5 Imaging extended objects

As soon as the individual apertures are not placed on a single steering mount, the entrance pupil of earth bound interferometers is different for objects with different declinations and varies with hour angle. Thus the part of the object complex visibility which is measured with the interferometer varies with time (see Fig.15).

Figure 15: Modulus of the object complex visibility of a uniform ellipse. The encircled areas are the parts of the object complex visibility picked up by VLTI's UT1 and UT3 for an object at declination -30$^{\circ }$ and hour angles -3, 0, and 3

From the object complex visibility we have to cut out the parts which are picked up by the entrance pupil of the interferometer. These are just the parts which lie in the uv-coverage belonging to the current array configuration for the observed object. The object complex visibility patches are now multiplied on their respective parts in the OTF, as illustrated in Fig.1.

For an earth-bound interferometer the baseline is changing with time with respect to the object. Thus the interferometer is sensitive to different parts of the object complex visibility. The projected baseline of an earth-bound interferometer can be calculated by

$$(u,v,w) = Rot(x,\delta)\,Rot(y,H)\,Rot(x,l)\,(x,y,z)$$ (10)

with $Rot(a,\alpha)$ being a rotation around axis a with angle $\alpha$, $\delta$ the declination of the object, H the hour angle of the observation, and l the latitude of the observatory. x is pointing to the east, y to the north, and z to zenith. u and v are coordinates in the (u,v)-plane and w is giving the sidereal optical path delay.

A value of the object complex visibility is estimated for every timestep of the aperture synthesis imaging process, i.e. for every orientation of the interferometer baseline with respect to the observed object. Using the Van Cittert Zernike theorem the object intensity distribution (in the sky) can be partially reconstructed from the measured visibilities.