1. Lecture 14. Radio Interferometry Talk at Nagoya University IMS Oct 2009 1/43

2. Talk at Nagoya University IMS Oct 2009 Radio Interferometry. Some of the approximations used in this talk: Bandwidth Δ f 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. 2/43

3. Talk at Nagoya University IMS Oct 2009 The fundamental quantity of interest: The phase difference between the signals received by 2 antennas. 3/43

4. Talk at Nagoya University IMS Oct 2009 Waves from a source at zenith: phase difference φ=0 4/43

5. Talk at Nagoya University IMS Oct 2009 Waves from an offset source: phase difference φ>0 5/43

6. Talk at Nagoya University IMS Oct 2009 θ Expression for the phase difference φ: Path difference d = D sin θ D=uλ θ θ Point source of flux density S watts m -2 Hz -1 at wavelength λ m. (Remember λ= c / f =2 πc / ω. ) 6/43

7. Talk at Nagoya University IMS Oct 2009 Phase φ  zenith angle θ. A: Correlate the signals from the 2 antennas. Q: How to measure phase? One definition of correlation: Zero-lag correlation (ie τ =0): 7/43

8. Talk at Nagoya University IMS Oct 2009 Zero-lag correlation between A and B: AB VA(t)VA(t) VB(t)VB(t) <> Low-pass filter 8/43

9. Talk at Nagoya University IMS Oct 2009 Zero-lag correlation between A and B: AB VA(t)VA(t) VB(t)VB(t) <> Low-pass filter <> Low-pass filter Δφ = π /2 Hilbert transform 9/43

10. Talk at Nagoya University IMS Oct 2009 Zero-lag correlation between A and B: AB VA(t)VA(t) VB(t)VB(t) <> Low-pass filter <> Low-pass filter Δφ = π /2 Hilbert transform Construct the complex zero-lag correlation between V A and V B : watts watts m -2 Hz -1 m 2 Hz 10/43

11. Talk at Nagoya University IMS Oct 2009 Integrate, substitute, change-of-variable: Replace S by I (θ) dθ and integrate over the sky: Substitute φ=2 πu sinθ: Change the variable to sinθ  l : (and make use of I(l) =0 for | l |>1) This is a Fourier transform! 11/43

12. Talk at Nagoya University IMS Oct 2009 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). Correct the distortion. –The result: Alas, life is usually not so simple… Courtesy Chris Carilli Cygnus A – VLA 6 cm 12/43

13. Talk at Nagoya University IMS Oct 2009 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 u i. Thus what we have is not V but Vs, where Back-transforming Vs gives I’ convolved with the so- called ‘dirty beam’ B : 13/43

14. Talk at Nagoya University IMS Oct 2009 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. 14/43

15. Talk at Nagoya University IMS Oct 2009 Problem: non-coplanar arrays. 2 1 3 There is now no common zenith – so there is no place in the sky from which signals arrive at the correlator in phase. Correlator 15/43

16. Talk at Nagoya University IMS Oct 2009 Solution: compensate for signal delays. Choose a direction of interest – this will be known as the phase centre. Calculate distances (in wavelengths) w j between each j th antenna and a projection plane normal to the phase centre. Delay each signal V ( t ) by -w j /f seconds. Signals from a source at the phase centre will then reach the correlator in phase. 16/43

17. Talk at Nagoya University IMS Oct 2009 2 1 3 Correlator Projection plane Phase centre Signals from source at phase centre reach the correlator in phase. w2w2 w3w3 w1w1 u 13 u 12 delay: t  t+w 2 /f delay: t  t+w 1 /f delay: t  t+w 3 /f 17/43

18. Talk at Nagoya University IMS Oct 2009 Non-coplanar imaging: The phase of a source at θ radians from the phase centre is Correlation of the lag-corrected signals V A and V B gives Approximates the original Fourier relation provided l << u/ Δ w. From the delays 18/43

19. Talk at Nagoya University IMS Oct 2009 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 in u and v coordinates. u v The visibility function is now written V(u,v). The ‘u-v plane’ 19/43

20. Talk at Nagoya University IMS Oct 2009 View from the phase centre 20/43