Aperture Synthesis Flashcards
fringe rate
dΦ/dt = 2π/λ Dcosδ cosH dH/dt
D = length
δ = declination
H = hour angle
dH/dt = 2π/86400
Γ(r) = 1/S ∫ B(α)exp(ikr-α)dΩ
< Ψ1 Ψ2> ∝ exp(-ikr.θ)
< Ψ1 Ψ2> ∝ S exp(-ikr.θ)
d < Ψ1 Ψ2> ∝ B(θ)dΩ exp(-ikr.θ)
<Ψ1 Ψ2> ∝ ∫sky B(θ) exp(-ikr.θ) dΩ
Γ(r) = < Ψ1 Ψ2*> , Γ(0) =1
1/ ∫ B(θ)dΩ = 1/S
giving result
Γ(r) = 1/S ∫ B(α)exp(ikr-α)dΩ
cross the meridian
H = 0
for east west baseline
dΦ/dt = 2π/λ Dcosδ cosH dH/dt
for north south baseline
D = (0,D,0)
D.θ(hat) = cosHcosδ
Φ = 2π/λ Dcosδ cosH
dΦ/dt = 2π/λ Dcosδ sinH dH/dt
resolution in right ascension
= λ/D
resolution in declination
= λ/Dsinδ
primary beam
beam of single dish
synthesized beam
effective beam of synthesized aperture
fringe rate can be attributed to doppler shift derivation
D = r2-r1
v1 = r1 x Ω
v2 = r2 x Ω
Doppler shift = Δv = v/c v.θ(hat) = 1/λ (Ω x D).θ(hat)
D = (D,0,0)
Ω = (0,0,Ω)
θ(hat) = (sinHcosδ,cosHcosδ,sinδ)
Δv1 - Δv2 = D/λ cosH dH/dt
Why can certain interferometers only image sources of high surface brightnesses
long baselines => needs lots of flux on small angular scales. Need high B
uv plane
plane perpendicular to the line of sight to the source
u and v measure the projected baseline of the interferometer onto the plane in units of wavelength
this is the fourier transform plane of the image
u-v tracks of δ= 90 and general δ diagram
semi-circular with D/λ
and elliptical with D/λ sinδ and D/λ
fringe rate derivation
tau = D/c cosθ
φ = 2πD/λ cosθ
dφ/dt = 2πD/λ sinθ
if the source is not directly overhead
we must consider the projected baseline in the plane perpendicular to the source direction. One may also need to preform path compensation
a point source at position so can be modelled as
a delta function, so that the sky brightness distribution is
B(s) = δ(s-so)
VCZ, correlated signal is
Γ(s) ∫ B(s) e^ikxs ds
= e^ikxso
the phase correlated signal, arg(Γ) = kxso = 2πxso/λ encodes the position of the source on the sky
fringe visibility definition
the fundamental measured quantity in a radio interferometer. it corresponds to the visibility of the young’s fringes that would be generated by an aperture comprising two spaced ‘holes’ corresponding to the radio dishes.
The normalised sky brightness is the FT of the fringe visibility as a function of antenna separation.
correlation definition
practical process of generating the fringe visibility from two antenna signals without actually generating the fringes. For two signals Ψ1 and Ψ2, the correlator generates an output proportional to the <Ψ1*Ψ2>. This number is proportional to the fringe visibility and is usually computed using dedicted hardware.
primary and synthesised beams definition
primary - beam of the individual antennas and defines the field of view of the interferometer.
synthesised - is the ‘point spread function’ of the whole interferometer, and corresponds to the dirty image that the interferometer you generate from a point source. It is the fourier transform of the u-v plane coverage.
