Transcript Lecture 1 - University of Cape Town

Lecture 16
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Calibration
Self-calibration
VLBI
Spectroscopic interferometry
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration
• So far we have been assuming that
geometry is the only source of phase
differences.
• But phase (and amplitude) may also vary
because of
– Imperfections and/or instabilities in the
electronics;
– Changes in gain originating in properties of
the primary beam;
– Changes in refraction and path length through
• the atmosphere (troposphere);
• the ionosphere.
Most important
for an observer.
• All these need to be calibrated out.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration - the troposphere
• Phase delay due to troposphere varies
with zenith angle α (approx as 1/cos α).
– Although for α -> 90°, curvature of the earth
stops this going to infinity.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration - the troposphere
• Components:
– ‘Dry air’ component:
• Varies slowly.
• Range in variation is small.
• Directly measurable (via pressure).
– ‘Wet’ component (ie, water vapour):
• Can vary quickly.
• Can vary over a large range.
• Unpredictable...
– although H2O radiometers can help.
• Typical addition to ‘optical path length’ is
over 2 metres.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration - the ionosphere
• Variation with zenith angle is very similar
to that for the troposphere.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration - the ionosphere
• Excess path length varies with frequency as
L  40
Ne
2
• where Ne is the electron column density in m-2.
Note that ΔL is negative because the refractive
index is negative (yes it is weird).
• A typical value is about 1 m.
• Varies by a factor of 5 between day and night;
• Also large variation during ‘solar storms’.
• The variation with ν means one can calculate it by
simultaneous measurements at different
frequencies.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration
• Methods of calibration include:
– Internal monitoring of antenna parameters (eg
pointing, gain), and corrections calculated from these.
– Infrequent observation (usually by observatory
support staff) of calibration sources to calibrate a host
of antenna parameters which vary only slowly with
time.
– Once-per-observing-session observation of a ‘primary
flux calibrator’.
– Frequent during-observation peeks at a ‘phase
calibrator’.
• The last two are the most important for an
observer to know about.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration – some maths.
• A measurement of visibility V~j,k between antennas j
and k is related to the ‘true’ visibility Vj,k by
~
V j ,k  G j ,kV j ,k   j ,k   j ,k
(Note: all these are complex numbers.)
• where Gj,k is the complex gain, εj,k is an offset and
ηj,k is noise.
• Gj,k can be resolved into a product gjg*k of
complex-valued antenna gains gj and gk, which
depend purely on the antennas j and k respectively,
times a so-called ‘closure error’ gj,k.
NASSP Masters 5003F - Computational Astronomy - 2009
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Calibration – some maths.
Since the correlation is a digital operation,
and thus not subject to analog-type errors,
most errors occur upstream of the
correlation, and are therefore antennaspecific.
Hence, the closure error gj,k is usually
close to 1+i0, and can be ignored, at least
at a first pass..
Similarly, the offset εj,k is usually small and
ignorable.
Noise ηj,k can be minimized through
integration over time.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration – some maths.
• Thus we can write, to first order
approximation,
~
i  j k 
V j ,k  V j ,k a j ak e
• where aj and ak are (real-valued)
amplitude errors and φj-φk is the (realvalued) phase-difference error.
• Note that both as and φs vary with time in
an unpredictable fashion. (φ much more
than a.)
• Most of this variation is due to random
changes in the atmosphere + ionosphere.
NASSP Masters 5003F - Computational Astronomy - 2009
Calibration – some maths.
• Note that there are only 2N quantities to
calculate (1 a and 1 φ per antenna) but we
have N(N-1) ~ N2 measurements (a real
and an imaginary value from each of N(N1)/2 correlations). This is a happy situation
whenever N>4 (as it is for all modern
arrays of any importance).
• This means that we don’t need
measurements of V~ from all baselines –
which turns out to be useful for reasons
shortly to be revealed.
NASSP Masters 5003F - Computational Astronomy - 2009
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The ideal calibrator
Position and structure:
1.
2.
3.
4.
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point-like;
not confused;
of exactly known position;
near the object of interest.
Flux:
5. strong;
6. doesn’t vary with time;
7. smooth, flat spectrum.
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Different situations impose differing
relative emphasis.
NASSP Masters 5003F - Computational Astronomy - 2009
The real calibrator
• Of course (just like with people) we have to
make do with calibrators which never meet all
these desirables at once, and sometimes
don’t meet any of them!
• Some trade-offs:
– To find a calibrator near our object, we might
have to accept one which is not as strong as
we’d like (n ~ S-5/2 rule).
– A compact calibrator is usually time-variable,
sometimes on timescales as short as a day.
– We may have to discard data from short
baselines (because the calibrators are confused)
and also from long ones (because they resolve
the calibrator).
NASSP Masters 5003F - Computational Astronomy - 2009
‘Goldilocks’ baselines
What short baselines see:
confused.
What long baselines see:
resolved.
What just-right BL see:
isolated points.
NASSP Masters 5003F - Computational Astronomy - 2009
A simulated observation – V(t) from 1 baseline:
Amplitudes shown here.
Telescopes are moving between the target (long cycle)
and the calibrator (short cycle).
NASSP Masters 5003F - Computational Astronomy - 2009
A simulated observation – V(t) from 1 baseline:
Phases shown here – all cycles.
Telescopes are moving between the target (long cycle)
and the calibrator (short cycle).
NASSP Masters 5003F - Computational Astronomy - 2009
A simulated observation – V(t) from 1 baseline:
Phases shown here cal cycles only.
Telescopes are moving between the target (long cycle)
and the calibrator (short cycle).
NASSP Masters 5003F - Computational Astronomy - 2009
Special calibrations:
• Spectral-line:
– requires band-pass calibration.
• Polarization:
– ‘leakage terms’;
• Most modern feeds have 2 detectors, of opposite
polarisation;
• but their mutual isolation is not perfect.
– variation in polarization across the beam;
• A kind of polarized spherical aberration.
– ionospheric Faraday rotation.
NASSP Masters 5003F - Computational Astronomy - 2009
Self-calibration
• Ordinary calibration using an offset source
relies on:
– interpolation between peeks at the cal;
– the assumption that the phase shift in the
direction of the cal is the same as towards the
target.
• Variations due to imperfection of these
assumptions is usually >> thermal noise.
– Effect of this depends on size of array:
• Small array: may degrade image a little.
• VLBI: completely impossible to calibrate this way!
• So - can we do better...?
NASSP Masters 5003F - Computational Astronomy - 2009
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Self-calibration
Suppose we knew I(l,m), the brightness
distribution of the target.
Fourier transform this to get a continuous
model visibility function V^(u,v).
Divide the measured visibility samples
V~j,k(u,v,t) for baseline j-k by this model V^.
Is not the result the V we’d get from a
point source, multiplied by the product of
the antenna ‘gains’ gjg*k (slide 8)?
Then just perform normal calibration of gj
and gk.
NASSP Masters 5003F - Computational Astronomy - 2009
Bootstrap
• But... I is what we are trying to find out!
• However – an iterative approach works well,
provided:
– the target is fairly bright;
– a starting model can be obtained.
• A quite separate approach utilizes redundant
spacings – 2 or more visibility samples which lie
very close to each other in the u-v plane.
• Westerbork was specifically designed to
facilitate this – lots of antennas are the same
distance apart.
NASSP Masters 5003F - Computational Astronomy - 2009
Very Long Baseline Interferometry (VLBI)
• Maths is the same as standard interferometry –
it is the practical details which tend to be
different.
• Antennas are thousands rather than tens of km
apart. Hence:
– Resolution can be 1 milliarcsecond (mas) or even
lower.
– It isn’t practical to lock all antennas to the same
frequency standard (LO). Each has its own...
• thus the LOs must be very stable (eg H maser). - $$$
– Data are stored on tape and correlated later.
– With the longer baselines, the array is proportionately
more sensitive to phase errors...
• so special calibration techniques are needed.
NASSP Masters 5003F - Computational Astronomy - 2009
VLBI – a typical image:
CSO 1943+546
Polatidis et al NewAR 43, 657 (1999)
Courtesy the EVN archive.
NASSP Masters 5003F - Computational Astronomy - 2009
Spectroscopy
• What physical environment is associated
with spectral lines?
– Isolated atoms (may be partially ionized).
• Spectral lines can be observed both in
emission and absorption.
• Information coming out of spectroscopic
interferometry:
– composition, column densities, temperature,
velocity (both ‘bulk’ or nett, and velocity
dispersion).
• Processing:
– each channel imaged and CLEANed
separately.
NASSP Masters 5003F - Computational Astronomy - 2009
Spectroscopy
• We are now measuring not just x and y but
also a 3rd axis, ν. Hence we deal not in
images but in data cubes.
– A challenge to handle
• Can be huge: 20482 x 1024 pixels = 16 GB
– A challenge to visualize
• Channel maps and movies
• Moment maps: intensity, velocity, width
– 3D rendering and visualization software…
– A challenge to analyse…
• Model for structure & kinematics
• Optical depth effects
• Excitation: collisions vs radiation
NASSP Masters 5003F - Computational Astronomy - 2009
M82 cube movie
Shows HI absorption – different velocities (therefore different Doppler-shifted
frequency) at different places.
Many thanks to Rob Beswick.
M82 HI (Wills, Pedlar et al)
Continuum plus line
NASSP Masters 5003F - Computational Astronomy - 2009
HI imaging
• Valuable for cosmology:
– Traces galaxies  large-scale structure.
– Also shows kinematics inside galaxies.
• But, it’s hard to get high resolution...
– because the brightness temperature of HI is
limited to about 100 K;
– bandwidth is limited;
– wavelength is relatively long;
– and, for an interferometer, Ae is not λ2/ΩA, as
is true for a single dish. Ae stays fixed (it’s a
property of the antenna).
• Thus smaller beam (ie ΩA) gets less flux. For HI,
practical limit to ΩA (and thus resolution) is ~1
arcsec.
NASSP Masters 5003F - Computational Astronomy - 2009
Another M82 movie, showing OH emission
and absorption, with continuum subtracted.
Many thanks to Rob Beswick.
M82 OH – 1665 & 1667 lines – masers & absorption
VLA A-array data (Argo et al 2007)
Two lines in band
Absorption in black
Masers in blue
Note ringing about the brightest 1667MHz maser.
NASSP Masters 5003F - Computational Astronomy - 2009