Newtonian vs General Relativistic light deflection angle

Question

The Newtonian prediction and GR prediction of light deflection angles are $\frac{2GM}{c^2 R}$ vs. $\frac{4GM}{c^2 R}$. One gives 0.8 arcsec and the other 1.7 arcsec for deflection around the Sun, and the observation confirmed it should be 1.7 arcsec as observed during the solar eclipse.

What value of "R" is used to get the predicted 0.8 or 1.7 arcsec? It seems solar radius is used, but why should it be? R is supposed to be distance between light ray and the sun, which could be at any distance from the Sun. Why do we assume the light is deflected right at the Sun's surface?

Answer 1

The $R$ in the formula is the closest approach of the light ray to the centre of the Sun. If a star were seen right at the limb of the Sun, then $R \simeq R_\odot$ and the GR deflection angle would be around 1.7 arcsec.

In practice stars are not viewed right at the limb of the Sun, so $R$ is larger and the deflection is smaller. When the deflection angle is this small, one just uses a radius corresponding to the projected separation on the sky - so $R=R_\odot$ corresponds to 15 arc minutes angular separation from the solar centre projected on the sky.

Note that the background star is very far away, so all rays reaching Earth are parallel. The ones that pass closer or further from the Sun, are not the ones that reach the observer on Earth. Basically you draw one ray from the star to the observer, going past the Sun. You are correct that you could draw an infinite number of rays that have differing values of $R$, but only one is correct according to General Relativity.

In the picture you have drawn, you have massively exaggerated the deflection angle, but the value of $R$ would be one that is the minimum distance between the light ray and the solar centre. It looks like $R \sim 4R_\odot$. The deflection angle would be 4 times less than if the star were viewed at the limb of the Sun. Possibly some pop-sci desriptions of the phenomenon are confusing. In practice, the deflections of stars at a variety of angular distances from the Sun are measured during an eclipse. One can then plot a straight-line graph of deflection versus the reciprocal of angular distance to extrapolate to what the maximum deflection would be for a hypothetical star at the limb of the Sun.

Edit: To follow up on that final point. Here are two diagrams taken from the original paper by Dyson et al. (1920). The first shows the positions of the stars, with respect to the Sun, that were used for the deflection measurement. Note that none of them are very close to the limb. The second graph is of the deflection angle versus the angular separation from the Sun (scaled so that the reciprocal of that angle is linear along the x-axis). You can see from this plot that the largest deflections actually measured were just over 1 arcsecond, not 1.7 arcseconds.

Picture (from photographic plates) of the stars used to measure the gravitational deflection due to the Sun in the 1919 eclipse (from Dyson et al. 1920).

Declection angle vs angular separation from the Sun for those stars. The x-axis is scaled so that the result should be a straight line. The dashed line shows the Newtonian prediction, thicker solid line is the GR prediction and the thinner solid line is the best fit.

Answer 2

If I interpret your question and the comments to Rob's answer correctly you are concerned that the ray trajectories would look like this (the deflection is greatly exaggerated for clarity):

and therefore that the ray reaching the observer is not the one with the maximum deflection. If I have misunderstood you ignore this answer!

Anyhow, you are quite correct that this is the case for a star directly behind the Sun i.e. one on the optic axis. However if we start at the observer and work backwards we get rays like this:

Assuming there is a dense enough distribution of stars there will be a ray that reaches the observer that grazes the Sun and has the maximum deflection, but the light from that ray was not initially parallel to the optic axis.