Once the baseline is fully calibrated (
) the exact source
coordinates are known from the
vector components. These components
are formally deduced from the differential of
![]() |
= | ![]() |
(17.9) |
![]() |
|||
![]() |
![]() |
(17.10) |
A | = | ![]() |
(17.11) |
B | = | ![]() |
(17.12) |
C | = | ![]() |
(17.13) |
Measurement of the phase at time intervals spanning a broad hour angle interval
allows us to determine the three unknowns A, B, and C, and hence
and
and the exact source position. Note that for sources close to the
equator, A and B alone cannot accurately give
. In the latter case, C
must be determined in order to obtain
; this requires to accurately
know the instrumental phase and that the baseline is not strictly oriented along the
E-W direction (in which case there is no polar baseline component).
A synthesis array with several, well calibrated, baseline orientations is thus a
powerful instrument to determine
. In practice, a least-squares
analysis is used to derive the unknowns
and
from the
measurements of many observed phases
(at hour angle Hi) relative to the
expected phase
. This is obtained by minimizing the quantity
with respect to A , B, and C where
. A complete analysis should give the variance of the
derived quantities
and
as well as the correlation
coefficient.
Of course we could solve for the exact source coordinates and baseline components simultaneously. However, measuring the baseline components requires to observe several quasars widely separated on the sky. At mm wavelengths where atmospheric phase noise is dominant this is best done in a rather short observing session whereas the source position measurements of often weak sources are better determined with long hour angle coverage. This is why baseline calibration is usually made in separate sessions with mm-wave connected-element arrays.
The equation giving the source coordinates can be reformulated in a more
compact manner by using the components u and v of the baseline projected in a
plane normal to the reference direction. With v directed toward the north and utoward the east, the phase difference is given by
![]() |
(17.14) |
u | = | ![]() |
(17.15) |
v | = | ![]() |
(17.16) |
![]() |
(17.17) |
u | = | ![]() |
(17.18) |
v | = | ![]() |
In order to derive the unknowns
and
the least-squares
analysis is now performed using the components ui, vi derived at hour angle
Hi. In the interesting case where the phase noise of each phase sample is
constant (this occurs when the thermal noise dominates and when the atmospheric
phase noise is ``frozen'') one can show that the error in the coordinates takes a
simple form. For a single baseline and for relatively high declination sources the
position error is of order
where
is the phase noise and np the number of individual phase
measurements. This result implies (as expected a priori) that lower formal
uncertainties are obtained with longer observing times and narrower synthesized
beams. Of course the position measurements are improved with several independent
interferometer baselines; the precision improves as the inverse of the square root
of n(n-1)/2 for n antennas in the array.
We have shown that for a well calibrated interferometer the least-squares fit analysis of the phase in the (u,v) plane can give accurate source coordinates. However, the exact source position could also be obtained in the Fourier transform plane by searching for the coordinates of the maximum brightness temperature in the source map. The results given by this method should of course be identical to those obtained in the (u,v) plane although the sensitivity to the data noise can be different.
Finally, it is interesting to remind that the polar component of the baseline does not appear in the equation of the fringe frequency which is deduced from the time derivative of the phase. There is thus less information in the fringe frequency than in the phase.