Advanced Topic in Astrophysics, Lecture 3
Radio Astronomy – Interferometer mapping Maria J. Rioja, ICRAR, UWA Outline •  The relationship between measured correlator
visibilities and the sky brightness.
•  Aperture synthesis for high quality images
•  Incomplete aperture
–  The problem with the “dirty image”
•  Deconvolution
–  CLEAN •  Some examples
Reminder…
Correlator ouput:
Spatial Coherence Function:
 
Vν ( r1, r2 ) =
 −i2 πν
∫ Iν (σ ) ⋅ e
  
( r1 − r2 )⋅σ / c

dσ
van-Cittert Zernicke theorem which states that the spatial coherence
function is the Fourier
€ Transform of the source brightness distribution
sYU-­‐ c
2 u
Simple case: One EW baseline, point-like polar source
sAUNc
2 u
(x,y)
λ
=(u,v)
(u,v)
Earth
=C
e t l e( p e
u  p l t uC u
l u

 
  
 
i2 πν ( r1 − r2 )⋅ sˆ / c
i2 π D ⋅ sˆ / λ
V12 ⇒ V ( r1, r2 ) = I ⋅ e
= I⋅e
, D ≡ r1 − r2
V12 ⇒ V (u1,v1 ) = I ⋅ e i2 π (u1 x +v1 y )
€
New coordinate system
(UV-Plane // XY-Planec
I(x,y)
2 u
Simple case: One EW baseline, extended polar source
σ
= (x,y)
sAUNc
(x,y)
2 u
σ
FT
Earth
Point source:
λ
=(u,v)
(u1,v1)
2 telescopes = 1 baseline -> 1 point in UV plane
V (u1,v1 ) = I ⋅ e i2 π (u1 x +v1 y )
+∞ +∞
Extended source:
V (u1,v1 ) =
∫∫
I(x, y) ⋅ e i2 π (u1 x +v1 y ) dxdy
−∞ −∞
Visibility V(u,v) is the 2-D Fourier Transform of the Sky Brightness I(x,y):
€
+∞ +∞
⇒ I (x, y) = ∫ ∫ V (u,v) ⋅ e i2 π (ux +vy ) dudv, I(x, y) = F −1[V (u,v)]
€
−∞ −∞
Ol t n
eCnA CAn
I(x)
Ol t n
niA u
V(u)
e
Examples of FT
Extended source
2-D Fourier Transform
More compact source
I(x,y)
I(x,y)
x
y
XY-Plane
V(u,v)
x
y
V(u,v)
u
v
UV-Plane
u
v
sYU-­‐ c
2 u
Simple case: Three EW baseline, extended polar source
σ
= (x,y)
sAUNc
(x,y)
σ
2 u
λ
=(u,v)
(u1,v1)(u ,v )
2 2
(u3,v3)
Earth
3 telescopes = 3 baselines  3 points in UV plane
N telescopes, N(N-1)/2 baselines
I(x,y)
UV-Plane
=I u C uue
1bp e u e
Interferometer Imaging: Earth Rotation Synthesis
2 u
(x,y)
2 u
σ
Earth
λ
=(u,v)
3 telescopes = 3 baselines  3 points in UV plane
N telescopes, N(N-1)/2 baselines
+ Earth Rotation  Many more points!
I(x,y) much better
+ many many more…
Effects of incomplete UV-coverage
•  We sample the Fourier plane at a discrete number of points:
•  Measured visibility is
• 
[S(u,v ) ⋅ V (u,v )]
So the inverse transform is:
€I(x,y)
V(u,v)
[S(u,v ) ⋅ V (u,v )]
B( x, y ) ⊗ I ( x, y )
What we measure is the “Dirty” Map:
€
€
with I(x,y) = “true map”; B (x,y) = “point spread function” or “dirty beam”
sAUNc
From a “Dirty” to “CLEAN” image,
V nC-­‐ B
V nC-­‐ B O
Deconvolve
“Clean” Image
V
uB O
(
t uNt N
O
O (
t uC l n
Au L t u
Deconvolution Algorithm: Classic CLEAN
•  Uses iterative algorithm to find sequence of point sources:
–  Find peak in “DIRTY” image –  Subtract a PSF, store component thus found –  If any significant points leZ, return to first step –  Convolve point components by “Clean” point spread funcUon •  Same width as dirty PSF but no sidelobes –  Add residuals image to obtain “restored” image Example
•  VLBA simulated observations of M87-like jet source
•  Will show:
–  UV coverage –  Visibility funcUon –  Point Spread FuncUon –  Dirty image –  Clean images Credit: T. Cornwell Original Source model
UV-coverage and Visibility Function
Point Spread Function (or “Dirty Beam”)
Original model and Dirty image
Classic CLEAN: 5000 and 20000 comps
Original model and best image
END
Next lessson:
- VLBI Technique, by Maria Rioja (3pm – 4pm)
Location: ICRAR/UWA
Bibliography •  “Imaging and deconvoluUon”, Tim Cornwell Synthesis imaging summer school, 2002, NRAO •  Other on-­‐line presentaUons in same summer school