Sensitivity. Bob Zavala US Naval Observatory. Outline

Transcription

1 Sensitivity Bob Zavala US Naval Observatory Tenth Synthesis Imaging Summer School University of New Mexico, June 13-20, 2006 Outline 2 What is Sensitivity? Antenna Performance Measures Interferometer Sensitivity Sensitivity of a Synthesis Image Other factors Summary 1

2 What is Sensitivity & Why Should You Care? 3 Measure of weakest detectable emission Important throughout research program Sound observing proposal Sensible error analysis in journal Expressed in units involving Janskys Unit for interferometer is Jansky (Jy) Unit for synthesis image is Jy beam 1 1 Jy = W m 2 Hz 1 = erg s 1 cm 2 Hz 1 Common current units: millijy, microjy Common future units: nanojy Measures of Antenna Performance 4 System Temperature Point at blank sky. What is received power P? Write P as equivalent temperature T of matched termination at receiver input Rayleigh-Jeans limit to Planck law P = k B T ν Boltzmann constant k B Observing bandwidth ν Amplify P by g 2 where g is voltage gain noise power P N = g 2 k B T sys ν T sys includes cosmic background, sky, receiver, ground Position and time dependence 2

3 Measures of Antenna Performance 5 Source (Antenna) Temperature T a Point at source of interest Received P = P n + P a T a source power P a = g 2 k B T a ν Measures of Antenna Performance 6 Gain Source power P a = g 2 k B T a ν Let T a = K S for source flux density S, constant K Then P a = g 2 k B K S ν (1) But source power also P a = ½ g 2 η a A S ν (2) Antenna area A, efficiency η a Rx accepts 1/2 radiation from unpolarized source (Malus Law) Equate (1), (2) and solve for K K = (η a A) / (2 k B ) = T a / S K is antenna s gain or sensitivity, unit Kelvin Jy 1 K is flux collection ability Big K s (Figure of merit). 3

4 Measures of Antenna Performance 7 System Equivalent Flux Density Antenna temperature T a = K S Source power P a = g 2 k B K S ν Express system temperature analogously Let T sys = K SEFD SEFD is system equivalent flux density, unit Jy System noise power P N = g 2 k B K SEFD ν SEFD measures overall antenna performance SEFD = T sys / K Small SEFD is better Interferometer Sensitivity 8 Real Correlator - 1 Simple correlator with single real output that is product of voltages from antennas j,i SEFD i = T sysi / K i and SEFD j = T sysj / K j Each antenna collects bandwidth ν Interferometer built from these antennas has Accumulation time τ acc, system efficiency η s Source, system noise powers imply sensitivity S ij Weak source limit S << SEFD 1 SEFDi SEFD j i Sij S << SEFD j ηs 2 ν τ acc 4

5 Interferometer Sensitivity 9 Real Correlator - 2 For SEFD i = SEFD j = SEFD drop subscripts 1 SEFD S η s 2 ν τ acc Units Jy Interferometer system efficiency η s Accounts for electronics, digital losses See instrument documentation Interferometer Sensitivity Abscissa spans 30 minutes. 10 Ordinate spans +/-300 millijy. Complex Correlator Delivers two channels Real S R, sensitivity S Imaginary S I, sensitivity S Eg: VLBA continuum Figure 9-1 at 8.4 GHz Observed scatter S R (t), S I (t) Predicted S = 69 millijy 1 SEFD S η 2 ν τ s acc Resembles observed scatter 5

6 Interferometer Sensitivity 11 Measured Amplitude Measured visibility amplitude S m = Standard deviation (s.d.) of S R or S I is S True visibility amplitude S Probability Pr(S m / S) Figure 9-2 S R S I Behavior with true S/ S High: Gaussian Zero: Rayleigh Low: Rice. S m gives biased estimate of S. Interferometer Sensitivity 12 Measured Phase Measured visibility phase φ m = arctan(s I /S R ) True visibility phase φ Probability Pr(φ φ m ) Figure 9-2 Behavior with true S/ S High: Gaussian Zero: Uniform Seek weak detection in phase, not in amplitude 6

7 Image Sensitivity 13 Single Polarization Simplest weighting case where visibility samples 1 SEFD Have same interferometer sensitivities S η s 2 ν τ Have same signal-to-noise ratios w Combined with natural weight (W=1), no taper (T=1) Image sensitivity is s.d. of mean of L samples, each with s.d. S, i.e., I m = S/ L N antennas, # of interferometers ½ N (N 1) # of accumulation times t int /τ acc L = ½ N (N 1) (t int /τ acc ) So I m 1 η s SEFD N ( N 1) ν t int acc Image Sensitivity Dual Polarizations 2 Eg: VLBA continuum Figure 9-3 at 8.4 GHz NGC5548 FOV 150 milliarcsec = 80 pc Wrobel 2000, ApJ, 531, Observed Stokes I, simplest weighting Gaussian noise I = 90 microjy beam 1 Predicted I = I m / 2 = S/ 2 L L = ½ N (N 1) (t int /τ acc ) Previous e.g. S Plus here L = 77,200 So s.d. I = 88 microjy beam -1 7

8 Image Sensitivity 15 Dual Polarizations - 1 Single-polarization image sensitivity I m Dual-polarization data image Stokes I,Q,U,V Gaussian noise in each image, 2 samples Mean zero, s.d. I = Q = U = V = I m / 2 Linearly polarized flux density P = 2 2 Q +U Rayleigh noise, P 0 Recall visibility amplitude Polarization position angle χ = ½ arctan(u/q) Recall visibility phase See Fig. 9.4 in text Other Sensitivity Factors 16 Beam size and T B Small synthesized beams require high T B VLBA sees non-thermal sources, Fig. 9.5 Very bright sources: High dynamic range Imaging (Monday) Confusion 8

9 Sensitivity 17 Summary 1 One antenna System temperature T sys Gain K Overall antenna performance is measured by system equivalent flux density SEFD Units Jy SEFD = T sys / K Sensitivity 18 Summary - 2 Connect two antennas to form interferometer Antennas have same SEFD, observing bandwidth ν Interferometer system efficiency η s Interferometer accumulation time τ acc Sensitivity of interferometer S Units Jy 1 SEFD η 2 ν τ s acc 9

10 Sensitivity 19 Summary - 3 Connect N antennas to form array Antennas have same SEFD, observing bandwidth ν Array has system efficiency η s Array integrates for time t int Form synthesis image of single polarization Sensitivity of synthesis image 1 SEFD I m η N ( N 1) ν t Units Jy beam 1 s int 10