aperture synthesis Flashcards
what is aperture synthesis
the practical way to make an image with an interferometer
preliminaries for aperture synthesis
take an east-west interferometer observing a source at declination delta and hour angle H
hour angle=
local siderial time - right ascension
interferometer baseline
D=(D,0,0)
path compensation
our target is not usually overhead, so we must introduce a physical delay into one arm of the interferometer to compensate
path compensation can be done by
switching in physical lengths of real cable or digitally, in memory
as the earth spins, the phase of the correlated signal
changes, and we get a fringe rate
what is fringe stopping
complex fringe visibility will be rotating at fringe rate in the complex plane and must be constantly rotated backwards to compensate
why do we need to path-compensate?
If bandwidth is deltav
⇒ noise signal evolves randomly on timescales of 1/deltav
⇒ must synchronise the two signals (in time) so that the excess path is
≪ c/deltav
This can be done coarsely with path compensation before fine
adjustment with fringe-stopping.
can we use more than two antennas?
yes - just take tehm in pairs
n antennas, how many baselines
n ways to choose first antenna
(n-1) to choose second
n(n-1) groupings but take away double counting gives
1/2 n(n-1) baseliens
multi-element interferometers are an efficient way to
generate many values of complex fringe visibility simultaneously
solution to only getting measurements in one direction
space the elements out on the ground in two dimensions
eg the very large array in new mexico:
y-shaped configuration
solution 2 for only getting measurements in one direction
let the Earth rotation supply different baseline projected lengths and orientations with respect to the source
“earth rotation aperture synthesis”
what is is that always determines resolution
the projection of the baseline perpendicular to the source direction
we should measure Γ(u,v) in what spatial plane
plane perpendicular to the source direction:
the (u,v) plane
u is parallel to
the right ascension direction
v is parallel to
the declination direction
because the sky brightness is real, the fourier transform Γ(u,v) is
hermitian conjugate
ie Γ(u,v)= Γ*(-u,-v)
so one half of the (u,v) plane measurements can be deduced form the other half - 12 hours observation needed
the synthesised aperture has a synthesised beam defining the
angular resolution of the interferometer
for an east-west line interferometer, the resolution in declination is
poor for low-dec sources
advantage of y-shaped interferometer
north-south spacings maintain resolution in declination even when dec=0
unlike conventional apertures, the synthesised beam of an imaging interferometer can be
very ‘dirty; with lots of strong sidelobes
if the (u,v)-plane sampling is sparse, we get a
heavily degraded (‘dirty’) image, missing many spatial frequencies
solution to sparse sampling
image reconstruction aka deconvolution
image reconstruction
an attempt to remove the effects of the convolving beam patttern
image reconstruction: the CLEAN algorithm
Disassemble the dirty image into many, weak, overlapping versions of the synthesised beam, then reconstruct with cleaner beams (e.g., Gaussians).
image reconstruction: the maximum entropy method
Determine the ‘least committal’
image consistent with the data. Use the configurational entropy of
any consistent image to quantify its prior probability, and progress
via Bayes Theorem to get the most probable representation of the
source
