Measurement of the Solar Gravitational Deflection of Radio Waves using Geodetic Very-Long-Baseline Interferometry Data, 1979–1999

We used very-long-baseline interferometry (VLBI) to measure the deflection by the Sun of radio waves emanating from distant compact radio sources. This bending is characterized in the parametrized post-Newtonian formalism by $\gamma$, which is unity in general relativity. Using a large geodetic VLBI data set, we obtained $\gamma ={0.9998}_3\pm {0.0004}_5$ (estimated standard error). We found no systematic biases from our analysis of subgroups of data.

Figure 1

Histogram of solar elongation angles associated with each source used in each experiment (see text). A source is counted once for each experiment in which it is observed. This histogram is divided into four large subgroups based on the season (see text) in which the observations were made. The lower portion of each seasonal bar represents sources with positive declination angles and the upper portion of each seasonal bar represents sources with negative declination angles. Overlaid on this plot is a dashed curve indicating the solar elongation angles expected if the sources were distributed uniformly on the sky and a solid line showing the relationship between solar elongation angle and predicted deflection angle (right-hand scale).

Figure 2

Distribution of VLBI sites and radio sources. Sites: Gray shaded circles represent the 36 sites used in 1–10 experiments, gray shaded diamonds represent the 32 sites used in 11–100 experiments, and gray shaded triangles represent the 19 sites used in more than 100 experiments. Sources: Black shaded circles represent the 413 sources used in 1–100 experiments, black shaded diamonds represent the 84 sources used in 101–500 experiments, black shaded triangles represent the 24 sources used in 501–1000 experiments, and black shaded hexagons represent the 20 sources used in more than 1000 experiments. Note that the VLBI sites are not uniformly distributed around Earth. During the 1980s, very few negative declination sources were observed and these sources were used in only a few experiments.

Figure 3

Differences between global estimates of $\gamma$ and GR’s prediction as a function of minimum source elevation angle, ${\epsilon }_{\min }$ (open triangles), and minimum solar elongation angle, ${\varphi }_{\min }$, for positive declination sources (black shaded circles) and negative declination sources (open circles). These deviations from GR are plotted as a function of the smallest source elevation or solar elongation angle contained in each subset of data. For example, the open circle plotted with an abscissa of $50°$ represents the deviation in $\gamma$ from GR’s prediction determined from negative declination sources with ${\varphi }_{\min }=50°$, i.e., the difference from zero in the value of this plotted point is determined solely by data with solar elongation angles between $50°$ and $180°$. “Ref.” refers to $\gamma -1$ for our reference solution (${\varphi }_{\min }\approx 0.7°$ and ${\epsilon }_{\min }\approx 5°$), for which there are no cutoffs. Error bars represent the statistical standard error scaled by 1.7, the ratio of the standard error that we estimated for the mean of our daily estimates of $\gamma$, and the statistical standard error from our (complete) global solution (see text).