Searching for Earth s Trajectory in the Cosmos

Transcription

1 Apeiron, Vol. 9, No. 4, Otober Searhing for Earth s Trajetory in the Cosmos Úlpio Nasimento (Researher retired from Laboratório Naional de Engenharia Civil LNEC). Av. de Roma, -5º-Dto., Lisboa, Portugal marionasimento@netabo.pt A reolletion is presented of a previous work ontaining a model of the refration mehanism, whih served as a basis for the analysis of Mihelson & Morley and Shamir & Fox experiments relating to the searh for the ether wind. It was subsequently assumed that the first produed a negative result due to lak of auray, as a result of the low refration index of the air (n = ). The seond, with arms onsisting of perspex rods with n = 1.49, made it possible to detet a 6.64km/s ether wind. In that work, an experiment was proposed with fibre-opti oils, in order to inrease auray. In the present doument, that proposal is again dealt with under the designation of fibre-opti tahymeter (FOT). It takes into aount the remarkable development ahieved in the last deade in the tehnology of fibre-opti gyrosopes (FOG). The Sagna effet, whih is the FOG priniple, is reviewed by analysing it in aordane with the refration mehanism model and by putting its preision into the equation. Referene is made to the trend toward using triaxial FOG and the same proedure is proposed for a triaxial FOT.

2 Apeiron, Vol. 9, No. 4, Otober It is proved that a gyro-tahymeter (FOGT) onsisting of a triaxial gyrosope and a triaxial tahymeter is able to detet movements related to Earth s rotation and with its translation around the Sun. In view of the fat that suh movements have a speifi daily and annual periodiity that haraterises them, it would appear possible to detet the veloity of 60km/s of Earth in the osmos, orresponding to the asymmetry of the fossil radiation deteted by the satellite COBE and in ase of ourrene of other movements, these will be deteted if they are within the preision range of the GT. 1 INTRODUCTION In a previous work [1], an analysis was made of the experiment in whih Fizeau, in 1851, proved how the veloity of propagation of light in the water varies when the latter is in motion. Also in that equation, he verified that suh variation is in agreement with the equation dedued by Fresnel in Nevertheless, onsidering that the hypothesis of partial dragging of the ether, from whih Fresnel dedued his equation, is poorly onvining, an alternative model of the refration mehanism was also proposed. In suh refration mehanism it is assumed that the light travels inside refrational bodies following zigzag ABCDE trajetories (fig. 1.1), under the sattering effet aused in the luminous radiation by the body s atoms. The veloity of propagation of light along eah side of the zigzag is onstant and equal to the veloity in the vauum. If the body is in motion under the effet of the inherent aberration, the sattering angle α hanges into α and the zigzag trajetory beomes A B C D E, maintaining however the side lengths (Fig. 1.1).

3 Apeiron, Vol. 9, No. 4, Otober Fig. 1.1 Zigzag trajetory of light on a refrational body. Fig. 1. shows that being the veloity of light in the vauum, due to sattering, its effetive veloity w inside the body at rest is os α. Also onsidering that the refration index n is the relation /w, there an be obtained: w = os α = 1.1 n thus: os α = n 1. If the body is in retilinear motion with veloity v, under the aberration effet, the apparent sattering angle beomes α and the veloity of propagation of light in the body hanges from w to w given by w ' = osα' 1.3 α being given by senα tgα ' = 1.4 w + v The differene of veloity of propagation due to the body s movement is given by: 1

4 Apeiron, Vol. 9, No. 4, Otober ( osα' osα) w ' w = 1.5 Fig In a body in motion with veloity v, under the aberration effet, the sattering angle hanges from α to α and the veloity of propagation from w to w. By omparing equation 1.5 with the famous Fresnel formula: 1 w' w = 1 v 1.6 n It an be verified that for low values of v, the results are almost equal [1]. The model was applied to Fizeau and Mihelson s experiment and it was onluded that the auray of the latter, onsidering that the refration index of the air is very low (1 = 1,0003), was not enough to detet the ether wind. The model was also used in the experiment arried out in 1969, by Shamir & Fox []. The latter was similar to Mihelson s experiment, but with a signifiant differene: the propagation of light on the arms of the apparatus, instead of ourring through air, was performed by means of perspex rods with a refration index n = Therefore, in that experiment an ether wind

5 Apeiron, Vol. 9, No. 4, Otober was deteted of 6.64 km/s, i.e., about % of the orbital veloity of Earth (30km/s). With suh satisfatory result, a modifiation in Shamir & Fox s interferometer was proposed, whih onsisted of replaing the arms of the apparatus by fibre-opti oils (Fig. 1.3b), so as to inrease its auray [1]. On the other hand, taking into aount the remarkable development ahieved in the 90 s, as regards the tehnology of fibreopti gyrosopes, it was onsidered that a study should be done on the feasibility of development of the fibre-opti gyro-tahymeter. This is preisely the subjet of the present work. Fig. 1.3 a) Shamir & Fox s interferometer F Light soure; P semi-transparent plate; R Perspex rods; E mirrors; I - interferometer. b) Fibre-opti tahymeter B1 and B fibre-opti oils

6 Apeiron, Vol. 9, No. 4, Otober 00 7 THE FIBRE-OPTIC GYROSCOPE (FOG).1 Conepts Sagna presented in 1914 a memoir about his interferometer [3]. Nevertheless, he had already published in 1910 the orresponding theory and, in 1913, the experimental results obtained. The Sagna interferometer onsists of the following (Fig..1): the luminous flow emitted by soure F is split into two by the semitransparent plate P, one beam proeeds through mirrors A, B, C and the other, ontinues in the opposite sense. They are subsequently reunited on P, whih is their point of departure to interferometer I. The mirrors and the plate form therefore a losed iruit, whih is travelled by light in opposite senses. When all the apparatus, inluding the light soure and the interferometer, is in rotation around point O, the path followed by one of the beams dereases omparatively with the other, thus produing a displaement of the orresponding interferene fringes [4]. Fig..1 Sagna interferometer.

7 Apeiron, Vol. 9, No. 4, Otober ω being the angular veloity of the rotation given to the interferometer (Fig..1), the veloity of P will be v = ωr. Thus, the time spent by the beam travelling on the side PA (in the same diretion as ω) will be: t AB R = = v / R ωr and the time spent by the beam ontinuing on the side PC (in the opposite sense to ω), will be: R t AD = + ωr being the veloity of light. The total time for the two paths in diret sense t and in opposite sense t will be therefore: and their differene be: 8R t = ωr t 8R = + ωr t = t t, having a series development, will 8R ω t =.1 Or, onsidering that the area limited by the iruit PABC is A = R : 4 Aω t =.

8 Apeiron, Vol. 9, No. 4, Otober τ and λ being the period and wavelength of monohromati light used, there an be obtained: λ τ =.3 and the relative development of the interferene fringes will be given by: or onsidering.: t =.4 τ 4 Aω =.5 λ That result has been experimentally tested. The Sagna interferometer is thus a gyrosope and Mihelson and Gayle used it, in 195, to measure the angular veloity of Earth [4]. As from the 80 s, the fibre-opti gyrosope has been developed on basis of the Sagna interferometer and it an be briefly desribed as follows [5]. Let us onsider in fig..a, an interferometer, in whih from the semi-transparent plate P, a luminous beam split into two and refleted on an infinitive number of mirrors and having followed in opposite senses a irular trajetory, returns subsequently in phase to the same point P. Let us assume now that the interferometer is in rotation in the vauum. Then, the light emitted in point P travels through those mirrors with the same veloity in opposite senses. However, during the time t v of that travel, the plate P will be displaed to P (Fig..b) and the ray of light that propagates in the same rotation sense as the interferometer has an inreased travel of l, whereas the ray propagated in the opposite sense has a dereased travel of v lv. There

9 Apeiron, Vol. 9, No. 4, Otober is therefore a travel differene l v between the two rays and whih orresponds to a time differene t v given by: t v πr + lv πr lv = v l = v.6 Fig.. Sagna effet on an ideal irular travel in vauum: a) motionless interferometer; b) interferometer in rotation. If the propagation of light, instead of ourring in the vauum with a refration index of n = 1, ours in a medium with n > 1, whih is the ase of a fibre-opti gyrosope, there is a very low phase differene, as will be demonstrated below, but whih is not null as some authors have onluded [5]. The demonstration of the nullity of suh phase differene is presented below. In a motionless fibre-opti iruit both luminous rays travel with a veloity w =.7 n and return to plate P at the end of time t m given by: πr πnr t = = = w m nt v.8

10 Apeiron, Vol. 9, No. 4, Otober When the interferometer is in rotation (Fig..3b) the plate P travels during time t m through the length l = Rω t = n l.9 m whih is n times higher than the length lv orresponding to vauum, as Fig..3 shows. In this ase however, the veloity of light is not equal in both senses. In aordane with the Fresnel formula, the veloities in the sense of rotation v ρ and in the opposite sense v r are given by: m v vρ = + α F Rω.10 n vr = α F Rω.11 n Rω being the tangential veloity of the fibre-opti and α F the Fresnel dragging oeffiient: t mf being therefore given by: t mf 1 α F = 1.1 n πr + n lv πr n lv = v v The expression above is approximately equal to: therefore, onsidering.1: t = t n 1 m r v m t v ( α ) F t =.15 This would serve to demonstrate that the Sagna effet is independent from the refration index of the medium [5].

11 Apeiron, Vol. 9, No. 4, Otober Fig..3 Sagna effet: a) in the vauum; b) in a fibre-opti. In order to prove otherwise, let us onsider again a fibre-opti iruit. When the iruit is at rest, the veloity of light, travelling through it, is given by: w = os α =.16 n α being the sattering angle at rest. When the iruit is in rotation, the veloities w r in the sense of the rotation and w r in the opposite sense will be given by: w r = osα ρ.17 w r = os.18 α ρ The orresponding sattering angles are given by: senα tgα r =.19 os α + ωr senα tgα r =.0 os α ωr The travel time differene will be therefore:

12 Fig..4 shows t Apeiron, Vol. 9, No. 4, Otober m πr + n lv πr n l = w w m r r v.1 t given by Eq..1 and also given by Eq..13 ( t mf), in aordane with R and ω for the refration indies n = 1 and n = Thus, the following equation an be dedued: 4πR ω t = fg ( n). f G (n) being a funtion of the refration index, whih, as fig..5 shows, assumes the following form: f G ( n) = n.3 Thus, the Eq.. an assume the following form: 4 Aω t = n.4 Fig..4 Travel time differene t in aordane with R and ω.

13 Apeiron, Vol. 9, No. 4, Otober Fig..5 Funtion fg(n) = n Whih is only different from Eq.. as regards the fator n. Thus, it an be onluded that the Eq..15, aording to whih, the Sagna effet is independent from the refration index of the medium [5], is only a first approah. In fat, it is proportional to the square of the refration index. In view of the fat that the length F of fibre-optis is given by: F = NπD.5 one will have: 1 A = FD.6 4 Thus, the Eq..4 an assume the form as follows: FDω t = n.7

14 . FOG Auray Apeiron, Vol. 9, No. 4, Otober On the basis of Eq..7, Fig..6 shows t, as a result of the angular veloity ω in deg/h, n = 1.45 being for F equal to 1m, 10 3 m and 10 6 m and D = 0.05m. The same Fig..6 indiates the zone A orresponding to the FOG urrently used in airrafts, whih require a preision ranging from 10 to 0.1 deg/h and whih ommonly use 100m to 00m of fibre in oils varying from 30 to 60mm diameter. It is also indiated the zone Hp of higher performane FOG s with 0.5 to Km of fibre in oils with 8 to 10 m diameter, with a preision of 0.01 deg/h [5]. As an be seen, values of t from to s orrespond to the preision above. This means that the present tehnology of FOG has embraed measurements of t from to s orresponding to angular veloities of 10 to 0.01 deg/h. Fig..6 Travel time differene as a funtion of ω and F.

15 Apeiron, Vol. 9, No. 4, Otober THE FIBRE-OPTIC TACHYMETER (FOT) Fig. 3.1 shows a shemati drawing of a FOT different from the one presented in fig. 1.3b. The differene is the fat that in the latter, the fibre-opti is not rolled in irles but rather in a mould formed by a flat zone, with a length l and a thikness D, and having almost ylindrial aps with a diameter equal to the thikness. Sine the tahymetri effet of a oil is proportional to its length: ( l D) L = N The length F of a fibre-opti of eah oil being given by: F = N l + πd 3. ( ) and D being very small in omparison with l, will be approximately: F L 3.3 Fig. 3.1 Shemati drawing of a FOT. Referene should be made of the fat that the length L of a FOT orresponds to the length of the arms of either Mihelson & Morley s or Shamir & Fox s interferometers. By assuming that the FOT travels at an absolute veloity v in the diretion of the oil B 1, the time t 1, spent by the luminous ray in the path P 1B1 P is

16 Apeiron, Vol. 9, No. 4, Otober 00 8 F 1 1 t = osα osα and the time t spent in the paths P 1BP F t = 3.5 w α, α and α being the sattering angles at rest, in the sense of v and in the opposite sense, respetively, given by Eq. 1.1 and 1.4; senα tgα = 3.6 w + v senα tgα = 3.7 w v The differene t will be therefore: t = t 1 t 3.8 In aordane with Fresnel formula it will be: tf = tf1 tf, Thus, 3. FOT Auray F 1 1 t = + F1 3.9 w + φfv w φ Fv F F t F = = 3.10 w W Fig. 3. shows how t varies in aordane with the length of the fibre F of eah oil and with the veloity v for oils of n = 1.45 and

17 Apeiron, Vol. 9, No. 4, Otober diameter D = 50mm. It an be thus dedued that the Eq. 3.8 an assume the form: Fv t = ft ( n) f T (n) being a funtion of the refration index, whih for n = 1.45 has the value f T (n) = In the same fig. 3., the point SF orresponds to Shamir & Fox s experiment, in whih, with n = 1.49 and F = 0.6, the veloity v = 6.64km/s was measured. The point MM represents Mihelson & Morley s experiment (n = , F = 60m) for the orbital veloity of Earth (v T = 30km/s). t of travel time in a FOT, in aordane with the Fig. 3. Differene veloity v and the length F of the fibre-opti of eah oil. SF: Shamir & Fox s experiment MM: Mihelson & Morley s experiment

18 Apeiron, Vol. 9, No. 4, Otober Fig. 3.3 shows that the funtion f T (n) has little variation when the veloity hanges from 1km/s to 30km/s. Thus, in a first approah, it an be onsidered independent from v and with the equation as follows:.95 f T ( n) = n Its values for some refration indies, inluding the one of the fibre-opti (1.45) and the one of the diamond (.417), are also inluded in the same figure..95 Fig. 3.3 Funtion f T ( n) = n 1.

19 Apeiron, Vol. 9, No. 4, Otober Fig. 4.1 Sattering angle α when the veloity v is not parallel to the axis OL. 4 THE FIBRE-OPTIC GYRO-TACHYMETER (FOGT) 4.1 The triaxial gyrosope In order to detet the rotations of a vehile, not only in relation to an axis with a ertain diretion, but also in relation to any diretion whatsoever, three gyrosopes are required, with the orresponding axis forming an orthogonal trihedron. Nevertheless, there is a tendeny to use triaxial gyrosopes with only one soure of light, instead of separate uniaxial gyrosopes. Suh triaxial gyrosopes, together with speial aelerometers, form the inertial measurement units (IMU) [5]. 4. The triaxial tahymeter Fig. 4.1 presents an overview of the problem referring to measurement in a FOT, when the absolute veloity v is not parallel to the diretion OL of the propagation of light. Thus, one should onsider the angular o-ordinates θ and h formed by v with a plan defined by OL and by a radius OP of the sattering one. This example refers to the ase presented by Chapter 3, whih replaed v

20 Apeiron, Vol. 9, No. 4, Otober in Eq. 3.6 and 3.7, by the omponent v osh osθ [1]. Thus, it is obtained: senα tgα = 4.1 w + v osh osθ Fig. 4. Three FOT s for determination of the absolute veloity v of Earth. FOT-EN in the horizontal plan FOT-EZ in the vertial plan direted to east FOT-NZ in the vertial plan direted to north Fig. 4. shows a shemati drawing of three FOT s, whih forms in the whole, an orthogonal trihedron.

21 Apeiron, Vol. 9, No. 4, Otober The arms of the first are direted to E and N, and it should detet the omponent of the veloity v given by v os h osθ. The arms of the seond are direted to E and Zenith (au Nadir), whih is intended to detet the omponent v sinh. Lastly, the arms of the third are direted to N and Zenith (au Nadir), whih is intended to detet the omponent v osh sinθ. Suh FOT s are able either to operate separately or to form a triaxial tahymeter, with the same light soure, similarly to the triaxial gyrosopes. 4.3 Searhing for Earth s trajetory Fig. 4.3 presents side-by-side the diagrams of FOT and FOG forming a fibre-opti gyro-tahymeter (FOGT). Fig. 4.4 shows the length l of eah FOT oil in aordane with the orresponding length F of the fibre-opti and of the number N of turns of the oil (Eq. 3.). As an be seen, with a preision of approximately within the range of the present FOG tehnology, the length F of the fibre-opti, whih is the neessary for a FOT to detet the orbital veloity of Earth ( v T = 30km/ s), is about a few ten metres. As Fig. 4.4 shows, that length an be obtained with oils having a length l of about a few metres, with N of only a few tens. In order to detet the veloity resulting from Earth s rotation at about 40º latitude ( v TR = 355m / s ) it would be required some oils with F of some kilometres, whih implies (Fig. 4.) Lengths l of about a few ten metres with N of a few hundreds.

22 Apeiron, Vol. 9, No. 4, Otober Fig. 4.3 Diagrams of a fibre-opti gyro-tahymeter (FOGT): VT = 30km/s Orbital veloity of Earth VTR = 355m/s veloity resulting from Earth s rotation at 40º latitude ωt = 15deg/h angular veloity of Earth s rotation ωto = 0.04 deg/h angular veloity in Earth s orbit around the Sun. ωp = 10-8 deg/h angular veloity due to the variation of the position of the Pole. Fig. 4.4 Length l of the FOT oil as a funtion of the length F of the fibre-opti.

23 Apeiron, Vol. 9, No. 4, Otober As regards the FOG, both the angular veloity of Earth s rotation ( ω T =15deg/ h ) and the angular veloity of Earth in its orbit around the Sun ( ω TO = 360 deg 365days / 4h 0.04deg/ h) are within the range of the present tehnology (Fig. 4.4). Nevertheless, the angular veloity related with the variation in the position of Earth s poles ( ω P ), whih is about ten metres per year [6] and to whih orresponds an angular veloity ω P of about 1E-8 deg/h [ ω P = ( 10/ 6.37E6)(180 / π )/365x4 1E 8deg/ h]. Therefore, they are far beyond the possibilities of the present tehnology, as Fig. 4.3 and 4.4 show. In fat, on the one hand, the FOGT has enough preision to detet the movements referring to Earth s rotation and to its orbit around the Sun. On the other hand, suh movements of the Earth have a daily and annual periodiity, whih learly identifies them. Thus, it seems that, if there are some movements other than those mentioned previously, these will be deteted, provided that their magnitude is within the preision range of the FOGT. This is the ase of the veloity of 60km/s of Earth in the osmos, whih orresponds to the asymmetry of fossil radiation deteted by the satellite COBE [7]. 5 CONCLUSIONS From the analysis performed on basis of the model of the refration mehanism proposed in a former work, the following onlusions an be presented: 1. The Sagna effet, whih is the priniple of the fibre-opti gyrosopes (FOG), is proportional to the square of the refration index of the fibre (Eq..7).

24 Apeiron, Vol. 9, No. 4, Otober The ether wind, whih is the priniple of the fibre-opti tahymeter, is a funtion of the refration index of the fibre (Eq. 3.1). 3. It seems that the detetion of the veloity of 60km/s of Earth in the osmos, orresponding to the asymmetry of the fossil radiation deteted by the satellite COBE is within the range of the fibre-opti gyro-tahymeter (FOGT). The onstrution of that apparatus is proposed on basis of the present tehnology of fibre-opti gyrosopes (FOG). Aknowledgements The Author wishes to thank to Prof. Ramalho Croa, from Fauldade de Ciênias de Lisboa, for his valuable ritial reading of this work.. And also to Maria Graça Tomé for the English translation of the doument and his son Mário for the informati help on this paper. Referenes [1] Nasimento, V., 1997 On the trail of Fresnel s Searh for an Ether Wind. Apeiron, Vol.5 Nr. 3-4, July-Otober P [] Shamir, J. & Fox, R. (1969) A new experimental test of speial relativity. Il Nuovo Cimento. Vol. LXII, Nr.. [3] Sagna, M.G., 1914 Effet Tourbillonnair optique. J. de Phys. 5ème série, t. IV. [4] Heht, E., 1987 Óptia. Translation from the English by José Manuel N.V. Rebordão. Fund. Calouste Gulbenkian. [5] Lefèvre, H.C., 1993 The fibre-opti gyrosope. Arteh House, In. Norwood, MH 006 [6] Siene & Vie, Nr. 986, Nov [7] Reeves, H., 1994 Últimas notíias do Cosmos. Translated from the Frenh by Ceília A. da Silva. Gradiva, 1995.