Basics of mm interferometry
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Basics of mm interferometry
Turku Summer School – June 2009
Sébastien MullerNordic ARCOnsala 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 H2
But H2 symmetric -> electric dipolar momentum is 0
Most abundant molecule after H2 is CO [CO/H2] ~ 10-4
First rotational transitions of CO in the mmCO(1-0) @115 GHzCO(2-1) @230 GHzCO(3-2) @345 GHz
E J=1,2,3 = 6, 17, 33 K Easily excited
CO is difficult to destroyhigh 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) = 1.82 1018 TBdv (K km/s)
- Molecular line emission:N(H2) (cm-2) = X 1020 TCOdv (K km/s) X =
0.5-3
Or use optically thin lines (13CO, C18O)
- Visual extinction:N(HI)+2N (H2) (cm-2) = 2 1021 AV (mag)
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Needs of angular resolution
Diameter @115GHz @230GHz @345GHz10m 65’’ 32’’ 22’’30m 22’’ 11’’ 7’’100m 7’’ 3’’ 2’’
1000m 0.6’’ 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 antennasAnd combine the signal they receive
-> Interferometry (Aperture synthesis)
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Mm antennas needGood surface accuracy
D APEX 12m <20 micronsIRAM-30m 30m 55 microns(GBT 100m300 microns)
PdBI 15m <50 micronsSMA 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 Bij as seen from the source.-The distances (u2 + v2) are refered to as spatial frequencies- Interferometers can access the spatial frequencies ONLY between Bmin and Bmax, the shortest and longest projected baselines respectively.
geometricaltime delay
source
baselineantenna
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 theSKY 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 possibleNeed to use some smart algorithms (e.g. CLEAN)
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Let’s take an easy example:
1DP = 1I(x) = Dirac function: S(x-x0)
S = constant
V(u) = FT(I) = Sexp(-i2ux0) -> this is a complex value
x0 x
I
u
SAmplitude
u
Phase
Slope = -2x0
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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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O2 H2O
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Calibration
Vobs = G Vtrue + N
Vobs = observed visibilities
Vtrue = true visibilies = FT(sky)
G = (complex) gainsusually can be decomposed into antenna-based terms:G = Gij= Gi x Gj*
N = noise
After calibration: Vcorr = G’ –1 Vobs
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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 Toolshttp://sma1.sma.hawaii.edu/
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From SMA Observer Center Toolshttp://sma1.sma.hawaii.edu/
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From SMA Observer Center Toolshttp://sma1.sma.hawaii.edu/
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Quasars usually variable ! -> need reliable flux calibrator
From SMA Observer Center Toolshttp://sma1.sma.hawaii.edu/
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Preparing a proposal
0) Search in ArchivesSMA: http://www.cfa.harvard.edu/cgi-bin/sma/smaarch.plPdBI: http://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=B/iramALMA …
1) Science justifications
-> Model(s) to interpret the data
2) Technical feasibility:
- Array configuration(s) (angular resolution, goals)
- Sensitivity use Time Estimator !Point source sensitivityBrightness sensitivity (extended sources)
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Array configuration
Compact DetectionMapping of extended regions
Intermediate Mapping
Extended High angular resolution mapping
Astrometry
Very-extended Size measurementsAstrometry
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PdBI
1 Jy = 10-26 W m-2 Hz-1
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For extended source:
Take into account the synthesized beam-> brightness sensitivity
T (K) = 2ln2c2/k2 x Flux density/majmin
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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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 techniqueHigh angular resolutionHigh 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 :Iobs(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
Iobs(x,y) = V(u,v) S(u,v) W(u,v) exp i2(ux+vy)
dudv
Iobs(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 Isou = (x,y) = Point source then
Iobs(x,y) = D(x,y)
That is : D is the image of a point source as seenby the interferometer.
~ Point Spread Function
Iobs(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 = 0 because S(0,0) = 0
And that : D(0,0) > 0 because 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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Iobs(x,y) is called DIRTY IMAGE
We want Iobs(x,y) I(x,y)
This includes the two key issues for imaging :
- Fourier Transform (to obtain Iobs from V and S)
- Deconvolution (to obtain I from Iobs)
Iobs(x,y) = P(x,y) I(x,y) ** D(x,y)