9 Geophysics and Radio-Astronomy: VLBI VeryLongBaseInterferometry

Transcription

1 9 Geophysis and Radio-Astronomy: VLBI VeryLongBaseInterferometry VLBI is an interferometry tehnique used in radio astronomy, in whih two or more signals, oming from the same astronomial objet, are reeived by antennas that are very distant from eah other, reorded and then orrelated in deferred time Fig. 9.1). Due to the very long distane between the reeivers and the fat that the resolution is proportional to that distane, a very high resolution an be obtained see, for instane [6]). In onventional interferometry tehniques, the signals reeived by the antennas are diretly transmitted via a physial link to the orrelator, whih produes the interferene fringe in real-time; the antennas are physially onneted to the orrelator. In VLBI, the reeived signals annot be transmitted diretly and in realtime to the orrelator; the propagation time flutuations in the physial links would ompletely anel the orrelation between them. On the ontrary, the signals are ombined in differed time; they are onverted to a lower standard frequeny IF) and reorded at eah telesope on magneti tape or hard disk, with a preise time base. The reorded Fig The priniple of VLBI

2 176 9 Very Long Base Interferometry signals are then sent to a orrelating entre, where they are synhronized, and due to the timing information, played together and ombined just as if they were oming in real-time from the antennas. The orrelated data an then, for instane, be turned into images using any appropriate software. This is possible only if the phase noise of the loal osillators that down onvert the signal frequeny does not blur the interferene fringe, and if the timestamps are aurate and stable during the duration of the experiment. In fat, only very stable atomi frequeny standards an meet these requirements. VLBI is most often performed at radio wavelengths and the following desription is limited to radio signals; however, the tehnique has been extended to optis. The priniple is very simple. Let AB be the baseline of an array of two antennas. It is the vetor position of one antenna B) with respet to the other A). s be a unit vetor in the diretion of the soure. The time interval τ AB between the arrival of a wave front) to the antennas is τ AB = B u, 9.1) where is the light veloity. The measurement of τ AB an provide one of the following types of information: the omponent of s along AB if this vetor is known, or the omponent of AB along s if this vetor is known. Consequently, the appliations of VLBI apply to the geodesi domain as well as the astronomi domain. If the unertainty on the measurement of τ AB is 1 ps s), 9.1) shows that if the position of the soure is perfetly known, the unertainty on the value of the baseline length is of the order of 1 mm, and if the baseline is perfetly known, the unertainty on the position of the soure is of the order of rd 10 3 arseond) for a baseline length of km. 9.1 Priniple of VLBI The following desription of astronomial interferometry is limited to 1D models but an easily be extended to the D model.

3 9.1 Priniple of VLBI Interferometry The appliation of interferene methods to provide better resolution in astronomial measurements in both optial and radio domains) is not a new onept see, for instane [93, 108]). The priniple is the following: Consider two optial or radio) reeivers A and B, separated by a distane D and reeiving the eletromagneti radiation emitted by a point soure whose diretion is at an angle α see Fig. 9.). Fig. 9.. The priniple of interferometri measurements Monohromati Plane Waves Consider in a first step that the inoming wave is plane and perfetly monohromati, the frequeny is ν, the wavelength is, thewavenumber is k = π and the amplitude is X. The soure is very far from Earth and its diretion is indiated by the unit vetor s. The diretion of the wave propagation is given by the unit vetor e = s. The equation of the wave is as follows: [ xt, r) =X exp πν t r e )]. 9.) The vetor r = OM orresponds to a point M in the viinity of the Earth. The origin O of r is the baryenter of the geoid, for instane. The two reeivers are loated at points A and B, respetively.

4 178 9 Very Long Base Interferometry Supposing that the wave front is not perturbed by the atmosphere, the antennas reeive the signals x A t) and x B t), [ x A t) =X exp jπν t r A e )] 9.3) and [ x B t) =X exp jπν t r B e )] 9.4) [ )] AB e = x A exp jπν 9.5) = x A exp [j AB k)]. 9.6) The vetor k is k = π e 9.7) = ke. 9.8) x A t) and x B t) an also be expressed as funtions of the baseline length D and the diretion of the soure α, x B t) =x A exp jπ D sinα) ) 9.9) = x A exp jkd sinα)). 9.10) Using the small angle approximation, whih is of ourse not neessary the soure position angle α being supposed small), x B t) =x A exp jkdα). 9.11) The two signals x A t) and x B t) are added to give xt), xt) =x A t)+x B t) 9.1) = x A [1 + exp jab k)] 9.13) = x A [1 + exp jkdα)] 9.14) =x A exp j kdα ) ) kdα os 9.15) =x A exp j AB k ) ) AB k os. 9.16) The output of the square law detetor is, onsequently, ) kdα yα) =4X os 9.17) ) AB k =4X os 9.18) =X [1 + osab k)]. 9.19)

5 9.1 Priniple of VLBI 179 This is the lassial interferene pattern for monohromati radiation. The entral fringe is obtained for α =0τ AB =0). For α 0,thevalueofτ AB an be measured by introduing in one of the arms of the interferometer a delay τ AB = ±τ AB, whih ompensates τ AB. Due to the rotation of the Earth, the value of α or AB) varies ontinuously and a series of onfigurations an onsequently be studied. Quasi-Monohromati Plane Wave In fat, the radiation emitted by the soure is never perfetly monohromati. Phase and amplitude flutuations our, [ xt, r) =Xt)exp jπν t r e )] 9.0) with Xt) =X 0 [1 + at)] exp[jφt)], 9.1) where at) represents the relative amplitude flutuations and φt) the phase flutuations. These flutuations are small, at) 1, 9.) dφt) dt πν. 9.3) The signals reeived by the two antennas are x A t) =X t r A e ) [ exp jπν t r A e )] 9.4) = X t τ A )exp[jπν t τ A )] 9.5) = X t A )expjπνt A ) 9.6) and x B t) =X t r B e ) [ exp jπν t r B e )] 9.7) = X t A τ AB )exp[jπν t A τ AB )]. 9.8) In these expressions, τ A = r A e,t A = t τ A,τ B = r B e,τ AB = τ B τ A = AB e. 9.9)

6 180 9 Very Long Base Interferometry The sum of the two signals x A t) and x B t) is x A t)+x B t) =X t A )exp[jπν t A )] + X t A τ AB )exp[jπν t A τ AB )] 9.30) =[X t A )+X t A τ AB )exp jπντ AB )] exp jπνt A ). 9.31) The output of the square law detetor is x A t)+x B t) = X t A ) + X t A τ AB ) 9.3) + X t A ) X t A τ AB )expjπντ AB ) 9.33) + X t A ) X t A τ AB )exp jπντ AB ). 9.34) This result is integrated over a time Δt, hosen muh longer than the period of the signals but muh shorter than the harateristi time of variation of the diretion α of the soure due to the Earth s rotation. The output yτ AB ) of the interferometer is, onsequently, yτ AB )= X t A ) + X t A τ AB ) + X t A ) X t A τ AB ) exp jπντ AB ) + X t A ) X t A τ AB ) exp jπντ AB ). 9.35) The mean value of the amplitude is onstant, X t A ) = X t A τ AB ) = X ) The mean values of the produts and X t A ) X t A τ AB ) X t A ) X t A τ AB ) are related to the autoorrelation funtion γ X t) of Xt), Consequently, γ X t) = X τ)xt + τ). 9.37) yτ AB )=X0 + γ X τ AB )expjπντ AB ) + γ X τ AB )exp jπντ AB ). 9.38) The following property of the autoorrelation funtion results from its definition the autoorrelation is a Hermitian operator): γ X t) =γx t). 9.39)

7 9.1 Priniple of VLBI 181 Consequently, yτ AB )=X 0 +R{γ X τ AB )expjπντ AB )}, 9.40) where Rz) is the real part of the omplex number z. Notie that γ X τ AB )expjπντ AB )=γ x τ AB ), 9.41) where γ x t) is the autoorrelation funtion of xt, r). Finally, yτ AB )=X 0 +R{γ x τ AB )}. 9.4) The onlusions are the following: 1. The useful information is ontained in the periodi part of yτ AB ),whih is the autoorrelation funtion γ x τ AB ) of the plane wave emitted by the point soure.. The time delay τ AB that onnets the baseline and the soure position appears in the value of γ x τ AB ). 3. The ratio of the periodi part of yτ AB ) to its onstant one is alled the omplex fringe visibility. It is proportional to the autoorrelation funtion of the plane wave. 4. The autoorrelation funtion γ x t) is maximal for t =0. This means that the fringe visibility is redued when the delay τ AB inreases, this is due to the limited oherene time of the radiation, related to its linewidth. The autoorrelation funtion is linked to the spetral density of the line the Wiener Khinhin theorem). If Ct) is the autoorrelation funtion of a time funtion ft) whose Fourier transform is F ν), thenct) is the Fourier transform of the absolute square of F ν), whih is the spetral density of ft). Consequently, the order of magnitude of the oherene time of the inoming wave is given by the inverse of its linewidth. This will ultimately limit the resolution of the observation. For instane, a linewidth of 1 khz gives an upper limit of only sforτ AB, orresponding to AB s 5 km and, for a baseline length of km, to α = rd. In fat, sine the signals are orrelated in deferred time, it is possible to shift one reord until the time differene is aneled and the orrelation funtion is maximal. The shift gives the value of the time differene τ AB. 5. The main part of the proessing of the signals reeived by any array of radio antennas is onsequently the alulation of their orrelation. Although the signals reeived by different antennas ome from the same soure, the quantity omputed is alled the ross-orrelation, taking into

8 18 9 Very Long Base Interferometry aount the fat that eah signal may have been modified in a different and non-orrelated way atmospheri perturbations, additive noise of the reeiver, et.). This alulation is made by a speialized data proessing system alled the orrelator. Extended Soure If the radiation soure is extended around the inoming diretion e 0, but is inoherent different points of the soure radiate independently), there is no interferene between the ontribution of the different points and their power ontributions are simply added. The following model is limited to a one-dimensional soure. It is easy to extend the model to the real ase of two-dimensional soures. The position of eah point of the soure is haraterized by the unit vetors e from the soure) or s = e toward the soure) or by the angle α between the perpendiular to the baseline and s. The sky brightness Bα) of the point in the diretion α is proportional to the square of the mean amplitude X 0 α) of the inoming radiation from that diretion and the total brightness is B t = α Bα)dα. The output y of the orrelator is the sum of the elementary rossorrelation funtions orresponding to all the points of the soure, y = γ x α, τ α )dα α = γ X α, τ α )expjπντ α ) dα. 9.43) α The integral is to be taken over the radio soure. Every point of the soure orresponds to a value of α, with α = α 0 + δα 9.44) δα 1, 9.45) where α 0 orresponds to an arbitrary referene point of the soure, τ α is the value of τ AB for the value α of the angle between the perpendiular to the baseline, and s τ α = D sin α = D sin α 0 D os α 0 δα, 9.46) πντ α = π D sin α 0 π D os α 0δα, 9.47)

9 9.1 Priniple of VLBI 183 where is the wavelength of the radiation. γ X δα, τ δα ) is the orrelation funtion of the sky brightness in the diretion α = α 0 + δα. Consequently, y = γ X δα, τ δα )exp jπ D sin α ) 0 δα exp jπ D os α ) 0 δα dδα). 9.48) This expression shows that the output of the orrelator is the Fourier transform of the funtion of δα, γ X,α0 δα, τ δα )=γ X δα, τ δα )exp jπ D sin α ) ) This funtion is losely related to the orrelation funtion of the sky brightness [117], ) ) D os α0 D os α0 y = F α [γ X,α0 α, τ α )]. 9.50) The onlusions are the following: 1. The orrelator gives the Fourier transform of the ross-orrelation funtion of the amplitude Xα, t AB ) of the signal emitted by the soure. This orrelation funtion is alulated for the delay t AB between the two reeivers.. The sky brightness an be alulated from this result if this Fourier transform is known for different sampled values of its parameter D os α0, i.e. for different values of D, the distane between the two reeivers involved in the alulation of the Fourier transform and/or different values of α 0. In the first ase, an array of reeivers is used, in the seond ase, the motion of the vetor AB due to the rotation of the Earth is used. Examples In the following simple examples 1. α 0 1: os α 0 =1and sin α 0 =0. Consequently, 9.48) simplifies to y = γ X δα, τ δα ) exp jπ D ) δα dδα) 9.51) δα and the output of the orrelator is the Fourier transform of the rossorrelation funtion γ X δα, τ δα ).. The linewidth of the radiation emitted by the soure is supposed to be narrow enough so that the ross-orrelation funtion γ X δα, τ δα ) is a monohromati wave) γ X δα, τ δα )=X0 δα). 9.5)

10 184 9 Very Long Base Interferometry The Retangular Sky Brightness Funtion The objet is entered at α 0 / 1 anditswidthisδα, { δα < α0 / Δα X0 0 δα) = δα > α 0 /+Δα, α 0/ Δα δα α 0 /+Δα. X ) The Fourier transform F α [γ X α, τ α )] ) D is ) D Y =X 0δα exp jπ D α ) 0 sin π D ) δα. 9.54) The omplex fringe visibility Γ D/) is ) D Γ =exp jπ D α 0 ) sin π D ) δα. 9.55) The modulus of the omplex visibility is onsequently maximal for small values of the ratio δα /D, i.e. for objets whose angular diameter is of the order of or smaller than /D; VLBI is used to observe very ompat soures. A Pair of Retangular Sky Brightness Funtions As a seond example, onsider a pair of retangular sky brightness funtions entered at ±α 0 / and having width Δα, δα < α 0 / Δα 0 α 0 /+Δα<δα<+α 0 / Δα, X0δα) = δα > +α 0 /+Δα 9.56) { α0 / Δα δα α 0 /+Δα X 0 +α 0 / Δα δα +α 0 /+Δα. In this ase, the Fourier transform Y D ) of the sky brightness funtion is ) =X π 0 δα sin D ) δα Y D [ exp jπ D α ) 0 =4X 0 δα sin π D δα ) os +exp π D α 0 jπ D +α )] ) ). 9.58)

11 The omplex fringe visibility Γ D ) is Γ 9.1 Priniple of VLBI 185 ) D = sin π D ) δα os π D α ) ) The onlusions are the following: 1. As in the previous example, the modulus of the omplex visibility is maximal for small values of the ratio δα /D.. The modulus of the omplex visibility is maximal for small values of the quantity πdα 0 )/), i.e. for D <πα ) The resolution of the interferometer is onsequently given by the ratio of the distane between the two reeivers to the wavelength of the radiation Proessing of the Signals The previous disussions show that proessing the signal reeived by the antennas allows one to produe an image of an astronomial objet aperture synthesis); preisely determine the relative position of the antennas if the emitting objet is distant and stable geodesy); preisely determine the position of a ground or spae radio soure if the positions of the antennas are known; determine the spetra of the radio emission. Proessing at Eah Antenna The data reeived by the antennas are proessed in the following way before being orrelated many steps of the proess, suh as amplifiation, filtering, et. are omitted in this shemati desription). 1. They are down onverted to a baseband signal by mixing them with a loal osillator. The auray and stability of this loal osillator must be onsistent with the phase shifts to be measured. Suppose we have an input signal xt) =X exp [jπνt + φ)

12 186 9 Very Long Base Interferometry and a loal osillator x LO = X LO exp [jπν LO t + φ LO )]. Mixing these two signals uses a non-linear operator, whih produes output omponents at various frequenies, the sum and differene of the multiples of the frequenies ν and ν LO. From these omponents, it is easy to selet, with a filter, the one whose frequeny is ν IF = ν ν OL this frequeny is alled the intermediate frequeny) and whose phase is φ φ LO these relations apply in the ase where ν LO <ν). The baseband is enteredonthisfrequenyν ν LO. The phase flutuations of the IF signal are onsequently the sum of that of the signal and of the loal osillator, x IF = KXX LO exp j[πν ν LO )t + φ φ LO ]. 9.61) This is not a problem if all the signals of the interferometer are down onverted using the same loal osillator, sine it is the phase differene between them that is the pertinent information. On the ontrary, in the ase of a VLBI, the signals from different antennas are down onverted using a different loal osillator, loated in the same station as the antenna; the phase of eah loal osillator must onsequently be very preisely defined.. The resulting signal is sampled and reorded in a digital media, along with a preise timestamp. 3. The reorded data are then sent to the orrelator to be further proessed. Delay Compensation Due to the delay between the two antennas whose signal are to be orrelated and to the finite linewidth of the line being studied, the fringe visibility is dereased see Set ). This an be ompensated, sine it is possible to shift the two reorded data to optimize the value of their ross-orrelation. Digital Correlator The orrelator is the masterpiee of VLBI signal proessing. Extensive desriptions an be found, for instane, in [7, 107]. In the ase of a digital proessing, the ross-orrelation of the disretetime proess funtion of fn) and gn) is easily omputed, γn) =f gn) = f p)gn + p). 9.6) p= A shemati blok diagram of a ross-orrelator is shown in Fig It uses memory to implement delays of a multiple of the sampling time T s, multipliers and aumulators. In fat, the summation does not extend from to + and the output of the devie is an estimator of the ross-orrelation.

13 9. Appliations of VLBI 187 Fig Shemati blok diagram of a ross-orrelator 9. Appliations of VLBI It was shown in Set. 9.1 that VLBI may have various appliations in astronomy position, spetra and imaging of astronomial objets) and in geodesy the relative position of the antennas, absolute position relative to referene astronomial objets, rotation of the Earth) Astronomy VLBI was developed first as a radio-astronomial tool and remains a powerful and high resolution tool for observing radio soures. It allows submilliarseond imaging [73, 60] and detetion [118] of extragalati objets. 9.. Geodesy In this kind of appliation, the astronomial soures are known and used as referenes to determine some parameters of the Earth.

14 188 9 Very Long Base Interferometry Rotation of the Earth Very distant quasars provide an inertial referene frame that is muh more aurate than the fundamental atalog of fix stars FK5 [16]. The antennas of a VLBI array are then in a situation that may be ompared to that of a differential GPS experiment; they reeive the signal emitted by the same soure. Nevertheless, in the ase of VLBI, the astronomi soures appear as a point-soure with no motion. There is onsequently no need to onstrut a model for their motion. Sine the radio telesopes are fixed on the rotating Earth, VLBI measures the orientation of the Earth in the inertial referene frame defined by these quasars as a funtion of time, monitoring the Earth rotation and orientation. It is onsequently possible to measure all the omponents of the Earth s rotation: the position of the Earth s spin axis in spae, the position of the Earth s spin axis relative to the Earth rust, and the veloity of the rotation, whih allows one to onnet the two time sales UT and UTC see Set. 7.1). This information allow one to perform orbit ontrols of satellites, inluding GPS satellites see, for instane [110, 95]). Monitoring of Plate Potions This appliation of VLBI, joined to the GPS tehnique, is well known. These spae geodeti tehniques allow the diret measurements of plate motions. Motions of a few m per year are learly visible see, for instane [49, 7, 47]). The results of these measurements are used in Earthquake researh. Preise Loalization on the Earth The preise measurement of the position of the VLBI and GPS stations allow one to maintain the realization of the International Terrestrial Referene System see, for instane [59, 88]).

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