lecture 12: the beautiful theory of interferometry
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Lecture 12: The beautiful theory of interferometry. First, some movies to illustrate the problem. Clues of what sort of maths to aim for. Geometrical calculation of phase. Visibility formulas. Simplifying approximations:. - PowerPoint PPT PresentationTRANSCRIPT
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NASSP Masters 5003F - Computational Astronomy - 2009
Lecture 12:The beautiful theory of interferometry.
• First, some movies to illustrate the problem.
• Clues of what sort of maths to aim for.
• Geometrical calculation of phase.
• Visibility formulas.
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NASSP Masters 5003F - Computational Astronomy - 2009
Simplifying approximations:
• I’ll assume a small bandwidth Δν, so signals can be approximated by sinusoids.
• I’ll neglect polarization.
• I’ll explain interferometry first in 2 dimensions, then extend this to 3.
• There are many sorts of interferometer but I’ll concentrate on aperture synthesis.
• The fundamental maths for a 2-antenna interferometer can be easily expanded to cater for more than 2.
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NASSP Masters 5003F - Computational Astronomy - 2009
Interferometry
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NASSP Masters 5003F - Computational Astronomy - 2009
Interferometry
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NASSP Masters 5003F - Computational Astronomy - 2009
Expression for the phase difference φ
θ
Pathdifference
d = D sinθ
sin22 ud
D=uλ
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NASSP Masters 5003F - Computational Astronomy - 2009
Clues..?
• Since φ is proportional to sin(θ), if we could measure φ, we could work out where the source is in the sky.
• From last lecture, we saw that correlating the signals from the two antennas gives us a number proportional to S exp(-iφ), where S is the flux density of the source in W Hz-1. Things are looking good!
• The fly in the ointment is...
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NASSP Masters 5003F - Computational Astronomy - 2009
The sky is full of sources.
The correlation returns anintensity-weighted averageof all their phases.
D=uλ
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NASSP Masters 5003F - Computational Astronomy - 2009
Formally speaking...• The correlation between the voltage signals
from the two antennas is
• We’ll call V the visibility function.• A is the variation in antenna efficiency with θ.• I is the quantity formerly known as B – ie the
brightness distribution. Its units (in this 2-dimensional model) are W m-2 rn-1 Hz-1
• To keep things simple(r) I’ve ignored any summation over frequency ν.
DiIAdDVyy ,exp21
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NASSP Masters 5003F - Computational Astronomy - 2009
Coordinate systems – everything is aligned with the phase centre.
b
w
u
For baselines:
2 1
sθ
l
(1-l2)1/2 Celestial sphere
dθ
=sinθ
sb
212ˆ2 lwul sb
The direction normal tothe plane of the antennasis called the phase centre.Normally the antennas arepointing that way, too.
=cosθ
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NASSP Masters 5003F - Computational Astronomy - 2009
Putting it all together:
• gives
• We want to complete the change of variable from θ to l. It’s not hard to show that
• so the final expression is
2
2
212exp,
lwuliIAdwuV
21cos l
dldld
1
1
2
212exp2exp
1, liwiul
l
lIlAdlwuV
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NASSP Masters 5003F - Computational Astronomy - 2009
Ahah!• Provided w=0, or in other words provided the
antennas all lie in a single plane normal to the phase centre,
• This is a Fourier transform! We just need to get a lot of samples of V at various u, then back-transform to get
• Trouble is, we can’t always keep w=0.
.2exp1 2
iul
l
lIlAdluV
21 l
lIlA
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NASSP Masters 5003F - Computational Astronomy - 2009
Non-coplanar arrays.
b1,2
2 1
sθ
l
(1-l2)1/2
dθ
3
b2,3
b1,3
Phase centre
w
u
For baselines:
u1,3
w1,3 = w2,3
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NASSP Masters 5003F - Computational Astronomy - 2009
• The factor of (1-l2)1/2 prevents the full expression for V´(u,w) from being a Fourier transform.
• But for small l, (1-l2)1/2 is close to 1, and varies only slowly. Let’s do a Taylor expansion of it:
• The full expression therefore becomes
(1-l2)1/2 = 1 – ½l2+O(l4).
2
2exp2exp
12exp, iwliul
l
lIlAdliwwuV
Non-coplanar arrays - small-field approximation:
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NASSP Masters 5003F - Computational Astronomy - 2009
Non-coplanar arrays - small-field approximation:
• For V(u)=V´(u,w)exp(2πiw), and πwl2<<1,
and we are back to the Fourier expression.
• V is like measuring V´ with a phantom antenna in the same plane as the others.
• So: for non-coplanar antennas, what matters is the projection length u of each baseline, projected on a plane normal to the ‘phase centre’.
iul
l
lIlAdluV 2exp
1 2
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NASSP Masters 5003F - Computational Astronomy - 2009
The phase centre• The location of the phase centre is controlled
as follows:1. Decide which direction you want to be the phase
centre.
2. That direction defines the orientation of the u-w axes.
3. Calculate wj,k for each j,kth baseline in that coordinate system.
4. Multiply each correlation Rj,k by the appropriate e2πiw factor. (Equivalent is to delay the leading signal by t=w/ν.)
5. After appropriate scaling, the result is a sample of the visibility function V(uj,k) appropriate to the chosen phase centre.
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NASSP Masters 5003F - Computational Astronomy - 2009
Now we go to 3 dimensions and N antennas.
• The coordinate axes in 3 dimensions are labelled (u,v,w) for baselines and (l,m,n) for source vectors.
• Each pair of antennas gives us a different projected baseline.
• N antennas give N(N-1)/2 baselines.
• Thus N antennas give N(N-1)/2 samples of the ‘coplanar’ visibility function
vmuliml
mlImlAdmdlvuV
2exp
1
,,,
22
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NASSP Masters 5003F - Computational Astronomy - 2009
An exampleThe full visibility function V(u,v)
(real part only shown). A familiar pattern of ‘sources’
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NASSP Masters 5003F - Computational Astronomy - 2009
Let’s observe this with three antennas:
u
v
Longitude
Latit
ude
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NASSP Masters 5003F - Computational Astronomy - 2009
‘Snapshot’ sampling of V is poor.
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NASSP Masters 5003F - Computational Astronomy - 2009
Aperture synthesis via the Earth’s rotation.
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NASSP Masters 5003F - Computational Astronomy - 2009
View from the phase centre