A Rotations and transformations Because we are living on a planet that rotates about its axis, whilst revolving around the Sun, rotations form the basis of many operations in celestial mechanics. In addition, coordinate transformations typically involve one or more rotations in 3D space. We will discuss the basic rotations in 2D and 3D in 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 counterclockwise (looking from positive ˆx ŷ to the origin) in a right-handed coordinate system using pre-multiplication. The first two items define an active transformation or alibi, which is used in the case where an object has moved in space (in a rotation about the origin) and the new position must be computed in the same, fixed coordinate system. The opposite case, where the coordinate system is rotated (CCW) rather than a point in item 1, is called a passive transformation or alias (the same would happen if we would rotate a point in a clockwise direction in item 2). In this case, the object does not move, but the coordinate system is rotated, and we wish to express the position of the same point in the new system. Hence, the passive transformation is used in coordinate transformations, where the position of an 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) is defined 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 ˆx-direction; 3. Bend your four fingers by 90, so that they point in the positive ŷ-direction. 4. When you stick out your thumb, it points in the positive ẑ-direction, perpendicular to the x y-plane (ẑ ˆx ŷ). 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 the transpose of R: R v = wr T. (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 a point represented by the vector v counterclockwise about an angle θ in a right-handed coordinate system using pre-multiplication, is given by ( ) cos θ sin θ v = R(θ) v; R(θ) =. (A.2.1) sin θ cos θ 18

19 Instead, we are interested in rotating the coordinate system rather than a point, in which case we will use the passive transformation ( ) cos θ sin θ v = S(θ) v; S(θ) = R(θ) T =. (A.2.2) sin θ cos θ A default 2D coordinate transformation is derived below. A.3 Derivation of a 2D coordinate transformation The passive rotation matrix for a default coordinate transformation in two dimensions can be derived as follows. We start with the coordinate system in the x y plane in which the coordinates for point P are (x, y). We want to express this position in the u v plane, which has been rotated over an angle 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 = x 1 x 2, y = y 1 + y 2, u = u 1 + u 2 and v = v 1 v 2. v y x 1 y x P x 2 v 2 v θ y y1 v v 1 u 1 u 2 y 2 x u u x Figure A.3.1: Derivation of the rotation of the coordinate system about an angle θ around the origin. 19

20 u: v: u 1 = x cos θ v 1 = y cos θ u 2 : y 2 = x tan θ v 2 : x 2 = y tan θ u 2 = y 1 sin θ = (y y 2 ) sin θ v 2 = x 1 sin θ = (x + x 2 ) sin θ = (y x tan θ) sin θ = (x + y tan θ) sin θ u = u 1 ( + u 2 = x cos θ ) + (y x tan θ) sin θ v = v 1( v 2 = y cos θ ) (x + y tan θ) sin θ = x 1 cos θ sin2 θ cos θ + y sin θ = y 1 cos θ sin2 θ cos θ x sin θ u = x cos θ + y sin θ v = x sin θ + y cos θ This gives the rotation ( ) u = v ( cos θ sin θ sin θ cos θ which contains the matrix S(θ) as given in Eq. A.2.2. ) ( ) x, (A.3.1) y A.4 3D rotations of coordinate systems A rotation about one of the main axes (ˆx, ŷ or ˆx) over a positive angle θ in three dimensions is defined as a counterclockwise rotation in a right-handed coordinate system, where the rotation axis points at the observer. As such, a 3D rotation about a main axis is basically a passive 2D rotation in the plane spanned by the two remaining axes, as given by Equation A.2.2: cos θ 0 sin θ cos θ sin θ 0 R x (θ)= 0 cos θ sin θ ; R y (θ)= ; R z (θ)= sin θ cos θ 0, 0 sin θ cos θ sin θ 0 cos θ (A.4.1) where R y may look like the transpose of what you expected, since the ˆx ẑ plane is a left-handed system when viewed from the positive ŷ-direction. In three or more dimensions, the multiplication of rotation matrices is not commutative. Hence, in a sequence of rotations, the order is important; the first rotation is indicated by the last matrix. For example, if we want to rotate about the ẑ-axis over an angle ϕ first, followed by a rotation over θ about the (new!) ˆx-axis, we should use v = R x (θ) R z (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 or two rotations in order to align one of the main axes with the desired rotation axis, and then perform the intended rotation. 20

21 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 that the two systems differ be a rotation about the location of the vernal (or spring) equinox, known as the (first) point of Aries. We choose our rectangular coordinates such that the x-axis points to that location in both systems, while the z-axis points to the ecliptical or celestial north pole: x cos l cos b x cos α cos δ y = sin l cos b ; y = sin α cos δ. (B.1.1) z sin b z sin δ The transformation from the ecliptical to the equatorial system is then given by a reverse rotation over the obliquity of the ecliptic ( ε) about the x-axis: cos α cos δ x cos l cos b cos l cos b sin α cos δ = R x ( ε) y = 0 cos ε sin ε sin l cos b = sin l cos b cos ε sin b sin ε, sin δ z 0 sin ε cos ε sin b sin l cos b sin ε + sin b cos ε (B.1.2) so that and tan α = sin α cos δ cos α cos δ = sin l cos b cos ε sin b sin ε cos l cos b = sin δ = sin l cos b sin ε + sin b cos ε. sin l cos ε tan b sin ε, (B.1.3) cos l (B.1.4) For the inverse transformation, from equatorial to ecliptical coordinates, we need to rotate in the opposite direction: Rx 1 ( ε) = R x (ε) = 0 cos ε sin ε. (B.1.5) 0 sin ε cos ε Hence, from which we find and cos l cos b cos α cos δ cos α cos δ sin l cos b = R x (ε) sin α cos δ = sin α cos δ cos ε + sin δ sin ε, sin b sin δ sin α cos δ sin ε + sin δ cos ε tan l = sin l cos b sin α cos δ cos ε + sin δ sin ε = = cos l cos b cos α cos δ sin b = sin δ cos ε sin α cos δ sin ε. (B.1.6) sin α cos ε + tan δ sin ε, (B.1.7) cos α See also Urban & Seidelmann (2012) Sect and their Equations and (B.1.8) 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, keeping the polar angle (latitude, declination, etc.) constant, we would intuitively expect that we can simply subtract the rotation angle from the phase angle. Indeed, this follows from the general 3D rotation matrices derived in Sect. A.4. 21

22 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 rotation angle θ is simply added to α. We expect the declination δ to be unchanged, but we will call the new corresponding coordinate d to see whether this is indeed the case. Hence we will transform the coordinates (α, δ) to (H, d). Expressing our input and output positions in Cartesian coordinates, and using the rotation matrix for a rotation about the z axis from Eq. A.4.1, we get x x cos( θ) sin( θ) 0 x y = R z ( θ) y = sin( θ) cos( θ) 0 y, z z z cos H cos d cos θ sin θ 0 cos( α) cos δ cos α cos δ cos θ + sin α cos δ sin θ sin H cos d = sin θ cos θ 0 sin( α) cos δ = cos α cos δ sin θ sin α cos δ cos θ. sin d sin δ sin δ Thus, and tan H = y cos α sin θ sin α cos θ x = cos α cos θ + sin α sin θ sin d = z = sin δ, so that we indeed find that H = α + θ and d = δ, as expected. sin(θ α) = = tan(θ α) cos(θ α) 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 with respect 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 the geographical location of the observer (λ obs, ϕ obs ) and, even for fixed objects like stars, the time. One can imagine that for an observer standing on the Earth s North Pole (ϕ obs = π/2 = 90 ), the equatorial plane of the Earth or the sky and the local horizontal plane coincide. Hence, the calculation of horizontal coordinates can be most easily be done from equatorial coordinates by a single rotation that takes into account the observer s latitude and perhaps a second rotation about the local vertical axis 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 northern hemisphere) and the positive x-axis points there, the rotation simplifies to a single rotation around the y-axis over an angle π/2 ϕ obs. We can find the corresponding rotation matrix from Eq. A.4.1: ( π ) cos ( π 2 R y 2 ϕ ϕ ( obs) 0 sin π 2 ϕ ) obs obs = sin ( π 2 ϕ ) obs 0 cos ( π 2 ϕ ) obs = sin ϕ obs 0 cos ϕ obs cos ϕ obs 0 sin ϕ obs. (B.3.1) Hence, the coordinate transformation becomes: cos A cos h sin ϕ obs 0 cos ϕ obs cos H cos δ cos H cos δ sin ϕ obs sin δ cos ϕ obs sin A cos h = sin H cos δ = sin H cos δ, sin h cos ϕ obs 0 sin ϕ obs sin δ cos H cos δ cos ϕ obs + sin δ sin ϕ obs 22

23 so that tan A = sin A cos h cos A cos h = sin H cos δ cos H cos δ sin ϕ obs sin δ cos ϕ obs = sin H cos H sin ϕ obs tan δ cos ϕ obs, (B.3.2) and sin h = cos H cos δ cos ϕ obs + sin δ sin ϕ obs. (B.3.3) For the inverse transformation, from horizontal to equatorial coordinates, we need the inverse matrix of Eq. B.3.1 (the same rotation, but in the other direction): cos H cos δ sin ϕ obs 0 cos ϕ obs cos A cos h cos A cos h sin ϕ obs + sin h cos ϕ obs sin H cos δ = sin A cos h = sin A cos h, sin δ cos ϕ obs 0 sin ϕ obs sin h cos A cos h cos ϕ obs + sin h sin ϕ obs so that tan H = sin H cos δ cos H cos δ = sin A cos h cos A cos h sin ϕ obs + sin h cos ϕ obs = sin A cos A sin ϕ obs + tanh cos ϕ obs, (B.3.4) and sin δ = sin h sin ϕ obs cos A cos h cos ϕ obs. (B.3.5) 23

24 References Bretagnon, P. & Francou, G. 1988, A&A, 202, 309 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 Gueymard, C. 1995, Professional Paper FSEC-PF Gueymard, C. A. 2001, Solar Energy, 71, 325 IERS. 2015, International Earth Rotation and Reference Systems Service Kovalevsky, J. & Seidelmann, P. K. 2004, Fundamentals of Astrometry libsufr , A library containing Some Useful Fortran Routines libthesky , A Fortran library to compute the positions of and events for celestial bodies with great accuracy 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 (3rd Edition) (University Science Books) Wikipedia. 2017, Rotation matrix Wikipedia, The Free Encyclopedia, [Online; accessed 18-Feb- 2017] matrix Young, A. T. 1994, ApOpt, 33,