Principles of Radio Interferometry Ast735: Submillimeter Astronomy IfA, University of Hawaii 1
Resources IRAM millimeter interferometry school hdp://www.iram- inshtute.org/en/content- page- 248-7- 67-248- 0-0.html IRAM mm school proceedings (from 2000) hdp://www.iram.fr/iramfr/is/is2002/ps_2/web.html copied and catenated at class website NRAO synthesis imaging workshop hdp://www.aoc.nrao.edu/events/synthesis/2012/ EssenHal Radio Astronomy hdp://www.cv.nrao.edu/course/astr534/era.shtml 2
Interferometry hdp://www.colorado.edu/physics/2000/applets/twoslitsa.html 3
Interferometry Because radio heterodyne techniques detect (and digihze) both the amplitude and phase, we can directly invert the interference padern to recover the source structure [OIR interferometers can bring two beams together but the detectors can only measure the amplitude of the fringes] 4
Interferometry: basic theory 5 Gueth, IRAM interferometry school
Interferometry: basic theory V 1 (t) = cos 2πν(t- τ g ) V 2 (t) = cos 2πνt Low pass filtered output R 12 = <V 1 V 2 > = V 1 V 2 cos 2πντ g = V 1 V 2 cos 2πb.s/λ 6 Gueth, IRAM interferometry school
Fringes R 12 = V 1 V 2 cos 2πb.s/λ Projected baseline in units of wavelength, as seen from source ALMA The output modulates as the source moves through the sky; the interference padern introduces structure into the singledish beam => increases resoluhon 7
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Interferometry: finite source size dr 12 = V 1 V 2 cos 2πb.s/λ = A(s) I(s) dω cos 2πb.s/λ 9 effechve collechng area source intensity (erg/s/cm 2 /ster)
VisibiliHes Where the complex visibility is defined as and measures the coherence of the source intensity 10
11 (Proof)
(Details) Delay tracking Bandwidth smearing Sky curvature CriHcally important to operahon of the instrument but not essenhal for you, the end- user, to know 12
The uv- plane v w u b = (u,v,w)λ PosiHons on the sky are defined by σ = (x,y), angular units from phase center, and solid angle dω = dxdy 13
The uv- plane VisibiliHes are the Fourier transform of the (antenna- weighted) sky brightness distribuhon. Hence we can obtain an image of the source by Fourier inversion of our measurements. 14
Fourier decomposihon 15 hdp://en.wikipedia.org/wiki/fourier_series
Fourier decomposihon Any funchon can be decomposed into a series of sine and cosine waves; we only need to know the amplitude of each harmonic (A n sin[nx] + B n cos[nx], n=0,1,2,...) A funchon can be approximated by the sum of a finite number of harmonics; the accuracy depends how many are used and how sharp the funchon is (sharp edges need more higher order harmonics) This allows very efficient data compression, e.g., MP3 for music, JPEG for images, etc. The implicahons for millimeter interferometry is that we can determine many of the salient features of an object from a relahvely sparsely sampled array 16
Example 2D Fourier Transform Pairs T(x,y) amp{v(u,v)} δ function constant elliptical Gaussian Gaussian elliptical Gaussian Gaussian narrow features transform into wide features (and vice-versa) Wilner NRAO presentation 17
Example 2D Fourier Transform Pairs T(x,y) amp{v(u,v)} disk Bessel Wilner NRAO presentation sharp edges result in many high spatial frequencies 18
Amplitude and Phase amplitude tells how much of a certain spatial frequency phase tells where this component is located T(x,y) V(u,v) amplitude phase Wilner NRAO presentation 19
amplitude tells how much of a certain spatial frequency phase tells where this component is located T(x,y) V(u,v) amplitude phase Wilner NRAO presentation 20
MulH- element interferometers 21 From EssenHal Radio Astronomy by Condon & Ransom www.cv.nrao.edu/course/astr534/era.shtml
Earth rotahon: aperture synthesis Baseline angle changes as sources moves through the sky => observahons fill in the visibility plane The intensity is real so the visibilihes are Hermi?an 22 i.e., we get two visibilihes from one measurement
An Example of (u,v) plane Sampling 2 configurations of 8 SMA antennas, 345 GHz, Dec. -24 dec Wilner NRAO presentation 23
Dirty Beam Shape and N Antennas Sampling in the uv-plane 2 Antennas Response to a point source Wilner NRAO presentation 24
Dirty Beam Shape and N Antennas Sampling in the uv-plane 3 Antennas Response to a point source Wilner NRAO presentation 25
Dirty Beam Shape and N Antennas Sampling in the uv-plane 4 Antennas Response to a point source Wilner NRAO presentation 26
Dirty Beam Shape and N Antennas Sampling in the uv-plane 5 Antennas Response to a point source Wilner NRAO presentation 27
Dirty Beam Shape and N Antennas Sampling in the uv-plane 6 Antennas Response to a point source Wilner NRAO presentation 28
Dirty Beam Shape and N Antennas Sampling in the uv-plane 7 Antennas Response to a point source Wilner NRAO presentation 29
Dirty Beam Shape and N Antennas Sampling in the uv-plane 8 Antennas Response to a point source Wilner NRAO presentation 30
Dirty Beam Shape and N Antennas Sampling in the uv-plane 8 Antennas x 6 samples Response to a point source Wilner NRAO presentation 31
Dirty Beam Shape and N Antennas Sampling in the uv-plane 8 Antennas x 30 samples Response to a point source Wilner NRAO presentation 32
Dirty Beam Shape and N Antennas Sampling in the uv-plane 8 Antennas x 60 samples Response to a point source Wilner NRAO presentation 33
Dirty Beam Shape and N Antennas Sampling in the uv-plane 8 Antennas x 120 samples Response to a point source Wilner NRAO presentation 34
Dirty Beam Shape and N Antennas Sampling in the uv-plane 8 Antennas x 240 samples Response to a point source Wilner NRAO presentation 35
Dirty Beam Shape and N Antennas Sampling in the uv-plane 8 Antennas x 480 samples Response to a point source Wilner NRAO presentation 36
Single dish versus interferomter Plambeck & Engargiola 2002 The interferometer passes the eye test but it doesn t produce a perfect image... 37
Resolving out extended emission 38Wilner NRAO presentation
Resolving out extended emission 39Wilner NRAO presentation
Resolving out extended emission 40Wilner NRAO presentation
Resolving out extended emission 41Wilner NRAO presentation
Imaging prachcalihes Interferometer field of view is the single- dish (primary) beam Incomplete uv- sampling, inner hole, finite size 42
Ring FT Bessel funchon Central hole => don t recover total flux miss large scale structure Finite size => finite resoluhon Incomplete sampling => reduced image fidelity 43
Ring FT Bessel funchon Large single dish Central hole => don t recover total flux miss large scale structure Finite size => finite resoluhon Incomplete sampling => reduced image fidelity 44
Ring FT Bessel funchon extended configurahons Central hole => don t recover total flux miss large scale structure Finite size => finite resolu>on Incomplete sampling => reduced image fidelity 45
Ring FT Bessel funchon Central hole => don t recover total flux miss large scale structure Finite size => finite resoluhon Incomplete sampling => reduced image fidelity more baselines, more sky rotahon 46
47 Interferometry provides imaging over a primary beam at the resoluhon of a synthesized beam
48 Interferometry provides imaging over a primary beam at the resoluhon of a synthesized beam plus spectra.
Next steps CalibraHon Image deconvoluhon 49