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Principles  of  Radio  Interferometry

Ast735:  Submillimeter  Astronomy

IfA,  University  of  Hawaii

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Resources

• IRAM  millimeter  interferometry  school– hDp://www.iram-­‐insHtute.org/EN/content-­‐page-­‐248-­‐7-­‐67-­‐248-­‐0-­‐0.html

• IRAM  mm  school  proceedings  (from  2000)– hDp://www.iram.fr/IRAMFR/IS/IS2002/ps_2/web.html

– copied  and  catenated  at  class  website

• NRAO  synthesis  imaging  workshop– hDp://www.aoc.nrao.edu/events/synthesis/2012/

• EssenHal  Radio  Astronomy– hDp://www.cv.nrao.edu/course/astr534/ERA.shtml

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Interferometry

hDp://www.colorado.edu/physics/2000/applets/twoslitsa.html

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Interferometry

Because  radio  heterodyne  techniques  detect  (and  digiHze)  both  the  amplitude  and  phase,  we  can  directly  invert  the  interference  paDern  to  recover  the  source  structure

[OIR  interferometers  can  bring  two  beams  together  but  the  detectors  can  only  measure  the  amplitude  of  the  fringes]4

Interferometry:  basic  theory

Gueth,  IRAM  interferometry  school5

Interferometry:  basic  theory

V1(t)  =  cos  2πν(t-­‐τg) V2(t)  =  cos  2πνt

Low  pass  filtered  output      R12  =  <V1V2>  =  V1V2cos  2πντg  =  V1V2cos  2πb.s/λ

6 Gueth,  IRAM  interferometry  school

Fringes

R12  =  V1V2cos  2πb.s/λ Projected  baseline  in  units  of  wavelength,  as  seen  from  source

The  output  modulates  as  the  source  moves  through  the  sky;  the  interference  paDern  introduces  structure  into  the  singledish  beam=>  increases  resoluHon  

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ALMA

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Interferometry:  finite  source  size

dR12  =  V1V2cos  2πb.s/λ  =  A(s)  I(s)  dΩ  cos  2πb.s/λ  

effecHve  collecHng  area source  intensity(erg/s/cm2/ster)9

VisibiliHes

Where  the  complex  visibility  is  defined  as

and  measures  the  coherence  of  the  source  intensity

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(Proof)

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(Details)

• Delay  tracking

• Bandwidth  smearing

• Sky  curvature

– CriHcally  important  to  operaHon  of  the  instrument  but  not  essenHal  for  you,  the  end-­‐user,  to  know

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The  uv-­‐plane

u

v

PosiHons  on  the  sky  are  defined  by  σ  =  (x,y),  angular  units  from  phase  center,and  solid  angle  dΩ  =  dxdy

w

b  =  (u,v,w)λ

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The  uv-­‐plane

VisibiliHes  are  the  Fourier  transform  of  the  (antenna-­‐weighted)  sky  brightness  distribuHon.  Hence  we  can  obtain  an  image  of    the  source  by  Fourier  inversion  of  our  measurements.

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Fourier  decomposiHon

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hDp://en.wikipedia.org/wiki/Fourier_series

Fourier  decomposiHon

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• Any  funcHon  can  be  decomposed  into  a  series  of  sine  and  cosine  waves;    we  only  need  to  know  the  amplitude  of  each  harmonic  (Ansin[nx]  +  Bncos[nx],  n=0,1,2,...)

• A  funcHon  can  be  approximated  by  the  sum  of  a  finite  number  of  harmonics;  the  accuracy  depends  how  many  are  used  and  how  “sharp”  the  funcHon  is  (sharp  edges  need  more  higher  order  harmonics)

• This  allows  very  efficient  data  compression,  e.g.,  MP3  for  music,  JPEG  for  images,  etc.

• The  implicaHons  for  millimeter  interferometry  is  that  we  can  determine  many  of  the  salient  features  of  an  object  from  a  relaHvely  sparsely  sampled  array

Example 2D Fourier Transform Pairs

T(x,y) amp{V(u,v)}

δ function constant

Gaussian Gaussian

narrow features transform into wide features (and vice-versa)

ellipticalGaussian

ellipticalGaussian

17Wilner NRAO presentation

Example 2D Fourier Transform Pairs

T(x,y) amp{V(u,v)}

disk Bessel

sharp edges result in many high spatial frequencies18Wilner NRAO presentation

Amplitude and Phase

• amplitude tells “how much” of a certain spatial frequency• phase tells “where” this component is located

T(x,y) V(u,v)

amplitude phase

19Wilner NRAO presentation

• amplitude tells “how much” of a certain spatial frequency• phase tells “where” this component is located

T(x,y) V(u,v)

amplitude phase

20Wilner NRAO presentation

MulH-­‐element  interferometers

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From  “EssenHal  Radio  Astronomy”  by  Condon  &  Ransomwww.cv.nrao.edu/course/astr534/ERA.shtml

Earth  rotaHon:  aperture  synthesis

Baseline  angle  changes  as  sources  moves  through  the  sky  =>  observaHons  “fill  in  the  visibility  plane”

The  intensity  is  real  so  the  visibiliHes  are  Hermi?an

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i.e.,  we  get  two  visibiliHes  from  one  measurement

An Example of (u,v) plane Sampling

• 2 configurations of 8 SMA antennas, 345 GHz, Dec. -24 dec

23Wilner NRAO presentation

Dirty Beam Shape and N Antennas

2 Antennas

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

3 Antennas

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

4 Antennas

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

5 Antennas

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

6 Antennas

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

7 Antennas

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

8 Antennas

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

8 Antennas x 6 samples

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

8 Antennas x 30 samples

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

8 Antennas x 60 samples

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

8 Antennas x 120 samples

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

8 Antennas x 240 samples

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Dirty Beam Shape and N Antennas

8 Antennas x 480 samples

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Sampling in the uv-plane Response to a point source

Wilner NRAO presentation

Single  dish  versus  interferomter

Plambeck  &  Engargiola  2002

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The  interferometer  passes  the  eye  test  but  it  doesn’t  produce  a  perfect  image...

“Resolving  out”  extended  emission

38Wilner NRAO presentation

“Resolving  out”  extended  emission

39Wilner NRAO presentation

“Resolving  out”  extended  emission

40Wilner NRAO presentation

“Resolving  out”  extended  emission

41Wilner NRAO presentation

Imaging  pracHcaliHes

Interferometer  field  of  view  is  the  single-­‐dish  

(primary)  beam

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Incomplete  uv-­‐sampling,  inner  hole,  finite  size

Ring Bessel  funcHonFT

Central  hole       =>  don’t  recover  total  flux             miss  large  scale  structureFinite  size         =>  finite  resoluHonIncomplete  sampling   =>  reduced  image  fidelity

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Ring Bessel  funcHonFT

Central  hole       =>  don’t  recover  total  flux             miss  large  scale  structureFinite  size         =>  finite  resoluHonIncomplete  sampling   =>  reduced  image  fidelity

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Large  single  dish

Ring Bessel  funcHonFT

Central  hole       =>  don’t  recover  total  flux             miss  large  scale  structureFinite  size         =>  finite  resolu>onIncomplete  sampling   =>  reduced  image  fidelity

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extended  configuraHons

Ring Bessel  funcHonFT

Central  hole       =>  don’t  recover  total  flux             miss  large  scale  structureFinite  size         =>  finite  resoluHonIncomplete  sampling  =>  reduced  image  fidelity

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more  baselines,  more  sky  rotaHon

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Interferometry  provides  imaging  over  a  primary  beam  at  the  resoluHon  of  a  synthesized  beam…

Interferometry  provides  imaging  over  a  primary  beam  at  the  resoluHon  of  a  synthesized  beam……plus  spectra.

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Next  steps

• CalibraHon

• Image  deconvoluHon