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Basics of mm interferometry
Turku Summer School – June 2009
Sébastien MullerNordic ARCOnsala Space Observatory, Sweden
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)
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
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)
Needs of angular resolution
Diameter @115GHz @230GHz @345GHz
10m 65’’ 32’’ 22’’
30m 22’’ 11’’ 7’’
100m 7’’ 3’’ 2’’
1000m 0.6’’ 0.3’’ 0.2’’
Resolution /D (theory of diffraction)
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)
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
Holography measurement
- 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
baseline
antenna
uv plane
Baseline, uv plane and spatial frequency
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
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)
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
x0x
I
u
S
Amplitude
u
Phase
Slope = -2x0
Illustration : dirty beam, dirty image and deconvolved (clean) image resulting in some interferometric
observations of a source model
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)
O2 H2O
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
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
Bandpass calibration
Phase calibration
Amplitude calibration
From SMA Observer Center Toolshttp://sma1.sma.hawaii.edu/
From SMA Observer Center Toolshttp://sma1.sma.hawaii.edu/
From SMA Observer Center Toolshttp://sma1.sma.hawaii.edu/
Quasars usually variable ! -> need reliable flux calibrator
From SMA Observer Center Toolshttp://sma1.sma.hawaii.edu/
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)
Array configuration
Compact DetectionMapping of extended regions
Intermediate Mapping
Extended High angular resolution mapping
Astrometry
Very-extended Size measurementsAstrometry
PdBI
1 Jy = 10-26 W m-2 Hz-1
For extended source:
Take into account the synthesized beam-> brightness sensitivity
T (K) = 2ln2c2/k2 x Flux density/majmin
Use Time Estimator !
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
Courtesy J. Pety
Courtesy J. Pety
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)
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
Mm interferometry: summary
- Essential to study the Cold Universe (SF)
- Astrophysics: temperature, density, kinematics …
- Robust techniqueHigh angular resolutionHigh spectral/velocity resolution
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
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 }
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)
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.
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)