A synthesis imaging radio instrument consists of a number of radio elements (radio dishes, dipoles, or other collectors of radio emission), which represent measurement points in u,v,w space. We need to describe how to convert an array of dishes on the ground to a set of points in u,v,w space.
E, N, U coordinates to x, y, z
The first step is to determine a consistent coordinate system. Antennas are typically measured in units such as meters along the ground. We will use a right-handed coordinate system of East, North, and Up (E, N, U). These coordinates are relative to the local horizon, however, and will change depending on where we are on the spherical Earth. It is convenient in astronomy to use a coordinate system aligned with the Earth's rotational axis, for which we will use coordinates (x, y, z) as shown in Figure 1. Conversion from (E, N, U) to (x, y, z) is done via a simple rotation matrix:
which yields the relations:
Baselines and Spatial Frequencies
Note that the baselines are differences of coordinates, i.e. for the baseline between two antennas we have a vector:
This vector difference in positions can point in any direction in space, but the part of the baseline that matters in calculating u,v,w is the component perpendicular to the direction (the phase center direction), which we called in Figure 2. Let us express the phase center direction as a unit vector , where is the hour angle (relative to the local meridian) and is the declination (relative to the celestial equator). Then .
Recall that the spatial frequencies u,v,w are just the distances expressed in wavelength units, so we can get the u,v,w coordinates from the baseline length expressed in wavelength units from the following coordinate transformation (see Thompson 1999 for details):
How baseline errors can contribute to the error in phase
The geometric phase difference at the phase center ( term in (1)) is:
where , geometric delay. We can see what can affect the geometric phase by taking the differential of this expression:
where we have used the relation between right ascension and hour angle: , so . Equation (2) shows how baseline errors and source position errors (, ) will affect the error in group delay (or yield an error in phase ). Note that a clock error is equivalent to a source position error .
If we have a source whose position is known, we can use Equation (2) to find the location of the antennas (this is called baseline determination). The error in antenna position is largely independent of the baseline lengths. For example, say that we can measure to within 1 degree at 5 GHz ( = 6 cm). Then we can measure , and to a precision of order (1 / 360) 6 cm ~ 1 / 60 cm even though = 5000 km or more (VLBI).
The time of day and location of the antennas must be known to relatively high accuracy -- needed for determining the geometric delay. A clock error of 1 s, or a baseline error of a few cm, will cause a serious phase shift of the source over, say, 10 minutes. At OVRO, using a GPS clock and measuring baselines with cosmic source calibration, we get a time accuracy of << 1 ms, and baseline errors of about 3 mm. Therefore, these effects are not serious over a short time interval, but may still be problematic over 8 hours. This is one reason that we do phase calibration observations every ~ 2 hours.
EOVSA Antenna Position Calibration
The positions of EOVSA antennas are determined using observations by the 27-m (Ant 14) low-frequency receiver (S and C band) of celestial radio sources during several observation runs in fall 2016. This document describes the procedure followed and the final? calibrated antenna positions.
For calibrator sources with locations with sufficient accuracy (we use caibrators from the VLA Calibrator Manual), and a good time-keeping accuracy at EOVSA (what is our time-keeping accuracy? --Bchen 19 November 2016) , and in Eq. 2 can be neglected. Hence Eq. 2 can be simplified to:
where is the intrinsic instrumental phase at the given baseline.
We use a two-step calibration to solve for the EOVSA baseline error as following:
Observing one strong and point-like calibrator for a sufficiently long time (at least several hours). Note it is important to observe for a long time in order to have sufficient variation of the phase vs. hour angle curve as determined by and . We use a function of the following form to fit the observed phases:
In a usual case where one has visibilities measured at many baselines