20.3 Determination of Source Coordinates and Errors

Once the baseline is fully calibrated ( $\delta{\vec {\cal B}} = 0$) the exact source coordinates are known from the $\delta{\vec k}$ vector components. These components are formally deduced from the differential of ${\vec k}_0$. In the right-handed equatorial system defined in Section 20.2 we obtain

$$\delta{\vec k} = ( - \sin(\delta) \sin(H) \Delta\delta - \cos(\delta)\cos(H) \Delta\alpha,$$ (20.9)

$$-\sin(\delta)\cos(H)\Delta\delta+\cos(\delta)\sin(H)\Delta\alpha,$$

$$\cos(\delta)\Delta\delta)$$

where $\Delta\alpha$ and $\Delta\delta$ are the right ascension and declination offsets in the equatorial system $(\Delta\alpha= -\Delta H)$. The phase difference is then a sinusoid in $H$

$$\frac{(\phi - \phi_r)\lambda}{2\pi} = {\vec {\cal B}}.\delta {\vec k} = C_1 \sin(H) + C_2 \cos(H) + C_3$$ (20.10)

where

$$C_1 = - {\cal B}_1 \sin(\delta)\Delta\delta + {\cal B}_2 \cos(\delta)\Delta\alpha$$ (20.11)

$$C_2 = - {\cal B}_2 \sin(\delta)\Delta\delta - {\cal B}_1 \cos(\delta)\Delta\alpha$$ (20.12)

$$C_3 = {\cal B}_3 \cos(\delta)\Delta\delta + \phi_{\mathrm{ins}}$$ (20.13)

and $C_3$ contains the instrumental phase $\phi_{\mathrm{ins}}$.

Measurement of the phase at time intervals spanning a broad hour angle interval allows us to determine the three unknowns $C_1$, $C_2$, and $C_3$, and hence $\Delta\alpha$ and $\Delta\delta$ and the exact source position. Note that for sources close to the equator, $C_1$ and $C_2$ alone cannot accurately give $\Delta\delta$. In the latter case, $C_3$ must be determined in order to obtain $\Delta\delta$; 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 $\delta{\vec k}$. In practice, a least-squares analysis is used to derive the unknowns $\Delta\alpha$ and $\Delta\delta$ from the measurements of many observed phases $\phi_i$ (at hour angle $H_i$) relative to the expected phase $\phi_r$. This is obtained by minimizing the quantity $\Sigma(\Delta\phi'_i - (C_1 \sin(H_i) + C_2 \cos(H_i) + C_3))^2$ with respect to $C_1$, $C_2$, and $C_3$ where $\Delta\phi'_i = (\phi_i - \phi_r)\lambda/2\pi$. A complete analysis should give the variance of the derived quantities $\Delta\alpha$ and $\Delta\delta$ 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 $u$ toward the east, the phase difference is given by

$$(\phi - \phi_r) = 2\pi(u \cos(\delta)\Delta\alpha + v \Delta\delta)$$ (20.14)

Comparing this formulation to the sinusoidal form of the phase difference we obtain

$$u = (- {\cal B}_1 \cos(H) + {\cal B}_2 \sin(H)) / \lambda$$ (20.15)

$$v = ({\cal B}_3 \cos(\delta) - \sin(\delta)({\cal B}_1 \sin(H) + {\cal B}_2 \cos(H))) / \lambda$$ (20.16)

Transforming the ${\cal B}_{1,2,3}$ into a system where the baseline is defined by its length ${\cal B} = ({\cal B}_1^2 + {\cal B}_2^2 + {\cal B}_3^2)^{0.5}$ and the declination $d$ and hour angle $h$ of the baseline vector (defined as intersecting the northern hemisphere) we obtain

$${\cal B}_1 = {\cal B} \cos(d) \sin(h), {\cal B}_2 = {\cal B} \cos(d) \cos(h), {\cal B}_3 = {\cal B} \sin(d)$$ (20.17)

and

$$u = (\cos(d) \sin (H - h)) {\cal B} / \lambda$$ (20.18)

$$v = (\cos(\delta)\sin(d) - \sin(\delta)\cos(d) \cos (H - h)) {\cal B} / \lambda$$

which shows that the locus of the projected baseline vector is an ellipse.

In order to derive the unknowns $\Delta\alpha$ and $\Delta\delta$ the least-squares analysis of the phase data is now performed using the components $u_i, v_i$ derived at hour angle $H_i$. 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 approximated by the equation

$$\sigma_{\alpha,\delta} = \Delta \theta \simeq \sigma_\phi / (2\pi \sqrt{n_p}({\cal B}/\lambda))$$ (20.19)

where $\sigma_\phi$ is the phase noise and $n_p$ 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.