Post on 12-Sep-2021
Astronomical Interferometryin a nutshell
1) Why Interferometry?2) Basic Theory3) Interferometers4) The influence of the atmosphere5) Image reconstruction
1. The need for resolution
The need for resolutionSize of BLRin nearby AGN
Schwarzschildradius of BH innearby AGN
Size of NLRin nearby AGN
8kpc ingalaxy at z=1
Galaxy clusterscales
Angularsize ofBetelgeuse
ExoplanetHD209458bstar->planet
The need for resolutionSize of BLRin nearby AGN
Schwarzschildradius of BH innearby AGN
Size of NLRin nearby AGN
8kpc ingalaxy at z=1
Galaxy clusterscales
Angularsize ofBetelgeuse
ExoplanetHD209458bstar->planet
Jodrell Bankat 5GHzGround-based
seeing limitVLT singletelescopediffraction limit
HSTSingleTelescopes visible light
Eye
The need for resolutionSize of BLRin nearby AGN
Schwarzschildradius of BH innearby AGN
Size of NLRin nearby AGN
8kpc ingalaxy at z=1
Galaxy clusterscales
Angularsize ofBetelgeuse
ExoplanetHD209458bstar->planet
Jodrell Bankat 5GHzGround-based
seeing limitVLT singletelescopediffraction limit
HSTSingletelescopes
VLA at5GHz
MERLIN at 5GHz
GlobalVLBI
SpaceVLBI VLTI (IR)
Interferometers
VLTI(optical)
ALMA
2. Basic theory of interferometry
Point source Fringes ofseparationλ/d
d
Young's slits revisited
Larger source
Source subtends anangle 0.4 λ/d
Fringes move by0.4 λ/d. Incoherentsources -> addintensities, fringesstart to add outdestructively
Define contrast |fringe visibility|=(Imax-Imin)/(Imax+Imin)
Still larger source
Source sizegets to λ/d
No fringes remain(cancellation). Littlefringing seen forlarger sources thanλ/d either.
Effect of slit size
Same size source,but smaller slit
Increased fringespacing, so fringesvisible again
Baseline length
Baseline length
Young's slits: summaryVisibility of interference fringes
•Decreases with increasing source size•Goes to zero when source size goes to λ/d•For given source size, increases for decreasing separation•For given source size and separation, increases with λ
It's a Fourier transform!
The fringe visibility of an interferometer gives informationabout the Fourier transform of the sky brightness distribution.
Long baselines record information about the small-scalestructure of the source but are INSENSITIVE to large-scalestructure (fringes wash out)
Short baselines record information about large-scale structureof the source but are INSENSITIVE to small-scale structure(resolution limit)
Van Cittert-Zernicke theorem
OPD – optical phase delay
The u-v plane
If we could measure FV for all u,v, transform > image
Location of the 8m Unit Telescopes (UTs) and the 1.8m Auxiliary Telescopes(ATs) of the ESO Very Large Telescope Interferometer (VLTI)
Earth rotation aperture synthesis (ERAS)
Over a day,can measuremany points inu-v plane witha single baseline
Locus is an ellipse;the longer the baseline,the larger the u-v (higherresolution)
b-vector plotted in brown
Image: A. Gunn/University of Manchester
Exact form of u-v track
D=declination of sourced=declination of point on sky pointed to by baseline
Resolution given by maximumextent of tracks
ERAS imaging of sources atdeclination D=0 is hard!
projected base line
[u,v] coordinates
Actual fringe visibility
Double sourceeach component 1Jy1Jy =10-26 W m-2 Hz-1
separation calculablefrom baseline length
Correlated flux vs. time
Limits on the field of view1. Finite range of wavelengths
Bigger range – smaller field of view (FT again)FOV = (λ/∆λ) x (λ/L) i.e. λ/∆λ resolution elementsCure – observe in multiple channels
Fringe pattern OK at field centre but different colours out of phase at higher relative delay
Limits on the field of view2. Too big integration time per data point3. Non-flat sky over large FOV
Rather technical, and only a problem for wide-field imaging
4. Primary beam
maximum field ofview set by thetelescope aperture
3. InterferometersRadio/Thermal Infrared
Non-photon-limited: electronic, relatively straightforward can “clone” and combine signals (heterodyne detection) “correlation” (multiplication+delay) can even record signals and combine later
Optical/Near-Infrared
Photon-limited case: use classical Michelson/Fizeau arrangements delay lines for manipulation cannot “clone” photons
Radio
VLA 30m-36kmMERLIN 6km-250kmEVN 250km-2300kmVLBA 250km-9000kmGlobal VLBI-12000km
Space VLBI-32000km
Very Large Array, NM, USA
VLBI (Very Long Baseline Interferometry)
Limited only by Earth size12000km baselines -> mas resolution
Atacama Large Millimetre Array - ALMA
- 30-950 GHz- max. baseline ~20km- molecules in galaxies at cosmological z- gas in Galactic star- forming regions
Chajnantor, Chile
Optical/IR interferometers
from Monnier 2003
Michelson's interferometer at the Mt. Wilson observatory. The angular diameters of 7 stars could be measured (Betelgeuse, Arcturus, Antares, Aldebaran, Ras Algethi (Alpha Herculis), Scheat (Beta Pegasi), and Mira).
Correcting the OPD difference – Delay Lines
fringe scanning: measurement of thefringe pattern by periodic modulationof the OPD
fringe
VINCI – VLTI Commissioning Instrument
What the atmosphere does
# Corrupts phase and amplitude of incoming signals
# Corruption is different for different telescopes/apertures
# Corruption changes with time (scales ms to mins)
# Corruption varies with position (<size of tel. for optical)
# Sources: water vapour..., ionosphere (low-frequency radio)
4. The real world - Dealing with the atmosphere
How bad is the problem? Phase fluctuations
Waveband Problem Phase variation timescale
Radio <300MHz ionosphere seconds-minutes few GHz water &c minutes >20GHz water sec (site dependent)mm water highly site dependentnear-IR atm cells ~100 millisecondsoptical 1-10 milliseconds
The shorter the wavelength, the more rapid the phasefluctuation and the harder the problem becomes.
Optical: Fried parameter ro – length scale of refractive index fluctuations Timescale = ro/wind velocity
VLA, 8.4GHz, phase fluctuations [degrees] over time
One approach: closure phase
Closure-phase mapping: COAST
Betelgeuse (Young et al. 2004,Proc. Nat. Astr. Meeting)
Capella (www.mrao.cam.ac.uk/telescopes/coast)
Self-calibration in action
Dirty map CLEANed map CLEANed map withphase selfcalibration
Phase calibration
Requirements
a<size of isoplanactic patcht <coherence time of atmosphere at this wavelength
Can nod back and forth, or have target and calibrator in samefield of view (FOV)
Phase calibration
Phase calibrator must be * bright (S:N in reasonable time/atmospheric
coherence time) * close (same wavefront distortion)
(cf. adaptive optics on single telescopes)
If isoplanatic patch is small * calibrator may not exist
Signal-to-noise (wave regime)
Radio interferometernoise level =
Tsys = system temperature, nb = number of baselines, T=integration time, ∆ν=bandwidth in Hz, A=area ofapertures, η=aperture efficiency
linearly as 1/A not (1/A)
In practice you rarely get to this!
1/2
Signal-to-noise (photons)
Fundamental limit set by n photon statistics
system limit is given by number ofphotons in coherence volume (multiplied by loss factors)
Mandatory adaptive optics on individual telescopes (need natural/artificial guide star for wavefront sensing)
Phase referencing for ultimate sensitivity (need bright star in isoplanatic patch)
1/2
Deconvolution
We want the image as a function of (x,y):
But instead we have the “dirty image”
where the sampling function S is 1 in the parts of the uvplane we've sampled and 0 where we haven't.
5. Making the image
Deconvolution (ctd)
We can use the convolution theorem to write
Where B is known as the dirty beam
and is the FT of the sampling function.
Problem is then one of deconvolution.
Hogbom CLEAN deconvolution
Brute-force iterative deconvolutionusing the dirty beam
Effectively reconstructs informationin unsampled parts of the u-vplane by assuming sky is sum ofpoint sources
Hogbom CLEAN in action
Dirty map Dirty beam
Hogbom CLEAN in action
Residual after 1 CLEAN (gain 0.5) CLEAN map (residual+CCs)after 100 CLEANs (gain 0.1)
Summary
Further readingPrinciples of Long Baseline Interferometry Mozurkevich, Michelson Summer School 2000
Synthesis imaging in radio astronomy ASP, Proc NRAO summer school
Optical interferometry in astronomy Monnier, Rep. Prog. Phys, 66, 789, 2003
An Introduction to Optical Stellar Interferometry Labeyrie et al., ISBN 0521828724 Optical Long Baseline Interferometry (OLBIN) website
N. Jackson, ELBA lecture; A. Glindemann, ESO-VLTI
try yourself the Virtual Radio Interferometer (VRI)!