1. Sensitivity & Polarization Huib Jan van Langevelde

2. IAC06 #4 15 mar 20062 Logistics  We pass the attendance list around  Please inform us why you cannot make specific dates  Courses will run 11:00 – 15:30/16:00  Coffee break 11:45 – 12:00  Lunch break 12:45 – 13:30  10 May collides with NAC  17 May is no option (no lecture room)  Proposal:  May 10: no lecture  May 17: Tour of Dwingeloo  May 24: last lecture (in Utrecht)  Most people should have their book  Have 3 more books (4 in fact)  Need 2 more payments…

3. IAC06 #4 15 mar 20063 Outline  Sensitivity  What is a good telescope, interferometer  Receiver temperatures, efficiencies and losses  What sources can I still detect  Flux versus brightness  Sensitivity in the images  Dynamic range limitations  Brightness vs flux, resolving out sources  Polarimetry, a brief introduction (by a non-expert)  Electric field and Stokes  Linear and circular feeds  Some calibration issues

4. IAC06 #4 15 mar 20064 Radio emission  What can you learn from radio emission  Morphology of course, see imaging lectures  Emission mechanisms, determine spectra  Thermal vs non-thermal emission  Will approximate black body  When optically thick  Non thermal S≈ν α  Spectral energy distribution  Positions, astrometry  Polarization properties  Magnetic fields, dust scattering  Measure of radio source strength is its flux density S ν  Unit of flux density: Jansky (Jy) = 10 −26 W Hz −1 m −2  And brightness when divided by ΔΩ e.g. Jy/beam, K

5. IAC06 #4 15 mar 20065 Antenna temperature  Received temperature can be associated with T  Power equivalent to a black body of temperature T filling the entire beam  In the Rayleigh-Jeans approximation (hν«kT):  Usually first an amplification factor G

6. IAC06 #4 15 mar 20066 Measures of Antenna Performance  So source power equals:  But source power is also equal to flux in dish:  Antenna area A, efficiency  a  Rx accepts 1/2 radiation from unpolarized source  Define scaling factor K  K is antenna’s gain or “sensitivity”, unit degree Jy  1

7. IAC06 #4 15 mar 20067 Measures of Antenna Performance  K measures antenna performance but no Tsys  Interested in detecting relative increase of noise  Define ‘Source Equivalent Flux Density’  Equals the flux of a source that doubles the system temperature  Typically 10 – 1000 Jy from large dishes to smaller telescopes  T sys has various contributions  T bg from 3K background and Galactic radio emission  T sky from the sky  T spill spilling over from the ground and other directions  T rx from the receiver electronics  And at times T cal from calibration signal

8. IAC06 #4 15 mar 20068 Interferometer Sensitivity  Simple correlator with single real output that is product of voltages from antennas j,i:  But for sensitivity we have to evaluate  For the math see the book  Assuming the noise contributions dominate over the sky flux and the correlated part is small:

9. IAC06 #4 15 mar 20069 Baseline sensitivity  Source, system noise powers imply sensitivity  S ij [Jy]  Derived for weak sources and noise dominating system  Sensitivity scales with root bandwidth  And accumulation time  acc  Noise will go down over time, but signal to noise will only increase if the signal can be added coherently  Big antennas give sensitive baselines even to small antennas  100m + 10m more sensitive than 2 25m  But contains more steel too  And the system efficiency  s accounts for losses in data handling

10. IAC06 #4 15 mar 200610 Quantization Losses

11. IAC06 #4 15 mar 200611 Interferometer Sensitivity  Complex correlator delivers similar noise on real and imaginary  Eg: VLBA continuum  A = 25m for both  η A = 0.75  T sys = 40K  SEFD = 301 Jy  BW = 16 MHz  η s = 0.56  t acc =2s  Predicted  S = 69 milliJy  Resembles observed scatter

12. IAC06 #4 15 mar 200612 Note amplitude scatter  Measured visibility amplitude:  Will approach standard deviation of both real and imaginary part  But cannot be zero when there is no signal  Called amplitude bias

13. IAC06 #4 15 mar 200613 And phase  Measured visibility phase  m = arctan(S I /S R )  Standard deviation approaches S/  S Gaussian for high signal to noise  Absence of signal: phase will be uniform  Seek weak detection in phase, not in amplitude

14. Real Imaginary Amplitude Phase

15. IAC06 #4 15 mar 200615 Averaging scalar vs vector  Vector averaging means “normal” complex adding  To be done in time or across spectrum  Calibrated data added up in phase  So for positions of synthesized beam in case there is uv spread  Uncalibrated data may cancel  Scalar averaging is taking average of amplitude  Biased to non-zero value, so be careful!  Could reveal spectral signature, even before calibration vector scalar Very little difference in case of reasonable signal to noise

16. IAC06 #4 15 mar 200616 Calibrator sensitivity  Previous example gave 69mJy noise in 2 sec and 16 MHz band on each baseline  So in 5 minutes 5mJy per baseline  So a 100mJy source gives S/N of 20  Sufficient to track amplitude and phase calibrator  In order to determine bandpass response  Need to determine relative response of 10kHz band  Maybe with S/N =100  Need a 10 Jy calibrator, observed for 24 minutes!

17. IAC06 #4 15 mar 200617 Image Sensitivity  Start with baseline sensitivity equal telescopes  Form image through Fourier transforming all these visibilities into image representation  Which is a linear operations  Every beam is sum of all visibilities  With appropriate phase term  Possibly modified by weighting scheme  W=1 for natural weighting  Image sensitivity from averaging all visibilities  Assuming identical antennas  These all have same noise

18. IAC06 #4 15 mar 200618 Image senitivity  Image sensitivity is expressed as standard deviation after taking the mean of L samples, each with standard deviation  S  N antennas, number of interferometers ½ N (N  1)  number of accumulation times t int /  acc  So L = ½ N (N  1) (t int /  acc )

19. IAC06 #4 15 mar 200619 Image Sensitivity  Same example  10 VLBA antennas  Observe for 1hr  69 mJy per baseline  4 IF bands  L = 77200  Predict 88 μJy/beam  Observed  Stokes I, simplest weighting  Gaussian noise  I = 90 μJy/beam

20. IAC06 #4 15 mar 200620 Detection thresholds  The formula above gives the detection statistics for point sources, unresolved  But note that you may expect 1 point S>3σ if you image 1000 beams  Map will contain maybe 1024x1024/3x3 = 100,000 beams!  And most maps which contain flux will be dynamic range limited  Calibration errors scatter small fraction of flux  Effort needed to reach 1:1000, possible 1:10,000-100,000  Estimates of the thermal noise may be obtained from Stokes V (RR-LL), has the same expected noise but much less signal  Or from difference of spectral channels

21. IAC06 #4 15 mar 200621 Stokes V to measure noise I map 7.8 mJy off sourceV map 1.5 mJy whole map

22. IAC06 #4 15 mar 200622 Most sensitive images reach μJy WSRT image of Spitzer deep field reaches 8.5 μJy, Morganti et al 2004 VLBI reaches 50 μJy range, still limited by recording and storage resources Garrett et al 2001

23. IAC06 #4 15 mar 200623 Position accuracy  Signal to noise also important for position accuracy  I peak = 2 milliJy beam  1  Gaussian noise  I = 90 microJy beam  1  Position error from sensitivity?  Gaussian beam  HPBW = 1.5 milliarcsec  Then  = 34 microarcsec  At this point other position errors dominate

24. IAC06 #4 15 mar 200624 Resolved, unresolved, resolved out  But this cannot be the whole story  VLBI combines all the telescopes and be the most sensitive  And it yields the best resolution  So why is not everybody doing VLBI all the time?  Longest and shortest baselines define minimum and maximum angular scale  Resolved sources: gaussian in the image, gaussian in the uv  Fitting in the image or uv-domain  Source structure inside shortest baseline resolve out  Interferometer cannot detect this  Note that polarization or absorption structure on smaller scales may be detectable  Brightness sensitivity is a useful way to look at this:

25. IAC06 #4 15 mar 200625 Brightness sensitivity:  Express the sensitivity of array in terms of T b  The sensitivity of an array with N identical dishes of size A is given by:  Convert to Intensity ΔI = ΔS/ΔΩ  Approximate the resolution as ΔΩ=πθ 2 ≈ (λ/L) 2  An use the Rayleigh-Jeans approximation  Not really valid in mm regime…

26. IAC06 #4 15 mar 200626 Brightness sensitivity  So one can derive:  Which expresses the brightness sensitivity in terms of the aperture filling factor ( A synt /ΣA tel )  So who wants the resolution, looses the sensitivity for cold sources  Often limited to:  Non-thermal emission will allow high resolution  VLBI:  Cold universe can be mapped at arcsec resolution at mm wavelength, where there is also lots of Δν

27. IAC06 #4 15 mar 200627 Brightness sensitivity  Sub-arcsec resolution requires 200000 λ baseline  Brightness sensitivity requires filled aperture  Thermal emission can be reached with ALMA  Highest possible brightness temperatures with VSOP VSOPALMA

29. IAC06 #4 15 mar 200629 Stokes components  Many ways to describe the polarization state  Using the Stokes paramters  I total intensity  Q, U two components of linear, 45 0 apart  V circular  Other important parameters derived from this  Φ the polarization angle  m fractional linear polarization  V fractional circular

30. IAC06 #4 15 mar 200630 Stokes parameters special cases  Linear Polarization  I = E 2 = S  Q = I cos 2   U = I sin 2   V = 0  Circular Polarization  I = S  Q = 0  U = 0  V = S (RCP) or –S (LCP) The Poincaré sphere provides a means for visualising the Stokes parameters. The point P represents a given polarization state and lies on the surface of the sphere whose radius is the total polarized power.

31. IAC06 #4 15 mar 200631 Antenna characteristics  All reflections swap circular polarization  Perfect response only on-axis  Otherwise inherent instrumental contribution  Two independent receivers necessary to capture all information  Could be linear or circular  Of course never perfect  In addition response rotates on sky when tracking for most telescopes

32. IAC06 #4 15 mar 200632 Interferometer response  Stokes can be formed from 4 circular cross-products  Or from linear feeds

33. IAC06 #4 15 mar 200633 Calibration  Receivers on both end have “leakage terms”, as well as unknown feed orientation:  Which then leads to a reasonably complicated response:  Calibration proceeds usually by concentrating on first order  strategies depend on type of feeds  Mixed feeds in principle possible in Measurement Equation

34. IAC06 #4 15 mar 200634 Calibration  For circular feeds:  Assume sources have hardly any circular, calibrate both hands  This is sufficient if you are after the √2 sensitivity  And do not require high dynamic range  Observe a source with known polarization  If unpolarized you observe the leakage  If polarized use a range of parallactic angles and solve for leakage as well as the source polarization  Leaves angle (ie phase difference at reference antenna) unknown, must be determined for calibrator by external means  Very hard to get circular calibration itself right…  Requires known unpolarized source  Or set of sources that on average will be unpolarized

35. IAC06 #4 15 mar 200635 For linear feeds  For linear feeds  Use polarized calibrator  Its response will be function of parallactic angle  Again orientation will be a problem, requires known source  Complications arise when doing wide field imaging especially for offset feeds (VLA)  Self-cal like approach trying to solve for polarized beam shape  In addition there is ionospheric Faraday rotation at low frequency  Attempt to solve with external calibration: GPS

36. IAC06 #4 15 mar 200636 Polarization data analysis  Making polarization images  follow general rules for imaging & deconvolution  image & deconvolve in I, Q, U, V (e.g. CLEAN, MEM)  note: Q, U, V will be positive and negative  in absence of CP, V image can be used as check  Polarization vector plots  Synchrotron emission is expected to be linearly polarized  use “electric vector position angle” (EVPA) calibrator to set angle (e.g. R-L phase difference)   = ½ tan-1 U/Q for E vectors ( B vectors ┴ E )  plot E vectors with length given by p  Faraday rotation: determine  vs. 2  Beam depolarization is diluting signature

37. IAC06 #4 15 mar 200637 Example: Cygnus A  VLA @ 8.5 GHz B-vectors Perley & Carilli (1996)

38. Left: rotation measure map of the weak emission line blazar BL Lac made using VLBA observations at 5, 8 and 15 GHz. The contours are the 5 GHz total intensity. The RM in the core (northernmost component) is enhanced with respect to the jet. (Reynolds et al. 2001 MNRAS 327, 1071). On the right is a plot of χ vs λ 2 for the core and 3 jet components.

39. IAC06 #4 15 mar 200639 SiO masers around evolved stars Trace outflowing material round old stars, very close to surface. Polarization can also arise from special orientation of amplification path, but now generally believed to be due to large scale magnetic field 48 GHz VLBA observations of UHer, Tx Cam, Diamond & Kemball 1998, 2000

40. IAC06 #4 15 mar 200640 Example: Zeeman effect