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)