Testing of General Relativity with Geodetic VLBI

The geodetic VLBI technique is capable of measuring the Sun's gravity light deflection from distant radio sources around the whole sky. This light deflection is equivalent to the conventional gravitational delay used for the reduction of geodetic VLBI data. While numerous tests based on a global set of VLBI data have shown that the parameter 'gamma' of the post-Newtonian approximation is equal to unity with a precision of about 0.02 percent, more detailed analysis reveals some systematic deviations depending on the angular elongation from the Sun. In this paper a limited set of VLBI observations near the Sun were adjusted to obtain the estimate of the parameter 'gamma' free of the elongation angle impact. The parameter 'gamma' is still found to be close to unity with precision of 0.06 percent, two subsets of VLBI data measured at short and long baselines produce some statistical inconsistency.


Introduction
In accordance with the predictions of general relativity the gravitational field of the Sun causes a light deflection of 1.75" at the Solar limb observed by optical facilities. In the case of the Very Long Baseline Interferometry (VLBI) technique the corresponding effect (known as gravitational delay) is traditionally formulated in a term of time delay and modelled as a function of the barycentre distance of two radio telescopes. A dependence on the baseline length and the elongation from the direction to the Solar system barycentre was not presented in the formula for the gravitational delay. Therefore, these two effects have been considered independently despite their common physical origin. A simple transformation between the deflection angle and the gravitational delay was recently developed to show the baseline length and the elongation angle explicitly.
The most accurate estimate of the post-Newtonian approximation parameter γ ( was obtained from analysis of data done by the Cassini spacecraft [1]. The current accuracy obtained with VLBI was obtained from analysis of a few millions of VLBI observations [2] collected and stored by the International VLBI Service (IVS) [3]. However it was found that the accuracy of the parameter γ estimate with geodetic VLBI observations is dominated by a small number of observations of radio sources near the Sun. This paper is emphasised on analysis of a limited set of near-Sun observations rather than a global set of all-sky data.

Basic equations
The conventional equation for the gravitational delay used for the reduction of geodetic VLBI data is given by where G is the gravitational constant, 1 and 2 are geocentirc distances from a body of mass M to both radio telescopes, s is the unit vector in the direction of the radio source, c is the speed of light and γ is the parameter of the post-Newtonian (PPN) formalism [4], equal to unity in GR. The positions of astronomical instruments on Earth are referenced to the solar system barycentre, and the measured delay is equal to the terrestrial time (TT) coordinate time interval between two events of the signal arrival at the first and second radio telescopes [5]. This equation is sufficient for a picosecond level of accuracy.
A deviation of the PPN parameter γ from unity is estimated to test general relativity. However, apart from the gravitational delay (1) another term including into the VLBI group delay model also comprises the parameter γ [5] (eq (11.9)).
where b is the baseline vector calculated as the difference between the two barycentre radiusvectors of two antennas Equation (2) is known to change the scale only and is used to be ignored, so only (1) is used for calculation of the partial derivative.
It was shown that the total effect of general relativity at picosec level accuracy of the PPN approximation for the grazing light is a sum of (1) and (2) The first term in (4) is linked to the light deflection angle for an arbitrary elongation from the Sun as follows The last two terms in (3) are pertinent only for two-station observational facilities as a standard VLBI baseline, and the additional light deflection is now proportional to the baseline length. The third term is responsible for an incremental deflection along the "source  deflecting body" direction, whereas the second term mimics the baseline parallax effect, i.e. non-zero angle for a radio source observed from two radio telescopes separated by a long baseline. This baseline parallax effect is ignorable for an extragalactic radio source as , however, it is exaggerated by the Solar gravitational field. Recalling that γ = 1 for general relativity, in the small-angle approximation GR  is given by and the last two terms present the baseline-dependent deflection referred to the second station. In accordance with (6), these two minor effects rapidly grow in the vicinity of the Sun (as 2 R b ) and their sum reaches 12 milliarcsec at a baseline of 10,000 km for the grazing light and 0.03 milliarcsec for an elongation of 5°. For a shorter baseline the magnitudes are proportionally smaller [6].

Data analysis
Since the light deflection at large elongation angles are likely to be affected by the numerical and astrometrical factors discussed at the previous section, we focus on the observations of reference radio sources near the Sun to reduce a possible bias or statistical degradation. The legacy VLBI data collected between 1993 and 2002 comprise a large amount of observations of radio sources close to the Sun (at 5° or less), however, there were no observations closer than 15° to the Sun between 2002 and 2011. As a result, standard IVS observations over that period do not contribute essentially to the improvement of the parameter γ.
In total we processed 53 sessions between 1993 and 2012, these included 58 approaches of radio sources at 5º or less from the Sun. The total number of single observations is 1581. The results of the data analysis are shown on the left panel in Fig. 4. All the single-day estimates of γ do not observations. This accuracy is about 30% better than for the combined solution that included all baselines.