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

Table C.1. Definition of the symbols used to explore the influence ofon-the-fly scanning on the measurement equation.A&A 517, A12 (2010)tel-00726959, version 1 - 31 Aug 2012Symbol & Definitionδt Integration timeˆα s Scanned angle averaged during δtû p Spatial frequency averaged during δtδα s Angular distance scanned during δtv slew Slew angular velocity of the telescopeA Primary beam apodizing functionB eff Effective primary beam resultingfrom OTF scanning: B eff (α) = {B ⋆ A}(α)ω earth Angular velocity of a spatial frequencydue to Earth rotationC.2. Effective primary beamAssuming that condition (C.3) is ensured, we can write Eq. (C.1)with the same form as Eq. (1) by the introduction of an effectiveprimary beam (B eff ); i.e.,V ( û p , ˆα s)=∫α pB eff(αp − ˆα s)I(αp)e−i2πα p û pdα p , (C.4)where∫ t0 +δt/2( ) 1B eff αp − ˆα s ≡ B { α p − α s (t) } dt.δt t 0 −δt/2Using the following change of variablesβ ≡ α s (t) − ˆα s , dβ = dα s(t)dβdt or dt =dtv slew (β) ,we derive( )∫B eff αp − ˆα s = B {( ) }α p − ˆα s − β A (β) dβwithA (β) ≡andδα s ≡( )1 βv slew (β) δt Π δα s∫ t0 +δt/2t 0 −δt/2βv slew (t)dt.(C.5)(C.6)(C.7)(C.8)(C.9)In these equations, v slew (β) is the slew angular velocity of thetelescope as a function of the sky position, δα s is the angulardistance covered during δt, A is an apodizing function, and Π(β)is the usual rectangle function, which reproduces the finite characterof the time integration.C.3. InterpretationThe form of the measurement equation is conserved when averagingthe visibility function over a finite integration time, aslong as the true primary beam is replaced by an effective primarybeam, which is the convolution of the true primary beamby an apodizing function. To go further, it is important to returnto the two dimensional case. Indeed, the convolution mustbe done along the slewing direction, resulting in an effective primarybeam elongated in a particular direction.In principle, the equations derived in Sect. 3 can be accommodatedjust by replacing the true primary beam by its effectivePage 20 of 21Fig. C.1. Assessement of the relative error implied by the use of thetrue primary beam instead of the effective primary beam when analyzinginterferometric on-the-fly data sets. Left: inverse Fourier transformof interferometer primary beam, B (i.e. the autocorrelation of the antennaillumination). Right: relative error as a function of sampling rateof the primary beam. The curves of different colors show the results atdifferent normalized uv distances (u/d prim ) from the center of B.associate. In practice, the probability to take into account the effectiveprimary beam is low because its shape varies with time.Indeed, it is often assumed that the sky is slewed along a straightline at constant angular velocity. Even in this simplest case, it isadvisable to slew along at least two perpendicular directions toaverage systematic errors, implying two different effective primarybeams. However, practical reasons may/will lead to complexscanning patterns: 1) the limitation of the acceleration whentrying to image a square region leads to spiral or Lissajous scanningpatterns; 2) the probable absence of derotators in futuremulti-beam receivers (B. Lazareff, private communication) impliesthe need to take into account the Earth rotation in the scanningpatterns of the off-axis pixels.C.4. Approximation accuracyIn the following, we thus ask what is the trade-off accuracy of usingthe true primary beam instead of the effective primary beam.The first point to mention is that using different scanning patternssomehow helps because the averaging process then makesthe bias less systematic. Following Holdaway & Foster (1994),we quantify the accuracy lost in the Fourier plane. Indeed, theEkers & Rots scheme tries to estimate missing sky brightnessFourier components from their measurements apodized by theFourier transform of the primary beam. In the Fourier space,the above convolution just translates into a product. The Fouriertransform of the apodizing function thus degrades the sensitivityof the measured visibility, V(u p , α s ), to spatial frequencies at theedges of the interval [u p − d prim , u p + d prim ]. To guide us in ourquantification of the accuracy lost, we now explore the simplestcase of linear scanning at constant velocity, where v slew (β) is constantand δα s = v slew δt. The Fourier transform of the apodizingfunction is then a sinc function:A (u) = sinc (u δα s ) .(C.10)The relative error implied by the use of the true primary beaminstead of the effective primary beam is thenB eff (u) − B (u)= 1 − 1B eff (u) A (u) = 1 − 1sinc (u δα s ) · (C.11)Figure C.1 shows this relative error as a function of the numberof samples per primary beam FWHM in the image plane (i.e.,