Transcript Basics of mm interferometry

Basics of mm interferometry
Sébastien Muller
Nordic ARC
Onsala Space Observatory, Sweden
Turku Summer School – June 2009
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 mm
CO(1-0) @115 GHz
CO(2-1) @230 GHz
CO(3-2) @345 GHz
Where the atmosphere
is relatively transparent
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)
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)
Or use optically thin lines (13CO, C18O)
- Visual extinction:
N(HI)+2N (H2) (cm-2) = 2 1021 AV (mag)
X = 0.5-3
Needs of angular resolution
Resolution  /D
(theory of diffraction)
Diameter
@115GHz
@230GHz
@345GHz
10m
65’’
32’’
22’’
30m
22’’
11’’
7’’
100m
7’’
3’’
2’’
1000m
0.6’’
0.3’’
0.2’’
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)
Mm antennas need
Good surface accuracy
D
APEX
12m <20 microns
IRAM-30m 30m 55 microns
(GBT
100m 300 microns)
PdBI
SMA
15m
6m
<50 microns
<20 microns
ALMA
12m
<25 microns
Holography measurement
Baseline, uv plane and spatial frequency
source
geometrical
time delay
uv plane
antenna
baseline
- 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.
Interferometers measure VISIBILITIES V
V(u,v) =
 P(x,y) I(x,y) exp –i2(ux+vy) dxdy
= FT { P I }
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 possible
Need to use some smart algorithms (e.g. CLEAN)
Let’s take an easy example:
I
1D
P=1
I(x) = Dirac function: S(x-x0)
S = constant
V(u) = FT(I) = Sexp(-i2ux0)
x
x0
-> this is a complex value
Phase
Amplitude
S
Slope = -2x0
u
u
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) gains
usually 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 Tools
http://sma1.sma.hawaii.edu/
From SMA Observer Center Tools
http://sma1.sma.hawaii.edu/
From SMA Observer Center Tools
http://sma1.sma.hawaii.edu/
Quasars usually variable !
-> need reliable flux calibrator
From SMA Observer Center Tools
http://sma1.sma.hawaii.edu/
Preparing a proposal
0) Search in Archives
SMA:
PdBI:
ALMA …
http://www.cfa.harvard.edu/cgi-bin/sma/smaarch.pl
http://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=B/iram
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)
Array configuration
Compact
Detection
Mapping of extended regions
Intermediate
Mapping
Extended
High angular resolution mapping
Astrometry
Very-extended
Size measurements
Astrometry
PdBI
1 Jy = 10-26 W m-2 Hz-1
For extended source:
Take into account the synthesized beam
-> brightness sensitivity
T (K) = 2ln2c2/k2 x Flux density/majmin
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 technique
High angular resolution
High 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 }
Iobs(x,y) =
 V(u,v) S(u,v) W(u,v) exp i2(ux+vy) dudv
Can be rewritten as :
Iobs(x,y) = P(x,y) I(x,y) ** D(x,y)
where
D(x,y) =
 S(u,v) W(u,v) exp i2(ux+vy) dudv
= FT { S W }
Iobs(x,y) = P(x,y) I(x,y) ** D(x,y)
If Isou = (x,y) = Point source then
Iobs(x,y) = D(x,y)
That is : D is the image of a point source as seen
by the interferometer.
~ Point Spread Function
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) = P(x,y) I(x,y) ** D(x,y)
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)