What is the proper procedure to compare observations of light bending around the sun to GR's predictions? I'm going through the history of GR and trying to understand the specifics of the experiments which validated the theory. There were a number of duplicate experiments which got consistent results.  I have a couple examples which I have focused on because of the availability and quality of papers.
Summary of the original Eddington experiment from 1919: https://arxiv.org/abs/astro-ph/0102462
A duplicate of the original experiment circa 1952: http://cdsads.u-strasbg.fr/pdf/1953AJ.....58...87V
A more precise experiment using radio waves (having a harder time following this one) https://arxiv.org/abs/1502.07395
The issue I have is that the computed deflection from GR's theory of ~1.75 arcseconds is at a radius which has the light skimming the sun, just ~5km over the service.  But the experiments can't observe stars that close to the sun (meaning close to the sun visually, so that their light would be skimming the sun on its approach to us).  In the 1952 experiment (which thankfully contains some data), the distance from the sun to the stars in question ranges from 2 to 8 solar radii.  At those distances (from the sun's core as the light passes by), the deflection from GR would be much less, ~.4 arcseconds at 4 solar radii.  And I can't seem to find any information about how they are using these observations and coming to conclusions of a deflection which aligns to GR's 1.75 arcsecond deflection value. This wouldn't be a barrier to validating GR of course, I'm just confused why they are comparing their results to a computed value which doesn't align to the specific distances for the observations.
For reference the equation from GR is:
$$\theta = \frac{4GM}{rc^2}$$
 A: The angular deflection for a ray that passes a distance $r$ from the Sun would be
$$
\theta = \frac{4GM}{rc^2} = \frac{4 GM}{r_\odot c^2} \frac{r_\odot}{r} \equiv \alpha \frac{r_\odot}{r}
$$
where $r_\odot$ is the radius of the sun and $\alpha \equiv 4 GM/r_\odot c^2 \approx 1.75''$.  As long as one can measure $r$ in units of solar radii, this means that one can indirectly measure $\alpha$.
In the original paper, Dyson et al. appear to have measured the displacements of several stars (seven for the Sobral data, five for the Principe data) and done a least-squares regression to find the value of $\alpha$ that is most consistent with the observations.  None of the individual stars were directly on the limb of the sun, but given the known relationship between $r$, $r_\odot$, and $\theta$, one can infer the value of $\alpha$ that best fits the data and compare it to the GR prediction of 1.75''.
A: The GR prediction for the bending of light that skims the sun is determined by the mass of the sun and the radius of the sun (for light skimming its surface).  The distance that the light source is from the sun has no real bearing on the results.
