20.2 The Phase Equation

The most important measurement for radio astrometry is that of the actual fringe phase of a connected-element interferometer (or similarly the group delay in VLBI). Let $\theta$ be the angle between the reference direction and the meridian plane of a given interferometer baseline. The phase is then defined by

$$\phi_r = 2 \pi {\cal B} \sin(\theta)/\lambda$$ (20.4)

If the point-like source of interest is offset by $\Delta\theta$ from the reference direction the total phase is

$$\phi = 2 \pi {\cal B} \sin (\theta+\Delta\theta)/ \lambda \simeq \phi_r + 2 \pi {\cal B} \cos(\theta) \Delta\theta / \lambda$$ (20.5)

It is thus clear that measuring an angle or an offset position on the celestial sphere becomes possible only when all phase calibration problems have been understood and solved.

Accounting for uncertainties in the baseline and source position vectors the actual phase is

$$\phi = 2 \pi ({\vec {\cal B}}+\delta{\vec {\cal B}}).({\vec k}_0+\delta{\vec k}) / \lambda$$ (20.6)

where ${\vec {\cal B}}$ is a first approximation of the baseline, ${\vec k}_0$ the tracking direction; ${\vec {\cal B}}+\delta{\vec {\cal B}}$ and ${\vec k}_0+\delta{\vec k}$ are the true baseline and source position vectors, respectively. The reference phase is given by

$$\phi_r = 2\pi{\vec {\cal B}}.{\vec k}_0 / \lambda$$ (20.7)

and, neglecting the term involving $\delta{\vec {\cal B}}.\delta {\vec k}$, we obtain

$$\phi - \phi_r = 2 \pi ({\vec {\cal B}}.\delta{\vec k} + \delta{\vec {\cal B}}.{\vec k}_0) / \lambda$$ (20.8)

We consider all vector projections in the right-handed equatorial system defined by the unit vectors $a_1$ ($H=6$ h, $\delta=0$), $a_2$ ($H=0$ h, $\delta=0$), $a_3$ ( $\delta = 90^\circ$). (Note that this system is not the Cartesian coordinate system used in [Thompson et al. 1986].) $H$ and $\delta$ are the hour angle and declination, respectively. In the equatorial system the baseline vector ${\vec {\cal B}}$ has components $( {\cal B_1, B_2, B_3} )$ and the components of the reference position ${\vec k}_0$ are given by $(\cos(\delta) \sin(H), \cos(\delta) \cos(H), \sin(\delta))$

The two limiting cases $\delta{\vec k} = 0$, and $\delta{\vec {\cal B}} = 0$ correspond to those where we either calibrate the baseline or determine the exact source position.

In the first case the source coordinates are perfectly known and by comparing the observed phase $\phi$ with the reference phase $\phi_r$ one determines $\delta{\vec {\cal B}}$ and hence the true baseline ${\vec {\cal B}}+\delta{\vec {\cal B}}$. The reference sources observed for baseline calibration are bright quasars or galactic nuclei whose absolute coordinates are accurately known. The most highly accurate source coordinates are those of the radio sources used to realize by VLBI the International Celestial Reference Frame (ICRF); distribution of coordinate errors are below one milliarcsecond. However, the ICRF catalogue is insufficient for phase and baseline calibrations of millimeter-wave arrays because most sources are not bright enough in the millimeter-wave domain. The IRAM calibration source list is thus a combination of several catalogues of compact radio sources. Today, the Plateau de Bure Interferometer catalogue of calibration sources is based mostly on compact radio sources from the Jodrell Bank - VLA Astrometric Survey (JVAS - [Patnaik et al 1992], [Browne et al. 1998], [Wilkinson et al. 1998]).