INTERFEROMETRY: II Nissim Kanekar (NCRA TIFR)

INTERFEROMETRY: II Nissim Kanekar (NCRA TIFR) WSRT GMRT VLA ATCA ALMA SKA MID

PLAN Introduction. The van Cittert Zernike theorem. A 2 element interferometer. The fringe pattern. 2 D and 3 D interferometers. Array configurations and U V coverage. The effect of bandwidth. References Synthesis Imaging in Radio Astronomy II Low Frequency Radio Astronomy Interferometry and Synthesis in Radio Astronomy (Taylor, Carilli, Perley) (Chengalur, Gupta, Dwarakanath) (Thomson, Moran, Svenson)

INTRODUCTION Radio interferometer: A set of radio antennas that observe simultaneously to effectively yield a telescope of much larger size, but with an incompletely filled aperture. The effective telescope size is roughly the largest antenna spacing. The antenna locations are where the effective telescope samples radiation from sources in the sky. These locations are shifted around by the rotation of the Earth, giving better sampling. Connected element interferometry: Baselines <~ 50 km. E.g. the Very Large Array (27 dishes, 35 km, 1 50 GHz), the Giant Metrewave Radio Telescope (30 dishes, 25 km, 0.14 1.4 GHz), the Atacama Large Millimeter Array (50 dishes, 16 km, 30 900 GHz). Very Long Baseline Interferometry (VLBI): Baselines ~ 10,000 km. The Very Long Baseline Array (0.3 100 GHz, 10 dishes, 8600 km).

THE VAN CITTERT ZERNIKE THEOREM E(r,t) is the field at r due to a sky element at R'. Since the sources are far away, we assume that the emission arises from the celestial sphere, at R. E(R,t R/c) For the quasi monochromatic field components, E (r) E (r) = E (R). e 2πi. R r /c R r R E(r, t) where we have assumed free space propagation, and that the electric field is a scalar. R' r The spatial correlation function of this field at two points r1, r2 is dv (r1,r2) = E (r1).e *(r2) = E (R1).E *(R2). e 2πi. R1 r1 /c. e2πi. R2 r2 /c R1 r1. R2 r2

THE VAN CITTERT ZERNIKE THEOREM We assume that the source field is not spatially coherent (critical!!). dv (r,r ) = E (R) 2. e 2πi. R r1 /c. e2πi. R r2 /c 1 2 R r1. R r2 We assume R ri, and define the unit vector s = R/ R. Integrating over all source elements, using ds = R 2 dω, R ri, and defining the observed intensity I (s) = E (s) 2, we obtain V (r1,r2) = I (s). e 2πi s.(r1 r2)/c dω = I (s). e 2πi s.b/c dω The right side depends on (r1,r2) only through the vector b = r1 r2! V (b) V (r1 r2) is called the spatial coherence function on the baseline, b = r1 r2.

THE VAN CITTERT ZERNIKE THEOREM w The baseline b has components λ(u,v,w). s and s0 are unit vectors, towards the source s n and the reference direction. σ = s s0. The (u,v,w) co ordinates of s are the direction cosines l = Cos(α), m = Cos (β), n = Cos(γ). Note that l2 + m2 + n2 = 1 l m b v u νb.s/c = (ul + vm + wn) = ul + vm + w.(1 l2 m2)1/2 dω = dl.dm/n = (1 l2 m2) 1/2 dl.dm νb.σ/c = ul + vm + w.{(1 l2 m2)1/2 1} For the above co ordinate system, 2πi[ul+vm+w{(1 l2 m2)1/2 1}] V (u,v,w) = I (l,m).e.(1 l2 m2) 1/2 dl.dm Note: This is not a Fourier transform relation!!!

SPECIAL CASES All measurements in a single plane, i.e. w 0. V (u,v,w) = I (l,m). e 2πi[ul+vm].(1 l2 m2) 1/2 dl.dm Fourier transform between V (u,v) and I (l,m) (1 l2 m2) 1/2! All sources in a small region: i.e. l, m 1 for non zero I (l,m). V (u,v) = I (l,m). e 2πi[ul+vm].(1 l2 m2) 1/2 dl.dm Fourier transform between V (u,v) and I (l,m) (1 l2 m2) 1/2! One measures V (u,v) and obtains the sky intensity I (l,m) by I (l,m) = V (u,v). e2πi[ul+vm] du.dv

THE PRIMARY AND SYNTHESIZED BEAMS V (u,v) is only measured at specific locations on the u v plane, given by the sampling function S(u,v) (0 where there's no data). The synthesized beam B(l,m) is the Fourier transform of S(u,v). B(l,m) = S(u,v). e2πi[ul+vm] du.dv (the PSF ). The dirty image ID (l,m) is the Fourier transform of [S(u,v)V (u,v)]. ID (l,m) = V (u,v). S(u,v).e2πi[ul+vm] du.dv Convolution theorem ID (l,m) = I * B V (u,v) is measured with antennas, with a reception pattern A (l,m). Thus, in reality, V (u,v) = A (l,m).i (l,m). e 2πi[ul+vm] dl.dm A (l,m) is the primary beam. For dishes, it's very directional, and so helps the assumption that the emission is from a small region.

RADIO INTERFEROMETRY The spatial correlation function V(r1, r2) E(r1).E*(r2) of the signals measured by antennas at r1 and r2 depends only on (r1 r2) and is the 2 D Fourier transform of the sky intensity distribution. Vν(u,v) Aν(l,m). Iν(l,m) e 2πi(ul + vm) dl dm (u,v) are components of (r1 r2); (l,m) are direction cosines in the sky. Measure cross correlations of the voltages determined at different antennas, as a function of their separation. Then carry out a 2 D Fourier transform to infer the sky intensity distribution! As the Earth rotates, the separation of a pair of antennas relative to the source direction will change! Each antenna pair measures V(u,v) at a changing (u,v) location with time, yielding a curve in the u v plane, and hence, better sampling of this plane! Earth rotation Aperture Synthesis

A 2 ELEMENT INTERFEROMETER s s τg = b.s/c V=V1Cos[ω(t τg)] Multiply Average θ b X An antenna V=V2Cos[ωt] [Cos(ωτg) + Cos(2ωt ωτg) ]/2 rc = [V1V2 Cos(ωτg)]/2 = [V1V2 Cos(2πνb.s/c)]/2 The received signal power in a bandwidth Δν from the source element dω is A(s).I(s).dΩ.Δν, where A(s) is the effective area. The correlator output from dω is proportional to the signal power and the cosine term drc = A(s).I(s).dΩ.Δν.Cos(2πνb.s/c)

A 2 ELEMENT INTERFEROMETER The monochromatic correlator output from the full sky is then rc = A(s). I(s). Cos(2πνb.s/c) dω But... Since the cosine function is even, the sky integral is only sensitive to the even part of I(s); the odd part is not included! This is a cosine correlator. For the odd part, introduce a 90 phase shift in one of the paths. This sine correlator output is rs = Δν A(s).I(s).Sin(2πνb.s/c) dω Define the visibility as V = (rc + irs) = A(s). I(s). e 2πiνb.s/c dω Measure the amplitude and phase of the correlator output, and infer the visibility amplitude V and phase φv by calibration! Thus, need a complex correlator to fully recover the sky intensity.

THE FRINGE PATTERN The cosine correlator output is rc = A(s).I(s). Cos(2πνb.s/c) dω Throws a sinusoidal fringe pattern on the sky, multiples the sky intensity by the fringe pattern, and then integrates over the sky. λ/b rad. Fringe orientation set by the baseline geometry. Fringe separation ( resolution!) set by the baseline length B. Short baselines Wide fringes. Long baselines Fine fringes. Source brightness + + + Fringe sign (e.g. Perley) Need a wide range of baseline lengths and orientations to fully sample the sky intensity distribution! Fine structure only detected with narrow fringes, while total flux density needs wide fringes.

THE 2 D INTERFEROMETER All measurements in a single plane, i.e. w = 0. V (u,v,w) = I (l,m). e 2πi[ul+vm].(1 l2 m2) 1/2 dl.dm Fourier transform between V (u,v) and I (l,m) (1 l2 m2) 1/2! Arrays like the GMRT and the VLA instantaneously have their antennas in one plane. As the Earth rotates, the plane changes. Can only use the above approximation for very short observations. Only East West arrays (e.g. WSRT, ATCA) have baselines that always lie in a plane. Earth rotation does not produce a baseline component towards the celestial pole (chosen for the w direction!). But... Full resolution only obtained towards the celestial pole! In other directions, resolution worse by Cos(δ). Zero resolution at the celestial equator!!!

THE 2 D INTERFEROMETER WSRT resolution at different declinations. δ~ 2 δ=34 (Carilli et al. 1996)

THE 3 D INTERFEROMETER w If all sources in a small sky region, i.e. l, m 1 for non zero A (l,m).i (l,m). Co ordinate system: u toward East, v toward North, w towards the direction of interest, s0. Positions on the sky have (l, m) co ordinates, direction cosines relative to the (u,v) axes. s n l m b u V (u,v,w) = A (l,m). I (l,m).e 2πi[ul+vm+w(n 1)].(1 l2 m2) 1/2 dl.dm If l, m 1 w[(1 l2 m2)1/2 1] w(l2 + m2)/2 0. If the phase error πw(l2 + m2) is ~ 1, then will incur large errors. The antenna pattern A (l,m) cuts off the sky intensity distribution! But, if the field of view is large and the sky is full of sources (e.g. at the low GMRT frequencies), must use 3 D corrections. v

GENERAL ISSUES Sensitivity Collecting area D (N = no. of antennas). Angular resolution ~ /L (where L is the longest baseline). For GMRT, L ~ 25 km 21cm angular resolution ~ 2''. For VLBA, L ~ 8,600 km 21cm angular resolution ~ 5 mas! If L1 and L2 are the longest and shortest baselines Information on the radio emission on angular scales between ( /L1) & ( /L2). The image of the sky is obtained via a 2 D Fourier transform of the cross correlations measured on the ground Requires that the antenna baselines cover the u v plane as uniformly as possible! Distribute the N antennas so as to obtain the best (most uniform) coverage of the 2 D u v plane, for all directions.

ARRAY CONFIGURATIONS Y shaped array optimal for ~ 20 40 antennas (VLA, GMRT). Random or spiral array for more than ~ 50 antennas. VLA: 27 antennas on rails. Moved every 4 months to optimize the coverage (1, 3.3, 11, 35 km). GMRT: 30 fixed antennas, in an optimal distribution (25 km).

ARRAY U V COVERAGE The set of all baseline vectors during the observations. Its 2 D FT is the synthesized beam. Earth rotation causes baselines to move in the u v plane. Observing a source for even a few hours hence gives a much better u v coverage and a cleaner beam. 5 min 1 hr δ=45 3 hr 12 hr (Image courtesy of Craig Walker and NRAO/AUI)

ARRAY U V COVERAGE: THE GMRT 5 min δ=45 3 hr δ=45 δ=45 1 hr 12 hr δ=45 12 hr δ=0 12 hr δ= 45 (Images courtesy of Jayaram Chengalur) Dense patches due to large number of central square antennas. Poor u v coverage at very southern declinations; poor v resolution.

THE EFFECT OF BANDWIDTH The correlator output visibility V (u,v) = A (s).i (s).e 2πiνb.s/c dω For a bandwidth Δν, and assuming the signal path, A (s), and I (s) are all independent of frequency, V(u,v) = A (s).i (s)dω. (1/Δν) e 2πiνb.s/c dν = [Sin(π.Δν.τg)/(π.Δν.τg)] A (s).i (s). e 2πiντg.dΩ. For finite bandwidth, the fringes are modulated by a sinc envelope! Full fringe amplitude only obtained for τg = 0 (i.e. the meridian). High attenuation if Δν.τg = Δν.(B/c).sinθ ~ (Δν/ν).(θ/θres) ~ 1. Problem solved by delay tracking, i.e. inserting a time delay in one of the signals to move the unattenuated region to the target. Accompanied by fringe stopping, to compensate for any extra geometric delay, if the delay tracking is not done at the RF.

A 2 ELEMENT INTERFEROMETER Incoming radiation Secondary Primary Receiver Secondary Primary Low noise amplifier Mixer Intermediate frequency Receiver Low noise amplifier Local oscillator Mixer Intermediate frequency Delay tracking, Fringe stopping Cross correlator