[tel-00726959, v1] Caractériser le milieu interstellaire ... - HAL - INRIA

J. Pety and N. Rodríguez-Fernández: Revisiting the theory of interferometric wide-field synthesistel-00726959, version 1 - 31 Aug 2012B.1. w-axis distortionWhen projection effects are taken into account, the measurementequation readsV ( )w, u p ,α s =∫B ( ) I ( )[α p√ )]α p − α s √ e −i2π α p u p +w(1−α 2 p −1dα p .α p1 − α 2 p(B.1)In this equation, we continue to work in 1 dimension for the skycosine direction (α p ), but we explicitly introduce the dependencealong the direction perpendicular to the sky plane. This dependenceappears in two ways,√which is handled in very differentways. First, the factor 1 − α 2 p can be absorbed into a generalizedsky brightness functionI ( ) I ( )α pα p ≡ √ · (B.2)1 − α 2 pAfter imaging and deconvolution, I(α p ) can be easily restoredfrom the deconvolved I(α p ) image. The second dependence appearsas an additional phase, which is written as)P ( α p ,w ) ( √≡ e −i2πw 1−α 2 p −1. (B.3)Thompson et al. (1986, Chap. 4) shows that this additional phasecan be neglected only if 8π θfield2 ≪ 1 or π λd max≪ 1.(B.4)4 θ syn dfield2The first form of the criterion indicates that the approximationgets worse at high spatial dynamic range (i.e., θ field /θ syn ≪ 1)while the second form indicates that the approximation getsworse at long wavelengths.B.2. uvw-unfacetingFor stop-and-go mosaicking, it is usual to delay-track at the centerof the primary beam for each pointing/field of the mosaic.This phase center is also the natural center of projection of eachpointing/field. Stop-and-go mosaicking thus naturally paves thecelestial sphere with as many tangent planes as there are pointings/fields;i.e., this observing scheme is somehow enforcing auvw-faceting scheme. In the framework of on-the-fly observationswith ALMA, D’Addario & Emerson (2000) indicate thatthe phase center will be modified between each on-the-fly scanwhile it will stay constant during each on-the-fly scan. This is acompromise between loss of coherence and technical possibilitiesof the phase-locked loop. Using this hypothesis, the maximumsky area covered by the on-the-fly scan must take intoaccount the maximum tolerable w-axis distortion.The easiest way to deal with such data is to image each pointing/fieldaround its phase center and then to reproject this imageonto the mosaic tangent plane as displayed in Fig. 5 of Saultet al. (1996b). These authors point out that this scheme impliesa typical w-axis distortion ɛ less thanɛ ≤ (1 − cos θ alias ) sin θ center ∼ 1 2 θ center θ 2 alias ,(B.5)8 In contrast to the convention used in this paper, the d max and d field unitis meter instead of unit of λ in the second form of the criterion, in orderto explicitely show the dependence on the wavelength.where θ center is the angle from the pointing/field center and θ aliasis the anti-aliasing scale defined in Sect. 4.2. In particular, ɛ is0 at the phase center of each pointing/field. In other words, thisscheme limits the magnitude of the w-axis distortion to its magnitudeon a size equal to the anti-aliasing scale (i.e., a few timethe primary beamwidth) instead of a size equal to the total mosaicfield of view. This scheme thus solves the projection effectas long as the w-axis distortion is negligible at sizes smaller thanor equal to the anti-aliasing scale. A natural name for this processingscheme is uvw-unfaceting because it is the combinationof a faceting observing mode (i.e., regular change of phase center)and a linear transform of the uv coordinates to derive a singlesine projection for the whole field of view.Saultetal.(1996b) also demonstrate that the reprojectionmay be done much more easily and quickly in the uvw space beforeimaging the visibilities because it is then just a simple transformationof the uv coordinates, followed by a multiplicationof the visibilities by a phase term. Finally, Sault et al. (1996b)note that it is the linear character of this uv coordinate transformwhich preserves the measurement Eq. (1). As the change of coordinateshappens before any other processing, it also conservesall the equations derived in the previous sections to implementthe wide-field synthesis.Appendix C: On-the-fly observing modeand effective primary beamUsual interferometric observing modes (including stop-and-gomosaicking) implies that the interferometer antennas observe afixed point of the sky during the integration time. Conversely,the on-the-fly observing mode implies that the antennas slew onthe sky during the integration time. This implies that the measurementEq. (1) must be written as (Holdaway & Foster 1994;Rodríguez-Fernández et al. 2009):V ( )û p , ˆα s =∫1t0⎡+δt/2 ∫⎢⎣ B { α p − α s (t) } I ( )α p e−i2πα p u p (t) dα p⎤⎥⎦ dt, (C.1)δt t 0 −δt/2 α pwhere δt is the integration time and û p and ˆα s are the mean spatialfrequency and direction cosine, defined as∫ t0 +δt/2∫ t0 +δt/2û p ≡ 1 u p (t)dt and ˆα s ≡ 1 α s (t)dt. (C.2)δt t 0 −δt/2δt t 0 −δt/2In this section, we analyze the consequences of the antenna slewingon the accuracy of the wide-field synthesis.C.1. Time averagingIn all interferometric observing modes, it is usual to adjust theintegration time so that u p (t) can be approximated as û p .Todothis, it is enough to ensure that u p (t) always varies less than the uvdistance associated with tolerable aliasing (d alias , see Sect. 4.2)during the integration time (δt)d aliasδt ≪d max ω earthordt1s ≪ 6900 ,θ alias /θ syn(C.3)where d max is the maximum baseline length, ω earth is the angularvelocity of a spatial frequency due to the Earth rotation(7.27×10 −5 rad s −1 ), θ alias and θ syn are respectively the minimumfield of view giving a tolerable aliasing and the synthesized beamangular values.Page 19 of 21