Eight Minutes and a Half Gabriele U. Varieschi, Lyla Marymunt University, Ls Angeles, CA It takes abut eight minutes and a half fr the light frm the Sun t reach us, therefre when we lk at a beautiful sunset and we see the Sun at the hrizn the Sun is actually nt there anymre, it s already belw the hrizn! The real sunset happened eight and a half minutes earlier! Similarly, at sunrise, the Sun seems t be at the hrizn, but is already up in the sky, due t the same time delay. I first heard this statement frm a friend, a frmer clleague in physics teaching wh fund it in different publicatins, ranging frm physics textbks, general science bks and astrnmical magazines. At first, the quted statement seems perfectly reasnable. After all if smething happens n the Sun, fr example if new sunspts were t develp n its surface r if fr sme strange reasn the Sun suddenly were t turn green r purple, we wuld actually bserve these events with a time delay f apprximately eight and a half minutes. Or ne can think f apparently similar situatins, such as bserving the lights f a mving car at night, with the car suddenly turning arund a crner and disappearing frm sight. The time delay fr the light t reach us wuld be practically minimal, but the car is effectively already arund the crner when its last light reaches us. A quick infrmal pll f sme f my students revealed that mst f them tend t agree with the statement, fr reasns similar t thse I just mentined. Hwever, a clser inspectin f the statement reveals the real hidden issue: a prblem f rtating vs. inertial reference systems. In this paper I will analyze this prblem and describe a simple experimental activity which can als be used as a demnstratin f the peculiar kinematical effects f rtating systems. The descriptin f the prblem T simplify the prblem, we can avid unnecessary cmplicatins by taking the Sun t be a pint-like surce f radiatin, s that there is n ambiguity in the meaning f sunrise r sunset: these events happen when we bserve the pint-like Sun exactly at the hrizn, i.e., alng a gemetrical tangent line t the Earth s surface, at ur pint f bservatin. Let s remve als any ptical effect frm the prblem: n light refractin in the atmsphere (let s remve the atmsphere altgether, if we prefer), n ther light bending f any srt due t gravitatinal fields. We can simply assume that the radiatin frm the Sun travels in perfectly straight lines at the speed f light in vacuum. Sunset and sunrise are essentially due t the daily rtatin f the Earth arund its axis, with a perid we can take t be exactly 4 hurs. Let s assume that the time delay fr the light t reach us frm the Sun is exactly 8 minutes (we dn t need the extra few secnds). An elementary calculatin shws that in 8 minutes the Earth rtates an angle f degrees, equivalent t an apparent rtatin f the Sun by the same amunt in the ppsite directin. If the statement quted at the beginning is crrect, we wuld cnclude that at
sunset the Sun is already belw the hrizn by an angle f tw degrees, a quite significant change in psitin. The rbital revlutin f the Earth arund the Sun will als affect the angle f rtatin, but its effect is minimal cmpared t the ne just discussed and will be neglected here. After all, if the Earth did nt rtate arund its axis, ur day wuld last ne full year and the apparent rtatin f the Sun in eight minutes wuld amunt t less than 0 secnds f a degree, a very minimal crrectin cmpared t the degrees calculated abve. Therefre, we can simply assume a fixed Sun, behaving like a pint-like surce f radiatin, with the emitted light traveling in straight lines and cnsider the Earth as a rtating sphere, with a perid f 4 hurs, receiving light frm the Sun, with the phtns taking abut 8 minutes t cmplete their trip. At this pint every physics student shuld knw hw t interpret the prblem. The fixed Sun reference frame can be assumed t be an inertial ne; the Sun is nt mving and is emitting prjectiles (phtns) which travel radially utward at cnstant speed. The Earth represents a rtating, nn-inertial system; therefre we shuld be careful when bserving and interpreting phenmena in such nn-inertial systems. The situatin as seen frm the inertial system f the Sun is extremely simple and is illustrated in Figs. a and b. In this reference system (x-y-z axis), the psitin f the Sun (S) is fixed, while the Earth bserver (pint T) is rtating tgether with the lcal directin f the hrizn, indicated in the figures by the blue dashed line, tangent t the Earth s surface at pint T. We chse t place the terrestrial bserver T n the equatr fr simplicity, but any ther latitude wuld give the same results, just with a radius R smaller than the equatrial radius (f curse at latitudes beynd the Arctic r Antarctic Circles we might have t wait a little lnger than usual t bserve a new sunrise r sunset). These figures shw the view frm the Nrth Ple, i.e., with the Earth rtating cunter-clckwise arund the z axis (perpendicular t the plane f the figures). We als indicate with x -y - z the rtating system, where the z and z axis (cming ut f the page) cincide with the axis f rtatin. Fig. a shws the situatin eight minutes befre sunset : at this time the last rays t be bserved n Earth depart frm the Sun, but the Sun is still well abve the hrizn. In fact, the angle between the hrizn directin at pint T and the directin f arrival f sunlight (dashed red line) crrespnds t the degrees angle calculated abve (and greatly exaggerated in Fig. a). In Fig. b we illustrate the sunset situatin. The light rays emitted eight minutes earlier finally arrive t the bserver at psitin T and their directin f arrival is aligned perfectly with the hrizn directin: at sunset the Sun psitin is precisely at the hrizn and nt belw it! Similar reasning wuld bviusly apply fr sunrise events, but we prefer t cntinue ding all ur examples just with sunsets. Althugh the riginal statement already appears t be incrrect, ne might argue that it was referring t an bserver n Earth, s we shuld analyze the prblem in the rtating reference system instead and illustrate the situatin als frm this pint f view. Fig. shws the path f the last light frm the Sun as seen frm the bserver rtating with the Earth. Frm this perspective the apparent rtatinal mtin f the Sun (with a 4 hur perid) is a clckwise mtin, when viewed frm the Nrth Ple.
Fr the nn-inertial bserver at psitin T, the path fllwed by the light is nt straight anymre, but is a spiral-like curve starting frm the riginal psitin f the Sun (pint A), eight minutes befre we bserve sunset, and ending with light reaching pint T eight minutes later, with a directin f mtin (given by the tangent t the path at pint T) cinciding with the hrizn directin, s that in ur rtating system we perceive the light at sunset as cming straight twards us frm the hrizn. And where is the Sun at sunset, frm ur rtating perspective? In thse eight minutes it tk the light t reach us at sunset, the Sun mved clckwise frm psitin A t psitin B, as seen in Fig., s it is exactly at the hrizn when we receive the last light. Any light emitted frm the Sun at psitin B (r frm any intermediate psitin between A and B) wuld nt reach the bserver at psitin T any mre. It wuld have t fllw a similar curved path reaching the Earth with a final directin f mtin belw the hrizn, therefre wuld nt be seen by the bserver at T. In any case, frm bth perspectives, the Sun is exactly at the hrizn at the precise mment f sunset r sunrise. A mre analytical apprach The analysis f the previus sectin can be made mre quantitative by using rtating crdinate systems. As illustrated in Fig. a and b, we cnsider the Sun as pint-like surce (S) lcated at a distance D f ne astrnmical unit, D = AU =.496 0 m, 6 and ur planet as a sphere f radius R = 6.378 0 m, equal t the Earth s equatrial 5 radius. The rati R / D = 4.6 0 is therefre very small and ur figures are bviusly nt up t scale. Fr ur prblem it is cnvenient t use a very particular set f units: distances will be measured in astrnmical units (AU), time in minutes (min) and angles in degrees ( ). Assuming that the time needed fr light t travel the Sun-Earth distance is exactly eight minutes, the speed f light c and the angular velcity ω f ur planet can be expressed by very simple numbers: 3 AU c = 8 min ; ω = 4 min, () s that, as already remarked, in eight minutes the Earth rtates an angle f tw degrees. We will als take t = 0 t be the time f sunset, i.e., when the Earth bserver receives the last light at psitin T. This light left the Sun eight minutes earlier, therefre at the initial time t0 = 8 min. At this time we will assume that the x -y -z system is rtated by an initial angle α 0 =, with respect t the fixed x-y-z system, arund the cmmn z, z axis. In this way, eight minutes later at time t = 0, the tw systems will cincide at sunset. The unifrm rtatin f the primed system is thus described by the angle f rtatin α: α = α 0 + ω( t t0) = + 4 ( t + 8) () and the equatin f mtin f the light prjectiles in the directin tangent t the Earth s surface, in the inertial system, describes a simple unifrm mtin in a straight line: x = D c( t t0) = 8 ( t + 8) (3) y = R 3
when expressed in the particular units we have intrduced abve. With this chice f units we can cmpare the expressins in Eqs. () and (3) and find a useful relatin between x and α: x = α. (4) At this pint we can determine the equatins f mtin f light in the rtating system x -y -z and shw that it is indeed fllwing the curved trajectry shwn in Fig.. The cnnectin between the primed and nn primed crdinates is given by a simple rtatin arund the z, z axis, by the angle α: x' = csα x + sinα y = α csα + Rsinα (5) y' = sinα x + csα y = α sinα + R csα where we used Eqs. (3) and (4) t express the crdinates x, y as a functin f a cmmn parameter α. These parametric equatins f mtin, when pltted fr α varying between α = (the initial angle) and α = 0, will reprduce the curve in Fig.. T plt this slutin it is actually easier t cnsider plane plar crdinates r, φ : r' = x' + y' = 4 α + R y' α sinα + R csα (6) ϕ' = arctan = arctan x' α csα + R sinα where again the parameter α varies frm t 0. It is easy t check frm Eq. (6) that at sunset ( α = 0 ) the light beam reaches the Earth bserver at T ( r ' = R and ϕ'= 90 fr dr α = 0 ) and that its directin f arrival is aligned with the hrizn directin ( dt' = 0 fr t = 0 ). Alternatively, ne can use the Cartesian cmpnents f the velcity in the primed crdinates, btained frm the time derivatives f Eq. (5), t study the directin f mtin at any pint alng the trajectry. It is als easy t check the direct cnnectin r r r r between the velcities in the tw systems, given by the classic relatin v = v' + ω and describe ur prblem as a classic Crilis Effect. The expressin in Eq. (6) describes a spiral-like curve, which is a typical result when a unifrm mtin in a straight line is seen in a unifrmly rtating frame f reference. Eq. (6) can be brught int the classic frm f an Archimedean spiral by neglecting the Earth radius R << AU, which is much smaller than the Sun-Earth distance. In general, an Archimedean spiral is the curve traced ut by a pint that mves at cnstant velcity v alng a rd that is rtating abut the rigin at a cnstant angular velcity ω. Its equatin in plar crdinates is r = a ϕ, a = v / ω > 0, < ϕ < +, cmpsed f tw different branches fr psitive r negative values f φ. In ur case, fr R=0, Eq. (6) reduces t r ' = x' + y' = α, ϕ' = α, since the α parameter has negative values, thus btaining the spiral curve r = ϕ', (7) ' v c /8 which beys the general expressin since in ur case a = ω = ω = / 4 =, in ur chice f units. Finally, it is pssible t reduce Eq. (6) t a mre cmpact expressin, withut any apprximatin. The first part f Eq. (6) can be slved fr α = r' R, where the 4
minus sign again is chsen because we use a negative α in the parameterizatin f ur curve. Using this last expressin and the secnd part f Eq. (6) we btain: tanϕ' α sinα + Rcsα R R sin ϕ' = = = sinα + csα. (8) ' r' r + tan ϕ' α + R 4 R R Since R / r', we can set sin β =, csβ = and use trignmetric relatins t r' rewrite Eq.(8) as sinϕ' = sin β csα csβ sinα = sin( β α), frm which we finally deduce: R ϕ ' = β α = arcsin + r' R (9) r' which is the mst cmpact expressin f the light trajectry in the rtating system, directly cnnecting the plar crdinates r and φ and will als reduce t the spiral f Archimedes fr R 0. r' A sunset-sunrise demnstratin The discussin presented abve suggests a very simple experiment which can be used as an effective in-class demnstratin f these rtatinal effects r even becme part f a mre structured labratry activity. Our experiment is an adaptatin f the standard Crilis effect Ball n rtating platfrm demnstratin, 4 cmbining tgether tw very basic pieces f equipment frm intrductry mechanics labs: a rtating platfrm and an inclined plane which serves as a prjectile launcher. A small metal ball is launched frm the inclined plane ver the rtating turntable and will represent ur beam f light traveling at almst cnstant velcity in the fixed frame f reference. 5 The Earth is represented n the turntable by a green circle (see Fig.3) and the prjectile is launched alng a tangential directin t the circle in the fixed frame f reference. This fixed directin is represented by a meter stick attached t the prjectile launcher (visible in the upper right crner f the figure). A small range circle, representing the Sun, is attached t the fixed inclined plane (its psitin is better identified by the red dts in the figure). A vide camera is munted n tp f the rtating platfrm and recrds the view f the rtating bserver, as seen in the x -y plane. We filmed the mtin f the prjectile in the rtating frame and then used vide editing sftware t prduce a strbscpic picture f the mtin. Figure 3 illustrates ne f the pictures we btained, shwing a clse resemblance t the spiral curve pltted in Fig. fllwing Eq. (6) r (9). The metal ball is launched when the Sun is at psitin A and then reaches the bserver at T when the Sun is apprximately at pint B, therefre aligned with the hrizn directin as expected. Althugh similar rtating platfrm demnstratins are described by many papers and articles, 6 we are nt aware they have ever been used t illustrate ur simple sunrisesunset prblem, which culd therefre represent a new way t intrduce the Crilis Effect and related tpics. 5
Relativistic analysis f the prblem A final pint needs t be cnsidered in the analysis f the riginal prblem. We have used cncepts f nn-relativistic kinematics f rtating systems t study the mtin f a beam f light. Shuld we cnsider crrectins arising frm special relativity? The relativistic treatment f rtating systems and their cnnectin t inertial nes is a tpic which is seldm discussed in standard relativity textbks. We fund a cmplete analysis in a classic bk by H. Arzeliès. 7 In a relativistic apprach, when a rigid bject is rtating at cnstant angular velcity ω, we need t ensure that all the linear velcities f all the pints f the bject d nt exceed the speed f light, v = rω < c, fr all the crdinates r f the bdy, and this is bviusly the case f ur rtating planet. Then, at any given instant, an infinitesimal element P f the bdy can be apprximately regarded as an inertial system mving with velcity v = rω with respect t the fixed system and standard Lrentz transfrmatins, length cntractin and time dilatin effects will apply. In particular, the radial crdinate, perpendicular t the instantaneus velcity f the element P, will nt be cntracted, i.e., r ' = r, but any tangential length will be affected. Fr example, the circumference length l is Lrentz cntracted: ' c c c v ( rω) ( r' ω) l = l / = πr / = πr ' / > πr ', (0) where, as befre, primed quantities refer t the rtating system as ppsed t nn-primed nes referring t the fixed system. Simple Euclidean gemetry wuld therefre nt apply in the rtating frame: the circumference t radius rati wuld be bigger than the standard π factr, as seen frm Eq. (0). On the cntrary, the derivatin f the light trajectry in rtating systems is nt affected at all by relativistic crrectins; this is mainly due t the invariance f the radial crdinates r ' = r mentined abve. An alternative derivatin f the light trajectry starts by nting that the equatin f mtin f ur beam f light is simply R = r sinϕ, using plar crdinates in the fixed system (see Fig. b) and the cnnectin between plar angles φ and φ is ϕ = ϕ' + α = ϕ' + α + ω( t ) = ϕ' + ωt = ϕ + t () 0 t 0 ' 4 The time crdinate in the last equatin can be eliminated by nting that in the inertial system the speed f light is fixed t c=/8 in ur units and therefre the distance traveled by the light in time t is: ct = ' R 8 t = r R = r. () The negative sign cmes frm the directin f the beam f light and we can exchange the r, r crdinates as mentined abve. Using Eqs. () and (), we can btain the trajectry in terms f r and φ : R = r sinϕ = r'sin( ϕ' + 4 t) = r'sin( ϕ' r' R ), (3) which, slved fr φ, gives exactly the same result f Eq. (9) and therefre represents the same slutin in ur riginal Eq. (6). 6
This alternative derivatin is ttally equivalent t the previus ne (and actually simpler), but we emphasize that it is als an exact slutin when relativistic effects are taken int accunt, i.e., it wuld be crrect even if the Earth were spinning very fast, with pints n its surface reaching relativistic speeds. We will leave all further details f the relativistic analysis t the cited textbk, 7 but we can cnclude that even in a fully relativistic treatment f the prblem the Sun at sunset r sunrise is precisely where we see it: at the hrizn and nt belw r abve it! Cnclusin An apparently simple questin abut phenmena that we witness every day, such as a beautiful sunrise r sunset, can be useful t intrduce in-class discussin n rtatinal frame f references, apparent mtin, Crilis Effects and even mre advanced tpics in relativity. A very simple demnstratin can be easily assembled fr further illustrating the prblem, which might als lead t a mre structured lab activity. Acknwledgments This research was supprted by an award frm Research Crpratin. The authr wuld like t acknwledge his friend Prf. G. Tnzig, wh reprted the riginal statement debated in this paper, and his clleague Dr. J. Phillips fr useful discussins. References. One f the riginal surces reprting the statement was: P. Carlsn, Du und die Natur, Ullstein, Berlin, 934; a German physics bk which was widely ppular in many cuntries and was cnsidered an educatinal mdel. A similar statement appeared als in translated versins f the riginal bk by A. Einstein and L. Infeld, The Evlutin f Physics, Simn & Schuster, New Yrk, 938. This prmpted several articles t appear in Eurpe, debating this ptential errr by the mst influential physicist f mdern times. Hwever, I persnally checked many different English editins f this (nce very ppular) bk and never fund such a statement. Einstein and Infeld simply refer t the time delay f sunlight as an example f events that happen in the past, but are bserved at later times. The statement abut the psitin f the Sun at sunset r sunrise was prbably intrduced by un-scrupulus bk translatrs. Similar incrrect statements can als be fund n sme astrnmical websites fr the general public.. The definitin f sunrise r sunset is generally related t the leading r trailing edge f the Sun passing the hrizn. This is different frm the s-called gemetric rise r set f a celestial bdy, which is when the center f the bject passes the hrizn and there is n atmspheric refractin. Due t the large apparent diameter f the Sun it can take up t ne r tw minutes, at mst latitudes, between the gemetric rise/set and the leading/trailing edge passing the 7
hrizn. The atmsphere actually bends light dwn tward the surface when a celestial bject is near the hrizn allwing us t see the Sun r ther stars befre they wuld nrmally be visible if there were n atmsphere. 3. In this way we are ver-estimating a little the speed f light in vacuum 8 c = AU / 8min = 3. 0 m / s, as ppsed t the standard value f 8 c =.9979458 0 m / s, but this des nt alter the physical explanatin f the prblem. 4. Physics Instructinal Resurce Assciatin (PIRA) demnstratin E30.8, Crilis effect - Crilis ball n turntable and references therein. See PIRA web page at http://www.wfu.edu/physics/pira/. Vide recrdings f this demnstratin and references are als available at the University f Maryland Physics Lecture Demnstratin Hme Page: http://www.physics.umd.edu/lecdem/services/dems/demsd5/d5-.htm. 5. Strictly speaking the rlling mtin f a sphere n a hrizntal turntable cannt be described just in terms f fictitius frces, due t the rtating frame f reference. Frictin is als playing a rle, but will nt change the trajectry significantly. See the discussin in: Rbert H. Rmer, Mtin f a sphere n a tilted turntable, Am. J. Phys. 49, 985 (Oct. 98). 6. See fr example: Rbert H. Jhns, Physics n a Rtating Reference Frame, Phys. Teach. 36, 78 (Mar. 998); Richard Andrew Secc, Crilis- Effect Demnstratin n an Overhead Prjectr, Phys. Teach. 37, 44 (Apr. 999); Andreas Wagner, et al., Multimedia in physics educatin: a vide fr the quantitative analysis f the centrifugal frce and the Crilis frce, Eur. J. Phys. 7, L7 (006). 7. H. Arzeliès, Relativistic Kinematics, Pergamn Press, Oxfrd, 966. See chapter IX The Rtating Disc- fr a cmplete relativistic analysis f light paths in rtating frames. PACS cdes: 0.55.+b; 45.40.Bb; 0.50My. Keywrds: sunset and sunrise; rtatinal kinematics; rtating reference frames; Crilis Effect. Gabriele U. Varieschi is an Assciate Prfessr f Physics at Lyla Marymunt University. He earned his Ph.D. in theretical particle physics frm the University f Califrnia at Ls Angeles. His research interests are in the area f astr-particle physics and csmlgy. Department f Physics, Lyla Marymunt University, LMU Drive, Ls Angeles, CA 90045; gvarieschi@lmu.edu 8
FIGURES: Fig.. The view frm the fixed inertial system: a) The Sun is emitting its last light eight minutes befre sunset (t = -8 min). At this time the Sun is well abve the hrizn directin, by an angle f tw degrees (greatly exaggerated in the figure). b) The situatin at sunset (t = 0 min). The hrizn directin fr the terrestrial bserver at T is nw aligned with the directin f the incming last light frm the Sun. 9
Fig.. The view frm the Earth s rtating system. The last light is emitted frm the Sun at psitin A (t = -8 min) and appears t be fllwing a spiral-like path twards the terrestrial bserver at pint T. Hwever, the light is perceived as cming frm the hrizn directin at sunset (t = 0 min). At the same time the Sun is at psitin B, as seen frm the rtating system, therefre perfectly aligned with the hrizn. The distances D and R in the figure are nt up t scale. 0
y A R T Hrizn S B O x Fig. 3. A strbscpic picture f the mtin prduced by ur experimental apparatus, as recrded by a vide camera rtating tgether with the turntable. The resulting curved trajectry f the metal sphere, frm pint A t pint T, resembles clsely the light path described in Fig.. The psitin f the Sun, in each frame f the vide recrding, is indicated by the red dts in the figure.