NASSP Masters 5003F - Computational Astronomy - 2009 Lecture 12: The beautiful theory of...

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.


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.


Interferometry


Expression for the phase difference φ

θ

Pathdifference

d = D sinθ

sin22 ud

D=uλ


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...


The sky is full of sources.

The correlation returns anintensity-weighted averageof all their phases.

D=uλ


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


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


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


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


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


• 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:


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


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.


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


An exampleThe full visibility function V(u,v)

(real part only shown). A familiar pattern of ‘sources’


Let’s observe this with three antennas:

u

v

Longitude

Latit

ude


‘Snapshot’ sampling of V is poor.


Aperture synthesis via the Earth’s rotation.


View from the phase centre

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