shown in Fig. 1 (right panel).
Quasars are distant galaxies that appear as bright point-like sources with their positions determined down to an accuracy of several tens of microarcseconds. The quasars are used as reference sources, and the fringe-fitting and self-calibration is performed on them. Thus, if the reference source is within a few degrees of the spacecraft, the delay, delay rate and phase errors of the spacecraft signal can be compensated for. Therefore, any residual error is due to the deviation of the spacecraft’s position from its previously calculated position.

In this way, the spacecraft’s position can be determined.

[Figure: Figure 1: Left: An interferometer with two antennas, separated by a baseline B and has a time delay between the wavefront reaching the receivers of τg. The fringe pattern is shown, with the Full Width Half Maximum of the correlated signal given by 1.22λ/B [6]. Right: Schematic of the VLBI tracking of spacecraft, with a quasar used for phase-referencing [13] The PRIDE VLBI phase-referencing observations are scheduled alternating between a quasar calibrator source and the spacecraft, spending up to a few minutes on each. This is unless the quasar is sufficiently close to the spacecraft as to be captured in the primary beam of the antenna, in which case the spacecraft is observed for the entire observation, known as in-beam phase referencing. The geometric delay is then measured by cross correlating the signals between pairs of antennas, which solves for the baseline and the position vectors of the spacecraft. The spacecraft’s position is found relative to the nearby quasar source, and ionospheric and atmospheric effects can be compensated for. Additionally, fringe fit scans of a strong quasar source are conducted at the start and end of observations to compensate for the clock offset and rates between stations.]

Following observations, in order to measure the relative positions of the spacecraft and the quasar source, the signal from each of these must be correlated separately. For the quasar source, a far-field delay model can be utilised to compensate for the geometric delay, as its wavefront can be assumed to be planar when arriving at each station (given they are typically billions of light years distant). For spacecraft communications in the Solar System at X-band the wavefronts have curvature, and a separate near-field delay model must be used. These are defined using the Barycentric Celestial Reference System (BCRS), a coordinate system where the origin is defined as the centre of mass of the Solar System.
Following correlation, the output of the broadband correlation of the quasar and spacecraft sources are stored as a FITS Interferometry Data Interchange (FITS-IDI) file. These files can be imported into the Astronomical Image Processing System (AIPS1), which allows the parameters necessary for calibration to be calculated. This enables the spacecraft’s position to be determined, either by solving the Fundamental Astrometric equation [14] or producing a
1 http://www.aips.nrao.edu/aips_faq.html radio image of the spacecraft, by comparing the offset of its apparent position to the centre of the image, which was the telescope’s pointing position.