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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

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


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

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

amplitude phase


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


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


3 Antennas

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4 Antennas

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5 Antennas

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6 Antennas

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7 Antennas

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8 Antennas

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8 Antennas x 6 samples

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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


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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Large single dish


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



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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

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