# Antenna Position

## Contents

## Obtaining *u,v,w* From An Antenna Array

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*,

*, and*

**North**

**Up***. 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*

**(E, N, U)***as shown in*

**(x, y, z)****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:

### How baseline errors can contribute to the error in phase

The geometric phase difference at the phase center (consider only 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.