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310
APPENDIX A
APERTURE SYNTHESIS
The method of interferometric aperture synthesis is
best understood by first considering the analogy with the
single-dish telescope. The voltage V produced at the horn
of this telescope by an element of .area of the dish dudv
due to radiation at frequency w from an element of sky
dxdy may be written as.
i [t+{-(x,y,t)] i2n[U.(X-x 0 ).+V(Y-y
V(u,v,x,y,t)=V(x,y)W(u,v)e e
where-the telescope is -pointed at (x%0, y ), u and v are
measured in wavelengths, is a random function which
varies rapidly in all coordinates, and W is a weighting
function which accounts for taper introduced by the horn
and which is zero outside the dish area. The total
voltage at the horn is the integral of this voltage over
the dish and the sky. This voltage is then detected by a
system which responds to the time average of the power.
Thus
311
R(x0 , y0) =
co ' oo • o.002 2r% ~S] dudvJ du'dv' dxdy 'dx'dy' T (x,y)
T(x',y)W(u,v)W (uv)exp[ 2i u(x x)-u'(x'-x0 )+v( y0)
-v (y' -y0) (-exp .il( (x, y 't.) - (x', ty.',t
where we introduce the brightness temperature of the sky
2T(x, y), where T(x, y) is proportional to V2(x, y). Since
i(x,yt)-g(x',y',t)]
2(e )= 6(x-x' y -y)
by our assumption of a random noise signal, the response
simplifies to
R y ) sfoou c -2Ti[(u-u')x +(v-v')y
R (x = 4dud fdu'dv'e 0
0 0• 0
W(u, v) W(u' , v')
.00
x f dxdy T(x,y)ei2n [(u-u' )x+ (v-v' )y]
(A-1)
F.
312
Transforming coordinates we obtain finally
00 -2ni(ux04yR(x 0 ,y.0 = ) dudV F(u,v)A(u,v)e
-oo
(A-2)unc in-i.*d,.ied a
where the complex fringe visibility function is defined as
2C i2rr (ux,+vy)F(u,v) 2 dxdy T(x,y)ei uv
and the autocorrelation of the weighting function is
co
A(u,v) f du'dv' W(u+u', v+v') W(u',v').
-co
Thus, the single dish responds to the Fourier transform of
the visibility function weighted by a sampling function.
Let us now consider the standard sampling theorem
(Bracewell 1958). We sample the visibility function at
regular intervals in u of 1/2 xc and v of 1/2 yc The
sampled visibility function may be written as
2V' (u,v) = III(2x mAu, 2y nAv) V(u,v)
c c
where the III (sha) function is defined as
S2- 00c co 2iII(xy) = 0 Z 6 (x-p, y-v).
If we Fourier transform the sampled visibility function we
obtain a map given by
- j'!M
313
2R(x, y) = 2III(x/2x, y/2y) * T(x, y)c c
where.. T(x, y) is the true brightness distribution and *
denotes a convolution. In other words, we now have -the
desired map repeated at intervals of 2x in x and 2y in
y. If
T(x, y) = 0 for jxj > x c or yl >yC ..
then the repetition does not cause the maps to overlap and
hence causes no error. Therefore, we must sample the
visibility function at intervals smaller than the inverse
diameter of the .source be ng observed.
Since this sampling interval is about 400 feet (for
the sources in this program at 21 cm) which is larger than
the diameter of almost all single-dish telescopes, .. ' all
information collected by the elements of a single dish is
redundant for such sources... We may remove this redundancy
or lack of resolution at reasonable cost if we sacrifice
collecting area. We use one small -telescope as the central
surface element of the aperture to be synthesized while
another movable telescope provides other surface elements
of the synthesized aperture (Figure Al). If the voltages
314
Ni
D
Figure Al. Geometry of two-element interferometer.
P
North South
Figure A2. Celestial sphere with source (Q) and pole ofbaseline (B) .
.'315
from .the two telescopes are multiplied together and
smoothed the output is2 iT(x, y) ( )
R(x, y) = V e (A-3)0
2nDwhere T(x, y) =-y sino(x,y) and V0 is the voltage pro-
duced by the source at each telescope. As the earth
rotates, sinG varies producing an approximately sinusoidal
output called fringes. The phase T may be expanded about
some angle 9 as0
2nD 2nD
T = sine + cost AG (A-4)
where 'the first term is the "expected fringe" which is
removed during reduction .and the second term is the "fringe
phase". For an extended source, the total response of the
interferometer is just the integral of the separate
responses of equation (A-3) to produce
or i2T (ux+vy)R(u,v) = JJ dxdy T(x,y)e
where the expected fringe has been removed and the phase
s cosO a.g has been expressed in rectangular coordinates.x 0
In other words, the interferometer directly measures the
visibility function.
316
The formulae for sin9 0 , u, and v may be derived as
follows. The three points indicated on the celestial
sphere in Figure A2 are the north celestial pole P, the
pole of the extended baseline B and the source .Q. The
hour angle and declination of the source are h and 6,..
respectively, while the corresponding quantities for the
baseline are H and A. The angle between Q and B is r/2-.-G
The law of cosines is used on the triangle PQB to give..
sinG = sinbsinA + cosbcosAcos (h-H) .
The projection of the baseline along constant longitude is
D .asinG D
S n cososin--sincosAcos•(h--H)
.and the projection of the baseline alongs constant latitude
u .D_1 D n - cos6sin(h-H).
Xcos6 ,h
The choice of signs for u and v is a matter of convention
while the factor (l/cosb) enters the equation for u so that
x may be expressed in units of angle rather than time. As
the source changes hour angle, the values of u and v describe
an ellipse in the (u, v) plane. To fully sample the visi-
bility function, we observe the source through the
I- -
317
available range of hour angle at a number of discrete
antenna separations D.
The Hermitian property of the .. visibility function
is quite useful and easily derived. Since the visibility
function and the source brightness are a Fourier-transform
pair we may write
• co. -i2m (ux+vy)
T(x, y) =. dudv V(u,v)e
Writing the visibility function as the sum of a real and
imaginary part we obtain ..the imaginary part of T as
Im(T(x, y)) =
y _.
cooa f dudv[I(uv)cos2n(ux+vy)-R(uiv)sin2n(ux+vy)]
Since this must be zero we may use the symmetry properties
of sines and cosines and the independence of R and I to
show that
1.(u, v) = R(-u, -v)
I(u, v) = - I(-u, -v)or
V(u, v) = V*(-u, -v) .
Therefore we need sample the visibility function over only
one-half of the (u, v) plane. This formalism is only a
more elegant way to express the fact that the only
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