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Modelling sundials: ancient and modern errors

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Three systems of celestial coordinates, the ecliptical, the equatorial and the horizontal, as well as their projection onto the shadow-receiving plane, define the geometry of sundials' construction. A new method is proposed to model planar
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  Modelling sundials: ancient and modern errors Irina Tupikova & Michael Soffel Lohrmann Observatory, TU Dresden ABSTRACT. Three systems of celestial coordinates, the ecliptical, the equatorial and the horizontal, as well as their projection onto the shadow-receiving plane, define the geom-etry of sundials’ construction. A new method is proposed to model planar sundials with arbitrarily oriented planes and shadow-casting parts based on a simple vector equation in combination with the application of a sequence of rotational matrices. This method allows one to draw the shadow maps for sundials with errors due to wrong determination of ge-ographical latitudes or erroneous construction. Applications for modelling some ancient sundials are considered and their shadow maps are discussed. Contents 1 Introduction 12 Mathematical solution 23 The sundial of Amphiareion 84 The Egyptian shadow-clock 135 Conclusion 21 1 Introduction According to J. Needham, 1 three systems of celestial coordinates - the equatorial, the ecliptical and thehorizontal - were used preferentially for surveying purposes by the Chinese, Greek and Arabic culturesrespectively. This point of view now appears to be oversimplified (Hipparchus’ system was, e.g., un-doubtedly equatorial); what one can, however, say with certainty is that the spatial thinking of thesecivilizations was impacted by their respective preferred coordinate systems. In particular, however, themanufacturing of sundials, which has been observed in all three cultures, requires working knowledge andunderstanding of the interplay between all three systems:- the equatorial system (because the visible daily motion of the Sun lies on a circle parallel to theequator),- the ecliptical system (because the annual motion of the Sun lies on the ecliptic),- the local horizontal system (because the horizon constrains the visible path of the Sun).The orientation of the equatorial plane relative to the local horizontal plane is given by the angle 90 ◦ − ϕ ,where  ϕ  is the geographical latitude of the location. Thus, the correct determination of the latitude of theplace where a sundial was to be installed was of crucial importance. Whereas the geographical latitudes 1 “ Astronomy in ancient and medieval China  ”, Phil. Trans. R. Soc. Lond. A.  276 , 67–82, 1974. 1  of famous localities were known, 2 the locations of smaller cities had to be guessed or extrapolated on thebasis of distances to the known cities. This was, in fact, the primary purpose of geographical mappingin antiquity.The inclination of the ecliptic relative to the plane of the celestial equator, approximately 23 . 5 ◦ , waswell known in antiquity. In the analemma construction of Vitruvius, 3 it was taken to be 24 ◦ for purelygeometrical reasons – the central angle over the side of a regular 15-gon was easy to construct by compassand straightedge and therefore seen as a convenient and elegant approximation. The correct orientationof a sundial relative to the north-south direction as well as the calibration of the daily and hourly curves,however, demanded the usage of astronomical methods.The local horizontal plane could be determined to a very high degree of precision with various technicaltools available in antiquity and used primarily in architecture. 4 The direction perpendicular to the localhorizon (that is, the local zenith direction) is even simpler to determine as the upward direction of theplumb line.Finally, the shadow map of a sundial depends on the position of the shadow-receiving plane whichshould be specified relative to the coordinate systems involved. The mathematical tools would therefore beapplied to the engineering solution based on the information provided by geographical and astronomicaldata. Because the visible daily motion of the Sun also determines the position of the hour lines onthe shadow receiving plane, we have chosen the equatorial plane as the primary reference plane forconstructing a mathematical model for the different kinds of sundials. The position of the Sun in themodern equatorial coordinate system is given by the declination  δ   relative to the equatorial plane andthe hour angle  h  measured towards the west from the meridian transition (that is, culmination) of theSun (see Fig. 1). NSNCP δ h Figure 1: Coordinates in equatorial system: declination  δ   and hour angle  h .The method will be applied to two different sundials: the equatorial sundial of Amphiareion and theold Egyptian shadow-casting instrument. 2 Mathematical solution Let us consider a standard equatorial sundial - that is, one with the shadow-receiving plane lying parallelto the celestial equator and the shadow-casting part (gnomon) parallel to the rotational axis of the Earth.The coordinate system used in the mathematical solution which will be proposed in the text is illustratedin Fig. 2. 2 The geographical latitudes were determined (and unambiguously defined in this way) as the ratio of the length of agnomon to that of its shadow at equinox, or as the ratio between the length of the longest day of the year and the shortestday. To find out the geographical longitudes, on the other hand, one requires either simultaneous astronomical observationsof eclipses at different locations or knowledge of the circumference of the Earth together with the directions and distancesbetween the localities. 3 On Architecture  , IX, 7. 4 See, e.g., Lewis M. J. T.,  Surveying Instruments of Greece and Rome  , Cambridge University Press, 2001. 2  EWNSxyzNCP 90° -  Figure 2: Orientation of an equatorial sundial relative to the horizontal plane. The gnomon is directedalong the rotational axis of the Earth towards the north celestial pole (NCP). The geographical latitudeof the location is  ϕ .We will now introduce a  left-handed   Cartesian coordinate system with the x-axis directed towards thesouth, the y-axis towards the west and the z-axis directed towards the northern celestial pole. Let usfurthermore introduce a unit vector in the gnomon’s direction, e g  =  001  , and a unit vector in the Sun’s direction defined in the standard way in terms of the Sun’s equatorialcoordinates as e s  =  cos δ  cos h cos δ  sin h sin δ   . While the orientation of the gnomon vector remains constant, the vector in the Sun’s direction will,naturally, change its orientation with time. The equation for a sunbeam that, coming from the Sun, goesthrough the gnomon’s tip can be written in vector form as x ( λ ; δ,h ) =  λ e s  + e g , where  λ  ∈ R  is a numerical parameter. The geometry of the problem is shown in Fig. 3. A sunbeam hits NSNCP δ e s e g x Figure 3: Coordinate system used to obtain a vector solution for a shadow model.the equator plane  z  = 0 when the  z -component of the vector  x  attains zero: λz s  +  z g  = 0 . 3  With  z s  = sin δ   and  z g  = 1, one obtains the corresponding value for  λ  : λ ∗ =  − 1sin δ . This value for  λ  allows us to calculate the  x - and  y -components of the end of the shadow in the equatorialplane as x ( λ ∗ ) =  λ ∗ cos δ  cos h  =  − cot δ  cos h,y ( λ ∗ ) =  λ ∗ cos δ  sin h  =  − cot δ  sin h. Combination of these two formulae gives the shadow’s equation x 2 +  y 2 = cot 2 δ, which describes a circle with radius  R  = cot δ   around the point  x  = 0 , y  = 0. The visualization 5 of thiswell-known result for an equatorial sundial placed at latitude 40 ◦ is shown in Fig. 4. At our latitudes, theFigure 4: Numerical results for an equatorial sundial mounted at latitude 40 ◦ . Left: summer side. Right:winter side.part of an equatorial sundial oriented towardsnorth (the summer side) exhibits circular shadow paths withlengths larger than corresponding semicircles, whereas the winter side (lying opposite) exhibits shadowpaths with lengths shorter than corresponding semicircles. This observation can be easily understoodwith the help of Fig. 5. The method described above can be easily adopted for planar sundials inclinedarbitrarily relative to the equatorial plane. A horizontal sundial, for instance, can be obtained with asimple rotation of an equatorial sundial around the y-axis by an angle  θ  =  − (90 ◦ − ϕ ) (Fig. 6). Thisrotation can be realized with the help of a rotational matrix  R y ( θ ), given, in our case, by R y ( θ ) =  cos θ  0 sin θ 0 1 0 − sin θ  0 cos θ  . Assuming that the gnomon is kept at a right angle to the shadow-receiving plane (normally, that will bethe case because right angles are the most simple to realize), the unit vector in the gnomon’s directionin the new coordinate system is given again by e 1 g  =  001  , 5 All the shadow maps in the text are normalized relative to the height of a gnomon adopted as unity. The 0-point liesat the basis of the gnomon. The computations are made with the help of the computer algebra system MAPLE 12; theprograms can be made available on request. 4  Horizondl90-  ϕϕ tan ϕ = d/l = ddVisible shadowsHorizonGGHorizondl90-  ϕϕ tan ϕ = d/l = ddVisible shadowsHorizonGG Figure 5: Visible shadow paths for an equatorial sundial mounted at latitude  ϕ  on the northern hemi-sphere. Left: summer side. Right: winter side. The distance  d  = tan ϕ  gives the displacement betweenthe line which bounds the visible shadow paths and the point  G  which lies at the basis of a gnomon withlength  l  = 1. The daily shadow curves are segments of circles with radius  R  = cot δ   around the basis of the gnomon. The Sun’s declination  δ   is assumed to remain constant over the course of a day. EWxyzNCP 90° - ϕ EWNSxyZ Figure 6: Transformation from an equatorial to a horizontal coordinate system by a rotation about they-axis by an angle of   − (90 ◦ − ϕ ).5
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