Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background. Vry littl, if any, ffort is put into dscribing th mathmatical background of why th quations work or whr thy com from. This documnt is to hlp thos studnts that hav a littl linar algbra knowldg s on mthod for driving thos common quations. Laying th Foundation: Pr-Coprnican Assumptions To start th drivation w will assum that w ar locatd at th cntr of th solar systm and that th sun orbits us at a constant angular vlocity and complts on full cycl vry hours. Lt s dfin our right-handd orthonormal basis vctors s, s 2, and s as th solar systm coordinat basis and also dfin θ such that th th sun s position is masurd clockwis from s. s s 2 θ In this imag, s is going out of th pag. This is st up as though w ar looking down on th north pol of a non-rotating arth with s 2 pointing wst pick a
longitud and s pointing up towards Polaris. For this stup, w assum that at midnight th sun is in th s dirction θ = and at noon th sun is at θ = π, and so on. So, for a hour day w dfin θ by θ = whr h is th local solar tim th tim that you would b using if not for daylight savings tim and timzons. By dfinition, h = 2 at noon and h = at midnight. This dfinition maks it asy to calculat th s, s 2 dirction of th sun at any givn tim making sur to tak into account our particular dfinition of θ cos θ s s = sin θ 2 whr s s is th vctor pointing to th sun in th solar systm coordinat basis. Th vctor in Equation 2 assums that th sun s dclination is zro which is tru during an quinox, but is not tru gnrally. Th sun s dclination δ is a minimum at th wintr solstic δ = 2.45 and maximum at th summr solstic δ = 2.45. Taking into account th dclination of th sun, w hav a polar coordinat systm in θ and δ. W can writ s s as a unit vctor in standard cartsian-from-polar form cos δ cos θ s s = cos δ sin θ sin δ Notic that w dfin it in such a way so that a positiv dclination in th summr givs a positiv s displacmnt. B sur to giv dclination in radians if your trigonomtric functions rquir it. For convninc, hr is a vry approximat quation for solar dclination in dgrs for a givn dat δ = 2.44 cos 6 N + 4 65 whr N is th day of th yar. Th maximum rror is approximatly dgr. Rotating to Account for Obsrvr Latitud W now nd to tak into account th fact that w ar locatd on th surfac of a clstial body that can b approximatd as a sphr. Bcaus w ar found at diffrnt latituds w nd to prform a basis transform on th s s vctor givn in Equation to giv us a vctor in our own basis for our givn latitud rprsntd by φ. 2
s s φ In this imag, 2 and s 2 ar pointing into th pag. From this imag w can radily s that w nd to rotat s s about s 2 which is qual to 2. Th gnral rotation tnsor for a rotation about 2 of angl α is givn by cos α sin α R = 5 sin α cos α For a prson locatd at th quator w would nd to rotat by α = π/2. For a prson at a givn latitud φ th rotation is α = π/2 φ. If w plug this dfinition of α into Equation 5 and simplify, w gt sin φ cos φ R = 6 cos φ sin φ Thn, w can gt our nw, rotatd s s vctor by sin φ cos φ cos δ cos θ s = R s s = cos δ sin θ cos φ sin φ sin δ Th vctor s can b writtn out only using basic trms as sin δ cos φ + cos δ sin φ cos s = cos δ sin sin δ sin φ cos δ cos φ cos which still rtains th unit-vctor proprty bcaus w simply rotatd a unit vctor. This is our hliostat vctor that always points towards th sun, vn at night 7 8
Altitud and Azimuth W can now calculat th altitud and azimuth of th sun for a givn hour. θ alt = sin s 9 s θ az = 2π tan 2 s Th factor of 2π in th dfinition of th azimuthal angl is to account for th fact that th arctangnt will giv positiv angl valus for countrclockwis angls whil azimuth is gnrally masurd in th clockwis dirction from north. Sunris and Sunst W can comput whn th sun will ris and st or find out if th sun will ris or st at all during a givn day by looking at Equation 8 and rcognizing that th sun is up whn s is positiv and ngativ whn th sun is down. To find whn th sun riss or sts, w st s = and solv for h: sin δ sin φ cos δ cos φ cos = tan δ tan φ = cos 2 h = 2π cos tan δ tan φ Bcaus of th arc-cosin w hav at most two possibl solutions on th domain of h on full day. Th two solutions ar givn by h ris = 2π cos tan δ tan φ 4 h st = 2π cos tan δ tan φ = 2π cos tan δ tan φ 5 Whn tan δ tan φ < th sun nvr riss and whn it is gratr than on th sun nvr sts. Casting Shadows If w want to dtrmin th lngth and dirction of a shadow cast by an objct w nd only dfin a vctor rprsnting th objct such as a prson standing or th gnomon of a sundial and th normal to th surfac onto which w want 4
to projct th shadow. For xampl, if w want to know about th shadow of a prson that is 8cm tall w dfin th objct vctor x as x = 6 8 and th normal to th plan as n =. Thn, th shadow vctor q is givn by Exampl q = x sn x s n 7 As an xampl, if a prson that livs at φ = 42 Chicago, IL and is 8cm tall gos outsid on th summr solstic δ = 2.45 at 2:pm h = 4, local solar tim th shadow will xtnd 49.5cm to th North and 96.cm to th East. Th closd-form solution for an objct of unit hight is sin δ cos φ + cos δ sin φ cos q = sin δ sin φ cos δ cos φ cos cos δ sin 8 Exampl 2 If w want to dtrmin th hour lins for a classical horizontal dial w nd to dtrmin th angl btwn th shadow of th gnomon and th noon mark. For this w nd to dfin cos φ x = 9 sin φ n = 2 For this spcial cas whr th gnomon is in lin with th axis of rotation of th arth and w ar only intrstd in th dirction not lngth of th shadow, w can assum that δ =. Thn th quation of q simplifis to cos φ q = tan φ tan 2π h 2 Th angl γ btwn th shadow and th noon mark can b found by tan γ = tan φ tan cos φ tan γ = sin φ tan 22 2 which is th sam quation that is givn in lmntary sundial books th ngativ sign can b ignord if all you want is th magnitud of th angl. 5