Lecture 14 Radio Interferometry Radio Interferometry Some of

Lecture 14. • Radio Interferometry

Radio Interferometry. Some of the approximations used in this talk: • • • Bandwidth Δf is small. Neglect polarization. 2 dimensions first; then 3 later. Aperture synthesis. 2 antennas first; later many. Many other complications and technicalities have been ignored.

The fundamental quantity of interest: The phase difference between the signals received by 2 antennas.

Waves from a source at zenith: phase difference φ=0

Waves from an offset source: phase difference φ>0

Expression for the phase difference φ: Point source of flux density S watts m-2 Hz-1 at wavelength λ m. (Remember λ=c/f=2πc/ω. ) θ θ Path difference d = D sinθ θ D=uλ

Phase φ zenith angle θ. Q: How to measure phase? A: Correlate the signals from the 2 antennas. One definition of correlation: Zero-lag correlation (ie τ=0):

Zero-lag correlation between A and B: A VA(t) B VB(t) <> Low-pass filter

Zero-lag correlation between A and B: A VA(t) B VB(t) <> Low-pass filter Δφ = π/2 Hilbert transform <> Low-pass filter

Zero-lag correlation between A and B: A VA(t) B VB(t) <> Low-pass filter Δφ = π/2 Hilbert transform <> Low-pass filter Construct the complex zero-lag correlation between VA and VB: watts m 2 Hz watts m-2 Hz-1

Integrate, substitute, change-of-variable: Replace S by I(θ)dθ and integrate over the sky: Substitute φ=2πu sinθ: Change the variable to sinθ l: (and make use of I(l)=0 for |l|>1) This is a Fourier transform!

Observing strategy: • Measure the visibility function V(u). – This is a complex-valued function of u, the antenna separation in wavelengths. – We set V(-u) = V*(u) (this ensures that I is real-valued). • Fourier back-transform V to obtain I’(l) = I(sinθ)/cosθ. – I’(l) is a distorted map of the sky brightness distribution I(θ). Cygnus A – VLA 6 cm • Correct the distortion. – The result: Alas, life is usually not so simple… Courtesy Chris Carilli

The sampling problem. • In practice, V cannot be measured at all antenna separations 0≤u<∞. • Even a continuous range of u values a≤u<b is impractical. • V is measured at a finite number of antenna separations ui. • Thus what we have is not V but Vs, where • Back-transforming Vs gives I’ convolved with the socalled ‘dirty beam’ B:

Ways to increase the number of samples: • Move one of the antennas. – Slow. • Add more antennas. – For N antennas, we have N(N-1)/2 combinations of pairs, or baselines.

Problem: non-coplanar arrays. There is now no common zenith – so there is no place in the sky from which signals arrive at the correlator in phase. 2 1 3 Correlator

Solution: compensate for signal delays. • Choose a direction of interest – this will be known as the phase centre. • Calculate distances (in wavelengths) wj between each jth antenna and a projection plane normal to the phase centre. • Delay each signal. V(t) by -wj/f seconds. • Signals from a source at the phase centre will then reach the correlator in phase.

Phase centre lane u 13 u 12 w 1 w 2 2 np Projectio w 3 1 delay: t t+w 2/f delay: t t+w 1/f 3 Correlator delay: t t+w 3/f Signals from source at phase centre reach the correlator in phase.

Non-coplanar imaging: • The phase of a source at θ radians from the phase centre is • Correlation of the lag-corrected signals VA and VB gives From the delays • Approximates the original Fourier relation provided l << u/Δw.

How to get even more samples: • Use of the Earth’s rotation: Aperture Synthesis. • We now go from 2 to 3 dimensions. Antenna separations projected onto the plane normal to the phase centre are given v in u and v coordinates. The ‘u-v plane’ • The visibility function is now written V(u, v). u

View from the phase centre