I know that Sir Arthur Eddington went to South American to photograph stars around an eclipse to see if they seemed to change position when the light past by the Sun relative to when the light didn't pass the Sun. My question is how many degrees did the stars appear off from normal, or if you prefer, how many degrees did the Sun bend the light from the stars that Eddington photographed. All I have been able to find thus far is :) "Sir Arthur Eddington went to South America, took pictures of the stars and showed that Einstein was right..." etc. etc. (:. Nowhere have been able to find even data that would allow me to determine the angle between the straight line the light followed from the stars to near the Sun and the straight line the light followed from the Sun to the Earth.
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1https://en.wikipedia.org/wiki/Eddington_experiment For light grazing the surface of the Sun, the approximate angular deflection is roughly 1.75 arcseconds. – PM 2Ring Jan 21 '24 at 04:21
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(short answer, order 1 arcsec) This is a really interesting question (to me at least) because there's this factor of 2 that I can't follow. Wikipedia's Gravitational lensing gives expressions for the deflection $\theta$ at the Sun's limb of both $4GM/c^2r$ and $2GM/c^2r$, and Dyson 1920 give results from 0.93 to 1.98 arcseconds. This needs a careful, well-informed answer in order to sort out which plates and results to discuss and where these factors of 2 are coming from. – uhoh Jan 21 '24 at 04:56
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@PM2Ring just fyi someone "double-checked" :-) Gravitational Starlight Deflection Measurements during the 21 August 2017 Total Solar Eclipse This is interesting too: The Royal Observatory Greenwich's General Relativity and the 1919 Solar Eclipse as is The 1919 eclipse results that verified general relativity and their later detractors: a story re-told – uhoh Jan 21 '24 at 05:09
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@PM2Ring and Gravitational Starlight Deflection Measurements during the 21 August 2017 Total Solar Eclipse – uhoh Jan 21 '24 at 05:12
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@uhoh As I mention here, the deflection angle (in radians) is approximately $\theta = \frac{2r_s}b$, where $b$ is the impact parameter and $r_s$ is the Schwarzschild radius $r_s=2GM/c^2$. You can use $r$, the distance of closest approach instead of $b$; it doesn't make much difference in the regime where that formula is valid. – PM 2Ring Jan 21 '24 at 05:57
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2@uhoh Note that GR predicts twice the deflection of Newtonian gravity (where we treat photons as particles of negligible non-zero mass). Since that approximate deflection equation is linear, we also get that the deflection is ~1 arcsec at 1.75 solar radii. The exact deflection calculation uses an incomplete elliptic integral of the first kind; that integral also arises in the exact solution of the motion of a simple pendulum. – PM 2Ring Jan 21 '24 at 05:59
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1@uhoh I suppose I should've said that the deflection equation is linear in 1/r. With Schwarzschild stuff, I tend to think in terms of $\frac{r_s}r$. ;) – PM 2Ring Jan 21 '24 at 16:52
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@PM2Ring OK, so it's unfortunate that for early experimental results, some observations agreed with $r_s/b$ and others agreed with $2r_s/b$. I'm looking at https://i.stack.imgur.com/1Qsto.png (from here) which looks like the plot on p. 332 of Dyson et al. 1920 which show the Einstein prediction of about 1 arcsecond at 30 arcminutes from the center of the Sun, not 1.8 arseconds. – uhoh Jan 21 '24 at 22:10
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@PM2Ring If that "Einstein" line is his 1911 prediction then that would make sense, but isn't that what's labeled "Newton" in the plot? This plot confuses the hell out of me! – uhoh Jan 21 '24 at 22:13
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@uhoh Yes, the deflection was almost 1 arcsec at 30 arcmin from the solar centre. Exactly as GR predicts. 30 arcmin is $b = 2R_\odot$. – ProfRob Jan 22 '24 at 07:07
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@ProfRob Ah, thank you for that! :-) – uhoh Jan 22 '24 at 07:46
1 Answers
As you can read here, Eddington led the expedition to the island of Principe off W Africa and did not go to South America.
The (General-Relativistic) prediction of the size of the deflection for stars measured close to the limb of the Sun is $1.75(R_\odot/r)$ arcseconds, where $R_\odot$ is the solar radius and $r$ is the projected radius of the star (i.e. as it appears) from the centre of the Sun.
Longair (2015) reviews the expeditions of 1919. He reports that the Principe results, were equivalent to a deflection (at the limb of the Sun) of $1.61 \pm 0.31$ arcseconds, based mainly on measurements of 2 stars. Looking at the original paper by Dyson (1920), the result appears to be quoted as $1.61 \pm 0.30$ arcsec and is based on measurements of 6 stars at approximately 0.5-1.5 degrees from the centre of the Sun (corresponding approximately to $2 < r/R_\odot < 6$), though not all stars were measured on all photographs, with measured deflections of up to about 1 arcsecond. In both cases, the uncertainty quoted is the "probable error"; a reanalysis of the data by Gilmore & Tausch-Pebody (2022) puts the deflection result at $1.61 \pm 0.45$ arcseconds, where the error bar is the more conbentional standard deviation.
The South American results were better, getting measurements of 7 stars at separations of 0.5-1.5 degrees from the centre of the Sun, with measured deflections of 1.0 -0.2 arcseconds respectively. Fitting a straight line to a graph of deflection vs $1/r$ (see the copyrighted Fig.2 in the Longair review) gave a deflection at the limb result of $1.98 \pm 0.12$ arcseconds (Gilmore & Tausch-Pebody report $1.98 \pm 0.18$ arcseconds).
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2There is also more recent discussion of the 1919 results (open access): https://royalsocietypublishing.org/doi/10.1098/rsnr.2020.0040 – Leos Ondra Jan 22 '24 at 10:19