Journal of Appled Matheatcs and Physcs, 6, 4, 68-78 Publshed Onlne June 6 n ScRes http://wwwscrporg/journal/jap http://dxdoorg/436/jap646 The Ipact of the Earth s Moeent through the Space on Measurng the Velocty of ght Mloš Čojanoć Independent Researcher, Montreal, Canada Receed 6 Aprl 6; accepted 6 June 6; publshed 9 June 6 Copyrght 6 by author and Scentfc Research Publshng Inc Ths work s lcensed under the Create Coons Attrbuton Internatonal cense (CC BY http://createcoonsorg/lcenses/by/4/ Abstract Goal of ths experent s bascally easurng the elocty of lght As usual we wll easure twoway elocty of lght (fro A to B and back In contrast to the slar experents we wll not assue that speeds of lght fro A to B and fro B to A are equal To achee ths we wll take nto account Earth s oeent through the space, rotaton around ts axs and apply least squares ethod for cosne functon, whch wll be explaned n Secton 9 Assung that drecton East- West s already known, one clock, a source of lght and a rror, s all equpent we need for ths experent Keywords Speed of ght, One Way Speed of ght, east Squares Method for Cosne Functon Introducton Obsere the planet Earth The Earth orbts the Sun For ths oton we wll jon the ector Sun orbts the center of the Mlky Way For ths oton we wll jon the ector In relaton to the center of the Mlky Way, we can jon to the Earth oeent su of ectors + It s also known that our Gala s ong relate to other galaxes (or to a pont n the space outsde the Mlky Way Gala Slarly, to ths oton we could jon the ector 3 Denote by the su of all these ectors = + + 3 + ( At the end of the su three ponts are left, because eentually there ay be soe other oeents In the perod of 4 h ectors, 3 can be taken as constants, whle the ector by akng a certan error How to cte ths paper: Čojanoć, M (6 The Ipact of the Earth s Moeent through the Space on Measurng the Velocty of ght Journal of Appled Matheatcs and Physcs, 4, 68-78 http://dxdoorg/436/jap646
M Čojanoć could also be taken as constant Thus for the Earth s oton through the space wthn 4 h, we can jon the constant ector The speed and drecton Earth orbts the Sun are known, and let represent ts aarage speed Suppose that soe approxate alues for ectors and 3 are known as well On the bass of these alues, let suppose that we hae nequalty Plannng an Experent = + + 3 ( Suppose that an arbtrary pont A s gen Earth rotaton axs wll be taken as the coordnate, and as the plane we wll take the plane passng through pont A and perpendcular to the axs In ths case t s natural to take secton of the plane and axs as the center of the coordnate syste In addton to pont A let the ponts B and D are gen ne AB les n the plane and parallel to the drecton of the Earth s rotaton Dstance AB wll be arked wth For the x axs, at soe ntal te t, we wll take the lne n the plane, parallel to AB The projecton of the ector n the plane denote by Due to the Earth s rotaton the drecton of AB wll be changed, so that t wll be changed the angle, arked by Φ, between the x axs (whch reaned fxed and the lne AB et at pont A we hae a clock and soe source of lght Suppose that speed of lght n the drecton AB s gen by equaton ( cab = c cos Φ ( Pont D wll be chosen so the lne AD s parralel to drecton South-orth Dstance AD s arked by Angle between lne AD and axs we wll denote by ϕ Angle ϕ actually represents attude of pont A on the Earth s surface, thus t reans unchanged durng the experent The projecton of the ector on axs denote by (actually + 3, because s perpendcular on axs Assue that the speed of sgnal n the drecton AD s gen by equaton ( cad = c cos ϕ ( where c represents elocty of lght n acuu for a body at rest Our a s to fnd the constant c, ectors and 3 Conductng an Experent In soe oent T we wll send sgnal fro pont A to pont B The angle between the axs x and s arked by Θ Once the sgnal arred at pont B t wll be reflected back to pont A Dfference between the te when the sgnal was beng sent fro pont A, and the te when the sgnal reached to the pont A s denoted by t At the sae te we wll send sgnal fro pont A to D and return back to pont A Dfference between the te when sgnal was beng sent and reached to pont A we wll denote by τ The sae procedure wll be wthn 4 h repeated ( > 4 tes, whereas the te between the two sets of consecute procedure to be sae and equal to 4 h/ In that way we wll get the seres {t } and {τ } { t } { τ } { } To the each t we can jon an angle α between x axs and lne AB In that way we get the seres,,,, ( { α } where α = Π Θ { } By assupton (3 the speed of the sgnal c n the drecton AB s equal to and n opposte drecton BA,,,, ( ( AB = ( Π Θ c c cos (3 69
M Čojanoć It follows that ( BA = + ( Π Θ c c cos (4 t = + c *cos Π Θ c+ cos Π Θ ( ( (5 c t = cos Π Θ c ( If we swap the roles of the ponts A and B, we would get the sae forula as n (6 Therefore t s copletely rreleant whether drecton of the ector s equal to drecton AB or BA We assue that for { } Θ [ Π Π ],,,,, It would be n prncple our experent ( c cos Π Θ > t > 4 Coputng the Values of c, and Θ In ths secton we wll deal only wth the easureents n drecton East-West et t s gen by (36 and denote the aerage speed c (fro pont A to pont B and back to A It follows that c can be wrtten as where e represents soe experental error Replacng we get n short for c c =, {,,, } ( t ( cos Π Θ = c + e c ( Π Θ = ( ( Π Θ + cos cos * * c = c cos( ( Π Θ + e c c ( ( { } c = B A cos Π Θ + e,,,, (4 (6 ( (3 = B c c (5 The coeffcents A, B and Θ wll be chosen so the su of squares has a nu alue A = c, where A (6 ( ( Θ = = + ( ( Π Θ S BA,, e c B A cos (7 7
M Čojanoć To achee our goal we are gong to apply Theore for k = For the sake of splcty we e only consdered cases when a *cos( * α and A Thus we hae B = c = c (8 = = ( α a sn = tg ( Θ = (9 a cos ( α ( α a cos Θ = A = ( { } a,,,,, = c c α = Π We ll ake a sall dgresson Fro ea t follows In the slar way we can get ( α ( c cos( α = c cos ( k α c cos( k α = c cos ( k α a cos k = c k a ( k α = c ( k α sn sn Generally we hae tg( x = tg( x Π tg( Θ = tg( Θ Π Fro (9 Functon Atan ( takes alues at nteral ( Π/, Π/ ( α ( α a sn Θ = Atan ( a cos Θ =Θ Π If we consder A as functon of Θ A ( Θ = A ( Θ Π = A ( Θ Fro (6 t folows that between the alues Θ and Θ we hae to choose that one for whch A > Fro (5 and (6 we can dere alues for c and c= B + A = c + A ( =± A c (3 We don t know exact drecton of ector, thus poste and negate alue are assgned to 5 Coparson between Two Methods In ths secton we wll ake coparson between the least squares ethod and the least squares ethod for cosne functon et consder { c } gen by (4 as the seres of utually ndependent easureents c et c represents the ean alue of seral { } ( If we apply east squares ethod, Varance V s gen by c = c ( 7
M Čojanoć and standard deaton σ by ( V = c c ( σ = (3 V Suppose that to the each c we joned the te when easureent took place, or rather the angle between the drecton of AB and ector Expected alue E (α for The east squares ethod for cosne functon s gen by where Denote a by et us fnd Varance V for ths ethod ( α = = cos( ( α Θ E y B A k (4 { } α = Π,,,, (5 a = c B = c c ( ( a A cos( k ( α ( c A cos( ( α V = c y = c + k Θ = + Θ a A a cos( k ( α A = + Θ + A = a + A a cos ( k ( α Θ + A A = fro (5 = a = ( c c ( k ( α + cos Θ (6 A V = V V V Standard deaton σ for ths ethod s gen by (7 σ = (8 V * A Fro (7 σ σ Fro (7 V ( c c * σ A If standard deaton σ s bgger then soe expected alue t eans ether our easureent are not accurate enough or our ethod (cure doesn t sut to our data 6 Analysys of South-orth Measureents In ths chapter we wll deal wth the seres { τ } gen by (3 Just to rend that τ represents te t takes for sgnal to trael fro A to D and back to A n drecton South- orth τ = c cos( ϕ + c+ cos( ϕ ( τ = c c cos ( ϕ ( et 7
M Čojanoć γ {,,, } = τ denote the aerage speed γ In that way we get the seres {γ } cos (3 ( ϕ γ = c + e c where e represents soe experental error Snce angle ϕ kept constant alue durng the experent we could apply east squares ethod to the seres gen by (4 et denote γ by ean alue of the seres {γ } We can calculate Varance V and standard deaton σ γ ( γ = (5 ( V = γ γ (6 V σ = (7 If standard deaton σ s bgger then soe expected alue we should declare the experent faled Cobnng equatons (4 and (5 we get c c γ (,cos cos = ± c γ ϕ (8 ( ϕ (4 We don t know exact drecton of ector, thus poste and negate alue were assgned to 7 Conclusons Fro (53 and (78 t follows that length of ector s gen by whle ector s gen by Recall (fro that ector can be wrtten also as = + ( = ± ± ( = + + (3 3 Suppose that durng one year the sae experents hae been repeated K tes In that way we wll get the seres { ( } K = (4 where ( represents length of ector gen by Equaton ( or (3 at -th try et ( + K and ( denote elocty at whch Earth orbts the Sun at ( K Suppose also that orgns of ectors ( + and ( { } + -th and -th try K,,,K lay on the daeter of Earth orbt around the Sun, so they are parallel but n oposte drectons Mean alue of the seral (3 s gen by Dependng on we wll consder followng cases: ( ( ( K = (5 73
M Čojanoć In other words s sgnfcantly less than what s n contradcton to our hypotess ( In ths case we hae to reject hypothess gen by (3 and declare that elocty of lght s not effected by Earth s oeent through the space Ths results s consstent wth soe other experents, for exaple wth Mchelson-Morley experent > Durng the experents n perod of one year s changng, whle + 3 s keepng the constant alue Recall that ector s perpendcular to axs Denote ector u by If we replace ( and ( + K u = + 3 (6 ( let proj ( represents orthogonal projecton of ector on plane a a (7 ( ( ( ( = proj = proj + + = proj + proj u = + u (8 3 ( ( ( = + u + u (9 ( K ( K ( K + = + + u + + u ( by ( represents aerage speed Earth orbts the Sun ( ( + K Fro (9 and ( we can get approxate alue for u ( We can for seral Mean alue ( ( K + + u ( {,,, K} ( u of the seral ( s gen by { ( } K u ( = K u = u ( K (3 = et fnd standard deaton σ for seral (3 If σ s bgger then soe expected alue we hae to declne our hypothess ( and declare the experent faled where u ( = ( at -th try For seral (5 ean alue u s gen by et standard deaton for seral (5 s arked by σ ( ( ( ( = proj = proj + + = proj + proj u = u (4 3 { ( } K u (5 = K u = u ( ( K = 74
M Čojanoć If σ s bgger then soe expected alue we hae to declne our hypothess ( and declare the experent faled Otherwse hypothess gen by (3 holds and we can conclude that elocty of lght depends on Earth s oeent through space In other words elocty of lght depends on the drecton n whch has been easured, what would be n contradcton wth Mchelson-Morley experent [] The speed that Solar syste oes n the space n ths case s gen by equaton u = u + u (6 ote that whle perforng the experent we cotted soe stakes It was not taken nto account the speed of Earth s rotaton Ths proble can be soled by conductng an experent at place closer to the Earth s poles, and thus the speed of Earth s rotaton taken as sall as we want On other hand ths would be counter-producte to our condtons for South-orth easureent Ideally, E-W experent should be perfored on the orth/south Pole and S- experent at soe place on equator In addton, wthn 4 h the Earth changes ts drecton and the speed at whch t reoles around the Sun We can t sole ths proble but we can assue that ths speed s relately sall coparng to total speed at whch Earth oes through the space 8 ea If, k are natural nubers ( <, < k < and Θ an arbtrary angle then j= (( j sn k Π Θ = ( Proof where QED j= (( j cos k Π Θ = ( Θ ( j k Π Θ* M cos( ( j k Π Θ + sn( ( j k Π Θ = e e = e j= j= (( * k * Π * ( k* Π * M = e = e = = (( k * Π * ( k = e, < * Π < Π 9 Theore east Squares Method for Cosne Functon Suppose we are gen the seres {c }, c >, {,,, } and there are at least two p, q thus c p <> c q et take arbtrary coeffcents B, A, Θ and for equatons Defne functon g(b, A, Θ by We wll proe that n case ( ( c = B A cos k Π Θ + e, < k < ( e ( cos( ( = c B+ A k α Θ ( (,, Θ = = + cos( ( Θ g BA e c B A k α ( A, functon g( has a nu alue at pont ( B, A,Θ B = c = c (3 = 75
M Čojanoć where a = c c, Proof et B, A and Θ hae arbtrary alues = ( k α a sn = tg ( k Θ = (4 a cos ( k α ( α a cos k k Θ = A = (5 α = Π, {,,, } ( ( BA,, Θ = B+ A cos( ( α Θ = ( ( c B + ( c c + A cos( k ( α Θ = ( c B + ( c B ( c c + A cos( k ( α Θ + ( c c + A cos( k ( α Θ = ( c B + ( c B ( c c + ( c B A cos( k ( α Θ + g ( c, A, Θ = ( c B + g ( c, A, Θ g c k thus we get ( Θ ( Θ g BA,, g c, A, (6 In that way we can reduce functon g( fro functon of three arables to fucton of two arables A and Θ, keepng coeffcent B fxed and equal to c ow we can wrte the functon g( n the for ( A, Θ = c + A cos( ( α Θ = + A cos( α Θ = a + A* a cos ( k ( α Θ + A cos ( k ( α Θ cos ( k ( α Θ = cos( k ( α Θ + ( ( ( ( g c k a k A A A (7 g, Θ = + a cos k Θ + a ( ( ( α In order to fnd nu for functon g(, frst we hae to fnd partal derates wth respect to A and Θ and crtcal pont (A, Θ g ( A, Θ g ( A, Θ =, = (8 A et us fnd the frst partal derates g = k A a sn ( k α k Θ = k A cos k Θ a sn k sn k Θ a cos k A = In ths case we would hae ( ( ( α ( ( α g = k A a sn ( k α k Θ = ( (9 76
M Čojanoć It s easy to proe that g( has nu at ( BA,, Θ = ( B = ( = B g g e c = = c = B c A a sn ( k α k Θ = ( ( α ( ( α cos k* Θ * a *sn k* sn k* Θ * a *cos k* = g = A + a cos( k α k Θ A = A + cos k Θ a cos k + sn k Θ a sn k ( ( ( α ( ( α ( ( α g a cos k Θ = A = A cos k Θ cos k + sn k Θ sn k = et us look at the Equatons ( and ( For A we wll consder three cases: ( α ( α a cos k =, a sn k = ( ( a ( α ( a ( α Fro ( t follows A = We wll reject ths posblty because A ( α ( α a cos k =, a sn k Fro ( t follows ( Θ = Θ = ±Π ( 3 a cos( k α Fro ( Fro ( cos k k A tg ( k ( ( ( k α ( k α sn k Θ a sn Θ = = cos k Θ a cos a cos k Θ = ( ( α ( for both cases ow we hae to fnd the second order partal derates of g( wth respect to A and Θ ( Θ g ga, = = > A A ( ( (3 g = k A a cos( k ( α Θ (4 ( ga, Θ = k A a cos( k ( α Θ A fro ( a cos( k ( α Θ = ( ga, Θ = k A a cos( k ( α Θ ( ga, Θ k A = 77
M Čojanoć ( ( g A, Θ g A, Θ A ( k ( α = = k a sn Θ (5 A ( ga, ( ga, Θ Θ = = k a sn ( k ( α Θ (6 A A ( ( α fro ( a sn k Θ = ( ga, ( ga, Θ Θ = = A A ( ga, ( ga, Θ Θ A A = = = ( k A ga, ( Θ ga, ( Θ k A A ( k Equatons gen by (3 and (8 are suffcent condtons for nu QED References [] Dtchburn, RW (99 ght Doer Publcatons Inc, ew York (7 = A > A (8 Subt your anuscrpt at: http://papersubssonscrporg/ 78