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Basics of mm interferometry Turku Summer School – June 2009 Sébastien Muller Nordic ARC Onsala Space Observatory, Sweden

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Interests of mm radioastronomy -> Cold Universe Giant Molecular Clouds -> COLD and DENSE phase Site of the STAR FORMATION -> Continuum emission of cold dust -> Molecular transitions - Diagnostics of the gas properties (temperature, density) - Kinematics (outflows, rotation)

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Interests of CO Molecular gas H 2 But H 2 symmetric -> electric dipolar momentum is 0 Most abundant molecule after H 2 is CO [CO/H 2 ] ~ First rotational transitions of CO in the mm GHz GHz GHz E J=1,2,3 = 6, 17, 33 K Easily excited CO is difficult to destroy high ionization potential (14eV) and dissociation energy (11 eV) Where the atmosphere is relatively transparent

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Handy formulae - HI line emission: N(HI) (cm -2 ) = T B dv (K km/s) - Molecular line emission: N(H 2 ) (cm -2 ) = X T CO dv (K km/s) X = Or use optically thin lines ( 13 CO, C 18 O) - Visual extinction: N(HI)+2N (H 2 ) (cm -2 ) = A V (mag)

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Needs of angular resolution 10m65’’32’’22’’ 30m22’’11’’7’’ 100m7’’3’’2’’ 1000m0.6’’0.3’’0.2’’ Resolution /D (theory of diffraction)

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Would need very large single-dish antennas BUT - Surface accuracy (few 10s of microns !) -> technically difficult and expensive ! - Small field of view (1 pixel) - Pointing accuracy (fraction of the beam) Let’s fill in a large collecting area with small antennas And combine the signal they receive -> Interferometry (Aperture synthesis)

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Mm antennas need Good surface accuracy D APEX 12m<20 microns IRAM-30m30m55 microns (GBT 100m300 microns) PdBI 15m<50 microns SMA 6m<20 microns ALMA 12m <25 microns

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Holography measurement

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- uv positions are the projection of the baseline vectors B ij as seen from the source. -The distances (u 2 + v 2 ) are refered to as spatial frequencies - Interferometers can access the spatial frequencies ONLY between B min and B max, the shortest and longest projected baselines respectively. geometrical time delay source baseline antenna uv plane Baseline, uv plane and spatial frequency

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V(u,v) = P(x,y) I(x,y) exp –i2 (ux+vy) dxdy = FT { P I } Interferometers measure VISIBILITIES V But astronomers want the SKY BRIGHTNESS DISTRIBUTION of the source : I(x,y) P(x,y) is the PRIMARY BEAM of the antennas - P has a finite support, so the field of view is limited - distorded source informations - P is in principle known ie. antenna characteristic

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I(x,y) P(x,y) = V(u,v) exp i2 (ux+vy) dudv Well, looks easy … BUT ! Interferometers have an irregular and limited uv sampling : - high spatial frequency (limit the resolution) - low spatial frequency (problem with wide field imaging) Incomplete sampling, non respect of the Nyquist’s criterion = LOSS of informations ! The direct deconvolution is not possible Need to use some smart algorithms (e.g. CLEAN)

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Let’s take an easy example: 1D P = 1 I(x) = Dirac function: S (x-x 0 ) S = constant V(u) = FT(I) = Sexp(-i2 ux 0 )-> this is a complex value x0x0 x I u S Amplitude u Phase Slope = -2 x 0

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Illustration : dirty beam, dirty image and deconvolved (clean) image resulting in some interferometric observations of a source model

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Atmosphere « The atmosphere is the worst part of an astronomical instrument » - emits thermally, thus add noise - absorbs incoming radiation - is turbulent ! (seeing) Changes in refractive index introduce phase delay Phase noise -> DECORRELATION (more on long baselines) exp(- 2 /2) - Main enemy is water vapor ( Scale height ~2 km)

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O2O2 H2OH2O

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Calibration V obs = G V true + N V obs = observed visibilities V true = true visibilies = FT(sky) G = (complex) gains usually can be decomposed into antenna-based terms: G = G ij = G i x G j * N = noise After calibration: V corr = G’ –1 V obs

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Calibration - Frequency-dependent response of the system Bandpass calibration -> Bright continuum source - Time-dependent response of the system Gain (phase and amplitude) -> Nearby quasars - Absolute flux scale calibration -> Flux calibrator

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Bandpass calibration

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Phase calibration

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Amplitude calibration

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From SMA Observer Center Tools

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From SMA Observer Center Tools

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From SMA Observer Center Tools

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Quasars usually variable ! -> need reliable flux calibrator From SMA Observer Center Tools

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Preparing a proposal 0) Search in Archives SMA: PdBI: ALMA … 1) Science justifications -> Model(s) to interpret the data 2) Technical feasibility: - Array configuration(s) (angular resolution, goals) - Sensitivity use Time Estimator ! Point source sensitivity Brightness sensitivity (extended sources)

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Array configuration CompactDetection Mapping of extended regions IntermediateMapping ExtendedHigh angular resolution mapping Astrometry Very-extendedSize measurements Astrometry

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PdBI 1 Jy = W m -2 Hz -1

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For extended source: Take into account the synthesized beam -> brightness sensitivity T (K) = 2ln2c 2 / k 2 x Flux density/ maj min Use Time Estimator !

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Short spacings V(u,v) = P(x,y) I(x,y) exp –i2 (ux+vy) dxdy V(0,0) = P(x,y) I(x,y) dxdy (Forget P), this is the total flux of the source And it is NOT measured by an interferometer ! -> Problem for extended sources !!! -> Try to fill in the short spacings

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Courtesy J. Pety

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Advantages of interferometers - High angular resolution - Large collecting area - Flatter baselines - Astrometry - Can filter out extended emission - Large field of view with independent pixels - Flexible angular resolution (different configuration)

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Disadvantages of interferometers - Require stable atmosphere - High altitude and ~flat site (usually difficult to access) - Lots of receivers to do - Complex correlator - Can filter out extended emission - Need time and different configuration to fill in the uv-plane

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Mm interferometry: summary - Essential to study the Cold Universe (SF) - Astrophysics: temperature, density, kinematics … - Robust technique High angular resolution High spectral/velocity resolution

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Let’s define - Sampling function S(u,v) = 1 at (u,v) points where visibilities are measured = 0 elsewhere - Weighting function W(u,v) = weights of the visibilities (arbitrary) We get : I obs (x,y) = V(u,v) S(u,v) W(u,v) exp i2 (ux+vy) dudv

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Due to the Fourier Transform properties : FT { A B } = FT { A } ** FT { B } Can be rewritten as : where I obs (x,y) = V(u,v) S(u,v) W(u,v) exp i2 (ux+vy) dudv I obs (x,y) = P(x,y) I(x,y) ** D(x,y) D(x,y) = S(u,v) W(u,v) exp i2 (ux+vy) dudv = FT { S W }

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If I sou = (x,y) = Point source then I obs (x,y) = D(x,y) That is : D is the image of a point source as seen by the interferometer. ~ Point Spread Function I obs (x,y) = P(x,y) I(x,y) ** D(x,y)

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D(x,y) = FT { S W } D(x,y) is called DIRTY BEAM This dirty beam depends on : - the uv sampling (uv coverage) S - the weighting function W Note that : D(x,y) dxdy = 0because S(0,0) = 0 And that : D(0,0) > 0because SW > 0 The dirty beam presents a positive peak at the center, surrounded by a complex pattern of positive and negative sidelobes, which depends on the uv coverage and the weighting function.

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I obs (x,y) is called DIRTY IMAGE We want I obs (x,y) I(x,y) This includes the two key issues for imaging : - Fourier Transform (to obtain I obs from V and S) - Deconvolution (to obtain I from I obs ) I obs (x,y) = P(x,y) I(x,y) ** D(x,y)

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