interferometry - Institute for AstronomyInterferometry...
Transcript of interferometry - Institute for AstronomyInterferometry...
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
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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
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• amplitude tells “how much” of a certain spatial frequency• phase tells “where” this component is located
T(x,y) V(u,v)
amplitude phase
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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
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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
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“Resolving out” extended emission
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“Resolving out” extended emission
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“Resolving out” extended emission
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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