Lecture 19

1. Lecture 19 • MEM • Calibration • Self-calibration • VLBI • Spectroscopic interferometry • Finishing with a couple of nice movies.

2. MEM – the Maximum Entropy Method. • References: (there are many more) • Cornwell T J, chapter 7, NRAO 1985 Synthesis Imaging Summer School. • Cornwell T J & Evans K F, Astron. Astrophys. 143, 77 (1985) • Notation: • Ij are the pixel values of the ‘true’ sky image. • Djrepresent the pixel values of the data (ie the dirty image). • Mj – an image representing a priori information. • Rj are the pixel values of the solution image. • (2-dimensional images are here represented by a single pixel index j just for brevity.)

3. MEM – the Maximum Entropy Method. Data Dj Prior information Mj Solution Rj ? ? …We need ways to measure the ‘distance’ between these various images.

4. MEM – the Maximum Entropy Method. Data Dj Prior information Mj Solution Rj ? Chi squared: …but this is not really valid because the adjacent pixels of the dirty image are not independent. We would need to include covariances σjk.

5. MEM – the Maximum Entropy Method. Data Dj Prior information Mj Solution Rj ? Chi squared: Better: where Vk are the measured visibilities and

6. MEM – the Maximum Entropy Method. Data Dj Prior information Mj Solution Rj Chi squared: ‘Entropy’: (Other choices are possible.)

7. MEM – the Maximum Entropy Method. Data Dj Prior information Mj Solution Rj Chi squared: ‘Entropy’: (Other choices are possible.) • The actual algorithm employed in interferometry maximizes H subject to the constraint that χ2 equals the expected value. • Constrained optimization can be done via the technique of Lagrange Multipliers (too advanced for this course). • Additional constraints (eg on the total image flux ΣRj) can easily be added.

8. MEM – the Maximum Entropy Method. • Choice of M? • Usually flat. • M can be adjusted during the procedure • as can other parameters. • MEM performs well on extended sources • complements CLEAN which is better for compact sources.

9. Calibration • So far we have been assuming that geometry is the only source of phase differences. • But phase (and amplitude) may also vary because of • Imperfections and/or instabilities in the electronics; • Changes in gain originating in properties of the primary beam; • Changes in refraction and path length through • the atmosphere (troposphere); • the ionosphere. • All these need to be calibrated out. Most important for an observer.

10. Calibration - the troposphere • Phase delay due to troposphere varies with zenith angle α (approx as 1/cos α). • Although for α -> 90°, curvature of the earth stops this going to infinity.

11. Calibration - the troposphere • Components: • ‘Dry air’ component: • Varies slowly. • Range in variation is small. • Directly measurable (via pressure). • ‘Wet’ component (ie, water vapour): • Can vary quickly. • Can vary over a large range. • Unpredictable... • although H2O radiometers can help. • Typical addition to ‘optical path length’ is over 2 metres.

12. Calibration - the ionosphere • Variation with zenith angle is very similar to that for the troposphere.

13. Calibration - the ionosphere • Excess path length varies with frequency as • where Ne is the electron column density in m-2. Note that ΔL is negative because the refractive index is negative (yes it is weird). • A typical value is about 1 m. • Varies by a factor of 5 between day and night; • Also large variation during ‘solar storms’. • The variation with ν means one can calculate it by simultaneous measurements at different frequencies.

14. Calibration • Methods of calibration include: • Internal monitoring of antenna parameters (eg pointing, gain), and corrections calculated from these. • Infrequent observation (usually by observatory support staff) of calibration sources to calibrate a host of antenna parameters which vary only slowly with time. • Once-per-observing-session observation of a ‘primary flux calibrator’. • Frequent during-observation peeks at a ‘phase calibrator’. • The last two are the most important for an observer to know about.

15. Calibration – some maths. • A measurement of visibility V~j,k between antennas j and k is related to the ‘true’ visibility Vj,k by • where Gj,k is the complex gain, εj,k is an offset and ηj,k is noise. • Gj,k can be resolved into a product gjg*k of complex-valued antenna gainsgj and gk, which depend purely on the antennas j and k respectively, times a so-called ‘closure error’ gj,k. (Note: all these are complex numbers.)

16. Calibration – some maths. • Since the correlation is a digital operation, and thus not subject to analog-type errors, most errors occur upstream of the correlation, and are therefore antenna-specific. • Hence, the closure error gj,k is usually close to 1+i0, and can be ignored, at least at a first pass.. • Similarly, the offset εj,k is usually small and ignorable. • Noise ηj,k can be minimized through integration over time.

17. Calibration – some maths. • Thus we can write, to first order approximation, • where aj and ak are (real-valued) amplitude errors and φj-φk is the (real-valued) phase-difference error. • Note that both as and φs vary with time in an unpredictable fashion. (φ much more than a.) • Most of this variation is due to random changes in the atmosphere + ionosphere.

18. Calibration – some maths. • Note that there are only 2N quantities to calculate (1 a and 1 φ per antenna) but we have N(N-1) ~ N2 measurements (a real and an imaginary value from each of N(N-1)/2 correlations). This is a happy situation whenever N>4 (as it is for all modern arrays of any importance). • This means that we don’t need measurements of V~ from all baselines – which turns out to be useful for reasons shortly to be revealed.

19. The ideal calibrator • Position and structure: • point-like; • not confused; • of exactly known position; • near the object of interest. • Flux: • strong; • doesn’t vary with time; • smooth, flat spectrum. • Different situations impose differing relative emphasis.

20. The real calibrator • Of course (just like with people) we have to make do with calibrators which never meet all these desirables at once, and sometimes don’t meet any of them! • Some trade-offs: • To find a calibrator near our object, we might have to accept one which is not as strong as we’d like (n ~ S-5/2 rule). • A compact calibrator is usually time-variable, sometimes on timescales as short as a day. • We may have to discard data from short baselines (because the calibrators are confused) and also from long ones (because they resolve the calibrator).

21. ‘Goldilocks’ baselines What short baselines see: confused. What long baselines see: resolved. What just-right BL see: isolated points.

22. A simulated observation – V(t) from 1 baseline: Amplitudes shown here. Telescopes are moving between the target (long cycle) and the calibrator (short cycle).

23. A simulated observation – V(t) from 1 baseline: Phases shown here – all cycles. Telescopes are moving between the target (long cycle) and the calibrator (short cycle).

24. A simulated observation – V(t) from 1 baseline: Phases shown here cal cycles only. Telescopes are moving between the target (long cycle) and the calibrator (short cycle).

25. Special calibrations: • Spectral-line: • requires band-pass calibration. • Polarization: • ‘leakage terms’; • Most modern feeds have 2 detectors, of opposite polarisation; • but their mutual isolation is not perfect. • variation in polarization across the beam; • A kind of polarized spherical aberration. • ionospheric Faraday rotation.

26. Self-calibration • Ordinary calibration using an offset source relies on: • interpolation between peeks at the cal; • the assumption that the phase shift in the direction of the cal is the same as towards the target. • Variations due to imperfection of these assumptions is usually >> thermal noise. • Effect of this depends on size of array: • Small array: may degrade image a little. • VLBI: completely impossible to calibrate this way! • So - can we do better...?

27. Self-calibration • Suppose we knew I(l,m), the brightness distribution of the target. • Fourier transform this to get a continuous model visibility function V^(u,v). • Divide the measured visibility samples V~j,k(u,v,t) for baseline j-k by this model V^. • Is not the result the V we’d get from a point source, multiplied by the product of the antenna ‘gains’ gjg*k (slide 8)? • Then just perform normal calibration of gj and gk.

28. Bootstrap • But... I is what we are trying to find out! • However – an iterative approach works well, provided: • the target is fairly bright; • a starting model can be obtained. • A quite separate approach utilizes redundant spacings – 2 or more visibility samples which lie very close to each other in the u-v plane. • Westerbork was specifically designed to facilitate this – lots of antennas are the same distance apart.

29. Very Long Baseline Interferometry (VLBI) • Maths is the same as standard interferometry – it is the practical details which tend to be different. • Antennas are thousands rather than tens of km apart. Hence: • Resolution can be 1 milliarcsecond (mas) or even lower. • It isn’t practical to lock all antennas to the same frequency standard (LO). Each has its own... • thus the LOs must be very stable (eg H maser). - $$$ • Data are stored on tape and correlated later. • With the longer baselines, the array is proportionately more sensitive to phase errors... • so special calibration techniques are needed.

30. VLBI – a typical image: CSO 1943+546 Polatidis et al NewAR 43, 657 (1999) Courtesy the EVN archive.

31. Spectroscopy • What physical environment is associated with spectral lines? • Isolated atoms (may be partially ionized). • Spectral lines can be observed both in emission and absorption. • Information coming out of spectroscopic interferometry: • composition, column densities, temperature, velocity (both ‘bulk’ or nett, and velocity dispersion). • Processing: • each channel imaged and CLEANed separately.

32. Spectroscopy • We are now measuring not just x and y but also a 3rd axis, ν. Hence we deal not in images but in data cubes. • A challenge to handle • Can be huge: 20482 x 1024 pixels = 16 GB • A challenge to visualize • Channel maps and movies • Moment maps: intensity, velocity, width • 3D rendering and visualization software… • A challenge to analyse… • Model for structure & kinematics • Optical depth effects • Excitation: collisions vs radiation

33. M82 cube movie Shows HI absorption – different velocities (therefore different Doppler-shifted frequency) at different places. Many thanks to Rob Beswick. Continuum plus line M82 HI (Wills, Pedlar et al)

34. HI imaging • Valuable for cosmology: • Traces galaxies  large-scale structure. • Also shows kinematics inside galaxies. • But, it’s hard to get high resolution... • because the brightness temperature of HI is limited to about 100 K; • bandwidth is limited; • wavelength is relatively long; • and, for an interferometer, Ae is notλ2/ΩA, as is true for a single dish. Ae stays fixed (it’s a property of the antennas). • Thus smaller beam (ie ΩA) gets less flux. For HI, practical limit to ΩA (and thus resolution) is ~1 arcsec.

35. Another M82 movie, showing OH emission and absorption, with continuum subtracted. Note ringing about the brightest 1667MHz maser. Many thanks to Rob Beswick. M82 OH – 1665 & 1667 lines – masers & absorption VLA A-array data (Argo et al 2007) Two lines in band Absorption in black Masers in blue