2.2.1 Source Size Effects

The signal power received from a sky area $d\Omega$ in direction is (see Fig.2.3 for notations) $A({\ensuremath{\mathbf s} }) I({\ensuremath{\mathbf s} }) d\Omega d\nu$over bandwidth $d\nu$, where $A({\ensuremath{\mathbf s} })$ is the antenna power pattern (assumed identical for both elements, more precisely $A({\ensuremath{\mathbf s} }) = A_i({\ensuremath{\mathbf s} }) A_j({\ensuremath{\mathbf s} })$ with A_i the voltage pattern of antenna i), and $I({\ensuremath{\mathbf s} })$ is the sky brightness distribution

 

$$A({\ensuremath{\mathbf s} }) I({\ensuremath{\mathbf s} }) d\Omega d\nu \cos(2 \pi \nu \tau_g)$$ dr = (2.13)

$$d\nu \int_{Sky} A({\ensuremath{\mathbf s} }) I({\ensuremath{\mathbf s} }) \cos(2 \pi \nu \ensuremath{\mathbf b} .\ensuremath{\mathbf s} /c) d\Omega$$ r = (2.14)

  

Figure 2.3: Position vectors used for the expression of the interferometer response to an extended source, schematically represented by the iso-contours of the sky brightness distribution.

Two implicit assumptions have been made in deriving Eq.2.14. We assumed incident plane waves, which implies that the source must be in the far field of the interferometer. We used a linear superposition of the incident waves, which implies that the source must be spatially incoherent. These assumptions are quite valid for most astronomical sources, but may be violated under special circumstances. For example VLBI observations of solar system objects would violate the first assumption, while observations of celestial masers could (in theory) violate the second one.

When the interferometer is tracking a source in direction ${\ensuremath{\mathbf s} _o}$, with ${\ensuremath{\mathbf s} } = {\ensuremath{\mathbf s} _o} + {\ensuremath{\mathbf \sigma} }$

 

$$d\nu \cos(2 \pi \nu \ensuremath{\mathbf b} .\ensuremath{\mathbf s... ...\cos (2 \pi \nu \ensuremath{\mathbf b} .\ensuremath{\mathbf \sigma} /c) d\Omega$$ r =

$$d\nu {\rm sin}(2 \pi \nu \ensuremath{\mathbf b} .\ensuremath{\mat... ...\sin (2 \pi \nu \ensuremath{\mathbf b} .\ensuremath{\mathbf \sigma} /c) d\Omega$$ - (2.15)

We define the Complex Visibility

$$V = \vert V\vert e^{i\Phi_V} = \int_{Sky} A(\ensuremath{\math... ...remath{\mathbf b} }.{\ensuremath{\mathbf \sigma} }/c)} d\Omega$$ (2.16)

which resembles a Fourier Transform...

We thus have

 

$$d\nu \left( \cos(2 \pi \nu \ensuremath{\mathbf b} .\ensuremath{\m... ...th{\mathbf b} .\ensuremath{\mathbf s} _o / c) \vert V\vert \sin(\Phi_V) \right)$$ r =

$$d\nu \vert V\vert \cos(2\pi \nu \tau_g - \Phi_V)$$ = (2.17)

i.e. the correlator output is proportional to the amplitude of the visibility, and contains a phase relation with the visibility.