Next: Astronomical Co-ordinate System
Up: Coordinate Systems
Previous: Coordinate Systems
Contents
As described in Chapter 4, the response of an
interferometer to quasi-monochromatic radiation from a point source
located at the phase center is given by
|
(10.1.1) |
where
is the geometrical delay,
is the direction which the antennas are tracking with respect
to the vertical direction, is the wavelength, is the
center frequency of the observing band and is the separation
between the antennas. As the antennas track the source, the geometrical
delay changes as a function of time.
This changing is exactly compensated with a computer controlled
delay and for a point source at the phase center, the output of the
interferometer is the amplitude of the fringe pattern.
For a source located at an angle
, for small ,
. Since fringe stopping compensates for ,
the response of the interferometer for a source away
from the phase center is
where
. If the phase center is
shifted by equivalent of , the interferometer will pick up
an extra phase of and the response will be sinusoidal instead
of co-sinusoidal. Hence, an interferometer responds to both even and
odd part of the brightness distribution. Interferometer response can
then be written in complex notation as
|
(10.1.2) |
Writing
, which is the projected separation
between the antennas in units of wavelength in the direction normal to
the phase center and
, we
get
|
(10.1.3) |
as the complex response of a two element interferometer for a point
source of unit flux located away from the phase center
given by the direction .
Usually the phase center coincides with the center of the field being
tracked by all the antennas. Let the normalized power reception
patter of antennas (which are assumed to be identical) at a particular
frequency be
and the surface brightness of an
extended source be represented by
. The response of
the interferometer to a point source located away from
the phase center would then be
. For an
extended source with a continuous surface brightness distribution, the
response is given by
|
(10.1.4) |
The above equation is a 1D Fourier transform relation between the
source brightness distribution and the output of the visibility
function . The integral is over the entire sky visible to the
antennas but is finite only for a range of limited by the antenna
primary reception pattern . In practice, is calculated as a
function of the source position in the sky, specified in astronomical
co-ordinate system, as seen by the observer on the surface of the
earth.
in the above equation is the direction of the elemental source
flux relative to the pointing center. then has the interpretation
of spatial frequency and represents the 1D spatial frequency
spectrum of the source.
Next: Astronomical Co-ordinate System
Up: Coordinate Systems
Previous: Coordinate Systems
Contents
NCRA-TIFR