1. NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 16 Calibration Self-calibration VLBI Spectroscopic interferometry

2. NASSP Masters 5003F - Computational Astronomy - 2009 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.

3. NASSP Masters 5003F - Computational Astronomy - 2009 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.

4. NASSP Masters 5003F - Computational Astronomy - 2009 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 H 2 O radiometers can help. Typical addition to ‘optical path length’ is over 2 metres.

5. NASSP Masters 5003F - Computational Astronomy - 2009 Calibration - the ionosphere Variation with zenith angle is very similar to that for the troposphere.

6. NASSP Masters 5003F - Computational Astronomy - 2009 Calibration - the ionosphere Excess path length varies with frequency as where N e 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.

7. NASSP Masters 5003F - Computational Astronomy - 2009 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.

8. NASSP Masters 5003F - Computational Astronomy - 2009 Calibration – some maths. A measurement of visibility V ~ j,k between antennas j and k is related to the ‘true’ visibility V j,k by where G j,k is the complex gain, ε j,k is an offset and η j,k is noise. G j,k can be resolved into a product g j g* k of complex-valued antenna gains g j and g k, which depend purely on the antennas j and k respectively, times a so-called ‘closure error’ g j,k. (Note: all these are complex numbers.)

9. NASSP Masters 5003F - Computational Astronomy - 2009 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 g j,k is usually close to 1+i0, and can be ignored, at least at a first pass.. Similarly, the offset ε j,k is usually small and ignorable. Noise η j,k can be minimized through integration over time.

10. NASSP Masters 5003F - Computational Astronomy - 2009 Calibration – some maths. Thus we can write, to first order approximation, where a j and a k are (real-valued) amplitude errors and φ j -φ k 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.

11. NASSP Masters 5003F - Computational Astronomy - 2009 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) ~ N 2 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.

12. NASSP Masters 5003F - Computational Astronomy - 2009 The ideal calibrator Position and structure: 1.point-like; 2.not confused; 3.of exactly known position; 4.near the object of interest. Flux: 5.strong; 6.doesn’t vary with time; 7.smooth, flat spectrum. Different situations impose differing relative emphasis.

13. NASSP Masters 5003F - Computational Astronomy - 2009 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).

14. NASSP Masters 5003F - Computational Astronomy - 2009 ‘Goldilocks’ baselines What short baselines see: confused. What long baselines see: resolved. What just-right BL see: isolated points.

15. NASSP Masters 5003F - Computational Astronomy - 2009 A simulated observation – V(t) from 1 baseline: Amplitudes shown here. Telescopes are moving between the target (long cycle) and the calibrator (short cycle).

16. NASSP Masters 5003F - Computational Astronomy - 2009 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).

17. NASSP Masters 5003F - Computational Astronomy - 2009 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).

18. NASSP Masters 5003F - Computational Astronomy - 2009 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.

19. NASSP Masters 5003F - Computational Astronomy - 2009 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...?

20. NASSP Masters 5003F - Computational Astronomy - 2009 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’ g j g* k (slide 8)? Then just perform normal calibration of g j and g k.

21. NASSP Masters 5003F - Computational Astronomy - 2009 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.

22. NASSP Masters 5003F - Computational Astronomy - 2009 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.

23. NASSP Masters 5003F - Computational Astronomy - 2009 VLBI – a typical image: CSO 1943+546 Polatidis et al NewAR 43, 657 (1999) Courtesy the EVN archive.

24. NASSP Masters 5003F - Computational Astronomy - 2009 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.

25. NASSP Masters 5003F - Computational Astronomy - 2009 Spectroscopy We are now measuring not just x and y but also a 3 rd axis, ν. Hence we deal not in images but in data cubes. –A challenge to handle Can be huge: 2048 2 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

26. NASSP Masters 5003F - Computational Astronomy - 2009 M82 HI (Wills, Pedlar et al) Continuum plus line M82 cube movie Shows HI absorption – different velocities (therefore different Doppler-shifted frequency) at different places. Many thanks to Rob Beswick.

27. NASSP Masters 5003F - Computational Astronomy - 2009 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, A e is not λ 2 / Ω A, as is true for a single dish. A e stays fixed (it’s a property of the antenna). Thus smaller beam (ie Ω A ) gets less flux. For HI, practical limit to Ω A (and thus resolution) is ~1 arcsec.

28. NASSP Masters 5003F - Computational Astronomy - 2009 Another M82 movie, showing OH emission and absorption, with continuum subtracted. M82 OH – 1665 & 1667 lines – masers & absorption VLA A-array data (Argo et al 2007) Note ringing about the brightest 1667MHz maser. Two lines in band Absorption in black Masers in blue Many thanks to Rob Beswick.